From 051e579779e1f725e705e59e90001a2def41ee0d Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Wed, 8 May 2019 22:08:26 +0800
Subject: [PATCH 01/61] Implement gyrotropic susceptibility class.

---
 src/meep.hpp           |  23 +++++-
 src/susceptibility.cpp | 171 +++++++++++++++++++++++++++++++++++++++--
 2 files changed, 188 insertions(+), 6 deletions(-)

diff --git a/src/meep.hpp b/src/meep.hpp
index 7689334e2..2fc6079be 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -223,7 +223,8 @@ class susceptibility {
 class lorentzian_susceptibility : public susceptibility {
 public:
   lorentzian_susceptibility(double omega_0, double gamma, bool no_omega_0_denominator = false)
-      : omega_0(omega_0), gamma(gamma), no_omega_0_denominator(no_omega_0_denominator) {}
+      : omega_0(omega_0), gamma(gamma), no_omega_0_denominator(no_omega_0_denominator),
+	have_gyrotropy(false) {}
   virtual susceptibility *clone() const { return new lorentzian_susceptibility(*this); }
   virtual ~lorentzian_susceptibility() {}
 
@@ -249,6 +250,7 @@ class lorentzian_susceptibility : public susceptibility {
 protected:
   double omega_0, gamma;
   bool no_omega_0_denominator;
+  bool have_gyrotropy; // whether to assign an extra slot for gyrotropy calculations
 };
 
 /* like a Lorentzian susceptibility, but the polarization equation
@@ -272,6 +274,25 @@ class noisy_lorentzian_susceptibility : public lorentzian_susceptibility {
   double noise_amp;
 };
 
+/* like a Lorentzian susceptibility, but with precession around a bias vector */
+class gyrotropic_susceptibility : public lorentzian_susceptibility {
+public:
+  gyrotropic_susceptibility(const vec &bias, double omega_0, double gamma,
+			    bool no_omega_0_denominator = true);
+  virtual susceptibility *clone() const { return new gyrotropic_susceptibility(*this); }
+
+  virtual void update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
+			realnum *W_prev[NUM_FIELD_COMPONENTS][2], double dt,
+			const grid_volume &gv, void *P_internal_data) const;
+
+  virtual void dump_params(h5file *h5f, size_t *start);
+  virtual int get_num_params() { return 7; }
+
+protected:
+  vec bias;
+  double gyro_tensor[3][3];
+};
+
 class multilevel_susceptibility : public susceptibility {
 public:
   multilevel_susceptibility() : L(0), T(0), Gamma(0), N0(0), alpha(0), omega(0), gamma(0) {}
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index d45034352..774357612 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -92,6 +92,7 @@ typedef struct {
   size_t ntot;
   realnum *P[NUM_FIELD_COMPONENTS][2];
   realnum *P_prev[NUM_FIELD_COMPONENTS][2];
+  realnum *P_tmp[NUM_FIELD_COMPONENTS][2]; // extra slot used for gyrotropic medium updating
   realnum data[1];
 } lorentzian_data;
 
@@ -100,8 +101,9 @@ typedef struct {
 void *lorentzian_susceptibility::new_internal_data(realnum *W[NUM_FIELD_COMPONENTS][2],
                                                    const grid_volume &gv) const {
   int num = 0;
+  int nslots = have_gyrotropy ? 3 : 2;
   FOR_COMPONENTS(c) DOCMP2 {
-    if (needs_P(c, cmp, W)) num += 2 * gv.ntot();
+    if (needs_P(c, cmp, W)) num += nslots * gv.ntot();
   }
   size_t sz = sizeof(lorentzian_data) + sizeof(realnum) * (num - 1);
   lorentzian_data *d = (lorentzian_data *)malloc(sz);
@@ -119,12 +121,18 @@ void lorentzian_susceptibility::init_internal_data(realnum *W[NUM_FIELD_COMPONEN
   size_t ntot = d->ntot = gv.ntot();
   realnum *P = d->data;
   realnum *P_prev = d->data + ntot;
+  realnum *P_tmp = have_gyrotropy ? P_prev + ntot : NULL;
+  int nslots = have_gyrotropy ? 3 : 2;
+
   FOR_COMPONENTS(c) DOCMP2 {
     if (needs_P(c, cmp, W)) {
       d->P[c][cmp] = P;
       d->P_prev[c][cmp] = P_prev;
-      P += 2 * ntot;
-      P_prev += 2 * ntot;
+      d->P_tmp[c][cmp] = P_tmp;
+
+      P += nslots * ntot;
+      P_prev += nslots * ntot;
+      if (have_gyrotropy) P_tmp += nslots * ntot;
     }
   }
 }
@@ -137,12 +145,17 @@ void *lorentzian_susceptibility::copy_internal_data(void *data) const {
   size_t ntot = d->ntot;
   realnum *P = dnew->data;
   realnum *P_prev = dnew->data + ntot;
+  realnum *P_tmp = have_gyrotropy ? P_prev + ntot : NULL;
+  int nslots = have_gyrotropy ? 3 : 2;
+
   FOR_COMPONENTS(c) DOCMP2 {
     if (d->P[c][cmp]) {
       dnew->P[c][cmp] = P;
       dnew->P_prev[c][cmp] = P_prev;
-      P += 2 * ntot;
-      P_prev += 2 * ntot;
+      dnew->P_tmp[c][cmp] = P_tmp;
+      P += nslots * ntot;
+      P_prev += nslots * ntot;
+      if (have_gyrotropy) P_tmp += nslots * ntot;
     }
   }
   return (void *)dnew;
@@ -321,4 +334,152 @@ void noisy_lorentzian_susceptibility::dump_params(h5file *h5f, size_t *start) {
   h5f->write_chunk(1, start, params_dims, params_data);
   *start += num_params;
 }
+
+
+gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double omega_0, double gamma,
+						     bool no_omega_0_denominator)
+  : lorentzian_susceptibility(omega_0, gamma, no_omega_0_denominator), bias(bias) {
+  have_gyrotropy = true;
+
+  // Precalculate g_{ij} = sum_k epsilon_{ijk} b_k, used in update_P.
+  memset(gyro_tensor, 0, 9 * sizeof(double));
+  gyro_tensor[X][Y] = bias.z(); gyro_tensor[Y][X] = -bias.z();
+  gyro_tensor[Y][Z] = bias.x(); gyro_tensor[Z][Y] = -bias.x();
+  gyro_tensor[Z][X] = bias.y(); gyro_tensor[X][Z] = -bias.y();
+}
+
+void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
+					 realnum *W_prev[NUM_FIELD_COMPONENTS][2], double dt,
+					 const grid_volume &gv, void *P_internal_data) const {
+  lorentzian_data *d = (lorentzian_data *)P_internal_data;
+  const double omega2pi = 2 * pi * omega_0, g2pi = gamma * 2 * pi;
+  const double omega0dtsqr = omega2pi * omega2pi * dt * dt;
+  const double gamma1 = (1 - g2pi * dt / 2);
+  const double omega0dtsqr_denom = no_omega_0_denominator ? 0 : omega0dtsqr;
+  (void)W_prev; // unused;
+
+  FOR_COMPONENTS(c) DOCMP2 {
+    if (d->P[c][cmp]) {
+      const direction d0 = component_direction(c);
+      const realnum *w = W[c][cmp], *s = sigma[c][d0];
+      if (w && s) {
+	if (d0 != X && d0 != Y && d0 != Z)
+	  abort("Cylindrical coordinates are not supported for gyrotropic media");
+
+        const realnum *p = d->P[c][cmp], *pp = d->P_prev[c][cmp];
+	realnum *rhs = d->P_tmp[c][cmp];
+
+        const ptrdiff_t is = gv.stride(d0) * (is_magnetic(c) ? -1 : +1);
+
+        direction d1 = cycle_direction(gv.dim, d0, 1);
+        component c1 = direction_component(c, d1);
+        ptrdiff_t is1 = gv.stride(d1) * (is_magnetic(c) ? -1 : +1);
+        const realnum *w1 = W[c1][cmp];
+        const realnum *s1 = w1 ? sigma[c][d1] : NULL;
+
+        direction d2 = cycle_direction(gv.dim, d0, 2);
+        component c2 = direction_component(c, d2);
+        ptrdiff_t is2 = gv.stride(d2) * (is_magnetic(c) ? -1 : +1);
+        const realnum *w2 = W[c2][cmp];
+        const realnum *s2 = w2 ? sigma[c][d2] : NULL;
+
+	// Off-diagonal polarization components
+        const realnum *pp1 = d->P_prev[c1][cmp];
+        const realnum *pp2 = d->P_prev[c2][cmp];
+	const realnum g1 = pi * dt * gyro_tensor[d0][d1];
+	const realnum g2 = pi * dt * gyro_tensor[d0][d2];
+
+        if (s2 && !s1) { // make s1 the non-NULL one if possible
+          SWAP(direction, d1, d2);
+          SWAP(component, c1, c2);
+          SWAP(ptrdiff_t, is1, is2);
+          SWAP(const realnum *, w1, w2);
+          SWAP(const realnum *, s1, s2);
+        }
+        if (s1 && s2) { // 3x3 anisotropic
+          LOOP_OVER_VOL_OWNED(gv, c, i) {
+            if (s[i] != 0) {
+              rhs[i] = (2 - omega0dtsqr_denom) * p[i]
+		+ omega0dtsqr * (s[i] * w[i]
+				 + OFFDIAG(s1, w1, is1, is)
+				 + OFFDIAG(s2, w2, is2, is))
+		- gamma1 * pp[i];
+	      if (pp1) rhs[i] -= g1 * pp1[i];
+	      if (pp2) rhs[i] -= g2 * pp2[i];
+            }
+          }
+        } else if (s1) { // 2x2 anisotropic
+          LOOP_OVER_VOL_OWNED(gv, c, i) {
+            if (s[i] != 0) {
+              rhs[i] = (2 - omega0dtsqr_denom) * p[i]
+		+ omega0dtsqr * (s[i] * w[i] + OFFDIAG(s1, w1, is1, is))
+		- gamma1 * pp[i];
+	      if (pp1) rhs[i] -= g1 * pp1[i];
+	      if (pp2) rhs[i] -= g2 * pp2[i];
+            }
+          }
+        } else { // isotropic
+          LOOP_OVER_VOL_OWNED(gv, c, i) {
+            rhs[i] = (2 - omega0dtsqr_denom) * p[i]
+	      + omega0dtsqr * (s[i] * w[i]) - gamma1 * pp[i];
+	    if (pp1) rhs[i] -= g1 * pp1[i];
+	    if (pp2) rhs[i] -= g2 * pp2[i];
+          }
+        }
+      }
+    }
+  }
+
+  // Perform 3x3 matrix inversion, exploiting skew symmetry
+  const double g2 = (1 + g2pi * dt / 2);
+  const double gx = pi * dt * gyro_tensor[Y][Z];
+  const double gy = pi * dt * gyro_tensor[Z][X];
+  const double gz = pi * dt * gyro_tensor[X][Y];
+  const double invdet = 1.0 / g2 / (g2*g2 + gx*gx + gy*gy + gz*gz);
+  double inv[3][3];
+
+  inv[X][X] = invdet * (g2*g2 + gx*gx);
+  inv[Y][Y] = invdet * (g2*g2 + gy*gy);
+  inv[Z][Z] = invdet * (g2*g2 + gz*gz);
+
+  inv[X][Y] = invdet * (gx*gy + g2*gz);
+  inv[Y][X] = invdet * (gy*gx - g2*gz);
+  inv[Z][X] = invdet * (gz*gx + g2*gy);
+  inv[X][Z] = invdet * (gx*gz - g2*gy);
+  inv[Y][Z] = invdet * (gy*gz + g2*gx);
+  inv[Z][Y] = invdet * (gz*gy - g2*gx);
+
+  FOR_COMPONENTS(c) DOCMP2 {
+    if (d->P[c][cmp]) {
+      const direction d0 = component_direction(c);
+      if (W[c][cmp] && sigma[c][d0]) {
+        realnum *p = d->P[c][cmp], *pp = d->P_prev[c][cmp], *rhs = d->P_tmp[c][cmp];
+        const direction d1  = cycle_direction(gv.dim, d0, 1);
+        const direction d2  = cycle_direction(gv.dim, d0, 2);
+        const component c1 = direction_component(c, d1);
+        const component c2 = direction_component(c, d2);
+        const realnum *rhs1 = W[c1][cmp] ? d->P_tmp[c1][cmp] : NULL;
+	const realnum *rhs2 = W[c2][cmp] ? d->P_tmp[c2][cmp] : NULL;
+
+	LOOP_OVER_VOL_OWNED(gv, c, i) {
+	  pp[i] = p[i];
+	  p[i]  = inv[d0][d0] * rhs[i];
+	  if (rhs1) p[i] += inv[d0][d1] * rhs1[i];
+	  if (rhs2) p[i] += inv[d0][d2] * rhs2[i];
+	}
+      }
+    }
+  }
+}
+
+void gyrotropic_susceptibility::dump_params(h5file *h5f, size_t *start) {
+  size_t num_params = 8;
+  size_t params_dims[1] = {num_params};
+  double params_data[] = {
+      7, (double)get_id(), bias.x(), bias.y(), bias.z(),
+      omega_0, gamma, (double)no_omega_0_denominator};
+  h5f->write_chunk(1, start, params_dims, params_data);
+  *start += num_params;
+}
+
 } // namespace meep

From c068942ea4c389d8d9b281f481dae1a956ed2622 Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Wed, 8 May 2019 22:08:52 +0800
Subject: [PATCH 02/61] Add Python and Scheme support for gyrotropic media.

---
 python/geom.py           | 12 ++++++++++++
 python/meep.i            |  2 ++
 python/typemap_utils.cpp | 14 ++++++++++++++
 scheme/meep.scm.in       |  4 ++++
 scheme/structure.cpp     | 10 ++++++++++
 src/material_data.hpp    |  1 +
 src/meepgeom.cpp         |  8 +++++++-
 7 files changed, 50 insertions(+), 1 deletion(-)

diff --git a/python/geom.py b/python/geom.py
index a3dc12715..e4d8fec14 100644
--- a/python/geom.py
+++ b/python/geom.py
@@ -280,6 +280,18 @@ def __init__(self, noise_amp=0.0, **kwargs):
         super(NoisyDrudeSusceptibility, self).__init__(**kwargs)
         self.noise_amp = noise_amp
 
+class GyrotropicLorentzianSusceptibility(LorentzianSusceptibility):
+
+    def __init__(self, bias=Vector3(0, 0, 1), **kwargs):
+        super(GyrotropicLorentzianSusceptibility, self).__init__(**kwargs)
+        self.bias = bias
+
+class GyrotropicDrudeSusceptibility(DrudeSusceptibility):
+
+    def __init__(self, bias=Vector3(), **kwargs):
+        super(GyrotropicDrudeSusceptibility, self).__init__(**kwargs)
+        self.bias = bias
+
 
 class MultilevelAtom(Susceptibility):
 
diff --git a/python/meep.i b/python/meep.i
index 12a020f52..998f13133 100644
--- a/python/meep.i
+++ b/python/meep.i
@@ -1371,6 +1371,8 @@ PyObject *_get_array_slice_dimensions(meep::fields *f, const meep::volume &where
         Ellipsoid,
         FreqRange,
         GeometricObject,
+        GyrotropicDrudeSusceptibility,
+        GyrotropicLorentzianSusceptibility,
         Lattice,
         LorentzianSusceptibility,
         Matrix,
diff --git a/python/typemap_utils.cpp b/python/typemap_utils.cpp
index dc0b2e253..a639370c1 100644
--- a/python/typemap_utils.cpp
+++ b/python/typemap_utils.cpp
@@ -469,6 +469,20 @@ static PyObject *susceptibility_to_py_obj(susceptibility_struct *s) {
       res = PyObject_Call(py_lorentz_class, args, NULL);
       Py_DECREF(py_lorentz_class);
     }
+  } else if (s->bias.x || s->bias.y || s->bias.z) {
+    if (s->drude) {
+      PyObject *py_gyrotropic_drude_class = PyObject_GetAttrString(geom_mod, "GyrotropicDrudeSusceptibility");
+      res = PyObject_Call(py_gyrotropic_drude_class, args, NULL);
+      Py_DECREF(py_gyrotropic_drude_class);
+    } else {
+      PyObject *py_gyrotropic_lorentz_class =
+	PyObject_GetAttrString(geom_mod, "GyrotropicLorentzianSusceptibility");
+      res = PyObject_Call(py_gyrotropic_lorentz_class, args, NULL);
+      Py_DECREF(py_gyrotropic_lorentz_class);
+    }
+    PyObject *py_bias = vec2py(vector3_to_vec(s->bias));
+    PyObject_SetAttrString(res, "bias", py_bias);
+    Py_DECREF(py_bias);
   } else {
     if (s->drude) {
       PyObject *py_noisy_drude_class = PyObject_GetAttrString(geom_mod, "NoisyDrudeSusceptibility");
diff --git a/scheme/meep.scm.in b/scheme/meep.scm.in
index a4ea946bd..cfec58b06 100644
--- a/scheme/meep.scm.in
+++ b/scheme/meep.scm.in
@@ -57,6 +57,10 @@
   (define-property noise-amp no-default 'number))
 (define-class noisy-drude-susceptibility drude-susceptibility
   (define-property noise-amp no-default 'number))
+(define-class gyrotropic-lorentzian-susceptibility lorentzian-susceptibility
+  (define-property bias no-default 'vector3))
+(define-class gyrotropic-drude-susceptibility drude-susceptibility
+  (define-property bias no-default 'vector3))
 
 (define polarizability lorentzian-susceptibility) ; backwards compat
 (define omega frequency) ; backwards compat
diff --git a/scheme/structure.cpp b/scheme/structure.cpp
index 43c8f49a2..068915971 100644
--- a/scheme/structure.cpp
+++ b/scheme/structure.cpp
@@ -1295,6 +1295,11 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
           master_printf("noisy lorentzian susceptibility: frequency=%g, gamma=%g, amp = %g\n",
                         d->frequency, d->gamma, nd->noise_amp);
           sus = new meep::noisy_lorentzian_susceptibility(nd->noise_amp, d->frequency, d->gamma);
+	} else if (d->which_subclass == lorentzian_susceptibility::GYROTROPIC_LORENTZIAN_SUSCEPTIBILITY) {
+          gyrotropic_lorentzian_susceptibility *gd = d->subclass.gyrotropic_lorentzian_susceptibility_data;
+          master_printf("gyrotropic lorentzian susceptibility: bias=(%g,%g,%g), frequency=%g, gamma=%g\n",
+			gd->bias.x, gd->bias.y, gd->bias.z, d->frequency, d->gamma);
+          sus = new meep::gyrotropic_susceptibility(vector3_to_vec(gd->bias), d->frequency, d->gamma);
         } else { // just a Lorentzian
           master_printf("lorentzian susceptibility: frequency=%g, gamma=%g\n", d->frequency,
                         d->gamma);
@@ -1310,6 +1315,11 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
                         d->frequency, d->gamma, nd->noise_amp);
           sus = new meep::noisy_lorentzian_susceptibility(nd->noise_amp, d->frequency, d->gamma,
                                                           true);
+	} else if (d->which_subclass == drude_susceptibility::GYROTROPIC_DRUDE_SUSCEPTIBILITY) {
+          gyrotropic_drude_susceptibility *gd = d->subclass.gyrotropic_drude_susceptibility_data;
+          master_printf("gyrotropic drude susceptibility: bias=(%g,%g,%g), frequency=%g, gamma=%g\n",
+			gd->bias.x, gd->bias.y, gd->bias.z, d->frequency, d->gamma);
+          sus = new meep::gyrotropic_susceptibility(vector3_to_vec(gd->bias), d->frequency, d->gamma, true);
         } else { // just a Drude
           master_printf("drude susceptibility: frequency=%g, gamma=%g\n", d->frequency, d->gamma);
           sus = new meep::lorentzian_susceptibility(d->frequency, d->gamma, true);
diff --git a/src/material_data.hpp b/src/material_data.hpp
index 0610d7fdb..352b6155e 100644
--- a/src/material_data.hpp
+++ b/src/material_data.hpp
@@ -59,6 +59,7 @@ struct transition {
 typedef struct susceptibility_struct {
   vector3 sigma_offdiag;
   vector3 sigma_diag;
+  vector3 bias;
   double frequency;
   double gamma;
   double noise_amp;
diff --git a/src/meepgeom.cpp b/src/meepgeom.cpp
index 3c2ad82dd..e1cb860b6 100644
--- a/src/meepgeom.cpp
+++ b/src/meepgeom.cpp
@@ -1363,6 +1363,7 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
     susceptibility *ss = &(p->user_s);
     if (ss->is_file) meep::abort("unknown susceptibility");
     bool noisy = (ss->noise_amp != 0.0);
+    bool gyrotropic = (ss->bias.x != 0.0 || ss->bias.y != 0.0 || ss->bias.z != 0.0);
     meep::susceptibility *sus;
 
     if (ss->transitions.size() != 0 || ss->initial_populations.size() != 0) {
@@ -1373,12 +1374,17 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
       if (noisy) {
         sus = new meep::noisy_lorentzian_susceptibility(ss->noise_amp, ss->frequency, ss->gamma,
                                                         ss->drude);
+      } else if (gyrotropic) {
+        sus = new meep::gyrotropic_susceptibility(vector3_to_vec(ss->bias), ss->frequency,
+						  ss->gamma, ss->drude);
       } else {
         sus = new meep::lorentzian_susceptibility(ss->frequency, ss->gamma, ss->drude);
       }
-      master_printf("%s%s susceptibility: frequency=%g, gamma=%g", noisy ? "noisy " : "",
+      master_printf("%s%s susceptibility: frequency=%g, gamma=%g",
+		    noisy ? "noisy " : gyrotropic ? "gyrotropic" : "",
                     ss->drude ? "drude" : "lorentzian", ss->frequency, ss->gamma);
       if (noisy) master_printf(" amp=%g ", ss->noise_amp);
+      if (gyrotropic) master_printf(" bias=(%g,%g,%g) ", ss->bias.x, ss->bias.y, ss->bias.z);
       master_printf("\n");
     }
 

From 374335841b66c1ce87c967e827ae6f8c1c3857ff Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Thu, 9 May 2019 17:13:34 +0800
Subject: [PATCH 03/61] Initialize bias vector in python susceptibility struct.

---
 python/typemap_utils.cpp | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/python/typemap_utils.cpp b/python/typemap_utils.cpp
index a639370c1..dea028d1d 100644
--- a/python/typemap_utils.cpp
+++ b/python/typemap_utils.cpp
@@ -330,6 +330,7 @@ static int py_susceptibility_to_susceptibility(PyObject *po, susceptibility_stru
   s->frequency = 0;
   s->gamma = 0;
   s->noise_amp = 0;
+  s->bias.x = s->bias.y = s->bias.z = 0;
   s->transitions.resize(0);
   s->initial_populations.resize(0);
 
@@ -345,6 +346,10 @@ static int py_susceptibility_to_susceptibility(PyObject *po, susceptibility_stru
     if (!get_attr_dbl(po, &s->noise_amp, "noise_amp")) { return 0; }
   }
 
+  if (PyObject_HasAttrString(po, "bias")) {
+    if (!get_attr_v3(po, &s->bias, "bias")) return 0;
+  }
+
   if (PyObject_HasAttrString(po, "transitions")) {
     // MultilevelAtom
     PyObject *py_trans = PyObject_GetAttrString(po, "transitions");

From 88ea6ca4ba400648c8991dce7b5cd1c2b6e1ce25 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Fri, 10 May 2019 10:36:47 +0800
Subject: [PATCH 04/61] Remove "bias" from gyrotropic_susceptibility; the
 information is already in gyro_tensor.

---
 src/meep.hpp           |  1 -
 src/susceptibility.cpp | 28 ++++++++++++++--------------
 2 files changed, 14 insertions(+), 15 deletions(-)

diff --git a/src/meep.hpp b/src/meep.hpp
index 2fc6079be..c03283e09 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -289,7 +289,6 @@ class gyrotropic_susceptibility : public lorentzian_susceptibility {
   virtual int get_num_params() { return 7; }
 
 protected:
-  vec bias;
   double gyro_tensor[3][3];
 };
 
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 774357612..3b7289c32 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -338,7 +338,7 @@ void noisy_lorentzian_susceptibility::dump_params(h5file *h5f, size_t *start) {
 
 gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double omega_0, double gamma,
 						     bool no_omega_0_denominator)
-  : lorentzian_susceptibility(omega_0, gamma, no_omega_0_denominator), bias(bias) {
+  : lorentzian_susceptibility(omega_0, gamma, no_omega_0_denominator) {
   have_gyrotropy = true;
 
   // Precalculate g_{ij} = sum_k epsilon_{ijk} b_k, used in update_P.
@@ -431,23 +431,22 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
   }
 
   // Perform 3x3 matrix inversion, exploiting skew symmetry
-  const double g2 = (1 + g2pi * dt / 2);
+  const double gd = (1 + g2pi * dt / 2);
   const double gx = pi * dt * gyro_tensor[Y][Z];
   const double gy = pi * dt * gyro_tensor[Z][X];
   const double gz = pi * dt * gyro_tensor[X][Y];
-  const double invdet = 1.0 / g2 / (g2*g2 + gx*gx + gy*gy + gz*gz);
+  const double invdet = 1.0 / gd / (gd*gd + gx*gx + gy*gy + gz*gz);
   double inv[3][3];
 
-  inv[X][X] = invdet * (g2*g2 + gx*gx);
-  inv[Y][Y] = invdet * (g2*g2 + gy*gy);
-  inv[Z][Z] = invdet * (g2*g2 + gz*gz);
-
-  inv[X][Y] = invdet * (gx*gy + g2*gz);
-  inv[Y][X] = invdet * (gy*gx - g2*gz);
-  inv[Z][X] = invdet * (gz*gx + g2*gy);
-  inv[X][Z] = invdet * (gx*gz - g2*gy);
-  inv[Y][Z] = invdet * (gy*gz + g2*gx);
-  inv[Z][Y] = invdet * (gz*gy - g2*gx);
+  inv[X][X] = invdet * (gd*gd + gx*gx);
+  inv[Y][Y] = invdet * (gd*gd + gy*gy);
+  inv[Z][Z] = invdet * (gd*gd + gz*gz);
+  inv[X][Y] = invdet * (gx*gy + gd*gz);
+  inv[Y][X] = invdet * (gy*gx - gd*gz);
+  inv[Z][X] = invdet * (gz*gx + gd*gy);
+  inv[X][Z] = invdet * (gx*gz - gd*gy);
+  inv[Y][Z] = invdet * (gy*gz + gd*gx);
+  inv[Z][Y] = invdet * (gz*gy - gd*gx);
 
   FOR_COMPONENTS(c) DOCMP2 {
     if (d->P[c][cmp]) {
@@ -475,8 +474,9 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 void gyrotropic_susceptibility::dump_params(h5file *h5f, size_t *start) {
   size_t num_params = 8;
   size_t params_dims[1] = {num_params};
+  double bias[] = { gyro_tensor[Y][Z], gyro_tensor[Z][X], gyro_tensor[X][Y] };
   double params_data[] = {
-      7, (double)get_id(), bias.x(), bias.y(), bias.z(),
+      7, (double)get_id(), bias[X], bias[Y], bias[Z],
       omega_0, gamma, (double)no_omega_0_denominator};
   h5f->write_chunk(1, start, params_dims, params_data);
   *start += num_params;

From 3115557e5cd2e1453e9d93e0cb2dbdc06ec603e9 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Wed, 22 May 2019 13:26:11 +0800
Subject: [PATCH 05/61] Add gyrotropic media to docs

---
 doc/docs/Materials.md             | 25 +++++++++++++++++++++++++
 doc/docs/Python_User_Interface.md |  8 ++++++++
 2 files changed, 33 insertions(+)

diff --git a/doc/docs/Materials.md b/doc/docs/Materials.md
index c71885402..882e1a76a 100644
--- a/doc/docs/Materials.md
+++ b/doc/docs/Materials.md
@@ -167,6 +167,31 @@ $$ \mathbf{E} \; (\textrm{SALT}) = \frac{2 |\theta|}{\hbar \sqrt{\gamma_\perp \g
 
 For a two level gain medium, $\gamma_\parallel = \gamma_{12} + \gamma_{21}$. For more details on applying SALT to atomic media with an arbitrary number of levels, see [Optics Express, Vol. 23, pp. 6455-77, 2015](https://www.osapublishing.org/oe/abstract.cfm?uri=oe-23-5-6455).
 
+Gyrotropic Media
+----------------
+
+(**Experimental feature**) Meep supports gyrotropic media, which break optical reciprocity and give rise to magneto-optical phenomena such as the [Faraday effect](https://en.wikipedia.org/wiki/Faraday_effect). Such materials are used in devices like [Faraday rotators](https://en.wikipedia.org/wiki/Faraday_rotator). The following discussion concerns a medium with gyrotropic ε (i.e., a gyroelectric medium). Gyromagnetic media can be treated by the usual substitution of ε with μ, **E** with **H**, etc.
+
+In a gyroelectric medium, the equation of motion of the polarization vector $\mathbf{P}_n$ is very similar to the [Drude-Lorentz](Materials.md#material-dispersion) case, but contains an additional precession term:
+
+$$\frac{d^2\mathbf{P}_n}{dt^2} + \gamma_n \frac{d\mathbf{P}_n}{dt} + \mathbf{b}_n \times \frac{d\mathbf{P}_n}{dt} + \omega_n^2 \mathbf{P}_n = \sigma_n(\mathbf{x}) \omega_n^2 \mathbf{E}$$
+
+(Optionally, the polarization may be of Drude form, in which case the final $\omega_n^2 \mathbf{P}_n$ term on the left-hand side is omitted; we will describe the Lorentz case.) The third term on the left-hand side breaks time-reversal symmetry and describes the gyrotropy of the medium, parameterized by the bias vector $\mathbf{b}_n$ (usually describing the effect of an applied static magnetic field). In the absence of damping ($\gamma = 0$) and coupling to the electric field ($\sigma_n = 0$), the equation of motion reduces to a precession around $\mathbf{b}_n$, with angular frequency equal to $|\mathbf{b}_n|$:
+
+$$\frac{d\mathbf{P}_n}{dt} = \mathbf{P}_n \times \mathbf{b}_n$$
+
+In the frequency domain, gyrotropy corresponds to the presence of imaginary off-diagonal components in the ε tensor (or the μ tensor for a gyromagnetic medium). Take the case $\mathbf{b} = b \hat{z}$.  When all fields have harmonic time-dependence $\exp(-i\omega t)$, the polarization equation of motion reduces to
+
+$$\mathbf{P}_n = \chi_n \mathbf{E}, \quad \chi_n = \frac{\omega_n^2}{\omega_n^2 - \omega^2 + i\omega\gamma_n} \begin{bmatrix}\xi & -i\eta & 0 \\ i\eta & \xi & 0 \\ 0 & 0 & 1 \end{bmatrix} \, \sigma_n(\mathbf{x})$$
+
+where
+
+$$\xi = \frac{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2}{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2 + \omega^2 b^2}\,, \quad \eta = \frac{\omega b (\omega_n^2 - \omega^2 - i\omega \gamma_n)}{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2 + \omega^2 b^2}$$
+
+It is common to operate in a limit where the gyrotropy is weak relative to the frequency detuning, but strong relative to the loss.  Let $\omega_n = \omega + \Delta\omega_n$, where $\gamma_n \ll b \ll \Delta \omega_n \ll \omega$.  Then the above susceptibility tensor simplifies to
+
+$$\chi_n \approx \frac{\omega}{2 \Delta\omega_n} \begin{bmatrix}1 & -i\eta & 0 \\ i\eta & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\, \sigma_n(\mathbf{x}), \quad \eta = \frac{b}{2\Delta\omega_n}$$
+
 Materials Library
 -----------------
 
diff --git a/doc/docs/Python_User_Interface.md b/doc/docs/Python_User_Interface.md
index 88b0050e6..0b0ef0cf1 100644
--- a/doc/docs/Python_User_Interface.md
+++ b/doc/docs/Python_User_Interface.md
@@ -392,6 +392,14 @@ The noise has root-mean square amplitude σ $\times$ `noise_amp`.
 
 This is a somewhat unusual polarizable medium, a Lorentzian susceptibility with a random noise term added into the damped-oscillator equation at each point. This can be used to directly model thermal radiation in both the [far field](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.93.213905) and the [near field](http://math.mit.edu/~stevenj/papers/RodriguezIl11.pdf). Note, however that it is more efficient to [compute far-field thermal radiation using Kirchhoff's law](http://www.simpetus.com/projects.html#meep_thermal_radiation) of radiation, which states that emissivity equals absorptivity. Near-field thermal radiation can usually be computed more efficiently using frequency-domain methods, e.g. via [SCUFF-EM](https://github.com/HomerReid/scuff-em), as described e.g. [here](http://doi.org/10.1103/PhysRevB.92.134202) or [here](http://doi.org/10.1103/PhysRevB.88.054305).
 
+### GyrotropicLorentzianSusceptibility or GyrotropicDrudeSusceptibility
+
+(**Experimental feature**) Specifies a single dispersive gyrotropic susceptibility of Lorentzian (damped harmonic oscillator) or Drude form. See [Material Dispersion](Materials.md#gyrotropic-media), with the same `sigma`, `frequency`, and `gamma` parameters, but with an additional 3-vector `bias`:
+
+**`bias` [`Vector3`]**
+—
+The gyrotropy vector.  The direction of this vector determines the orientation of the gyrotropic response, and the magnitude equals the precession frequency $|\mathbf{b}_n|/2\pi$.
+
 ### Vector3
 
 Properties:

From c0b5939bc20d69d97f809b3fafbe09ea00f8a561 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Wed, 22 May 2019 13:29:25 +0800
Subject: [PATCH 06/61] Minor copyedit

---
 doc/docs/Python_User_Interface.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/doc/docs/Python_User_Interface.md b/doc/docs/Python_User_Interface.md
index 0b0ef0cf1..11f3b3f28 100644
--- a/doc/docs/Python_User_Interface.md
+++ b/doc/docs/Python_User_Interface.md
@@ -394,7 +394,7 @@ This is a somewhat unusual polarizable medium, a Lorentzian susceptibility with
 
 ### GyrotropicLorentzianSusceptibility or GyrotropicDrudeSusceptibility
 
-(**Experimental feature**) Specifies a single dispersive gyrotropic susceptibility of Lorentzian (damped harmonic oscillator) or Drude form. See [Material Dispersion](Materials.md#gyrotropic-media), with the same `sigma`, `frequency`, and `gamma` parameters, but with an additional 3-vector `bias`:
+(**Experimental feature**) Specifies a single dispersive gyrotropic susceptibility of Lorentzian (damped harmonic oscillator) or Drude form. See [Material Dispersion](Materials.md#gyrotropic-media). The parameters `sigma`, `frequency`, and `gamma` have the usual meanings, and there is an additional 3-vector `bias`:
 
 **`bias` [`Vector3`]**
 —

From 9e595d0f53d49fe273a56409123821e84220aa58 Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Wed, 22 May 2019 20:36:26 +0800
Subject: [PATCH 07/61] Fix logic in py_susceptibility_to_susceptibility

---
 python/typemap_utils.cpp | 22 +++++++++++-----------
 1 file changed, 11 insertions(+), 11 deletions(-)

diff --git a/python/typemap_utils.cpp b/python/typemap_utils.cpp
index dea028d1d..e5d569bae 100644
--- a/python/typemap_utils.cpp
+++ b/python/typemap_utils.cpp
@@ -464,17 +464,7 @@ static PyObject *susceptibility_to_py_obj(susceptibility_struct *s) {
   PyObject *res;
   PyObject *args = PyTuple_New(0);
 
-  if (s->noise_amp == 0) {
-    if (s->drude) {
-      PyObject *py_drude_class = PyObject_GetAttrString(geom_mod, "DrudeSusceptibility");
-      res = PyObject_Call(py_drude_class, args, NULL);
-      Py_DECREF(py_drude_class);
-    } else {
-      PyObject *py_lorentz_class = PyObject_GetAttrString(geom_mod, "LorentzianSusceptibility");
-      res = PyObject_Call(py_lorentz_class, args, NULL);
-      Py_DECREF(py_lorentz_class);
-    }
-  } else if (s->bias.x || s->bias.y || s->bias.z) {
+  if (s->bias.x || s->bias.y || s->bias.z) {
     if (s->drude) {
       PyObject *py_gyrotropic_drude_class = PyObject_GetAttrString(geom_mod, "GyrotropicDrudeSusceptibility");
       res = PyObject_Call(py_gyrotropic_drude_class, args, NULL);
@@ -488,6 +478,16 @@ static PyObject *susceptibility_to_py_obj(susceptibility_struct *s) {
     PyObject *py_bias = vec2py(vector3_to_vec(s->bias));
     PyObject_SetAttrString(res, "bias", py_bias);
     Py_DECREF(py_bias);
+  } else if (s->noise_amp == 0) {
+    if (s->drude) {
+      PyObject *py_drude_class = PyObject_GetAttrString(geom_mod, "DrudeSusceptibility");
+      res = PyObject_Call(py_drude_class, args, NULL);
+      Py_DECREF(py_drude_class);
+    } else {
+      PyObject *py_lorentz_class = PyObject_GetAttrString(geom_mod, "LorentzianSusceptibility");
+      res = PyObject_Call(py_lorentz_class, args, NULL);
+      Py_DECREF(py_lorentz_class);
+    }
   } else {
     if (s->drude) {
       PyObject *py_noisy_drude_class = PyObject_GetAttrString(geom_mod, "NoisyDrudeSusceptibility");

From 074ecf097bc2c35491d71e3312a5e9421fc05f04 Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Thu, 23 May 2019 00:07:42 +0800
Subject: [PATCH 08/61] In add_susceptibilities, always pass a 3-vector as
 gyrotropic bias

---
 src/meepgeom.cpp | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/src/meepgeom.cpp b/src/meepgeom.cpp
index e1cb860b6..d42c2aec6 100644
--- a/src/meepgeom.cpp
+++ b/src/meepgeom.cpp
@@ -1375,16 +1375,16 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
         sus = new meep::noisy_lorentzian_susceptibility(ss->noise_amp, ss->frequency, ss->gamma,
                                                         ss->drude);
       } else if (gyrotropic) {
-        sus = new meep::gyrotropic_susceptibility(vector3_to_vec(ss->bias), ss->frequency,
+        sus = new meep::gyrotropic_susceptibility(meep::vec(ss->bias.x, ss->bias.y, ss->bias.z), ss->frequency,
 						  ss->gamma, ss->drude);
       } else {
         sus = new meep::lorentzian_susceptibility(ss->frequency, ss->gamma, ss->drude);
       }
       master_printf("%s%s susceptibility: frequency=%g, gamma=%g",
-		    noisy ? "noisy " : gyrotropic ? "gyrotropic" : "",
+		    noisy ? "noisy " : gyrotropic ? "gyrotropic " : "",
                     ss->drude ? "drude" : "lorentzian", ss->frequency, ss->gamma);
-      if (noisy) master_printf(" amp=%g ", ss->noise_amp);
-      if (gyrotropic) master_printf(" bias=(%g,%g,%g) ", ss->bias.x, ss->bias.y, ss->bias.z);
+      if (noisy) master_printf(", amp=%g ", ss->noise_amp);
+      if (gyrotropic) master_printf(", bias=(%g,%g,%g) ", ss->bias.x, ss->bias.y, ss->bias.z);
       master_printf("\n");
     }
 

From 579b76d331fe8739cbd72e6f24453bfe4e8b6e24 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Thu, 23 May 2019 09:17:32 +0800
Subject: [PATCH 09/61] First try at gyrotropic media tutorial

---
 doc/docs/Materials.md                         |   4 +-
 doc/docs/Python_Tutorials/Gyrotropic_Media.md | 121 ++++++++++++++++++
 2 files changed, 123 insertions(+), 2 deletions(-)
 create mode 100644 doc/docs/Python_Tutorials/Gyrotropic_Media.md

diff --git a/doc/docs/Materials.md b/doc/docs/Materials.md
index 882e1a76a..12c03b61b 100644
--- a/doc/docs/Materials.md
+++ b/doc/docs/Materials.md
@@ -182,11 +182,11 @@ $$\frac{d\mathbf{P}_n}{dt} = \mathbf{P}_n \times \mathbf{b}_n$$
 
 In the frequency domain, gyrotropy corresponds to the presence of imaginary off-diagonal components in the ε tensor (or the μ tensor for a gyromagnetic medium). Take the case $\mathbf{b} = b \hat{z}$.  When all fields have harmonic time-dependence $\exp(-i\omega t)$, the polarization equation of motion reduces to
 
-$$\mathbf{P}_n = \chi_n \mathbf{E}, \quad \chi_n = \frac{\omega_n^2}{\omega_n^2 - \omega^2 + i\omega\gamma_n} \begin{bmatrix}\xi & -i\eta & 0 \\ i\eta & \xi & 0 \\ 0 & 0 & 1 \end{bmatrix} \, \sigma_n(\mathbf{x})$$
+$$\mathbf{P}_n = \chi_n \mathbf{E}, \quad \chi_n = \frac{\omega_n^2}{\omega_n^2 - \omega^2 - i\omega\gamma_n} \begin{bmatrix}\xi & -i\eta & 0 \\ i\eta & \xi & 0 \\ 0 & 0 & 1 \end{bmatrix} \, \sigma_n(\mathbf{x})$$
 
 where
 
-$$\xi = \frac{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2}{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2 + \omega^2 b^2}\,, \quad \eta = \frac{\omega b (\omega_n^2 - \omega^2 - i\omega \gamma_n)}{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2 + \omega^2 b^2}$$
+$$\xi = \frac{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2}{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2 - \omega^2 b^2}\,, \quad \eta = \frac{\omega b (\omega_n^2 - \omega^2 - i\omega \gamma_n)}{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2 - \omega^2 b^2}$$
 
 It is common to operate in a limit where the gyrotropy is weak relative to the frequency detuning, but strong relative to the loss.  Let $\omega_n = \omega + \Delta\omega_n$, where $\gamma_n \ll b \ll \Delta \omega_n \ll \omega$.  Then the above susceptibility tensor simplifies to
 
diff --git a/doc/docs/Python_Tutorials/Gyrotropic_Media.md b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
new file mode 100644
index 000000000..b67fb0afc
--- /dev/null
+++ b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
@@ -0,0 +1,121 @@
+---
+# Gyrotropic media
+---
+
+In this example, we will perform simulations with a gyrotropic medium. See [Materials](../Materials.md#gyrotropic-media) for more information on how gyrotropy is supported.
+
+[TOC]
+
+### Dispersion Relation for a Gyrotropic Medium
+
+Here, we model a uniform gyrotroelectric medium with bias vector $\mathbf{b} = b \hat{z}$. For TE modes, the electric field lies in the *x*-*y* plane and magnetic field pointing along *z*, with the wavevector $\mathbf{k}$ in the *x*-*y* plane.  The dispersion relation can be shown to be
+
+$$\omega = v(\omega) |\mathbf{k}|, \quad v^2 = \mu^{-1} \left[\epsilon^{-1}\right]_{xx}$$
+
+The ε tensor is given by
+
+$$\epsilon = \epsilon_\infty + \frac{\omega_n^2}{\omega_n^2 - \omega^2 - i\omega\gamma_n} \begin{bmatrix}\xi & -i\eta & 0 \\ i\eta & \xi & 0 \\ 0 & 0 & 1 \end{bmatrix} \, \sigma_n(\mathbf{x}),$$
+
+with
+
+$$\xi = \frac{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2}{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2 + \omega^2 b^2}\,, \quad \eta = \frac{\omega b (\omega_n^2 - \omega^2 - i\omega \gamma_n)}{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2 + \omega^2 b^2}$$
+
+We choose the material parameters $\epsilon_\infty = 1$, $f_n = 1.2$, $\gamma_n = 10^{-5}$, and $\sigma_n = 0.1$, with $|\mathbf{b}_n|$ ranging from zero to $10^{-2}$.
+
+
+
+
+From the dispersion relation ω(k), we will compute the numerical ε(ω) via the formula:
+
+$$\varepsilon(\omega) = \left( \frac{ck}{\omega} \right) ^2$$
+
+We will then compare this with the analytical ε(ω) that we specified. The simulation script is in [material-dispersion.py](https://github.com/NanoComp/meep/blob/master/python/examples/material-dispersion.py).
+
+Since this is a uniform medium, our computational cell can actually be of *zero* size (i.e. one pixel), where we will use Bloch-periodic boundary conditions to specify the wavevector *k*.
+
+```py
+cell = mp.Vector3()
+resolution = 20
+```
+
+We will then fill all space with an artificial dispersive material:
+
+```py
+susceptibilities = [mp.LorentzianSusceptibility(frequency=1.1, gamma=1e-5, sigma=0.5),
+                    mp.LorentzianSusceptibility(frequency=0.5, gamma=0.1, sigma=2e-5)]
+
+default_material = mp.Medium(epsilon=2.25, E_susceptibilities=susceptibilities)
+```
+
+corresponding to the dielectric function:
+
+$$\varepsilon(\omega) = \varepsilon(2\pi f) = 2.25 + \frac{1.1^2 \cdot 0.5}{1.1^2 - f^2 -if \cdot 10^{-5}/2\pi} + \frac{0.5^2 \cdot 2\cdot 10^{-5}}{0.5^2 - f^2 -if \cdot 0.1 / 2\pi}$$
+
+The real and imaginary parts of this dielectric function ε(ω) are plotted below:
+
+<center>
+![](../images/Material-dispersion-eps.png)
+</center>
+
+We can see that the f=1.1 resonance causes a large change in both the real and imaginary parts of ε around that frequency. In fact, there is a range of frequencies from 1.1 to 1.2161 where ε is *negative*. In this range, no propagating modes exist &mdash; it is actually a kind of electromagnetic band gap associated with polariton resonances in a material. For more information on the physics of such materials, see e.g. Chapter 14 of [Introduction to Solid State Physics](http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000803.html) by C. Kittel.
+
+On the other hand, the f=0.5 resonance, because the `sigma` numerator is so small, causes very little change in the real part of ε. Nevertheless, it generates a clear peak in the *imaginary* part of ε, corresponding to a resonant absorption peak.
+
+Now, we'll set up the rest of the simulation. We'll specify a broadband $E_z$-polarized Gaussian source, create a list of *k* wavevectors that we want to compute ω(k) over, and compute the associated frequencies by using the `k_points` function:
+
+```py
+fcen = 1.0
+df = 2.0
+
+sources = [mp.Source(mp.GaussianSource(fcen, fwidth=df), component=mp.Ez, center=mp.Vector3())]
+
+kmin = 0.3
+kmax = 2.2
+k_interp = 99
+
+kpts = mp.interpolate(k_interp, [mp.Vector3(kmin), mp.Vector3(kmax)])
+
+sim = mp.Simulation(cell_size=cell, geometry=[], sources=sources, default_material=default_material, resolution=resolution)
+
+all_freqs = sim.run_k_points(200, kpts)  # a list of lists of frequencies
+```
+
+The `run_k_points` function returns a *list of lists* of frequencies &mdash; one list of complex frequencies for each *k* point &mdash; which we store in the `all_freqs` variable. Finally, we want to loop over this list and print out the corresponding ε via the ratio (ck/ω)$^2$ as described above. To do this, we will use Python's `zip` function which combines multiple lists into one:
+
+```py
+for fs, kx in zip(all_freqs, [v.x for v in kpts]):
+    for f in fs:
+            print("eps:, {.6f}, {.6f}, {.6f}".format(f.real, f.imag, (kx / f)**2))
+```
+
+Alternatively we could just read all of the frequencies into Python or Octave/Matlab and compute the ratios there. After running the program with
+
+```sh
+unix% python -u material-dispersion.py | tee material-dispersion.out
+```
+
+we can then `grep` for the frequencies and the computed dielectric function, and plot it. First, let's plot the dispersion relation ω(k) for the real part of ω:
+
+<center>
+![](../images/Material-dispersion-bands.png)
+</center>
+
+The red circles are the computed points from Meep, whereas the blue line is the analytical band diagram from the specified ε(ω). As you can see, we get *two* bands at each *k*, separated by a polaritonic gap (shaded yellow). This dispersion relation can be thought of as the interaction (anti-crossing) between the light line of the ambient ε=2.25 material (dashed black line) and the horizontal line corresponding to the phonon resonance.
+
+Similarly, the computed and analytical real parts of the dielectric function are given by:
+
+<center>
+![](../images/Material-dispersion-epsre.png)
+</center>
+
+which shows excellent agreement between the analytical (blue line) and numerical (red circles) calculations. The imaginary part, however, is more subtle:
+
+<center>
+![](../images/Material-dispersion-epsim.png)
+</center>
+
+The blue line is the analytical calculation from above and the red circles are the numerical value from Meep &mdash; why is the agreement so poor? There is nothing wrong with Meep, and this is *not* a numerical error. The problem is simply that we are comparing apples and oranges.
+
+The blue line is the analytical calculation of ε(ω) for a *real* frequency ω which corresponds to solutions with a *complex* wavevector *k*, whereas Meep is computing ε at a *complex* ω for a *real* wavevector *k*. So, the correct comparison is to plug Meep's *complex* ω into the analytical formula for ε(ω), which results in the green lines on the graph that fall almost on top of the red circles.
+
+Why did our comparison of the *real* part of ε look so good, then? The reason is that ε(ω) at real and complex values of ω are closely related by the analytic properties of ε. In particular, because ε is an analytic function on the real-ω axis, adding a *small* imaginary part to ω as we are doing here does not change ε by much. The losses are small for all of the computed *k* points. The change was only significant for the imaginary ε because the imaginary ε was small to begin with.

From afd8583485670b0c0dccba2b72e2c77066827527 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Thu, 23 May 2019 13:37:39 +0800
Subject: [PATCH 10/61] Fix errors in gyrotropy formulas in notes

---
 doc/docs/Materials.md | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/doc/docs/Materials.md b/doc/docs/Materials.md
index 12c03b61b..d6316aa3b 100644
--- a/doc/docs/Materials.md
+++ b/doc/docs/Materials.md
@@ -176,21 +176,21 @@ In a gyroelectric medium, the equation of motion of the polarization vector $\ma
 
 $$\frac{d^2\mathbf{P}_n}{dt^2} + \gamma_n \frac{d\mathbf{P}_n}{dt} + \mathbf{b}_n \times \frac{d\mathbf{P}_n}{dt} + \omega_n^2 \mathbf{P}_n = \sigma_n(\mathbf{x}) \omega_n^2 \mathbf{E}$$
 
-(Optionally, the polarization may be of Drude form, in which case the final $\omega_n^2 \mathbf{P}_n$ term on the left-hand side is omitted; we will describe the Lorentz case.) The third term on the left-hand side breaks time-reversal symmetry and describes the gyrotropy of the medium, parameterized by the bias vector $\mathbf{b}_n$ (usually describing the effect of an applied static magnetic field). In the absence of damping ($\gamma = 0$) and coupling to the electric field ($\sigma_n = 0$), the equation of motion reduces to a precession around $\mathbf{b}_n$, with angular frequency equal to $|\mathbf{b}_n|$:
+(Optionally, the polarization may be of Drude form, in which case the $\omega_n^2 \mathbf{P}_n$ term on the left-hand side is omitted.) The third term on the left-hand side breaks time-reversal symmetry and describes the gyrotropy of the medium; it is parameterized by the bias vector $\mathbf{b}_n$, usually describing the effect of an applied static magnetic field. In the $\gamma_n = \omega_n = 0$ limit, the equation of motion reduces to a precession around $\mathbf{b}_n$, with angular frequency equal to $|\mathbf{b}_n|$:
 
 $$\frac{d\mathbf{P}_n}{dt} = \mathbf{P}_n \times \mathbf{b}_n$$
 
-In the frequency domain, gyrotropy corresponds to the presence of imaginary off-diagonal components in the ε tensor (or the μ tensor for a gyromagnetic medium). Take the case $\mathbf{b} = b \hat{z}$.  When all fields have harmonic time-dependence $\exp(-i\omega t)$, the polarization equation of motion reduces to
+In the frequency domain, a gyroelectric medium has imaginary off-diagonal components in the ε tensor. Take the case $\mathbf{b} = b \hat{z}$. When all fields have harmonic time-dependence $\exp(-i\omega t)$, the polarization equation of motion reduces to
 
-$$\mathbf{P}_n = \chi_n \mathbf{E}, \quad \chi_n = \frac{\omega_n^2}{\omega_n^2 - \omega^2 - i\omega\gamma_n} \begin{bmatrix}\xi & -i\eta & 0 \\ i\eta & \xi & 0 \\ 0 & 0 & 1 \end{bmatrix} \, \sigma_n(\mathbf{x})$$
+$$\mathbf{P}_n = \chi_n \mathbf{E}, \quad \chi_n = \omega_n^2 \begin{bmatrix}\xi_\perp & -i\eta & 0 \\ i\eta & \xi_\perp & 0 \\ 0 & 0 & \xi_\parallel \end{bmatrix} \, \sigma_n(\mathbf{x})$$
 
 where
 
-$$\xi = \frac{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2}{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2 - \omega^2 b^2}\,, \quad \eta = \frac{\omega b (\omega_n^2 - \omega^2 - i\omega \gamma_n)}{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2 - \omega^2 b^2}$$
+$$\xi_\perp = \frac{\Delta_n}{\Delta_n^2 - \omega^2 b^2}\,,\;\;\; \xi_\parallel = \frac{1}{\Delta_n}, \;\;\; \eta = \frac{\omega b}{\Delta_n^2 - \omega^2 b^2}, \;\;\;\Delta_n \equiv \omega_n^2 - \omega^2 - i\omega\gamma_n$$
 
-It is common to operate in a limit where the gyrotropy is weak relative to the frequency detuning, but strong relative to the loss.  Let $\omega_n = \omega + \Delta\omega_n$, where $\gamma_n \ll b \ll \Delta \omega_n \ll \omega$.  Then the above susceptibility tensor simplifies to
+In many applications, the gyrotropy is weak relative to the frequency detuning but strong relative to the loss.  For this case, we can define $\omega_n = \omega + \Delta\omega_n$, and take $\gamma_n \ll b \ll \Delta \omega_n \ll \omega$. In this limit, the susceptibility tensor reduces to
 
-$$\chi_n \approx \frac{\omega}{2 \Delta\omega_n} \begin{bmatrix}1 & -i\eta & 0 \\ i\eta & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\, \sigma_n(\mathbf{x}), \quad \eta = \frac{b}{2\Delta\omega_n}$$
+$$\chi_n \approx \frac{\omega}{2\Delta\omega_n} \begin{bmatrix}1 & -ib/2\Delta\omega_n & 0 \\ ib/2\Delta\omega_n & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} \sigma_n(\mathbf{x})$$
 
 Materials Library
 -----------------

From 69a302b8691bbf1d88936e30ebe58822b05e70dc Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Thu, 23 May 2019 23:08:54 +0800
Subject: [PATCH 11/61] Re-implemention of gyrotropy using LLG equation

---
 doc/docs/Materials.md    |  20 +++--
 python/geom.py           |  14 +---
 python/meep.i            |  15 ++--
 python/typemap_utils.cpp |  24 +++---
 scheme/meep.scm.in       |   7 +-
 scheme/structure.cpp     |  18 ++---
 src/material_data.hpp    |   1 +
 src/meep.hpp             |  12 ++-
 src/meepgeom.cpp         |  11 ++-
 src/susceptibility.cpp   | 165 +++++++++++++++------------------------
 10 files changed, 118 insertions(+), 169 deletions(-)

diff --git a/doc/docs/Materials.md b/doc/docs/Materials.md
index d6316aa3b..0205127b2 100644
--- a/doc/docs/Materials.md
+++ b/doc/docs/Materials.md
@@ -172,25 +172,23 @@ Gyrotropic Media
 
 (**Experimental feature**) Meep supports gyrotropic media, which break optical reciprocity and give rise to magneto-optical phenomena such as the [Faraday effect](https://en.wikipedia.org/wiki/Faraday_effect). Such materials are used in devices like [Faraday rotators](https://en.wikipedia.org/wiki/Faraday_rotator). The following discussion concerns a medium with gyrotropic ε (i.e., a gyroelectric medium). Gyromagnetic media can be treated by the usual substitution of ε with μ, **E** with **H**, etc.
 
-In a gyroelectric medium, the equation of motion of the polarization vector $\mathbf{P}_n$ is very similar to the [Drude-Lorentz](Materials.md#material-dispersion) case, but contains an additional precession term:
+In a gyroelectric medium, the equation of motion of the polarization vector $\mathbf{P}_n$ is a variant of the [Landau-Lifshitz-Gilbert equation](https://en.wikipedia.org/wiki/Landau%E2%80%93Lifshitz%E2%80%93Gilbert_equation):
 
-$$\frac{d^2\mathbf{P}_n}{dt^2} + \gamma_n \frac{d\mathbf{P}_n}{dt} + \mathbf{b}_n \times \frac{d\mathbf{P}_n}{dt} + \omega_n^2 \mathbf{P}_n = \sigma_n(\mathbf{x}) \omega_n^2 \mathbf{E}$$
+$$\frac{d\mathbf{P}_n}{dt} = \mathbf{b}_n \times \left( - \sigma_n \mathbf{E} + \omega_n \mathbf{P}_n + \alpha_n \frac{d\mathbf{P}_n}{dt} \right) - \gamma_n \mathbf{P}_n$$
 
-(Optionally, the polarization may be of Drude form, in which case the $\omega_n^2 \mathbf{P}_n$ term on the left-hand side is omitted.) The third term on the left-hand side breaks time-reversal symmetry and describes the gyrotropy of the medium; it is parameterized by the bias vector $\mathbf{b}_n$, usually describing the effect of an applied static magnetic field. In the $\gamma_n = \omega_n = 0$ limit, the equation of motion reduces to a precession around $\mathbf{b}_n$, with angular frequency equal to $|\mathbf{b}_n|$:
+Although this is completely different from the damped harmonic oscillator equation used for [Drude-Lorentz susceptibility](Materials.md#material-dispersion), the constants σ$_n$, ω$_n$, and γ$_n$ play rather analogous roles: σ$_n$ couples the polarization to the electric field, ω$_n$ is the angular frequency of precession, and γ$_n$ is a damping factor. There is also a second damping factor α$_n$, and a "bias vector" **b**$_n$ which typically describes the direction of an applied static magnetic field. The bias vector is assumed to have unit length.
 
-$$\frac{d\mathbf{P}_n}{dt} = \mathbf{P}_n \times \mathbf{b}_n$$
+In the frequency domain, the ε tensor acquires off-diagonal components representing a gyrotropic response. Take the case $\mathbf{b}_n = \hat{\mathbf{z}}$, and assume that all fields have harmonic time-dependence $\exp(-i\omega t)$. The polarization equation then reduces to
 
-In the frequency domain, a gyroelectric medium has imaginary off-diagonal components in the ε tensor. Take the case $\mathbf{b} = b \hat{z}$. When all fields have harmonic time-dependence $\exp(-i\omega t)$, the polarization equation of motion reduces to
+$$\mathbf{P}_n = \chi_n \mathbf{E}, \quad \chi_n = \frac{\sigma_n}{(\omega_n - i \omega \alpha_n)^2 - (\omega + i \gamma_n)^2} \begin{bmatrix} \omega_n - i \omega \alpha_n & -i(\omega + i \gamma) & 0 \\ i(\omega + i \gamma) & \omega_n - i \omega \alpha_n & 0 \\ 0 & 0 & 0\end{bmatrix}$$
 
-$$\mathbf{P}_n = \chi_n \mathbf{E}, \quad \chi_n = \omega_n^2 \begin{bmatrix}\xi_\perp & -i\eta & 0 \\ i\eta & \xi_\perp & 0 \\ 0 & 0 & \xi_\parallel \end{bmatrix} \, \sigma_n(\mathbf{x})$$
+Hence, the dielectric function is
 
-where
-
-$$\xi_\perp = \frac{\Delta_n}{\Delta_n^2 - \omega^2 b^2}\,,\;\;\; \xi_\parallel = \frac{1}{\Delta_n}, \;\;\; \eta = \frac{\omega b}{\Delta_n^2 - \omega^2 b^2}, \;\;\;\Delta_n \equiv \omega_n^2 - \omega^2 - i\omega\gamma_n$$
+$$\epsilon = \begin{bmatrix} \epsilon_\perp & -i \eta & 0 \\ i\eta & \epsilon_\perp & 0 \\ 0 & 0 & \epsilon_\infty\end{bmatrix}$$
 
-In many applications, the gyrotropy is weak relative to the frequency detuning but strong relative to the loss.  For this case, we can define $\omega_n = \omega + \Delta\omega_n$, and take $\gamma_n \ll b \ll \Delta \omega_n \ll \omega$. In this limit, the susceptibility tensor reduces to
+where
 
-$$\chi_n \approx \frac{\omega}{2\Delta\omega_n} \begin{bmatrix}1 & -ib/2\Delta\omega_n & 0 \\ ib/2\Delta\omega_n & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} \sigma_n(\mathbf{x})$$
+$$\epsilon_\perp = \epsilon_\infty + \frac{\sigma_n (\omega_n - i\omega\alpha_n)}{(\omega_n - i \omega \alpha_n)^2 - (\omega + i \gamma_n)^2}, \;\;\; \eta = \frac{\sigma_n (\omega + i\gamma_n)}{(\omega_n - i \omega \alpha_n)^2 - (\omega + i \gamma_n)^2}$$
 
 Materials Library
 -----------------
diff --git a/python/geom.py b/python/geom.py
index e4d8fec14..4101c53e4 100644
--- a/python/geom.py
+++ b/python/geom.py
@@ -280,18 +280,12 @@ def __init__(self, noise_amp=0.0, **kwargs):
         super(NoisyDrudeSusceptibility, self).__init__(**kwargs)
         self.noise_amp = noise_amp
 
-class GyrotropicLorentzianSusceptibility(LorentzianSusceptibility):
+class GyrotropicSusceptibility(LorentzianSusceptibility):
 
-    def __init__(self, bias=Vector3(0, 0, 1), **kwargs):
-        super(GyrotropicLorentzianSusceptibility, self).__init__(**kwargs)
+    def __init__(self, bias=Vector3(0, 0, 1), alpha=0.0, **kwargs):
+        super(GyrotropicSusceptibility, self).__init__(**kwargs)
         self.bias = bias
-
-class GyrotropicDrudeSusceptibility(DrudeSusceptibility):
-
-    def __init__(self, bias=Vector3(), **kwargs):
-        super(GyrotropicDrudeSusceptibility, self).__init__(**kwargs)
-        self.bias = bias
-
+        self.alpha = alpha
 
 class MultilevelAtom(Susceptibility):
 
diff --git a/python/meep.i b/python/meep.i
index 998f13133..d984185d2 100644
--- a/python/meep.i
+++ b/python/meep.i
@@ -749,21 +749,21 @@ meep::volume_list *make_volume_list(const meep::volume &v, int c,
 }
 
 %typemap(in) const meep::src_time & {
-    PyObject *swig_obj = NULL;
+    PyObject *obj = NULL;
     void *tmp_ptr = 0;
     int tmp_res = 0;
 
     if(PyObject_IsInstance($input, py_source_time_object())) {
-        swig_obj = PyObject_GetAttrString($input, "swigobj");
+        obj = PyObject_GetAttrString($input, "swigobj");
     } else if(PyObject_IsInstance($input, py_meep_src_time_object())) {
-        swig_obj = $input;
-        Py_XINCREF(swig_obj);
+        obj = $input;
+        Py_XINCREF(obj);
     } else {
       meep::abort("Expected a meep.source.SourceTime or a meep.src_time\n");
     }
 
-    tmp_res = SWIG_ConvertPtr(swig_obj, &tmp_ptr, $1_descriptor, 0);
-    Py_XDECREF(swig_obj);
+    tmp_res = SWIG_ConvertPtr(obj, &tmp_ptr, $1_descriptor, 0);
+    Py_XDECREF(obj);
 
     if(!SWIG_IsOK(tmp_res)) {
         SWIG_exception_fail(SWIG_ArgError(tmp_res), "Couldn't convert Python object to meep::src_time");
@@ -1371,8 +1371,7 @@ PyObject *_get_array_slice_dimensions(meep::fields *f, const meep::volume &where
         Ellipsoid,
         FreqRange,
         GeometricObject,
-        GyrotropicDrudeSusceptibility,
-        GyrotropicLorentzianSusceptibility,
+        GyrotropicSusceptibility,
         Lattice,
         LorentzianSusceptibility,
         Matrix,
diff --git a/python/typemap_utils.cpp b/python/typemap_utils.cpp
index e5d569bae..60d9a776a 100644
--- a/python/typemap_utils.cpp
+++ b/python/typemap_utils.cpp
@@ -331,6 +331,7 @@ static int py_susceptibility_to_susceptibility(PyObject *po, susceptibility_stru
   s->gamma = 0;
   s->noise_amp = 0;
   s->bias.x = s->bias.y = s->bias.z = 0;
+  s->alpha = 0;
   s->transitions.resize(0);
   s->initial_populations.resize(0);
 
@@ -350,6 +351,10 @@ static int py_susceptibility_to_susceptibility(PyObject *po, susceptibility_stru
     if (!get_attr_v3(po, &s->bias, "bias")) return 0;
   }
 
+  if (PyObject_HasAttrString(po, "alpha")) {
+    if (!get_attr_dbl(po, &s->alpha, "alpha")) { return 0; }
+  }
+
   if (PyObject_HasAttrString(po, "transitions")) {
     // MultilevelAtom
     PyObject *py_trans = PyObject_GetAttrString(po, "transitions");
@@ -465,19 +470,18 @@ static PyObject *susceptibility_to_py_obj(susceptibility_struct *s) {
   PyObject *args = PyTuple_New(0);
 
   if (s->bias.x || s->bias.y || s->bias.z) {
-    if (s->drude) {
-      PyObject *py_gyrotropic_drude_class = PyObject_GetAttrString(geom_mod, "GyrotropicDrudeSusceptibility");
-      res = PyObject_Call(py_gyrotropic_drude_class, args, NULL);
-      Py_DECREF(py_gyrotropic_drude_class);
-    } else {
-      PyObject *py_gyrotropic_lorentz_class =
-	PyObject_GetAttrString(geom_mod, "GyrotropicLorentzianSusceptibility");
-      res = PyObject_Call(py_gyrotropic_lorentz_class, args, NULL);
-      Py_DECREF(py_gyrotropic_lorentz_class);
-    }
+    PyObject *py_gyrotropic_class = PyObject_GetAttrString(geom_mod, "GyrotropicSusceptibility");
+    res = PyObject_Call(py_gyrotropic_class, args, NULL);
+    Py_DECREF(py_gyrotropic_class);
+
     PyObject *py_bias = vec2py(vector3_to_vec(s->bias));
     PyObject_SetAttrString(res, "bias", py_bias);
     Py_DECREF(py_bias);
+
+    PyObject *py_alpha = PyFloat_FromDouble(s->alpha);
+    PyObject_SetAttrString(res, "alpha", py_alpha);
+    Py_DECREF(py_alpha);
+
   } else if (s->noise_amp == 0) {
     if (s->drude) {
       PyObject *py_drude_class = PyObject_GetAttrString(geom_mod, "DrudeSusceptibility");
diff --git a/scheme/meep.scm.in b/scheme/meep.scm.in
index cfec58b06..06f5df939 100644
--- a/scheme/meep.scm.in
+++ b/scheme/meep.scm.in
@@ -57,10 +57,9 @@
   (define-property noise-amp no-default 'number))
 (define-class noisy-drude-susceptibility drude-susceptibility
   (define-property noise-amp no-default 'number))
-(define-class gyrotropic-lorentzian-susceptibility lorentzian-susceptibility
-  (define-property bias no-default 'vector3))
-(define-class gyrotropic-drude-susceptibility drude-susceptibility
-  (define-property bias no-default 'vector3))
+(define-class gyrotropic-susceptibility lorentzian-susceptibility
+  (define-property bias no-default 'vector3)
+  (define-property alpha no-default 'number))
 
 (define polarizability lorentzian-susceptibility) ; backwards compat
 (define omega frequency) ; backwards compat
diff --git a/scheme/structure.cpp b/scheme/structure.cpp
index 068915971..be5b99789 100644
--- a/scheme/structure.cpp
+++ b/scheme/structure.cpp
@@ -1295,11 +1295,12 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
           master_printf("noisy lorentzian susceptibility: frequency=%g, gamma=%g, amp = %g\n",
                         d->frequency, d->gamma, nd->noise_amp);
           sus = new meep::noisy_lorentzian_susceptibility(nd->noise_amp, d->frequency, d->gamma);
-	} else if (d->which_subclass == lorentzian_susceptibility::GYROTROPIC_LORENTZIAN_SUSCEPTIBILITY) {
-          gyrotropic_lorentzian_susceptibility *gd = d->subclass.gyrotropic_lorentzian_susceptibility_data;
-          master_printf("gyrotropic lorentzian susceptibility: bias=(%g,%g,%g), frequency=%g, gamma=%g\n",
-			gd->bias.x, gd->bias.y, gd->bias.z, d->frequency, d->gamma);
-          sus = new meep::gyrotropic_susceptibility(vector3_to_vec(gd->bias), d->frequency, d->gamma);
+	} else if (d->which_subclass == lorentzian_susceptibility::GYROTROPIC_SUSCEPTIBILITY) {
+          gyrotropic_susceptibility *gd = d->subclass.gyrotropic_susceptibility_data;
+          master_printf("gyrotropic lorentzian susceptibility: bias=(%g,%g,%g), alpha=%g, frequency=%g, gamma=%g\n",
+			gd->bias.x, gd->bias.y, gd->bias.z, gd->alpha, d->frequency, d->gamma);
+          sus = new meep::gyrotropic_susceptibility(vector3_to_vec(gd->bias), gd->alpha,
+                                                    d->frequency, d->gamma);
         } else { // just a Lorentzian
           master_printf("lorentzian susceptibility: frequency=%g, gamma=%g\n", d->frequency,
                         d->gamma);
@@ -1315,12 +1316,7 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
                         d->frequency, d->gamma, nd->noise_amp);
           sus = new meep::noisy_lorentzian_susceptibility(nd->noise_amp, d->frequency, d->gamma,
                                                           true);
-	} else if (d->which_subclass == drude_susceptibility::GYROTROPIC_DRUDE_SUSCEPTIBILITY) {
-          gyrotropic_drude_susceptibility *gd = d->subclass.gyrotropic_drude_susceptibility_data;
-          master_printf("gyrotropic drude susceptibility: bias=(%g,%g,%g), frequency=%g, gamma=%g\n",
-			gd->bias.x, gd->bias.y, gd->bias.z, d->frequency, d->gamma);
-          sus = new meep::gyrotropic_susceptibility(vector3_to_vec(gd->bias), d->frequency, d->gamma, true);
-        } else { // just a Drude
+	} else { // just a Drude
           master_printf("drude susceptibility: frequency=%g, gamma=%g\n", d->frequency, d->gamma);
           sus = new meep::lorentzian_susceptibility(d->frequency, d->gamma, true);
         }
diff --git a/src/material_data.hpp b/src/material_data.hpp
index 352b6155e..ec5c998c4 100644
--- a/src/material_data.hpp
+++ b/src/material_data.hpp
@@ -60,6 +60,7 @@ typedef struct susceptibility_struct {
   vector3 sigma_offdiag;
   vector3 sigma_diag;
   vector3 bias;
+  double alpha;
   double frequency;
   double gamma;
   double noise_amp;
diff --git a/src/meep.hpp b/src/meep.hpp
index c03283e09..0e0a401dd 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -223,8 +223,7 @@ class susceptibility {
 class lorentzian_susceptibility : public susceptibility {
 public:
   lorentzian_susceptibility(double omega_0, double gamma, bool no_omega_0_denominator = false)
-      : omega_0(omega_0), gamma(gamma), no_omega_0_denominator(no_omega_0_denominator),
-	have_gyrotropy(false) {}
+      : omega_0(omega_0), gamma(gamma), no_omega_0_denominator(no_omega_0_denominator) {}
   virtual susceptibility *clone() const { return new lorentzian_susceptibility(*this); }
   virtual ~lorentzian_susceptibility() {}
 
@@ -250,7 +249,6 @@ class lorentzian_susceptibility : public susceptibility {
 protected:
   double omega_0, gamma;
   bool no_omega_0_denominator;
-  bool have_gyrotropy; // whether to assign an extra slot for gyrotropy calculations
 };
 
 /* like a Lorentzian susceptibility, but the polarization equation
@@ -274,11 +272,10 @@ class noisy_lorentzian_susceptibility : public lorentzian_susceptibility {
   double noise_amp;
 };
 
-/* like a Lorentzian susceptibility, but with precession around a bias vector */
+/* gyrotropic susceptibility */
 class gyrotropic_susceptibility : public lorentzian_susceptibility {
 public:
-  gyrotropic_susceptibility(const vec &bias, double omega_0, double gamma,
-			    bool no_omega_0_denominator = true);
+  gyrotropic_susceptibility(const vec &bias, double alpha, double omega_0, double gamma);
   virtual susceptibility *clone() const { return new gyrotropic_susceptibility(*this); }
 
   virtual void update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
@@ -286,9 +283,10 @@ class gyrotropic_susceptibility : public lorentzian_susceptibility {
 			const grid_volume &gv, void *P_internal_data) const;
 
   virtual void dump_params(h5file *h5f, size_t *start);
-  virtual int get_num_params() { return 7; }
+  virtual int get_num_params() { return 8; }
 
 protected:
+  double alpha;
   double gyro_tensor[3][3];
 };
 
diff --git a/src/meepgeom.cpp b/src/meepgeom.cpp
index d42c2aec6..a135396c7 100644
--- a/src/meepgeom.cpp
+++ b/src/meepgeom.cpp
@@ -1375,16 +1375,19 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
         sus = new meep::noisy_lorentzian_susceptibility(ss->noise_amp, ss->frequency, ss->gamma,
                                                         ss->drude);
       } else if (gyrotropic) {
-        sus = new meep::gyrotropic_susceptibility(meep::vec(ss->bias.x, ss->bias.y, ss->bias.z), ss->frequency,
-						  ss->gamma, ss->drude);
+        sus = new meep::gyrotropic_susceptibility(meep::vec(ss->bias.x, ss->bias.y, ss->bias.z),
+						  ss->alpha, ss->frequency, ss->gamma);
       } else {
         sus = new meep::lorentzian_susceptibility(ss->frequency, ss->gamma, ss->drude);
       }
       master_printf("%s%s susceptibility: frequency=%g, gamma=%g",
 		    noisy ? "noisy " : gyrotropic ? "gyrotropic " : "",
-                    ss->drude ? "drude" : "lorentzian", ss->frequency, ss->gamma);
+		    gyrotropic ? "" : ss->drude ? "drude" : "lorentzian",
+		    ss->frequency, ss->gamma);
       if (noisy) master_printf(", amp=%g ", ss->noise_amp);
-      if (gyrotropic) master_printf(", bias=(%g,%g,%g) ", ss->bias.x, ss->bias.y, ss->bias.z);
+      if (gyrotropic) master_printf(", bias=(%g,%g,%g), alpha=%g ",
+				    ss->bias.x, ss->bias.y, ss->bias.z,
+				    ss->alpha);
       master_printf("\n");
     }
 
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 3b7289c32..76296aea8 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -92,7 +92,6 @@ typedef struct {
   size_t ntot;
   realnum *P[NUM_FIELD_COMPONENTS][2];
   realnum *P_prev[NUM_FIELD_COMPONENTS][2];
-  realnum *P_tmp[NUM_FIELD_COMPONENTS][2]; // extra slot used for gyrotropic medium updating
   realnum data[1];
 } lorentzian_data;
 
@@ -101,9 +100,8 @@ typedef struct {
 void *lorentzian_susceptibility::new_internal_data(realnum *W[NUM_FIELD_COMPONENTS][2],
                                                    const grid_volume &gv) const {
   int num = 0;
-  int nslots = have_gyrotropy ? 3 : 2;
   FOR_COMPONENTS(c) DOCMP2 {
-    if (needs_P(c, cmp, W)) num += nslots * gv.ntot();
+    if (needs_P(c, cmp, W)) num += 2 * gv.ntot();
   }
   size_t sz = sizeof(lorentzian_data) + sizeof(realnum) * (num - 1);
   lorentzian_data *d = (lorentzian_data *)malloc(sz);
@@ -121,18 +119,12 @@ void lorentzian_susceptibility::init_internal_data(realnum *W[NUM_FIELD_COMPONEN
   size_t ntot = d->ntot = gv.ntot();
   realnum *P = d->data;
   realnum *P_prev = d->data + ntot;
-  realnum *P_tmp = have_gyrotropy ? P_prev + ntot : NULL;
-  int nslots = have_gyrotropy ? 3 : 2;
-
   FOR_COMPONENTS(c) DOCMP2 {
     if (needs_P(c, cmp, W)) {
       d->P[c][cmp] = P;
       d->P_prev[c][cmp] = P_prev;
-      d->P_tmp[c][cmp] = P_tmp;
-
-      P += nslots * ntot;
-      P_prev += nslots * ntot;
-      if (have_gyrotropy) P_tmp += nslots * ntot;
+      P += 2 * ntot;
+      P_prev += 2 * ntot;
     }
   }
 }
@@ -145,17 +137,12 @@ void *lorentzian_susceptibility::copy_internal_data(void *data) const {
   size_t ntot = d->ntot;
   realnum *P = dnew->data;
   realnum *P_prev = dnew->data + ntot;
-  realnum *P_tmp = have_gyrotropy ? P_prev + ntot : NULL;
-  int nslots = have_gyrotropy ? 3 : 2;
-
   FOR_COMPONENTS(c) DOCMP2 {
     if (d->P[c][cmp]) {
       dnew->P[c][cmp] = P;
       dnew->P_prev[c][cmp] = P_prev;
-      dnew->P_tmp[c][cmp] = P_tmp;
-      P += nslots * ntot;
-      P_prev += nslots * ntot;
-      if (have_gyrotropy) P_tmp += nslots * ntot;
+      P += 2 * ntot;
+      P_prev += 2 * ntot;
     }
   }
   return (void *)dnew;
@@ -336,16 +323,14 @@ void noisy_lorentzian_susceptibility::dump_params(h5file *h5f, size_t *start) {
 }
 
 
-gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double omega_0, double gamma,
-						     bool no_omega_0_denominator)
-  : lorentzian_susceptibility(omega_0, gamma, no_omega_0_denominator) {
-  have_gyrotropy = true;
+gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double alpha, double omega_0, double gamma)
+  : lorentzian_susceptibility(omega_0, gamma, false), alpha(alpha) {
 
-  // Precalculate g_{ij} = sum_k epsilon_{ijk} b_k, used in update_P.
+  const vec bn = bias / fmax(abs(bias), 1e-10); // avoid division by zero
   memset(gyro_tensor, 0, 9 * sizeof(double));
-  gyro_tensor[X][Y] = bias.z(); gyro_tensor[Y][X] = -bias.z();
-  gyro_tensor[Y][Z] = bias.x(); gyro_tensor[Z][Y] = -bias.x();
-  gyro_tensor[Z][X] = bias.y(); gyro_tensor[X][Z] = -bias.y();
+  gyro_tensor[X][Y] = bn.z(); gyro_tensor[Y][X] = -bn.z();
+  gyro_tensor[Y][Z] = bn.x(); gyro_tensor[Z][Y] = -bn.x();
+  gyro_tensor[Z][X] = bn.y(); gyro_tensor[X][Z] = -bn.y();
 }
 
 void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
@@ -353,9 +338,12 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 					 const grid_volume &gv, void *P_internal_data) const {
   lorentzian_data *d = (lorentzian_data *)P_internal_data;
   const double omega2pi = 2 * pi * omega_0, g2pi = gamma * 2 * pi;
-  const double omega0dtsqr = omega2pi * omega2pi * dt * dt;
-  const double gamma1 = (1 - g2pi * dt / 2);
-  const double omega0dtsqr_denom = no_omega_0_denominator ? 0 : omega0dtsqr;
+  const double alpha2pi = 2 * pi * alpha;
+  const double ua = 1 - 0.5 * g2pi * dt;
+  const double va = alpha2pi - 0.5 * omega2pi * dt;
+  const double ub = 1 + 0.5 * g2pi * dt;
+  const double vb = alpha2pi + 0.5 * omega2pi * dt;
+
   (void)W_prev; // unused;
 
   FOR_COMPONENTS(c) DOCMP2 {
@@ -366,105 +354,74 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 	if (d0 != X && d0 != Y && d0 != Z)
 	  abort("Cylindrical coordinates are not supported for gyrotropic media");
 
-        const realnum *p = d->P[c][cmp], *pp = d->P_prev[c][cmp];
-	realnum *rhs = d->P_tmp[c][cmp];
-
+        const realnum *p = d->P[c][cmp];
+	realnum *pp = d->P_prev[c][cmp];
         const ptrdiff_t is = gv.stride(d0) * (is_magnetic(c) ? -1 : +1);
 
         direction d1 = cycle_direction(gv.dim, d0, 1);
-        component c1 = direction_component(c, d1);
-        ptrdiff_t is1 = gv.stride(d1) * (is_magnetic(c) ? -1 : +1);
-        const realnum *w1 = W[c1][cmp];
-        const realnum *s1 = w1 ? sigma[c][d1] : NULL;
-
         direction d2 = cycle_direction(gv.dim, d0, 2);
+
+        component c1 = direction_component(c, d1);
         component c2 = direction_component(c, d2);
-        ptrdiff_t is2 = gv.stride(d2) * (is_magnetic(c) ? -1 : +1);
+
+        const realnum *w1 = W[c1][cmp];
         const realnum *w2 = W[c2][cmp];
-        const realnum *s2 = w2 ? sigma[c][d2] : NULL;
 
 	// Off-diagonal polarization components
-        const realnum *pp1 = d->P_prev[c1][cmp];
-        const realnum *pp2 = d->P_prev[c2][cmp];
-	const realnum g1 = pi * dt * gyro_tensor[d0][d1];
-	const realnum g2 = pi * dt * gyro_tensor[d0][d2];
+        const realnum *p1 = d->P[c1][cmp];
+        const realnum *p2 = d->P[c2][cmp];
 
-        if (s2 && !s1) { // make s1 the non-NULL one if possible
-          SWAP(direction, d1, d2);
-          SWAP(component, c1, c2);
-          SWAP(ptrdiff_t, is1, is2);
-          SWAP(const realnum *, w1, w2);
-          SWAP(const realnum *, s1, s2);
-        }
-        if (s1 && s2) { // 3x3 anisotropic
-          LOOP_OVER_VOL_OWNED(gv, c, i) {
-            if (s[i] != 0) {
-              rhs[i] = (2 - omega0dtsqr_denom) * p[i]
-		+ omega0dtsqr * (s[i] * w[i]
-				 + OFFDIAG(s1, w1, is1, is)
-				 + OFFDIAG(s2, w2, is2, is))
-		- gamma1 * pp[i];
-	      if (pp1) rhs[i] -= g1 * pp1[i];
-	      if (pp2) rhs[i] -= g2 * pp2[i];
-            }
-          }
-        } else if (s1) { // 2x2 anisotropic
-          LOOP_OVER_VOL_OWNED(gv, c, i) {
-            if (s[i] != 0) {
-              rhs[i] = (2 - omega0dtsqr_denom) * p[i]
-		+ omega0dtsqr * (s[i] * w[i] + OFFDIAG(s1, w1, is1, is))
-		- gamma1 * pp[i];
-	      if (pp1) rhs[i] -= g1 * pp1[i];
-	      if (pp2) rhs[i] -= g2 * pp2[i];
-            }
-          }
-        } else { // isotropic
-          LOOP_OVER_VOL_OWNED(gv, c, i) {
-            rhs[i] = (2 - omega0dtsqr_denom) * p[i]
-	      + omega0dtsqr * (s[i] * w[i]) - gamma1 * pp[i];
-	    if (pp1) rhs[i] -= g1 * pp1[i];
-	    if (pp2) rhs[i] -= g2 * pp2[i];
-          }
-        }
+	const realnum vab1 = va * gyro_tensor[d0][d1];
+	const realnum vab2 = va * gyro_tensor[d0][d2];
+
+	const realnum ndt1 = dt * gyro_tensor[d0][d1];
+	const realnum ndt2 = dt * gyro_tensor[d0][d2];
+
+	LOOP_OVER_VOL_OWNED(gv, c, i) {
+	  pp[i] = ua * p[i];
+	  if (p1) pp[i] += vab1 * p1[i];
+	  if (p2) pp[i] += vab2 * p2[i];
+	  if (w1) pp[i] += ndt1 * s[i] * w1[i];
+	  if (w2) pp[i] += ndt2 * s[i] * w2[i];
+	}
       }
     }
   }
 
   // Perform 3x3 matrix inversion, exploiting skew symmetry
-  const double gd = (1 + g2pi * dt / 2);
-  const double gx = pi * dt * gyro_tensor[Y][Z];
-  const double gy = pi * dt * gyro_tensor[Z][X];
-  const double gz = pi * dt * gyro_tensor[X][Y];
-  const double invdet = 1.0 / gd / (gd*gd + gx*gx + gy*gy + gz*gz);
+  const double gx = vb * gyro_tensor[Y][Z];
+  const double gy = vb * gyro_tensor[Z][X];
+  const double gz = vb * gyro_tensor[X][Y];
+  const double invdet = 1.0 / ub / (ub*ub + gx*gx + gy*gy + gz*gz);
   double inv[3][3];
 
-  inv[X][X] = invdet * (gd*gd + gx*gx);
-  inv[Y][Y] = invdet * (gd*gd + gy*gy);
-  inv[Z][Z] = invdet * (gd*gd + gz*gz);
-  inv[X][Y] = invdet * (gx*gy + gd*gz);
-  inv[Y][X] = invdet * (gy*gx - gd*gz);
-  inv[Z][X] = invdet * (gz*gx + gd*gy);
-  inv[X][Z] = invdet * (gx*gz - gd*gy);
-  inv[Y][Z] = invdet * (gy*gz + gd*gx);
-  inv[Z][Y] = invdet * (gz*gy - gd*gx);
+  inv[X][X] = invdet * (ub*ub + gx*gx);
+  inv[Y][Y] = invdet * (ub*ub + gy*gy);
+  inv[Z][Z] = invdet * (ub*ub + gz*gz);
+  inv[X][Y] = invdet * (gx*gy - ub*gz);
+  inv[Y][X] = invdet * (gy*gx + ub*gz);
+  inv[Z][X] = invdet * (gz*gx - ub*gy);
+  inv[X][Z] = invdet * (gx*gz + ub*gy);
+  inv[Y][Z] = invdet * (gy*gz - ub*gx);
+  inv[Z][Y] = invdet * (gz*gy + ub*gx);
 
   FOR_COMPONENTS(c) DOCMP2 {
     if (d->P[c][cmp]) {
       const direction d0 = component_direction(c);
       if (W[c][cmp] && sigma[c][d0]) {
-        realnum *p = d->P[c][cmp], *pp = d->P_prev[c][cmp], *rhs = d->P_tmp[c][cmp];
+        realnum *p = d->P[c][cmp];
+	const realnum *pp = d->P_prev[c][cmp];
         const direction d1  = cycle_direction(gv.dim, d0, 1);
         const direction d2  = cycle_direction(gv.dim, d0, 2);
         const component c1 = direction_component(c, d1);
         const component c2 = direction_component(c, d2);
-        const realnum *rhs1 = W[c1][cmp] ? d->P_tmp[c1][cmp] : NULL;
-	const realnum *rhs2 = W[c2][cmp] ? d->P_tmp[c2][cmp] : NULL;
+        const realnum *pp1 = W[c1][cmp] ? d->P_prev[c1][cmp] : NULL;
+	const realnum *pp2 = W[c2][cmp] ? d->P_prev[c2][cmp] : NULL;
 
 	LOOP_OVER_VOL_OWNED(gv, c, i) {
-	  pp[i] = p[i];
-	  p[i]  = inv[d0][d0] * rhs[i];
-	  if (rhs1) p[i] += inv[d0][d1] * rhs1[i];
-	  if (rhs2) p[i] += inv[d0][d2] * rhs2[i];
+	  p[i] = inv[d0][d0] * pp[i];
+	  if (pp1) p[i] += inv[d0][d1] * pp1[i];
+	  if (pp2) p[i] += inv[d0][d2] * pp2[i];
 	}
       }
     }
@@ -472,12 +429,12 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 }
 
 void gyrotropic_susceptibility::dump_params(h5file *h5f, size_t *start) {
-  size_t num_params = 8;
+  size_t num_params = 9;
   size_t params_dims[1] = {num_params};
   double bias[] = { gyro_tensor[Y][Z], gyro_tensor[Z][X], gyro_tensor[X][Y] };
   double params_data[] = {
-      7, (double)get_id(), bias[X], bias[Y], bias[Z],
-      omega_0, gamma, (double)no_omega_0_denominator};
+      8, (double)get_id(), bias[X], bias[Y], bias[Z],
+      alpha, omega_0, gamma, (double)no_omega_0_denominator};
   h5f->write_chunk(1, start, params_dims, params_data);
   *start += num_params;
 }

From 2481e16b7265e30d0dee00989a00e0d9ff193484 Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Thu, 23 May 2019 23:49:29 +0800
Subject: [PATCH 12/61] Tweak handling of sigma tensor in gyrotropic case

---
 src/susceptibility.cpp | 9 +++++----
 1 file changed, 5 insertions(+), 4 deletions(-)

diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 76296aea8..895793d5b 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -356,7 +356,6 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 
         const realnum *p = d->P[c][cmp];
 	realnum *pp = d->P_prev[c][cmp];
-        const ptrdiff_t is = gv.stride(d0) * (is_magnetic(c) ? -1 : +1);
 
         direction d1 = cycle_direction(gv.dim, d0, 1);
         direction d2 = cycle_direction(gv.dim, d0, 2);
@@ -367,7 +366,9 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
         const realnum *w1 = W[c1][cmp];
         const realnum *w2 = W[c2][cmp];
 
-	// Off-diagonal polarization components
+        const realnum *s1 = w1 ? sigma[c1][d1] : NULL;
+        const realnum *s2 = w2 ? sigma[c2][d2] : NULL;
+
         const realnum *p1 = d->P[c1][cmp];
         const realnum *p2 = d->P[c2][cmp];
 
@@ -381,8 +382,8 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 	  pp[i] = ua * p[i];
 	  if (p1) pp[i] += vab1 * p1[i];
 	  if (p2) pp[i] += vab2 * p2[i];
-	  if (w1) pp[i] += ndt1 * s[i] * w1[i];
-	  if (w2) pp[i] += ndt2 * s[i] * w2[i];
+	  if (w1) pp[i] += ndt1 * s1[i] * w1[i];
+	  if (w2) pp[i] += ndt2 * s2[i] * w2[i];
 	}
       }
     }

From d6cc4854dea95e10eaa51ab0d83cdb1e09763133 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Fri, 24 May 2019 09:18:42 +0800
Subject: [PATCH 13/61] Update Python doc.

---
 doc/docs/Python_User_Interface.md | 20 ++++++++++++++++++--
 1 file changed, 18 insertions(+), 2 deletions(-)

diff --git a/doc/docs/Python_User_Interface.md b/doc/docs/Python_User_Interface.md
index 11f3b3f28..75bb1ae69 100644
--- a/doc/docs/Python_User_Interface.md
+++ b/doc/docs/Python_User_Interface.md
@@ -394,11 +394,27 @@ This is a somewhat unusual polarizable medium, a Lorentzian susceptibility with
 
 ### GyrotropicLorentzianSusceptibility or GyrotropicDrudeSusceptibility
 
-(**Experimental feature**) Specifies a single dispersive gyrotropic susceptibility of Lorentzian (damped harmonic oscillator) or Drude form. See [Material Dispersion](Materials.md#gyrotropic-media). The parameters `sigma`, `frequency`, and `gamma` have the usual meanings, and there is an additional 3-vector `bias`:
+(**Experimental feature**) Specifies a single dispersive [gyrotropic susceptibility](Materials.md#gyrotropic-media) governed by the Landau-Lifshitz-Gilbert equation. Note that the parameters `sigma`, `frequency`, and `gamma` play different roles compared to the Lorentzian or Drude case.
+
+**`sigma` [`number`]**
+—
+The scale factor σ specifying the coupling to the driving field. In magnetic ferrites, this is equal to the Larmor precession frequency at magnetic saturation.
+
+**`frequency` [`number`]**
+—
+The frequency of gyrotropic precession (Larmor frequency), $f_n = \omega_n / 2\pi$.
+
+**`gamma` [`number`]**
+—
+The loss rate $\gamma_n / 2\pi$ in the off-diagonal response.
+
+**`alpha` [`number`]**
+—
+The loss rate $\alpha_n / 2\pi$ in the diagonal response.
 
 **`bias` [`Vector3`]**
 —
-The gyrotropy vector.  The direction of this vector determines the orientation of the gyrotropic response, and the magnitude equals the precession frequency $|\mathbf{b}_n|/2\pi$.
+Gyrotropy vector describing the direction of the static biasing magnetic field. The magnitude is ignored.
 
 ### Vector3
 

From 9267c353ba3efa9ffbcc3a59b625d4e48d228bed Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Fri, 24 May 2019 09:28:31 +0800
Subject: [PATCH 14/61] Drop 2pi factor from gyrotropic sigma

---
 doc/docs/Python_User_Interface.md | 4 ++--
 src/susceptibility.cpp            | 6 +++---
 2 files changed, 5 insertions(+), 5 deletions(-)

diff --git a/doc/docs/Python_User_Interface.md b/doc/docs/Python_User_Interface.md
index 75bb1ae69..0dc977f35 100644
--- a/doc/docs/Python_User_Interface.md
+++ b/doc/docs/Python_User_Interface.md
@@ -398,11 +398,11 @@ This is a somewhat unusual polarizable medium, a Lorentzian susceptibility with
 
 **`sigma` [`number`]**
 —
-The scale factor σ specifying the coupling to the driving field. In magnetic ferrites, this is equal to the Larmor precession frequency at magnetic saturation.
+The coupling factor $\sigma_n / 2\pi$ between the polarization and the driving field. In magnetic ferrites, this is the Larmor precession frequency at the saturation field.
 
 **`frequency` [`number`]**
 —
-The frequency of gyrotropic precession (Larmor frequency), $f_n = \omega_n / 2\pi$.
+The frequency of gyrotropic precession, $f_n = \omega_n / 2\pi$. In magnetic ferrites, this is the Larmor precession frequency.
 
 **`gamma` [`number`]**
 —
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 895793d5b..d8614c1dd 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -349,7 +349,7 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
   FOR_COMPONENTS(c) DOCMP2 {
     if (d->P[c][cmp]) {
       const direction d0 = component_direction(c);
-      const realnum *w = W[c][cmp], *s = sigma[c][d0];
+      const realnum *w = W[c][cmp], *s = 2*pi*sigma[c][d0];
       if (w && s) {
 	if (d0 != X && d0 != Y && d0 != Z)
 	  abort("Cylindrical coordinates are not supported for gyrotropic media");
@@ -366,8 +366,8 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
         const realnum *w1 = W[c1][cmp];
         const realnum *w2 = W[c2][cmp];
 
-        const realnum *s1 = w1 ? sigma[c1][d1] : NULL;
-        const realnum *s2 = w2 ? sigma[c2][d2] : NULL;
+        const realnum *s1 = w1 ? 2*pi*sigma[c1][d1] : NULL;
+        const realnum *s2 = w2 ? 2*pi*sigma[c2][d2] : NULL;
 
         const realnum *p1 = d->P[c1][cmp];
         const realnum *p2 = d->P[c2][cmp];

From 77812cd426a49c0bca39612ba09125a602a6cda5 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Fri, 24 May 2019 11:45:16 +0800
Subject: [PATCH 15/61] Doc updates

---
 doc/docs/Materials.md                         |  2 +-
 doc/docs/Python_Tutorials/Gyrotropic_Media.md | 33 ++++++++++++-------
 doc/docs/Scheme_User_Interface.md             | 24 ++++++++++++++
 3 files changed, 47 insertions(+), 12 deletions(-)

diff --git a/doc/docs/Materials.md b/doc/docs/Materials.md
index 0205127b2..c4f60bee8 100644
--- a/doc/docs/Materials.md
+++ b/doc/docs/Materials.md
@@ -176,7 +176,7 @@ In a gyroelectric medium, the equation of motion of the polarization vector $\ma
 
 $$\frac{d\mathbf{P}_n}{dt} = \mathbf{b}_n \times \left( - \sigma_n \mathbf{E} + \omega_n \mathbf{P}_n + \alpha_n \frac{d\mathbf{P}_n}{dt} \right) - \gamma_n \mathbf{P}_n$$
 
-Although this is completely different from the damped harmonic oscillator equation used for [Drude-Lorentz susceptibility](Materials.md#material-dispersion), the constants σ$_n$, ω$_n$, and γ$_n$ play rather analogous roles: σ$_n$ couples the polarization to the electric field, ω$_n$ is the angular frequency of precession, and γ$_n$ is a damping factor. There is also a second damping factor α$_n$, and a "bias vector" **b**$_n$ which typically describes the direction of an applied static magnetic field. The bias vector is assumed to have unit length.
+Although this is completely different from the damped harmonic oscillator equation used for [Drude-Lorentz susceptibility](Materials.md#material-dispersion), the constants σ$_n$, ω$_n$, and γ$_n$ play analogous roles: σ$_n$ couples the polarization to the electric field, ω$_n$ is the angular frequency of precession, and γ$_n$ is a damping factor. There is also a second damping factor α$_n$, and a "bias vector" **b**$_n$ which typically describes the direction of an applied static magnetic field. The bias vector is assumed to have unit length.
 
 In the frequency domain, the ε tensor acquires off-diagonal components representing a gyrotropic response. Take the case $\mathbf{b}_n = \hat{\mathbf{z}}$, and assume that all fields have harmonic time-dependence $\exp(-i\omega t)$. The polarization equation then reduces to
 
diff --git a/doc/docs/Python_Tutorials/Gyrotropic_Media.md b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
index b67fb0afc..532f506d5 100644
--- a/doc/docs/Python_Tutorials/Gyrotropic_Media.md
+++ b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
@@ -6,25 +6,39 @@ In this example, we will perform simulations with a gyrotropic medium. See [Mate
 
 [TOC]
 
-### Dispersion Relation for a Gyrotropic Medium
+## Electromagnetic wave propagation in a ferrite
 
-Here, we model a uniform gyrotroelectric medium with bias vector $\mathbf{b} = b \hat{z}$. For TE modes, the electric field lies in the *x*-*y* plane and magnetic field pointing along *z*, with the wavevector $\mathbf{k}$ in the *x*-*y* plane.  The dispersion relation can be shown to be
+### Material properties
 
-$$\omega = v(\omega) |\mathbf{k}|, \quad v^2 = \mu^{-1} \left[\epsilon^{-1}\right]_{xx}$$
+Ferrites are ceramic materials exhibiting strong gyromagnetic resonances, commonly used in microwave engineering to construct isolators (one-way valves) and circulators. In the presence of a biasing magnetic field $\mathbf{H} = H_0 \hat{\mathbf{z}}$, the permeability tensor of a ferrite material has the form
 
-The ε tensor is given by
+$$\mu = \begin{bmatrix} \mu_\perp & -i \kappa & 0 \\ i\kappa & \mu_\perp & 0 \\ 0 & 0 & 1\end{bmatrix}$$
 
-$$\epsilon = \epsilon_\infty + \frac{\omega_n^2}{\omega_n^2 - \omega^2 - i\omega\gamma_n} \begin{bmatrix}\xi & -i\eta & 0 \\ i\eta & \xi & 0 \\ 0 & 0 & 1 \end{bmatrix} \, \sigma_n(\mathbf{x}),$$
+where
 
-with
+$$\mu_\perp = 1 + \frac{\omega_m (\omega_0 - i\omega\alpha)}{(\omega_0 - i \omega \alpha)^2 - (\omega + i \gamma)^2}, \;\;\; \kappa = \frac{\omega_m (\omega + i\gamma)}{(\omega_0 - i \omega \alpha)^2 - (\omega + i \gamma)^2}$$
 
-$$\xi = \frac{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2}{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2 + \omega^2 b^2}\,, \quad \eta = \frac{\omega b (\omega_n^2 - \omega^2 - i\omega \gamma_n)}{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2 + \omega^2 b^2}$$
+This permeability can be derived from the [Landau-Lifshitz-Gilbert equation](https://en.wikipedia.org/wiki/Landau%E2%80%93Lifshitz%E2%80%93Gilbert_equation) describing a magnetic dipole in a magnetic field. The Larmor precession frequency is $\omega_0 = \gamma H_0$ where $\gamma$ is the gyromagnetic ratio of the electron. The Larmor precession frequency at saturation field is $\omega_m = \gamma M_s$ where $M_s$ is the saturation magnetization. This precisely matches the permeability tensor derived from [Meep's gyrotropy model](../Materials.md#gyrotropic-media), with the identification $\omega_0 \leftrightarrow \omega_n$ and $\omega_m \leftrightarrow \sigma_n$.
 
-We choose the material parameters $\epsilon_\infty = 1$, $f_n = 1.2$, $\gamma_n = 10^{-5}$, and $\sigma_n = 0.1$, with $|\mathbf{b}_n|$ ranging from zero to $10^{-2}$.
+In Meep units, we will take $\omega_0/2\pi = 1$ and $\omega_m/2\pi = 1.2$, with $\alpha/2\pi = \gamma/2\pi = 10^{-3}$ and $\epsilon = 15$. The diagonal and off-diagonal components of the permeability tensor are plotted below:
 
+<center>
+![](../images/Ferrite-mu.png)
+</center>
+
+### Dispersion relation
+
+Consider a uniform ferrite medium with bias field in the *z* direction. The gyrotropy acts upon the *x* and *y* components of the magnetic field, so we focus on TM plane waves that have wavevector $\mathbf{k}$ and magnetic fields in the *x*-*y* plane, and electric fields parallel to *z*. From Maxwell's equations, we derive the dispersion relation
+
+$$\omega^2 = \epsilon^{-1} \left[\mu^{-1}\right]_{\perp} |\mathbf{k}|^2$$
 
+where $[\mu^{-1}\right]_{\perp} = [\mu^{-1}\right]_{xx} = [\mu^{-1}\right]_{yy}$.
 
 
+
+
+### FIXME
+
 From the dispersion relation ω(k), we will compute the numerical ε(ω) via the formula:
 
 $$\varepsilon(\omega) = \left( \frac{ck}{\omega} \right) ^2$$
@@ -53,9 +67,6 @@ $$\varepsilon(\omega) = \varepsilon(2\pi f) = 2.25 + \frac{1.1^2 \cdot 0.5}{1.1^
 
 The real and imaginary parts of this dielectric function ε(ω) are plotted below:
 
-<center>
-![](../images/Material-dispersion-eps.png)
-</center>
 
 We can see that the f=1.1 resonance causes a large change in both the real and imaginary parts of ε around that frequency. In fact, there is a range of frequencies from 1.1 to 1.2161 where ε is *negative*. In this range, no propagating modes exist &mdash; it is actually a kind of electromagnetic band gap associated with polariton resonances in a material. For more information on the physics of such materials, see e.g. Chapter 14 of [Introduction to Solid State Physics](http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000803.html) by C. Kittel.
 
diff --git a/doc/docs/Scheme_User_Interface.md b/doc/docs/Scheme_User_Interface.md
index c422c2945..85f47b7cc 100644
--- a/doc/docs/Scheme_User_Interface.md
+++ b/doc/docs/Scheme_User_Interface.md
@@ -339,6 +339,30 @@ Specifies a single dispersive susceptibility of Lorentzian (damped harmonic osci
 —
 The noise has root-mean square amplitude σ $\times$ `noise-amp`.
 
+### gyrotropic-susceptibility
+
+(**Experimental feature**) Specifies a single dispersive [gyrotropic susceptibility](Materials.md#gyrotropic-media) governed by the Landau-Lifshitz-Gilbert equation. Note that the parameters `sigma`, `frequency`, and `gamma` play different roles compared to the Lorentzian or Drude case.
+
+**`sigma` [`number`]**
+—
+The coupling factor $\sigma_n / 2\pi$ between the polarization and the driving field. In magnetic ferrites, this is the Larmor precession frequency at the saturation field.
+
+**`frequency` [`number`]**
+—
+The frequency of gyrotropic precession, $f_n = \omega_n / 2\pi$. In magnetic ferrites, this is the Larmor precession frequency.
+
+**`gamma` [`number`]**
+—
+The loss rate $\gamma_n / 2\pi$ in the off-diagonal response.
+
+**`alpha` [`number`]**
+—
+The loss rate $\alpha_n / 2\pi$ in the diagonal response.
+
+**`bias` [`vector3`]**
+—
+Gyrotropy vector describing the direction of the static biasing magnetic field. The magnitude is ignored.
+
 ### geometric-object
 
 This class, and its descendants, are used to specify the solid geometric objects that form the dielectric structure being simulated. The base class is:

From 14454cf9fdf2c4d6a7016e094526b7cc668e2098 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Fri, 24 May 2019 11:51:24 +0800
Subject: [PATCH 16/61] Fix last change to update_P

---
 src/susceptibility.cpp | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index d8614c1dd..b73e6fb7a 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -349,8 +349,8 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
   FOR_COMPONENTS(c) DOCMP2 {
     if (d->P[c][cmp]) {
       const direction d0 = component_direction(c);
-      const realnum *w = W[c][cmp], *s = 2*pi*sigma[c][d0];
-      if (w && s) {
+      const realnum *w = W[c][cmp];
+      if (w) {
 	if (d0 != X && d0 != Y && d0 != Z)
 	  abort("Cylindrical coordinates are not supported for gyrotropic media");
 
@@ -366,8 +366,8 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
         const realnum *w1 = W[c1][cmp];
         const realnum *w2 = W[c2][cmp];
 
-        const realnum *s1 = w1 ? 2*pi*sigma[c1][d1] : NULL;
-        const realnum *s2 = w2 ? 2*pi*sigma[c2][d2] : NULL;
+        const realnum *s1 = w1 ? sigma[c1][d1] : NULL;
+        const realnum *s2 = w2 ? sigma[c2][d2] : NULL;
 
         const realnum *p1 = d->P[c1][cmp];
         const realnum *p2 = d->P[c2][cmp];
@@ -375,8 +375,8 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 	const realnum vab1 = va * gyro_tensor[d0][d1];
 	const realnum vab2 = va * gyro_tensor[d0][d2];
 
-	const realnum ndt1 = dt * gyro_tensor[d0][d1];
-	const realnum ndt2 = dt * gyro_tensor[d0][d2];
+	const realnum ndt1 = 2*pi*dt * gyro_tensor[d0][d1];
+	const realnum ndt2 = 2*pi*dt * gyro_tensor[d0][d2];
 
 	LOOP_OVER_VOL_OWNED(gv, c, i) {
 	  pp[i] = ua * p[i];

From 5e0f3719de95161e45205b6cdcc868676a116a43 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Fri, 24 May 2019 14:41:54 +0800
Subject: [PATCH 17/61] Fix printf typo

---
 src/meepgeom.cpp | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/meepgeom.cpp b/src/meepgeom.cpp
index a135396c7..bc825a3e2 100644
--- a/src/meepgeom.cpp
+++ b/src/meepgeom.cpp
@@ -1381,7 +1381,7 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
         sus = new meep::lorentzian_susceptibility(ss->frequency, ss->gamma, ss->drude);
       }
       master_printf("%s%s susceptibility: frequency=%g, gamma=%g",
-		    noisy ? "noisy " : gyrotropic ? "gyrotropic " : "",
+		    noisy ? "noisy " : gyrotropic ? "gyrotropic" : "",
 		    gyrotropic ? "" : ss->drude ? "drude" : "lorentzian",
 		    ss->frequency, ss->gamma);
       if (noisy) master_printf(", amp=%g ", ss->noise_amp);

From 722c9c6b78dbf8e20ae99f12975d3a8129cacd8d Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Fri, 24 May 2019 14:50:20 +0800
Subject: [PATCH 18/61] Remove spurious 2pi factor in alpha (which is not a
 rate)

---
 doc/docs/Python_Tutorials/Gyrotropic_Media.md | 2 +-
 doc/docs/Python_User_Interface.md             | 2 +-
 doc/docs/Scheme_User_Interface.md             | 2 +-
 src/susceptibility.cpp                        | 5 ++---
 4 files changed, 5 insertions(+), 6 deletions(-)

diff --git a/doc/docs/Python_Tutorials/Gyrotropic_Media.md b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
index 532f506d5..4902f4d55 100644
--- a/doc/docs/Python_Tutorials/Gyrotropic_Media.md
+++ b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
@@ -20,7 +20,7 @@ $$\mu_\perp = 1 + \frac{\omega_m (\omega_0 - i\omega\alpha)}{(\omega_0 - i \omeg
 
 This permeability can be derived from the [Landau-Lifshitz-Gilbert equation](https://en.wikipedia.org/wiki/Landau%E2%80%93Lifshitz%E2%80%93Gilbert_equation) describing a magnetic dipole in a magnetic field. The Larmor precession frequency is $\omega_0 = \gamma H_0$ where $\gamma$ is the gyromagnetic ratio of the electron. The Larmor precession frequency at saturation field is $\omega_m = \gamma M_s$ where $M_s$ is the saturation magnetization. This precisely matches the permeability tensor derived from [Meep's gyrotropy model](../Materials.md#gyrotropic-media), with the identification $\omega_0 \leftrightarrow \omega_n$ and $\omega_m \leftrightarrow \sigma_n$.
 
-In Meep units, we will take $\omega_0/2\pi = 1$ and $\omega_m/2\pi = 1.2$, with $\alpha/2\pi = \gamma/2\pi = 10^{-3}$ and $\epsilon = 15$. The diagonal and off-diagonal components of the permeability tensor are plotted below:
+We will take $\omega_0/2\pi = 1$, $\omega_m/2\pi = 1.2$, $\alpha = \gamma/2\pi = 10^{-3}$, and $\epsilon = 15$. The diagonal and off-diagonal components of the permeability tensor are plotted below:
 
 <center>
 ![](../images/Ferrite-mu.png)
diff --git a/doc/docs/Python_User_Interface.md b/doc/docs/Python_User_Interface.md
index 0dc977f35..8cc35d1aa 100644
--- a/doc/docs/Python_User_Interface.md
+++ b/doc/docs/Python_User_Interface.md
@@ -410,7 +410,7 @@ The loss rate $\gamma_n / 2\pi$ in the off-diagonal response.
 
 **`alpha` [`number`]**
 —
-The loss rate $\alpha_n / 2\pi$ in the diagonal response.
+The loss factor $\alpha_n$ in the diagonal response (note that there is no 2π factor).
 
 **`bias` [`Vector3`]**
 —
diff --git a/doc/docs/Scheme_User_Interface.md b/doc/docs/Scheme_User_Interface.md
index 85f47b7cc..1117430af 100644
--- a/doc/docs/Scheme_User_Interface.md
+++ b/doc/docs/Scheme_User_Interface.md
@@ -357,7 +357,7 @@ The loss rate $\gamma_n / 2\pi$ in the off-diagonal response.
 
 **`alpha` [`number`]**
 —
-The loss rate $\alpha_n / 2\pi$ in the diagonal response.
+The loss factor $\alpha_n$ in the diagonal response (note that there is no 2π factor).
 
 **`bias` [`vector3`]**
 —
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index b73e6fb7a..582e52787 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -338,11 +338,10 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 					 const grid_volume &gv, void *P_internal_data) const {
   lorentzian_data *d = (lorentzian_data *)P_internal_data;
   const double omega2pi = 2 * pi * omega_0, g2pi = gamma * 2 * pi;
-  const double alpha2pi = 2 * pi * alpha;
   const double ua = 1 - 0.5 * g2pi * dt;
-  const double va = alpha2pi - 0.5 * omega2pi * dt;
+  const double va = alpha - 0.5 * omega2pi * dt;
   const double ub = 1 + 0.5 * g2pi * dt;
-  const double vb = alpha2pi + 0.5 * omega2pi * dt;
+  const double vb = alpha + 0.5 * omega2pi * dt;
 
   (void)W_prev; // unused;
 

From e091b5e54de9c8ebcf050477cfd2b28530dee31f Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Fri, 24 May 2019 17:42:08 +0800
Subject: [PATCH 19/61] Minor code tweak

---
 src/susceptibility.cpp | 10 ++++------
 1 file changed, 4 insertions(+), 6 deletions(-)

diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 582e52787..4a9b50e2e 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -337,12 +337,10 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 					 realnum *W_prev[NUM_FIELD_COMPONENTS][2], double dt,
 					 const grid_volume &gv, void *P_internal_data) const {
   lorentzian_data *d = (lorentzian_data *)P_internal_data;
-  const double omega2pi = 2 * pi * omega_0, g2pi = gamma * 2 * pi;
-  const double ua = 1 - 0.5 * g2pi * dt;
-  const double va = alpha - 0.5 * omega2pi * dt;
-  const double ub = 1 + 0.5 * g2pi * dt;
-  const double vb = alpha + 0.5 * omega2pi * dt;
-
+  const double ua = 1 - pi * gamma * dt;
+  const double ub = 1 + pi * gamma * dt;
+  const double va = alpha - pi * omega_0 * dt;
+  const double vb = alpha + pi * omega_0 * dt;
   (void)W_prev; // unused;
 
   FOR_COMPONENTS(c) DOCMP2 {

From 104fff9322589b4a09b733d8e2846f5e9465b570 Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Sat, 25 May 2019 17:25:41 +0800
Subject: [PATCH 20/61] Use a central-difference scheme for the LLG dynamics,
 which seems slighly more stable...

---
 src/meep.hpp           |   4 +-
 src/susceptibility.cpp | 126 +++++++++++++++++++++++------------------
 2 files changed, 73 insertions(+), 57 deletions(-)

diff --git a/src/meep.hpp b/src/meep.hpp
index 0e0a401dd..4669a974d 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -223,7 +223,8 @@ class susceptibility {
 class lorentzian_susceptibility : public susceptibility {
 public:
   lorentzian_susceptibility(double omega_0, double gamma, bool no_omega_0_denominator = false)
-      : omega_0(omega_0), gamma(gamma), no_omega_0_denominator(no_omega_0_denominator) {}
+    : omega_0(omega_0), gamma(gamma), no_omega_0_denominator(no_omega_0_denominator),
+      have_gyrotropy(false) {}
   virtual susceptibility *clone() const { return new lorentzian_susceptibility(*this); }
   virtual ~lorentzian_susceptibility() {}
 
@@ -249,6 +250,7 @@ class lorentzian_susceptibility : public susceptibility {
 protected:
   double omega_0, gamma;
   bool no_omega_0_denominator;
+  bool have_gyrotropy; // whether to assign an extra slot for gyrotropy calculations
 };
 
 /* like a Lorentzian susceptibility, but the polarization equation
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 4a9b50e2e..038afa62f 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -92,6 +92,7 @@ typedef struct {
   size_t ntot;
   realnum *P[NUM_FIELD_COMPONENTS][2];
   realnum *P_prev[NUM_FIELD_COMPONENTS][2];
+  realnum *P_tmp[NUM_FIELD_COMPONENTS][2]; // extra slot used for gyrotropic medium updating
   realnum data[1];
 } lorentzian_data;
 
@@ -100,8 +101,9 @@ typedef struct {
 void *lorentzian_susceptibility::new_internal_data(realnum *W[NUM_FIELD_COMPONENTS][2],
                                                    const grid_volume &gv) const {
   int num = 0;
+  int nslots = have_gyrotropy ? 3 : 2;
   FOR_COMPONENTS(c) DOCMP2 {
-    if (needs_P(c, cmp, W)) num += 2 * gv.ntot();
+    if (needs_P(c, cmp, W)) num += nslots * gv.ntot();
   }
   size_t sz = sizeof(lorentzian_data) + sizeof(realnum) * (num - 1);
   lorentzian_data *d = (lorentzian_data *)malloc(sz);
@@ -119,12 +121,18 @@ void lorentzian_susceptibility::init_internal_data(realnum *W[NUM_FIELD_COMPONEN
   size_t ntot = d->ntot = gv.ntot();
   realnum *P = d->data;
   realnum *P_prev = d->data + ntot;
+  realnum *P_tmp = have_gyrotropy ? P_prev + ntot : NULL;
+  int nslots = have_gyrotropy ? 3 : 2;
+
   FOR_COMPONENTS(c) DOCMP2 {
     if (needs_P(c, cmp, W)) {
       d->P[c][cmp] = P;
       d->P_prev[c][cmp] = P_prev;
-      P += 2 * ntot;
-      P_prev += 2 * ntot;
+      d->P_tmp[c][cmp] = P_tmp;
+
+      P += nslots * ntot;
+      P_prev += nslots * ntot;
+      if (have_gyrotropy) P_tmp += nslots * ntot;
     }
   }
 }
@@ -137,12 +145,17 @@ void *lorentzian_susceptibility::copy_internal_data(void *data) const {
   size_t ntot = d->ntot;
   realnum *P = dnew->data;
   realnum *P_prev = dnew->data + ntot;
+  realnum *P_tmp = have_gyrotropy ? P_prev + ntot : NULL;
+  int nslots = have_gyrotropy ? 3 : 2;
+
   FOR_COMPONENTS(c) DOCMP2 {
     if (d->P[c][cmp]) {
       dnew->P[c][cmp] = P;
       dnew->P_prev[c][cmp] = P_prev;
-      P += 2 * ntot;
-      P_prev += 2 * ntot;
+      dnew->P_tmp[c][cmp] = P_tmp;
+      P += nslots * ntot;
+      P_prev += nslots * ntot;
+      if (have_gyrotropy) P_tmp += nslots * ntot;
     }
   }
   return (void *)dnew;
@@ -325,6 +338,7 @@ void noisy_lorentzian_susceptibility::dump_params(h5file *h5f, size_t *start) {
 
 gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double alpha, double omega_0, double gamma)
   : lorentzian_susceptibility(omega_0, gamma, false), alpha(alpha) {
+  have_gyrotropy = true;
 
   const vec bn = bias / fmax(abs(bias), 1e-10); // avoid division by zero
   memset(gyro_tensor, 0, 9 * sizeof(double));
@@ -334,13 +348,11 @@ gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double alp
 }
 
 void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
-					 realnum *W_prev[NUM_FIELD_COMPONENTS][2], double dt,
-					 const grid_volume &gv, void *P_internal_data) const {
+                                         realnum *W_prev[NUM_FIELD_COMPONENTS][2], double dt,
+                                         const grid_volume &gv, void *P_internal_data) const {
   lorentzian_data *d = (lorentzian_data *)P_internal_data;
-  const double ua = 1 - pi * gamma * dt;
-  const double ub = 1 + pi * gamma * dt;
-  const double va = alpha - pi * omega_0 * dt;
-  const double vb = alpha + pi * omega_0 * dt;
+  const realnum w4pit = 4*pi*dt*omega_0;
+  const realnum g4pit = 4*pi*dt*gamma;
   (void)W_prev; // unused;
 
   FOR_COMPONENTS(c) DOCMP2 {
@@ -348,79 +360,81 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
       const direction d0 = component_direction(c);
       const realnum *w = W[c][cmp];
       if (w) {
-	if (d0 != X && d0 != Y && d0 != Z)
-	  abort("Cylindrical coordinates are not supported for gyrotropic media");
+        if (d0 != X && d0 != Y && d0 != Z)
+          abort("Cylindrical coordinates are not supported for gyrotropic media");
 
         const realnum *p = d->P[c][cmp];
-	realnum *pp = d->P_prev[c][cmp];
+        const realnum *pp = d->P_prev[c][cmp];
+        realnum *ptmp = d->P_tmp[c][cmp];
 
         direction d1 = cycle_direction(gv.dim, d0, 1);
         direction d2 = cycle_direction(gv.dim, d0, 2);
-
         component c1 = direction_component(c, d1);
         component c2 = direction_component(c, d2);
 
         const realnum *w1 = W[c1][cmp];
         const realnum *w2 = W[c2][cmp];
-
         const realnum *s1 = w1 ? sigma[c1][d1] : NULL;
         const realnum *s2 = w2 ? sigma[c2][d2] : NULL;
-
-        const realnum *p1 = d->P[c1][cmp];
-        const realnum *p2 = d->P[c2][cmp];
-
-	const realnum vab1 = va * gyro_tensor[d0][d1];
-	const realnum vab2 = va * gyro_tensor[d0][d2];
-
-	const realnum ndt1 = 2*pi*dt * gyro_tensor[d0][d1];
-	const realnum ndt2 = 2*pi*dt * gyro_tensor[d0][d2];
-
-	LOOP_OVER_VOL_OWNED(gv, c, i) {
-	  pp[i] = ua * p[i];
-	  if (p1) pp[i] += vab1 * p1[i];
-	  if (p2) pp[i] += vab2 * p2[i];
-	  if (w1) pp[i] += ndt1 * s1[i] * w1[i];
-	  if (w2) pp[i] += ndt2 * s2[i] * w2[i];
-	}
+        const realnum *p1 = d->P[c1][cmp], *pp1 = d->P_prev[c1][cmp];
+        const realnum *p2 = d->P[c2][cmp], *pp2 = d->P_prev[c2][cmp];
+
+        const realnum ab1 = alpha * gyro_tensor[d0][d1];
+        const realnum ab2 = alpha * gyro_tensor[d0][d2];
+        const realnum wb1 = w4pit * gyro_tensor[d0][d1];
+        const realnum wb2 = w4pit * gyro_tensor[d0][d2];
+        const realnum bt1 = 4*pi*dt * gyro_tensor[d0][d1];
+        const realnum bt2 = 4*pi*dt * gyro_tensor[d0][d2];
+
+        LOOP_OVER_VOL_OWNED(gv, c, i) {
+          ptmp[i] = pp[i] - g4pit * p[i];
+          if (pp1) ptmp[i] += ab1 * pp1[i];
+          if (pp2) ptmp[i] += ab2 * pp2[i];
+          if (p1)  ptmp[i] -= wb1 * p1[i];
+          if (p2)  ptmp[i] -= wb2 * p2[i];
+          if (s1)  ptmp[i] += bt1 * s1[i] * w1[i];
+          if (s2)  ptmp[i] += bt2 * s2[i] * w2[i];
+        }
       }
     }
   }
 
   // Perform 3x3 matrix inversion, exploiting skew symmetry
-  const double gx = vb * gyro_tensor[Y][Z];
-  const double gy = vb * gyro_tensor[Z][X];
-  const double gz = vb * gyro_tensor[X][Y];
-  const double invdet = 1.0 / ub / (ub*ub + gx*gx + gy*gy + gz*gz);
+  const double ax = alpha * gyro_tensor[Y][Z];
+  const double ay = alpha * gyro_tensor[Z][X];
+  const double az = alpha * gyro_tensor[X][Y];
+  const double invdet = 1.0 / (1 + ax*ax + ay*ay + az*az);
   double inv[3][3];
 
-  inv[X][X] = invdet * (ub*ub + gx*gx);
-  inv[Y][Y] = invdet * (ub*ub + gy*gy);
-  inv[Z][Z] = invdet * (ub*ub + gz*gz);
-  inv[X][Y] = invdet * (gx*gy - ub*gz);
-  inv[Y][X] = invdet * (gy*gx + ub*gz);
-  inv[Z][X] = invdet * (gz*gx - ub*gy);
-  inv[X][Z] = invdet * (gx*gz + ub*gy);
-  inv[Y][Z] = invdet * (gy*gz - ub*gx);
-  inv[Z][Y] = invdet * (gz*gy + ub*gx);
+  inv[X][X] = invdet * (1 + ax*ax);
+  inv[Y][Y] = invdet * (1 + ay*ay);
+  inv[Z][Z] = invdet * (1 + az*az);
+  inv[X][Y] = invdet * (ax*ay - az);
+  inv[Y][X] = invdet * (ay*ax + az);
+  inv[Z][X] = invdet * (az*ax - ay);
+  inv[X][Z] = invdet * (ax*az + ay);
+  inv[Y][Z] = invdet * (ay*az - ax);
+  inv[Z][Y] = invdet * (az*ay + ax);
 
   FOR_COMPONENTS(c) DOCMP2 {
     if (d->P[c][cmp]) {
       const direction d0 = component_direction(c);
       if (W[c][cmp] && sigma[c][d0]) {
-        realnum *p = d->P[c][cmp];
-	const realnum *pp = d->P_prev[c][cmp];
+        realnum *p = d->P[c][cmp], *pp = d->P_prev[c][cmp];
+        const realnum *ptmp = d->P_tmp[c][cmp];
         const direction d1  = cycle_direction(gv.dim, d0, 1);
         const direction d2  = cycle_direction(gv.dim, d0, 2);
         const component c1 = direction_component(c, d1);
         const component c2 = direction_component(c, d2);
-        const realnum *pp1 = W[c1][cmp] ? d->P_prev[c1][cmp] : NULL;
-	const realnum *pp2 = W[c2][cmp] ? d->P_prev[c2][cmp] : NULL;
-
-	LOOP_OVER_VOL_OWNED(gv, c, i) {
-	  p[i] = inv[d0][d0] * pp[i];
-	  if (pp1) p[i] += inv[d0][d1] * pp1[i];
-	  if (pp2) p[i] += inv[d0][d2] * pp2[i];
-	}
+        const realnum *ptmp1 = W[c1][cmp] ? d->P_tmp[c1][cmp] : NULL;
+        const realnum *ptmp2 = W[c2][cmp] ? d->P_tmp[c2][cmp] : NULL;
+
+        LOOP_OVER_VOL_OWNED(gv, c, i) {
+          pp[i] = p[i];
+          p[i] = inv[d0][d0] * ptmp[i];
+          if (ptmp1) p[i] += inv[d0][d1] * ptmp1[i];
+          if (ptmp2) p[i] += inv[d0][d2] * ptmp2[i];
+        }
       }
     }
   }

From 054c5f5d3253d3be561b750b32c6e2c328b467c7 Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Sun, 26 May 2019 16:51:32 +0800
Subject: [PATCH 21/61] Try implementing the full nonlinear LLG equation

---
 src/meep.hpp           |  3 +-
 src/susceptibility.cpp | 95 ++++++++++++++++++++----------------------
 2 files changed, 47 insertions(+), 51 deletions(-)

diff --git a/src/meep.hpp b/src/meep.hpp
index 4669a974d..ae82ad616 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -289,7 +289,8 @@ class gyrotropic_susceptibility : public lorentzian_susceptibility {
 
 protected:
   double alpha;
-  double gyro_tensor[3][3];
+  double bvec[3];
+  int sgn[3][3];
 };
 
 class multilevel_susceptibility : public susceptibility {
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 038afa62f..609a35bd6 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -92,7 +92,7 @@ typedef struct {
   size_t ntot;
   realnum *P[NUM_FIELD_COMPONENTS][2];
   realnum *P_prev[NUM_FIELD_COMPONENTS][2];
-  realnum *P_tmp[NUM_FIELD_COMPONENTS][2]; // extra slot used for gyrotropic medium updating
+  realnum *P_tmp[NUM_FIELD_COMPONENTS][2]; // extra slot for gyrotropic medium updating
   realnum data[1];
 } lorentzian_data;
 
@@ -340,11 +340,12 @@ gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double alp
   : lorentzian_susceptibility(omega_0, gamma, false), alpha(alpha) {
   have_gyrotropy = true;
 
-  const vec bn = bias / fmax(abs(bias), 1e-10); // avoid division by zero
-  memset(gyro_tensor, 0, 9 * sizeof(double));
-  gyro_tensor[X][Y] = bn.z(); gyro_tensor[Y][X] = -bn.z();
-  gyro_tensor[Y][Z] = bn.x(); gyro_tensor[Z][Y] = -bn.x();
-  gyro_tensor[Z][X] = bn.y(); gyro_tensor[X][Z] = -bn.y();
+  const vec b = bias / fmax(abs(bias), 1e-10); // avoid division by zero
+  bvec[0] = b.x(); bvec[1] = b.y(); bvec[2] = b.z();
+
+  memset(sgn, 0, 9 * sizeof(int));
+  sgn[X][Y] = sgn[Y][Z] = sgn[Z][X] = 1;
+  sgn[X][Z] = sgn[Z][Y] = sgn[Y][X] = -1;
 }
 
 void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
@@ -352,7 +353,9 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
                                          const grid_volume &gv, void *P_internal_data) const {
   lorentzian_data *d = (lorentzian_data *)P_internal_data;
   const realnum w4pit = 4*pi*dt*omega_0;
-  const realnum g4pit = 4*pi*dt*gamma;
+  const realnum g2pi  = 2*pi*gamma;
+  const realnum g2pisq = g2pi*g2pi;
+  const realnum pi4dt = 4*pi*dt;
   (void)W_prev; // unused;
 
   FOR_COMPONENTS(c) DOCMP2 {
@@ -361,79 +364,72 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
       const realnum *w = W[c][cmp];
       if (w) {
         if (d0 != X && d0 != Y && d0 != Z)
-          abort("Cylindrical coordinates are not supported for gyrotropic media");
+          abort("Cylindrical coordinates not supported for gyrotropic media\n");
 
-        const realnum *p = d->P[c][cmp];
         const realnum *pp = d->P_prev[c][cmp];
         realnum *ptmp = d->P_tmp[c][cmp];
-
         direction d1 = cycle_direction(gv.dim, d0, 1);
         direction d2 = cycle_direction(gv.dim, d0, 2);
         component c1 = direction_component(c, d1);
         component c2 = direction_component(c, d2);
-
         const realnum *w1 = W[c1][cmp];
         const realnum *w2 = W[c2][cmp];
         const realnum *s1 = w1 ? sigma[c1][d1] : NULL;
         const realnum *s2 = w2 ? sigma[c2][d2] : NULL;
         const realnum *p1 = d->P[c1][cmp], *pp1 = d->P_prev[c1][cmp];
         const realnum *p2 = d->P[c2][cmp], *pp2 = d->P_prev[c2][cmp];
+        const realnum wb1 = w4pit * sgn[d0][d1] * bvec[c2];
+        const realnum wb2 = w4pit * sgn[d0][d2] * bvec[c1];
+	const int sgn1 = sgn[d0][d1];
+	const int sgn2 = sgn[d0][d2];
 
-        const realnum ab1 = alpha * gyro_tensor[d0][d1];
-        const realnum ab2 = alpha * gyro_tensor[d0][d2];
-        const realnum wb1 = w4pit * gyro_tensor[d0][d1];
-        const realnum wb2 = w4pit * gyro_tensor[d0][d2];
-        const realnum bt1 = 4*pi*dt * gyro_tensor[d0][d1];
-        const realnum bt2 = 4*pi*dt * gyro_tensor[d0][d2];
+	if (!pp1 || !pp2 || !p1 || !p2 || !s1 || !s2)
+	  abort("Gyrotropic media require 3D fields\n");
 
         LOOP_OVER_VOL_OWNED(gv, c, i) {
-          ptmp[i] = pp[i] - g4pit * p[i];
-          if (pp1) ptmp[i] += ab1 * pp1[i];
-          if (pp2) ptmp[i] += ab2 * pp2[i];
-          if (p1)  ptmp[i] -= wb1 * p1[i];
-          if (p2)  ptmp[i] -= wb2 * p2[i];
-          if (s1)  ptmp[i] += bt1 * s1[i] * w1[i];
-          if (s2)  ptmp[i] += bt2 * s2[i] * w2[i];
+          ptmp[i] = pp[i] - wb1 * p1[i] - wb2 * p2[i]
+	    - sgn1 * p1[i] * (pi4dt*s2[i]*w2[i] + g2pi * pp2[i])
+	    - sgn2 * p2[i] * (pi4dt*s1[i]*w1[i] + g2pi * pp1[i]);
         }
       }
     }
   }
 
-  // Perform 3x3 matrix inversion, exploiting skew symmetry
-  const double ax = alpha * gyro_tensor[Y][Z];
-  const double ay = alpha * gyro_tensor[Z][X];
-  const double az = alpha * gyro_tensor[X][Y];
-  const double invdet = 1.0 / (1 + ax*ax + ay*ay + az*az);
-  double inv[3][3];
-
-  inv[X][X] = invdet * (1 + ax*ax);
-  inv[Y][Y] = invdet * (1 + ay*ay);
-  inv[Z][Z] = invdet * (1 + az*az);
-  inv[X][Y] = invdet * (ax*ay - az);
-  inv[Y][X] = invdet * (ay*ax + az);
-  inv[Z][X] = invdet * (az*ax - ay);
-  inv[X][Z] = invdet * (ax*az + ay);
-  inv[Y][Z] = invdet * (ay*az - ax);
-  inv[Z][Y] = invdet * (az*ay + ax);
+  FOR_COMPONENTS(c) DOCMP2 {   // Overwrite P_prev
+    if (d->P[c][cmp]) {
+      if (W[c][cmp] && sigma[c][component_direction(c)]) {
+        const realnum *p = d->P[c][cmp];
+	realnum *pp = d->P_prev[c][cmp];
+        LOOP_OVER_VOL_OWNED(gv, c, i) {
+          pp[i] = p[i];
+	}
+      }
+    }
+  }
 
-  FOR_COMPONENTS(c) DOCMP2 {
+  FOR_COMPONENTS(c) DOCMP2 {  // Perform matrix inversion
     if (d->P[c][cmp]) {
       const direction d0 = component_direction(c);
       if (W[c][cmp] && sigma[c][d0]) {
-        realnum *p = d->P[c][cmp], *pp = d->P_prev[c][cmp];
+        realnum *p = d->P[c][cmp];
+        const realnum *pp = d->P_prev[c][cmp];
         const realnum *ptmp = d->P_tmp[c][cmp];
         const direction d1  = cycle_direction(gv.dim, d0, 1);
         const direction d2  = cycle_direction(gv.dim, d0, 2);
         const component c1 = direction_component(c, d1);
         const component c2 = direction_component(c, d2);
-        const realnum *ptmp1 = W[c1][cmp] ? d->P_tmp[c1][cmp] : NULL;
-        const realnum *ptmp2 = W[c2][cmp] ? d->P_tmp[c2][cmp] : NULL;
+        const realnum *pp1 = d->P_prev[c1][cmp];
+        const realnum *pp2 = d->P_prev[c2][cmp];
+        const realnum *ptmp1 = d->P_tmp[c1][cmp];
+        const realnum *ptmp2 = d->P_tmp[c2][cmp];
+	const int sgn1 = sgn[d0][d1];
+	const int sgn2 = sgn[d0][d2];
 
         LOOP_OVER_VOL_OWNED(gv, c, i) {
-          pp[i] = p[i];
-          p[i] = inv[d0][d0] * ptmp[i];
-          if (ptmp1) p[i] += inv[d0][d1] * ptmp1[i];
-          if (ptmp2) p[i] += inv[d0][d2] * ptmp2[i];
+          p[i] = 1/(1 + g2pisq*(pp[i]*pp[i] + pp1[i]*pp1[i] + pp2[i]*pp2[i]))
+	    * ((1 + g2pisq * pp[i]*pp[i]) * ptmp[i]
+	       + (g2pisq*pp[i]*pp1[i] - g2pi*sgn1*pp2[i]) * ptmp1[i]
+	       + (g2pisq*pp[i]*pp2[i] - g2pi*sgn2*pp1[i]) * ptmp2[i]);
         }
       }
     }
@@ -443,9 +439,8 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 void gyrotropic_susceptibility::dump_params(h5file *h5f, size_t *start) {
   size_t num_params = 9;
   size_t params_dims[1] = {num_params};
-  double bias[] = { gyro_tensor[Y][Z], gyro_tensor[Z][X], gyro_tensor[X][Y] };
   double params_data[] = {
-      8, (double)get_id(), bias[X], bias[Y], bias[Z],
+      8, (double)get_id(), bvec[0], bvec[1], bvec[2],
       alpha, omega_0, gamma, (double)no_omega_0_denominator};
   h5f->write_chunk(1, start, params_dims, params_data);
   *start += num_params;

From 15636743b8d6d104d63e9fe0c9835818efcf836f Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Mon, 27 May 2019 13:52:22 +0800
Subject: [PATCH 22/61] Add implicit static polarization to gyrotropy
 implementation

---
 python/geom.py           |  3 +-
 python/typemap_utils.cpp | 10 -----
 scheme/meep.scm.in       |  3 +-
 scheme/structure.cpp     |  7 ++--
 src/material_data.hpp    |  1 -
 src/meep.hpp             | 10 ++---
 src/meepgeom.cpp         |  7 ++--
 src/susceptibility.cpp   | 81 +++++++++++++++++++++-------------------
 8 files changed, 55 insertions(+), 67 deletions(-)

diff --git a/python/geom.py b/python/geom.py
index 4101c53e4..4ef0be335 100644
--- a/python/geom.py
+++ b/python/geom.py
@@ -282,10 +282,9 @@ def __init__(self, noise_amp=0.0, **kwargs):
 
 class GyrotropicSusceptibility(LorentzianSusceptibility):
 
-    def __init__(self, bias=Vector3(0, 0, 1), alpha=0.0, **kwargs):
+    def __init__(self, bias=Vector3(0, 0, 1), **kwargs):
         super(GyrotropicSusceptibility, self).__init__(**kwargs)
         self.bias = bias
-        self.alpha = alpha
 
 class MultilevelAtom(Susceptibility):
 
diff --git a/python/typemap_utils.cpp b/python/typemap_utils.cpp
index 60d9a776a..302d78c35 100644
--- a/python/typemap_utils.cpp
+++ b/python/typemap_utils.cpp
@@ -331,7 +331,6 @@ static int py_susceptibility_to_susceptibility(PyObject *po, susceptibility_stru
   s->gamma = 0;
   s->noise_amp = 0;
   s->bias.x = s->bias.y = s->bias.z = 0;
-  s->alpha = 0;
   s->transitions.resize(0);
   s->initial_populations.resize(0);
 
@@ -351,10 +350,6 @@ static int py_susceptibility_to_susceptibility(PyObject *po, susceptibility_stru
     if (!get_attr_v3(po, &s->bias, "bias")) return 0;
   }
 
-  if (PyObject_HasAttrString(po, "alpha")) {
-    if (!get_attr_dbl(po, &s->alpha, "alpha")) { return 0; }
-  }
-
   if (PyObject_HasAttrString(po, "transitions")) {
     // MultilevelAtom
     PyObject *py_trans = PyObject_GetAttrString(po, "transitions");
@@ -477,11 +472,6 @@ static PyObject *susceptibility_to_py_obj(susceptibility_struct *s) {
     PyObject *py_bias = vec2py(vector3_to_vec(s->bias));
     PyObject_SetAttrString(res, "bias", py_bias);
     Py_DECREF(py_bias);
-
-    PyObject *py_alpha = PyFloat_FromDouble(s->alpha);
-    PyObject_SetAttrString(res, "alpha", py_alpha);
-    Py_DECREF(py_alpha);
-
   } else if (s->noise_amp == 0) {
     if (s->drude) {
       PyObject *py_drude_class = PyObject_GetAttrString(geom_mod, "DrudeSusceptibility");
diff --git a/scheme/meep.scm.in b/scheme/meep.scm.in
index 06f5df939..857559f97 100644
--- a/scheme/meep.scm.in
+++ b/scheme/meep.scm.in
@@ -58,8 +58,7 @@
 (define-class noisy-drude-susceptibility drude-susceptibility
   (define-property noise-amp no-default 'number))
 (define-class gyrotropic-susceptibility lorentzian-susceptibility
-  (define-property bias no-default 'vector3)
-  (define-property alpha no-default 'number))
+  (define-property bias no-default 'vector3))
 
 (define polarizability lorentzian-susceptibility) ; backwards compat
 (define omega frequency) ; backwards compat
diff --git a/scheme/structure.cpp b/scheme/structure.cpp
index be5b99789..8f6bdf16f 100644
--- a/scheme/structure.cpp
+++ b/scheme/structure.cpp
@@ -1297,10 +1297,9 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
           sus = new meep::noisy_lorentzian_susceptibility(nd->noise_amp, d->frequency, d->gamma);
 	} else if (d->which_subclass == lorentzian_susceptibility::GYROTROPIC_SUSCEPTIBILITY) {
           gyrotropic_susceptibility *gd = d->subclass.gyrotropic_susceptibility_data;
-          master_printf("gyrotropic lorentzian susceptibility: bias=(%g,%g,%g), alpha=%g, frequency=%g, gamma=%g\n",
-			gd->bias.x, gd->bias.y, gd->bias.z, gd->alpha, d->frequency, d->gamma);
-          sus = new meep::gyrotropic_susceptibility(vector3_to_vec(gd->bias), gd->alpha,
-                                                    d->frequency, d->gamma);
+          master_printf("gyrotropic lorentzian susceptibility: bias=(%g,%g,%g), frequency=%g, gamma=%g\n",
+			gd->bias.x, gd->bias.y, gd->bias.z, d->frequency, d->gamma);
+          sus = new meep::gyrotropic_susceptibility(vector3_to_vec(gd->bias), d->frequency, d->gamma);
         } else { // just a Lorentzian
           master_printf("lorentzian susceptibility: frequency=%g, gamma=%g\n", d->frequency,
                         d->gamma);
diff --git a/src/material_data.hpp b/src/material_data.hpp
index ec5c998c4..352b6155e 100644
--- a/src/material_data.hpp
+++ b/src/material_data.hpp
@@ -60,7 +60,6 @@ typedef struct susceptibility_struct {
   vector3 sigma_offdiag;
   vector3 sigma_diag;
   vector3 bias;
-  double alpha;
   double frequency;
   double gamma;
   double noise_amp;
diff --git a/src/meep.hpp b/src/meep.hpp
index ae82ad616..1fc6ca2e0 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -277,7 +277,7 @@ class noisy_lorentzian_susceptibility : public lorentzian_susceptibility {
 /* gyrotropic susceptibility */
 class gyrotropic_susceptibility : public lorentzian_susceptibility {
 public:
-  gyrotropic_susceptibility(const vec &bias, double alpha, double omega_0, double gamma);
+  gyrotropic_susceptibility(const vec &bias, double omega_0, double gamma);
   virtual susceptibility *clone() const { return new gyrotropic_susceptibility(*this); }
 
   virtual void update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
@@ -285,12 +285,12 @@ class gyrotropic_susceptibility : public lorentzian_susceptibility {
 			const grid_volume &gv, void *P_internal_data) const;
 
   virtual void dump_params(h5file *h5f, size_t *start);
-  virtual int get_num_params() { return 8; }
+  virtual int get_num_params() { return 7; }
 
 protected:
-  double alpha;
-  double bvec[3];
-  int sgn[3][3];
+  double psat;     // conserved magnitude of the polarization vector
+  double bvec[3];  // unit vector pointing along direction of gyrotropic bias
+  int sgn[3][3];   // antisymmetric tensor: sgn[X][Y] = 1, sgn[Z][Y] = -1, etc.
 };
 
 class multilevel_susceptibility : public susceptibility {
diff --git a/src/meepgeom.cpp b/src/meepgeom.cpp
index bc825a3e2..d30a32adf 100644
--- a/src/meepgeom.cpp
+++ b/src/meepgeom.cpp
@@ -1376,7 +1376,7 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
                                                         ss->drude);
       } else if (gyrotropic) {
         sus = new meep::gyrotropic_susceptibility(meep::vec(ss->bias.x, ss->bias.y, ss->bias.z),
-						  ss->alpha, ss->frequency, ss->gamma);
+						  ss->frequency, ss->gamma);
       } else {
         sus = new meep::lorentzian_susceptibility(ss->frequency, ss->gamma, ss->drude);
       }
@@ -1385,9 +1385,8 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
 		    gyrotropic ? "" : ss->drude ? "drude" : "lorentzian",
 		    ss->frequency, ss->gamma);
       if (noisy) master_printf(", amp=%g ", ss->noise_amp);
-      if (gyrotropic) master_printf(", bias=(%g,%g,%g), alpha=%g ",
-				    ss->bias.x, ss->bias.y, ss->bias.z,
-				    ss->alpha);
+      if (gyrotropic) master_printf(", bias=(%g,%g,%g) ",
+				    ss->bias.x, ss->bias.y, ss->bias.z);
       master_printf("\n");
     }
 
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 609a35bd6..06c80cbf8 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -336,11 +336,13 @@ void noisy_lorentzian_susceptibility::dump_params(h5file *h5f, size_t *start) {
 }
 
 
-gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double alpha, double omega_0, double gamma)
-  : lorentzian_susceptibility(omega_0, gamma, false), alpha(alpha) {
+gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double omega_0, double gamma)
+  : lorentzian_susceptibility(omega_0, gamma, false) {
   have_gyrotropy = true;
 
-  const vec b = bias / fmax(abs(bias), 1e-10); // avoid division by zero
+  psat = abs(bias);
+
+  const vec b = (psat != 0.0 ? bias/psat : vec(0,0,1));
   bvec[0] = b.x(); bvec[1] = b.y(); bvec[2] = b.z();
 
   memset(sgn, 0, 9 * sizeof(int));
@@ -354,8 +356,7 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
   lorentzian_data *d = (lorentzian_data *)P_internal_data;
   const realnum w4pit = 4*pi*dt*omega_0;
   const realnum g2pi  = 2*pi*gamma;
-  const realnum g2pisq = g2pi*g2pi;
-  const realnum pi4dt = 4*pi*dt;
+  const realnum dt4pi = 4*pi*dt;
   (void)W_prev; // unused;
 
   FOR_COMPONENTS(c) DOCMP2 {
@@ -366,31 +367,30 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
         if (d0 != X && d0 != Y && d0 != Z)
           abort("Cylindrical coordinates not supported for gyrotropic media\n");
 
+        const direction d1 = cycle_direction(gv.dim, d0, 1);
+        const direction d2 = cycle_direction(gv.dim, d0, 2);
+        const component c1 = direction_component(c, d1);
+        const component c2 = direction_component(c, d2);
+
         const realnum *pp = d->P_prev[c][cmp];
-        realnum *ptmp = d->P_tmp[c][cmp];
-        direction d1 = cycle_direction(gv.dim, d0, 1);
-        direction d2 = cycle_direction(gv.dim, d0, 2);
-        component c1 = direction_component(c, d1);
-        component c2 = direction_component(c, d2);
-        const realnum *w1 = W[c1][cmp];
-        const realnum *w2 = W[c2][cmp];
-        const realnum *s1 = w1 ? sigma[c1][d1] : NULL;
-        const realnum *s2 = w2 ? sigma[c2][d2] : NULL;
         const realnum *p1 = d->P[c1][cmp], *pp1 = d->P_prev[c1][cmp];
         const realnum *p2 = d->P[c2][cmp], *pp2 = d->P_prev[c2][cmp];
-        const realnum wb1 = w4pit * sgn[d0][d1] * bvec[c2];
-        const realnum wb2 = w4pit * sgn[d0][d2] * bvec[c1];
-	const int sgn1 = sgn[d0][d1];
-	const int sgn2 = sgn[d0][d2];
+        realnum *ptmp = d->P_tmp[c][cmp];
+
+        const realnum *w1 = W[c1][cmp], *s1 = w1 ? sigma[c1][d1] : NULL;
+        const realnum *w2 = W[c2][cmp], *s2 = w2 ? sigma[c2][d2] : NULL;
+        const realnum pb1 = psat  * bvec[d1], pb2 = psat  * bvec[d2];
+        const realnum wb1 = w4pit * bvec[d1], wb2 = w4pit * bvec[d2];
+	const int sgn1 = sgn[d0][d1], sgn2 = sgn[d0][d2];
 
 	if (!pp1 || !pp2 || !p1 || !p2 || !s1 || !s2)
 	  abort("Gyrotropic media require 3D fields\n");
 
         LOOP_OVER_VOL_OWNED(gv, c, i) {
-          ptmp[i] = pp[i] - wb1 * p1[i] - wb2 * p2[i]
-	    - sgn1 * p1[i] * (pi4dt*s2[i]*w2[i] + g2pi * pp2[i])
-	    - sgn2 * p2[i] * (pi4dt*s1[i]*w1[i] + g2pi * pp1[i]);
-        }
+          ptmp[i] = pp[i]
+	    + sgn1 * ((g2pi * pp1[i] + dt4pi*s1[i]*w1[i]) * (pb2+p2[i]) + wb1*p2[i])
+	    + sgn2 * ((g2pi * pp2[i] + dt4pi*s2[i]*w2[i]) * (pb1+p1[i]) + wb2*p1[i]);
+	}
       }
     }
   }
@@ -411,25 +411,28 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
     if (d->P[c][cmp]) {
       const direction d0 = component_direction(c);
       if (W[c][cmp] && sigma[c][d0]) {
-        realnum *p = d->P[c][cmp];
-        const realnum *pp = d->P_prev[c][cmp];
-        const realnum *ptmp = d->P_tmp[c][cmp];
         const direction d1  = cycle_direction(gv.dim, d0, 1);
         const direction d2  = cycle_direction(gv.dim, d0, 2);
         const component c1 = direction_component(c, d1);
         const component c2 = direction_component(c, d2);
-        const realnum *pp1 = d->P_prev[c1][cmp];
-        const realnum *pp2 = d->P_prev[c2][cmp];
-        const realnum *ptmp1 = d->P_tmp[c1][cmp];
-        const realnum *ptmp2 = d->P_tmp[c2][cmp];
-	const int sgn1 = sgn[d0][d1];
-	const int sgn2 = sgn[d0][d2];
 
-        LOOP_OVER_VOL_OWNED(gv, c, i) {
-          p[i] = 1/(1 + g2pisq*(pp[i]*pp[i] + pp1[i]*pp1[i] + pp2[i]*pp2[i]))
-	    * ((1 + g2pisq * pp[i]*pp[i]) * ptmp[i]
-	       + (g2pisq*pp[i]*pp1[i] - g2pi*sgn1*pp2[i]) * ptmp1[i]
-	       + (g2pisq*pp[i]*pp2[i] - g2pi*sgn2*pp1[i]) * ptmp2[i]);
+        realnum *p = d->P[c][cmp];
+        const realnum *pp = d->P_prev[c][cmp], *ptmp = d->P_tmp[c][cmp];
+        const realnum *pp1 = d->P_prev[c1][cmp], *ptmp1 = d->P_tmp[c1][cmp];
+        const realnum *pp2 = d->P_prev[c2][cmp], *ptmp2 = d->P_tmp[c2][cmp];
+	const realnum pb0 = psat * bvec[d0];
+        const realnum pb1 = psat * bvec[d1], pb2 = psat  * bvec[d2];
+	const int sgn1 = sgn[d0][d1], sgn2 = sgn[d0][d2];
+	realnum gp0, gp1, gp2;
+
+	LOOP_OVER_VOL_OWNED(gv, c, i) {
+	  gp0 = g2pi * (pb0 + pp[i]);
+	  gp1 = g2pi * (pb1 + pp1[i]);
+	  gp2 = g2pi * (pb2 + pp2[i]);
+          p[i] = 1/(1 + gp0*gp0 + gp1*gp1 + gp2*gp2)
+	    * ((1 + gp0*gp0) * ptmp[i]
+	       + (gp0*gp1 - sgn1*gp2) * ptmp1[i]
+	       + (gp0*gp2 - sgn2*gp1) * ptmp2[i]);
         }
       }
     }
@@ -437,11 +440,11 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 }
 
 void gyrotropic_susceptibility::dump_params(h5file *h5f, size_t *start) {
-  size_t num_params = 9;
+  size_t num_params = 8;
   size_t params_dims[1] = {num_params};
   double params_data[] = {
-      8, (double)get_id(), bvec[0], bvec[1], bvec[2],
-      alpha, omega_0, gamma, (double)no_omega_0_denominator};
+      7, (double)get_id(), bvec[0], bvec[1], bvec[2],
+      omega_0, gamma, (double)no_omega_0_denominator};
   h5f->write_chunk(1, start, params_dims, params_data);
   *start += num_params;
 }

From a7b1e2795ee528be9ef87f36af747eb26b83d89e Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Mon, 27 May 2019 16:18:41 +0800
Subject: [PATCH 23/61] Put static P back in in subtract_P

---
 src/meep.hpp           |  3 +++
 src/susceptibility.cpp | 24 ++++++++++++++++++++++--
 2 files changed, 25 insertions(+), 2 deletions(-)

diff --git a/src/meep.hpp b/src/meep.hpp
index 1fc6ca2e0..57a84b11e 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -284,6 +284,9 @@ class gyrotropic_susceptibility : public lorentzian_susceptibility {
 			realnum *W_prev[NUM_FIELD_COMPONENTS][2], double dt,
 			const grid_volume &gv, void *P_internal_data) const;
 
+  virtual void subtract_P(field_type ft, realnum *f_minus_p[NUM_FIELD_COMPONENTS][2],
+                          void *P_internal_data) const;
+
   virtual void dump_params(h5file *h5f, size_t *start);
   virtual int get_num_params() { return 7; }
 
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 06c80cbf8..1c28e87cf 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -399,10 +399,10 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
     if (d->P[c][cmp]) {
       if (W[c][cmp] && sigma[c][component_direction(c)]) {
         const realnum *p = d->P[c][cmp];
-	realnum *pp = d->P_prev[c][cmp];
+        realnum *pp = d->P_prev[c][cmp];
         LOOP_OVER_VOL_OWNED(gv, c, i) {
           pp[i] = p[i];
-	}
+        }
       }
     }
   }
@@ -439,6 +439,26 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
   }
 }
 
+void gyrotropic_susceptibility::subtract_P(field_type ft,
+                                           realnum *f_minus_p[NUM_FIELD_COMPONENTS][2],
+                                           void *P_internal_data) const {
+  lorentzian_data *d = (lorentzian_data *)P_internal_data;
+  field_type ft2 = ft == E_stuff ? D_stuff : B_stuff; // for sources etc.
+  size_t ntot = d->ntot;
+  FOR_FT_COMPONENTS(ft, ec) DOCMP2 {
+    if (d->P[ec][cmp]) {
+      component dc = field_type_component(ft2, ec);
+      if (f_minus_p[dc][cmp]) {
+	const realnum pb = psat * bvec[component_direction(ec)];
+        realnum *p = d->P[ec][cmp];
+        realnum *fmp = f_minus_p[dc][cmp];
+        for (size_t i = 0; i < ntot; ++i)
+          fmp[i] -= pb + p[i];
+      }
+    }
+  }
+}
+
 void gyrotropic_susceptibility::dump_params(h5file *h5f, size_t *start) {
   size_t num_params = 8;
   size_t params_dims[1] = {num_params};

From 1dc5912cf2081d70ea6554c0300f0d392e2bddbf Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Mon, 27 May 2019 17:36:59 +0800
Subject: [PATCH 24/61] Add gyrotropy example

---
 python/examples/gyrotropic-dispersion.py | 61 ++++++++++++++++++++++++
 1 file changed, 61 insertions(+)
 create mode 100644 python/examples/gyrotropic-dispersion.py

diff --git a/python/examples/gyrotropic-dispersion.py b/python/examples/gyrotropic-dispersion.py
new file mode 100644
index 000000000..09f0fde06
--- /dev/null
+++ b/python/examples/gyrotropic-dispersion.py
@@ -0,0 +1,61 @@
+import numpy as np
+from numpy.linalg import inv
+import matplotlib.pyplot as plt
+
+## Material parameters
+sn = 0.0002
+fn = 0.5
+gn = 1e-5
+epsn = 15
+psat = 100
+
+def meep_sim(gn, sn, epsn, tmax):
+
+    ## For mu ~ 1 and epsn ~ 15, wavelength will be ~0.26.  Set dx and dt to ~0.01
+    resolution = 100
+
+    cellsize = [0.01, 0.01, 7.0] # Don't set to zero size: need 3D fields
+    fsrc = 0.8
+    src_z = -1.75
+    pml_depth = 1.5
+
+    import meep as mp
+
+    cell = mp.Vector3(*cellsize)
+
+    ## Define gyromagnetic material biased along z direction
+    susc = [mp.GyrotropicSusceptibility(frequency=fn, gamma=gn, sigma=sn,
+                                        bias=mp.Vector3(0, 0, psat))]
+    mat = mp.Medium(epsilon=epsn, mu=1, H_susceptibilities=susc)
+
+    pml_layers = [mp.PML(thickness=pml_depth, direction=mp.Z)]
+
+    sources = [mp.Source(mp.ContinuousSource(frequency=fsrc),
+                         component=mp.Ex, center=mp.Vector3(0, 0, src_z))]
+
+    sim = mp.Simulation(cell_size=cell, geometry=[], sources=sources,
+                        k_point=mp.Vector3(), # Periodic BCs in x and y
+                        boundary_layers=pml_layers,
+                        default_material=mat, resolution=resolution)
+
+    sim.run(until=tmax)
+
+    ex_data = sim.get_array(center=mp.Vector3(), size=mp.Vector3(0, 0, cellsize[2]), component=mp.Ex)
+    ey_data = sim.get_array(center=mp.Vector3(), size=mp.Vector3(0, 0, cellsize[2]), component=mp.Ey)
+
+    L = cellsize[2]/2
+    z = np.linspace(-L, L, len(ex_data))
+    plt.title('t = {}'.format(tmax))
+    plt.plot(z, ex_data, label='Ex')
+    plt.plot(z, ey_data, label='Ey')
+    plt.xlim(-L, L)
+    plt.xlabel('z')
+    plt.legend()
+
+plt.figure(2, figsize=(8,8))
+plt.subplot(2,1,1)
+meep_sim(gn, sn, epsn, 200)
+plt.subplot(2,1,2)
+meep_sim(gn, sn, epsn, 500)
+plt.tight_layout()
+plt.show()

From 8dd19881195fea5a807288d8634734aad23f8d20 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Tue, 28 May 2019 11:42:18 +0800
Subject: [PATCH 25/61] Fix; use LOOP_OVER_VOL instead of LOOP_OVER_VOL_OWNED
 to ensure updating of off-diagonal components

---
 python/meep.i          |  12 ++--
 src/susceptibility.cpp | 142 ++++++++++++++++++++---------------------
 2 files changed, 75 insertions(+), 79 deletions(-)

diff --git a/python/meep.i b/python/meep.i
index 998f13133..3f576efd9 100644
--- a/python/meep.i
+++ b/python/meep.i
@@ -749,21 +749,21 @@ meep::volume_list *make_volume_list(const meep::volume &v, int c,
 }
 
 %typemap(in) const meep::src_time & {
-    PyObject *swig_obj = NULL;
+    PyObject *swig_obj_1 = NULL;
     void *tmp_ptr = 0;
     int tmp_res = 0;
 
     if(PyObject_IsInstance($input, py_source_time_object())) {
-        swig_obj = PyObject_GetAttrString($input, "swigobj");
+        swig_obj_1 = PyObject_GetAttrString($input, "swigobj");
     } else if(PyObject_IsInstance($input, py_meep_src_time_object())) {
-        swig_obj = $input;
-        Py_XINCREF(swig_obj);
+        swig_obj_1 = $input;
+        Py_XINCREF(swig_obj_1);
     } else {
       meep::abort("Expected a meep.source.SourceTime or a meep.src_time\n");
     }
 
-    tmp_res = SWIG_ConvertPtr(swig_obj, &tmp_ptr, $1_descriptor, 0);
-    Py_XDECREF(swig_obj);
+    tmp_res = SWIG_ConvertPtr(swig_obj_1, &tmp_ptr, $1_descriptor, 0);
+    Py_XDECREF(swig_obj_1);
 
     if(!SWIG_IsOK(tmp_res)) {
         SWIG_exception_fail(SWIG_ArgError(tmp_res), "Couldn't convert Python object to meep::src_time");
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 3b7289c32..4c65f1844 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -362,70 +362,66 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
     if (d->P[c][cmp]) {
       const direction d0 = component_direction(c);
       const realnum *w = W[c][cmp], *s = sigma[c][d0];
-      if (w && s) {
-	if (d0 != X && d0 != Y && d0 != Z)
-	  abort("Cylindrical coordinates are not supported for gyrotropic media");
 
-        const realnum *p = d->P[c][cmp], *pp = d->P_prev[c][cmp];
-	realnum *rhs = d->P_tmp[c][cmp];
+      if (!w || !s || (d0 != X && d0 != Y && d0 != Z))
+	abort("Non-3D fields not supported for gyrotropic media\n");
 
-        const ptrdiff_t is = gv.stride(d0) * (is_magnetic(c) ? -1 : +1);
+      const realnum *p = d->P[c][cmp], *pp = d->P_prev[c][cmp];
+      realnum *rhs = d->P_tmp[c][cmp];
+      const ptrdiff_t is = gv.stride(d0) * (is_magnetic(c) ? -1 : +1);
 
-        direction d1 = cycle_direction(gv.dim, d0, 1);
-        component c1 = direction_component(c, d1);
-        ptrdiff_t is1 = gv.stride(d1) * (is_magnetic(c) ? -1 : +1);
-        const realnum *w1 = W[c1][cmp];
-        const realnum *s1 = w1 ? sigma[c][d1] : NULL;
+      direction d1 = cycle_direction(gv.dim, d0, 1);
+      direction d2 = cycle_direction(gv.dim, d0, 2);
+      component c1 = direction_component(c, d1);
+      component c2 = direction_component(c, d2);
+      ptrdiff_t is1 = gv.stride(d1) * (is_magnetic(c) ? -1 : +1);
+      ptrdiff_t is2 = gv.stride(d2) * (is_magnetic(c) ? -1 : +1);
 
-        direction d2 = cycle_direction(gv.dim, d0, 2);
-        component c2 = direction_component(c, d2);
-        ptrdiff_t is2 = gv.stride(d2) * (is_magnetic(c) ? -1 : +1);
-        const realnum *w2 = W[c2][cmp];
-        const realnum *s2 = w2 ? sigma[c][d2] : NULL;
+      const realnum *w1 = W[c1][cmp], *s1 = w1 ? sigma[c][d1] : NULL;
+      const realnum *w2 = W[c2][cmp], *s2 = w2 ? sigma[c][d2] : NULL;
+      const realnum *pp1 = d->P_prev[c1][cmp], g1 = pi * dt * gyro_tensor[d0][d1];
+      const realnum *pp2 = d->P_prev[c2][cmp], g2 = pi * dt * gyro_tensor[d0][d2];
 
-	// Off-diagonal polarization components
-        const realnum *pp1 = d->P_prev[c1][cmp];
-        const realnum *pp2 = d->P_prev[c2][cmp];
-	const realnum g1 = pi * dt * gyro_tensor[d0][d1];
-	const realnum g2 = pi * dt * gyro_tensor[d0][d2];
+      const realnum *p1 = d->P[c1][cmp];
+      const realnum *p2 = d->P[c2][cmp];
 
-        if (s2 && !s1) { // make s1 the non-NULL one if possible
-          SWAP(direction, d1, d2);
-          SWAP(component, c1, c2);
-          SWAP(ptrdiff_t, is1, is2);
-          SWAP(const realnum *, w1, w2);
-          SWAP(const realnum *, s1, s2);
-        }
-        if (s1 && s2) { // 3x3 anisotropic
-          LOOP_OVER_VOL_OWNED(gv, c, i) {
-            if (s[i] != 0) {
-              rhs[i] = (2 - omega0dtsqr_denom) * p[i]
-		+ omega0dtsqr * (s[i] * w[i]
-				 + OFFDIAG(s1, w1, is1, is)
-				 + OFFDIAG(s2, w2, is2, is))
-		- gamma1 * pp[i];
-	      if (pp1) rhs[i] -= g1 * pp1[i];
-	      if (pp2) rhs[i] -= g2 * pp2[i];
-            }
-          }
-        } else if (s1) { // 2x2 anisotropic
-          LOOP_OVER_VOL_OWNED(gv, c, i) {
-            if (s[i] != 0) {
-              rhs[i] = (2 - omega0dtsqr_denom) * p[i]
-		+ omega0dtsqr * (s[i] * w[i] + OFFDIAG(s1, w1, is1, is))
-		- gamma1 * pp[i];
-	      if (pp1) rhs[i] -= g1 * pp1[i];
-	      if (pp2) rhs[i] -= g2 * pp2[i];
-            }
-          }
-        } else { // isotropic
-          LOOP_OVER_VOL_OWNED(gv, c, i) {
-            rhs[i] = (2 - omega0dtsqr_denom) * p[i]
-	      + omega0dtsqr * (s[i] * w[i]) - gamma1 * pp[i];
-	    if (pp1) rhs[i] -= g1 * pp1[i];
-	    if (pp2) rhs[i] -= g2 * pp2[i];
-          }
-        }
+      realnum *rhs1 = d->P_tmp[c1][cmp];
+      realnum *rhs2 = d->P_tmp[c2][cmp];
+
+      if (!pp1 || !pp2)
+	abort("Non-3D fields not supported for gyrotropic media\n");
+
+      if (s2 && !s1) { // make s1 the non-NULL one if possible
+        SWAP(direction, d1, d2);
+        SWAP(component, c1, c2);
+        SWAP(ptrdiff_t, is1, is2);
+        SWAP(const realnum *, w1, w2);
+        SWAP(const realnum *, s1, s2);
+      }
+      if (s1 && s2) { // 3x3 anisotropic
+	LOOP_OVER_VOL(gv, c, i) {
+	  if (s[i] != 0) {
+	    rhs[i] = (2 - omega0dtsqr_denom) * p[i]
+	      + omega0dtsqr * (s[i] * w[i]
+			       + OFFDIAG(s1, w1, is1, is)
+			       + OFFDIAG(s2, w2, is2, is))
+	      - gamma1 * pp[i] - g1 * pp1[i] - g2 * pp2[i];
+	  }
+	}
+      } else if (s1) { // 2x2 anisotropic
+	LOOP_OVER_VOL(gv, c, i) {
+	  if (s[i] != 0) {
+	    rhs[i] = (2 - omega0dtsqr_denom) * p[i]
+	      + omega0dtsqr * (s[i] * w[i] + OFFDIAG(s1, w1, is1, is))
+	      - gamma1 * pp[i] - g1 * pp1[i] - g2 * pp2[i];
+	  }
+	}
+      } else { // isotropic
+	LOOP_OVER_VOL(gv, c, i) {
+	  rhs[i] = (2 - omega0dtsqr_denom) * p[i]
+	    + omega0dtsqr * (s[i] * w[i]) - gamma1 * pp[i]
+	    - g1 * pp1[i] - g2 * pp2[i];
+	}
       }
     }
   }
@@ -451,21 +447,21 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
   FOR_COMPONENTS(c) DOCMP2 {
     if (d->P[c][cmp]) {
       const direction d0 = component_direction(c);
-      if (W[c][cmp] && sigma[c][d0]) {
-        realnum *p = d->P[c][cmp], *pp = d->P_prev[c][cmp], *rhs = d->P_tmp[c][cmp];
-        const direction d1  = cycle_direction(gv.dim, d0, 1);
-        const direction d2  = cycle_direction(gv.dim, d0, 2);
-        const component c1 = direction_component(c, d1);
-        const component c2 = direction_component(c, d2);
-        const realnum *rhs1 = W[c1][cmp] ? d->P_tmp[c1][cmp] : NULL;
-	const realnum *rhs2 = W[c2][cmp] ? d->P_tmp[c2][cmp] : NULL;
-
-	LOOP_OVER_VOL_OWNED(gv, c, i) {
-	  pp[i] = p[i];
-	  p[i]  = inv[d0][d0] * rhs[i];
-	  if (rhs1) p[i] += inv[d0][d1] * rhs1[i];
-	  if (rhs2) p[i] += inv[d0][d2] * rhs2[i];
-	}
+      realnum *p = d->P[c][cmp], *pp = d->P_prev[c][cmp];
+      const realnum *rhs = d->P_tmp[c][cmp];
+      const direction d1  = cycle_direction(gv.dim, d0, 1);
+      const direction d2  = cycle_direction(gv.dim, d0, 2);
+      const component c1 = direction_component(c, d1);
+      const component c2 = direction_component(c, d2);
+      const realnum *rhs1 = d->P_tmp[c1][cmp];
+      const realnum *rhs2 = d->P_tmp[c2][cmp];
+
+      if (!rhs || !rhs1 || !rhs2)
+	abort("Internal error in gyrotropy code\n");
+
+      LOOP_OVER_VOL(gv, c, i) {
+	pp[i] = p[i];
+	p[i]  = inv[d0][d0] * rhs[i] + inv[d0][d1] * rhs1[i] + inv[d0][d2] * rhs2[i];
       }
     }
   }

From 0842ad65e2d696729caa3bc1852fda2f78bb19ef Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Tue, 28 May 2019 13:11:51 +0800
Subject: [PATCH 26/61] Clamp the magnitude of the LLG polarization vector.

---
 python/examples/gyrotropic-dispersion.py |  7 +-
 src/susceptibility.cpp                   | 83 +++++++++++++++++-------
 2 files changed, 61 insertions(+), 29 deletions(-)

diff --git a/python/examples/gyrotropic-dispersion.py b/python/examples/gyrotropic-dispersion.py
index 09f0fde06..f4939976f 100644
--- a/python/examples/gyrotropic-dispersion.py
+++ b/python/examples/gyrotropic-dispersion.py
@@ -14,10 +14,10 @@ def meep_sim(gn, sn, epsn, tmax):
     ## For mu ~ 1 and epsn ~ 15, wavelength will be ~0.26.  Set dx and dt to ~0.01
     resolution = 100
 
-    cellsize = [0.01, 0.01, 7.0] # Don't set to zero size: need 3D fields
+    cellsize = [0.01, 0.01, 5.0] # Don't set to zero size: need 3D fields
     fsrc = 0.8
-    src_z = -1.75
-    pml_depth = 1.5
+    src_z = -1.8
+    pml_depth = 0.25
 
     import meep as mp
 
@@ -58,4 +58,3 @@ def meep_sim(gn, sn, epsn, tmax):
 plt.subplot(2,1,2)
 meep_sim(gn, sn, epsn, 500)
 plt.tight_layout()
-plt.show()
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 1c28e87cf..656137533 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -363,34 +363,33 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
     if (d->P[c][cmp]) {
       const direction d0 = component_direction(c);
       const realnum *w = W[c][cmp];
-      if (w) {
-        if (d0 != X && d0 != Y && d0 != Z)
-          abort("Cylindrical coordinates not supported for gyrotropic media\n");
 
-        const direction d1 = cycle_direction(gv.dim, d0, 1);
-        const direction d2 = cycle_direction(gv.dim, d0, 2);
-        const component c1 = direction_component(c, d1);
-        const component c2 = direction_component(c, d2);
+      if (!w || (d0 != X && d0 != Y && d0 != Z))
+	abort("Cylindrical coordinates not supported for gyrotropic media\n");
 
-        const realnum *pp = d->P_prev[c][cmp];
-        const realnum *p1 = d->P[c1][cmp], *pp1 = d->P_prev[c1][cmp];
-        const realnum *p2 = d->P[c2][cmp], *pp2 = d->P_prev[c2][cmp];
-        realnum *ptmp = d->P_tmp[c][cmp];
+      const direction d1 = cycle_direction(gv.dim, d0, 1);
+      const direction d2 = cycle_direction(gv.dim, d0, 2);
+      const component c1 = direction_component(c, d1);
+      const component c2 = direction_component(c, d2);
 
-        const realnum *w1 = W[c1][cmp], *s1 = w1 ? sigma[c1][d1] : NULL;
-        const realnum *w2 = W[c2][cmp], *s2 = w2 ? sigma[c2][d2] : NULL;
-        const realnum pb1 = psat  * bvec[d1], pb2 = psat  * bvec[d2];
-        const realnum wb1 = w4pit * bvec[d1], wb2 = w4pit * bvec[d2];
-	const int sgn1 = sgn[d0][d1], sgn2 = sgn[d0][d2];
+      const realnum *pp = d->P_prev[c][cmp];
+      const realnum *p1 = d->P[c1][cmp], *pp1 = d->P_prev[c1][cmp];
+      const realnum *p2 = d->P[c2][cmp], *pp2 = d->P_prev[c2][cmp];
+      realnum *ptmp = d->P_tmp[c][cmp];
+
+      const realnum *w1 = W[c1][cmp], *s1 = w1 ? sigma[c1][d1] : NULL;
+      const realnum *w2 = W[c2][cmp], *s2 = w2 ? sigma[c2][d2] : NULL;
+      const realnum pb1 = psat  * bvec[d1], pb2 = psat  * bvec[d2];
+      const realnum wb1 = w4pit * bvec[d1], wb2 = w4pit * bvec[d2];
+      const int sgn1 = sgn[d0][d1], sgn2 = sgn[d0][d2];
 
-	if (!pp1 || !pp2 || !p1 || !p2 || !s1 || !s2)
-	  abort("Gyrotropic media require 3D fields\n");
+      if (!pp1 || !pp2 || !p1 || !p2 || !s1 || !s2)
+	abort("Gyrotropic media require 3D fields\n");
 
-        LOOP_OVER_VOL_OWNED(gv, c, i) {
-          ptmp[i] = pp[i]
-	    + sgn1 * ((g2pi * pp1[i] + dt4pi*s1[i]*w1[i]) * (pb2+p2[i]) + wb1*p2[i])
-	    + sgn2 * ((g2pi * pp2[i] + dt4pi*s2[i]*w2[i]) * (pb1+p1[i]) + wb2*p1[i]);
-	}
+      LOOP_OVER_VOL(gv, c, i) {
+	ptmp[i] = pp[i]
+	  + sgn1 * ((g2pi * pp1[i] + dt4pi*s1[i]*w1[i]) * (pb2+p2[i]) + wb1*p2[i])
+	  + sgn2 * ((g2pi * pp2[i] + dt4pi*s2[i]*w2[i]) * (pb1+p1[i]) + wb2*p1[i]);
       }
     }
   }
@@ -400,7 +399,7 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
       if (W[c][cmp] && sigma[c][component_direction(c)]) {
         const realnum *p = d->P[c][cmp];
         realnum *pp = d->P_prev[c][cmp];
-        LOOP_OVER_VOL_OWNED(gv, c, i) {
+        LOOP_OVER_VOL(gv, c, i) {
           pp[i] = p[i];
         }
       }
@@ -425,7 +424,7 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 	const int sgn1 = sgn[d0][d1], sgn2 = sgn[d0][d2];
 	realnum gp0, gp1, gp2;
 
-	LOOP_OVER_VOL_OWNED(gv, c, i) {
+	LOOP_OVER_VOL(gv, c, i) {
 	  gp0 = g2pi * (pb0 + pp[i]);
 	  gp1 = g2pi * (pb1 + pp1[i]);
 	  gp2 = g2pi * (pb2 + pp2[i]);
@@ -437,6 +436,40 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
       }
     }
   }
+
+  // FIXME: hack to clamp the magnitude of the total polarization vector at psat.
+  FOR_COMPONENTS(c) DOCMP2 {
+    if (d->P[c][cmp]) {
+      const direction d0 = component_direction(c);
+      const direction d1  = cycle_direction(gv.dim, d0, 1);
+      const direction d2  = cycle_direction(gv.dim, d0, 2);
+      const component c1 = direction_component(c, d1);
+      const component c2 = direction_component(c, d2);
+
+      const realnum *p = d->P[c][cmp];
+      const realnum *p1 = d->P_prev[c1][cmp], *p2 = d->P_prev[c2][cmp];
+      const realnum pb0 = psat * bvec[d0];
+      const realnum pb1 = psat * bvec[d1], pb2 = psat  * bvec[d2];
+      realnum *ptmp = d->P_tmp[c][cmp];
+      realnum ptot0, ptot1, ptot2;
+
+      LOOP_OVER_VOL(gv, c, i) {
+	ptot0 = pb0 + p[i];
+	ptot1 = pb1 + p1[i];
+	ptot2 = pb2 + p2[i];
+	ptmp[i] = ptot0*psat/sqrt(ptot0*ptot0+ptot1*ptot1+ptot2*ptot2) - pb0;
+      }
+    }
+  }
+  FOR_COMPONENTS(c) DOCMP2 {
+    if (d->P[c][cmp]) {
+      realnum *p = d->P[c][cmp];
+      const realnum *ptmp = d->P_tmp[c][cmp];
+      LOOP_OVER_VOL(gv, c, i) {
+	p[i] = ptmp[i];
+      }
+    }
+  }
 }
 
 void gyrotropic_susceptibility::subtract_P(field_type ft,

From 9364b6f3ffc18a41a2cd67e8a693686ac39592bc Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Tue, 28 May 2019 13:34:12 +0800
Subject: [PATCH 27/61] Revert inadvertent unrelated change to meep.i

---
 python/meep.i | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/python/meep.i b/python/meep.i
index 3f576efd9..998f13133 100644
--- a/python/meep.i
+++ b/python/meep.i
@@ -749,21 +749,21 @@ meep::volume_list *make_volume_list(const meep::volume &v, int c,
 }
 
 %typemap(in) const meep::src_time & {
-    PyObject *swig_obj_1 = NULL;
+    PyObject *swig_obj = NULL;
     void *tmp_ptr = 0;
     int tmp_res = 0;
 
     if(PyObject_IsInstance($input, py_source_time_object())) {
-        swig_obj_1 = PyObject_GetAttrString($input, "swigobj");
+        swig_obj = PyObject_GetAttrString($input, "swigobj");
     } else if(PyObject_IsInstance($input, py_meep_src_time_object())) {
-        swig_obj_1 = $input;
-        Py_XINCREF(swig_obj_1);
+        swig_obj = $input;
+        Py_XINCREF(swig_obj);
     } else {
       meep::abort("Expected a meep.source.SourceTime or a meep.src_time\n");
     }
 
-    tmp_res = SWIG_ConvertPtr(swig_obj_1, &tmp_ptr, $1_descriptor, 0);
-    Py_XDECREF(swig_obj_1);
+    tmp_res = SWIG_ConvertPtr(swig_obj, &tmp_ptr, $1_descriptor, 0);
+    Py_XDECREF(swig_obj);
 
     if(!SWIG_IsOK(tmp_res)) {
         SWIG_exception_fail(SWIG_ArgError(tmp_res), "Couldn't convert Python object to meep::src_time");

From 18831d288f5e1b457713cb3b051d4fcf2162f569 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Tue, 28 May 2019 15:15:29 +0800
Subject: [PATCH 28/61] Minor code cleanup

---
 python/geom.py         |  2 +-
 src/susceptibility.cpp | 29 ++++++++++-------------------
 2 files changed, 11 insertions(+), 20 deletions(-)

diff --git a/python/geom.py b/python/geom.py
index e4d8fec14..9b8744f6c 100644
--- a/python/geom.py
+++ b/python/geom.py
@@ -282,7 +282,7 @@ def __init__(self, noise_amp=0.0, **kwargs):
 
 class GyrotropicLorentzianSusceptibility(LorentzianSusceptibility):
 
-    def __init__(self, bias=Vector3(0, 0, 1), **kwargs):
+    def __init__(self, bias=Vector3(), **kwargs):
         super(GyrotropicLorentzianSusceptibility, self).__init__(**kwargs)
         self.bias = bias
 
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 4c65f1844..3d67c709b 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -364,7 +364,7 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
       const realnum *w = W[c][cmp], *s = sigma[c][d0];
 
       if (!w || !s || (d0 != X && d0 != Y && d0 != Z))
-	abort("Non-3D fields not supported for gyrotropic media\n");
+	abort("Gyrotropic media require 3D Cartesian fields\n");
 
       const realnum *p = d->P[c][cmp], *pp = d->P_prev[c][cmp];
       realnum *rhs = d->P_tmp[c][cmp];
@@ -379,17 +379,13 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 
       const realnum *w1 = W[c1][cmp], *s1 = w1 ? sigma[c][d1] : NULL;
       const realnum *w2 = W[c2][cmp], *s2 = w2 ? sigma[c][d2] : NULL;
-      const realnum *pp1 = d->P_prev[c1][cmp], g1 = pi * dt * gyro_tensor[d0][d1];
-      const realnum *pp2 = d->P_prev[c2][cmp], g2 = pi * dt * gyro_tensor[d0][d2];
-
-      const realnum *p1 = d->P[c1][cmp];
-      const realnum *p2 = d->P[c2][cmp];
-
-      realnum *rhs1 = d->P_tmp[c1][cmp];
-      realnum *rhs2 = d->P_tmp[c2][cmp];
+      const realnum *pp1 = d->P_prev[c1][cmp], g1 = pi*dt*gyro_tensor[d0][d1];
+      const realnum *pp2 = d->P_prev[c2][cmp], g2 = pi*dt*gyro_tensor[d0][d2];
+      const realnum *p1 = d->P[c1][cmp], *p2 = d->P[c2][cmp];
+      realnum *rhs1 = d->P_tmp[c1][cmp], *rhs2 = d->P_tmp[c2][cmp];
 
       if (!pp1 || !pp2)
-	abort("Non-3D fields not supported for gyrotropic media\n");
+	abort("Gyrotropic media require 3D Cartesian fields\n");
 
       if (s2 && !s1) { // make s1 the non-NULL one if possible
         SWAP(direction, d1, d2);
@@ -448,20 +444,15 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
     if (d->P[c][cmp]) {
       const direction d0 = component_direction(c);
       realnum *p = d->P[c][cmp], *pp = d->P_prev[c][cmp];
-      const realnum *rhs = d->P_tmp[c][cmp];
+      const realnum *rhs0 = d->P_tmp[c][cmp];
       const direction d1  = cycle_direction(gv.dim, d0, 1);
       const direction d2  = cycle_direction(gv.dim, d0, 2);
-      const component c1 = direction_component(c, d1);
-      const component c2 = direction_component(c, d2);
-      const realnum *rhs1 = d->P_tmp[c1][cmp];
-      const realnum *rhs2 = d->P_tmp[c2][cmp];
-
-      if (!rhs || !rhs1 || !rhs2)
-	abort("Internal error in gyrotropy code\n");
+      const realnum *rhs1 = d->P_tmp[direction_component(c, d1)][cmp];
+      const realnum *rhs2 = d->P_tmp[direction_component(c, d2)][cmp];
 
       LOOP_OVER_VOL(gv, c, i) {
 	pp[i] = p[i];
-	p[i]  = inv[d0][d0] * rhs[i] + inv[d0][d1] * rhs1[i] + inv[d0][d2] * rhs2[i];
+	p[i]  = inv[d0][d0] * rhs0[i] + inv[d0][d1] * rhs1[i] + inv[d0][d2] * rhs2[i];
       }
     }
   }

From 006365d251639bb0868b6bac0a3ba0c2ec9dcbf8 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Wed, 29 May 2019 10:57:36 +0800
Subject: [PATCH 29/61] Flag "needs_W_notowned" for gyrotropic media

---
 src/meep.hpp | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/src/meep.hpp b/src/meep.hpp
index c03283e09..4fffd168f 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -287,6 +287,11 @@ class gyrotropic_susceptibility : public lorentzian_susceptibility {
 
   virtual void dump_params(h5file *h5f, size_t *start);
   virtual int get_num_params() { return 7; }
+  virtual bool needs_W_notowned(component c, realnum *W[NUM_FIELD_COMPONENTS][2]) const {
+    (void)c;
+    (void)W;
+    return true;
+  }
 
 protected:
   double gyro_tensor[3][3];

From 021cf485c2d17b37a89793fa623d7eabbc3dc4bf Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Thu, 30 May 2019 12:55:08 +0800
Subject: [PATCH 30/61] Update gyrotropic P components explicitly; don't use
 LOOP_OVER_VOL_OWNED

---
 src/susceptibility.cpp | 112 +++++++++++++++--------------------------
 1 file changed, 41 insertions(+), 71 deletions(-)

diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 3d67c709b..323284b3e 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -355,7 +355,7 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
   const double omega2pi = 2 * pi * omega_0, g2pi = gamma * 2 * pi;
   const double omega0dtsqr = omega2pi * omega2pi * dt * dt;
   const double gamma1 = (1 - g2pi * dt / 2);
-  const double omega0dtsqr_denom = no_omega_0_denominator ? 0 : omega0dtsqr;
+  const double diagfac = 2 - (no_omega_0_denominator ? 0 : omega0dtsqr);
   (void)W_prev; // unused;
 
   FOR_COMPONENTS(c) DOCMP2 {
@@ -364,60 +364,29 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
       const realnum *w = W[c][cmp], *s = sigma[c][d0];
 
       if (!w || !s || (d0 != X && d0 != Y && d0 != Z))
-	abort("Gyrotropic media require 3D Cartesian fields\n");
-
-      const realnum *p = d->P[c][cmp], *pp = d->P_prev[c][cmp];
-      realnum *rhs = d->P_tmp[c][cmp];
-      const ptrdiff_t is = gv.stride(d0) * (is_magnetic(c) ? -1 : +1);
-
-      direction d1 = cycle_direction(gv.dim, d0, 1);
-      direction d2 = cycle_direction(gv.dim, d0, 2);
-      component c1 = direction_component(c, d1);
-      component c2 = direction_component(c, d2);
-      ptrdiff_t is1 = gv.stride(d1) * (is_magnetic(c) ? -1 : +1);
-      ptrdiff_t is2 = gv.stride(d2) * (is_magnetic(c) ? -1 : +1);
+	abort("gyrotropic media require 3D Cartesian fields\n");
 
+      const direction d1 = cycle_direction(gv.dim, d0, 1);
+      const direction d2 = cycle_direction(gv.dim, d0, 2);
+      const component c1 = direction_component(c, d1);
+      const component c2 = direction_component(c, d2);
       const realnum *w1 = W[c1][cmp], *s1 = w1 ? sigma[c][d1] : NULL;
       const realnum *w2 = W[c2][cmp], *s2 = w2 ? sigma[c][d2] : NULL;
-      const realnum *pp1 = d->P_prev[c1][cmp], g1 = pi*dt*gyro_tensor[d0][d1];
-      const realnum *pp2 = d->P_prev[c2][cmp], g2 = pi*dt*gyro_tensor[d0][d2];
-      const realnum *p1 = d->P[c1][cmp], *p2 = d->P[c2][cmp];
-      realnum *rhs1 = d->P_tmp[c1][cmp], *rhs2 = d->P_tmp[c2][cmp];
-
-      if (!pp1 || !pp2)
-	abort("Gyrotropic media require 3D Cartesian fields\n");
-
-      if (s2 && !s1) { // make s1 the non-NULL one if possible
-        SWAP(direction, d1, d2);
-        SWAP(component, c1, c2);
-        SWAP(ptrdiff_t, is1, is2);
-        SWAP(const realnum *, w1, w2);
-        SWAP(const realnum *, s1, s2);
-      }
-      if (s1 && s2) { // 3x3 anisotropic
-	LOOP_OVER_VOL(gv, c, i) {
-	  if (s[i] != 0) {
-	    rhs[i] = (2 - omega0dtsqr_denom) * p[i]
-	      + omega0dtsqr * (s[i] * w[i]
-			       + OFFDIAG(s1, w1, is1, is)
-			       + OFFDIAG(s2, w2, is2, is))
-	      - gamma1 * pp[i] - g1 * pp1[i] - g2 * pp2[i];
-	  }
-	}
-      } else if (s1) { // 2x2 anisotropic
-	LOOP_OVER_VOL(gv, c, i) {
-	  if (s[i] != 0) {
-	    rhs[i] = (2 - omega0dtsqr_denom) * p[i]
-	      + omega0dtsqr * (s[i] * w[i] + OFFDIAG(s1, w1, is1, is))
-	      - gamma1 * pp[i] - g1 * pp1[i] - g2 * pp2[i];
-	  }
-	}
-      } else { // isotropic
-	LOOP_OVER_VOL(gv, c, i) {
-	  rhs[i] = (2 - omega0dtsqr_denom) * p[i]
-	    + omega0dtsqr * (s[i] * w[i]) - gamma1 * pp[i]
-	    - g1 * pp1[i] - g2 * pp2[i];
-	}
+      const realnum *p0 = d->P[c][cmp],  *pp0 = d->P_prev[c][cmp];
+      const realnum *p1 = d->P[c1][cmp], *pp1 = d->P_prev[c1][cmp];
+      const realnum *p2 = d->P[c2][cmp], *pp2 = d->P_prev[c2][cmp];
+      realnum *rhs0 = d->P_tmp[c][cmp], *rhs1 = d->P_tmp[c1][cmp], *rhs2 = d->P_tmp[c2][cmp];
+      const realnum g01 = pi*dt*gyro_tensor[d0][d1], g10 = pi*dt*gyro_tensor[d1][d0];
+      const realnum g12 = pi*dt*gyro_tensor[d1][d2], g21 = pi*dt*gyro_tensor[d2][d1];
+      const realnum g20 = pi*dt*gyro_tensor[d2][d0], g02 = pi*dt*gyro_tensor[d0][d2];
+
+      if (!pp1 || !pp2) abort("gyrotropic media require 3D Cartesian fields\n");
+      if (s1 || s2) abort("gyrotropic media do not support tensor sigma\n");
+
+      LOOP_OVER_VOL_OWNED(gv, c, i) {
+	rhs0[i] = diagfac*p0[i] - g01*pp1[i] - g02*pp2[i] - gamma1*pp0[i] + omega0dtsqr*s[i]*w[i];
+	rhs1[i] = diagfac*p1[i] - g12*pp2[i] - g10*pp0[i] - gamma1*pp1[i];
+	rhs2[i] = diagfac*p2[i] - g21*pp1[i] - g20*pp0[i] - gamma1*pp2[i];
       }
     }
   }
@@ -428,31 +397,32 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
   const double gy = pi * dt * gyro_tensor[Z][X];
   const double gz = pi * dt * gyro_tensor[X][Y];
   const double invdet = 1.0 / gd / (gd*gd + gx*gx + gy*gy + gz*gz);
-  double inv[3][3];
-
-  inv[X][X] = invdet * (gd*gd + gx*gx);
-  inv[Y][Y] = invdet * (gd*gd + gy*gy);
-  inv[Z][Z] = invdet * (gd*gd + gz*gz);
-  inv[X][Y] = invdet * (gx*gy + gd*gz);
-  inv[Y][X] = invdet * (gy*gx - gd*gz);
-  inv[Z][X] = invdet * (gz*gx + gd*gy);
-  inv[X][Z] = invdet * (gx*gz - gd*gy);
-  inv[Y][Z] = invdet * (gy*gz + gd*gx);
-  inv[Z][Y] = invdet * (gz*gy - gd*gx);
+  const double inv[3][3]
+    = {{ invdet*(gd*gd+gx*gx), invdet*(gx*gy+gd*gz), invdet*(gx*gz-gd*gy) },
+       { invdet*(gy*gx-gd*gz), invdet*(gd*gd+gy*gy), invdet*(gy*gz+gd*gx) },
+       { invdet*(gz*gx+gd*gy), invdet*(gz*gy-gd*gx), invdet*(gd*gd+gz*gz) }};
 
   FOR_COMPONENTS(c) DOCMP2 {
     if (d->P[c][cmp]) {
       const direction d0 = component_direction(c);
-      realnum *p = d->P[c][cmp], *pp = d->P_prev[c][cmp];
-      const realnum *rhs0 = d->P_tmp[c][cmp];
       const direction d1  = cycle_direction(gv.dim, d0, 1);
       const direction d2  = cycle_direction(gv.dim, d0, 2);
-      const realnum *rhs1 = d->P_tmp[direction_component(c, d1)][cmp];
-      const realnum *rhs2 = d->P_tmp[direction_component(c, d2)][cmp];
-
-      LOOP_OVER_VOL(gv, c, i) {
-	pp[i] = p[i];
-	p[i]  = inv[d0][d0] * rhs0[i] + inv[d0][d1] * rhs1[i] + inv[d0][d2] * rhs2[i];
+      const component c1 = direction_component(c, d1);
+      const component c2 = direction_component(c, d2);
+      realnum *p0 = d->P[c][cmp],  *pp0 = d->P_prev[c][cmp];
+      realnum *p1 = d->P[c1][cmp], *pp1 = d->P_prev[c1][cmp];
+      realnum *p2 = d->P[c2][cmp], *pp2 = d->P_prev[c2][cmp];
+      const realnum *rhs0 = d->P_tmp[c][cmp];
+      const realnum *rhs1 = d->P_tmp[c1][cmp];
+      const realnum *rhs2 = d->P_tmp[c2][cmp];
+
+      LOOP_OVER_VOL_OWNED(gv, c, i) {
+	pp0[i] = p0[i];
+	pp1[i] = p1[i];
+	pp2[i] = p2[i];
+	p0[i]  = inv[d0][d0] * rhs0[i] + inv[d0][d1] * rhs1[i] + inv[d0][d2] * rhs2[i];
+	p1[i]  = inv[d1][d1] * rhs1[i] + inv[d1][d2] * rhs2[i] + inv[d1][d0] * rhs0[i];
+	p2[i]  = inv[d2][d2] * rhs2[i] + inv[d2][d0] * rhs0[i] + inv[d2][d1] * rhs1[i];
       }
     }
   }

From 164b5a822983d122a623e91dd73bb7c415c69e94 Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Thu, 30 May 2019 21:54:32 +0800
Subject: [PATCH 31/61] Enable needs_P on all components for gyrotropic media

---
 src/meep.hpp           |  1 +
 src/susceptibility.cpp | 45 ++++++++++++++++++++++--------------------
 2 files changed, 25 insertions(+), 21 deletions(-)

diff --git a/src/meep.hpp b/src/meep.hpp
index 4fffd168f..68a69d3f9 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -281,6 +281,7 @@ class gyrotropic_susceptibility : public lorentzian_susceptibility {
 			    bool no_omega_0_denominator = true);
   virtual susceptibility *clone() const { return new gyrotropic_susceptibility(*this); }
 
+  bool needs_P(component c, int cmp, realnum *W[NUM_FIELD_COMPONENTS][2]) const;
   virtual void update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 			realnum *W_prev[NUM_FIELD_COMPONENTS][2], double dt,
 			const grid_volume &gv, void *P_internal_data) const;
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 323284b3e..1d6815d10 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -348,6 +348,17 @@ gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double ome
   gyro_tensor[Z][X] = bias.y(); gyro_tensor[X][Z] = -bias.y();
 }
 
+bool gyrotropic_susceptibility::needs_P(component c, int cmp, realnum *W[NUM_FIELD_COMPONENTS][2]) const {
+  if (!is_electric(c) && !is_magnetic(c)) return false;
+  direction d0 = component_direction(c);
+  if ((d0 == X || d0 == Y || d0 == Z) && sigma[c][d0]) {
+    FOR_DIRECTIONS(d) {
+      if (W[direction_component(c, d)][cmp]) return true;
+    }
+  }
+  return false;
+}
+
 void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 					 realnum *W_prev[NUM_FIELD_COMPONENTS][2], double dt,
 					 const grid_volume &gv, void *P_internal_data) const {
@@ -370,23 +381,21 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
       const direction d2 = cycle_direction(gv.dim, d0, 2);
       const component c1 = direction_component(c, d1);
       const component c2 = direction_component(c, d2);
-      const realnum *w1 = W[c1][cmp], *s1 = w1 ? sigma[c][d1] : NULL;
-      const realnum *w2 = W[c2][cmp], *s2 = w2 ? sigma[c][d2] : NULL;
-      const realnum *p0 = d->P[c][cmp],  *pp0 = d->P_prev[c][cmp];
-      const realnum *p1 = d->P[c1][cmp], *pp1 = d->P_prev[c1][cmp];
-      const realnum *p2 = d->P[c2][cmp], *pp2 = d->P_prev[c2][cmp];
-      realnum *rhs0 = d->P_tmp[c][cmp], *rhs1 = d->P_tmp[c1][cmp], *rhs2 = d->P_tmp[c2][cmp];
-      const realnum g01 = pi*dt*gyro_tensor[d0][d1], g10 = pi*dt*gyro_tensor[d1][d0];
-      const realnum g12 = pi*dt*gyro_tensor[d1][d2], g21 = pi*dt*gyro_tensor[d2][d1];
-      const realnum g20 = pi*dt*gyro_tensor[d2][d0], g02 = pi*dt*gyro_tensor[d0][d2];
+      const realnum *s1 = W[c1][cmp] ? sigma[c][d1] : NULL;
+      const realnum *s2 = W[c2][cmp] ? sigma[c][d2] : NULL;
+      const realnum *p = d->P[c][cmp];
+      const realnum *pp0 = d->P_prev[c][cmp];
+      const realnum *pp1 = d->P_prev[c1][cmp];
+      const realnum *pp2 = d->P_prev[c2][cmp];
+      realnum *rhs = d->P_tmp[c][cmp];
+      const realnum g1 = pi*dt*gyro_tensor[d0][d1], g2 = pi*dt*gyro_tensor[d0][d2];
 
       if (!pp1 || !pp2) abort("gyrotropic media require 3D Cartesian fields\n");
       if (s1 || s2) abort("gyrotropic media do not support tensor sigma\n");
 
       LOOP_OVER_VOL_OWNED(gv, c, i) {
-	rhs0[i] = diagfac*p0[i] - g01*pp1[i] - g02*pp2[i] - gamma1*pp0[i] + omega0dtsqr*s[i]*w[i];
-	rhs1[i] = diagfac*p1[i] - g12*pp2[i] - g10*pp0[i] - gamma1*pp1[i];
-	rhs2[i] = diagfac*p2[i] - g21*pp1[i] - g20*pp0[i] - gamma1*pp2[i];
+	rhs[i] = diagfac*p[i] - g1*pp1[i] - g2*pp2[i] - gamma1*pp0[i]
+	  + omega0dtsqr*s[i]*w[i];
       }
     }
   }
@@ -409,20 +418,14 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
       const direction d2  = cycle_direction(gv.dim, d0, 2);
       const component c1 = direction_component(c, d1);
       const component c2 = direction_component(c, d2);
-      realnum *p0 = d->P[c][cmp],  *pp0 = d->P_prev[c][cmp];
-      realnum *p1 = d->P[c1][cmp], *pp1 = d->P_prev[c1][cmp];
-      realnum *p2 = d->P[c2][cmp], *pp2 = d->P_prev[c2][cmp];
+      realnum *p = d->P[c][cmp],  *pp = d->P_prev[c][cmp];
       const realnum *rhs0 = d->P_tmp[c][cmp];
       const realnum *rhs1 = d->P_tmp[c1][cmp];
       const realnum *rhs2 = d->P_tmp[c2][cmp];
 
       LOOP_OVER_VOL_OWNED(gv, c, i) {
-	pp0[i] = p0[i];
-	pp1[i] = p1[i];
-	pp2[i] = p2[i];
-	p0[i]  = inv[d0][d0] * rhs0[i] + inv[d0][d1] * rhs1[i] + inv[d0][d2] * rhs2[i];
-	p1[i]  = inv[d1][d1] * rhs1[i] + inv[d1][d2] * rhs2[i] + inv[d1][d0] * rhs0[i];
-	p2[i]  = inv[d2][d2] * rhs2[i] + inv[d2][d0] * rhs0[i] + inv[d2][d1] * rhs1[i];
+	pp[i] = p[i];
+	p[i]  = inv[d0][d0] * rhs0[i] + inv[d0][d1] * rhs1[i] + inv[d0][d2] * rhs2[i];
       }
     }
   }

From 6515785d14d3cde4cf1ec2bf3d5e04ba225304e2 Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Thu, 30 May 2019 22:59:37 +0800
Subject: [PATCH 32/61] Fix gyrotropy scheme to track 9 polarization components
 per unit cell.

---
 python/meep.i          |  12 +--
 src/meep.hpp           |  11 ++-
 src/susceptibility.cpp | 184 +++++++++++++++++++++++++++++------------
 3 files changed, 147 insertions(+), 60 deletions(-)

diff --git a/python/meep.i b/python/meep.i
index 998f13133..3f576efd9 100644
--- a/python/meep.i
+++ b/python/meep.i
@@ -749,21 +749,21 @@ meep::volume_list *make_volume_list(const meep::volume &v, int c,
 }
 
 %typemap(in) const meep::src_time & {
-    PyObject *swig_obj = NULL;
+    PyObject *swig_obj_1 = NULL;
     void *tmp_ptr = 0;
     int tmp_res = 0;
 
     if(PyObject_IsInstance($input, py_source_time_object())) {
-        swig_obj = PyObject_GetAttrString($input, "swigobj");
+        swig_obj_1 = PyObject_GetAttrString($input, "swigobj");
     } else if(PyObject_IsInstance($input, py_meep_src_time_object())) {
-        swig_obj = $input;
-        Py_XINCREF(swig_obj);
+        swig_obj_1 = $input;
+        Py_XINCREF(swig_obj_1);
     } else {
       meep::abort("Expected a meep.source.SourceTime or a meep.src_time\n");
     }
 
-    tmp_res = SWIG_ConvertPtr(swig_obj, &tmp_ptr, $1_descriptor, 0);
-    Py_XDECREF(swig_obj);
+    tmp_res = SWIG_ConvertPtr(swig_obj_1, &tmp_ptr, $1_descriptor, 0);
+    Py_XDECREF(swig_obj_1);
 
     if(!SWIG_IsOK(tmp_res)) {
         SWIG_exception_fail(SWIG_ArgError(tmp_res), "Couldn't convert Python object to meep::src_time");
diff --git a/src/meep.hpp b/src/meep.hpp
index 68a69d3f9..79caf4480 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -223,8 +223,7 @@ class susceptibility {
 class lorentzian_susceptibility : public susceptibility {
 public:
   lorentzian_susceptibility(double omega_0, double gamma, bool no_omega_0_denominator = false)
-      : omega_0(omega_0), gamma(gamma), no_omega_0_denominator(no_omega_0_denominator),
-	have_gyrotropy(false) {}
+      : omega_0(omega_0), gamma(gamma), no_omega_0_denominator(no_omega_0_denominator) {}
   virtual susceptibility *clone() const { return new lorentzian_susceptibility(*this); }
   virtual ~lorentzian_susceptibility() {}
 
@@ -250,7 +249,6 @@ class lorentzian_susceptibility : public susceptibility {
 protected:
   double omega_0, gamma;
   bool no_omega_0_denominator;
-  bool have_gyrotropy; // whether to assign an extra slot for gyrotropy calculations
 };
 
 /* like a Lorentzian susceptibility, but the polarization equation
@@ -281,10 +279,17 @@ class gyrotropic_susceptibility : public lorentzian_susceptibility {
 			    bool no_omega_0_denominator = true);
   virtual susceptibility *clone() const { return new gyrotropic_susceptibility(*this); }
 
+  void *new_internal_data(realnum *W[NUM_FIELD_COMPONENTS][2], const grid_volume &gv) const;
+  void init_internal_data(realnum *W[NUM_FIELD_COMPONENTS][2], double dt,
+			  const grid_volume &gv, void *data) const;
+  void *copy_internal_data(void *data) const;
+
   bool needs_P(component c, int cmp, realnum *W[NUM_FIELD_COMPONENTS][2]) const;
   virtual void update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 			realnum *W_prev[NUM_FIELD_COMPONENTS][2], double dt,
 			const grid_volume &gv, void *P_internal_data) const;
+  void subtract_P(field_type ft, realnum *f_minus_p[NUM_FIELD_COMPONENTS][2],
+		  void *P_internal_data) const;
 
   virtual void dump_params(h5file *h5f, size_t *start);
   virtual int get_num_params() { return 7; }
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 1d6815d10..4cb5a07dc 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -92,7 +92,6 @@ typedef struct {
   size_t ntot;
   realnum *P[NUM_FIELD_COMPONENTS][2];
   realnum *P_prev[NUM_FIELD_COMPONENTS][2];
-  realnum *P_tmp[NUM_FIELD_COMPONENTS][2]; // extra slot used for gyrotropic medium updating
   realnum data[1];
 } lorentzian_data;
 
@@ -101,9 +100,8 @@ typedef struct {
 void *lorentzian_susceptibility::new_internal_data(realnum *W[NUM_FIELD_COMPONENTS][2],
                                                    const grid_volume &gv) const {
   int num = 0;
-  int nslots = have_gyrotropy ? 3 : 2;
   FOR_COMPONENTS(c) DOCMP2 {
-    if (needs_P(c, cmp, W)) num += nslots * gv.ntot();
+    if (needs_P(c, cmp, W)) num += 2 * gv.ntot();
   }
   size_t sz = sizeof(lorentzian_data) + sizeof(realnum) * (num - 1);
   lorentzian_data *d = (lorentzian_data *)malloc(sz);
@@ -121,18 +119,12 @@ void lorentzian_susceptibility::init_internal_data(realnum *W[NUM_FIELD_COMPONEN
   size_t ntot = d->ntot = gv.ntot();
   realnum *P = d->data;
   realnum *P_prev = d->data + ntot;
-  realnum *P_tmp = have_gyrotropy ? P_prev + ntot : NULL;
-  int nslots = have_gyrotropy ? 3 : 2;
-
   FOR_COMPONENTS(c) DOCMP2 {
     if (needs_P(c, cmp, W)) {
       d->P[c][cmp] = P;
       d->P_prev[c][cmp] = P_prev;
-      d->P_tmp[c][cmp] = P_tmp;
-
-      P += nslots * ntot;
-      P_prev += nslots * ntot;
-      if (have_gyrotropy) P_tmp += nslots * ntot;
+      P += 2 * ntot;
+      P_prev += 2 * ntot;
     }
   }
 }
@@ -145,17 +137,12 @@ void *lorentzian_susceptibility::copy_internal_data(void *data) const {
   size_t ntot = d->ntot;
   realnum *P = dnew->data;
   realnum *P_prev = dnew->data + ntot;
-  realnum *P_tmp = have_gyrotropy ? P_prev + ntot : NULL;
-  int nslots = have_gyrotropy ? 3 : 2;
-
   FOR_COMPONENTS(c) DOCMP2 {
     if (d->P[c][cmp]) {
       dnew->P[c][cmp] = P;
       dnew->P_prev[c][cmp] = P_prev;
-      dnew->P_tmp[c][cmp] = P_tmp;
-      P += nslots * ntot;
-      P_prev += nslots * ntot;
-      if (have_gyrotropy) P_tmp += nslots * ntot;
+      P += 2 * ntot;
+      P_prev += 2 * ntot;
     }
   }
   return (void *)dnew;
@@ -339,8 +326,6 @@ void noisy_lorentzian_susceptibility::dump_params(h5file *h5f, size_t *start) {
 gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double omega_0, double gamma,
 						     bool no_omega_0_denominator)
   : lorentzian_susceptibility(omega_0, gamma, no_omega_0_denominator) {
-  have_gyrotropy = true;
-
   // Precalculate g_{ij} = sum_k epsilon_{ijk} b_k, used in update_P.
   memset(gyro_tensor, 0, 9 * sizeof(double));
   gyro_tensor[X][Y] = bias.z(); gyro_tensor[Y][X] = -bias.z();
@@ -348,6 +333,73 @@ gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double ome
   gyro_tensor[Z][X] = bias.y(); gyro_tensor[X][Z] = -bias.y();
 }
 
+/* To implement gyrotropic susceptibilities, we track three
+   polarization components (e.g. Px, Py, Pz) on EACH of the Yee cell's
+   three driving field positions (e.g., Ex, Ey, and Ez), i.e. 9
+   numbers per cell.  This takes 3x the memory and runtime compared to
+   Lorentzian susceptibility.  The advantage is that during update_P,
+   we can directly access the value of P at each update point without
+   averaging.  */
+
+typedef struct {
+  size_t sz_data;
+  size_t ntot;
+  realnum *P[NUM_FIELD_COMPONENTS][2][3];
+  realnum *P_prev[NUM_FIELD_COMPONENTS][2][3];
+  realnum *P_tmp[NUM_FIELD_COMPONENTS][2][3];
+  realnum data[1];
+} gyrotropy_data;
+
+void *gyrotropic_susceptibility::new_internal_data(realnum *W[NUM_FIELD_COMPONENTS][2],
+                                                   const grid_volume &gv) const {
+  int num = 0;
+  FOR_COMPONENTS(c) DOCMP2 {
+    if (needs_P(c, cmp, W)) num += 9 * gv.ntot();
+  }
+  size_t sz = sizeof(gyrotropy_data) + sizeof(realnum) * (num - 1);
+  gyrotropy_data *d = (gyrotropy_data *)malloc(sz);
+  d->sz_data = sz;
+  return (void *)d;
+}
+
+void gyrotropic_susceptibility::init_internal_data(realnum *W[NUM_FIELD_COMPONENTS][2], double dt,
+                                                   const grid_volume &gv, void *data) const {
+  (void)dt; // unused
+  gyrotropy_data *d = (gyrotropy_data *)data;
+  size_t sz_data = d->sz_data;
+  memset(d, 0, sz_data);
+  d->sz_data = sz_data;
+  d->ntot = gv.ntot();
+  realnum *p = d->data;
+  FOR_COMPONENTS(c) DOCMP2 {
+    if (needs_P(c, cmp, W)) {
+      for (int dd = X; dd < R; dd++) {
+	d->P[c][cmp][dd] = p;      p += d->ntot;
+	d->P_prev[c][cmp][dd] = p; p += d->ntot;
+	d->P_tmp[c][cmp][dd] = p;  p += d->ntot;
+      }
+    }
+  }
+}
+
+void *gyrotropic_susceptibility::copy_internal_data(void *data) const {
+  gyrotropy_data *d = (gyrotropy_data *)data;
+  if (!d) return 0;
+  gyrotropy_data *dnew = (gyrotropy_data *)malloc(d->sz_data);
+  memcpy(dnew, d, d->sz_data);
+  realnum *p = dnew->data;
+  FOR_COMPONENTS(c) DOCMP2 {
+    if (d->P[c][cmp][0]) {
+      for (int dd = X; dd < R; dd++) {
+	dnew->P[c][cmp][dd] = p;      p += d->ntot;
+	dnew->P_prev[c][cmp][dd] = p; p += d->ntot;
+	dnew->P_tmp[c][cmp][dd] = p;  p += d->ntot;
+      }
+    }
+  }
+  return (void *)dnew;
+}
+
 bool gyrotropic_susceptibility::needs_P(component c, int cmp, realnum *W[NUM_FIELD_COMPONENTS][2]) const {
   if (!is_electric(c) && !is_magnetic(c)) return false;
   direction d0 = component_direction(c);
@@ -362,49 +414,58 @@ bool gyrotropic_susceptibility::needs_P(component c, int cmp, realnum *W[NUM_FIE
 void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 					 realnum *W_prev[NUM_FIELD_COMPONENTS][2], double dt,
 					 const grid_volume &gv, void *P_internal_data) const {
-  lorentzian_data *d = (lorentzian_data *)P_internal_data;
+  gyrotropy_data *d = (gyrotropy_data *)P_internal_data;
   const double omega2pi = 2 * pi * omega_0, g2pi = gamma * 2 * pi;
   const double omega0dtsqr = omega2pi * omega2pi * dt * dt;
   const double gamma1 = (1 - g2pi * dt / 2);
   const double diagfac = 2 - (no_omega_0_denominator ? 0 : omega0dtsqr);
+  const double pt = pi*dt;
   (void)W_prev; // unused;
 
   FOR_COMPONENTS(c) DOCMP2 {
-    if (d->P[c][cmp]) {
+    if (d->P[c][cmp][0]) {
       const direction d0 = component_direction(c);
-      const realnum *w = W[c][cmp], *s = sigma[c][d0];
+      const realnum *w0 = W[c][cmp], *s = sigma[c][d0];
 
-      if (!w || !s || (d0 != X && d0 != Y && d0 != Z))
+      if (!w0 || !s || (d0 != X && d0 != Y && d0 != Z))
 	abort("gyrotropic media require 3D Cartesian fields\n");
 
       const direction d1 = cycle_direction(gv.dim, d0, 1);
       const direction d2 = cycle_direction(gv.dim, d0, 2);
-      const component c1 = direction_component(c, d1);
-      const component c2 = direction_component(c, d2);
-      const realnum *s1 = W[c1][cmp] ? sigma[c][d1] : NULL;
-      const realnum *s2 = W[c2][cmp] ? sigma[c][d2] : NULL;
-      const realnum *p = d->P[c][cmp];
-      const realnum *pp0 = d->P_prev[c][cmp];
-      const realnum *pp1 = d->P_prev[c1][cmp];
-      const realnum *pp2 = d->P_prev[c2][cmp];
-      realnum *rhs = d->P_tmp[c][cmp];
-      const realnum g1 = pi*dt*gyro_tensor[d0][d1], g2 = pi*dt*gyro_tensor[d0][d2];
-
-      if (!pp1 || !pp2) abort("gyrotropic media require 3D Cartesian fields\n");
-      if (s1 || s2) abort("gyrotropic media do not support tensor sigma\n");
+      const realnum *w1 = W[direction_component(c, d1)][cmp];
+      const realnum *w2 = W[direction_component(c, d2)][cmp];
+      const realnum *p0 = d->P[c][cmp][d0], *pp0 = d->P_prev[c][cmp][d0];
+      const realnum *p1 = d->P[c][cmp][d1], *pp1 = d->P_prev[c][cmp][d1];
+      const realnum *p2 = d->P[c][cmp][d2], *pp2 = d->P_prev[c][cmp][d2];
+      realnum *rhs0 = d->P_tmp[c][cmp][d0];
+      realnum *rhs1 = d->P_tmp[c][cmp][d1];
+      realnum *rhs2 = d->P_tmp[c][cmp][d2];
+      const ptrdiff_t is  = gv.stride(d0) * (is_magnetic(c) ? -1 : +1);
+      const ptrdiff_t is1 = gv.stride(d1) * (is_magnetic(c) ? -1 : +1);
+      const ptrdiff_t is2 = gv.stride(d2) * (is_magnetic(c) ? -1 : +1);
+
+      if (!pp1 || !pp2 || !w1 || !w2)
+	abort("gyrotropic media require 3D Cartesian fields\n");
+
+      if (sigma[c][d1] || sigma[c][d2])
+	abort("gyrotropic media do not support tensor sigma\n");
 
       LOOP_OVER_VOL_OWNED(gv, c, i) {
-	rhs[i] = diagfac*p[i] - g1*pp1[i] - g2*pp2[i] - gamma1*pp0[i]
-	  + omega0dtsqr*s[i]*w[i];
+	rhs0[i] = diagfac*p0[i] - gamma1*pp0[i] + omega0dtsqr*s[i]*w0[i]
+	  - pt*gyro_tensor[d0][d1]*pp1[i] - pt*gyro_tensor[d0][d2]*pp2[i];
+	rhs1[i] = diagfac*p1[i] - gamma1*pp1[i] + omega0dtsqr*s[i]*OFFDIAG(s,w1,is1,is)
+	  - pt*gyro_tensor[d1][d0]*pp0[i] - pt*gyro_tensor[d1][d2]*pp2[i];
+	rhs2[i] = diagfac*p2[i] - gamma1*pp2[i] + omega0dtsqr*s[i]*OFFDIAG(s,w2,is2,is)
+	  - pt*gyro_tensor[d2][d1]*pp1[i] - pt*gyro_tensor[d2][d0]*pp0[i];
       }
     }
   }
 
   // Perform 3x3 matrix inversion, exploiting skew symmetry
   const double gd = (1 + g2pi * dt / 2);
-  const double gx = pi * dt * gyro_tensor[Y][Z];
-  const double gy = pi * dt * gyro_tensor[Z][X];
-  const double gz = pi * dt * gyro_tensor[X][Y];
+  const double gx = pt * gyro_tensor[Y][Z];
+  const double gy = pt * gyro_tensor[Z][X];
+  const double gz = pt * gyro_tensor[X][Y];
   const double invdet = 1.0 / gd / (gd*gd + gx*gx + gy*gy + gz*gz);
   const double inv[3][3]
     = {{ invdet*(gd*gd+gx*gx), invdet*(gx*gy+gd*gz), invdet*(gx*gz-gd*gy) },
@@ -412,20 +473,41 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
        { invdet*(gz*gx+gd*gy), invdet*(gz*gy-gd*gx), invdet*(gd*gd+gz*gz) }};
 
   FOR_COMPONENTS(c) DOCMP2 {
-    if (d->P[c][cmp]) {
+    if (d->P[c][cmp][0]) {
       const direction d0 = component_direction(c);
       const direction d1  = cycle_direction(gv.dim, d0, 1);
       const direction d2  = cycle_direction(gv.dim, d0, 2);
-      const component c1 = direction_component(c, d1);
-      const component c2 = direction_component(c, d2);
-      realnum *p = d->P[c][cmp],  *pp = d->P_prev[c][cmp];
-      const realnum *rhs0 = d->P_tmp[c][cmp];
-      const realnum *rhs1 = d->P_tmp[c1][cmp];
-      const realnum *rhs2 = d->P_tmp[c2][cmp];
+      realnum *p0 = d->P[c][cmp][d0], *pp0 = d->P_prev[c][cmp][d0];
+      realnum *p1 = d->P[c][cmp][d1], *pp1 = d->P_prev[c][cmp][d1];
+      realnum *p2 = d->P[c][cmp][d2], *pp2 = d->P_prev[c][cmp][d2];
+      const realnum *rhs0 = d->P_tmp[c][cmp][d0];
+      const realnum *rhs1 = d->P_tmp[c][cmp][d1];
+      const realnum *rhs2 = d->P_tmp[c][cmp][d2];
 
       LOOP_OVER_VOL_OWNED(gv, c, i) {
-	pp[i] = p[i];
-	p[i]  = inv[d0][d0] * rhs0[i] + inv[d0][d1] * rhs1[i] + inv[d0][d2] * rhs2[i];
+	pp0[i] = p0[i];	pp1[i] = p1[i];	pp2[i] = p2[i];
+	p0[i]  = inv[d0][d0] * rhs0[i] + inv[d0][d1] * rhs1[i] + inv[d0][d2] * rhs2[i];
+	p1[i]  = inv[d1][d0] * rhs0[i] + inv[d1][d1] * rhs1[i] + inv[d1][d2] * rhs2[i];
+	p2[i]  = inv[d2][d0] * rhs0[i] + inv[d2][d1] * rhs1[i] + inv[d2][d2] * rhs2[i];
+      }
+    }
+  }
+}
+
+void gyrotropic_susceptibility::subtract_P(field_type ft,
+                                           realnum *f_minus_p[NUM_FIELD_COMPONENTS][2],
+                                           void *P_internal_data) const {
+  gyrotropy_data *d = (gyrotropy_data *)P_internal_data;
+  field_type ft2 = ft == E_stuff ? D_stuff : B_stuff; // for sources etc.
+  size_t ntot = d->ntot;
+  FOR_FT_COMPONENTS(ft, ec) DOCMP2 {
+    if (d->P[ec][cmp][0]) {
+      component dc = field_type_component(ft2, ec);
+      if (f_minus_p[dc][cmp]) {
+        realnum *p = d->P[ec][cmp][component_direction(ec)];
+        realnum *fmp = f_minus_p[dc][cmp];
+        for (size_t i = 0; i < ntot; ++i)
+          fmp[i] -= p[i];
       }
     }
   }

From be5ae68e0c7e9c13b0ff67aef2f3a582e07a7e0a Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Thu, 30 May 2019 23:00:23 +0800
Subject: [PATCH 33/61] Revert unrelated last change to meep.i

---
 python/meep.i | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/python/meep.i b/python/meep.i
index 3f576efd9..998f13133 100644
--- a/python/meep.i
+++ b/python/meep.i
@@ -749,21 +749,21 @@ meep::volume_list *make_volume_list(const meep::volume &v, int c,
 }
 
 %typemap(in) const meep::src_time & {
-    PyObject *swig_obj_1 = NULL;
+    PyObject *swig_obj = NULL;
     void *tmp_ptr = 0;
     int tmp_res = 0;
 
     if(PyObject_IsInstance($input, py_source_time_object())) {
-        swig_obj_1 = PyObject_GetAttrString($input, "swigobj");
+        swig_obj = PyObject_GetAttrString($input, "swigobj");
     } else if(PyObject_IsInstance($input, py_meep_src_time_object())) {
-        swig_obj_1 = $input;
-        Py_XINCREF(swig_obj_1);
+        swig_obj = $input;
+        Py_XINCREF(swig_obj);
     } else {
       meep::abort("Expected a meep.source.SourceTime or a meep.src_time\n");
     }
 
-    tmp_res = SWIG_ConvertPtr(swig_obj_1, &tmp_ptr, $1_descriptor, 0);
-    Py_XDECREF(swig_obj_1);
+    tmp_res = SWIG_ConvertPtr(swig_obj, &tmp_ptr, $1_descriptor, 0);
+    Py_XDECREF(swig_obj);
 
     if(!SWIG_IsOK(tmp_res)) {
         SWIG_exception_fail(SWIG_ArgError(tmp_res), "Couldn't convert Python object to meep::src_time");

From 795fef522dd4c729c466e3fe9e2bbb62564c31da Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Fri, 31 May 2019 11:57:21 +0800
Subject: [PATCH 34/61] Avoiding need for allocation of P_tmp in
 gyrotropy_data.

---
 src/susceptibility.cpp | 65 ++++++++++++++----------------------------
 1 file changed, 22 insertions(+), 43 deletions(-)

diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 4cb5a07dc..2040cf260 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -346,7 +346,6 @@ typedef struct {
   size_t ntot;
   realnum *P[NUM_FIELD_COMPONENTS][2][3];
   realnum *P_prev[NUM_FIELD_COMPONENTS][2][3];
-  realnum *P_tmp[NUM_FIELD_COMPONENTS][2][3];
   realnum data[1];
 } gyrotropy_data;
 
@@ -354,7 +353,7 @@ void *gyrotropic_susceptibility::new_internal_data(realnum *W[NUM_FIELD_COMPONEN
                                                    const grid_volume &gv) const {
   int num = 0;
   FOR_COMPONENTS(c) DOCMP2 {
-    if (needs_P(c, cmp, W)) num += 9 * gv.ntot();
+    if (needs_P(c, cmp, W)) num += 6 * gv.ntot();
   }
   size_t sz = sizeof(gyrotropy_data) + sizeof(realnum) * (num - 1);
   gyrotropy_data *d = (gyrotropy_data *)malloc(sz);
@@ -376,7 +375,6 @@ void gyrotropic_susceptibility::init_internal_data(realnum *W[NUM_FIELD_COMPONEN
       for (int dd = X; dd < R; dd++) {
 	d->P[c][cmp][dd] = p;      p += d->ntot;
 	d->P_prev[c][cmp][dd] = p; p += d->ntot;
-	d->P_tmp[c][cmp][dd] = p;  p += d->ntot;
       }
     }
   }
@@ -393,7 +391,6 @@ void *gyrotropic_susceptibility::copy_internal_data(void *data) const {
       for (int dd = X; dd < R; dd++) {
 	dnew->P[c][cmp][dd] = p;      p += d->ntot;
 	dnew->P_prev[c][cmp][dd] = p; p += d->ntot;
-	dnew->P_tmp[c][cmp][dd] = p;  p += d->ntot;
       }
     }
   }
@@ -422,6 +419,17 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
   const double pt = pi*dt;
   (void)W_prev; // unused;
 
+  // Precalculate 3x3 matrix inverse, exploiting skew symmetry
+  const double gd = (1 + g2pi * dt / 2);
+  const double gx = pt * gyro_tensor[Y][Z];
+  const double gy = pt * gyro_tensor[Z][X];
+  const double gz = pt * gyro_tensor[X][Y];
+  const double invdet = 1.0 / gd / (gd*gd + gx*gx + gy*gy + gz*gz);
+  const double inv[3][3]
+    = {{ invdet*(gd*gd+gx*gx), invdet*(gx*gy+gd*gz), invdet*(gx*gz-gd*gy) },
+       { invdet*(gy*gx-gd*gz), invdet*(gd*gd+gy*gy), invdet*(gy*gz+gd*gx) },
+       { invdet*(gz*gx+gd*gy), invdet*(gz*gy-gd*gx), invdet*(gd*gd+gz*gz) }};
+
   FOR_COMPONENTS(c) DOCMP2 {
     if (d->P[c][cmp][0]) {
       const direction d0 = component_direction(c);
@@ -434,15 +442,13 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
       const direction d2 = cycle_direction(gv.dim, d0, 2);
       const realnum *w1 = W[direction_component(c, d1)][cmp];
       const realnum *w2 = W[direction_component(c, d2)][cmp];
-      const realnum *p0 = d->P[c][cmp][d0], *pp0 = d->P_prev[c][cmp][d0];
-      const realnum *p1 = d->P[c][cmp][d1], *pp1 = d->P_prev[c][cmp][d1];
-      const realnum *p2 = d->P[c][cmp][d2], *pp2 = d->P_prev[c][cmp][d2];
-      realnum *rhs0 = d->P_tmp[c][cmp][d0];
-      realnum *rhs1 = d->P_tmp[c][cmp][d1];
-      realnum *rhs2 = d->P_tmp[c][cmp][d2];
+      realnum *p0 = d->P[c][cmp][d0], *pp0 = d->P_prev[c][cmp][d0];
+      realnum *p1 = d->P[c][cmp][d1], *pp1 = d->P_prev[c][cmp][d1];
+      realnum *p2 = d->P[c][cmp][d2], *pp2 = d->P_prev[c][cmp][d2];
       const ptrdiff_t is  = gv.stride(d0) * (is_magnetic(c) ? -1 : +1);
       const ptrdiff_t is1 = gv.stride(d1) * (is_magnetic(c) ? -1 : +1);
       const ptrdiff_t is2 = gv.stride(d2) * (is_magnetic(c) ? -1 : +1);
+      realnum rhs0, rhs1, rhs2;
 
       if (!pp1 || !pp2 || !w1 || !w2)
 	abort("gyrotropic media require 3D Cartesian fields\n");
@@ -451,44 +457,17 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 	abort("gyrotropic media do not support tensor sigma\n");
 
       LOOP_OVER_VOL_OWNED(gv, c, i) {
-	rhs0[i] = diagfac*p0[i] - gamma1*pp0[i] + omega0dtsqr*s[i]*w0[i]
+	rhs0 = diagfac*p0[i] - gamma1*pp0[i] + omega0dtsqr*s[i]*w0[i]
 	  - pt*gyro_tensor[d0][d1]*pp1[i] - pt*gyro_tensor[d0][d2]*pp2[i];
-	rhs1[i] = diagfac*p1[i] - gamma1*pp1[i] + omega0dtsqr*s[i]*OFFDIAG(s,w1,is1,is)
+	rhs1 = diagfac*p1[i] - gamma1*pp1[i] + omega0dtsqr*s[i]*OFFDIAG(s,w1,is1,is)
 	  - pt*gyro_tensor[d1][d0]*pp0[i] - pt*gyro_tensor[d1][d2]*pp2[i];
-	rhs2[i] = diagfac*p2[i] - gamma1*pp2[i] + omega0dtsqr*s[i]*OFFDIAG(s,w2,is2,is)
+	rhs2 = diagfac*p2[i] - gamma1*pp2[i] + omega0dtsqr*s[i]*OFFDIAG(s,w2,is2,is)
 	  - pt*gyro_tensor[d2][d1]*pp1[i] - pt*gyro_tensor[d2][d0]*pp0[i];
-      }
-    }
-  }
-
-  // Perform 3x3 matrix inversion, exploiting skew symmetry
-  const double gd = (1 + g2pi * dt / 2);
-  const double gx = pt * gyro_tensor[Y][Z];
-  const double gy = pt * gyro_tensor[Z][X];
-  const double gz = pt * gyro_tensor[X][Y];
-  const double invdet = 1.0 / gd / (gd*gd + gx*gx + gy*gy + gz*gz);
-  const double inv[3][3]
-    = {{ invdet*(gd*gd+gx*gx), invdet*(gx*gy+gd*gz), invdet*(gx*gz-gd*gy) },
-       { invdet*(gy*gx-gd*gz), invdet*(gd*gd+gy*gy), invdet*(gy*gz+gd*gx) },
-       { invdet*(gz*gx+gd*gy), invdet*(gz*gy-gd*gx), invdet*(gd*gd+gz*gz) }};
 
-  FOR_COMPONENTS(c) DOCMP2 {
-    if (d->P[c][cmp][0]) {
-      const direction d0 = component_direction(c);
-      const direction d1  = cycle_direction(gv.dim, d0, 1);
-      const direction d2  = cycle_direction(gv.dim, d0, 2);
-      realnum *p0 = d->P[c][cmp][d0], *pp0 = d->P_prev[c][cmp][d0];
-      realnum *p1 = d->P[c][cmp][d1], *pp1 = d->P_prev[c][cmp][d1];
-      realnum *p2 = d->P[c][cmp][d2], *pp2 = d->P_prev[c][cmp][d2];
-      const realnum *rhs0 = d->P_tmp[c][cmp][d0];
-      const realnum *rhs1 = d->P_tmp[c][cmp][d1];
-      const realnum *rhs2 = d->P_tmp[c][cmp][d2];
-
-      LOOP_OVER_VOL_OWNED(gv, c, i) {
 	pp0[i] = p0[i];	pp1[i] = p1[i];	pp2[i] = p2[i];
-	p0[i]  = inv[d0][d0] * rhs0[i] + inv[d0][d1] * rhs1[i] + inv[d0][d2] * rhs2[i];
-	p1[i]  = inv[d1][d0] * rhs0[i] + inv[d1][d1] * rhs1[i] + inv[d1][d2] * rhs2[i];
-	p2[i]  = inv[d2][d0] * rhs0[i] + inv[d2][d1] * rhs1[i] + inv[d2][d2] * rhs2[i];
+	p0[i]  = inv[d0][d0] * rhs0 + inv[d0][d1] * rhs1 + inv[d0][d2] * rhs2;
+	p1[i]  = inv[d1][d0] * rhs0 + inv[d1][d1] * rhs1 + inv[d1][d2] * rhs2;
+	p2[i]  = inv[d2][d0] * rhs0 + inv[d2][d1] * rhs1 + inv[d2][d2] * rhs2;
       }
     }
   }

From 83f42e1c0216d415d7bbb294aa917b3e7a405ec3 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Fri, 31 May 2019 12:32:19 +0800
Subject: [PATCH 35/61] Implement num_cinternal_notowned stuff for gyrotropic
 media

---
 src/meep.hpp           |  4 ++++
 src/susceptibility.cpp | 13 +++++++++++++
 2 files changed, 17 insertions(+)

diff --git a/src/meep.hpp b/src/meep.hpp
index 79caf4480..507558d0e 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -291,6 +291,10 @@ class gyrotropic_susceptibility : public lorentzian_susceptibility {
   void subtract_P(field_type ft, realnum *f_minus_p[NUM_FIELD_COMPONENTS][2],
 		  void *P_internal_data) const;
 
+  int num_cinternal_notowned_needed(component c, void *P_internal_data) const;
+  realnum *cinternal_notowned_ptr(int inotowned, component c, int cmp,
+				  int n, void *P_internal_data) const;
+
   virtual void dump_params(h5file *h5f, size_t *start);
   virtual int get_num_params() { return 7; }
   virtual bool needs_W_notowned(component c, realnum *W[NUM_FIELD_COMPONENTS][2]) const {
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 2040cf260..257f200ee 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -492,6 +492,19 @@ void gyrotropic_susceptibility::subtract_P(field_type ft,
   }
 }
 
+int gyrotropic_susceptibility::num_cinternal_notowned_needed(component c,
+                                                             void *P_internal_data) const {
+  gyrotropy_data *d = (gyrotropy_data *)P_internal_data;
+  return d->P[c][0][0] ? 3 : 0;
+}
+
+realnum *gyrotropic_susceptibility::cinternal_notowned_ptr(int inotowned, component c, int cmp,
+                                                           int n, void *P_internal_data) const {
+  gyrotropy_data *d = (gyrotropy_data *)P_internal_data;
+  if (!d || !d->P[c][cmp][inotowned]) return NULL;
+  return d->P[c][cmp][inotowned] + n;
+}
+
 void gyrotropic_susceptibility::dump_params(h5file *h5f, size_t *start) {
   size_t num_params = 8;
   size_t params_dims[1] = {num_params};

From 7fa88a22622bb33b252a95c366c9bbcf3152db2e Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Fri, 31 May 2019 16:28:57 +0800
Subject: [PATCH 36/61] Update documentation for gyrotropic media, and relax
 some minor restrictions.

---
 doc/docs/Materials.md                         |  10 +-
 doc/docs/Python_Tutorials/Gyrotropic_Media.md | 162 +++++++++---------
 .../images/Faraday-rotation-comparison.png    | Bin 0 -> 97734 bytes
 doc/docs/images/Faraday-rotation.png          | Bin 0 -> 71606 bytes
 python/examples/faraday-rotation.py           |  68 ++++++++
 src/susceptibility.cpp                        |  15 +-
 6 files changed, 155 insertions(+), 100 deletions(-)
 create mode 100644 doc/docs/images/Faraday-rotation-comparison.png
 create mode 100644 doc/docs/images/Faraday-rotation.png
 create mode 100644 python/examples/faraday-rotation.py

diff --git a/doc/docs/Materials.md b/doc/docs/Materials.md
index d6316aa3b..1d9a553e5 100644
--- a/doc/docs/Materials.md
+++ b/doc/docs/Materials.md
@@ -180,17 +180,13 @@ $$\frac{d^2\mathbf{P}_n}{dt^2} + \gamma_n \frac{d\mathbf{P}_n}{dt} + \mathbf{b}_
 
 $$\frac{d\mathbf{P}_n}{dt} = \mathbf{P}_n \times \mathbf{b}_n$$
 
-In the frequency domain, a gyroelectric medium has imaginary off-diagonal components in the ε tensor. Take the case $\mathbf{b} = b \hat{z}$. When all fields have harmonic time-dependence $\exp(-i\omega t)$, the polarization equation of motion reduces to
+In the frequency domain, a gyroelectric medium has skew-symmetric off-diagonal components in the ε tensor. Take the case $\mathbf{b} = b \hat{z}$. When all fields have harmonic time-dependence $\exp(-i\omega t)$, the polarization equation of motion reduces to
 
-$$\mathbf{P}_n = \chi_n \mathbf{E}, \quad \chi_n = \omega_n^2 \begin{bmatrix}\xi_\perp & -i\eta & 0 \\ i\eta & \xi_\perp & 0 \\ 0 & 0 & \xi_\parallel \end{bmatrix} \, \sigma_n(\mathbf{x})$$
+$$\mathbf{P}_n =  \begin{bmatrix}\chi_\perp & -i\eta & 0 \\ i\eta & \chi_\perp & 0 \\ 0 & 0 & \chi_\parallel \end{bmatrix} \mathbf{E}$$
 
 where
 
-$$\xi_\perp = \frac{\Delta_n}{\Delta_n^2 - \omega^2 b^2}\,,\;\;\; \xi_\parallel = \frac{1}{\Delta_n}, \;\;\; \eta = \frac{\omega b}{\Delta_n^2 - \omega^2 b^2}, \;\;\;\Delta_n \equiv \omega_n^2 - \omega^2 - i\omega\gamma_n$$
-
-In many applications, the gyrotropy is weak relative to the frequency detuning but strong relative to the loss.  For this case, we can define $\omega_n = \omega + \Delta\omega_n$, and take $\gamma_n \ll b \ll \Delta \omega_n \ll \omega$. In this limit, the susceptibility tensor reduces to
-
-$$\chi_n \approx \frac{\omega}{2\Delta\omega_n} \begin{bmatrix}1 & -ib/2\Delta\omega_n & 0 \\ ib/2\Delta\omega_n & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} \sigma_n(\mathbf{x})$$
+$$\chi_\perp = \frac{\omega_n^2 \Delta_n}{\Delta_n^2 - \omega^2 b^2}\,\sigma_n(\mathbf{x}),\;\;\; \chi_\parallel = \frac{\omega_n^2}{\Delta_n}\,\sigma_n(\mathbf{x}), \;\;\; \eta = \frac{\omega_n^2 \omega b}{\Delta_n^2 - \omega^2 b^2}\,\sigma_n(\mathbf{x}), \;\;\;\Delta_n \equiv \omega_n^2 - \omega^2 - i\omega\gamma_n$$
 
 Materials Library
 -----------------
diff --git a/doc/docs/Python_Tutorials/Gyrotropic_Media.md b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
index b67fb0afc..baf6a25dd 100644
--- a/doc/docs/Python_Tutorials/Gyrotropic_Media.md
+++ b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
@@ -2,120 +2,116 @@
 # Gyrotropic media
 ---
 
-In this example, we will perform simulations with a gyrotropic medium. See [Materials](../Materials.md#gyrotropic-media) for more information on how gyrotropy is supported.
+In this example, we will perform simulations with gyrotropic media. See [Materials](../Materials.md#gyrotropic-media) for more information on how gyrotropy is supported.
 
-[TOC]
+### Faraday Rotation
 
-### Dispersion Relation for a Gyrotropic Medium
+Consider a uniform gyrotroelectric medium with bias vector $\mathbf{b} = b \hat{z}$. According to the [gyrotropic Lorentzian model implemented in Meep](../Materials.md#gyrotropic-media), the *x* and *y* components of the dielectric function are
 
-Here, we model a uniform gyrotroelectric medium with bias vector $\mathbf{b} = b \hat{z}$. For TE modes, the electric field lies in the *x*-*y* plane and magnetic field pointing along *z*, with the wavevector $\mathbf{k}$ in the *x*-*y* plane.  The dispersion relation can be shown to be
+$$\epsilon = \begin{bmatrix}\epsilon_\perp & -i\eta \\ i\eta & \epsilon_\perp \end{bmatrix}$$
 
-$$\omega = v(\omega) |\mathbf{k}|, \quad v^2 = \mu^{-1} \left[\epsilon^{-1}\right]_{xx}$$
+where
 
-The ε tensor is given by
+$$\epsilon_\perp = \epsilon_\infty + \frac{\omega_n^2 \Delta_n}{\Delta_n^2 - \omega^2 b^2}\,\sigma_n(\mathbf{x}),\;\;\; \eta = \frac{\omega_n^2 \omega b}{\Delta_n^2 - \omega^2 b^2}\,\sigma_n(\mathbf{x}), \;\;\;\Delta_n \equiv \omega_n^2 - \omega^2 - i\omega\gamma_n$$
 
-$$\epsilon = \epsilon_\infty + \frac{\omega_n^2}{\omega_n^2 - \omega^2 - i\omega\gamma_n} \begin{bmatrix}\xi & -i\eta & 0 \\ i\eta & \xi & 0 \\ 0 & 0 & 1 \end{bmatrix} \, \sigma_n(\mathbf{x}),$$
+The skew-symmetric off-diagonal components in ε give rise to [Faraday rotation](https://en.wikipedia.org/wiki/Faraday_effect): when a plane wave linearly polarized along *x* is launched along the gyrotropy axis *z*, the polarization vector will precess around the gyrotropy axis as the wave propagates. This is the principle behind [Faraday rotators](https://en.wikipedia.org/wiki/Faraday_rotator), devices that act as one-way valves for light.
 
-with
+A plane wave undergoing Faraday rotation can be described by the complex ansatz
 
-$$\xi = \frac{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2}{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2 + \omega^2 b^2}\,, \quad \eta = \frac{\omega b (\omega_n^2 - \omega^2 - i\omega \gamma_n)}{(\omega_n^2 - \omega^2 - i\omega \gamma_n)^2 + \omega^2 b^2}$$
+$$\begin{bmatrix}E_x \\ E_y\end{bmatrix} = E_0 \begin{bmatrix}\cos(\kappa_c z) \\ \sin(\kappa_c z)\end{bmatrix} e^{i(kz-\omega t)}$$
 
-We choose the material parameters $\epsilon_\infty = 1$, $f_n = 1.2$, $\gamma_n = 10^{-5}$, and $\sigma_n = 0.1$, with $|\mathbf{b}_n|$ ranging from zero to $10^{-2}$.
+where $\kappa_c$ is the Faraday rotation, in radians, per unit of propagation distance. Substituting this into Maxwell's equations, with the above gyroelectric dielectric function, yields
 
+$$|\kappa_c| = \frac{1}{2} \omega^2 \mu \sqrt{\epsilon_\perp - \sqrt{\epsilon_\perp^2 - \eta^2}}$$
 
+We model this phenomenon in the simulation script [faraday-rotation.py](https://github.com/NanoComp/meep/blob/master/python/examples/faraday-rotation.py). First, we define a gyroelectric material:
 
+```import meep as mp
 
-From the dispersion relation ω(k), we will compute the numerical ε(ω) via the formula:
+## Define a gyroelectric medium
+f0 = 1.0
+gamma = 1e-4
+epsn = 2.0
+b0 = 1.05
+sigma = 0.2
 
-$$\varepsilon(\omega) = \left( \frac{ck}{\omega} \right) ^2$$
-
-We will then compare this with the analytical ε(ω) that we specified. The simulation script is in [material-dispersion.py](https://github.com/NanoComp/meep/blob/master/python/examples/material-dispersion.py).
-
-Since this is a uniform medium, our computational cell can actually be of *zero* size (i.e. one pixel), where we will use Bloch-periodic boundary conditions to specify the wavevector *k*.
-
-```py
-cell = mp.Vector3()
-resolution = 20
-```
-
-We will then fill all space with an artificial dispersive material:
-
-```py
-susceptibilities = [mp.LorentzianSusceptibility(frequency=1.1, gamma=1e-5, sigma=0.5),
-                    mp.LorentzianSusceptibility(frequency=0.5, gamma=0.1, sigma=2e-5)]
-
-default_material = mp.Medium(epsilon=2.25, E_susceptibilities=susceptibilities)
+susc = [mp.GyrotropicLorentzianSusceptibility(frequency=f0, gamma=gamma, sigma=sigma,
+                                              bias=mp.Vector3(0, 0, b0))]
+mat = mp.Medium(epsilon=epsn, mu=1, E_susceptibilities=susc)
 ```
 
-corresponding to the dielectric function:
-
-$$\varepsilon(\omega) = \varepsilon(2\pi f) = 2.25 + \frac{1.1^2 \cdot 0.5}{1.1^2 - f^2 -if \cdot 10^{-5}/2\pi} + \frac{0.5^2 \cdot 2\cdot 10^{-5}}{0.5^2 - f^2 -if \cdot 0.1 / 2\pi}$$
-
-The real and imaginary parts of this dielectric function ε(ω) are plotted below:
-
-<center>
-![](../images/Material-dispersion-eps.png)
-</center>
-
-We can see that the f=1.1 resonance causes a large change in both the real and imaginary parts of ε around that frequency. In fact, there is a range of frequencies from 1.1 to 1.2161 where ε is *negative*. In this range, no propagating modes exist &mdash; it is actually a kind of electromagnetic band gap associated with polariton resonances in a material. For more information on the physics of such materials, see e.g. Chapter 14 of [Introduction to Solid State Physics](http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000803.html) by C. Kittel.
-
-On the other hand, the f=0.5 resonance, because the `sigma` numerator is so small, causes very little change in the real part of ε. Nevertheless, it generates a clear peak in the *imaginary* part of ε, corresponding to a resonant absorption peak.
+The `GyrotropicLorentzianSusceptibility` object has a `bias` argument that takes a `Vector3` specifying the gyrotropy vector. In this case, the vector points along *z*, and its magnitude (which specifies the precession frequency) is determined by the variable `b0`. The other arguments play the same role as in an ordinary (non-gyrotropic) [Lorentzian susceptibility](Material_Dispersion.md).
 
-Now, we'll set up the rest of the simulation. We'll specify a broadband $E_z$-polarized Gaussian source, create a list of *k* wavevectors that we want to compute ω(k) over, and compute the associated frequencies by using the `k_points` function:
+Next, we set up and run the Meep simulation.
 
-```py
-fcen = 1.0
-df = 2.0
+```tmax = 100
+L = 20.0
+cell = mp.Vector3(0, 0, L)
+fsrc, src_z = 0.8, -8.5
+pml_layers = [mp.PML(thickness=1.0, direction=mp.Z)]
 
-sources = [mp.Source(mp.GaussianSource(fcen, fwidth=df), component=mp.Ez, center=mp.Vector3())]
+sources = [mp.Source(mp.ContinuousSource(frequency=fsrc),
+                     component=mp.Ex, center=mp.Vector3(0, 0, src_z))]
 
-kmin = 0.3
-kmax = 2.2
-k_interp = 99
-
-kpts = mp.interpolate(k_interp, [mp.Vector3(kmin), mp.Vector3(kmax)])
-
-sim = mp.Simulation(cell_size=cell, geometry=[], sources=sources, default_material=default_material, resolution=resolution)
-
-all_freqs = sim.run_k_points(200, kpts)  # a list of lists of frequencies
+sim = mp.Simulation(cell_size=cell, geometry=[], sources=sources,
+                    k_point=mp.Vector3(),   # Periodic boundary conditions
+                    boundary_layers=pml_layers,
+                    default_material=mat, resolution=50)
+sim.run(until=tmax)
 ```
 
-The `run_k_points` function returns a *list of lists* of frequencies &mdash; one list of complex frequencies for each *k* point &mdash; which we store in the `all_freqs` variable. Finally, we want to loop over this list and print out the corresponding ε via the ratio (ck/ω)$^2$ as described above. To do this, we will use Python's `zip` function which combines multiple lists into one:
+The simulation cell is one pixel wide in the *x* and *y* directions, with periodic boundary conditions. PMLs are placed in the *z* direction. A `ContinuousSource` emits a wave whose electric field is initially polarized along *x*. We then plot the *x* and *y* components of the electric field versus *z*:
 
-```py
-for fs, kx in zip(all_freqs, [v.x for v in kpts]):
-    for f in fs:
-            print("eps:, {.6f}, {.6f}, {.6f}".format(f.real, f.imag, (kx / f)**2))
-```
+```import numpy as np
+import matplotlib.pyplot as plt
 
-Alternatively we could just read all of the frequencies into Python or Octave/Matlab and compute the ratios there. After running the program with
+ex_data = sim.get_array(center=mp.Vector3(), size=mp.Vector3(0, 0, L), component=mp.Ex)
+ey_data = sim.get_array(center=mp.Vector3(), size=mp.Vector3(0, 0, L), component=mp.Ey)
 
-```sh
-unix% python -u material-dispersion.py | tee material-dispersion.out
+z = np.linspace(-L/2, L/2, len(ex_data))
+plt.figure(1)
+plt.plot(z, ex_data, label='Ex')
+plt.plot(z, ey_data, label='Ey')
+plt.xlim(-L/2, L/2)
+plt.xlabel('z')
+plt.legend()
+plt.show()
 ```
 
-we can then `grep` for the frequencies and the computed dielectric function, and plot it. First, let's plot the dispersion relation ω(k) for the real part of ω:
-
 <center>
-![](../images/Material-dispersion-bands.png)
+![](../images/Faraday-rotation.png)
 </center>
 
-The red circles are the computed points from Meep, whereas the blue line is the analytical band diagram from the specified ε(ω). As you can see, we get *two* bands at each *k*, separated by a polaritonic gap (shaded yellow). This dispersion relation can be thought of as the interaction (anti-crossing) between the light line of the ambient ε=2.25 material (dashed black line) and the horizontal line corresponding to the phonon resonance.
-
-Similarly, the computed and analytical real parts of the dielectric function are given by:
-
-<center>
-![](../images/Material-dispersion-epsre.png)
-</center>
+We see that the wave indeed rotates in the *x*-*y* plane as it travels. This can be compared quantitatively to the above ansatz for a wave undergoing Faraday rotation, using the material parameters to calculate the rotation rate $\kappa_c$:
+
+```dfsq = (f0*2 - 1j*f0*gamma - fsrc**2)
+eperp = epsn + sigma * f0**2 * dfsq / (dfsq**2 + (fsrc*b0)**2)
+eta = sigma * f0**2 * fsrc * b0 / (dfsq**2 + (fsrc*b0)**2)
+
+k_gyro = 2*np.pi*fsrc * np.sqrt(0.5*(eperp - np.sqrt(eperp**2 - eta**2)))
+Ex_theory = 0.37 * np.cos(k_gyro * (z - src_z))    # amplitude estimated by eye
+Ey_theory = 0.37 * np.sin(k_gyro * (z - src_z))
+
+plt.figure(2)
+plt.subplot(2,1,1)
+plt.plot(z, ex_data, label='Ex (MEEP)')
+plt.plot(z, Ex_theory, 'k--')
+plt.plot(z, -Ex_theory, 'k--', label='Ex (theory)')
+plt.xlim(-L/2, L/2); plt.xlabel('z')
+plt.legend()
+
+plt.subplot(2,1,2)
+plt.plot(z, ey_data, label='Ey (MEEP)')
+plt.plot(z, Ey_theory, 'k--')
+plt.plot(z, -Ey_theory, 'k--', label='Ey (theory)')
+plt.xlim(-L/2, L/2); plt.xlabel('z')
+plt.legend()
+plt.tight_layout()
+plt.show()
+```
 
-which shows excellent agreement between the analytical (blue line) and numerical (red circles) calculations. The imaginary part, however, is more subtle:
+The results are in excellent agreement:
 
 <center>
-![](../images/Material-dispersion-epsim.png)
+![](../images/Faraday-rotation-comparison.png)
 </center>
-
-The blue line is the analytical calculation from above and the red circles are the numerical value from Meep &mdash; why is the agreement so poor? There is nothing wrong with Meep, and this is *not* a numerical error. The problem is simply that we are comparing apples and oranges.
-
-The blue line is the analytical calculation of ε(ω) for a *real* frequency ω which corresponds to solutions with a *complex* wavevector *k*, whereas Meep is computing ε at a *complex* ω for a *real* wavevector *k*. So, the correct comparison is to plug Meep's *complex* ω into the analytical formula for ε(ω), which results in the green lines on the graph that fall almost on top of the red circles.
-
-Why did our comparison of the *real* part of ε look so good, then? The reason is that ε(ω) at real and complex values of ω are closely related by the analytic properties of ε. In particular, because ε is an analytic function on the real-ω axis, adding a *small* imaginary part to ω as we are doing here does not change ε by much. The losses are small for all of the computed *k* points. The change was only significant for the imaginary ε because the imaginary ε was small to begin with.
diff --git a/doc/docs/images/Faraday-rotation-comparison.png b/doc/docs/images/Faraday-rotation-comparison.png
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diff --git a/python/examples/faraday-rotation.py b/python/examples/faraday-rotation.py
new file mode 100644
index 000000000..9b415fabd
--- /dev/null
+++ b/python/examples/faraday-rotation.py
@@ -0,0 +1,68 @@
+import meep as mp
+
+## Define a gyroelectric medium
+f0 = 1.0
+gamma = 1e-4
+epsn = 2.0
+b0 = 1.05
+sn = 0.2
+
+susc = [mp.GyrotropicLorentzianSusceptibility(frequency=f0, gamma=gamma, sigma=sn,
+                                              bias=mp.Vector3(0, 0, b0))]
+mat = mp.Medium(epsilon=epsn, mu=1, E_susceptibilities=susc)
+
+## Set up and run the Meep simulation:
+tmax = 100
+L = 20.0
+cell = mp.Vector3(0, 0, L)
+fsrc, src_z = 0.8, -8.5
+pml_layers = [mp.PML(thickness=1.0, direction=mp.Z)]
+
+sources = [mp.Source(mp.ContinuousSource(frequency=fsrc),
+                     component=mp.Ex, center=mp.Vector3(0, 0, src_z))]
+
+sim = mp.Simulation(cell_size=cell, geometry=[], sources=sources,
+                    k_point=mp.Vector3(),   # Periodic boundary conditions
+                    boundary_layers=pml_layers,
+                    default_material=mat, resolution=50)
+sim.run(until=tmax)
+
+## Plot results:
+import numpy as np
+import matplotlib.pyplot as plt
+
+ex_data = sim.get_array(center=mp.Vector3(), size=mp.Vector3(0, 0, L), component=mp.Ex)
+ey_data = sim.get_array(center=mp.Vector3(), size=mp.Vector3(0, 0, L), component=mp.Ey)
+
+z = np.linspace(-L/2, L/2, len(ex_data))
+plt.figure(1)
+plt.plot(z, ex_data, label='Ex')
+plt.plot(z, ey_data, label='Ey')
+plt.xlim(-L/2, L/2); plt.xlabel('z')
+plt.legend()
+
+## Comparison with analytic result:
+dfsq = (f0*2 - 1j*f0*gamma - fsrc**2)
+eperp = epsn + sn * f0**2 * dfsq / (dfsq**2 + (fsrc*b0)**2)
+eta = sn * f0**2 * fsrc * b0 / (dfsq**2 + (fsrc*b0)**2)
+
+k_gyro = 2*np.pi*fsrc * np.sqrt(0.5*(eperp - np.sqrt(eperp**2 - eta**2)))
+Ex_theory = 0.37 * np.cos(k_gyro * (z - src_z))
+Ey_theory = 0.37 * np.sin(k_gyro * (z - src_z))
+
+plt.figure(2)
+plt.subplot(2,1,1)
+plt.plot(z, ex_data, label='Ex (MEEP)')
+plt.plot(z, Ex_theory, 'k--')
+plt.plot(z, -Ex_theory, 'k--', label='Ex (theory)')
+plt.xlim(-L/2, L/2); plt.xlabel('z')
+plt.legend()
+
+plt.subplot(2,1,2)
+plt.plot(z, ey_data, label='Ey (MEEP)')
+plt.plot(z, Ey_theory, 'k--')
+plt.plot(z, -Ey_theory, 'k--', label='Ey (theory)')
+plt.xlim(-L/2, L/2); plt.xlabel('z')
+plt.legend()
+plt.tight_layout()
+plt.show()
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 257f200ee..e0ef45d9a 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -400,12 +400,7 @@ void *gyrotropic_susceptibility::copy_internal_data(void *data) const {
 bool gyrotropic_susceptibility::needs_P(component c, int cmp, realnum *W[NUM_FIELD_COMPONENTS][2]) const {
   if (!is_electric(c) && !is_magnetic(c)) return false;
   direction d0 = component_direction(c);
-  if ((d0 == X || d0 == Y || d0 == Z) && sigma[c][d0]) {
-    FOR_DIRECTIONS(d) {
-      if (W[direction_component(c, d)][cmp]) return true;
-    }
-  }
-  return false;
+  return (d0 == X || d0 == Y || d0 == Z) && sigma[c][d0] && W[c][cmp];
 }
 
 void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
@@ -450,18 +445,18 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
       const ptrdiff_t is2 = gv.stride(d2) * (is_magnetic(c) ? -1 : +1);
       realnum rhs0, rhs1, rhs2;
 
-      if (!pp1 || !pp2 || !w1 || !w2)
+      if (!pp1 || !pp2)
 	abort("gyrotropic media require 3D Cartesian fields\n");
 
       if (sigma[c][d1] || sigma[c][d2])
-	abort("gyrotropic media do not support tensor sigma\n");
+	abort("gyrotropic media do not support anisotropic sigma\n");
 
       LOOP_OVER_VOL_OWNED(gv, c, i) {
 	rhs0 = diagfac*p0[i] - gamma1*pp0[i] + omega0dtsqr*s[i]*w0[i]
 	  - pt*gyro_tensor[d0][d1]*pp1[i] - pt*gyro_tensor[d0][d2]*pp2[i];
-	rhs1 = diagfac*p1[i] - gamma1*pp1[i] + omega0dtsqr*s[i]*OFFDIAG(s,w1,is1,is)
+	rhs1 = diagfac*p1[i] - gamma1*pp1[i] + (w1 ? omega0dtsqr*s[i]*OFFDIAG(s,w1,is1,is) : 0)
 	  - pt*gyro_tensor[d1][d0]*pp0[i] - pt*gyro_tensor[d1][d2]*pp2[i];
-	rhs2 = diagfac*p2[i] - gamma1*pp2[i] + omega0dtsqr*s[i]*OFFDIAG(s,w2,is2,is)
+	rhs2 = diagfac*p2[i] - gamma1*pp2[i] + (w2 ? omega0dtsqr*s[i]*OFFDIAG(s,w2,is2,is) : 0)
 	  - pt*gyro_tensor[d2][d1]*pp1[i] - pt*gyro_tensor[d2][d0]*pp0[i];
 
 	pp0[i] = p0[i];	pp1[i] = p1[i];	pp2[i] = p2[i];

From 3028e304aaee4ab6b5965e8bf2009496c1d99e2d Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Fri, 31 May 2019 17:51:30 +0800
Subject: [PATCH 37/61] Add virtual keywords to gyrotropic_susceptibility
 methods

---
 src/meep.hpp | 20 ++++++++++----------
 1 file changed, 10 insertions(+), 10 deletions(-)

diff --git a/src/meep.hpp b/src/meep.hpp
index 507558d0e..10c9420fd 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -279,21 +279,21 @@ class gyrotropic_susceptibility : public lorentzian_susceptibility {
 			    bool no_omega_0_denominator = true);
   virtual susceptibility *clone() const { return new gyrotropic_susceptibility(*this); }
 
-  void *new_internal_data(realnum *W[NUM_FIELD_COMPONENTS][2], const grid_volume &gv) const;
-  void init_internal_data(realnum *W[NUM_FIELD_COMPONENTS][2], double dt,
-			  const grid_volume &gv, void *data) const;
-  void *copy_internal_data(void *data) const;
+  virtual void *new_internal_data(realnum *W[NUM_FIELD_COMPONENTS][2], const grid_volume &gv) const;
+  virtual void init_internal_data(realnum *W[NUM_FIELD_COMPONENTS][2], double dt,
+				  const grid_volume &gv, void *data) const;
+  virtual void *copy_internal_data(void *data) const;
 
-  bool needs_P(component c, int cmp, realnum *W[NUM_FIELD_COMPONENTS][2]) const;
+  virtual bool needs_P(component c, int cmp, realnum *W[NUM_FIELD_COMPONENTS][2]) const;
   virtual void update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 			realnum *W_prev[NUM_FIELD_COMPONENTS][2], double dt,
 			const grid_volume &gv, void *P_internal_data) const;
-  void subtract_P(field_type ft, realnum *f_minus_p[NUM_FIELD_COMPONENTS][2],
-		  void *P_internal_data) const;
+  virtual void subtract_P(field_type ft, realnum *f_minus_p[NUM_FIELD_COMPONENTS][2],
+			  void *P_internal_data) const;
 
-  int num_cinternal_notowned_needed(component c, void *P_internal_data) const;
-  realnum *cinternal_notowned_ptr(int inotowned, component c, int cmp,
-				  int n, void *P_internal_data) const;
+  virtual int num_cinternal_notowned_needed(component c, void *P_internal_data) const;
+  virtual realnum *cinternal_notowned_ptr(int inotowned, component c, int cmp,
+					  int n, void *P_internal_data) const;
 
   virtual void dump_params(h5file *h5f, size_t *start);
   virtual int get_num_params() { return 7; }

From 1cc846e20283bcb186a68397c6be7b0698c17299 Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Sun, 2 Jun 2019 10:52:17 +0800
Subject: [PATCH 38/61] Merge latest changes from master

---
 doc/docs/Python_User_Interface.md | 16 ++++++++++------
 python/geom.py                    |  8 +++++++-
 python/meep.i                     |  3 ++-
 python/typemap_utils.cpp          | 13 ++++++++++---
 scheme/meep.scm.in                |  4 +++-
 scheme/structure.cpp              | 13 +++++++++----
 src/meep.hpp                      | 10 +++-------
 src/meepgeom.cpp                  | 12 +++++-------
 8 files changed, 49 insertions(+), 30 deletions(-)

diff --git a/doc/docs/Python_User_Interface.md b/doc/docs/Python_User_Interface.md
index 8cc35d1aa..458ea9733 100644
--- a/doc/docs/Python_User_Interface.md
+++ b/doc/docs/Python_User_Interface.md
@@ -394,7 +394,15 @@ This is a somewhat unusual polarizable medium, a Lorentzian susceptibility with
 
 ### GyrotropicLorentzianSusceptibility or GyrotropicDrudeSusceptibility
 
-(**Experimental feature**) Specifies a single dispersive [gyrotropic susceptibility](Materials.md#gyrotropic-media) governed by the Landau-Lifshitz-Gilbert equation. Note that the parameters `sigma`, `frequency`, and `gamma` play different roles compared to the Lorentzian or Drude case.
+(**Experimental feature**) Specifies a single dispersive gyrotropic susceptibility. See [Material Dispersion](Materials.md#gyrotropic-media).
+
+For gyrotropic media of Lorentzian (damped harmonic oscillator) or Drude form, the parameters `sigma`, `frequency`, and `gamma` have the same meanings as in `LorentzianSusceptibility` and `DrudeSusceptibility`, but with an additional 3-vector `bias`:
+
+**`bias` [`Vector3`]**
+—
+The gyrotropy vector.  The direction of this vector determines the orientation of the gyrotropic response, and the magnitude equals the precession frequency $|\mathbf{b}_n|/2\pi$.
+
+For gyrotropic media of Landau-Lifshitz-Gilbert form, the parameters `sigma`, `frequency`, and `gamma` play different roles, and there is an additiona `alpha` parameter:
 
 **`sigma` [`number`]**
 —
@@ -402,7 +410,7 @@ The coupling factor $\sigma_n / 2\pi$ between the polarization and the driving f
 
 **`frequency` [`number`]**
 —
-The frequency of gyrotropic precession, $f_n = \omega_n / 2\pi$. In magnetic ferrites, this is the Larmor precession frequency.
+The Larmor precession frequency, $f_n = \omega_n / 2\pi$.
 
 **`gamma` [`number`]**
 —
@@ -412,10 +420,6 @@ The loss rate $\gamma_n / 2\pi$ in the off-diagonal response.
 —
 The loss factor $\alpha_n$ in the diagonal response (note that there is no 2π factor).
 
-**`bias` [`Vector3`]**
-—
-Gyrotropy vector describing the direction of the static biasing magnetic field. The magnitude is ignored.
-
 ### Vector3
 
 Properties:
diff --git a/python/geom.py b/python/geom.py
index 203289a1b..e52aea387 100644
--- a/python/geom.py
+++ b/python/geom.py
@@ -280,12 +280,18 @@ def __init__(self, noise_amp=0.0, **kwargs):
         super(NoisyDrudeSusceptibility, self).__init__(**kwargs)
         self.noise_amp = noise_amp
 
-class GyrotropicSusceptibility(LorentzianSusceptibility):
+class GyrotropicLorentzianSusceptibility(LorentzianSusceptibility):
 
     def __init__(self, bias=Vector3(), **kwargs):
         super(GyrotropicLorentzianSusceptibility, self).__init__(**kwargs)
         self.bias = bias
 
+class GyrotropicDrudeSusceptibility(DrudeSusceptibility):
+
+    def __init__(self, bias=Vector3(), **kwargs):
+        super(GyrotropicDrudeSusceptibility, self).__init__(**kwargs)
+        self.bias = bias
+
 class MultilevelAtom(Susceptibility):
 
     def __init__(self, initial_populations=[], transitions=[], **kwargs):
diff --git a/python/meep.i b/python/meep.i
index d984185d2..98223c364 100644
--- a/python/meep.i
+++ b/python/meep.i
@@ -1371,7 +1371,8 @@ PyObject *_get_array_slice_dimensions(meep::fields *f, const meep::volume &where
         Ellipsoid,
         FreqRange,
         GeometricObject,
-        GyrotropicSusceptibility,
+	GyrotropicDrudeSusceptibility,
+	GyrotropicLorentzianSusceptibility,
         Lattice,
         LorentzianSusceptibility,
         Matrix,
diff --git a/python/typemap_utils.cpp b/python/typemap_utils.cpp
index 302d78c35..f8fe8327b 100644
--- a/python/typemap_utils.cpp
+++ b/python/typemap_utils.cpp
@@ -465,9 +465,16 @@ static PyObject *susceptibility_to_py_obj(susceptibility_struct *s) {
   PyObject *args = PyTuple_New(0);
 
   if (s->bias.x || s->bias.y || s->bias.z) {
-    PyObject *py_gyrotropic_class = PyObject_GetAttrString(geom_mod, "GyrotropicSusceptibility");
-    res = PyObject_Call(py_gyrotropic_class, args, NULL);
-    Py_DECREF(py_gyrotropic_class);
+    if (s->drude) {
+      PyObject *py_gyrotropic_drude_class = PyObject_GetAttrString(geom_mod, "GyrotropicDrudeSusceptibility");
+      res = PyObject_Call(py_gyrotropic_drude_class, args, NULL);
+      Py_DECREF(py_gyrotropic_drude_class);
+    } else {
+      PyObject *py_gyrotropic_lorentz_class =
+       PyObject_GetAttrString(geom_mod, "GyrotropicLorentzianSusceptibility");
+      res = PyObject_Call(py_gyrotropic_lorentz_class, args, NULL);
+      Py_DECREF(py_gyrotropic_lorentz_class);
+    }
 
     PyObject *py_bias = vec2py(vector3_to_vec(s->bias));
     PyObject_SetAttrString(res, "bias", py_bias);
diff --git a/scheme/meep.scm.in b/scheme/meep.scm.in
index 857559f97..cfec58b06 100644
--- a/scheme/meep.scm.in
+++ b/scheme/meep.scm.in
@@ -57,7 +57,9 @@
   (define-property noise-amp no-default 'number))
 (define-class noisy-drude-susceptibility drude-susceptibility
   (define-property noise-amp no-default 'number))
-(define-class gyrotropic-susceptibility lorentzian-susceptibility
+(define-class gyrotropic-lorentzian-susceptibility lorentzian-susceptibility
+  (define-property bias no-default 'vector3))
+(define-class gyrotropic-drude-susceptibility drude-susceptibility
   (define-property bias no-default 'vector3))
 
 (define polarizability lorentzian-susceptibility) ; backwards compat
diff --git a/scheme/structure.cpp b/scheme/structure.cpp
index 8f6bdf16f..85a601965 100644
--- a/scheme/structure.cpp
+++ b/scheme/structure.cpp
@@ -1295,10 +1295,10 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
           master_printf("noisy lorentzian susceptibility: frequency=%g, gamma=%g, amp = %g\n",
                         d->frequency, d->gamma, nd->noise_amp);
           sus = new meep::noisy_lorentzian_susceptibility(nd->noise_amp, d->frequency, d->gamma);
-	} else if (d->which_subclass == lorentzian_susceptibility::GYROTROPIC_SUSCEPTIBILITY) {
-          gyrotropic_susceptibility *gd = d->subclass.gyrotropic_susceptibility_data;
+        } else if (d->which_subclass == lorentzian_susceptibility::GYROTROPIC_LORENTZIAN_SUSCEPTIBILITY) {
+          gyrotropic_lorentzian_susceptibility *gd = d->subclass.gyrotropic_lorentzian_susceptibility_data;
           master_printf("gyrotropic lorentzian susceptibility: bias=(%g,%g,%g), frequency=%g, gamma=%g\n",
-			gd->bias.x, gd->bias.y, gd->bias.z, d->frequency, d->gamma);
+                        gd->bias.x, gd->bias.y, gd->bias.z, d->frequency, d->gamma);
           sus = new meep::gyrotropic_susceptibility(vector3_to_vec(gd->bias), d->frequency, d->gamma);
         } else { // just a Lorentzian
           master_printf("lorentzian susceptibility: frequency=%g, gamma=%g\n", d->frequency,
@@ -1315,7 +1315,12 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
                         d->frequency, d->gamma, nd->noise_amp);
           sus = new meep::noisy_lorentzian_susceptibility(nd->noise_amp, d->frequency, d->gamma,
                                                           true);
-	} else { // just a Drude
+        } else if (d->which_subclass == drude_susceptibility::GYROTROPIC_DRUDE_SUSCEPTIBILITY) {
+          gyrotropic_drude_susceptibility *gd = d->subclass.gyrotropic_drude_susceptibility_data;
+          master_printf("gyrotropic drude susceptibility: bias=(%g,%g,%g), frequency=%g, gamma=%g\n",
+                        gd->bias.x, gd->bias.y, gd->bias.z, d->frequency, d->gamma);
+          sus = new meep::gyrotropic_susceptibility(vector3_to_vec(gd->bias), d->frequency, d->gamma, true);
+        } else { // just a Drude
           master_printf("drude susceptibility: frequency=%g, gamma=%g\n", d->frequency, d->gamma);
           sus = new meep::lorentzian_susceptibility(d->frequency, d->gamma, true);
         }
diff --git a/src/meep.hpp b/src/meep.hpp
index 9ca175a65..e1a1767a9 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -275,7 +275,8 @@ class noisy_lorentzian_susceptibility : public lorentzian_susceptibility {
 /* gyrotropic susceptibility */
 class gyrotropic_susceptibility : public lorentzian_susceptibility {
 public:
-  gyrotropic_susceptibility(const vec &bias, double omega_0, double gamma);
+  gyrotropic_susceptibility(const vec &bias, double omega_0, double gamma,
+                            bool no_omega_0_denominator = true);
   virtual susceptibility *clone() const { return new gyrotropic_susceptibility(*this); }
 
   virtual void *new_internal_data(realnum *W[NUM_FIELD_COMPONENTS][2], const grid_volume &gv) const;
@@ -294,9 +295,6 @@ class gyrotropic_susceptibility : public lorentzian_susceptibility {
   virtual realnum *cinternal_notowned_ptr(int inotowned, component c, int cmp,
 					  int n, void *P_internal_data) const;
 
-  virtual void subtract_P(field_type ft, realnum *f_minus_p[NUM_FIELD_COMPONENTS][2],
-                          void *P_internal_data) const;
-
   virtual void dump_params(h5file *h5f, size_t *start);
   virtual int get_num_params() { return 7; }
   virtual bool needs_W_notowned(component c, realnum *W[NUM_FIELD_COMPONENTS][2]) const {
@@ -306,9 +304,7 @@ class gyrotropic_susceptibility : public lorentzian_susceptibility {
   }
 
 protected:
-  double psat;     // conserved magnitude of the polarization vector
-  double bvec[3];  // unit vector pointing along direction of gyrotropic bias
-  int sgn[3][3];   // antisymmetric tensor: sgn[X][Y] = 1, sgn[Z][Y] = -1, etc.
+  double gyro_tensor[3][3];
 };
 
 class multilevel_susceptibility : public susceptibility {
diff --git a/src/meepgeom.cpp b/src/meepgeom.cpp
index d30a32adf..a259669b5 100644
--- a/src/meepgeom.cpp
+++ b/src/meepgeom.cpp
@@ -1375,18 +1375,16 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
         sus = new meep::noisy_lorentzian_susceptibility(ss->noise_amp, ss->frequency, ss->gamma,
                                                         ss->drude);
       } else if (gyrotropic) {
-        sus = new meep::gyrotropic_susceptibility(meep::vec(ss->bias.x, ss->bias.y, ss->bias.z),
-						  ss->frequency, ss->gamma);
+        sus = new meep::gyrotropic_susceptibility(meep::vec(ss->bias.x, ss->bias.y, ss->bias.z), ss->frequency,
+                                                  ss->gamma, ss->drude);
       } else {
         sus = new meep::lorentzian_susceptibility(ss->frequency, ss->gamma, ss->drude);
       }
       master_printf("%s%s susceptibility: frequency=%g, gamma=%g",
-		    noisy ? "noisy " : gyrotropic ? "gyrotropic" : "",
-		    gyrotropic ? "" : ss->drude ? "drude" : "lorentzian",
-		    ss->frequency, ss->gamma);
+                    noisy ? "noisy " : gyrotropic ? "gyrotropic " : "",
+                    ss->drude ? "drude" : "lorentzian", ss->frequency, ss->gamma);
       if (noisy) master_printf(", amp=%g ", ss->noise_amp);
-      if (gyrotropic) master_printf(", bias=(%g,%g,%g) ",
-				    ss->bias.x, ss->bias.y, ss->bias.z);
+      if (gyrotropic) master_printf(", bias=(%g,%g,%g) ", ss->bias.x, ss->bias.y, ss->bias.z);
       master_printf("\n");
     }
 

From 2951befe1e1919931d8bd36074933d7add28c0ec Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Sun, 2 Jun 2019 10:53:14 +0800
Subject: [PATCH 39/61] Remove gyrotropic-dispersion.py (incomplete attempt)

---
 python/examples/gyrotropic-dispersion.py | 60 ------------------------
 1 file changed, 60 deletions(-)
 delete mode 100644 python/examples/gyrotropic-dispersion.py

diff --git a/python/examples/gyrotropic-dispersion.py b/python/examples/gyrotropic-dispersion.py
deleted file mode 100644
index f4939976f..000000000
--- a/python/examples/gyrotropic-dispersion.py
+++ /dev/null
@@ -1,60 +0,0 @@
-import numpy as np
-from numpy.linalg import inv
-import matplotlib.pyplot as plt
-
-## Material parameters
-sn = 0.0002
-fn = 0.5
-gn = 1e-5
-epsn = 15
-psat = 100
-
-def meep_sim(gn, sn, epsn, tmax):
-
-    ## For mu ~ 1 and epsn ~ 15, wavelength will be ~0.26.  Set dx and dt to ~0.01
-    resolution = 100
-
-    cellsize = [0.01, 0.01, 5.0] # Don't set to zero size: need 3D fields
-    fsrc = 0.8
-    src_z = -1.8
-    pml_depth = 0.25
-
-    import meep as mp
-
-    cell = mp.Vector3(*cellsize)
-
-    ## Define gyromagnetic material biased along z direction
-    susc = [mp.GyrotropicSusceptibility(frequency=fn, gamma=gn, sigma=sn,
-                                        bias=mp.Vector3(0, 0, psat))]
-    mat = mp.Medium(epsilon=epsn, mu=1, H_susceptibilities=susc)
-
-    pml_layers = [mp.PML(thickness=pml_depth, direction=mp.Z)]
-
-    sources = [mp.Source(mp.ContinuousSource(frequency=fsrc),
-                         component=mp.Ex, center=mp.Vector3(0, 0, src_z))]
-
-    sim = mp.Simulation(cell_size=cell, geometry=[], sources=sources,
-                        k_point=mp.Vector3(), # Periodic BCs in x and y
-                        boundary_layers=pml_layers,
-                        default_material=mat, resolution=resolution)
-
-    sim.run(until=tmax)
-
-    ex_data = sim.get_array(center=mp.Vector3(), size=mp.Vector3(0, 0, cellsize[2]), component=mp.Ex)
-    ey_data = sim.get_array(center=mp.Vector3(), size=mp.Vector3(0, 0, cellsize[2]), component=mp.Ey)
-
-    L = cellsize[2]/2
-    z = np.linspace(-L, L, len(ex_data))
-    plt.title('t = {}'.format(tmax))
-    plt.plot(z, ex_data, label='Ex')
-    plt.plot(z, ey_data, label='Ey')
-    plt.xlim(-L, L)
-    plt.xlabel('z')
-    plt.legend()
-
-plt.figure(2, figsize=(8,8))
-plt.subplot(2,1,1)
-meep_sim(gn, sn, epsn, 200)
-plt.subplot(2,1,2)
-meep_sim(gn, sn, epsn, 500)
-plt.tight_layout()

From d29cbe83bba2892e7c4fbfefa395184965dd16b5 Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Sun, 2 Jun 2019 10:57:05 +0800
Subject: [PATCH 40/61] Complete merge

---
 python/geom.py           |  3 ++-
 python/meep.i            | 16 ++++++++--------
 python/typemap_utils.cpp |  1 -
 src/susceptibility.cpp   |  3 ++-
 4 files changed, 12 insertions(+), 11 deletions(-)

diff --git a/python/geom.py b/python/geom.py
index e52aea387..002ecc5ff 100644
--- a/python/geom.py
+++ b/python/geom.py
@@ -292,7 +292,8 @@ def __init__(self, bias=Vector3(), **kwargs):
         super(GyrotropicDrudeSusceptibility, self).__init__(**kwargs)
         self.bias = bias
 
-class MultilevelAtom(Susceptibility):
+
+        class MultilevelAtom(Susceptibility):
 
     def __init__(self, initial_populations=[], transitions=[], **kwargs):
         super(MultilevelAtom, self).__init__(**kwargs)
diff --git a/python/meep.i b/python/meep.i
index 98223c364..998f13133 100644
--- a/python/meep.i
+++ b/python/meep.i
@@ -749,21 +749,21 @@ meep::volume_list *make_volume_list(const meep::volume &v, int c,
 }
 
 %typemap(in) const meep::src_time & {
-    PyObject *obj = NULL;
+    PyObject *swig_obj = NULL;
     void *tmp_ptr = 0;
     int tmp_res = 0;
 
     if(PyObject_IsInstance($input, py_source_time_object())) {
-        obj = PyObject_GetAttrString($input, "swigobj");
+        swig_obj = PyObject_GetAttrString($input, "swigobj");
     } else if(PyObject_IsInstance($input, py_meep_src_time_object())) {
-        obj = $input;
-        Py_XINCREF(obj);
+        swig_obj = $input;
+        Py_XINCREF(swig_obj);
     } else {
       meep::abort("Expected a meep.source.SourceTime or a meep.src_time\n");
     }
 
-    tmp_res = SWIG_ConvertPtr(obj, &tmp_ptr, $1_descriptor, 0);
-    Py_XDECREF(obj);
+    tmp_res = SWIG_ConvertPtr(swig_obj, &tmp_ptr, $1_descriptor, 0);
+    Py_XDECREF(swig_obj);
 
     if(!SWIG_IsOK(tmp_res)) {
         SWIG_exception_fail(SWIG_ArgError(tmp_res), "Couldn't convert Python object to meep::src_time");
@@ -1371,8 +1371,8 @@ PyObject *_get_array_slice_dimensions(meep::fields *f, const meep::volume &where
         Ellipsoid,
         FreqRange,
         GeometricObject,
-	GyrotropicDrudeSusceptibility,
-	GyrotropicLorentzianSusceptibility,
+        GyrotropicDrudeSusceptibility,
+        GyrotropicLorentzianSusceptibility,
         Lattice,
         LorentzianSusceptibility,
         Matrix,
diff --git a/python/typemap_utils.cpp b/python/typemap_utils.cpp
index f8fe8327b..5c74107c6 100644
--- a/python/typemap_utils.cpp
+++ b/python/typemap_utils.cpp
@@ -475,7 +475,6 @@ static PyObject *susceptibility_to_py_obj(susceptibility_struct *s) {
       res = PyObject_Call(py_gyrotropic_lorentz_class, args, NULL);
       Py_DECREF(py_gyrotropic_lorentz_class);
     }
-
     PyObject *py_bias = vec2py(vector3_to_vec(s->bias));
     PyObject_SetAttrString(res, "bias", py_bias);
     Py_DECREF(py_bias);
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 50aebbda5..e0ef45d9a 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -503,8 +503,9 @@ realnum *gyrotropic_susceptibility::cinternal_notowned_ptr(int inotowned, compon
 void gyrotropic_susceptibility::dump_params(h5file *h5f, size_t *start) {
   size_t num_params = 8;
   size_t params_dims[1] = {num_params};
+  double bias[] = { gyro_tensor[Y][Z], gyro_tensor[Z][X], gyro_tensor[X][Y] };
   double params_data[] = {
-      7, (double)get_id(), bvec[0], bvec[1], bvec[2],
+      7, (double)get_id(), bias[X], bias[Y], bias[Z],
       omega_0, gamma, (double)no_omega_0_denominator};
   h5f->write_chunk(1, start, params_dims, params_data);
   *start += num_params;

From 2c6df3347fe8b3e50b098571e67834543908bc50 Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Sun, 2 Jun 2019 11:19:04 +0800
Subject: [PATCH 41/61] Update Materials.md to discuss both Lorentzian and LLG
 gyrotropic models

---
 doc/docs/Materials.md | 20 +++++++++++++-------
 1 file changed, 13 insertions(+), 7 deletions(-)

diff --git a/doc/docs/Materials.md b/doc/docs/Materials.md
index 67d8e4482..fda5ebbcc 100644
--- a/doc/docs/Materials.md
+++ b/doc/docs/Materials.md
@@ -170,23 +170,29 @@ For a two level gain medium, $\gamma_\parallel = \gamma_{12} + \gamma_{21}$. For
 Gyrotropic Media
 ----------------
 
-(**Experimental feature**) Meep supports gyrotropic media, which break optical reciprocity and give rise to magneto-optical phenomena such as the [Faraday effect](https://en.wikipedia.org/wiki/Faraday_effect). Such materials are used in devices like [Faraday rotators](https://en.wikipedia.org/wiki/Faraday_rotator). The following discussion concerns a medium with gyrotropic ε (i.e., a gyroelectric medium). Gyromagnetic media can be treated by the usual substitution of ε with μ, **E** with **H**, etc.
+(**Experimental feature**) Meep supports gyrotropic media, which break optical reciprocity and give rise to magneto-optical phenomena such as the [Faraday effect](https://en.wikipedia.org/wiki/Faraday_effect). Such materials are used in devices like [Faraday rotators](https://en.wikipedia.org/wiki/Faraday_rotator). The following discussion concerns media with gyrotropic ε (i.e., gyroelectric media). Gyromagnetic media, which have gyrotropic μ, can be treated by the usual substitution of ε with μ, **E** with **H**, etc.
 
-In a gyroelectric medium, the equation of motion of the polarization vector $\mathbf{P}_n$ is a variant of the [Landau-Lifshitz-Gilbert equation](https://en.wikipedia.org/wiki/Landau%E2%80%93Lifshitz%E2%80%93Gilbert_equation):
+In a gyrotropic medium, the polarization vector $\mathbf{P}_n$ undergoes precession around a preferred direction. There are two available equations of motion for $\mathbf{P}_n$. The first is a [Drude-Lorentz](Materials.md#material-dispersion) model with an additional precession term:
+
+$$\frac{d^2\mathbf{P}_n}{dt^2} + \gamma_n \frac{d\mathbf{P}_n}{dt} + \mathbf{b}_n \times \frac{d\mathbf{P}_n}{dt} + \omega_n^2 \mathbf{P}_n = \sigma_n(\mathbf{x}) \omega_n^2 \mathbf{E}$$
+
+(The above equation shows the Lorentz case; for the Drude case, the $\omega_n^2 \mathbf{P}_n$ term on the left-hand side is omitted.) The third term on the left-hand side breaks time-reversal symmetry and is responsible for the gyrotropy; it is parameterized by the bias vector $\mathbf{b}_n$, usually describing the effect of a static magnetic field applied to the material. In the $\gamma_n = \omega_n = 0$ limit, the equation of motion reduces to a precession around $\mathbf{b}_n$, with angular frequency equal to $|\mathbf{b}_n|$:
+
+$$\frac{d\mathbf{P}_n}{dt} = \mathbf{P}_n \times \mathbf{b}_n$$
+
+The second type of gyrotropic equation of motion is a linearized [Landau-Lifshitz-Gilbert equation](https://en.wikipedia.org/wiki/Landau%E2%80%93Lifshitz%E2%80%93Gilbert_equation):
 
 $$\frac{d\mathbf{P}_n}{dt} = \mathbf{b}_n \times \left( - \sigma_n \mathbf{E} + \omega_n \mathbf{P}_n + \alpha_n \frac{d\mathbf{P}_n}{dt} \right) - \gamma_n \mathbf{P}_n$$
 
-Although this is completely different from the damped harmonic oscillator equation used for [Drude-Lorentz susceptibility](Materials.md#material-dispersion), the constants σ$_n$, ω$_n$, and γ$_n$ play analogous roles: σ$_n$ couples the polarization to the electric field, ω$_n$ is the angular frequency of precession, and γ$_n$ is a damping factor. There is also a second damping factor α$_n$, and a "bias vector" **b**$_n$ which typically describes the direction of an applied static magnetic field. The bias vector is assumed to have unit length.
+The Landau-Lifshitz-Gilbert equation describes the precessional motion of a point magnetic dipole in a magnetic field; when it is used to model magnetic susceptibility, it accurately describes the behavior of gyromagnetic materials such as ferrites. Although this equation of motion is completely different from the [Drude-Lorentz equation](Materials.md#material-dispersion), the constants σ$_n$, ω$_n$, and γ$_n$ play analogous roles: σ$_n$ couples the polarization to the electric field, ω$_n$ is the angular frequency of precession, and γ$_n$ is a damping factor. There is also a second damping factor α$_n$, and a bias vector **b**$_n$.
 
-In the frequency domain, a gyroelectric medium has skew-symmetric off-diagonal components in the ε tensor. Take the case $\mathbf{b} = b \hat{z}$. When all fields have harmonic time-dependence $\exp(-i\omega t)$, the polarization equation of motion reduces to
+In the frequency domain, a gyroelectric medium exhibits skew-symmetric off-diagonal components in its ε tensor. Take the case $\mathbf{b} = b \hat{z}$. When all fields have harmonic time-dependence $\exp(-i\omega t)$, the polarization equation of motion reduces to
 
 $$\mathbf{P}_n =  \begin{bmatrix}\chi_\perp & -i\eta & 0 \\ i\eta & \chi_\perp & 0 \\ 0 & 0 & \chi_\parallel \end{bmatrix} \mathbf{E}$$
 
 For the gyrotropic Lorentzian model,
 
-In the frequency domain, the ε tensor acquires off-diagonal components representing a gyrotropic response. Take the case $\mathbf{b}_n = \hat{\mathbf{z}}$, and assume that all fields have harmonic time-dependence $\exp(-i\omega t)$. The polarization equation then reduces to
-
-$$\chi_\perp = \frac{\omega_n^2 \Delta_n}{\Delta_n^2 - \omega^2 b^2}\,\sigma_n(\mathbf{x}),\;\;\; \chi_\parallel = \frac{\omega_n^2}{\Delta_n}\,\sigma_n(\mathbf{x}), \;\;\; \eta = \frac{\omega_n^2 \omega b}{\Delta_n^2 - \omega^2 b^2}\,\sigma_n(\mathbf{x}), \;\;\;\Delta_n \equiv \omega_n^2 - \omega^2 - i\omega\gamma_n$$
+$$\chi_\perp = \frac{\omega_n^2 \Delta_n \sigma_n}{\Delta_n^2 - \omega^2 b^2},\;\;\; \chi_\parallel = \frac{\omega_n^2 \sigma_n}{\Delta_n}, \;\;\; \eta = \frac{\omega_n^2 \omega b \sigma_n}{\Delta_n^2 - \omega^2 b^2}, \;\;\;\Delta_n \equiv \omega_n^2 - \omega^2 - i\omega\gamma_n$$
 
 And for the Landau-Lifshitz-Gilbert model,
 

From 67e60dee9b2d181627b4258eef4d83ea8a6b432d Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Sun, 2 Jun 2019 16:47:54 +0800
Subject: [PATCH 42/61] Introduce a new gyrotropy_model enum type, to allow for
 the LLG model.

---
 scheme/structure.cpp   | 12 ++++++------
 src/meep.hpp           |  8 ++++++--
 src/meepgeom.cpp       |  6 ++++--
 src/susceptibility.cpp |  8 ++++----
 4 files changed, 20 insertions(+), 14 deletions(-)

diff --git a/scheme/structure.cpp b/scheme/structure.cpp
index 85a601965..59f042348 100644
--- a/scheme/structure.cpp
+++ b/scheme/structure.cpp
@@ -1299,10 +1299,10 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
           gyrotropic_lorentzian_susceptibility *gd = d->subclass.gyrotropic_lorentzian_susceptibility_data;
           master_printf("gyrotropic lorentzian susceptibility: bias=(%g,%g,%g), frequency=%g, gamma=%g\n",
                         gd->bias.x, gd->bias.y, gd->bias.z, d->frequency, d->gamma);
-          sus = new meep::gyrotropic_susceptibility(vector3_to_vec(gd->bias), d->frequency, d->gamma);
+          sus = new meep::gyrotropic_susceptibility(vector3_to_vec(gd->bias), d->frequency, d->gamma,
+                                                    meep::GYROTROPIC_LORENTZIAN);
         } else { // just a Lorentzian
-          master_printf("lorentzian susceptibility: frequency=%g, gamma=%g\n", d->frequency,
-                        d->gamma);
+          master_printf("lorentzian susceptibility: frequency=%g, gamma=%g\n", d->frequency, d->gamma);
           sus = new meep::lorentzian_susceptibility(d->frequency, d->gamma);
         }
         break;
@@ -1313,13 +1313,13 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
           noisy_drude_susceptibility *nd = d->subclass.noisy_drude_susceptibility_data;
           master_printf("noisy drude susceptibility: frequency=%g, gamma=%g, amp = %g\n",
                         d->frequency, d->gamma, nd->noise_amp);
-          sus = new meep::noisy_lorentzian_susceptibility(nd->noise_amp, d->frequency, d->gamma,
-                                                          true);
+          sus = new meep::noisy_lorentzian_susceptibility(nd->noise_amp, d->frequency, d->gamma, true);
         } else if (d->which_subclass == drude_susceptibility::GYROTROPIC_DRUDE_SUSCEPTIBILITY) {
           gyrotropic_drude_susceptibility *gd = d->subclass.gyrotropic_drude_susceptibility_data;
           master_printf("gyrotropic drude susceptibility: bias=(%g,%g,%g), frequency=%g, gamma=%g\n",
                         gd->bias.x, gd->bias.y, gd->bias.z, d->frequency, d->gamma);
-          sus = new meep::gyrotropic_susceptibility(vector3_to_vec(gd->bias), d->frequency, d->gamma, true);
+          sus = new meep::gyrotropic_susceptibility(vector3_to_vec(gd->bias), d->frequency, d->gamma,
+                                                    meep::GYROTROPIC_DRUDE);
         } else { // just a Drude
           master_printf("drude susceptibility: frequency=%g, gamma=%g\n", d->frequency, d->gamma);
           sus = new meep::lorentzian_susceptibility(d->frequency, d->gamma, true);
diff --git a/src/meep.hpp b/src/meep.hpp
index e1a1767a9..bc566dfc7 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -272,11 +272,13 @@ class noisy_lorentzian_susceptibility : public lorentzian_susceptibility {
   double noise_amp;
 };
 
+typedef enum { GYROTROPIC_LORENTZIAN, GYROTROPIC_DRUDE, LANDAU_LIFSHITZ_GILBERT } gyrotropy_model;
+
 /* gyrotropic susceptibility */
-class gyrotropic_susceptibility : public lorentzian_susceptibility {
+class gyrotropic_susceptibility : public susceptibility {
 public:
   gyrotropic_susceptibility(const vec &bias, double omega_0, double gamma,
-                            bool no_omega_0_denominator = true);
+                            gyrotropy_model model=GYROTROPIC_LORENTZIAN);
   virtual susceptibility *clone() const { return new gyrotropic_susceptibility(*this); }
 
   virtual void *new_internal_data(realnum *W[NUM_FIELD_COMPONENTS][2], const grid_volume &gv) const;
@@ -305,6 +307,8 @@ class gyrotropic_susceptibility : public lorentzian_susceptibility {
 
 protected:
   double gyro_tensor[3][3];
+  double omega_0, gamma;
+  gyrotropy_model model;
 };
 
 class multilevel_susceptibility : public susceptibility {
diff --git a/src/meepgeom.cpp b/src/meepgeom.cpp
index a259669b5..ddbc52632 100644
--- a/src/meepgeom.cpp
+++ b/src/meepgeom.cpp
@@ -1375,8 +1375,10 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
         sus = new meep::noisy_lorentzian_susceptibility(ss->noise_amp, ss->frequency, ss->gamma,
                                                         ss->drude);
       } else if (gyrotropic) {
-        sus = new meep::gyrotropic_susceptibility(meep::vec(ss->bias.x, ss->bias.y, ss->bias.z), ss->frequency,
-                                                  ss->gamma, ss->drude);
+	meep::gyrotropy_model model
+	  = ss->drude ? meep::GYROTROPIC_DRUDE : meep::GYROTROPIC_LORENTZIAN;
+        sus = new meep::gyrotropic_susceptibility(meep::vec(ss->bias.x, ss->bias.y, ss->bias.z),
+						  ss->frequency, ss->gamma, model);
       } else {
         sus = new meep::lorentzian_susceptibility(ss->frequency, ss->gamma, ss->drude);
       }
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index e0ef45d9a..d6a565316 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -324,8 +324,8 @@ void noisy_lorentzian_susceptibility::dump_params(h5file *h5f, size_t *start) {
 
 
 gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double omega_0, double gamma,
-						     bool no_omega_0_denominator)
-  : lorentzian_susceptibility(omega_0, gamma, no_omega_0_denominator) {
+						     gyrotropy_model model)
+  : omega_0(omega_0), gamma(gamma), model(model) {
   // Precalculate g_{ij} = sum_k epsilon_{ijk} b_k, used in update_P.
   memset(gyro_tensor, 0, 9 * sizeof(double));
   gyro_tensor[X][Y] = bias.z(); gyro_tensor[Y][X] = -bias.z();
@@ -410,7 +410,7 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
   const double omega2pi = 2 * pi * omega_0, g2pi = gamma * 2 * pi;
   const double omega0dtsqr = omega2pi * omega2pi * dt * dt;
   const double gamma1 = (1 - g2pi * dt / 2);
-  const double diagfac = 2 - (no_omega_0_denominator ? 0 : omega0dtsqr);
+  const double diagfac = 2 - (model == GYROTROPIC_DRUDE ? 0 : omega0dtsqr);
   const double pt = pi*dt;
   (void)W_prev; // unused;
 
@@ -506,7 +506,7 @@ void gyrotropic_susceptibility::dump_params(h5file *h5f, size_t *start) {
   double bias[] = { gyro_tensor[Y][Z], gyro_tensor[Z][X], gyro_tensor[X][Y] };
   double params_data[] = {
       7, (double)get_id(), bias[X], bias[Y], bias[Z],
-      omega_0, gamma, (double)no_omega_0_denominator};
+      omega_0, gamma, (double)model};
   h5f->write_chunk(1, start, params_dims, params_data);
   *start += num_params;
 }

From 566e32c142057e3cb901abcba651af80371ed769 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Mon, 3 Jun 2019 15:51:36 +0800
Subject: [PATCH 43/61] More plumbing to provide support for
 Landau-Lifshitz-Gilbert type gyrotropy model

---
 python/geom.py           |  10 +++-
 python/meep.i            |   1 +
 python/typemap_utils.cpp |  15 ++++-
 scheme/meep.scm.in       |   6 ++
 scheme/structure.cpp     |  12 +++-
 src/material_data.hpp    |   2 +
 src/meep.hpp             |   6 +-
 src/meepgeom.cpp         |  15 +++--
 src/susceptibility.cpp   | 123 ++++++++++++++++++++++-----------------
 9 files changed, 122 insertions(+), 68 deletions(-)

diff --git a/python/geom.py b/python/geom.py
index 002ecc5ff..e3dc6ea61 100644
--- a/python/geom.py
+++ b/python/geom.py
@@ -292,8 +292,16 @@ def __init__(self, bias=Vector3(), **kwargs):
         super(GyrotropicDrudeSusceptibility, self).__init__(**kwargs)
         self.bias = bias
 
+class GyrotropicSaturatedSusceptibility(Susceptibility):
 
-        class MultilevelAtom(Susceptibility):
+    def __init__(self, bias=Vector3(), frequency=0.0, gamma=0.0, alpha=0.0, **kwargs):
+        super(GyrotropicSaturatedSusceptibility, self).__init__(**kwargs)
+        self.frequency = frequency
+        self.gamma = gamma
+        self.bias = bias
+        self.alpha = alpha
+
+class MultilevelAtom(Susceptibility):
 
     def __init__(self, initial_populations=[], transitions=[], **kwargs):
         super(MultilevelAtom, self).__init__(**kwargs)
diff --git a/python/meep.i b/python/meep.i
index 998f13133..af274db9d 100644
--- a/python/meep.i
+++ b/python/meep.i
@@ -1373,6 +1373,7 @@ PyObject *_get_array_slice_dimensions(meep::fields *f, const meep::volume &where
         GeometricObject,
         GyrotropicDrudeSusceptibility,
         GyrotropicLorentzianSusceptibility,
+        GyrotropicSaturatedSusceptibility,
         Lattice,
         LorentzianSusceptibility,
         Matrix,
diff --git a/python/typemap_utils.cpp b/python/typemap_utils.cpp
index 5c74107c6..04f472347 100644
--- a/python/typemap_utils.cpp
+++ b/python/typemap_utils.cpp
@@ -329,8 +329,10 @@ static int py_susceptibility_to_susceptibility(PyObject *po, susceptibility_stru
 
   s->frequency = 0;
   s->gamma = 0;
+  s->alpha = 0;
   s->noise_amp = 0;
   s->bias.x = s->bias.y = s->bias.z = 0;
+  s->saturated_gyrotropy = false;
   s->transitions.resize(0);
   s->initial_populations.resize(0);
 
@@ -350,6 +352,11 @@ static int py_susceptibility_to_susceptibility(PyObject *po, susceptibility_stru
     if (!get_attr_v3(po, &s->bias, "bias")) return 0;
   }
 
+  if (PyObject_HasAttrString(po, "alpha")) {
+    s->saturated_gyrotropy = true;
+    if (!get_attr_dbl(po, &s->alpha, "alpha")) { return 0; }
+  }
+
   if (PyObject_HasAttrString(po, "transitions")) {
     // MultilevelAtom
     PyObject *py_trans = PyObject_GetAttrString(po, "transitions");
@@ -464,8 +471,12 @@ static PyObject *susceptibility_to_py_obj(susceptibility_struct *s) {
   PyObject *res;
   PyObject *args = PyTuple_New(0);
 
-  if (s->bias.x || s->bias.y || s->bias.z) {
-    if (s->drude) {
+  if (s->saturated_gyrotropy || s->bias.x || s->bias.y || s->bias.z) {
+    if (s->saturated_gyrotropy) {
+      PyObject *py_gyrotropic_class = PyObject_GetAttrString(geom_mod, "GyrotropicSaturatedSusceptibility");
+      res = PyObject_Call(py_gyrotropic_class, args, NULL);
+      Py_DECREF(py_gyrotropic_class);
+    } else if (s->drude) {
       PyObject *py_gyrotropic_drude_class = PyObject_GetAttrString(geom_mod, "GyrotropicDrudeSusceptibility");
       res = PyObject_Call(py_gyrotropic_drude_class, args, NULL);
       Py_DECREF(py_gyrotropic_drude_class);
diff --git a/scheme/meep.scm.in b/scheme/meep.scm.in
index cfec58b06..7c312c83a 100644
--- a/scheme/meep.scm.in
+++ b/scheme/meep.scm.in
@@ -62,6 +62,12 @@
 (define-class gyrotropic-drude-susceptibility drude-susceptibility
   (define-property bias no-default 'vector3))
 
+(define-class gyrotropic-saturated-susceptibility susceptibility
+  (define-property frequency no-default 'number)
+  (define-property gamma no-default 'number)
+  (define-property bias no-default 'vector3)
+  (define-property alpha no-default 'number))
+
 (define polarizability lorentzian-susceptibility) ; backwards compat
 (define omega frequency) ; backwards compat
 (define (sigma x) (sigma-diag x x x))
diff --git a/scheme/structure.cpp b/scheme/structure.cpp
index 59f042348..d71626bcf 100644
--- a/scheme/structure.cpp
+++ b/scheme/structure.cpp
@@ -1300,7 +1300,7 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
           master_printf("gyrotropic lorentzian susceptibility: bias=(%g,%g,%g), frequency=%g, gamma=%g\n",
                         gd->bias.x, gd->bias.y, gd->bias.z, d->frequency, d->gamma);
           sus = new meep::gyrotropic_susceptibility(vector3_to_vec(gd->bias), d->frequency, d->gamma,
-                                                    meep::GYROTROPIC_LORENTZIAN);
+                                                    0.0, meep::GYROTROPIC_LORENTZIAN);
         } else { // just a Lorentzian
           master_printf("lorentzian susceptibility: frequency=%g, gamma=%g\n", d->frequency, d->gamma);
           sus = new meep::lorentzian_susceptibility(d->frequency, d->gamma);
@@ -1319,13 +1319,21 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
           master_printf("gyrotropic drude susceptibility: bias=(%g,%g,%g), frequency=%g, gamma=%g\n",
                         gd->bias.x, gd->bias.y, gd->bias.z, d->frequency, d->gamma);
           sus = new meep::gyrotropic_susceptibility(vector3_to_vec(gd->bias), d->frequency, d->gamma,
-                                                    meep::GYROTROPIC_DRUDE);
+                                                    0.0, meep::GYROTROPIC_DRUDE);
         } else { // just a Drude
           master_printf("drude susceptibility: frequency=%g, gamma=%g\n", d->frequency, d->gamma);
           sus = new meep::lorentzian_susceptibility(d->frequency, d->gamma, true);
         }
         break;
       }
+      case susceptibility::GYROTROPIC_SATURATED_SUSCEPTIBILITY: {
+	gyrotropic_saturated_susceptibility *d = p->user_s.subclass.gyrotropic_saturated_susceptibility_data;
+	master_printf("gyrotropic Landau-Lifshitz-Gilbert-type susceptibility: bias=(%g,%g,%g), frequency=%g, gamma=%g, alpha=%g\n",
+                      d->bias.x, d->bias.y, d->bias.z, d->frequency, d->gamma, d->alpha);
+	sus = new meep::gyrotropic_susceptibility(vector3_to_vec(d->bias), d->frequency, d->gamma,
+                                                  d->alpha, meep::GYROTROPIC_SATURATED);
+	break;
+      }
       case susceptibility::MULTILEVEL_ATOM: {
         multilevel_atom *d = p->user_s.subclass.multilevel_atom_data;
         sus = make_multilevel_sus(d);
diff --git a/src/material_data.hpp b/src/material_data.hpp
index 352b6155e..20c6f9823 100644
--- a/src/material_data.hpp
+++ b/src/material_data.hpp
@@ -62,8 +62,10 @@ typedef struct susceptibility_struct {
   vector3 bias;
   double frequency;
   double gamma;
+  double alpha;
   double noise_amp;
   bool drude;
+  bool saturated_gyrotropy;
   bool is_file;
   std::vector<transition> transitions;
   std::vector<double> initial_populations;
diff --git a/src/meep.hpp b/src/meep.hpp
index bc566dfc7..c1fd244d5 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -272,12 +272,12 @@ class noisy_lorentzian_susceptibility : public lorentzian_susceptibility {
   double noise_amp;
 };
 
-typedef enum { GYROTROPIC_LORENTZIAN, GYROTROPIC_DRUDE, LANDAU_LIFSHITZ_GILBERT } gyrotropy_model;
+typedef enum { GYROTROPIC_LORENTZIAN, GYROTROPIC_DRUDE, GYROTROPIC_SATURATED } gyrotropy_model;
 
 /* gyrotropic susceptibility */
 class gyrotropic_susceptibility : public susceptibility {
 public:
-  gyrotropic_susceptibility(const vec &bias, double omega_0, double gamma,
+  gyrotropic_susceptibility(const vec &bias, double omega_0, double gamma, double alpha=0.0,
                             gyrotropy_model model=GYROTROPIC_LORENTZIAN);
   virtual susceptibility *clone() const { return new gyrotropic_susceptibility(*this); }
 
@@ -307,7 +307,7 @@ class gyrotropic_susceptibility : public susceptibility {
 
 protected:
   double gyro_tensor[3][3];
-  double omega_0, gamma;
+  double omega_0, gamma, alpha;
   gyrotropy_model model;
 };
 
diff --git a/src/meepgeom.cpp b/src/meepgeom.cpp
index ddbc52632..efab7a6d3 100644
--- a/src/meepgeom.cpp
+++ b/src/meepgeom.cpp
@@ -1375,18 +1375,23 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
         sus = new meep::noisy_lorentzian_susceptibility(ss->noise_amp, ss->frequency, ss->gamma,
                                                         ss->drude);
       } else if (gyrotropic) {
-	meep::gyrotropy_model model
-	  = ss->drude ? meep::GYROTROPIC_DRUDE : meep::GYROTROPIC_LORENTZIAN;
+        meep::gyrotropy_model model
+          = ss->saturated_gyrotropy ? meep::GYROTROPIC_SATURATED
+          : ss->drude ? meep::GYROTROPIC_DRUDE : meep::GYROTROPIC_LORENTZIAN;
         sus = new meep::gyrotropic_susceptibility(meep::vec(ss->bias.x, ss->bias.y, ss->bias.z),
-						  ss->frequency, ss->gamma, model);
+                                                  ss->frequency, ss->gamma, ss->alpha, model);
       } else {
         sus = new meep::lorentzian_susceptibility(ss->frequency, ss->gamma, ss->drude);
       }
       master_printf("%s%s susceptibility: frequency=%g, gamma=%g",
                     noisy ? "noisy " : gyrotropic ? "gyrotropic " : "",
-                    ss->drude ? "drude" : "lorentzian", ss->frequency, ss->gamma);
+                    ss->saturated_gyrotropy ? "Landau-Lifshitz-Gilbert-type"
+                    : ss->drude ? "drude" : "lorentzian", ss->frequency, ss->gamma);
       if (noisy) master_printf(", amp=%g ", ss->noise_amp);
-      if (gyrotropic) master_printf(", bias=(%g,%g,%g) ", ss->bias.x, ss->bias.y, ss->bias.z);
+      if (gyrotropic) {
+        if (ss->saturated_gyrotropy) master_printf(", alpha=%g", ss->alpha);
+        master_printf(", bias=(%g,%g,%g)", ss->bias.x, ss->bias.y, ss->bias.z);
+      }
       master_printf("\n");
     }
 
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index d6a565316..a36cb94e4 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -324,8 +324,8 @@ void noisy_lorentzian_susceptibility::dump_params(h5file *h5f, size_t *start) {
 
 
 gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double omega_0, double gamma,
-						     gyrotropy_model model)
-  : omega_0(omega_0), gamma(gamma), model(model) {
+						     double alpha, gyrotropy_model model)
+  : omega_0(omega_0), gamma(gamma), alpha(alpha), model(model) {
   // Precalculate g_{ij} = sum_k epsilon_{ijk} b_k, used in update_P.
   memset(gyro_tensor, 0, 9 * sizeof(double));
   gyro_tensor[X][Y] = bias.z(); gyro_tensor[Y][X] = -bias.z();
@@ -408,63 +408,76 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 					 const grid_volume &gv, void *P_internal_data) const {
   gyrotropy_data *d = (gyrotropy_data *)P_internal_data;
   const double omega2pi = 2 * pi * omega_0, g2pi = gamma * 2 * pi;
-  const double omega0dtsqr = omega2pi * omega2pi * dt * dt;
-  const double gamma1 = (1 - g2pi * dt / 2);
-  const double diagfac = 2 - (model == GYROTROPIC_DRUDE ? 0 : omega0dtsqr);
-  const double pt = pi*dt;
   (void)W_prev; // unused;
 
-  // Precalculate 3x3 matrix inverse, exploiting skew symmetry
-  const double gd = (1 + g2pi * dt / 2);
-  const double gx = pt * gyro_tensor[Y][Z];
-  const double gy = pt * gyro_tensor[Z][X];
-  const double gz = pt * gyro_tensor[X][Y];
-  const double invdet = 1.0 / gd / (gd*gd + gx*gx + gy*gy + gz*gz);
-  const double inv[3][3]
-    = {{ invdet*(gd*gd+gx*gx), invdet*(gx*gy+gd*gz), invdet*(gx*gz-gd*gy) },
-       { invdet*(gy*gx-gd*gz), invdet*(gd*gd+gy*gy), invdet*(gy*gz+gd*gx) },
-       { invdet*(gz*gx+gd*gy), invdet*(gz*gy-gd*gx), invdet*(gd*gd+gz*gz) }};
-
-  FOR_COMPONENTS(c) DOCMP2 {
-    if (d->P[c][cmp][0]) {
-      const direction d0 = component_direction(c);
-      const realnum *w0 = W[c][cmp], *s = sigma[c][d0];
-
-      if (!w0 || !s || (d0 != X && d0 != Y && d0 != Z))
-	abort("gyrotropic media require 3D Cartesian fields\n");
-
-      const direction d1 = cycle_direction(gv.dim, d0, 1);
-      const direction d2 = cycle_direction(gv.dim, d0, 2);
-      const realnum *w1 = W[direction_component(c, d1)][cmp];
-      const realnum *w2 = W[direction_component(c, d2)][cmp];
-      realnum *p0 = d->P[c][cmp][d0], *pp0 = d->P_prev[c][cmp][d0];
-      realnum *p1 = d->P[c][cmp][d1], *pp1 = d->P_prev[c][cmp][d1];
-      realnum *p2 = d->P[c][cmp][d2], *pp2 = d->P_prev[c][cmp][d2];
-      const ptrdiff_t is  = gv.stride(d0) * (is_magnetic(c) ? -1 : +1);
-      const ptrdiff_t is1 = gv.stride(d1) * (is_magnetic(c) ? -1 : +1);
-      const ptrdiff_t is2 = gv.stride(d2) * (is_magnetic(c) ? -1 : +1);
-      realnum rhs0, rhs1, rhs2;
-
-      if (!pp1 || !pp2)
-	abort("gyrotropic media require 3D Cartesian fields\n");
-
-      if (sigma[c][d1] || sigma[c][d2])
-	abort("gyrotropic media do not support anisotropic sigma\n");
-
-      LOOP_OVER_VOL_OWNED(gv, c, i) {
-	rhs0 = diagfac*p0[i] - gamma1*pp0[i] + omega0dtsqr*s[i]*w0[i]
-	  - pt*gyro_tensor[d0][d1]*pp1[i] - pt*gyro_tensor[d0][d2]*pp2[i];
-	rhs1 = diagfac*p1[i] - gamma1*pp1[i] + (w1 ? omega0dtsqr*s[i]*OFFDIAG(s,w1,is1,is) : 0)
-	  - pt*gyro_tensor[d1][d0]*pp0[i] - pt*gyro_tensor[d1][d2]*pp2[i];
-	rhs2 = diagfac*p2[i] - gamma1*pp2[i] + (w2 ? omega0dtsqr*s[i]*OFFDIAG(s,w2,is2,is) : 0)
-	  - pt*gyro_tensor[d2][d1]*pp1[i] - pt*gyro_tensor[d2][d0]*pp0[i];
-
-	pp0[i] = p0[i];	pp1[i] = p1[i];	pp2[i] = p2[i];
-	p0[i]  = inv[d0][d0] * rhs0 + inv[d0][d1] * rhs1 + inv[d0][d2] * rhs2;
-	p1[i]  = inv[d1][d0] * rhs0 + inv[d1][d1] * rhs1 + inv[d1][d2] * rhs2;
-	p2[i]  = inv[d2][d0] * rhs0 + inv[d2][d1] * rhs1 + inv[d2][d2] * rhs2;
+  switch (model) {
+  case GYROTROPIC_LORENTZIAN:
+  case GYROTROPIC_DRUDE:
+    {
+      const double omega0dtsqr = omega2pi * omega2pi * dt * dt;
+      const double gamma1 = (1 - g2pi * dt / 2);
+      const double diagfac = 2 - (model == GYROTROPIC_DRUDE ? 0 : omega0dtsqr);
+      const double pt = pi*dt;
+
+      // Precalculate 3x3 matrix inverse, exploiting skew symmetry
+      const double gd = (1 + g2pi * dt / 2);
+      const double gx = pt * gyro_tensor[Y][Z];
+      const double gy = pt * gyro_tensor[Z][X];
+      const double gz = pt * gyro_tensor[X][Y];
+      const double invdet = 1.0 / gd / (gd*gd + gx*gx + gy*gy + gz*gz);
+      const double inv[3][3]
+	= {{ invdet*(gd*gd+gx*gx), invdet*(gx*gy+gd*gz), invdet*(gx*gz-gd*gy) },
+	   { invdet*(gy*gx-gd*gz), invdet*(gd*gd+gy*gy), invdet*(gy*gz+gd*gx) },
+	   { invdet*(gz*gx+gd*gy), invdet*(gz*gy-gd*gx), invdet*(gd*gd+gz*gz) }};
+
+      FOR_COMPONENTS(c) DOCMP2 {
+	if (d->P[c][cmp][0]) {
+	  const direction d0 = component_direction(c);
+	  const realnum *w0 = W[c][cmp], *s = sigma[c][d0];
+
+	  if (!w0 || !s || (d0 != X && d0 != Y && d0 != Z))
+	    abort("gyrotropic media require 3D Cartesian fields\n");
+
+	  const direction d1 = cycle_direction(gv.dim, d0, 1);
+	  const direction d2 = cycle_direction(gv.dim, d0, 2);
+	  const realnum *w1 = W[direction_component(c, d1)][cmp];
+	  const realnum *w2 = W[direction_component(c, d2)][cmp];
+	  realnum *p0 = d->P[c][cmp][d0], *pp0 = d->P_prev[c][cmp][d0];
+	  realnum *p1 = d->P[c][cmp][d1], *pp1 = d->P_prev[c][cmp][d1];
+	  realnum *p2 = d->P[c][cmp][d2], *pp2 = d->P_prev[c][cmp][d2];
+	  const ptrdiff_t is  = gv.stride(d0) * (is_magnetic(c) ? -1 : +1);
+	  const ptrdiff_t is1 = gv.stride(d1) * (is_magnetic(c) ? -1 : +1);
+	  const ptrdiff_t is2 = gv.stride(d2) * (is_magnetic(c) ? -1 : +1);
+	  realnum rhs0, rhs1, rhs2;
+
+	  if (!pp1 || !pp2)
+	    abort("gyrotropic media require 3D Cartesian fields\n");
+	  if (sigma[c][d1] || sigma[c][d2])
+	    abort("gyrotropic media do not support anisotropic sigma\n");
+
+	  LOOP_OVER_VOL_OWNED(gv, c, i) {
+	    rhs0 = diagfac*p0[i] - gamma1*pp0[i] + omega0dtsqr*s[i]*w0[i]
+	      - pt*gyro_tensor[d0][d1]*pp1[i] - pt*gyro_tensor[d0][d2]*pp2[i];
+	    rhs1 = diagfac*p1[i] - gamma1*pp1[i] + (w1 ? omega0dtsqr*s[i]*OFFDIAG(s,w1,is1,is) : 0)
+	      - pt*gyro_tensor[d1][d0]*pp0[i] - pt*gyro_tensor[d1][d2]*pp2[i];
+	    rhs2 = diagfac*p2[i] - gamma1*pp2[i] + (w2 ? omega0dtsqr*s[i]*OFFDIAG(s,w2,is2,is) : 0)
+	      - pt*gyro_tensor[d2][d1]*pp1[i] - pt*gyro_tensor[d2][d0]*pp0[i];
+
+	    pp0[i] = p0[i];	pp1[i] = p1[i];	pp2[i] = p2[i];
+	    p0[i]  = inv[d0][d0] * rhs0 + inv[d0][d1] * rhs1 + inv[d0][d2] * rhs2;
+	    p1[i]  = inv[d1][d0] * rhs0 + inv[d1][d1] * rhs1 + inv[d1][d2] * rhs2;
+	    p2[i]  = inv[d2][d0] * rhs0 + inv[d2][d1] * rhs1 + inv[d2][d2] * rhs2;
+	  }
+	}
       }
     }
+    break;
+
+  case GYROTROPIC_SATURATED:
+    {
+      abort("Landau-Liftshitz-Gilbert type gyrotropy is not yet implemented\n");
+    }
+    break;
   }
 }
 

From 6b76cccff10d4c60411b5f58dac803b0ffacd0e1 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Tue, 4 Jun 2019 11:03:45 +0800
Subject: [PATCH 44/61] Fix typo in susceptibility update equation

---
 python/examples/faraday-rotation.py | 16 ++++++++--------
 src/susceptibility.cpp              |  7 +++++--
 2 files changed, 13 insertions(+), 10 deletions(-)

diff --git a/python/examples/faraday-rotation.py b/python/examples/faraday-rotation.py
index 9b415fabd..28a74f75c 100644
--- a/python/examples/faraday-rotation.py
+++ b/python/examples/faraday-rotation.py
@@ -2,10 +2,10 @@
 
 ## Define a gyroelectric medium
 f0 = 1.0
-gamma = 1e-4
-epsn = 2.0
-b0 = 1.05
-sn = 0.2
+gamma = 1e-6
+epsn = 1.5
+b0 = 0.15
+sn = 0.1
 
 susc = [mp.GyrotropicLorentzianSusceptibility(frequency=f0, gamma=gamma, sigma=sn,
                                               bias=mp.Vector3(0, 0, b0))]
@@ -24,7 +24,7 @@
 sim = mp.Simulation(cell_size=cell, geometry=[], sources=sources,
                     k_point=mp.Vector3(),   # Periodic boundary conditions
                     boundary_layers=pml_layers,
-                    default_material=mat, resolution=50)
+                    default_material=mat, resolution=100)
 sim.run(until=tmax)
 
 ## Plot results:
@@ -42,13 +42,13 @@
 plt.legend()
 
 ## Comparison with analytic result:
-dfsq = (f0*2 - 1j*f0*gamma - fsrc**2)
+dfsq = (f0**2 - 1j*fsrc*gamma - fsrc**2)
 eperp = epsn + sn * f0**2 * dfsq / (dfsq**2 + (fsrc*b0)**2)
 eta = sn * f0**2 * fsrc * b0 / (dfsq**2 + (fsrc*b0)**2)
 
 k_gyro = 2*np.pi*fsrc * np.sqrt(0.5*(eperp - np.sqrt(eperp**2 - eta**2)))
-Ex_theory = 0.37 * np.cos(k_gyro * (z - src_z))
-Ey_theory = 0.37 * np.sin(k_gyro * (z - src_z))
+Ex_theory = 0.37 * np.cos(k_gyro * (z - src_z)).real
+Ey_theory = 0.37 * np.sin(k_gyro * (z - src_z)).real
 
 plt.figure(2)
 plt.subplot(2,1,1)
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index e0ef45d9a..2165652e2 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -403,6 +403,9 @@ bool gyrotropic_susceptibility::needs_P(component c, int cmp, realnum *W[NUM_FIE
   return (d0 == X || d0 == Y || d0 == Z) && sigma[c][d0] && W[c][cmp];
 }
 
+// Similar to the OFFDIAG macro, but without averaging sigma.
+#define OFFDIAGW(g, sx, s) (0.25 * (g[i] + g[i - sx] + g[i + s] + g[i + s - sx]))
+
 void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 					 realnum *W_prev[NUM_FIELD_COMPONENTS][2], double dt,
 					 const grid_volume &gv, void *P_internal_data) const {
@@ -454,9 +457,9 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
       LOOP_OVER_VOL_OWNED(gv, c, i) {
 	rhs0 = diagfac*p0[i] - gamma1*pp0[i] + omega0dtsqr*s[i]*w0[i]
 	  - pt*gyro_tensor[d0][d1]*pp1[i] - pt*gyro_tensor[d0][d2]*pp2[i];
-	rhs1 = diagfac*p1[i] - gamma1*pp1[i] + (w1 ? omega0dtsqr*s[i]*OFFDIAG(s,w1,is1,is) : 0)
+	rhs1 = diagfac*p1[i] - gamma1*pp1[i] + (w1 ? omega0dtsqr*s[i]*OFFDIAGW(w1,is1,is) : 0)
 	  - pt*gyro_tensor[d1][d0]*pp0[i] - pt*gyro_tensor[d1][d2]*pp2[i];
-	rhs2 = diagfac*p2[i] - gamma1*pp2[i] + (w2 ? omega0dtsqr*s[i]*OFFDIAG(s,w2,is2,is) : 0)
+	rhs2 = diagfac*p2[i] - gamma1*pp2[i] + (w2 ? omega0dtsqr*s[i]*OFFDIAGW(w2,is2,is) : 0)
 	  - pt*gyro_tensor[d2][d1]*pp1[i] - pt*gyro_tensor[d2][d0]*pp0[i];
 
 	pp0[i] = p0[i];	pp1[i] = p1[i];	pp2[i] = p2[i];

From aa32c6ded27f3fe0c144478693fb5d5a6648a219 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Tue, 4 Jun 2019 11:12:00 +0800
Subject: [PATCH 45/61] Fix typos in Faraday rotation formula in docs

---
 doc/docs/Python_Tutorials/Gyrotropic_Media.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/doc/docs/Python_Tutorials/Gyrotropic_Media.md b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
index baf6a25dd..dcccb64b0 100644
--- a/doc/docs/Python_Tutorials/Gyrotropic_Media.md
+++ b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
@@ -22,7 +22,7 @@ $$\begin{bmatrix}E_x \\ E_y\end{bmatrix} = E_0 \begin{bmatrix}\cos(\kappa_c z) \
 
 where $\kappa_c$ is the Faraday rotation, in radians, per unit of propagation distance. Substituting this into Maxwell's equations, with the above gyroelectric dielectric function, yields
 
-$$|\kappa_c| = \frac{1}{2} \omega^2 \mu \sqrt{\epsilon_\perp - \sqrt{\epsilon_\perp^2 - \eta^2}}$$
+$$|\kappa_c| = \omega \sqrt{\frac{\mu}{2} \, \left(\epsilon_\perp - \sqrt{\epsilon_\perp^2 - \eta^2}\right)}$$
 
 We model this phenomenon in the simulation script [faraday-rotation.py](https://github.com/NanoComp/meep/blob/master/python/examples/faraday-rotation.py). First, we define a gyroelectric material:
 

From 3934533a6b079d6d330eaa1a17dab129ae64bc9a Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Tue, 4 Jun 2019 11:21:42 +0800
Subject: [PATCH 46/61] Merge from master

---
 src/susceptibility.cpp | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index b9b808153..f054e8067 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -461,9 +461,9 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 	  LOOP_OVER_VOL_OWNED(gv, c, i) {
 	    rhs0 = diagfac*p0[i] - gamma1*pp0[i] + omega0dtsqr*s[i]*w0[i]
 	      - pt*gyro_tensor[d0][d1]*pp1[i] - pt*gyro_tensor[d0][d2]*pp2[i];
-	    rhs1 = diagfac*p1[i] - gamma1*pp1[i] + (w1 ? omega0dtsqr*s[i]*OFFDIAG(s,w1,is1,is) : 0)
+	    rhs1 = diagfac*p1[i] - gamma1*pp1[i] + (w1 ? omega0dtsqr*s[i]*OFFDIAGW(w1,is1,is) : 0)
 	      - pt*gyro_tensor[d1][d0]*pp0[i] - pt*gyro_tensor[d1][d2]*pp2[i];
-	    rhs2 = diagfac*p2[i] - gamma1*pp2[i] + (w2 ? omega0dtsqr*s[i]*OFFDIAG(s,w2,is2,is) : 0)
+	    rhs2 = diagfac*p2[i] - gamma1*pp2[i] + (w2 ? omega0dtsqr*s[i]*OFFDIAGW(w2,is2,is) : 0)
 	      - pt*gyro_tensor[d2][d1]*pp1[i] - pt*gyro_tensor[d2][d0]*pp0[i];
 
 	    pp0[i] = p0[i];	pp1[i] = p1[i];	pp2[i] = p2[i];

From c499a468190d5d9a2b70499def021fc6eee8ab6c Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Tue, 4 Jun 2019 11:44:02 +0800
Subject: [PATCH 47/61] Reimplement linearized-LLG updating equations

---
 src/susceptibility.cpp | 81 ++++++++++++++++++++++++++++++++++++------
 1 file changed, 71 insertions(+), 10 deletions(-)

diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index f054e8067..8c5108ed0 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -410,20 +410,21 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 					 realnum *W_prev[NUM_FIELD_COMPONENTS][2], double dt,
 					 const grid_volume &gv, void *P_internal_data) const {
   gyrotropy_data *d = (gyrotropy_data *)P_internal_data;
-  const double omega2pi = 2 * pi * omega_0, g2pi = gamma * 2 * pi;
+  const double omega2pidt = 2 * pi * omega_0 * dt;
+  const double g2pidt = 2 * pi * gamma * dt;
   (void)W_prev; // unused;
 
   switch (model) {
   case GYROTROPIC_LORENTZIAN:
   case GYROTROPIC_DRUDE:
     {
-      const double omega0dtsqr = omega2pi * omega2pi * dt * dt;
-      const double gamma1 = (1 - g2pi * dt / 2);
-      const double diagfac = 2 - (model == GYROTROPIC_DRUDE ? 0 : omega0dtsqr);
+      const double omega0dtsqr = omega2pidt * omega2pidt;
+      const double gamma1 = (1 - g2pidt / 2);
+      const double diag = 2 - (model == GYROTROPIC_DRUDE ? 0 : omega0dtsqr);
       const double pt = pi*dt;
 
       // Precalculate 3x3 matrix inverse, exploiting skew symmetry
-      const double gd = (1 + g2pi * dt / 2);
+      const double gd = (1 + g2pidt / 2);
       const double gx = pt * gyro_tensor[Y][Z];
       const double gy = pt * gyro_tensor[Z][X];
       const double gz = pt * gyro_tensor[X][Y];
@@ -459,14 +460,14 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 	    abort("gyrotropic media do not support anisotropic sigma\n");
 
 	  LOOP_OVER_VOL_OWNED(gv, c, i) {
-	    rhs0 = diagfac*p0[i] - gamma1*pp0[i] + omega0dtsqr*s[i]*w0[i]
+	    rhs0 = diag*p0[i] - gamma1*pp0[i] + omega0dtsqr*s[i]*w0[i]
 	      - pt*gyro_tensor[d0][d1]*pp1[i] - pt*gyro_tensor[d0][d2]*pp2[i];
-	    rhs1 = diagfac*p1[i] - gamma1*pp1[i] + (w1 ? omega0dtsqr*s[i]*OFFDIAGW(w1,is1,is) : 0)
+	    rhs1 = diag*p1[i] - gamma1*pp1[i] + (w1 ? omega0dtsqr*s[i]*OFFDIAGW(w1,is1,is) : 0)
 	      - pt*gyro_tensor[d1][d0]*pp0[i] - pt*gyro_tensor[d1][d2]*pp2[i];
-	    rhs2 = diagfac*p2[i] - gamma1*pp2[i] + (w2 ? omega0dtsqr*s[i]*OFFDIAGW(w2,is2,is) : 0)
+	    rhs2 = diag*p2[i] - gamma1*pp2[i] + (w2 ? omega0dtsqr*s[i]*OFFDIAGW(w2,is2,is) : 0)
 	      - pt*gyro_tensor[d2][d1]*pp1[i] - pt*gyro_tensor[d2][d0]*pp0[i];
 
-	    pp0[i] = p0[i];	pp1[i] = p1[i];	pp2[i] = p2[i];
+	    pp0[i] = p0[i]; pp1[i] = p1[i]; pp2[i] = p2[i];
 	    p0[i]  = inv[d0][d0] * rhs0 + inv[d0][d1] * rhs1 + inv[d0][d2] * rhs2;
 	    p1[i]  = inv[d1][d0] * rhs0 + inv[d1][d1] * rhs1 + inv[d1][d2] * rhs2;
 	    p2[i]  = inv[d2][d0] * rhs0 + inv[d2][d1] * rhs1 + inv[d2][d2] * rhs2;
@@ -478,7 +479,67 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 
   case GYROTROPIC_SATURATED:
     {
-      abort("Landau-Liftshitz-Gilbert type gyrotropy is not yet implemented\n");
+      const double dwfac = -2*pi*omega2pidt, alppi = pi*alpha;
+      const double dt2pi = 2*pi*dt;
+
+      // Precalculate 3x3 matrix inverse, exploiting skew symmetry
+      const double gd = 0.5;
+      const double gx = -0.5 * alpha * gyro_tensor[Y][Z];
+      const double gy = -0.5 * alpha * gyro_tensor[Z][X];
+      const double gz = -0.5 * alpha * gyro_tensor[X][Y];
+      const double invdet = 1.0 / gd / (gd*gd + gx*gx + gy*gy + gz*gz);
+      const double inv[3][3]
+	= {{ invdet*(gd*gd+gx*gx), invdet*(gx*gy+gd*gz), invdet*(gx*gz-gd*gy) },
+	   { invdet*(gy*gx-gd*gz), invdet*(gd*gd+gy*gy), invdet*(gy*gz+gd*gx) },
+	   { invdet*(gz*gx+gd*gy), invdet*(gz*gy-gd*gx), invdet*(gd*gd+gz*gz) }};
+
+      FOR_COMPONENTS(c) DOCMP2 {
+	if (d->P[c][cmp][0]) {
+	  const direction d0 = component_direction(c);
+	  const realnum *w0 = W[c][cmp], *s = sigma[c][d0];
+
+	  if (!w0 || !s || (d0 != X && d0 != Y && d0 != Z))
+	    abort("gyrotropic media require 3D Cartesian fields\n");
+
+	  const direction d1 = cycle_direction(gv.dim, d0, 1);
+	  const direction d2 = cycle_direction(gv.dim, d0, 2);
+	  const realnum *w1 = W[direction_component(c, d1)][cmp];
+	  const realnum *w2 = W[direction_component(c, d2)][cmp];
+	  realnum *p0 = d->P[c][cmp][d0], *pp0 = d->P_prev[c][cmp][d0];
+	  realnum *p1 = d->P[c][cmp][d1], *pp1 = d->P_prev[c][cmp][d1];
+	  realnum *p2 = d->P[c][cmp][d2], *pp2 = d->P_prev[c][cmp][d2];
+	  const ptrdiff_t is  = gv.stride(d0) * (is_magnetic(c) ? -1 : +1);
+	  const ptrdiff_t is1 = gv.stride(d1) * (is_magnetic(c) ? -1 : +1);
+	  const ptrdiff_t is2 = gv.stride(d2) * (is_magnetic(c) ? -1 : +1);
+	  realnum rhs0, rhs1, rhs2;
+
+	  if (!pp1 || !pp2)
+	    abort("gyrotropic media require 3D Cartesian fields\n");
+	  if (sigma[c][d1] || sigma[c][d2])
+	    abort("gyrotropic media do not support anisotropic sigma\n");
+
+	  LOOP_OVER_VOL_OWNED(gv, c, i) {
+	    rhs0 = 0.5*pp0[i] - g2pidt*p0[i]
+	      + gyro_tensor[d0][d1]*(dwfac*p1[i] + alppi*pp1[i]
+				     + (w1 ? dt2pi*s[i]*OFFDIAGW(w1,is1,is) : 0))
+	      + gyro_tensor[d0][d2]*(dwfac*p2[i] + alppi*pp2[i]
+				     + (w2 ? dt2pi*s[i]*OFFDIAGW(w2,is2,is) : 0));
+	    rhs1 = 0.5*pp1[i] - g2pidt*p1[i]
+	      + gyro_tensor[d1][d2]*(dwfac*p2[i] + alppi*pp2[i]
+				     + (w2 ? dt2pi*s[i]*OFFDIAGW(w2,is2,is) : 0))
+	      + gyro_tensor[d1][d0]*(dwfac*p0[i] + alppi*pp0[i] + dt2pi*s[i]*w0[i]);
+	    rhs2 = 0.5*pp2[i] - g2pidt*p2[i]
+	      + gyro_tensor[d2][d0]*(dwfac*p0[i] + alppi*pp0[i] + dt2pi*s[i]*w0[i])
+	      + gyro_tensor[d2][d1]*(dwfac*p1[i] + alppi*pp1[i]
+				     + (w1 ? dt2pi*s[i]*OFFDIAGW(w1,is1,is) : 0));
+
+	    pp0[i] = p0[i]; pp1[i] = p1[i]; pp2[i] = p2[i];
+	    p0[i]  = inv[d0][d0] * rhs0 + inv[d0][d1] * rhs1 + inv[d0][d2] * rhs2;
+	    p1[i]  = inv[d1][d0] * rhs0 + inv[d1][d1] * rhs1 + inv[d1][d2] * rhs2;
+	    p2[i]  = inv[d2][d0] * rhs0 + inv[d2][d1] * rhs1 + inv[d2][d2] * rhs2;
+	  }
+	}
+      }
     }
     break;
   }

From 545da4a0e30aba636de7d4b9450b5e16eb89d49c Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Tue, 4 Jun 2019 15:48:58 +0800
Subject: [PATCH 48/61] Fix typo

---
 src/susceptibility.cpp | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 8c5108ed0..a3a42335d 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -484,9 +484,9 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 
       // Precalculate 3x3 matrix inverse, exploiting skew symmetry
       const double gd = 0.5;
-      const double gx = -0.5 * alpha * gyro_tensor[Y][Z];
-      const double gy = -0.5 * alpha * gyro_tensor[Z][X];
-      const double gz = -0.5 * alpha * gyro_tensor[X][Y];
+      const double gx = -pi * alpha * gyro_tensor[Y][Z];
+      const double gy = -pi * alpha * gyro_tensor[Z][X];
+      const double gz = -pi * alpha * gyro_tensor[X][Y];
       const double invdet = 1.0 / gd / (gd*gd + gx*gx + gy*gy + gz*gz);
       const double inv[3][3]
 	= {{ invdet*(gd*gd+gx*gx), invdet*(gx*gy+gd*gz), invdet*(gx*gz-gd*gy) },

From c6576ecb4a79825070f8af0325e6e576d47e7e46 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Tue, 4 Jun 2019 17:33:54 +0800
Subject: [PATCH 49/61] Minor code clarification

---
 src/susceptibility.cpp | 42 ++++++++++++++++++------------------------
 1 file changed, 18 insertions(+), 24 deletions(-)

diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index a3a42335d..1528c57ee 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -452,7 +452,7 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 	  const ptrdiff_t is  = gv.stride(d0) * (is_magnetic(c) ? -1 : +1);
 	  const ptrdiff_t is1 = gv.stride(d1) * (is_magnetic(c) ? -1 : +1);
 	  const ptrdiff_t is2 = gv.stride(d2) * (is_magnetic(c) ? -1 : +1);
-	  realnum rhs0, rhs1, rhs2;
+	  realnum r0, r1, r2;
 
 	  if (!pp1 || !pp2)
 	    abort("gyrotropic media require 3D Cartesian fields\n");
@@ -460,17 +460,17 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 	    abort("gyrotropic media do not support anisotropic sigma\n");
 
 	  LOOP_OVER_VOL_OWNED(gv, c, i) {
-	    rhs0 = diag*p0[i] - gamma1*pp0[i] + omega0dtsqr*s[i]*w0[i]
+	    r0 = diag*p0[i] - gamma1*pp0[i] + omega0dtsqr*s[i]*w0[i]
 	      - pt*gyro_tensor[d0][d1]*pp1[i] - pt*gyro_tensor[d0][d2]*pp2[i];
-	    rhs1 = diag*p1[i] - gamma1*pp1[i] + (w1 ? omega0dtsqr*s[i]*OFFDIAGW(w1,is1,is) : 0)
+	    r1 = diag*p1[i] - gamma1*pp1[i] + (w1 ? omega0dtsqr*s[i]*OFFDIAGW(w1,is1,is) : 0)
 	      - pt*gyro_tensor[d1][d0]*pp0[i] - pt*gyro_tensor[d1][d2]*pp2[i];
-	    rhs2 = diag*p2[i] - gamma1*pp2[i] + (w2 ? omega0dtsqr*s[i]*OFFDIAGW(w2,is2,is) : 0)
+	    r2 = diag*p2[i] - gamma1*pp2[i] + (w2 ? omega0dtsqr*s[i]*OFFDIAGW(w2,is2,is) : 0)
 	      - pt*gyro_tensor[d2][d1]*pp1[i] - pt*gyro_tensor[d2][d0]*pp0[i];
 
 	    pp0[i] = p0[i]; pp1[i] = p1[i]; pp2[i] = p2[i];
-	    p0[i]  = inv[d0][d0] * rhs0 + inv[d0][d1] * rhs1 + inv[d0][d2] * rhs2;
-	    p1[i]  = inv[d1][d0] * rhs0 + inv[d1][d1] * rhs1 + inv[d1][d2] * rhs2;
-	    p2[i]  = inv[d2][d0] * rhs0 + inv[d2][d1] * rhs1 + inv[d2][d2] * rhs2;
+	    p0[i]  = inv[d0][d0] * r0 + inv[d0][d1] * r1 + inv[d0][d2] * r2;
+	    p1[i]  = inv[d1][d0] * r0 + inv[d1][d1] * r1 + inv[d1][d2] * r2;
+	    p2[i]  = inv[d2][d0] * r0 + inv[d2][d1] * r1 + inv[d2][d2] * r2;
 	  }
 	}
       }
@@ -511,7 +511,7 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 	  const ptrdiff_t is  = gv.stride(d0) * (is_magnetic(c) ? -1 : +1);
 	  const ptrdiff_t is1 = gv.stride(d1) * (is_magnetic(c) ? -1 : +1);
 	  const ptrdiff_t is2 = gv.stride(d2) * (is_magnetic(c) ? -1 : +1);
-	  realnum rhs0, rhs1, rhs2;
+	  realnum r0, r1, r2, q0, q1, q2;
 
 	  if (!pp1 || !pp2)
 	    abort("gyrotropic media require 3D Cartesian fields\n");
@@ -519,24 +519,18 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 	    abort("gyrotropic media do not support anisotropic sigma\n");
 
 	  LOOP_OVER_VOL_OWNED(gv, c, i) {
-	    rhs0 = 0.5*pp0[i] - g2pidt*p0[i]
-	      + gyro_tensor[d0][d1]*(dwfac*p1[i] + alppi*pp1[i]
-				     + (w1 ? dt2pi*s[i]*OFFDIAGW(w1,is1,is) : 0))
-	      + gyro_tensor[d0][d2]*(dwfac*p2[i] + alppi*pp2[i]
-				     + (w2 ? dt2pi*s[i]*OFFDIAGW(w2,is2,is) : 0));
-	    rhs1 = 0.5*pp1[i] - g2pidt*p1[i]
-	      + gyro_tensor[d1][d2]*(dwfac*p2[i] + alppi*pp2[i]
-				     + (w2 ? dt2pi*s[i]*OFFDIAGW(w2,is2,is) : 0))
-	      + gyro_tensor[d1][d0]*(dwfac*p0[i] + alppi*pp0[i] + dt2pi*s[i]*w0[i]);
-	    rhs2 = 0.5*pp2[i] - g2pidt*p2[i]
-	      + gyro_tensor[d2][d0]*(dwfac*p0[i] + alppi*pp0[i] + dt2pi*s[i]*w0[i])
-	      + gyro_tensor[d2][d1]*(dwfac*p1[i] + alppi*pp1[i]
-				     + (w1 ? dt2pi*s[i]*OFFDIAGW(w1,is1,is) : 0));
+	    q0 = dwfac*p0[i] + alppi*pp0[i] + dt2pi*s[i]*w0[i];
+	    q1 = dwfac*p1[i] + alppi*pp1[i] + (w1 ? dt2pi*s[i]*OFFDIAGW(w1,is1,is) : 0);
+	    q2 = dwfac*p2[i] + alppi*pp2[i] + (w2 ? dt2pi*s[i]*OFFDIAGW(w2,is2,is) : 0);
+
+	    r0 = 0.5*pp0[i] - g2pidt*p0[i] + gyro_tensor[d0][d1]*q1 + gyro_tensor[d0][d2]*q2;
+	    r1 = 0.5*pp1[i] - g2pidt*p1[i] + gyro_tensor[d1][d2]*q2 + gyro_tensor[d1][d0]*q0;
+	    r2 = 0.5*pp2[i] - g2pidt*p2[i] + gyro_tensor[d2][d0]*q0 + gyro_tensor[d2][d1]*q1;
 
 	    pp0[i] = p0[i]; pp1[i] = p1[i]; pp2[i] = p2[i];
-	    p0[i]  = inv[d0][d0] * rhs0 + inv[d0][d1] * rhs1 + inv[d0][d2] * rhs2;
-	    p1[i]  = inv[d1][d0] * rhs0 + inv[d1][d1] * rhs1 + inv[d1][d2] * rhs2;
-	    p2[i]  = inv[d2][d0] * rhs0 + inv[d2][d1] * rhs1 + inv[d2][d2] * rhs2;
+	    p0[i]  = inv[d0][d0] * r0 + inv[d0][d1] * r1 + inv[d0][d2] * r2;
+	    p1[i]  = inv[d1][d0] * r0 + inv[d1][d1] * r1 + inv[d1][d2] * r2;
+	    p2[i]  = inv[d2][d0] * r0 + inv[d2][d1] * r1 + inv[d2][d2] * r2;
 	  }
 	}
       }

From aedb897184580d0734e6c59ba42b7899d68d8172 Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Wed, 5 Jun 2019 11:05:34 +0800
Subject: [PATCH 50/61] Fix Faraday rotation example

---
 doc/docs/Materials.md                         |   6 ++--
 doc/docs/Python_Tutorials/Gyrotropic_Media.md |  32 +++++++++---------
 doc/docs/Python_User_Interface.md             |   2 +-
 .../images/Faraday-rotation-comparison.png    | Bin 97734 -> 100658 bytes
 doc/docs/images/Faraday-rotation.png          | Bin 71606 -> 76081 bytes
 python/examples/faraday-rotation.py           |  14 ++++----
 6 files changed, 28 insertions(+), 26 deletions(-)

diff --git a/doc/docs/Materials.md b/doc/docs/Materials.md
index 1d9a553e5..aeb33576d 100644
--- a/doc/docs/Materials.md
+++ b/doc/docs/Materials.md
@@ -174,12 +174,14 @@ Gyrotropic Media
 
 In a gyroelectric medium, the equation of motion of the polarization vector $\mathbf{P}_n$ is very similar to the [Drude-Lorentz](Materials.md#material-dispersion) case, but contains an additional precession term:
 
-$$\frac{d^2\mathbf{P}_n}{dt^2} + \gamma_n \frac{d\mathbf{P}_n}{dt} + \mathbf{b}_n \times \frac{d\mathbf{P}_n}{dt} + \omega_n^2 \mathbf{P}_n = \sigma_n(\mathbf{x}) \omega_n^2 \mathbf{E}$$
+$$\frac{d^2\mathbf{P}_n}{dt^2} + \gamma_n \frac{d\mathbf{P}_n}{dt} - \frac{d\mathbf{P}_n}{dt} \times \mathbf{b}_n + \omega_n^2 \mathbf{P}_n = \sigma_n(\mathbf{x}) \omega_n^2 \mathbf{E}$$
 
-(Optionally, the polarization may be of Drude form, in which case the $\omega_n^2 \mathbf{P}_n$ term on the left-hand side is omitted.) The third term on the left-hand side breaks time-reversal symmetry and describes the gyrotropy of the medium; it is parameterized by the bias vector $\mathbf{b}_n$, usually describing the effect of an applied static magnetic field. In the $\gamma_n = \omega_n = 0$ limit, the equation of motion reduces to a precession around $\mathbf{b}_n$, with angular frequency equal to $|\mathbf{b}_n|$:
+(Optionally, the polarization may be of Drude form, in which case the $\omega_n^2 \mathbf{P}_n$ term on the left is omitted.) The third term on the left side is responsible for the gyrotropy of the medium. This term breaks time-reversal symmetry and typically describes the effect of a static external magnetic field on the motion of electrons within the material. The bias vector $\mathbf{b}_n$ characterizes the direction and magnitude of the external magnetic field. In the $\gamma_n = \omega_n = 0$ limit, the equation of motion reduces to a precession around $\mathbf{b}_n$:
 
 $$\frac{d\mathbf{P}_n}{dt} = \mathbf{P}_n \times \mathbf{b}_n$$
 
+Hence, the magnitude of the bias vector is the angular frequency of the gyrotropic precession.
+
 In the frequency domain, a gyroelectric medium has skew-symmetric off-diagonal components in the ε tensor. Take the case $\mathbf{b} = b \hat{z}$. When all fields have harmonic time-dependence $\exp(-i\omega t)$, the polarization equation of motion reduces to
 
 $$\mathbf{P}_n =  \begin{bmatrix}\chi_\perp & -i\eta & 0 \\ i\eta & \chi_\perp & 0 \\ 0 & 0 & \chi_\parallel \end{bmatrix} \mathbf{E}$$
diff --git a/doc/docs/Python_Tutorials/Gyrotropic_Media.md b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
index dcccb64b0..568eb163b 100644
--- a/doc/docs/Python_Tutorials/Gyrotropic_Media.md
+++ b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
@@ -6,11 +6,11 @@ In this example, we will perform simulations with gyrotropic media. See [Materia
 
 ### Faraday Rotation
 
-Consider a uniform gyrotroelectric medium with bias vector $\mathbf{b} = b \hat{z}$. According to the [gyrotropic Lorentzian model implemented in Meep](../Materials.md#gyrotropic-media), the *x* and *y* components of the dielectric function are
+Consider a uniform gyroelectric medium with bias vector $\mathbf{b} = b \hat{z}$. In the frequency domain, the *x* and *y* components of the dielectric function are
 
 $$\epsilon = \begin{bmatrix}\epsilon_\perp & -i\eta \\ i\eta & \epsilon_\perp \end{bmatrix}$$
 
-where
+In the [gyrotropic Lorentzian model](../Materials.md#gyrotropic-media), the tensor components are
 
 $$\epsilon_\perp = \epsilon_\infty + \frac{\omega_n^2 \Delta_n}{\Delta_n^2 - \omega^2 b^2}\,\sigma_n(\mathbf{x}),\;\;\; \eta = \frac{\omega_n^2 \omega b}{\Delta_n^2 - \omega^2 b^2}\,\sigma_n(\mathbf{x}), \;\;\;\Delta_n \equiv \omega_n^2 - \omega^2 - i\omega\gamma_n$$
 
@@ -20,7 +20,7 @@ A plane wave undergoing Faraday rotation can be described by the complex ansatz
 
 $$\begin{bmatrix}E_x \\ E_y\end{bmatrix} = E_0 \begin{bmatrix}\cos(\kappa_c z) \\ \sin(\kappa_c z)\end{bmatrix} e^{i(kz-\omega t)}$$
 
-where $\kappa_c$ is the Faraday rotation, in radians, per unit of propagation distance. Substituting this into Maxwell's equations, with the above gyroelectric dielectric function, yields
+where $\kappa_c$ is the Faraday rotation (in radians) per unit of propagation distance. Substituting this into the frequency domain Maxwell's equations, with the above dielectric tensor, yields
 
 $$|\kappa_c| = \omega \sqrt{\frac{\mu}{2} \, \left(\epsilon_\perp - \sqrt{\epsilon_\perp^2 - \eta^2}\right)}$$
 
@@ -30,10 +30,10 @@ We model this phenomenon in the simulation script [faraday-rotation.py](https://
 
 ## Define a gyroelectric medium
 f0 = 1.0
-gamma = 1e-4
-epsn = 2.0
-b0 = 1.05
-sigma = 0.2
+gamma = 1e-6
+epsn = 1.5
+b0 = 0.15
+sn = 0.1
 
 susc = [mp.GyrotropicLorentzianSusceptibility(frequency=f0, gamma=gamma, sigma=sigma,
                                               bias=mp.Vector3(0, 0, b0))]
@@ -84,28 +84,28 @@ plt.show()
 
 We see that the wave indeed rotates in the *x*-*y* plane as it travels. This can be compared quantitatively to the above ansatz for a wave undergoing Faraday rotation, using the material parameters to calculate the rotation rate $\kappa_c$:
 
-```dfsq = (f0*2 - 1j*f0*gamma - fsrc**2)
-eperp = epsn + sigma * f0**2 * dfsq / (dfsq**2 + (fsrc*b0)**2)
-eta = sigma * f0**2 * fsrc * b0 / (dfsq**2 + (fsrc*b0)**2)
+```dfsq = (f0**2 - 1j*fsrc*gamma - fsrc**2)
+eperp = epsn + sn * f0**2 * dfsq / (dfsq**2 - (fsrc*b0)**2)
+eta = sn * f0**2 * fsrc * b0 / (dfsq**2 - (fsrc*b0)**2)
 
 k_gyro = 2*np.pi*fsrc * np.sqrt(0.5*(eperp - np.sqrt(eperp**2 - eta**2)))
-Ex_theory = 0.37 * np.cos(k_gyro * (z - src_z))    # amplitude estimated by eye
-Ey_theory = 0.37 * np.sin(k_gyro * (z - src_z))
+Ex_theory = 0.37 * np.cos(k_gyro * (z - src_z)).real
+Ey_theory = 0.37 * np.sin(k_gyro * (z - src_z)).real
 
 plt.figure(2)
 plt.subplot(2,1,1)
 plt.plot(z, ex_data, label='Ex (MEEP)')
 plt.plot(z, Ex_theory, 'k--')
-plt.plot(z, -Ex_theory, 'k--', label='Ex (theory)')
+plt.plot(z, -Ex_theory, 'k--', label='Ex envelope (theory)')
 plt.xlim(-L/2, L/2); plt.xlabel('z')
-plt.legend()
+plt.legend(loc='lower right')
 
 plt.subplot(2,1,2)
 plt.plot(z, ey_data, label='Ey (MEEP)')
 plt.plot(z, Ey_theory, 'k--')
-plt.plot(z, -Ey_theory, 'k--', label='Ey (theory)')
+plt.plot(z, -Ey_theory, 'k--', label='Ey envelope (theory)')
 plt.xlim(-L/2, L/2); plt.xlabel('z')
-plt.legend()
+plt.legend(loc='lower right')
 plt.tight_layout()
 plt.show()
 ```
diff --git a/doc/docs/Python_User_Interface.md b/doc/docs/Python_User_Interface.md
index 11f3b3f28..f36b4dafe 100644
--- a/doc/docs/Python_User_Interface.md
+++ b/doc/docs/Python_User_Interface.md
@@ -394,7 +394,7 @@ This is a somewhat unusual polarizable medium, a Lorentzian susceptibility with
 
 ### GyrotropicLorentzianSusceptibility or GyrotropicDrudeSusceptibility
 
-(**Experimental feature**) Specifies a single dispersive gyrotropic susceptibility of Lorentzian (damped harmonic oscillator) or Drude form. See [Material Dispersion](Materials.md#gyrotropic-media). The parameters `sigma`, `frequency`, and `gamma` have the usual meanings, and there is an additional 3-vector `bias`:
+(**Experimental feature**) Specifies a single dispersive gyrotropic susceptibility of Lorentzian (damped harmonic oscillator) or Drude form. See [Material Dispersion](Materials.md#gyrotropic-media). Its parameters are `sigma`, `frequency`, and `gamma`, which have the usual meanings, and an additional 3-vector `bias`:
 
 **`bias` [`Vector3`]**
 —
diff --git a/doc/docs/images/Faraday-rotation-comparison.png b/doc/docs/images/Faraday-rotation-comparison.png
index eb8e33972b828d7a9061de356630b01d78de93f9..9cf49a63fab4128aa310d3bb0452fd5968a4519f 100644
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diff --git a/doc/docs/images/Faraday-rotation.png b/doc/docs/images/Faraday-rotation.png
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diff --git a/python/examples/faraday-rotation.py b/python/examples/faraday-rotation.py
index 28a74f75c..16e17b697 100644
--- a/python/examples/faraday-rotation.py
+++ b/python/examples/faraday-rotation.py
@@ -24,7 +24,7 @@
 sim = mp.Simulation(cell_size=cell, geometry=[], sources=sources,
                     k_point=mp.Vector3(),   # Periodic boundary conditions
                     boundary_layers=pml_layers,
-                    default_material=mat, resolution=100)
+                    default_material=mat, resolution=50)
 sim.run(until=tmax)
 
 ## Plot results:
@@ -43,8 +43,8 @@
 
 ## Comparison with analytic result:
 dfsq = (f0**2 - 1j*fsrc*gamma - fsrc**2)
-eperp = epsn + sn * f0**2 * dfsq / (dfsq**2 + (fsrc*b0)**2)
-eta = sn * f0**2 * fsrc * b0 / (dfsq**2 + (fsrc*b0)**2)
+eperp = epsn + sn * f0**2 * dfsq / (dfsq**2 - (fsrc*b0)**2)
+eta = sn * f0**2 * fsrc * b0 / (dfsq**2 - (fsrc*b0)**2)
 
 k_gyro = 2*np.pi*fsrc * np.sqrt(0.5*(eperp - np.sqrt(eperp**2 - eta**2)))
 Ex_theory = 0.37 * np.cos(k_gyro * (z - src_z)).real
@@ -54,15 +54,15 @@
 plt.subplot(2,1,1)
 plt.plot(z, ex_data, label='Ex (MEEP)')
 plt.plot(z, Ex_theory, 'k--')
-plt.plot(z, -Ex_theory, 'k--', label='Ex (theory)')
+plt.plot(z, -Ex_theory, 'k--', label='Ex envelope (theory)')
 plt.xlim(-L/2, L/2); plt.xlabel('z')
-plt.legend()
+plt.legend(loc='lower right')
 
 plt.subplot(2,1,2)
 plt.plot(z, ey_data, label='Ey (MEEP)')
 plt.plot(z, Ey_theory, 'k--')
-plt.plot(z, -Ey_theory, 'k--', label='Ey (theory)')
+plt.plot(z, -Ey_theory, 'k--', label='Ey envelope (theory)')
 plt.xlim(-L/2, L/2); plt.xlabel('z')
-plt.legend()
+plt.legend(loc='lower right')
 plt.tight_layout()
 plt.show()

From a39cea1bde1b152cef30d55807715a47c5e907af Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Wed, 5 Jun 2019 16:14:18 +0800
Subject: [PATCH 51/61] For Landau-Lifshitz-Gilbert model, ignore the magnitude
 of the bias vector.

---
 src/susceptibility.cpp | 24 ++++++++++++------------
 1 file changed, 12 insertions(+), 12 deletions(-)

diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 1528c57ee..7d3256be9 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -327,10 +327,13 @@ gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double ome
 						     double alpha, gyrotropy_model model)
   : omega_0(omega_0), gamma(gamma), alpha(alpha), model(model) {
   // Precalculate g_{ij} = sum_k epsilon_{ijk} b_k, used in update_P.
+
+  // Ignore |b| for Landau-Lifshitz-Gilbert gyrotropy model:
+  const vec b = (model == GYROTROPIC_SATURATED) ? bias/abs(bias) : bias;
   memset(gyro_tensor, 0, 9 * sizeof(double));
-  gyro_tensor[X][Y] = bias.z(); gyro_tensor[Y][X] = -bias.z();
-  gyro_tensor[Y][Z] = bias.x(); gyro_tensor[Z][Y] = -bias.x();
-  gyro_tensor[Z][X] = bias.y(); gyro_tensor[X][Z] = -bias.y();
+  gyro_tensor[X][Y] = b.z(); gyro_tensor[Y][X] = -b.z();
+  gyro_tensor[Y][Z] = b.x(); gyro_tensor[Z][Y] = -b.x();
+  gyro_tensor[Z][X] = b.y(); gyro_tensor[X][Z] = -b.y();
 }
 
 /* To implement gyrotropic susceptibilities, we track three
@@ -479,14 +482,11 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 
   case GYROTROPIC_SATURATED:
     {
-      const double dwfac = -2*pi*omega2pidt, alppi = pi*alpha;
-      const double dt2pi = 2*pi*dt;
-
       // Precalculate 3x3 matrix inverse, exploiting skew symmetry
       const double gd = 0.5;
-      const double gx = -pi * alpha * gyro_tensor[Y][Z];
-      const double gy = -pi * alpha * gyro_tensor[Z][X];
-      const double gz = -pi * alpha * gyro_tensor[X][Y];
+      const double gx = -0.5 * alpha * gyro_tensor[Y][Z];
+      const double gy = -0.5 * alpha * gyro_tensor[Z][X];
+      const double gz = -0.5 * alpha * gyro_tensor[X][Y];
       const double invdet = 1.0 / gd / (gd*gd + gx*gx + gy*gy + gz*gz);
       const double inv[3][3]
 	= {{ invdet*(gd*gd+gx*gx), invdet*(gx*gy+gd*gz), invdet*(gx*gz-gd*gy) },
@@ -519,9 +519,9 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 	    abort("gyrotropic media do not support anisotropic sigma\n");
 
 	  LOOP_OVER_VOL_OWNED(gv, c, i) {
-	    q0 = dwfac*p0[i] + alppi*pp0[i] + dt2pi*s[i]*w0[i];
-	    q1 = dwfac*p1[i] + alppi*pp1[i] + (w1 ? dt2pi*s[i]*OFFDIAGW(w1,is1,is) : 0);
-	    q2 = dwfac*p2[i] + alppi*pp2[i] + (w2 ? dt2pi*s[i]*OFFDIAGW(w2,is2,is) : 0);
+	    q0 = -omega2pidt*p0[i] + 0.5*alpha*pp0[i] + dt*s[i]*w0[i];
+	    q1 = -omega2pidt*p1[i] + 0.5*alpha*pp1[i] + dt*s[i]*(w1 ? OFFDIAGW(w1,is1,is) : 0);
+	    q2 = -omega2pidt*p2[i] + 0.5*alpha*pp2[i] + dt*s[i]*(w2 ? OFFDIAGW(w2,is2,is) : 0);
 
 	    r0 = 0.5*pp0[i] - g2pidt*p0[i] + gyro_tensor[d0][d1]*q1 + gyro_tensor[d0][d2]*q2;
 	    r1 = 0.5*pp1[i] - g2pidt*p1[i] + gyro_tensor[d1][d2]*q2 + gyro_tensor[d1][d0]*q0;

From 01d9f98969e0e33d6e8de1d5573c7c664587dadd Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Wed, 5 Jun 2019 17:25:18 +0800
Subject: [PATCH 52/61] Fix minor hiccup in docs.

---
 doc/docs/Python_User_Interface.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/doc/docs/Python_User_Interface.md b/doc/docs/Python_User_Interface.md
index 8bb257980..a15e95e97 100644
--- a/doc/docs/Python_User_Interface.md
+++ b/doc/docs/Python_User_Interface.md
@@ -406,7 +406,7 @@ The gyrotropy vector.  The direction of this vector determines the orientation o
 
 **`sigma` [`number`]**
 —
-The coupling factor $\sigma_n / 2\pi$ between the polarization and the driving field. In magnetic ferrites, this is the Larmor precession frequency at the saturation field if `bias` is a unit vector.
+The coupling factor $\sigma_n / 2\pi$ between the polarization and the driving field. In magnetic ferrites, this is the Larmor precession frequency at the saturation field.
 
 **`frequency` [`number`]**
 —

From d2ab1740fb6ab5306bddbf42115ecf2b8043562e Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Thu, 6 Jun 2019 10:45:14 +0800
Subject: [PATCH 53/61] Support dumping and undumping of gyrotropic
 susceptibilities

---
 python/typemap_utils.cpp |  3 +++
 src/meep.hpp             |  2 +-
 src/meepgeom.cpp         | 13 +++++++++----
 src/structure_dump.cpp   | 26 ++++++++++++++++++++++++++
 src/susceptibility.cpp   |  6 +++---
 5 files changed, 42 insertions(+), 8 deletions(-)

diff --git a/python/typemap_utils.cpp b/python/typemap_utils.cpp
index 04f472347..6e3d7029b 100644
--- a/python/typemap_utils.cpp
+++ b/python/typemap_utils.cpp
@@ -476,6 +476,9 @@ static PyObject *susceptibility_to_py_obj(susceptibility_struct *s) {
       PyObject *py_gyrotropic_class = PyObject_GetAttrString(geom_mod, "GyrotropicSaturatedSusceptibility");
       res = PyObject_Call(py_gyrotropic_class, args, NULL);
       Py_DECREF(py_gyrotropic_class);
+      PyObject *py_alpha = PyFloat_FromDouble(s->alpha);
+      PyObject_SetAttrString(res, "alpha", py_alpha);
+      Py_DECREF(py_alpha);
     } else if (s->drude) {
       PyObject *py_gyrotropic_drude_class = PyObject_GetAttrString(geom_mod, "GyrotropicDrudeSusceptibility");
       res = PyObject_Call(py_gyrotropic_drude_class, args, NULL);
diff --git a/src/meep.hpp b/src/meep.hpp
index c1fd244d5..e387006cb 100644
--- a/src/meep.hpp
+++ b/src/meep.hpp
@@ -298,7 +298,7 @@ class gyrotropic_susceptibility : public susceptibility {
 					  int n, void *P_internal_data) const;
 
   virtual void dump_params(h5file *h5f, size_t *start);
-  virtual int get_num_params() { return 7; }
+  virtual int get_num_params() { return 8; }
   virtual bool needs_W_notowned(component c, realnum *W[NUM_FIELD_COMPONENTS][2]) const {
     (void)c;
     (void)W;
diff --git a/src/meepgeom.cpp b/src/meepgeom.cpp
index efab7a6d3..a85a1250d 100644
--- a/src/meepgeom.cpp
+++ b/src/meepgeom.cpp
@@ -38,9 +38,11 @@ void check_offdiag(medium_struct *m) {
 
 bool susceptibility_equal(const susceptibility &s1, const susceptibility &s2) {
   return (vector3_equal(s1.sigma_diag, s2.sigma_diag) &&
-          vector3_equal(s1.sigma_offdiag, s2.sigma_offdiag) && s1.frequency == s2.frequency &&
-          s1.gamma == s2.gamma && s1.noise_amp == s2.noise_amp && s1.drude == s2.drude &&
-          s1.is_file == s2.is_file);
+          vector3_equal(s1.sigma_offdiag, s2.sigma_offdiag) &&
+          vector3_equal(s1.bias, s2.bias) && s1.frequency == s2.frequency &&
+          s1.gamma == s2.gamma && s1.alpha == s2.alpha &&
+          s1.noise_amp == s2.noise_amp && s1.drude == s2.drude &&
+          s1.saturated_gyrotropy == s2.saturated_gyrotropy && s1.is_file == s2.is_file);
 }
 
 bool susceptibility_list_equal(const susceptibility_list &s1, const susceptibility_list &s2) {
@@ -1178,10 +1180,12 @@ double geom_epsilon::conductivity(meep::component c, const meep::vec &r) {
    (must be updated manually, re-copying from ctl-io.cpp), if we
    add new susceptibility subclasses) */
 static bool susceptibility_equiv(const susceptibility *o0, const susceptibility *o) {
+  if (!vector3_equal(o0->bias, o->bias)) return 0;
   if (o0->frequency != o->frequency) return 0;
   if (o0->gamma != o->gamma) return 0;
   if (o0->noise_amp != o->noise_amp) return 0;
   if (o0->drude != o->drude) return 0;
+  if (o0->saturated_gyrotropy != o->saturated_gyrotropy) return 0;
   if (o0->is_file != o->is_file) return 0;
 
   if (o0->transitions != o->transitions) return 0;
@@ -1363,7 +1367,8 @@ void geom_epsilon::add_susceptibilities(meep::field_type ft, meep::structure *s)
     susceptibility *ss = &(p->user_s);
     if (ss->is_file) meep::abort("unknown susceptibility");
     bool noisy = (ss->noise_amp != 0.0);
-    bool gyrotropic = (ss->bias.x != 0.0 || ss->bias.y != 0.0 || ss->bias.z != 0.0);
+    bool gyrotropic = (ss->saturated_gyrotropy || ss->bias.x != 0.0
+                       || ss->bias.y != 0.0 || ss->bias.z != 0.0);
     meep::susceptibility *sus;
 
     if (ss->transitions.size() != 0 || ss->initial_populations.size() != 0) {
diff --git a/src/structure_dump.cpp b/src/structure_dump.cpp
index b7cd91eff..7b9ca14e0 100644
--- a/src/structure_dump.cpp
+++ b/src/structure_dump.cpp
@@ -362,6 +362,32 @@ susceptibility *make_sus_list_from_params(h5file *file, int rank, size_t dims[3]
         res = sus;
       }
       if (sus->next) sus = sus->next;
+    } else if (num_params == 8) {
+      // This is a gyrotropic_susceptibility and the next 8 values in the dataset
+      // are id, bias.x, bias.y, bias.z, omega_0, gamma, alpha, and model.
+      size_t gyro_susc_dims[3] = {8, 0, 0};
+      realnum gyro_susc_params[8];
+      file->read_chunk(rank, &start, gyro_susc_dims, gyro_susc_params);
+      start += gyro_susc_dims[0];
+
+      int id = (int)gyro_susc_params[0];
+      const vec bias(gyro_susc_params[1], gyro_susc_params[2], gyro_susc_params[3]);
+      const double omega_0 = gyro_susc_params[4];
+      const double gamma = gyro_susc_params[5];
+      const double alpha = gyro_susc_params[6];
+      const gyrotropy_model model = (gyrotropy_model)gyro_susc_params[7];
+      if (sus) {
+        sus->next =
+            new gyrotropic_susceptibility(bias, omega_0, gamma, alpha, model);
+        sus->next->ntot = ntot;
+        sus->next->set_id(id);
+      } else {
+        sus = new gyrotropic_susceptibility(bias, omega_0, gamma, alpha, model);
+        sus->ntot = ntot;
+        sus->set_id(id);
+        res = sus;
+      }
+      if (sus->next) sus = sus->next;
     } else {
       abort("Invalid number of susceptibility parameters in structure::load");
     }
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 7d3256be9..9b7c2b2cb 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -572,12 +572,12 @@ realnum *gyrotropic_susceptibility::cinternal_notowned_ptr(int inotowned, compon
 }
 
 void gyrotropic_susceptibility::dump_params(h5file *h5f, size_t *start) {
-  size_t num_params = 8;
+  size_t num_params = 9;
   size_t params_dims[1] = {num_params};
   double bias[] = { gyro_tensor[Y][Z], gyro_tensor[Z][X], gyro_tensor[X][Y] };
   double params_data[] = {
-      7, (double)get_id(), bias[X], bias[Y], bias[Z],
-      omega_0, gamma, (double)model};
+      8, (double)get_id(), bias[X], bias[Y], bias[Z],
+      omega_0, gamma, alpha, (double)model};
   h5f->write_chunk(1, start, params_dims, params_data);
   *start += num_params;
 }

From 329ab8952daf042fc882fc8d0b1b699700f50e14 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Thu, 6 Jun 2019 11:22:42 +0800
Subject: [PATCH 54/61] Doc updates and minor tweaks accompanying last merge

---
 doc/docs/Python_User_Interface.md |  2 +-
 doc/docs/Scheme_User_Interface.md | 18 +++++++++++++-----
 src/susceptibility.cpp            |  3 +--
 3 files changed, 15 insertions(+), 8 deletions(-)

diff --git a/doc/docs/Python_User_Interface.md b/doc/docs/Python_User_Interface.md
index a15e95e97..4611437ff 100644
--- a/doc/docs/Python_User_Interface.md
+++ b/doc/docs/Python_User_Interface.md
@@ -422,7 +422,7 @@ The loss factor $\alpha_n$ in the diagonal response. Note that this parameter is
 
 **`bias` [`Vector3`]**
 —
-Vector specifying the orientation of the gyrotropic response. The magnitude is ignored (unlike the `GyrotropicLorentzianSusceptibility` or `GyrotropicDrudeSusceptibility` case); the relevant precession frequencies are determined by `sigma` and `frequency`, described above.
+Vector specifying the orientation of the gyrotropic response. Unlike the similarly-named `bias` parameter `GyrotropicLorentzianSusceptibility` or `GyrotropicDrudeSusceptibility`, the magnitude is ignored; the relevant precession frequencies are instead determined by `sigma` and `frequency`, described above.
 
 ### Vector3
 
diff --git a/doc/docs/Scheme_User_Interface.md b/doc/docs/Scheme_User_Interface.md
index 1117430af..c03a86483 100644
--- a/doc/docs/Scheme_User_Interface.md
+++ b/doc/docs/Scheme_User_Interface.md
@@ -339,9 +339,17 @@ Specifies a single dispersive susceptibility of Lorentzian (damped harmonic osci
 —
 The noise has root-mean square amplitude σ $\times$ `noise-amp`.
 
-### gyrotropic-susceptibility
+### gyrotropic-lorentzian-susceptibility or gyrotropic-drude-susceptibility
 
-(**Experimental feature**) Specifies a single dispersive [gyrotropic susceptibility](Materials.md#gyrotropic-media) governed by the Landau-Lifshitz-Gilbert equation. Note that the parameters `sigma`, `frequency`, and `gamma` play different roles compared to the Lorentzian or Drude case.
+(**Experimental feature**) Specifies a single dispersive gyrotropic susceptibility of Lorentzian (damped harmonic oscillator) or Drude form. See [Material Dispersion](Materials.md#gyrotropic-media). Its parameters are `sigma`, `frequency`, and `gamma`, which have the usual meanings, and an additional 3-vector `bias`:
+
+**`bias` [`vector3`]**
+—
+The gyrotropy vector.  The direction of this vector determines the orientation of the gyrotropic response, and the magnitude equals the precession frequency $|\mathbf{b}_n|/2\pi$.
+
+### gyrotropic-saturated-susceptibility
+
+(**Experimental feature**) Specifies a single dispersive gyrotropic susceptibility governed by a linearized Landau-Lifshitz-Gilbert equation. See [Material Dispersion](Materials.md#gyrotropic-media). This class takes parameters `sigma`, `frequency`, and `gamma`, whose meanings are different from the Lorentzian and Drude case. It also takes a 3-vector `bias` parameter and an `alpha` parameter:
 
 **`sigma` [`number`]**
 —
@@ -349,7 +357,7 @@ The coupling factor $\sigma_n / 2\pi$ between the polarization and the driving f
 
 **`frequency` [`number`]**
 —
-The frequency of gyrotropic precession, $f_n = \omega_n / 2\pi$. In magnetic ferrites, this is the Larmor precession frequency.
+The Larmor precession frequency, $f_n = \omega_n / 2\pi$.
 
 **`gamma` [`number`]**
 —
@@ -357,11 +365,11 @@ The loss rate $\gamma_n / 2\pi$ in the off-diagonal response.
 
 **`alpha` [`number`]**
 —
-The loss factor $\alpha_n$ in the diagonal response (note that there is no 2π factor).
+The loss factor $\alpha_n$ in the diagonal response. Note that this parameter is dimensionless and contains no 2π factor.
 
 **`bias` [`vector3`]**
 —
-Gyrotropy vector describing the direction of the static biasing magnetic field. The magnitude is ignored.
+Vector specifying the orientation of the gyrotropic response. Unlike the similarly-named `bias` parameter in `gyrotropic-lorentzian-susceptibility` or `gyrotropic-drude-susceptibility`, the magnitude is ignored; the relevant precession frequencies are instead determined by `sigma` and `frequency`, described above.
 
 ### geometric-object
 
diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 9b7c2b2cb..60663ccff 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -327,8 +327,7 @@ gyrotropic_susceptibility::gyrotropic_susceptibility(const vec &bias, double ome
 						     double alpha, gyrotropy_model model)
   : omega_0(omega_0), gamma(gamma), alpha(alpha), model(model) {
   // Precalculate g_{ij} = sum_k epsilon_{ijk} b_k, used in update_P.
-
-  // Ignore |b| for Landau-Lifshitz-Gilbert gyrotropy model:
+  // Ignore |b| for Landau-Lifshitz-Gilbert gyrotropy model.
   const vec b = (model == GYROTROPIC_SATURATED) ? bias/abs(bias) : bias;
   memset(gyro_tensor, 0, 9 * sizeof(double));
   gyro_tensor[X][Y] = b.z(); gyro_tensor[Y][X] = -b.z();

From 256aa892d2e9a2078aa54612c9f87faa732ceb0e Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Wed, 12 Jun 2019 10:03:18 +0800
Subject: [PATCH 55/61] Translate Faraday rotation tutorial from Python to
 Scheme

---
 doc/docs/Python_Tutorials/Gyrotropic_Media.md | 46 ++++------
 doc/docs/Scheme_Tutorials/Gyrotropic_Media.md | 88 +++++++++++++++++++
 python/examples/faraday-rotation.py           | 14 +--
 scheme/examples/faraday-rotation.ctl          | 40 +++++++++
 4 files changed, 151 insertions(+), 37 deletions(-)
 create mode 100644 doc/docs/Scheme_Tutorials/Gyrotropic_Media.md
 create mode 100644 scheme/examples/faraday-rotation.ctl

diff --git a/doc/docs/Python_Tutorials/Gyrotropic_Media.md b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
index a9835f6df..c63313a65 100644
--- a/doc/docs/Python_Tutorials/Gyrotropic_Media.md
+++ b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
@@ -8,14 +8,10 @@ In this example, we will perform simulations with gyrotropic media. See [Materia
 
 ### Faraday Rotation
 
-Consider a uniform gyroelectric medium with bias vector $\mathbf{b} = b \hat{z}$. In the frequency domain, the *x* and *y* components of the dielectric function are
+Consider a uniform gyroelectric medium with bias vector $\mathbf{b} = b \hat{z}$. In the frequency domain, the *x* and *y* components of the dielectric function have the form
 
 $$\epsilon = \begin{bmatrix}\epsilon_\perp & -i\eta \\ i\eta & \epsilon_\perp \end{bmatrix}$$
 
-In the [gyrotropic Lorentzian model](../Materials.md#gyrotropic-media), the tensor components are
-
-$$\epsilon_\perp = \epsilon_\infty + \frac{\omega_n^2 \Delta_n}{\Delta_n^2 - \omega^2 b^2}\,\sigma_n(\mathbf{x}),\;\;\; \eta = \frac{\omega_n^2 \omega b}{\Delta_n^2 - \omega^2 b^2}\,\sigma_n(\mathbf{x}), \;\;\;\Delta_n \equiv \omega_n^2 - \omega^2 - i\omega\gamma_n$$
-
 The skew-symmetric off-diagonal components in ε give rise to [Faraday rotation](https://en.wikipedia.org/wiki/Faraday_effect): when a plane wave linearly polarized along *x* is launched along the gyrotropy axis *z*, the polarization vector will precess around the gyrotropy axis as the wave propagates. This is the principle behind [Faraday rotators](https://en.wikipedia.org/wiki/Faraday_rotator), devices that act as one-way valves for light.
 
 A plane wave undergoing Faraday rotation can be described by the complex ansatz
@@ -30,12 +26,12 @@ We model this phenomenon in the simulation script [faraday-rotation.py](https://
 
 ```import meep as mp
 
-## Define a gyroelectric medium
-f0 = 1.0
-gamma = 1e-6
-epsn = 1.5
-b0 = 0.15
-sn = 0.1
+## Parameters for a gyrotropic Lorentzian medium
+epsn = 1.5    # background permittivity
+f0   = 1.0    # natural frequency
+gamm = 1e-6   # damping rate
+sn   = 0.1    # sigma parameter
+b0   = 0.15   # magnitude of bias vector
 
 susc = [mp.GyrotropicLorentzianSusceptibility(frequency=f0, gamma=gamma, sigma=sigma,
                                               bias=mp.Vector3(0, 0, b0))]
@@ -84,7 +80,13 @@ plt.show()
 ![](../images/Faraday-rotation.png)
 </center>
 
-We see that the wave indeed rotates in the *x*-*y* plane as it travels. This can be compared quantitatively to the above ansatz for a wave undergoing Faraday rotation, using the material parameters to calculate the rotation rate $\kappa_c$:
+We see that the wave indeed rotates in the *x*-*y* plane as it travels.
+
+Moreover, we can compare the Faraday rotation rate in these simulation results to theoretical predictions. In the [gyrotropic Lorentzian model](../Materials.md#gyrotropic-media), the ε tensor components are given by
+
+$$\epsilon_\perp = \epsilon_\infty + \frac{\omega_n^2 \Delta_n}{\Delta_n^2 - \omega^2 b^2}\,\sigma_n(\mathbf{x}),\;\;\; \eta = \frac{\omega_n^2 \omega b}{\Delta_n^2 - \omega^2 b^2}\,\sigma_n(\mathbf{x}), \;\;\;\Delta_n \equiv \omega_n^2 - \omega^2 - i\omega\gamma_n$$
+
+From these expressions, we can calculate the rotation rate $\kappa_c$ at the operating frequency, and hence find the $\mathbf{E}_x$ and $\mathbf{E}_y$ field envelopes for the complex ansatz given at the top of this section.
 
 ```dfsq = (f0**2 - 1j*fsrc*gamma - fsrc**2)
 eperp = epsn + sn * f0**2 * dfsq / (dfsq**2 - (fsrc*b0)**2)
@@ -112,26 +114,8 @@ plt.tight_layout()
 plt.show()
 ```
 
-The results are in excellent agreement:
+As shown in the figure below, the results are in excellent agreement:
 
 <center>
 ![](../images/Faraday-rotation-comparison.png)
 </center>
-
-## Microwave ferrites
-
-Ferrites are ceramic materials exhibiting strong gyromagnetic resonances, commonly used in microwave engineering to construct isolators (one-way valves) and circulators. In the presence of a biasing magnetic field $\mathbf{H} = H_0 \hat{\mathbf{z}}$, the permeability tensor of a ferrite material has the form
-
-$$\mu = \begin{bmatrix} \mu_\perp & -i \kappa & 0 \\ i\kappa & \mu_\perp & 0 \\ 0 & 0 & 1\end{bmatrix}$$
-
-where
-
-$$\mu_\perp = 1 + \frac{\omega_m (\omega_0 - i\omega\alpha)}{(\omega_0 - i \omega \alpha)^2 - (\omega + i \gamma)^2}, \;\;\; \kappa = \frac{\omega_m (\omega + i\gamma)}{(\omega_0 - i \omega \alpha)^2 - (\omega + i \gamma)^2}$$
-
-This permeability can be derived from the [Landau-Lifshitz-Gilbert equation](https://en.wikipedia.org/wiki/Landau%E2%80%93Lifshitz%E2%80%93Gilbert_equation) describing a magnetic dipole in a magnetic field. The Larmor precession frequency is $\omega_0 = \gamma H_0$ where $\gamma$ is the gyromagnetic ratio of the electron. The Larmor precession frequency at saturation field is $\omega_m = \gamma M_s$ where $M_s$ is the saturation magnetization. This precisely matches the permeability tensor derived from [Meep's gyrotropy model](../Materials.md#gyrotropic-media), with the identification $\omega_0 \leftrightarrow \omega_n$ and $\omega_m \leftrightarrow \sigma_n$.
-
-We will take $\omega_0/2\pi = 1$, $\omega_m/2\pi = 1.2$, $\alpha = \gamma/2\pi = 10^{-3}$, and $\epsilon = 15$. The diagonal and off-diagonal components of the permeability tensor are plotted below:
-
-<center>
-![](../images/Ferrite-mu.png)
-</center>
diff --git a/doc/docs/Scheme_Tutorials/Gyrotropic_Media.md b/doc/docs/Scheme_Tutorials/Gyrotropic_Media.md
new file mode 100644
index 000000000..235db22d7
--- /dev/null
+++ b/doc/docs/Scheme_Tutorials/Gyrotropic_Media.md
@@ -0,0 +1,88 @@
+---
+# Gyrotropic media
+---
+
+In this example, we will perform simulations with gyrotropic media. See [Materials](../Materials.md#gyrotropic-media) for more information on how gyrotropy is supported.
+
+[TOC]
+
+### Faraday Rotation
+
+Consider a uniform gyroelectric medium with bias vector $\mathbf{b} = b \hat{z}$. In the frequency domain, the *x* and *y* components of the dielectric function have the form
+
+$$\epsilon = \begin{bmatrix}\epsilon_\perp & -i\eta \\ i\eta & \epsilon_\perp \end{bmatrix}$$
+
+The skew-symmetric off-diagonal components in ε give rise to [Faraday rotation](https://en.wikipedia.org/wiki/Faraday_effect): when a plane wave linearly polarized along *x* is launched along the gyrotropy axis *z*, the polarization vector will precess around the gyrotropy axis as the wave propagates. This is the principle behind [Faraday rotators](https://en.wikipedia.org/wiki/Faraday_rotator), devices that act as one-way valves for light.
+
+A plane wave undergoing Faraday rotation can be described by the complex ansatz
+
+$$\begin{bmatrix}E_x \\ E_y\end{bmatrix} = E_0 \begin{bmatrix}\cos(\kappa_c z) \\ \sin(\kappa_c z)\end{bmatrix} e^{i(kz-\omega t)}$$
+
+where $\kappa_c$ is the Faraday rotation (in radians) per unit of propagation distance. Substituting this into the frequency domain Maxwell's equations, with the above dielectric tensor, yields
+
+$$|\kappa_c| = \omega \sqrt{\frac{\mu}{2} \, \left(\epsilon_\perp - \sqrt{\epsilon_\perp^2 - \eta^2}\right)}$$
+
+We model this phenomenon in the simulation script [faraday-rotation.ctl](https://github.com/NanoComp/meep/blob/master/scheme/examples/faraday-rotation.ctl). First, we define a gyroelectric material:
+
+```(define-param epsn 1.5)    ; background permittivity
+(define-param f0 1.0)      ; natural frequency
+(define-param g0 1e-6)     ; damping rate
+(define-param sn 0.1)      ; sigma parameter
+(define-param b0 0.15)     ; magnitude of bias vector
+
+(set! default-material
+      (make dielectric
+	(epsilon epsn)
+	(E-susceptibilities
+	 (make gyrotropic-lorentzian-susceptibility
+	   (frequency f0)
+	   (sigma sn)
+	   (gamma g0)
+	   (bias (vector3 0 0 b0))))))
+```
+
+The `gyrotropic-lorentzian-susceptibility` object has a `bias` argument that takes a `vector3` specifying the gyrotropy vector. In this case, the vector points along *z*, and its magnitude (which specifies the precession frequency) is determined by the variable `b0`. The other arguments play the same role as in an ordinary (non-gyrotropic) [Lorentzian susceptibility](Material_Dispersion.md).
+
+Next, we set up and run the Meep simulation.
+
+```(define-param tmax 100)
+(define-param L 20.0)
+(define-param fsrc 0.8)
+(define-param src-z -8.5)
+(set-param! resolution 50)
+
+(set! geometry-lattice (make lattice (size 0 0 L)))
+
+(set! pml-layers (list (make pml (thickness 1.0) (direction Z))))
+
+(set! sources (list
+	       (make source
+		 (src (make continuous-src (frequency fsrc)))
+		 (component Ex)
+		 (center (vector3 0 0 src-z)))))
+
+(run-until tmax
+	   (to-appended "efields"
+			(at-end output-efield-x)
+			(at-end output-efield-y)))
+```
+
+The simulation cell is one pixel wide in the *x* and *y* directions, with periodic boundary conditions. PMLs are placed in the *z* direction. A `ContinuousSource` emits a wave whose electric field is initially polarized along *x*.
+
+After running the simulation, the `ex` and `ey` datasets in `faraday-rotation-efields.h5` contain the values of $\mathbf{E}_x$ and $\mathbf{E}_y$. These are plotted against *z* in the figure below:
+
+<center>
+![](../images/Faraday-rotation.png)
+</center>
+
+We see that the wave indeed rotates in the *x*-*y* plane as it travels.
+
+Moreover, we can compare the Faraday rotation rate in these simulation results to theoretical predictions. In the [gyrotropic Lorentzian model](../Materials.md#gyrotropic-media), the ε tensor components are given by
+
+$$\epsilon_\perp = \epsilon_\infty + \frac{\omega_n^2 \Delta_n}{\Delta_n^2 - \omega^2 b^2}\,\sigma_n(\mathbf{x}),\;\;\; \eta = \frac{\omega_n^2 \omega b}{\Delta_n^2 - \omega^2 b^2}\,\sigma_n(\mathbf{x}), \;\;\;\Delta_n \equiv \omega_n^2 - \omega^2 - i\omega\gamma_n$$
+
+From these expressions, we can calculate the rotation rate $\kappa_c$ at the operating frequency, and hence find the $\mathbf{E}_x$ and $\mathbf{E}_y$ field envelopes for the complex ansatz given at the top of this section. As shown in the figure below, the results are in excellent agreement:
+
+<center>
+![](../images/Faraday-rotation-comparison.png)
+</center>
diff --git a/python/examples/faraday-rotation.py b/python/examples/faraday-rotation.py
index 16e17b697..9bba6a35e 100644
--- a/python/examples/faraday-rotation.py
+++ b/python/examples/faraday-rotation.py
@@ -1,11 +1,13 @@
+# From the Meep tutorial: plotting Faraday rotation of a linearly polarized plane wave
+
 import meep as mp
 
-## Define a gyroelectric medium
-f0 = 1.0
-gamma = 1e-6
-epsn = 1.5
-b0 = 0.15
-sn = 0.1
+## Parameters for a gyrotropic Lorentzian medium
+epsn = 1.5    # background permittivity
+f0   = 1.0    # natural frequency
+gamm = 1e-6   # damping rate
+sn   = 0.1    # sigma parameter
+b0   = 0.15   # magnitude of bias vector
 
 susc = [mp.GyrotropicLorentzianSusceptibility(frequency=f0, gamma=gamma, sigma=sn,
                                               bias=mp.Vector3(0, 0, b0))]
diff --git a/scheme/examples/faraday-rotation.ctl b/scheme/examples/faraday-rotation.ctl
new file mode 100644
index 000000000..bd287eda3
--- /dev/null
+++ b/scheme/examples/faraday-rotation.ctl
@@ -0,0 +1,40 @@
+;; From the Meep tutorial: plotting Faraday rotation of a linearly polarized plane wave
+
+;; Parameters for a gyrotropic Lorentzian medium
+(define-param epsn 1.5)    ; background permittivity
+(define-param f0 1.0)      ; natural frequency
+(define-param g0 1e-6)     ; damping rate
+(define-param sn 0.1)      ; sigma parameter
+(define-param b0 0.15)     ; magnitude of bias vector
+
+(set! default-material
+      (make dielectric
+	(epsilon epsn)
+	(E-susceptibilities
+	 (make gyrotropic-lorentzian-susceptibility
+	   (frequency f0)
+	   (sigma sn)
+	   (gamma g0)
+	   (bias (vector3 0 0 b0))))))
+
+;; Set up and run the Meep simulation:
+(define-param tmax 100)
+(define-param L 20.0)
+(define-param fsrc 0.8)
+(define-param src-z -8.5)
+(set-param! resolution 50)
+
+(set! geometry-lattice (make lattice (size 0 0 L)))
+
+(set! pml-layers (list (make pml (thickness 1.0) (direction Z))))
+
+(set! sources (list
+	       (make source
+		 (src (make continuous-src (frequency fsrc)))
+		 (component Ex)
+		 (center (vector3 0 0 src-z)))))
+
+(run-until tmax
+	   (to-appended "efields"
+			(at-end output-efield-x)
+			(at-end output-efield-y)))

From 9b7c6b7385d160fd6a5cdba45f3dd206c0321d69 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Wed, 12 Jun 2019 10:08:42 +0800
Subject: [PATCH 56/61] Fix typo in last change

---
 python/examples/faraday-rotation.py | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/python/examples/faraday-rotation.py b/python/examples/faraday-rotation.py
index 9bba6a35e..b09d1fcec 100644
--- a/python/examples/faraday-rotation.py
+++ b/python/examples/faraday-rotation.py
@@ -3,11 +3,11 @@
 import meep as mp
 
 ## Parameters for a gyrotropic Lorentzian medium
-epsn = 1.5    # background permittivity
-f0   = 1.0    # natural frequency
-gamm = 1e-6   # damping rate
-sn   = 0.1    # sigma parameter
-b0   = 0.15   # magnitude of bias vector
+epsn  = 1.5    # background permittivity
+f0    = 1.0    # natural frequency
+gamma = 1e-6   # damping rate
+sn    = 0.1    # sigma parameter
+b0    = 0.15   # magnitude of bias vector
 
 susc = [mp.GyrotropicLorentzianSusceptibility(frequency=f0, gamma=gamma, sigma=sn,
                                               bias=mp.Vector3(0, 0, b0))]

From 0bc91b22e18fd2f038d4754c0b6debedccbb4843 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Wed, 12 Jun 2019 10:16:42 +0800
Subject: [PATCH 57/61] Fix missing 2pi factor in gyrotropic LLG
 susceptibility's sigma parameter

---
 src/susceptibility.cpp | 8 +++++---
 1 file changed, 5 insertions(+), 3 deletions(-)

diff --git a/src/susceptibility.cpp b/src/susceptibility.cpp
index 60663ccff..f63332095 100644
--- a/src/susceptibility.cpp
+++ b/src/susceptibility.cpp
@@ -481,6 +481,8 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 
   case GYROTROPIC_SATURATED:
     {
+      const double dt2pi = 2*pi*dt;
+
       // Precalculate 3x3 matrix inverse, exploiting skew symmetry
       const double gd = 0.5;
       const double gx = -0.5 * alpha * gyro_tensor[Y][Z];
@@ -518,9 +520,9 @@ void gyrotropic_susceptibility::update_P(realnum *W[NUM_FIELD_COMPONENTS][2],
 	    abort("gyrotropic media do not support anisotropic sigma\n");
 
 	  LOOP_OVER_VOL_OWNED(gv, c, i) {
-	    q0 = -omega2pidt*p0[i] + 0.5*alpha*pp0[i] + dt*s[i]*w0[i];
-	    q1 = -omega2pidt*p1[i] + 0.5*alpha*pp1[i] + dt*s[i]*(w1 ? OFFDIAGW(w1,is1,is) : 0);
-	    q2 = -omega2pidt*p2[i] + 0.5*alpha*pp2[i] + dt*s[i]*(w2 ? OFFDIAGW(w2,is2,is) : 0);
+	    q0 = -omega2pidt*p0[i] + 0.5*alpha*pp0[i] + dt2pi*s[i]*w0[i];
+	    q1 = -omega2pidt*p1[i] + 0.5*alpha*pp1[i] + dt2pi*s[i]*(w1 ? OFFDIAGW(w1,is1,is) : 0);
+	    q2 = -omega2pidt*p2[i] + 0.5*alpha*pp2[i] + dt2pi*s[i]*(w2 ? OFFDIAGW(w2,is2,is) : 0);
 
 	    r0 = 0.5*pp0[i] - g2pidt*p0[i] + gyro_tensor[d0][d1]*q1 + gyro_tensor[d0][d2]*q2;
 	    r1 = 0.5*pp1[i] - g2pidt*p1[i] + gyro_tensor[d1][d2]*q2 + gyro_tensor[d1][d0]*q0;

From 2ee30762dd47d835cfd94a42f5d7961f14f75be4 Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Wed, 12 Jun 2019 13:57:24 +0800
Subject: [PATCH 58/61] Minor fixes for gyrotropy documentation

---
 doc/docs/Materials.md                         | 14 +++----
 doc/docs/Python_Tutorials/Gyrotropic_Media.md | 20 +++++----
 doc/docs/Python_User_Interface.md             |  8 ++--
 doc/docs/Scheme_Tutorials/Gyrotropic_Media.md | 42 ++++++++++---------
 doc/docs/Scheme_User_Interface.md             |  8 ++--
 mkdocs.yml                                    |  2 +
 6 files changed, 51 insertions(+), 43 deletions(-)

diff --git a/doc/docs/Materials.md b/doc/docs/Materials.md
index 1a28f7a5d..5ea4a16d2 100644
--- a/doc/docs/Materials.md
+++ b/doc/docs/Materials.md
@@ -174,7 +174,7 @@ Gyrotropic Media
 
 In a gyrotropic medium, the polarization vector undergoes precession around a preferred direction. In the frequency domain, this corresponds to the presence of skew-symmetric off-diagonal components in the ε tensor (for a gyroelectric medium) or the μ tensor (for a gyromagnetic medium). Two different gyrotropy models are supported:
 
-### Gyrotropic Drude-Lorentz model
+### Gyrotropic Drude-Lorentz Model
 
 The first gyrotropy model is a [Drude-Lorentz](Materials.md#material-dispersion) model with an additional precession, which is intended to describe gyroelectric materials. The polarization equation is
 
@@ -186,7 +186,7 @@ $$\frac{d\mathbf{P}_n}{dt} = \mathbf{P}_n \times \mathbf{b}_n$$
 
 Hence, the magnitude of the bias vector is the angular frequency of the gyrotropic precession induced by the external field.
 
-### Gyrotropic saturated dipole (linearized Landau-Lifshitz-Gilbert) model
+### Gyrotropic Saturated Dipole (Linearized Landau-Lifshitz-Gilbert) Model
 
 The second gyrotropy model is a linearized [Landau-Lifshitz-Gilbert equation](https://en.wikipedia.org/wiki/Landau%E2%80%93Lifshitz%E2%80%93Gilbert_equation), suitable for modeling gyromagnetic materials such as ferrites. Its polarization equation of motion is
 
@@ -194,21 +194,21 @@ $$\frac{d\mathbf{P}_n}{dt} = \mathbf{b}_n \times \left( - \sigma_n \mathbf{E} +
 
 Note: although the above equation is written in terms of electric susceptibilities, this model is typically used for magnetic susceptibilities. Meep places no restriction on the field type that either gyrotropy model can be applied to. As usual, electric and magnetic susceptibilities can be swapped by substituting ε with μ, **E** with **H**, etc.
 
-The Landau-Lifshitz-Gilbert equation describes the precessional motion of a saturated point magnetic dipole in a magnetic field. In the above equation, the variable $P_n$ represents the linearized deviation of the polarization from its static equilibrium value (assumed to be much larger and aligned parallel to $\mathbf{b}_n$). Note that this equation of motion is completely different from the [Drude-Lorentz equation](Materials.md#material-dispersion), though the constants σ$_n$, ω$_n$, and γ$_n$ play analogous roles (σ$_n$ couples the polarization to the driving field, ω$_n$ is the angular frequency of precession, and γ$_n$ is a damping factor).
+The Landau-Lifshitz-Gilbert equation describes the precessional motion of a saturated point magnetic dipole in a magnetic field. In the above equation, the variable $\mathbf{P}_n$ represents the linearized deviation of the polarization from its static equilibrium value (assumed to be much larger and aligned parallel to $\mathbf{b}_n$). Note that this equation of motion is completely different from the [Drude-Lorentz equation](Materials.md#material-dispersion), though the constants σ$_n$, ω$_n$, and γ$_n$ play analogous roles (σ$_n$ couples the polarization to the driving field, ω$_n$ is the angular frequency of precession, and γ$_n$ is a damping factor).
 
 In this model, $\mathbf{b}_n$ is taken to be a unit vector (i.e., its magnitude is ignored).
 
-### Frequency domain susceptibility tensors
+### Frequency Domain Susceptibility Tensors
 
-Suppose $\mathbf{b} = b \hat{z}$, and let all fields have harmonic time-dependence $\exp(-i\omega t)$. Then
+Suppose $\mathbf{b} = b \hat{z}$, and let all fields have harmonic time-dependence $\exp(-i\omega t)$. Then $\mathbf{P}_n$ is related to the applied field $\mathbf{E}$ by
 
 $$\mathbf{P}_n =  \begin{bmatrix}\chi_\perp & -i\eta & 0 \\ i\eta & \chi_\perp & 0 \\ 0 & 0 & \chi_\parallel \end{bmatrix} \mathbf{E}$$
 
-For the gyrotropic Lorentzian model,
+For the [gyrotropic Lorentzian model](Materials.md#gyrotropic-drude-lorentz-model), the components of the susceptibility tensor are
 
 $$\chi_\perp = \frac{\omega_n^2 \Delta_n \sigma_n}{\Delta_n^2 - \omega^2 b^2},\;\;\; \chi_\parallel = \frac{\omega_n^2 \sigma_n}{\Delta_n}, \;\;\; \eta = \frac{\omega_n^2 \omega b \sigma_n}{\Delta_n^2 - \omega^2 b^2}, \;\;\;\Delta_n \equiv \omega_n^2 - \omega^2 - i\omega\gamma_n$$
 
-And for the gyrotropic saturated dipole (linearized Landau-Lifshitz-Gilbert) model,
+And for the [gyrotropic saturated dipole (linearized Landau-Lifshitz-Gilbert) model](Materials.md#gyrotropic-saturated-dipole-linearized-landau-lifshitz-gilbert-model),
 
 $$\chi_\perp = \frac{\sigma_n (\omega_n - i \omega \alpha_n)}{(\omega_n - i \omega \alpha_n)^2 - (\omega + i \gamma_n)^2}, \;\;\; \chi_\parallel = 0, \;\;\;  \eta = \frac{\sigma_n (\omega + i \gamma)}{(\omega_n - i \omega \alpha_n)^2 - (\omega + i \gamma_n)^2}$$
 
diff --git a/doc/docs/Python_Tutorials/Gyrotropic_Media.md b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
index c63313a65..cb16d053d 100644
--- a/doc/docs/Python_Tutorials/Gyrotropic_Media.md
+++ b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
@@ -1,5 +1,5 @@
 ---
-# Gyrotropic media
+# Gyrotropic Media
 ---
 
 In this example, we will perform simulations with gyrotropic media. See [Materials](../Materials.md#gyrotropic-media) for more information on how gyrotropy is supported.
@@ -8,11 +8,11 @@ In this example, we will perform simulations with gyrotropic media. See [Materia
 
 ### Faraday Rotation
 
-Consider a uniform gyroelectric medium with bias vector $\mathbf{b} = b \hat{z}$. In the frequency domain, the *x* and *y* components of the dielectric function have the form
+Consider a uniform gyroelectric medium with bias vector $\mathbf{b} = b \hat{z}$. In the frequency domain, the *x* and *y* components of the dielectric tensor have the form
 
 $$\epsilon = \begin{bmatrix}\epsilon_\perp & -i\eta \\ i\eta & \epsilon_\perp \end{bmatrix}$$
 
-The skew-symmetric off-diagonal components in ε give rise to [Faraday rotation](https://en.wikipedia.org/wiki/Faraday_effect): when a plane wave linearly polarized along *x* is launched along the gyrotropy axis *z*, the polarization vector will precess around the gyrotropy axis as the wave propagates. This is the principle behind [Faraday rotators](https://en.wikipedia.org/wiki/Faraday_rotator), devices that act as one-way valves for light.
+The skew-symmetric off-diagonal components give rise to [Faraday rotation](https://en.wikipedia.org/wiki/Faraday_effect): when a plane wave linearly polarized along *x* is launched along the gyrotropy axis *z*, the polarization vector will precess around the gyrotropy axis as the wave propagates. This is the principle behind [Faraday rotators](https://en.wikipedia.org/wiki/Faraday_rotator), devices that act as one-way valves for light.
 
 A plane wave undergoing Faraday rotation can be described by the complex ansatz
 
@@ -24,7 +24,8 @@ $$|\kappa_c| = \omega \sqrt{\frac{\mu}{2} \, \left(\epsilon_\perp - \sqrt{\epsil
 
 We model this phenomenon in the simulation script [faraday-rotation.py](https://github.com/NanoComp/meep/blob/master/python/examples/faraday-rotation.py). First, we define a gyroelectric material:
 
-```import meep as mp
+```python
+import meep as mp
 
 ## Parameters for a gyrotropic Lorentzian medium
 epsn = 1.5    # background permittivity
@@ -42,7 +43,8 @@ The `GyrotropicLorentzianSusceptibility` object has a `bias` argument that takes
 
 Next, we set up and run the Meep simulation.
 
-```tmax = 100
+```python
+tmax = 100
 L = 20.0
 cell = mp.Vector3(0, 0, L)
 fsrc, src_z = 0.8, -8.5
@@ -58,9 +60,10 @@ sim = mp.Simulation(cell_size=cell, geometry=[], sources=sources,
 sim.run(until=tmax)
 ```
 
-The simulation cell is one pixel wide in the *x* and *y* directions, with periodic boundary conditions. PMLs are placed in the *z* direction. A `ContinuousSource` emits a wave whose electric field is initially polarized along *x*. We then plot the *x* and *y* components of the electric field versus *z*:
+The simulation cell is one pixel wide in the *x* and *y* directions, with periodic boundary conditions. [PMLs](../Perfectly_Matched_Layer.md) are placed in the *z* direction. A `ContinuousSource` emits a wave whose electric field is initially polarized along *x*. We then plot the *x* and *y* components of the electric field versus *z*:
 
-```import numpy as np
+```python
+import numpy as np
 import matplotlib.pyplot as plt
 
 ex_data = sim.get_array(center=mp.Vector3(), size=mp.Vector3(0, 0, L), component=mp.Ex)
@@ -88,7 +91,8 @@ $$\epsilon_\perp = \epsilon_\infty + \frac{\omega_n^2 \Delta_n}{\Delta_n^2 - \om
 
 From these expressions, we can calculate the rotation rate $\kappa_c$ at the operating frequency, and hence find the $\mathbf{E}_x$ and $\mathbf{E}_y$ field envelopes for the complex ansatz given at the top of this section.
 
-```dfsq = (f0**2 - 1j*fsrc*gamma - fsrc**2)
+```python
+dfsq = (f0**2 - 1j*fsrc*gamma - fsrc**2)
 eperp = epsn + sn * f0**2 * dfsq / (dfsq**2 - (fsrc*b0)**2)
 eta = sn * f0**2 * fsrc * b0 / (dfsq**2 - (fsrc*b0)**2)
 
diff --git a/doc/docs/Python_User_Interface.md b/doc/docs/Python_User_Interface.md
index 4611437ff..d8a7c86fe 100644
--- a/doc/docs/Python_User_Interface.md
+++ b/doc/docs/Python_User_Interface.md
@@ -394,15 +394,15 @@ This is a somewhat unusual polarizable medium, a Lorentzian susceptibility with
 
 ### GyrotropicLorentzianSusceptibility or GyrotropicDrudeSusceptibility
 
-(**Experimental feature**) Specifies a single dispersive gyrotropic susceptibility of Lorentzian (damped harmonic oscillator) or Drude form. See [Material Dispersion](Materials.md#gyrotropic-media). Its parameters are `sigma`, `frequency`, and `gamma`, which have the usual meanings, and an additional 3-vector `bias`:
+(**Experimental feature**) Specifies a single dispersive [gyrotropic susceptibility](Materials.md#gyrotropic-media) of [Lorentzian (damped harmonic oscillator) or Drude form](Materials.md#gyrotropic-drude-lorentz-model). Its parameters are `sigma`, `frequency`, and `gamma`, which have the [usual meanings](#susceptibility), and an additional 3-vector `bias`:
 
 **`bias` [`Vector3`]**
 —
-The gyrotropy vector.  The direction of this vector determines the orientation of the gyrotropic response, and the magnitude equals the precession frequency $|\mathbf{b}_n|/2\pi$.
+The gyrotropy vector.  Its direction determines the orientation of the gyrotropic response, and the magnitude is the precession frequency $|\mathbf{b}_n|/2\pi$.
 
 ### GyrotropicSaturatedSusceptibility
 
-(**Experimental feature**) Specifies a single dispersive gyrotropic susceptibility governed by a linearized Landau-Lifshitz-Gilbert equation. See [Material Dispersion](Materials.md#gyrotropic-media). This class takes parameters `sigma`, `frequency`, and `gamma`, whose meanings are different from the Lorentzian and Drude case. It also takes a 3-vector `bias` parameter and an `alpha` parameter:
+(**Experimental feature**) Specifies a single dispersive [gyrotropic susceptibility](Materials.md#gyrotropic-media) governed by a [linearized Landau-Lifshitz-Gilbert equation](Materials.md#gyrotropic-saturated-dipole-linearized-landau-lifshitz-gilbert-model). This class takes parameters `sigma`, `frequency`, and `gamma`, whose meanings are different from the Lorentzian and Drude case. It also takes a 3-vector `bias` parameter and an `alpha` parameter:
 
 **`sigma` [`number`]**
 —
@@ -422,7 +422,7 @@ The loss factor $\alpha_n$ in the diagonal response. Note that this parameter is
 
 **`bias` [`Vector3`]**
 —
-Vector specifying the orientation of the gyrotropic response. Unlike the similarly-named `bias` parameter `GyrotropicLorentzianSusceptibility` or `GyrotropicDrudeSusceptibility`, the magnitude is ignored; the relevant precession frequencies are instead determined by `sigma` and `frequency`, described above.
+Vector specifying the orientation of the gyrotropic response. Unlike the similarly-named `bias` parameter for the [gyrotropic Lorentzian/Drude susceptibilities](#gyrotropiclorentziansusceptibility-or-gyrotropicdrudesusceptibility), the magnitude is ignored; instead, the relevant precession frequencies are determined by the `sigma` and `frequency` parameters.
 
 ### Vector3
 
diff --git a/doc/docs/Scheme_Tutorials/Gyrotropic_Media.md b/doc/docs/Scheme_Tutorials/Gyrotropic_Media.md
index 235db22d7..0164be888 100644
--- a/doc/docs/Scheme_Tutorials/Gyrotropic_Media.md
+++ b/doc/docs/Scheme_Tutorials/Gyrotropic_Media.md
@@ -1,5 +1,5 @@
 ---
-# Gyrotropic media
+# Gyrotropic Media
 ---
 
 In this example, we will perform simulations with gyrotropic media. See [Materials](../Materials.md#gyrotropic-media) for more information on how gyrotropy is supported.
@@ -8,11 +8,11 @@ In this example, we will perform simulations with gyrotropic media. See [Materia
 
 ### Faraday Rotation
 
-Consider a uniform gyroelectric medium with bias vector $\mathbf{b} = b \hat{z}$. In the frequency domain, the *x* and *y* components of the dielectric function have the form
+Consider a uniform gyroelectric medium with bias vector $\mathbf{b} = b \hat{z}$. In the frequency domain, the *x* and *y* components of the dielectric tensor have the form
 
 $$\epsilon = \begin{bmatrix}\epsilon_\perp & -i\eta \\ i\eta & \epsilon_\perp \end{bmatrix}$$
 
-The skew-symmetric off-diagonal components in ε give rise to [Faraday rotation](https://en.wikipedia.org/wiki/Faraday_effect): when a plane wave linearly polarized along *x* is launched along the gyrotropy axis *z*, the polarization vector will precess around the gyrotropy axis as the wave propagates. This is the principle behind [Faraday rotators](https://en.wikipedia.org/wiki/Faraday_rotator), devices that act as one-way valves for light.
+The skew-symmetric off-diagonal components give rise to [Faraday rotation](https://en.wikipedia.org/wiki/Faraday_effect): when a plane wave linearly polarized along *x* is launched along the gyrotropy axis *z*, the polarization vector will precess around the gyrotropy axis as the wave propagates. This is the principle behind [Faraday rotators](https://en.wikipedia.org/wiki/Faraday_rotator), devices that act as one-way valves for light.
 
 A plane wave undergoing Faraday rotation can be described by the complex ansatz
 
@@ -24,7 +24,8 @@ $$|\kappa_c| = \omega \sqrt{\frac{\mu}{2} \, \left(\epsilon_\perp - \sqrt{\epsil
 
 We model this phenomenon in the simulation script [faraday-rotation.ctl](https://github.com/NanoComp/meep/blob/master/scheme/examples/faraday-rotation.ctl). First, we define a gyroelectric material:
 
-```(define-param epsn 1.5)    ; background permittivity
+```scm
+(define-param epsn 1.5)    ; background permittivity
 (define-param f0 1.0)      ; natural frequency
 (define-param g0 1e-6)     ; damping rate
 (define-param sn 0.1)      ; sigma parameter
@@ -32,20 +33,21 @@ We model this phenomenon in the simulation script [faraday-rotation.ctl](https:/
 
 (set! default-material
       (make dielectric
-	(epsilon epsn)
-	(E-susceptibilities
-	 (make gyrotropic-lorentzian-susceptibility
-	   (frequency f0)
-	   (sigma sn)
-	   (gamma g0)
-	   (bias (vector3 0 0 b0))))))
+            (epsilon epsn)
+            (E-susceptibilities
+             (make gyrotropic-lorentzian-susceptibility
+                   (frequency f0)
+                   (sigma sn)
+                   (gamma g0)
+                   (bias (vector3 0 0 b0))))))
 ```
 
 The `gyrotropic-lorentzian-susceptibility` object has a `bias` argument that takes a `vector3` specifying the gyrotropy vector. In this case, the vector points along *z*, and its magnitude (which specifies the precession frequency) is determined by the variable `b0`. The other arguments play the same role as in an ordinary (non-gyrotropic) [Lorentzian susceptibility](Material_Dispersion.md).
 
 Next, we set up and run the Meep simulation.
 
-```(define-param tmax 100)
+```scm
+(define-param tmax 100)
 (define-param L 20.0)
 (define-param fsrc 0.8)
 (define-param src-z -8.5)
@@ -56,18 +58,18 @@ Next, we set up and run the Meep simulation.
 (set! pml-layers (list (make pml (thickness 1.0) (direction Z))))
 
 (set! sources (list
-	       (make source
-		 (src (make continuous-src (frequency fsrc)))
-		 (component Ex)
-		 (center (vector3 0 0 src-z)))))
+               (make source
+                     (src (make continuous-src (frequency fsrc)))
+                     (component Ex)
+                     (center (vector3 0 0 src-z)))))
 
 (run-until tmax
-	   (to-appended "efields"
-			(at-end output-efield-x)
-			(at-end output-efield-y)))
+           (to-appended "efields"
+                        (at-end output-efield-x)
+                        (at-end output-efield-y)))
 ```
 
-The simulation cell is one pixel wide in the *x* and *y* directions, with periodic boundary conditions. PMLs are placed in the *z* direction. A `ContinuousSource` emits a wave whose electric field is initially polarized along *x*.
+The simulation cell is one pixel wide in the *x* and *y* directions, with periodic boundary conditions. [PMLs](../Perfectly_Matched_Layer.md) are placed in the *z* direction. A `ContinuousSource` emits a wave whose electric field is initially polarized along *x*.
 
 After running the simulation, the `ex` and `ey` datasets in `faraday-rotation-efields.h5` contain the values of $\mathbf{E}_x$ and $\mathbf{E}_y$. These are plotted against *z* in the figure below:
 
diff --git a/doc/docs/Scheme_User_Interface.md b/doc/docs/Scheme_User_Interface.md
index c03a86483..871327c48 100644
--- a/doc/docs/Scheme_User_Interface.md
+++ b/doc/docs/Scheme_User_Interface.md
@@ -341,15 +341,15 @@ The noise has root-mean square amplitude σ $\times$ `noise-amp`.
 
 ### gyrotropic-lorentzian-susceptibility or gyrotropic-drude-susceptibility
 
-(**Experimental feature**) Specifies a single dispersive gyrotropic susceptibility of Lorentzian (damped harmonic oscillator) or Drude form. See [Material Dispersion](Materials.md#gyrotropic-media). Its parameters are `sigma`, `frequency`, and `gamma`, which have the usual meanings, and an additional 3-vector `bias`:
+(**Experimental feature**) Specifies a single dispersive [gyrotropic susceptibility](Materials.md#gyrotropic-media) of [Lorentzian (damped harmonic oscillator) or Drude form](Materials.md#gyrotropic-drude-lorentz-model). Its parameters are `sigma`, `frequency`, and `gamma`, which have the [usual meanings](#susceptibility), and an additional 3-vector `bias`:
 
 **`bias` [`vector3`]**
 —
-The gyrotropy vector.  The direction of this vector determines the orientation of the gyrotropic response, and the magnitude equals the precession frequency $|\mathbf{b}_n|/2\pi$.
+The gyrotropy vector.  Its direction determines the orientation of the gyrotropic response, and the magnitude is the precession frequency $|\mathbf{b}_n|/2\pi$.
 
 ### gyrotropic-saturated-susceptibility
 
-(**Experimental feature**) Specifies a single dispersive gyrotropic susceptibility governed by a linearized Landau-Lifshitz-Gilbert equation. See [Material Dispersion](Materials.md#gyrotropic-media). This class takes parameters `sigma`, `frequency`, and `gamma`, whose meanings are different from the Lorentzian and Drude case. It also takes a 3-vector `bias` parameter and an `alpha` parameter:
+(**Experimental feature**) Specifies a single dispersive [gyrotropic susceptibility](Materials.md#gyrotropic-media) governed by a [linearized Landau-Lifshitz-Gilbert equation](Materials.md#gyrotropic-saturated-dipole-linearized-landau-lifshitz-gilbert-model). This class takes parameters `sigma`, `frequency`, and `gamma`, whose meanings are different from the Lorentzian and Drude case. It also takes a 3-vector `bias` parameter and an `alpha` parameter:
 
 **`sigma` [`number`]**
 —
@@ -369,7 +369,7 @@ The loss factor $\alpha_n$ in the diagonal response. Note that this parameter is
 
 **`bias` [`vector3`]**
 —
-Vector specifying the orientation of the gyrotropic response. Unlike the similarly-named `bias` parameter in `gyrotropic-lorentzian-susceptibility` or `gyrotropic-drude-susceptibility`, the magnitude is ignored; the relevant precession frequencies are instead determined by `sigma` and `frequency`, described above.
+Vector specifying the orientation of the gyrotropic response. Unlike the similarly-named `bias` parameter for the [gyrotropic Lorentzian/Drude susceptibilities](#gyrotropiclorentziansusceptibility-or-gyrotropicdrudesusceptibility), the magnitude is ignored; instead, the relevant precession frequencies are determined by the `sigma` and `frequency` parameters.
 
 ### geometric-object
 
diff --git a/mkdocs.yml b/mkdocs.yml
index b3bbe07dc..fc67d59c7 100644
--- a/mkdocs.yml
+++ b/mkdocs.yml
@@ -56,6 +56,7 @@ pages:
     - 'Tutorial/Near-to-Far Field Spectra': 'Python_Tutorials/Near_to_Far_Field_Spectra.md'
     - 'Tutorial/Local Density of States': 'Python_Tutorials/Local_Density_of_States.md'
     - 'Tutorial/Optical Forces': 'Python_Tutorials/Optical_Forces.md'
+    - 'Tutorial/Gyrotropic Media': 'Python_Tutorials/Gyrotropic_Media.md'
     - 'Tutorial/Multilevel Atomic Susceptibility': 'Python_Tutorials/Multilevel_Atomic_Susceptibility.md'
     - 'Tutorial/Frequency-Domain Solver': 'Python_Tutorials/Frequency_Domain_Solver.md'
     - 'Tutorial/Eigenmode Source': 'Python_Tutorials/Eigenmode_Source.md'
@@ -72,6 +73,7 @@ pages:
     - 'Tutorial/Near-to-Far Field Spectra': 'Scheme_Tutorials/Near_to_Far_Field_Spectra.md'
     - 'Tutorial/Local Density of States': 'Scheme_Tutorials/Local_Density_of_States.md'
     - 'Tutorial/Optical Forces': 'Scheme_Tutorials/Optical_Forces.md'
+    - 'Tutorial/Gyrotropic Media': 'Scheme_Tutorials/Gyrotropic_Media.md'
     - 'Tutorial/Multilevel Atomic Susceptibility': 'Scheme_Tutorials/Multilevel_Atomic_Susceptibility.md'
     - 'Tutorial/Frequency-Domain Solver': 'Scheme_Tutorials/Frequency_Domain_Solver.md'
     - 'Tutorial/Eigenmode Source': 'Scheme_Tutorials/Eigenmode_Source.md'

From 356a5070bbc09bf5bfbe2f4819b51baf056e665d Mon Sep 17 00:00:00 2001
From: Chong Yidong <cyd@gnu.org>
Date: Thu, 13 Jun 2019 15:52:53 +0800
Subject: [PATCH 59/61] Add Faraday rotation unit test

---
 doc/docs/Python_Tutorials/Gyrotropic_Media.md |  5 +-
 python/examples/faraday-rotation.py           |  5 +-
 python/tests/faraday_rotation.py              | 94 +++++++++++++++++++
 3 files changed, 98 insertions(+), 6 deletions(-)
 create mode 100644 python/tests/faraday_rotation.py

diff --git a/doc/docs/Python_Tutorials/Gyrotropic_Media.md b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
index cb16d053d..a10250f05 100644
--- a/doc/docs/Python_Tutorials/Gyrotropic_Media.md
+++ b/doc/docs/Python_Tutorials/Gyrotropic_Media.md
@@ -54,7 +54,6 @@ sources = [mp.Source(mp.ContinuousSource(frequency=fsrc),
                      component=mp.Ex, center=mp.Vector3(0, 0, src_z))]
 
 sim = mp.Simulation(cell_size=cell, geometry=[], sources=sources,
-                    k_point=mp.Vector3(),   # Periodic boundary conditions
                     boundary_layers=pml_layers,
                     default_material=mat, resolution=50)
 sim.run(until=tmax)
@@ -66,8 +65,8 @@ The simulation cell is one pixel wide in the *x* and *y* directions, with period
 import numpy as np
 import matplotlib.pyplot as plt
 
-ex_data = sim.get_array(center=mp.Vector3(), size=mp.Vector3(0, 0, L), component=mp.Ex)
-ey_data = sim.get_array(center=mp.Vector3(), size=mp.Vector3(0, 0, L), component=mp.Ey)
+ex_data = sim.get_efield_x().real
+ey_data = sim.get_efield_y().real
 
 z = np.linspace(-L/2, L/2, len(ex_data))
 plt.figure(1)
diff --git a/python/examples/faraday-rotation.py b/python/examples/faraday-rotation.py
index b09d1fcec..449c18b10 100644
--- a/python/examples/faraday-rotation.py
+++ b/python/examples/faraday-rotation.py
@@ -24,7 +24,6 @@
                      component=mp.Ex, center=mp.Vector3(0, 0, src_z))]
 
 sim = mp.Simulation(cell_size=cell, geometry=[], sources=sources,
-                    k_point=mp.Vector3(),   # Periodic boundary conditions
                     boundary_layers=pml_layers,
                     default_material=mat, resolution=50)
 sim.run(until=tmax)
@@ -33,8 +32,8 @@
 import numpy as np
 import matplotlib.pyplot as plt
 
-ex_data = sim.get_array(center=mp.Vector3(), size=mp.Vector3(0, 0, L), component=mp.Ex)
-ey_data = sim.get_array(center=mp.Vector3(), size=mp.Vector3(0, 0, L), component=mp.Ey)
+ex_data = sim.get_efield_x().real
+ey_data = sim.get_efield_y().real
 
 z = np.linspace(-L/2, L/2, len(ex_data))
 plt.figure(1)
diff --git a/python/tests/faraday_rotation.py b/python/tests/faraday_rotation.py
new file mode 100644
index 000000000..61ec9712e
--- /dev/null
+++ b/python/tests/faraday_rotation.py
@@ -0,0 +1,94 @@
+from __future__ import division
+
+import unittest
+import numpy as np
+import meep as mp
+
+## Farady rotation rate for gyrotropic Lorentzian medium
+def kgyro_lorentzian(freq, epsn, f0, gamma, sigma, b0):
+    dfsq = (f0**2 - 1j*freq*gamma - freq**2)
+    eperp = epsn + sigma * f0**2 * dfsq / (dfsq**2 - (freq*b0)**2)
+    eta = sigma * f0**2 * freq * b0 / (dfsq**2 - (freq*b0)**2)
+    return 2*np.pi*freq * np.sqrt(0.5*(eperp - np.sqrt(eperp**2 - eta**2)))
+
+## Farady rotation rate for gyrotropic Drude medium
+def kgyro_drude(freq, epsn, f0, gamma, sigma, b0):
+    dfsq = - 1j*freq*gamma - freq**2
+    eperp = epsn + sigma * f0**2 * dfsq / (dfsq**2 - (freq*b0)**2)
+    eta = sigma * f0**2 * freq * b0 / (dfsq**2 - (freq*b0)**2)
+    return 2*np.pi*freq * np.sqrt(0.5*(eperp - np.sqrt(eperp**2 - eta**2)))
+
+## Farady rotation rate for Landau-Lifshitz-Gilbert medium
+def kgyro_llg(freq, epsn, f0, gamma, sigma, alpha):
+    df1 = f0 - 1j*freq*alpha
+    df2 = freq + 1j*gamma
+    eperp = epsn + sigma * df1/(df1**2 - df2**2)
+    eta = sigma * df2 / (df1**2 - df2**2)
+    return 2*np.pi*freq * np.sqrt(0.5*(eperp - np.sqrt(eperp**2 - eta**2)))
+
+class TestFaradayRotation(unittest.TestCase):
+    ## Simulate a linearly polarized plane wave traveling along the gyrotropy axis.
+    ## Extract Faraday rotation angle by comparing the Ex and Ey amplitudes, and
+    ## compare to the theoretical result predicted by rotation rate KPRED
+    ## up to relative tolerance RTOL.
+    def check_rotation(self, mat, L, fsrc, zsrc, resolution, tmax, zout, kpred, rtol):
+        cell = mp.Vector3(0, 0, L)
+        pml_layers = [mp.PML(thickness=1.0, direction=mp.Z)]
+        sources = [mp.Source(mp.ContinuousSource(frequency=fsrc),
+                             component=mp.Ex, center=mp.Vector3(0, 0, zsrc))]
+
+        self.sim = mp.Simulation(cell_size=cell, geometry=[], sources=sources,
+                                 boundary_layers=pml_layers,
+                                 default_material=mat, resolution=resolution)
+
+        record_vol = mp.Volume(center=mp.Vector3(0, 0, zout))
+        record_Ex, record_Ey, record_t = [], [], []
+
+        def record_ex_ey(sim):
+            record_Ex.append(sim.get_array(vol=record_vol, component=mp.Ex))
+            record_Ey.append(sim.get_array(vol=record_vol, component=mp.Ey))
+            record_t.append(sim.meep_time())
+
+        self.sim.run(mp.after_time(0.5*tmax, mp.at_every(1e-6, record_ex_ey)), until=tmax)
+
+        ex_rel = np.amax(abs(np.fft.fft(record_Ex)))
+        ey_rel = np.amax(abs(np.fft.fft(record_Ey)))
+        result = np.arctan2(ey_rel, ex_rel)
+
+        Ex_theory = np.abs(np.cos(kpred * (zout - zsrc)).real)
+        Ey_theory = np.abs(np.sin(kpred * (zout - zsrc)).real)
+        expected = np.arctan2(Ey_theory, Ex_theory)
+
+        np.testing.assert_allclose(expected, result, rtol=rtol)
+
+    def test_faraday_rotation(self):
+        L, zsrc, zout = 12.0, -4.5, 4.0
+        freq, tmax = 0.8, 300.0
+        resolution = 24
+
+        ## Test gyrotropic lorentzian (2% tolerance)
+        epsn, f0, gamma, sn, b0  = 1.5, 1.0, 1e-3, 0.1, 0.15
+        susc = [mp.GyrotropicLorentzianSusceptibility(frequency=f0, gamma=gamma, sigma=sn,
+                                                      bias=mp.Vector3(0, 0, b0))]
+        mat = mp.Medium(epsilon=epsn, mu=1, E_susceptibilities=susc)
+        k = kgyro_lorentzian(freq, epsn, f0, gamma, sn, b0)
+        self.check_rotation(mat, L, freq, zsrc, resolution, tmax, zout, k, 0.02)
+
+        ## Test gyrotropic Drude medium (2% tolerance)
+        susc = [mp.GyrotropicDrudeSusceptibility(frequency=f0, gamma=gamma, sigma=sn,
+                                                 bias=mp.Vector3(0, 0, b0))]
+        mat = mp.Medium(epsilon=epsn, mu=1, E_susceptibilities=susc)
+        k = kgyro_drude(freq, epsn, f0, gamma, sn, b0)
+        self.check_rotation(mat, L, freq, zsrc, resolution, tmax, zout, k, 0.02)
+
+        ## Test Landau-Lifshitz-Gilbert medium (5% tolerance)
+        alpha = 1e-5
+        susc = [mp.GyrotropicSaturatedSusceptibility(frequency=f0, gamma=gamma, sigma=sn,
+                                                     alpha=alpha,
+                                                     bias=mp.Vector3(0, 0, 1.0))]
+        mat = mp.Medium(epsilon=epsn, mu=1, E_susceptibilities=susc)
+        k = kgyro_llg(freq, epsn, f0, gamma, sn, alpha)
+        self.check_rotation(mat, L, freq, zsrc, resolution, tmax, zout, k, 0.05)
+
+if __name__ == '__main__':
+    unittest.main()

From 5742b8937c15c8805bb0c1dfd4472d16f4cd562d Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Thu, 13 Jun 2019 21:02:52 +0800
Subject: [PATCH 60/61] Use absolute tolerance (in degrees) for Faraday
 rotation unit test

---
 python/tests/faraday_rotation.py | 33 +++++++++++++++++++-------------
 1 file changed, 20 insertions(+), 13 deletions(-)

diff --git a/python/tests/faraday_rotation.py b/python/tests/faraday_rotation.py
index 61ec9712e..140868b5b 100644
--- a/python/tests/faraday_rotation.py
+++ b/python/tests/faraday_rotation.py
@@ -29,9 +29,9 @@ def kgyro_llg(freq, epsn, f0, gamma, sigma, alpha):
 class TestFaradayRotation(unittest.TestCase):
     ## Simulate a linearly polarized plane wave traveling along the gyrotropy axis.
     ## Extract Faraday rotation angle by comparing the Ex and Ey amplitudes, and
-    ## compare to the theoretical result predicted by rotation rate KPRED
-    ## up to relative tolerance RTOL.
-    def check_rotation(self, mat, L, fsrc, zsrc, resolution, tmax, zout, kpred, rtol):
+    ## compare to a theoretical result corresponding to rotation rate KPRED.
+    ## The default acceptable tolerance TOL is 1.5 degrees.
+    def check_rotation(self, mat, L, fsrc, zsrc, resolution, tmax, zout, kpred, tol=1.5):
         cell = mp.Vector3(0, 0, L)
         pml_layers = [mp.PML(thickness=1.0, direction=mp.Z)]
         sources = [mp.Source(mp.ContinuousSource(frequency=fsrc),
@@ -53,42 +53,49 @@ def record_ex_ey(sim):
 
         ex_rel = np.amax(abs(np.fft.fft(record_Ex)))
         ey_rel = np.amax(abs(np.fft.fft(record_Ey)))
-        result = np.arctan2(ey_rel, ex_rel)
+        result = np.arctan2(ey_rel, ex_rel) * 180/np.pi
 
         Ex_theory = np.abs(np.cos(kpred * (zout - zsrc)).real)
         Ey_theory = np.abs(np.sin(kpred * (zout - zsrc)).real)
-        expected = np.arctan2(Ey_theory, Ex_theory)
+        expected = np.arctan2(Ey_theory, Ex_theory) * 180/np.pi
 
-        np.testing.assert_allclose(expected, result, rtol=rtol)
+        print("Rotation angle (in degrees): {}, expected {}\n".format(result, expected))
+        np.testing.assert_allclose(expected, result, atol=tol)
 
     def test_faraday_rotation(self):
         L, zsrc, zout = 12.0, -4.5, 4.0
-        freq, tmax = 0.8, 300.0
+        freq, tmax = 0.8, 100.0
         resolution = 24
 
-        ## Test gyrotropic lorentzian (2% tolerance)
+        ## Test gyrotropic Lorentzian medium
         epsn, f0, gamma, sn, b0  = 1.5, 1.0, 1e-3, 0.1, 0.15
         susc = [mp.GyrotropicLorentzianSusceptibility(frequency=f0, gamma=gamma, sigma=sn,
                                                       bias=mp.Vector3(0, 0, b0))]
         mat = mp.Medium(epsilon=epsn, mu=1, E_susceptibilities=susc)
         k = kgyro_lorentzian(freq, epsn, f0, gamma, sn, b0)
-        self.check_rotation(mat, L, freq, zsrc, resolution, tmax, zout, k, 0.02)
+        print('=' * 24)
+        print("Testing Faraday rotation for gyrotropic Lorentzian model...")
+        self.check_rotation(mat, L, freq, zsrc, resolution, tmax, zout, k)
 
-        ## Test gyrotropic Drude medium (2% tolerance)
+        ## Test gyrotropic Drude medium
         susc = [mp.GyrotropicDrudeSusceptibility(frequency=f0, gamma=gamma, sigma=sn,
                                                  bias=mp.Vector3(0, 0, b0))]
         mat = mp.Medium(epsilon=epsn, mu=1, E_susceptibilities=susc)
         k = kgyro_drude(freq, epsn, f0, gamma, sn, b0)
-        self.check_rotation(mat, L, freq, zsrc, resolution, tmax, zout, k, 0.02)
+        print('=' * 24)
+        print("Testing Faraday rotation for gyrotropic Drude model...")
+        self.check_rotation(mat, L, freq, zsrc, resolution, tmax, zout, k)
 
-        ## Test Landau-Lifshitz-Gilbert medium (5% tolerance)
+        ## Test Landau-Lifshitz-Gilbert medium
         alpha = 1e-5
         susc = [mp.GyrotropicSaturatedSusceptibility(frequency=f0, gamma=gamma, sigma=sn,
                                                      alpha=alpha,
                                                      bias=mp.Vector3(0, 0, 1.0))]
         mat = mp.Medium(epsilon=epsn, mu=1, E_susceptibilities=susc)
         k = kgyro_llg(freq, epsn, f0, gamma, sn, alpha)
-        self.check_rotation(mat, L, freq, zsrc, resolution, tmax, zout, k, 0.05)
+        print('=' * 24)
+        print("Testing Faraday rotation for Landau-Lifshitz-Gilbert model...")
+        self.check_rotation(mat, L, freq, zsrc, resolution, tmax, zout, k)
 
 if __name__ == '__main__':
     unittest.main()

From f649858fa579ad7bfe0b5e7f5d6ec2407b7215ba Mon Sep 17 00:00:00 2001
From: Chong Yidong <seewhydee@gmail.com>
Date: Thu, 13 Jun 2019 23:31:32 +0800
Subject: [PATCH 61/61] Add faraday rotation test to python/Makefile.am

---
 python/Makefile.am | 1 +
 1 file changed, 1 insertion(+)

diff --git a/python/Makefile.am b/python/Makefile.am
index fd1197209..9e9ad2999 100644
--- a/python/Makefile.am
+++ b/python/Makefile.am
@@ -43,6 +43,7 @@ TESTS =                                   \
     $(TEST_DIR)/cyl_ellipsoid.py          \
     $(TEST_DIR)/dft_energy.py             \
     $(TEST_DIR)/dft_fields.py             \
+    $(TEST_DIR)/faraday_rotation.py       \
     $(TEST_DIR)/field_functions.py        \
     $(TEST_DIR)/force.py                  \
     $(TEST_DIR)/fragment_stats.py         \