Python tutorial example for broadband eigenmode sources (#1238)

* Python tutorial example for broadband eigenmode sources

* Update Eigenmode_Source.md

* Update Eigenmode_Source.md

* Update Eigenmode_Source.md

Co-authored-by: Steven G. Johnson <stevenj@mit.edu>

doc/docs/Python_Tutorials/Eigenmode_Source.md

@@ -94,6 +94,110 @@ else:
 
 Note that in `EigenModeSource` as well as `get_eigenmode_coefficients`, the `direction` property must be set to `NO_DIRECTION` for a non-zero `eig_kpoint` which specifies the waveguide axis.
 
+### What Happens When the Source Time Profile is a Pulse?
+
+The eigenmode source launches a fixed spatial mode profile specified by either its frequency (`eig_match_freq=True`) or wavevector (`eig_kpoint`) multiplied by the time profile.  When the time profile of the source has a finite bandwidth, e.g. a [Gaussian pulse](Python_User_Interface.md#gaussiansource) (which is typical for calculations involving [Fourier-transformed fields](FAQ.md#for-calculations-involving-fourier-transformed-fields-why-should-the-source-be-a-pulse-rather-than-a-continuous-wave) such as [mode coefficients or S-parameters](Python_Tutorials/GDSII_Import.md)), then the frequency-dependence (dispersion) of the true modal pattern means that the eigenmode source does not match the desired mode exactly over the whole bandwidth.  This is described in section 4.2.2 of our ["Electromagnetic Wave Source Conditions"](https://arxiv.org/abs/1301.5366) review article.   A more accurate mode profile may be obtained by adding multiple narrow-band eigenmode sources at the same position at several frequencies across the bandwidth, but this has the disadvantage that the runtime increases as you add more frequency points due to the narrower source bandwidths.  However, a *single* broadband eigenmode source is often sufficient for most practical applications (excepting cases with extreme modal dispersion, e.g. near a cutoff frequency).
+
+This can be demonstrated by computing the error in a broadband eigenmode source via the backward-propagating and scattered power (i.e., any fields which are not forward-propagating waveguide modes) for the single and multi mode ridge waveguides.
+
+```py
+import meep as mp
+import numpy as np
+import matplotlib.pyplot as plt
+
+resolution = 50 # pixels/μm
+
+sxy = 10
+cell_size = mp.Vector3(sxy,sxy)
+
+dpml = 1
+pml_layers = [mp.PML(thickness=dpml)]
+
+w = 1.0 # width of waveguide
+
+geometry = [mp.Block(center=mp.Vector3(),
+                     size=mp.Vector3(mp.inf,w,mp.inf),
+                     material=mp.Medium(epsilon=12))]
+
+fsrc = 0.35 # frequency of eigenmode source
+kx = 1.2    # initial guess for wavevector in x-direction of eigenmode
+bnum = 1    # band number of eigenmode
+
+df = 0.1    # frequency width
+nfreq = 51  # number of frequencies
+
+symmetries = [mp.Mirror(mp.Y)]
+
+sources = [mp.EigenModeSource(src=mp.GaussianSource(fsrc,fwidth=df),
+                              center=mp.Vector3(0,0),
+                              size=mp.Vector3(0,sxy),
+                              direction=mp.NO_DIRECTION,
+                              eig_parity=mp.EVEN_Y+mp.ODD_Z,
+                              eig_kpoint=mp.Vector3(kx),
+                              eig_band=bnum,
+                              eig_match_freq=True)]
+
+sim = mp.Simulation(cell_size=cell_size,
+                    resolution=resolution,
+                    boundary_layers=pml_layers,
+                    sources=sources,
+                    geometry=geometry,
+                    symmetries=symmetries)
+
+flux_mon_tp = sim.add_flux(fsrc,df,nfreq,mp.FluxRegion(center=mp.Vector3(0,+0.5*sxy-dpml),size=mp.Vector3(sxy-2*dpml,0),weight=+1))
+flux_mon_bt = sim.add_flux(fsrc,df,nfreq,mp.FluxRegion(center=mp.Vector3(0,-0.5*sxy+dpml),size=mp.Vector3(sxy-2*dpml,0),weight=-1))
+flux_mon_rt = sim.add_flux(fsrc,df,nfreq,mp.FluxRegion(center=mp.Vector3(+0.5*sxy-dpml,0),size=mp.Vector3(0,sxy-2*dpml),weight=+1))
+flux_mon_lt = sim.add_flux(fsrc,df,nfreq,mp.FluxRegion(center=mp.Vector3(-0.5*sxy+dpml,0),size=mp.Vector3(0,sxy-2*dpml),weight=-1))
+
+sim.run(until_after_sources=50)
+
+freqs = mp.get_flux_freqs(flux_mon_tp)
+flux_tp = np.asarray(mp.get_fluxes(flux_mon_tp))
+flux_bt = np.asarray(mp.get_fluxes(flux_mon_bt))
+flux_rt = np.asarray(mp.get_fluxes(flux_mon_rt))
+flux_lt = np.asarray(mp.get_fluxes(flux_mon_lt))
+
+res = sim.get_eigenmode_coefficients(flux_mon_rt,[bnum],eig_parity=mp.EVEN_Y+mp.ODD_Z)
+coeffs = res.alpha[0,:,0]
+guided = np.power(np.abs(coeffs),2)
+
+flux_total = flux_tp + flux_bt + flux_rt + flux_lt
+back = flux_lt / flux_total
+scat = (flux_total - guided) / flux_total
+
+if mp.am_master():
+    fig = plt.figure()
+    plt.subplot(1,2,1)
+    plt.plot(freqs,back,'bo-')
+    plt.xlabel('frequency')
+    plt.ylabel('backward power (fraction of total power)')
+    plt.subplot(1,2,2)
+    plt.plot(freqs,scat,'ro-')
+    plt.xlabel('frequency')
+    plt.ylabel('scattered power (fraction of total power)')
+    fig.subplots_adjust(wspace=0.5, hspace=0)
+    fig.suptitle("multi mode waveguide with pulsed eigenmode source\n center frequency = {}, band = {} (mode B)".format(fsrc,bnum))
+    plt.savefig('multi_mode_eigsource_B.png',dpi=150,bbox_inches='tight')
+```
+
+Results are shown for the single mode waveguide with one eigenmode **A** (band 1) and multi mode waveguide with two eigenmodes **A** (higher-order mode, band 2) and **B** (fundamental mode, band 1), all with a center frequency of `0.35`.
+
+<center>
+![](../images/single_mode_eigsource_pulse.png)
+</center>
+
+<center>
+![](../images/multi_mode_eigsource_pulse_A.png)
+</center>
+
+<center>
+![](../images/multi_mode_eigsource_pulse_B.png)
+</center>
+
+These results demonstrate that in all cases the error is nearly 0 at the center frequency and increases roughly quadratically away from the center frequency. The error tends to be smallest for single-mode waveguides because a localized source excitation couples most strongly into guided modes. Note that in this case the maximum error is ~1% for a source bandwidth that is 67% of its center frequency.  For the multi-mode waveguide, a much larger scattering loss is obtained for the higher-order mode **A** at frequencies below the center frequency, but this is simply because that mode ceases to be guided around a frequency `≈ 0.3`, and the mode pattern changes dramatically as this cutoff is approached.
+
+Another thing to keep in mind is that, even if the modes are imperfectly launched (some power leaks into radiation or into the backward direction), correct result can still be obtained by the standard procedure of normalizing against a separate straight-waveguide run (and subtracting off fields before computing reflected fluxes), as explained in the [Meep Introduction](Introduction.md#transmittancereflectance-spectra).   More accurate mode launching may be required for multi-mode waveguide systems, however, as power coupled into an undesired *guided* mode will not typically decay away.
+
 ### Oblique Waveguides
 
 The eigenmode source can also be used to launch modes in an oblique/rotated waveguide. The results are shown in the two figures below for the single- and multi-mode case. There is one subtlety: for mode **A** in the multi-mode case, the `bnum` parameter is set to 3 rather than 2. This is because a non-zero rotation angle breaks the symmetry in the $y$-direction which therefore precludes the use of `EVEN_Y` in `eig_parity`. Without any parity specified for the $y$-direction, the second band corresponds to *odd* modes. This is why we must select the third band which contains even modes.