From fd4b14dae4e5386cdd4608079df2d68768269410 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Mon, 11 May 2020 12:06:48 -0700 Subject: [PATCH 1/2] update Jupyter notebook and Scheme version of optical forces tutorial --- doc/docs/Python_Tutorials/Optical_Forces.md | 55 +++++----- doc/docs/Scheme_Tutorials/Optical_Forces.md | 64 ++++++----- python/examples/parallel-wvgs-force.ipynb | 115 +++++++++----------- python/examples/parallel-wvgs-force.py | 29 +++-- python/examples/parallel-wvgs-mpb.py | 6 +- scheme/examples/parallel-wvgs-force.ctl | 38 +++---- 6 files changed, 140 insertions(+), 167 deletions(-) diff --git a/doc/docs/Python_Tutorials/Optical_Forces.md b/doc/docs/Python_Tutorials/Optical_Forces.md index 82e45a67e..d4f9dfa08 100644 --- a/doc/docs/Python_Tutorials/Optical_Forces.md +++ b/doc/docs/Python_Tutorials/Optical_Forces.md @@ -14,20 +14,20 @@ $$F=-\frac{1}{\omega}\frac{d\omega}{ds}\Bigg\vert_\vec{k}U,$$ where $\omega$ is the mode frequency of the coupled waveguide system, $s$ is the separation distance between the parallel waveguides, $k$ is the conserved wave vector, and $U$ is the total energy of the electromagnetic fields. By convention, negative and positive values correspond to attractive and repulsive forces, respectively. For more details, see [Optics Letters, Vol. 30, pp. 3042-4, 2005](https://www.osapublishing.org/ol/abstract.cfm?uri=ol-30-22-3042). This expression has been shown to be mathematically equivalent to the Maxwell stress tensor in [Optics Express, Vol. 17, pp. 18116-35, 2009](http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-20-18116). We will verify this result in this tutorial. In this particular example, only the fundamental mode with odd mirror-symmetry in $y$ shows the bidirectional force. -It is convenient to normalize the force in order to work with [dimensionless quantities](../Introduction.md#units-in-meep). Since the total transmitted power in the waveguide per unit length is $P=v_gU/L$ where $v_g$ is the group velocity, $U$ is the total energy of the electromagnetic fields (same as before), and $L$ is the waveguide length, we focus instead on the force per unit length per unit power $(F/L)(ac/P)$ where $a$ is the waveguide width/height and $c$ is the speed of light. This dimensionless quantity can be computed in a single simulation. +It is convenient to [normalize the force](../Python_User_Interface.md#force-spectra) in order to work with [dimensionless quantities](../Introduction.md#units-in-meep). Since the total transmitted power in the waveguide per unit length is $P=v_gU/L$ where $v_g$ is the group velocity, $U$ is the total energy of the electromagnetic fields (same as before), and $L$ is the waveguide length, we focus instead on the force per unit length per unit power $(F/L)(ac/P)$ where $a$ is the waveguide width/height and $c$ is the speed of light. This dimensionless quantity can be computed in a single simulation. The gradient force $F$ can be computed using two different methods: (1) using MPB, compute the frequency $\omega$ and group velocity $v_g$ for a given mode over a range of separation distances $s$ and then use a centered [finite-difference](https://en.wikipedia.org/wiki/Finite_difference) scheme to evaluate $F$ using the formula from above, and (2) using Meep, directly compute both the gradient force $F$ and the transmitted power $P$ over the same range of separation distances $s$. This tutorial verifies that (1) and (2) produce equivalent results. The simulation script is in [examples/parallel-wvgs-force.py](https://github.com/NanoComp/meep/blob/master/python/examples/parallel-wvgs-force.py). The notebook is [examples/parallel-wvgs-force.ipynb](https://nbviewer.jupyter.org/github/NanoComp/meep/blob/master/python/examples/parallel-wvgs-force.ipynb). -The main component of the Meep script is the function `parallel_waveguide` which computes the force $F$ and transmitted power $P$ given the waveguide separation distance `s` and parity of the waveguide mode `xodd=True/False`. The eigenmode frequency $\omega$ is computed first using `get_eigenmode` in order to specify the frequency of the `add_force` and `add_flux` monitors. An [`EigenModeSource`](../Python_User_Interface.md#eigenmodesource) is then used to launch the guided mode. Alternatively, a constant-amplitude point/area source can be used to launch the mode but this is less efficient than the [eigenmode source](Eigenmode_Source.md#index-guided-modes-in-a-ridge-waveguide). The waveguide has width/height of $a=1$ μm and a fixed propagation wavevector of $\pi/a$. +The main component of the Meep script is the function `parallel_waveguide` which computes the force $F$ and transmitted power $P$ given the waveguide separation distance `s` and relative phase of the waveguide modes `xodd=True/False`. The eigenmode frequency $\omega$ is computed first using `get_eigenmode` in order to specify the frequency of the `add_force` and `add_flux` monitors. An [`EigenModeSource`](../Python_User_Interface.md#eigenmodesource) with `eig_match_freq=False` is then used to launch the guided mode. Alternatively, a constant-amplitude point/area source can be used to launch the mode but this is less efficient as demonstrated in [Tutorial/Eigenmode Source/Index-Guided Modes in a Ridge Waveguide](Eigenmode_Source.md#index-guided-modes-in-a-ridge-waveguide). The waveguide has width/height of $a=1$ μm and a fixed propagation wavevector of $\pi/a$. ```py import meep as mp import numpy as np import matplotlib.pyplot as plt -resolution = 30 # pixels/μm +resolution = 40 # pixels/μm Si = mp.Medium(index=3.45) @@ -42,8 +42,6 @@ a = 1.0 # waveguide width/height k_point = mp.Vector3(z=0.5) -fcen = 0.2 - def parallel_waveguide(s,xodd): geometry = [mp.Block(center=mp.Vector3(-0.5*(s+a)), size=mp.Vector3(a,a,mp.inf), @@ -58,11 +56,12 @@ def parallel_waveguide(s,xodd): sim = mp.Simulation(resolution=resolution, cell_size=cell, geometry=geometry, + boundary_layers=pml_layers, symmetries=symmetries, k_point=k_point) sim.init_sim() - EigenmodeData = sim.get_eigenmode(fcen, + EigenmodeData = sim.get_eigenmode(0.22, mp.Z, mp.Volume(center=mp.Vector3(), size=mp.Vector3(sx,sy)), 2 if xodd else 1, @@ -70,8 +69,8 @@ def parallel_waveguide(s,xodd): match_frequency=False, parity=mp.ODD_Y) - f = EigenmodeData.freq - print("freq-{}:, {}, {}".format("xodd" if xodd else "xeven", s, f)) + fcen = EigenmodeData.freq + print("freq:, {}, {}, {}".format("xodd" if xodd else "xeven", s, fcen)) sim.reset_meep() @@ -86,32 +85,25 @@ def parallel_waveguide(s,xodd): sim.change_sources(eig_sources) flux_reg = mp.FluxRegion(direction=mp.Z, center=mp.Vector3(), size=mp.Vector3(sx,sy)) - wvg_flux = sim.add_flux(f, 0, 1, flux_reg) + wvg_flux = sim.add_flux(fcen, 0, 1, flux_reg) - force_reg1 = mp.ForceRegion(mp.Vector3(0.5*s), direction=mp.X, weight=1, size=mp.Vector3(y=a)) - force_reg2 = mp.ForceRegion(mp.Vector3(0.5*s+a), direction=mp.X, weight=-1, size=mp.Vector3(y=a)) - wvg_force = sim.add_force(f, 0, 1, force_reg1, force_reg2) + force_reg1 = mp.ForceRegion(mp.Vector3(0.49*s), direction=mp.X, weight=1, size=mp.Vector3(y=sy)) + force_reg2 = mp.ForceRegion(mp.Vector3(0.5*s+1.01*a), direction=mp.X, weight=-1, size=mp.Vector3(y=sy)) + wvg_force = sim.add_force(fcen, 0, 1, force_reg1, force_reg2) - sim.run(until_after_sources=1000) + sim.run(until_after_sources=1500) flux = mp.get_fluxes(wvg_flux)[0] force = mp.get_forces(wvg_force)[0] - print("flux-{}:, {}, {}".format("xodd" if xodd else "xeven", s, flux)) - print("force-{}:, {}, {}".format("xodd" if xodd else "xeven", s, force)) - - sim.reset_meep() - return flux, force + print("data:, {}, {}, {}, {}, {}".format("xodd" if xodd else "xeven", s, flux, force, -force/flux)) - flux = mp.get_fluxes(wvg_flux)[0] - force = mp.get_forces(wvg_force)[0] - sim.reset_meep() return flux, force ``` -There are four important items to note: (1) a single flux surface is used to compute the Poynting flux in $z$ which spans the entire non-PML region of the cell. This is because in the limit of small waveguide separation distance, two separate flux surfaces for each waveguide would overlap and result in overcounting. The total power through a single flux surface need, by symmetry, only be halved in order to determine the value for just one of the two waveguides. (2) Instead of defining a closed, four-sided "box" surrounding the waveguides, the Maxwell stress tensor is computed using just two line monitors oriented in $y$ (to obtain the force in the perpendicular $x$ direction) with `weight` values of `+1`/`-1` to correctly sum the total force. By symmetry, the force in the $y$ direction is zero and need not be computed. (3) Since the `parity`/`eig_parity` parameter of `get_eigenmode`/`EigenModeSource` can only be specified using the $y$ and/or $z$ directions (but *not* $x$, the waveguide separation axis), the `band`/`eig_band` parameter must be set to `1`/`2` to distinguish modes with even/odd $x$-mirror symmetry. (4) A 2d `cell_size` in $xy$ combined with a `k_point` containing a non-zero $z$ component results in a [2d simulation (by default)](../2d_Cell_Special_kz.md). +There are four important items to note: (1) a single flux surface is used to compute the Poynting flux in $z$ which spans the entire non-PML region of the cell. This is because in the limit of small waveguide separation distance, two separate flux surfaces for each waveguide would overlap and result in overcounting. The total power through a single flux surface need, by symmetry, only be halved in order to determine the value for a single waveguide. (2) Instead of defining a closed, four-sided "box" surrounding the waveguides, the Maxwell stress tensor is computed using just two line monitors oriented in $y$ (to obtain the force in the perpendicular $x$ direction) with `weight` values of `+1`/`-1` to correctly sum the total force. The force monitors are placed in the vacuum region adjacent to the waveguide rather than on its surface so that the fields are [second-order accurate](../Subpixel_Smoothing.md). By symmetry, the force in the $y$ direction is zero and need not be computed. (3) Since the `parity`/`eig_parity` parameter of `get_eigenmode`/`EigenModeSource` can only be specified using the $y$ and/or $z$ directions (but *not* $x$, the waveguide separation axis), the `band`/`eig_band` parameter must be set to `1`/`2` to distinguish modes with even/odd $x$-mirror symmetry. (4) A 2d `cell_size` in $xy$ combined with a `k_point` containing a non-zero $z$ component results in a [2d simulation (which is the default)](../2d_Cell_Special_kz.md). -In this example, the fields never decay away to zero. Choosing a runtime is therefore arbitrary but requires some care. A sufficiently long runtime is necessary to obtain the [steady-state response](../FAQ.md#how-do-i-compute-the-steady-state-fields). However, an excessively long runtime will lead to large values for the Fourier-transformed fields used to compute both the flux and the Maxwell stress tensor. Large floating-point numbers may contain [roundoff errors](https://en.wikipedia.org/wiki/Round-off_error) and produce inaccurate results. +In this example, the fields of the guided mode never decay away to zero. [Choosing a runtime](../FAQ.md#checking-convergence) is therefore somewhat arbitrary but requires some care. A sufficiently long runtime is necessary to obtain the [steady-state response](../FAQ.md#how-do-i-compute-the-steady-state-fields). However, an excessively long runtime will lead to large values for the Fourier-transformed fields used to compute both the flux and the Maxwell stress tensor. Large floating-point numbers may contain [roundoff errors](https://en.wikipedia.org/wiki/Round-off_error) and produce inaccurate results. The simulation is run over the range of separation distances $s$ from 0.05 to 1.00 μm in increments of 0.05 μm. The results are compared with those from MPB. This is shown in the top figure. The two methods show good agreement. @@ -119,7 +111,6 @@ The simulation is run over the range of separation distances $s$ from 0.05 to 1. s = np.arange(0.05,1.05,0.05) fluxes_odd = np.zeros(s.size) forces_odd = np.zeros(s.size) - fluxes_even = np.zeros(s.size) forces_even = np.zeros(s.size) @@ -128,16 +119,22 @@ for k in range(len(s)): fluxes_even[k], forces_even[k] = parallel_waveguide(s[k],False) plt.figure(dpi=150) -plt.plot(s,-forces_odd/fluxes_odd,'rs',label='anti symmetric') +plt.plot(s,-forces_odd/fluxes_odd,'rs',label='anti-symmetric') plt.plot(s,-forces_even/fluxes_even,'bo',label='symmetric') plt.grid(True) -plt.xlabel('waveguide separation s/a') +plt.xlabel('waveguide separation s/a') plt.ylabel('optical force (F/L)(ac/P)') plt.legend(loc='upper right') plt.show() ``` -The MPB simulation is in [examples/parallel-wvgs-mpb.py](https://github.com/NanoComp/meep/blob/master/python/examples/parallel-wvgs-mpb.py). There are important differences related to the coordinate dimensions between the MPB and Meep scripts. In the MPB script, the 2d cell is defined in the $yz$ plane, the waveguide propagation axis is $x$, and the waveguide separation axis is $y$. As a consequence, the `num_bands` parameter is always `1` since the $y$ parity of the mode can be defined explicitly (i.e., `run_yodd_zodd` vs. `run_yeven_zodd`). This is different from the Meep script since Meep requires that a 2d cell be defined in the $xy$ plane. MPB has no such requirement. +The following figure shows the $E_y$ mode profile at a waveguide separation distance of 0.1 μm. This figure was generated using the [`plot2D`](../Python_User_Interface.md#data-visualization) routine and shows the source/flux region (red hatches), force monitors (blue lines), and PMLs (green hatches) surrounding the cell. From the force spectra shown above, at this separation distance the anti-symmetric mode is repulsive whereas the symmetric mode is attractive. + +
+![](../images/parallel_wvgs_s0.1.png) +
+ +The MPB simulation is in [examples/parallel-wvgs-mpb.py](https://github.com/NanoComp/meep/blob/master/python/examples/parallel-wvgs-mpb.py). There are important differences related to the coordinate dimensions between the MPB and Meep scripts. In the MPB script, the 2d cell is defined using the $yz$ plane, the waveguide propagation axis is $x$, and the waveguide separation axis is $y$. As a consequence, the `num_bands` parameter is always `1` since the $y$ parity of the mode can be defined explicitly (i.e., `run_yodd_zodd` vs. `run_yeven_zodd`). This is different from the Meep script since Meep requires that a 2d cell be defined in the $xy$ plane. MPB has no such requirement. ```py import meep as mp @@ -181,7 +178,7 @@ def parallel_waveguide(s,yodd): return f,vg -ss = np.arange(0.05,1.05,0.05) +ss = np.arange(0.025,1.075,0.05) f_odd = np.zeros(len(ss)) vg_odd = np.zeros(len(ss)) @@ -204,7 +201,7 @@ force_odd = compute_force(f_odd,vg_odd) force_even = compute_force(f_even,vg_even) plt.figure(dpi=200) -plt.plot(ss[:-1],force_odd,'b-',label='antisymmetric') +plt.plot(ss[:-1],force_odd,'b-',label='anti-symmetric') plt.plot(ss[:-1],force_even,'r-',label='symmetric') plt.xlabel("waveguide separation s/a") plt.ylabel("optical force (F/L)(ac/P)") diff --git a/doc/docs/Scheme_Tutorials/Optical_Forces.md b/doc/docs/Scheme_Tutorials/Optical_Forces.md index 4a6533cd5..98977a857 100644 --- a/doc/docs/Scheme_Tutorials/Optical_Forces.md +++ b/doc/docs/Scheme_Tutorials/Optical_Forces.md @@ -2,27 +2,27 @@ # Optical Forces --- -This tutorial demonstrates Meep's ability to compute classical forces via the [Maxwell stress tensor](https://en.wikipedia.org/wiki/Maxwell_stress_tensor) (MST). Also demonstrated is the [eigenmode source](../Scheme_User_Interface.md#eigenmode-source). The geometry consists of two identical, parallel, silicon waveguides with square cross section in vacuum. A schematic of the geometry is shown below. Due to the parallel orientation of the waveguides, the two modes can be chosen to be either symmetric or anti-symmetric with respect to a mirror-symmetry plane between them. As the two waveguides are brought closer and closer together, their modes increasingly couple and give rise to a gradient force that is *transverse* to the waveguide axis. This is different from [radiation pressure](https://en.wikipedia.org/wiki/Radiation_pressure) which involves momentum exchange between photons and is *longitudinal* in nature. An interesting phenomena that occurs for this coupled system is that the force can be tuned to be either attractive or repulsive depending on the relative phase of the modes. This tutorial will demonstrate this effect. +This tutorial demonstrates Meep's ability to compute classical forces via the [Maxwell stress tensor](https://en.wikipedia.org/wiki/Maxwell_stress_tensor). The geometry consists of two identical, parallel, silicon waveguides with square cross section in vacuum. A schematic of the geometry is shown below. Due to the parallel orientation of the waveguides (propagation axis along $z$ and separation in $x$), the two modes can be chosen to be either symmetric or anti-symmetric with respect to an $x$ mirror-symmetry plane between them. As the two waveguides are brought closer and closer together, their modes couple and give rise to a gradient force that is *transverse* to the waveguide axis (i.e., in the $x$ direction). This is different from [radiation pressure](https://en.wikipedia.org/wiki/Radiation_pressure) which involves momentum exchange between photons and is longitudinal in nature (i.e., along the $z$ direction). An interesting phenomena that occurs for this coupled waveguide system is that the force can be tuned to be either attractive or repulsive depending on the relative phase of the two modes. This tutorial will demonstrate this effect.
![](../images/Waveguide_forces.png)
-The gradient force on each waveguide arising from the evanescent coupling of the two waveguide modes can be computed analytically: +The gradient force $F$ on each waveguide arising from the evanescent coupling of the two waveguide modes can be computed analytically: $$F=-\frac{1}{\omega}\frac{d\omega}{ds}\Bigg\vert_\vec{k}U,$$ -where ω is the mode frequency of the coupled-waveguide system, $s$ is the separation distance between the parallel waveguides, $k$ is the conserved wave vector and $U$ is the total energy of the electromagnetic fields. By convention, negative and positive values correspond to attractive and repulsive forces, respectively. For more details, see [Optics Letters, Vol. 30, pp. 3042-4, 2005](https://www.osapublishing.org/ol/abstract.cfm?uri=ol-30-22-3042). This expression has been shown to be mathematically equivalent to the MST in [Optics Express, Vol. 17, pp. 18116-35, 2009](http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-20-18116). We will verify this result in this tutorial. +where $\omega$ is the mode frequency of the coupled waveguide system, $s$ is the separation distance between the parallel waveguides, $k$ is the conserved wave vector, and $U$ is the total energy of the electromagnetic fields. By convention, negative and positive values correspond to attractive and repulsive forces, respectively. For more details, see [Optics Letters, Vol. 30, pp. 3042-4, 2005](https://www.osapublishing.org/ol/abstract.cfm?uri=ol-30-22-3042). This expression has been shown to be mathematically equivalent to the Maxwell stress tensor in [Optics Express, Vol. 17, pp. 18116-35, 2009](http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-20-18116). We will verify this result in this tutorial. In this particular example, only the fundamental mode with odd mirror-symmetry in $y$ shows the bidirectional force. -It is convenient to normalize the force in order to work with dimensionless quantities. Since the total power transmitted through the waveguide is $P=v_gU/L$ where $v_g$ is the group velocity, $L$ is the waveguide length, and $U$ is defined as before, we focus instead on the force per unit length per unit power $(F/L)(ac/P)$ where $a$ is the waveguide width and $c$ is the speed of light. This dimensionless quantity enables us to compute both the flux and the force in a single simulation. +It is convenient to [normalize the force](../Scheme_User_Interface.md#force-spectra) in order to work with [dimensionless quantities](../Introduction.md#units-in-meep). Since the total transmitted power in the waveguide per unit length is $P=v_gU/L$ where $v_g$ is the group velocity, $U$ is the total energy of the electromagnetic fields (same as before), and $L$ is the waveguide length, we focus instead on the force per unit length per unit power $(F/L)(ac/P)$ where $a$ is the waveguide width/height and $c$ is the speed of light. This dimensionless quantity can be computed in a single simulation. -We can compute the gradient force using two different methods and verify that they are equivalent: (1) using MPB, we compute the frequency and group velocity for a given mode over a range of separation distances and then use a [finite-difference](https://en.wikipedia.org/wiki/Finite_difference) scheme to numerically evaluate the formula from above, and (2) using Meep, we directly compute both the gradient force and the power transmitted through the waveguide for the guided mode over the same range of separation distances. In this particular example, we consider just the fundamental `ODD-Y` mode which shows the bidirectional force. The range of separation distances is from 0.02 to 1.02 μm in increments of 0.02 μm. +The gradient force $F$ can be computed using two different methods: (1) using MPB, compute the frequency $\omega$ and group velocity $v_g$ for a given mode over a range of separation distances $s$ and then use a centered [finite-difference](https://en.wikipedia.org/wiki/Finite_difference) scheme to evaluate $F$ using the formula from above, and (2) using Meep, directly compute both the gradient force $F$ and the transmitted power $P$ over the same range of separation distances $s$. This tutorial verifies that (1) and (2) produce equivalent results. The simulation script is in [examples/parallel-wvgs-force.ctl](https://github.com/NanoComp/meep/blob/master/scheme/examples/parallel-wvgs-force.ctl). ```scm -(set-param! resolution 30) ; pixels/um +(set-param! resolution 40) ; pixels/um (define Si (make medium (index 3.45))) @@ -58,7 +58,7 @@ Next, we set the Bloch-periodic boundary condition for the mode with wavevector (set! k-point (vector3 0 0 0.5)) ``` -Since we do not know apriori what the mode frequency is for a given waveguide separation distance, a preliminary run is required to find this out using [`Harminv`](../Scheme_User_Interface.md#harminv) and a broadband pulsed source. Since the propagating mode never decays away, the runtime is chosen arbitrarily as 200 time units after the pulsed sources have turned off. +Since we do not know apriori what the mode frequency is for a given waveguide separation distance, a preliminary run is required to find this out using [`Harminv`](../Scheme_User_Interface.md#harminv) and a broadband pulsed source. Since the guided mode never decays away, the runtime is chosen arbitrarily as 200 time units after the pulsed sources have turned off. ```scm (define-param fcen 0.22) @@ -77,51 +77,49 @@ Since we do not know apriori what the mode frequency is for a given waveguide se (print "freq:, " s ", " f "\n") ``` -Once we have determined the mode frequency, we then replace the `source` with [`eigenmode-source`](../Scheme_User_Interface.md#eigenmodesource) to perform the main simulation: compute (1) the force on each waveguide due to the mode coupling and (2) the power in the mode. The `eigenmode-source` invokes [MPB](https://mpb.readthedocs.io) to compute the given mode of interest. The mode profile is then imported into Meep for use as the initial source amplitude. This enables a more efficient mode excitation than simply using a point or area source with constant amplitude. For more details on the eigenmode source feature, refer to Section 4.2 ("Incident Fields and Equivalent Currents") in [Chapter 4](http://arxiv.org/abs/arXiv:1301.5366) ("Electromagnetic Wave Source Conditions") of the book [Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology](https://www.amazon.com/Advances-FDTD-Computational-Electrodynamics-Nanotechnology/dp/1608071707). +Once we have determined the mode frequency, we then replace the `source` with [`eigenmode-source`](../Scheme_User_Interface.md#eigenmode-source) to perform the main simulation: compute (1) the force on each waveguide due to the mode coupling and (2) the power in the mode. An `eigenmode-source` with `eig-match-freq?` set to `false` is then used to launch the guided mode. Alternatively, a constant-amplitude point/area source can be used to launch the mode but this is less efficient as demonstrated in [Tutorial/Eigenmode Source/Index-Guided Modes in a Ridge Waveguide](Eigenmode_Source.md#index-guided-modes-in-a-ridge-waveguide). ```scm (reset-meep) (change-sources! (list (make eigenmode-source (src (make gaussian-src (frequency f) (fwidth df))) - (size a a 0) - (center (* -0.5 (+ s a)) 0) - (direction Z) + (size sx sy 0) + (center 0 0 0) + (eig-band (if xodd? 2 1)) (eig-kpoint k-point) - (eig-match-freq? true) - (eig-parity ODD-Y)) - (make eigenmode-source - (src (make gaussian-src (frequency f) (fwidth df))) - (size a a 0) - (center (* 0.5 (+ s a)) 0) - (direction Z) - (eig-kpoint k-point) - (eig-match-freq? true) - (eig-parity ODD-Y) - (amplitude (if xodd? -1.0 1.0))))) + (eig-match-freq? false) + (eig-parity ODD-Y)))) (define wvg-flux (add-flux f 0 1 - (make flux-region (direction Z) (center 0 0) - (size (* 1.2 (+ (* 2 a) s)) (* 1.2 a) 0)))) + (make flux-region (direction Z) + (center 0 0) (size sx sy 0)))) + (define wvg-force (add-force f 0 1 - (make force-region (direction X) (weight +1.0) - (center (* 0.5 s) 0) (size 0 a)) - (make force-region (direction X) (weight -1.0) - (center (+ (* 0.5 s) a) 0) (size 0 a)))) + (make force-region (direction X) (weight +1) + (center (* 0.5 s) 0) (size 0 sy)) + (make force-region (direction X) (weight -1) + (center (+ (* 0.5 s) a) 0) (size 0 sy)))) -(run-sources+ 5000) +(run-sources+ 1500) (display-fluxes wvg-flux) (display-forces wvg-force) ``` -There are three important items to note in the script: (1) We have defined a single flux surface to compute the Poynting flux in $z$ which spans an area slightly larger than both waveguides rather than two separate flux surfaces (one for each waveguide). This is because in the limit of small separation, two flux surfaces overlap whereas the total power through a single flux surface need, by symmetry, only be halved in order to determine the value for just one of the two waveguides. (2) Instead of defining a closed, four-sided "box" surrounding the waveguides for computing the MST, we chose instead to compute the MST along just two $y$-oriented lines (to obtain the force in the $x$ direction) with different `weight` values to correctly sum the total force. By symmetry, we need not consider the force in the $y$ direction. (3) A 2d `geometry-lattice` in $xy$ combined with a `k-point` containing a non-zero $z$ component results in a [2d simulation (by default)](../2d_Cell_Special_kz.md). Since `eig-match-freq?` is `true` for the 2d `eigenmode-source`, its `direction` property must be set to $z$. +There are four important items to note: (1) a single flux surface is used to compute the Poynting flux in $z$ which spans the entire non-PML region of the cell. This is because in the limit of small waveguide separation distance, two separate flux surfaces for each waveguide would overlap and result in overcounting. The total power through a single flux surface need, by symmetry, only be halved in order to determine the value for a single waveguide. (2) Instead of defining a closed, four-sided "box" surrounding the waveguides, the Maxwell stress tensor is computed using just two line monitors oriented in $y$ (to obtain the force in the perpendicular $x$ direction) with `weight` values of `+1`/`-1` to correctly sum the total force. The force monitors are placed in the vacuum region adjacent to the waveguide rather than on its surface so that the fields are [second-order accurate](../Subpixel_Smoothing.md). By symmetry, the force in the $y$ direction is zero and need not be computed. (3) Since the `eig-parity` parameter of `eigenmode-source` can only be specified using the $y$ and/or $z$ directions (but *not* $x$, the waveguide separation axis), the `eig_band` parameter must be set to `1`/`2` to distinguish modes with even/odd $x$-mirror symmetry. (4) A 2d `cell-size` in $xy$ combined with a `k-point` containing a non-zero $z$ component results in a [2d simulation (which is the default)](../2d_Cell_Special_kz.md). -Choosing a suitable runtime requires some care. A large runtime is necessary to obtain the steady-state response but this will also lead to large values for the discrete Fourier-transformed fields used to compute both the flux and the MST. These large values may contain [roundoff errors](https://en.wikipedia.org/wiki/Round-off_error). +In this example, the fields of the guided mode never decay away to zero. [Choosing a runtime](../FAQ.md#checking-convergence) is therefore somewhat arbitrary but requires some care. A sufficiently long runtime is necessary to obtain the [steady-state response](../FAQ.md#how-do-i-compute-the-steady-state-fields). However, an excessively long runtime will lead to large values for the Fourier-transformed fields used to compute both the flux and the Maxwell stress tensor. Large floating-point numbers may contain [roundoff errors](https://en.wikipedia.org/wiki/Round-off_error) and produce inaccurate results. -We run this simulation over the range of separation distances and compare the results to those obtained from MPB. This is shown in the figure above. The two methods show good agreement. +The simulation is run over the range of separation distances $s$ from 0.05 to 1.00 μm in increments of 0.05 μm. The results are compared with those from MPB. This is shown in the top figure. The two methods show good agreement. + +The following figure shows the $E_y$ mode profile at a waveguide separation distance of 0.1 μm. This figure shows the source/flux region (red hatches), force monitors (blue lines), and PMLs (green hatches) surrounding the cell. From the force spectra shown above, at this separation distance the anti-symmetric mode is repulsive whereas the symmetric mode is attractive. + +
+![](../images/parallel_wvgs_s0.1.png) +
-The MPB simulation is in [examples/parallel-wvgs-mpb.ctl](https://github.com/NanoComp/meep/blob/master/scheme/examples/parallel-wvgs-mpb.ctl). Note: since MPB permits symmetries only in the $y$ and $z$ directions, the coordinate axes used in the MPB script to define the waveguide geometry are different than those in the Meep script. In MPB, the propagating axis is $x$ whereas in Meep it is $z$. +The MPB simulation is in [examples/parallel-wvgs-mpb.ctl](https://github.com/NanoComp/meep/blob/master/scheme/examples/parallel-wvgs-mpb.ctl). There are important differences related to the coordinate dimensions between the MPB and Meep scripts. In the MPB script, the 2d cell is defined using the $yz$ plane, the waveguide propagation axis is $x$, and the waveguide separation axis is $y$. As a consequence, the `num-bands` parameter is always `1` since the $y$ parity of the mode can be defined explicitly (i.e., `run-yodd-zodd` vs. `run-yeven-zodd`). This is different from the Meep script since Meep requires that a 2d cell be defined in the $xy$ plane. MPB has no such requirement. ```scm (set-param! resolution 128) ; pixels/μm diff --git a/python/examples/parallel-wvgs-force.ipynb b/python/examples/parallel-wvgs-force.ipynb index 545ce45ff..b3bdbff88 100644 --- a/python/examples/parallel-wvgs-force.ipynb +++ b/python/examples/parallel-wvgs-force.ipynb @@ -6,21 +6,21 @@ "source": [ "## Optical Forces\n", "\n", - "This tutorial demonstrates Meep's ability to compute classical forces via the [Maxwell stress tensor](https://en.wikipedia.org/wiki/Maxwell_stress_tensor) (MST). The geometry consists of two identical, parallel, silicon waveguides with square cross section in vacuum. A schematic of the geometry is shown below. Due to the parallel orientation of the waveguides, the two modes can be chosen to be either symmetric or anti-symmetric with respect to an $x$ mirror-symmetry plane between them. As the two waveguides are brought closer and closer together, their modes increasingly couple and give rise to a gradient force that is *transverse* to the waveguide axis (i.e., in the $x$ direction). This is different from [radiation pressure](https://en.wikipedia.org/wiki/Radiation_pressure) which involves momentum exchange between photons and is *longitudinal* in nature. An interesting phenomena that occurs for this coupled system is that the force can be tuned to be either attractive or repulsive depending on the relative phase of the modes. This tutorial will demonstrate this effect.\n", + "This tutorial demonstrates Meep's ability to compute classical forces via the [Maxwell stress tensor](https://en.wikipedia.org/wiki/Maxwell_stress_tensor). The geometry consists of two identical, parallel, silicon waveguides with square cross section in vacuum. A schematic of the geometry is shown below. Due to the parallel orientation of the waveguides (propagation axis along $z$ and separation in $x$), the two modes can be chosen to be either symmetric or anti-symmetric with respect to an $x$ mirror-symmetry plane between them. As the two waveguides are brought closer and closer together, their modes couple and give rise to a gradient force that is *transverse* to the waveguide axis (i.e., in the $x$ direction). This is different from [radiation pressure](https://en.wikipedia.org/wiki/Radiation_pressure) which involves momentum exchange between photons and is longitudinal in nature (i.e., along the $z$ direction). An interesting phenomena that occurs for this coupled waveguide system is that the force can be tuned to be either attractive or repulsive depending on the relative phase of the two modes. This tutorial will demonstrate this effect.\n", "\n", "![](https://meep.readthedocs.io/en/latest/images/Waveguide_forces.png)\n", "\n", - "The gradient force on each waveguide arising from the evanescent coupling of the two waveguide modes can be computed analytically:\n", + "The gradient force $F$ on each waveguide arising from the evanescent coupling of the two waveguide modes can be computed analytically:\n", "\n", "$$F=-\\frac{1}{\\omega}\\frac{d\\omega}{ds}\\Bigg\\vert_\\vec{k}U,$$\n", "\n", - "where $ω$ is the mode frequency of the coupled-waveguide system, $s$ is the separation distance between the parallel waveguides, $k$ is the conserved wave vector and $U$ is the total energy of the electromagnetic fields. By convention, negative and positive values correspond to attractive and repulsive forces, respectively. For more details, see [Optics Letters, Vol. 30, pp. 3042-4, 2005](https://www.osapublishing.org/ol/abstract.cfm?uri=ol-30-22-3042). This expression has been shown to be mathematically equivalent to the MST in [Optics Express, Vol. 17, pp. 18116-35, 2009](http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-20-18116). We will verify this result in this tutorial. Note: in this particular example, only the fundamental `ODD_Y` mode shows the bidirectional force.\n", + "where $\\omega$ is the mode frequency of the coupled waveguide system, $s$ is the separation distance between the parallel waveguides, $k$ is the conserved wave vector, and $U$ is the total energy of the electromagnetic fields. By convention, negative and positive values correspond to attractive and repulsive forces, respectively. For more details, see [Optics Letters, Vol. 30, pp. 3042-4, 2005](https://www.osapublishing.org/ol/abstract.cfm?uri=ol-30-22-3042). This expression has been shown to be mathematically equivalent to the Maxwell stress tensor in [Optics Express, Vol. 17, pp. 18116-35, 2009](http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-20-18116). We will verify this result in this tutorial. In this particular example, only the fundamental mode with odd mirror-symmetry in $y$ shows the bidirectional force.\n", "\n", - "It is convenient to normalize the force in order to work with dimensionless quantities. Since the total power transmitted through the waveguide is $P=v_gU/L$ where $v_g$ is the group velocity, $L$ is the waveguide length, and $U$ is defined as before, we focus instead on the force per unit length per unit power $(F/L)(ac/P)$ where $a$ is the waveguide width and $c$ is the speed of light. This dimensionless quantity enables us to compute both the flux and the force in a single simulation.\n", + "It is convenient to [normalize the force](https://meep.readthedocs.io/en/latest/Python_User_Interface/#force-spectra) in order to work with [dimensionless quantities](https://meep.readthedocs.io/en/latest/Introduction/#units-in-meep). Since the total transmitted power in the waveguide per unit length is $P=v_gU/L$ where $v_g$ is the group velocity, $U$ is the total energy of the electromagnetic fields (same as before), and $L$ is the waveguide length, we focus instead on the force per unit length per unit power $(F/L)(ac/P)$ where $a$ is the waveguide width/height and $c$ is the speed of light. This dimensionless quantity can be computed in a single simulation.\n", "\n", - "The gradient force can be computed using two different methods: (1) using MPB, compute the frequency and group velocity for a given mode over a range of separation distances and then use a centered [finite-difference](https://en.wikipedia.org/wiki/Finite_difference) scheme to numerically evaluate the formula from above, and (2) using Meep, directly compute both the gradient force and the power transmitted through the waveguide for the guided mode over the same range of separation distances. This tutorial verifies that (1) and (2) produce equivalent results.\n", + "The gradient force $F$ can be computed using two different methods: (1) using MPB, compute the frequency $\\omega$ and group velocity $v_g$ for a given mode over a range of separation distances $s$ and then use a centered [finite-difference](https://en.wikipedia.org/wiki/Finite_difference) scheme to evaluate $F$ using the formula from above, and (2) using Meep, directly compute both the gradient force $F$ and the transmitted power $P$ over the same range of separation distances $s$. This tutorial verifies that (1) and (2) produce equivalent results.\n", "\n", - "The main component of the script is the function `parallel_waveguide` which computes the Poynting flux and the force given the waveguide separation distance `s` and parity of the waveguide mode `xodd`. Since the eigenmode frequency is not known apriori, a preliminary [`Harminv`](../Python_User_Interface.md#harminv) run is required using a broadband pulsed source. The propagating mode never decays away and the runtime is therefore chosen arbitrarily as 200 time units after the pulsed sources have turned off. Once we have determined the eigenmode frequency, we then replace the `Source` with [`EigenModeSource`](https://meep.readthedocs.io/en/latest/Python_User_Interface/#eigenmodesource) to compute: (1) the force on each waveguide due to the mode coupling and (2) the power in the mode. The [eigenmode source](https://meep.readthedocs.io/en/latest/Python_Tutorials/Eigenmode_Source/) enables a more efficient mode excitation than simply using a constant-amplitude point/area source." + "The main component of the Meep script is the function `parallel_waveguide` which computes the force $F$ and transmitted power $P$ given the waveguide separation distance `s` and relative phase of the waveguide modes `xodd=True/False`. The eigenmode frequency $\\omega$ is computed first using `get_eigenmode` in order to specify the frequency of the `add_force` and `add_flux` monitors. An [`EigenModeSource`](https://meep.readthedocs.io/en/latest/Python_User_Interface/#eigenmodesource) with `eig_match_freq=False` is then used to launch the guided mode. Alternatively, a constant-amplitude point/area source can be used to launch the mode but this is less efficient as demonstrated in [Tutorial/Eigenmode Source/Index-Guided Modes in a Ridge Waveguide](https://meep.readthedocs.io/en/latest/Python_Tutorials/Eigenmode_Source/#index-guided-modes-in-a-ridge-waveguide). The waveguide has width/height of $a=1$ μm and a fixed propagation wavevector of $\\pi/a$." ] }, { @@ -33,24 +33,21 @@ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", - "resolution = 30 # pixels/μm\n", - " \n", + "resolution = 40 # pixels/μm\n", + "\n", "Si = mp.Medium(index=3.45)\n", "\n", "dpml = 1.0\n", "pml_layers = [mp.PML(dpml)]\n", - " \n", + "\n", "sx = 5\n", "sy = 3\n", "cell = mp.Vector3(sx+2*dpml,sy+2*dpml,0)\n", "\n", - "a = 1.0 # waveguide width\n", + "a = 1.0 # waveguide width/height\n", "\n", "k_point = mp.Vector3(z=0.5)\n", "\n", - "fcen = 0.22\n", - "df = 0.06\n", - "\n", "def parallel_waveguide(s,xodd):\n", " geometry = [mp.Block(center=mp.Vector3(-0.5*(s+a)),\n", " size=mp.Vector3(a,a,mp.inf),\n", @@ -59,65 +56,52 @@ " size=mp.Vector3(a,a,mp.inf),\n", " material=Si)]\n", "\n", - " symmetries = [mp.Mirror(mp.X, phase=-1.0 if xodd else 1.0),\n", - " mp.Mirror(mp.Y, phase=-1.0)]\n", - "\n", - " sources = [mp.Source(src=mp.GaussianSource(fcen, fwidth=df),\n", - " component=mp.Ey,\n", - " center=mp.Vector3(-0.5*(s+a)),\n", - " size=mp.Vector3(a,a)),\n", - " mp.Source(src=mp.GaussianSource(fcen, fwidth=df),\n", - " component=mp.Ey,\n", - " center=mp.Vector3(0.5*(s+a)),\n", - " size=mp.Vector3(a,a),\n", - " amplitude=-1.0 if xodd else 1.0)]\n", + " symmetries = [mp.Mirror(mp.X, phase=-1 if xodd else 1),\n", + " mp.Mirror(mp.Y, phase=-1)]\n", "\n", " sim = mp.Simulation(resolution=resolution,\n", " cell_size=cell,\n", - " boundary_layers=pml_layers,\n", " geometry=geometry,\n", + " boundary_layers=pml_layers,\n", " symmetries=symmetries,\n", - " k_point=k_point,\n", - " sources=sources)\n", - "\n", - " h = mp.Harminv(mp.Ey, mp.Vector3(0.5*(s+a)), fcen, df)\n", - "\n", - " sim.run(mp.after_sources(h), until_after_sources=200)\n", + " k_point=k_point)\n", + " \n", + " sim.init_sim()\n", + " EigenmodeData = sim.get_eigenmode(0.22,\n", + " mp.Z,\n", + " mp.Volume(center=mp.Vector3(), size=mp.Vector3(sx,sy)),\n", + " 2 if xodd else 1,\n", + " k_point,\n", + " match_frequency=False,\n", + " parity=mp.ODD_Y)\n", "\n", - " f = h.modes[0].freq\n", - " print(\"freq:, {}, {}\".format(s, f))\n", + " fcen = EigenmodeData.freq\n", + " print(\"freq:, {}, {}, {}\".format(\"xodd\" if xodd else \"xeven\", s, fcen))\n", "\n", " sim.reset_meep()\n", "\n", - " eig_sources = [mp.EigenModeSource(src=mp.GaussianSource(f, fwidth=df),\n", - " size=mp.Vector3(a,a),\n", - " center=mp.Vector3(-0.5*(s+a)),\n", - " direction=mp.Z,\n", - " eig_kpoint=k_point,\n", - " eig_match_freq=True,\n", - " eig_parity=mp.ODD_Y),\n", - " mp.EigenModeSource(src=mp.GaussianSource(f, fwidth=df),\n", - " size=mp.Vector3(a,a),\n", - " center=mp.Vector3(0.5*(s+a)),\n", - " direction=mp.Z,\n", + " eig_sources = [mp.EigenModeSource(src=mp.GaussianSource(fcen, fwidth=0.1*fcen),\n", + " size=mp.Vector3(sx,sy),\n", + " center=mp.Vector3(),\n", + " eig_band=2 if xodd else 1,\n", " eig_kpoint=k_point,\n", - " eig_match_freq=True,\n", - " eig_parity=mp.ODD_Y,\n", - " amplitude=-1.0 if xodd else 1.0)]\n", + " eig_match_freq=False,\n", + " eig_parity=mp.ODD_Y)]\n", "\n", " sim.change_sources(eig_sources)\n", "\n", - " flux_reg = mp.FluxRegion(direction=mp.Z, center=mp.Vector3(), size=mp.Vector3(1.2*(2*a+s),1.2*a))\n", - " wvg_flux = sim.add_flux(f, 0, 1, flux_reg)\n", + " flux_reg = mp.FluxRegion(direction=mp.Z, center=mp.Vector3(), size=mp.Vector3(sx,sy))\n", + " wvg_flux = sim.add_flux(fcen, 0, 1, flux_reg)\n", "\n", - " force_reg1 = mp.ForceRegion(mp.Vector3(0.5*s), direction=mp.X, weight=1.0, size=mp.Vector3(y=a))\n", - " force_reg2 = mp.ForceRegion(mp.Vector3(0.5*s+a), direction=mp.X, weight=-1.0, size=mp.Vector3(y=a))\n", - " wvg_force = sim.add_force(f, 0, 1, force_reg1, force_reg2)\n", + " force_reg1 = mp.ForceRegion(mp.Vector3(0.49*s), direction=mp.X, weight=1, size=mp.Vector3(y=sy))\n", + " force_reg2 = mp.ForceRegion(mp.Vector3(0.5*s+1.01*a), direction=mp.X, weight=-1, size=mp.Vector3(y=sy))\n", + " wvg_force = sim.add_force(fcen, 0, 1, force_reg1, force_reg2)\n", "\n", - " sim.run(until_after_sources=500)\n", + " sim.run(until_after_sources=1500)\n", "\n", " flux = mp.get_fluxes(wvg_flux)[0]\n", " force = mp.get_forces(wvg_force)[0]\n", + " print(\"data:, {}, {}, {}, {}, {}\".format(\"xodd\" if xodd else \"xeven\", s, flux, force, -force/flux))\n", " \n", " sim.reset_meep()\n", " return flux, force" @@ -127,11 +111,11 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "There are three important items to note in `parallel_waveguide`: (1) a single flux surface is used to compute the Poynting flux in $z$ which spans an area slightly larger than both waveguides rather than two separate flux surfaces (one for each waveguide). This is because in the limit of small separation, two flux surfaces overlap whereas the total power through a single flux surface need, by symmetry, only be halved in order to determine the value for just one of the two waveguides. (2) Instead of defining a closed, four-sided \"box\" surrounding the waveguides, the MST is computed along just two $y$-oriented lines (to obtain the force in the $x$ direction) with different `weight` values to correctly sum the total force. By symmetry, the force in the $y$ direction is zero and need not be computed. (3) A 2d `cell_size` in $xy$ combined with a `k_point` containing a non-zero $z$ component results in a [2d simulation (by default)](https://meep.readthedocs.io/en/latest/2d_Cell_Special_kz/). Since `eig_match_freq=True` for the 2d `EigenModeSource`, its `direction` property must be set to $z$. \n", + "There are four important items to note: (1) a single flux surface is used to compute the Poynting flux in $z$ which spans the entire non-PML region of the cell. This is because in the limit of small waveguide separation distance, two separate flux surfaces for each waveguide would overlap and result in overcounting. The total power through a single flux surface need, by symmetry, only be halved in order to determine the value for a single waveguide. (2) Instead of defining a closed, four-sided \"box\" surrounding the waveguides, the Maxwell stress tensor is computed using just two line monitors oriented in $y$ (to obtain the force in the perpendicular $x$ direction) with `weight` values of `+1`/`-1` to correctly sum the total force. The force monitors are placed in the vacuum region adjacent to the waveguide rather than on its surface so that the fields are [second-order accurate](https://meep.readthedocs.io/en/latest/Subpixel_Smoothing/). By symmetry, the force in the $y$ direction is zero and need not be computed. (3) Since the `parity`/`eig_parity` parameter of `get_eigenmode`/`EigenModeSource` can only be specified using the $y$ and/or $z$ directions (but *not* $x$, the waveguide separation axis), the `band`/`eig_band` parameter must be set to `1`/`2` to distinguish modes with even/odd $x$-mirror symmetry. (4) A 2d `cell_size` in $xy$ combined with a `k_point` containing a non-zero $z$ component results in a [2d simulation (which is the default)](https://meep.readthedocs.io/en/latest/2d_Cell_Special_kz/).\n", "\n", - "Choosing a suitable runtime requires some care. A large runtime is necessary to obtain the steady-state response but this will also lead to large values for the discrete Fourier-transformed fields used to compute both the flux and the MST. Large floating-point numbers may contain [roundoff errors](https://en.wikipedia.org/wiki/Round-off_error).\n", + "In this example, the fields of the guided mode never decay away to zero. [Choosing a runtime](https://meep.readthedocs.io/en/latest/FAQ/#checking-convergence) is therefore somewhat arbitrary but requires some care. A sufficiently long runtime is necessary to obtain the [steady-state response](https://meep.readthedocs.io/en/latest/FAQ/#how-do-i-compute-the-steady-state-fields). However, an excessively long runtime will lead to large values for the Fourier-transformed fields used to compute both the flux and the Maxwell stress tensor. Large floating-point numbers may contain [roundoff errors](https://en.wikipedia.org/wiki/Round-off_error) and produce inaccurate results.\n", "\n", - "The simulation is run over the range of separation distances from 0.1 to 1.00 μm in increments of 0.1 μm. The results are compared with those from MPB. This is shown in the figure above. The two methods show good agreement." + "The simulation is run over the range of separation distances $s$ from 0.1 to 1.0 μm in increments of 0.1 μm. The results are compared with those from MPB. This is shown in the top figure. The two methods show good agreement." ] }, { @@ -2338,7 +2322,6 @@ "s = np.arange(0.1,1.1,0.1)\n", "fluxes_odd = np.zeros(s.size)\n", "forces_odd = np.zeros(s.size)\n", - "\n", "fluxes_even = np.zeros(s.size)\n", "forces_even = np.zeros(s.size)\n", "\n", @@ -2367,7 +2350,7 @@ ], "source": [ "plt.figure(dpi=150)\n", - "plt.plot(s,-forces_odd/fluxes_odd,'rs',label='anti symmetric')\n", + "plt.plot(s,-forces_odd/fluxes_odd,'rs',label='anti-symmetric')\n", "plt.plot(s,-forces_even/fluxes_even,'bo',label='symmetric')\n", "plt.grid(True)\n", "plt.xlabel('waveguide separation s/a')\n", @@ -2380,7 +2363,13 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "The MPB simulation is below. Note: since MPB permits symmetries only in the $y$ and $z$ directions, the coordinate axes used in the MPB script to define the waveguide geometry are different than those in the Meep script. In MPB, the propagating axis is $x$ whereas in Meep it is $z$." + "The following figure shows the $E_y$ mode profile at a waveguide separation distance of 0.1 μm. This figure was generated using the [`plot2D`](https://meep.readthedocs.io/en/latest/Python_User_Interface/#data-visualization) routine and shows the source/flux region (red hatches), force monitors (blue lines), and PMLs (green hatches) surrounding the cell. From the force spectra shown above, at this separation distance the anti-symmetric mode is repulsive whereas the symmetric mode is attractive.\n", + "\n", + "
\n", + "![](https://meep.readthedocs.io/en/latest/images/parallel_wvgs_s0.1.png)\n", + "
\n", + "\n", + "The MPB simulation is in [examples/parallel-wvgs-mpb.py](https://github.com/NanoComp/meep/blob/master/python/examples/parallel-wvgs-mpb.py). There are important differences related to the coordinate dimensions between the MPB and Meep scripts. In the MPB script, the 2d cell is defined using the $yz$ plane, the waveguide propagation axis is $x$,and the waveguide separation axis is $y$. As a consequence, the `num_bands` parameter is always `1` since the $y$ parity of the mode can be defined explicitly (i.e., `run_yodd_zodd` vs. `run_yeven_zodd`). This is different from the Meep script since Meep requires that a 2d cell be defined in the $xy$ plane. MPB has no such requirement." ] }, { @@ -3232,7 +3221,7 @@ "\n", " return f,vg\n", "\n", - "ss = np.arange(0.1,1.1,0.1)\n", + "ss = np.arange(0.05,1.15,0.1)\n", "\n", "f_odd = np.zeros(len(ss))\n", "vg_odd = np.zeros(len(ss))\n", @@ -3249,7 +3238,7 @@ " f_avg = 0.5*(f[:-1]+f[1:])\n", " df = f[1:]-f[:-1]\n", " vg_avg = 0.5*(vg[:-1]+vg[1:])\n", - " return np.multiply(np.multiply(-1/f_avg,df/ds), 1/vg_avg)\n", + " return -1/f_avg * df/ds * 1/vg_avg\n", "\n", "force_odd = compute_force(f_odd,vg_odd)\n", "force_even = compute_force(f_even,vg_even)" @@ -3275,7 +3264,7 @@ ], "source": [ "plt.figure(dpi=200)\n", - "plt.plot(ss[:-1],force_odd,'b-',label='antisymmetric')\n", + "plt.plot(ss[:-1],force_odd,'b-',label='anti-symmetric')\n", "plt.plot(ss[:-1],force_even,'r-',label='symmetric')\n", "plt.xlabel(\"waveguide separation s/a\")\n", "plt.ylabel(\"optical force (F/L)(ac/P)\")\n", diff --git a/python/examples/parallel-wvgs-force.py b/python/examples/parallel-wvgs-force.py index ff78e3b93..c7a3e2a42 100644 --- a/python/examples/parallel-wvgs-force.py +++ b/python/examples/parallel-wvgs-force.py @@ -2,7 +2,7 @@ import numpy as np import matplotlib.pyplot as plt -resolution = 30 # pixels/μm +resolution = 40 # pixels/μm Si = mp.Medium(index=3.45) @@ -17,8 +17,6 @@ k_point = mp.Vector3(z=0.5) -fcen = 0.2 - def parallel_waveguide(s,xodd): geometry = [mp.Block(center=mp.Vector3(-0.5*(s+a)), size=mp.Vector3(a,a,mp.inf), @@ -33,11 +31,12 @@ def parallel_waveguide(s,xodd): sim = mp.Simulation(resolution=resolution, cell_size=cell, geometry=geometry, + boundary_layers=pml_layers, symmetries=symmetries, k_point=k_point) sim.init_sim() - EigenmodeData = sim.get_eigenmode(fcen, + EigenmodeData = sim.get_eigenmode(0.22, mp.Z, mp.Volume(center=mp.Vector3(), size=mp.Vector3(sx,sy)), 2 if xodd else 1, @@ -45,8 +44,8 @@ def parallel_waveguide(s,xodd): match_frequency=False, parity=mp.ODD_Y) - f = EigenmodeData.freq - print("freq-{}:, {}, {}".format("xodd" if xodd else "xeven", s, f)) + fcen = EigenmodeData.freq + print("freq:, {}, {}, {}".format("xodd" if xodd else "xeven", s, fcen)) sim.reset_meep() @@ -61,18 +60,17 @@ def parallel_waveguide(s,xodd): sim.change_sources(eig_sources) flux_reg = mp.FluxRegion(direction=mp.Z, center=mp.Vector3(), size=mp.Vector3(sx,sy)) - wvg_flux = sim.add_flux(f, 0, 1, flux_reg) + wvg_flux = sim.add_flux(fcen, 0, 1, flux_reg) - force_reg1 = mp.ForceRegion(mp.Vector3(0.5*s), direction=mp.X, weight=1, size=mp.Vector3(y=a)) - force_reg2 = mp.ForceRegion(mp.Vector3(0.5*s+a), direction=mp.X, weight=-1, size=mp.Vector3(y=a)) - wvg_force = sim.add_force(f, 0, 1, force_reg1, force_reg2) + force_reg1 = mp.ForceRegion(mp.Vector3(0.49*s), direction=mp.X, weight=1, size=mp.Vector3(y=sy)) + force_reg2 = mp.ForceRegion(mp.Vector3(0.5*s+1.01*a), direction=mp.X, weight=-1, size=mp.Vector3(y=sy)) + wvg_force = sim.add_force(fcen, 0, 1, force_reg1, force_reg2) - sim.run(until_after_sources=5000) + sim.run(until_after_sources=1500) flux = mp.get_fluxes(wvg_flux)[0] force = mp.get_forces(wvg_force)[0] - print("flux-{}:, {}, {}".format("xodd" if xodd else "xeven", s, flux)) - print("force-{}:, {}, {}".format("xodd" if xodd else "xeven", s, force)) + print("data:, {}, {}, {}, {}, {}".format("xodd" if xodd else "xeven", s, flux, force, -force/flux)) sim.reset_meep() return flux, force @@ -81,7 +79,6 @@ def parallel_waveguide(s,xodd): s = np.arange(0.05,1.05,0.05) fluxes_odd = np.zeros(s.size) forces_odd = np.zeros(s.size) - fluxes_even = np.zeros(s.size) forces_even = np.zeros(s.size) @@ -90,10 +87,10 @@ def parallel_waveguide(s,xodd): fluxes_even[k], forces_even[k] = parallel_waveguide(s[k],False) plt.figure(dpi=150) -plt.plot(s,-forces_odd/fluxes_odd,'rs',label='anti symmetric') +plt.plot(s,-forces_odd/fluxes_odd,'rs',label='anti-symmetric') plt.plot(s,-forces_even/fluxes_even,'bo',label='symmetric') plt.grid(True) -plt.xlabel('waveguide separation s/a') +plt.xlabel('waveguide separation s/a') plt.ylabel('optical force (F/L)(ac/P)') plt.legend(loc='upper right') plt.show() diff --git a/python/examples/parallel-wvgs-mpb.py b/python/examples/parallel-wvgs-mpb.py index e4e09fe60..1436b908c 100644 --- a/python/examples/parallel-wvgs-mpb.py +++ b/python/examples/parallel-wvgs-mpb.py @@ -41,7 +41,7 @@ def parallel_waveguide(s,yodd): return f,vg -ss = np.arange(0.05,1.05,0.05) +ss = np.arange(0.025,1.075,0.05) f_odd = np.zeros(len(ss)) vg_odd = np.zeros(len(ss)) @@ -58,13 +58,13 @@ def compute_force(f,vg): f_avg = 0.5*(f[:-1]+f[1:]) df = f[1:]-f[:-1] vg_avg = 0.5*(vg[:-1]+vg[1:]) - return np.multiply(np.multiply(-1/f_avg,df/ds), 1/vg_avg) + return -1/f_avg * df/ds * 1/vg_avg force_odd = compute_force(f_odd,vg_odd) force_even = compute_force(f_even,vg_even) plt.figure(dpi=200) -plt.plot(ss[:-1],force_odd,'b-',label='antisymmetric') +plt.plot(ss[:-1],force_odd,'b-',label='anti-symmetric') plt.plot(ss[:-1],force_even,'r-',label='symmetric') plt.xlabel("waveguide separation s/a") plt.ylabel("optical force (F/L)(ac/P)") diff --git a/scheme/examples/parallel-wvgs-force.ctl b/scheme/examples/parallel-wvgs-force.ctl index 0f76cd2f0..74614618e 100644 --- a/scheme/examples/parallel-wvgs-force.ctl +++ b/scheme/examples/parallel-wvgs-force.ctl @@ -1,4 +1,4 @@ -(set-param! resolution 30) ; pixels/um +(set-param! resolution 40) ; pixels/um (define Si (make medium (index 3.45))) @@ -10,7 +10,7 @@ (set! geometry-lattice (make lattice (size (+ sx (* 2 dpml)) (+ sy (* 2 dpml)) no-size))) -(define-param a 1.0) ; waveguide width +(define-param a 1.0) ; waveguide width/height (define-param s 1.0) ; waveguide separation distance (set! geometry (list (make block (center (* -0.5 (+ s a)) 0) @@ -44,32 +44,24 @@ (change-sources! (list (make eigenmode-source (src (make gaussian-src (frequency f) (fwidth df))) - (size a a 0) - (center (* -0.5 (+ s a)) 0) - (direction Z) + (size sx sy 0) + (center 0 0 0) + (eig-band (if xodd? 2 1)) (eig-kpoint k-point) - (eig-match-freq? true) - (eig-parity ODD-Y)) - (make eigenmode-source - (src (make gaussian-src (frequency f) (fwidth df))) - (size a a 0) - (center (* 0.5 (+ s a)) 0) - (direction Z) - (eig-kpoint k-point) - (eig-match-freq? true) - (eig-parity ODD-Y) - (amplitude (if xodd? -1.0 1.0))))) + (eig-match-freq? false) + (eig-parity ODD-Y)))) (define wvg-flux (add-flux f 0 1 - (make flux-region (direction Z) (center 0 0) - (size (* 1.2 (+ (* 2 a) s)) (* 1.2 a) 0)))) + (make flux-region (direction Z) + (center 0 0) (size sx sy 0)))) + (define wvg-force (add-force f 0 1 - (make force-region (direction X) (weight +1.0) - (center (* 0.5 s) 0) (size 0 a)) - (make force-region (direction X) (weight -1.0) - (center (+ (* 0.5 s) a) 0) (size 0 a)))) + (make force-region (direction X) (weight +1) + (center (* 0.5 s) 0) (size 0 sy)) + (make force-region (direction X) (weight -1) + (center (+ (* 0.5 s) a) 0) (size 0 sy)))) -(run-sources+ 5000) +(run-sources+ 1500) (display-fluxes wvg-flux) (display-forces wvg-force) From 2ec83aa478e69e60d7c20f9d45317d6ab303cd58 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Mon, 11 May 2020 12:11:57 -0700 Subject: [PATCH 2/2] new figure showing mode profile --- doc/docs/images/parallel_wvgs_s0.1.png | Bin 0 -> 349971 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 doc/docs/images/parallel_wvgs_s0.1.png diff --git a/doc/docs/images/parallel_wvgs_s0.1.png b/doc/docs/images/parallel_wvgs_s0.1.png new file mode 100644 index 0000000000000000000000000000000000000000..85a916e1af23a2521627c9f3c1e708ce3b82f387 GIT binary patch literal 349971 zcma&NWmJ{j_ceUz?(RlXx=T`!mO6BIcY}0ENJ+P%fRuDdHwZ{dcS=i1y_@^@f4;xh z7!Kjowa?yb%{Av-ix@SP*BGdzs1OJQLqT5VEd&A!g+So?kP*Qn&HL$>;6HeGDFsbr 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