Custom Source

This tutorial demonstrates using a custom source to define a source with an arbitrary time profile.

Stochastic Dipole Emission in Light Emitting Diodes

In addition to the two source time profiles of a continuous wave (CW) and finite-bandwidth pulse, Meep supports a custom source for defining an arbitrary time profile. This feature can be used to model spatially incoherent random (i.e., white noise) dipole emission in a light-emitting diode (LED), spontaneously recombining excitonic emission in an organic light-emitting diode (OLED), as well as near-field heat transfer. Such incoherent emission processes are very different from coherent emission by a single collection of sources with a fixed phase relation, and more complicated modeling techniques are required.

This tutorial example involves computing the radiated Poynting flux from $N=10$ dipole emitters of a 2d LED-like periodic structure with a thin (1d) light-emitting layer. A schematic of the unit cell geometry and simulation layout is shown below. A silver (Ag) back reflector is used to direct nearly all the flux upwards into the $+y$ direction. PML is used to terminate the air region at the top of the cell. (Note: PML is not necessary at the bottom of the cell due to the Ag layer which is effectively a lossless mirror with skin depth of a few nanometers at a wavelength of 1 μm.) The emitting layer is placed within a lossless dielectric substrate with wavelength-independent refractive index of 3.45.

Exploiting Linear Time Invariance of the Materials and Geometry

One can take two different approaches to computing the radiated flux based on the type of emitter: (1) random or (2) deterministic. In Method 1 (brute-force Monte Carlo), each emitter is a white-noise dipole: every timestep for every dipole is an independent random number. A single run involves all $N$ dipoles which are modeled using a CustomSource. The stochastic results for the radiated flux are averaged over multiple trials/iterations via Monte Carlo sampling. Method 2 exploits the property of linear time-invariance of the materials/geometry and involves a sequence of $N$ separate runs each with a single deterministic dipole (i.e., pulse time profile, GaussianSource) at different positions in the emitting layer. Because dipoles at different positions are uncorrelated, the radiated flux from the ensemble is simply the average of all the individual iterations. (The interference terms between different dipoles integrate to zero when averaging over all possible phases.) The two approaches converge towards identical results, but Method 1 is more computationally expensive than Method 2 due to the much larger number of trials/iterations ($\gg N$) required to attain low noise variance. (Even more sophisticated deterministic methods exist to reduce the number of separate simulations, especially at high resolutions; for example, replacing the point-dipole sources with a rapidly converging set of smooth basis functions as demonstrated below, or fancier methods that exploit trace-estimation methods and/or transform volumetric sources to surface sources.)

In principle, to compute the total power emitted upwards in all directions, we must also average over all possible Bloch wavevectors $k_x \in [-\pi/s_x,+\pi/s_x]$ (e.g. see this paper; this is also called an "array-scanning" method). For simplicity, however, in this tutorial we only compute the average for $k_x=0$ (i.e., the portion of the power in all diffraction orders for $k_x=0$, including normal emission). Conversely, if one is only interested in the emitted power in a single direction, e.g. normal emission, then it turns out to be possible to compute the net effect using only a single "reciprocal" simulation as reviewed in Yao (2022) in a general setting or in Jansen (2010) for the LED case in particular.

Note regarding normalization: To directly compare the results for the radiated flux from the two methods, one might scale the spectrum from Method 2 in post processing to correct for the difference in spectrum between a Gaussian pulse and white noise. However, it is usually more convenient to nondimensionalize the results (for both methods) by dividing the flux spectrum for the textured surface with a reference spectrum computed by the same method, for example emission from a flat surface or in a homogeneous medium. This way, the details of the source spectra cancel automatically, and the nondimensionalized results can be compared as-is without any tricky scaling calculations. This kind of nondimensionalized comparison is useful to determine the emission enhancement (or suppression) of one structure relative to another as well as the light-extraction efficiency (the ratio of the radiated flux to the total flux emitted by the dipoles). In order to compute the absolute (not relative) light emission by a particular structure, using either Method 1 or Method 2, one would need to rescale the output (thanks to linearity) to convert the input spectrum (from white noise or Gaussian) to the actual emission spectrum (e.g. determined from the gain spectrum of a light-emitting diode).

The simulation script is in examples/stochastic_emitter.py. The notebook is examples/stochastic_emitter.ipynb.

import meep as mp
from meep.materials import Ag
import numpy as np

import argparse
parser = argparse.ArgumentParser()
parser.add_argument('-nr', type=int, default=20, help='number of random trials (method 1)')
parser.add_argument('-nd', type=int, default=10, help='number of dipoles')
parser.add_argument('-nf', type=int, default=500, help='number of frequencies')
parser.add_argument('-textured', action='store_true', default=False, help='flat (default) or textured surface')
parser.add_argument('-method', type=int, choices=[1,2], default=1, help='type of method: (1) random dipoles with nr trials or (2) single dipole with 1 run per dipole')
args = parser.parse_args()

resolution = args.res

dpml = 1.0
dair = 1.0
hrod = 0.7
wrod = 0.5
dsub = 5.0
dAg = 0.5

sx = 1.1
sy = dpml+dair+hrod+dsub+dAg

cell_size = mp.Vector3(sx,sy)

pml_layers = [mp.PML(direction=mp.Y,
thickness=dpml,
side=mp.High)]

fcen = 1.0
df = 0.2
nfreq = args.nf
ndipole = args.nd
ntrial = args.nr
run_time = 2*nfreq/df

geometry = [mp.Block(material=mp.Medium(index=3.45),
center=mp.Vector3(0,0.5*sy-dpml-dair-hrod-0.5*dsub),
size=mp.Vector3(mp.inf,dsub,mp.inf)),
mp.Block(material=Ag,
center=mp.Vector3(0,-0.5*sy+0.5*dAg),
size=mp.Vector3(mp.inf,dAg,mp.inf))]

if args.textured:
geometry.append(mp.Block(material=mp.Medium(index=3.45),
center=mp.Vector3(0,0.5*sy-dpml-dair-0.5*hrod),
size=mp.Vector3(wrod,hrod,mp.inf)))

def compute_flux(m=1,n=0):
if m == 1:
sources = []
for n in range(ndipole):
sources.append(mp.Source(mp.CustomSource(src_func=lambda t: np.random.randn()),
component=mp.Ez,
center=mp.Vector3(sx*(-0.5+n/ndipole),-0.5*sy+dAg+0.5*dsub)))
else:
sources = [mp.Source(mp.GaussianSource(fcen,fwidth=df),
component=mp.Ez,
center=mp.Vector3(sx*(-0.5+n/ndipole),-0.5*sy+dAg+0.5*dsub))]

sim = mp.Simulation(cell_size=cell_size,
resolution=resolution,
k_point=mp.Vector3(),
boundary_layers=pml_layers,
geometry=geometry,
sources=sources)

df,
nfreq,
mp.FluxRegion(center=mp.Vector3(0,0.5*sy-dpml),size=mp.Vector3(sx)))

sim.run(until=run_time)

flux = mp.get_fluxes(flux_mon)
freqs = mp.get_flux_freqs(flux_mon)

return freqs, flux

if args.method == 1:
fluxes = np.zeros((nfreq,ntrial))
for t in range(ntrial):
freqs, fluxes[:,t] = compute_flux(m=1)
else:
fluxes = np.zeros((nfreq,ndipole))
for d in range(ndipole):
freqs, fluxes[:,d] = compute_flux(m=2,n=d)

if mp.am_master():
with open('method{}_{}_res{}_nfreq{}_ndipole{}.npz'.format(args.method,"textured" if args.textured else "flat",resolution,nfreq,ndipole),'wb') as f:
np.savez(f,freqs=freqs,fluxes=fluxes)

There are five items to note in this script. (1) The frequency discretization of the flux spectrum must be sufficiently fine to resolve noisy features. In this example, the frequency range is 0.9 to 1.1 with spacing of 0.0004. (2) The runtime must be long enough for the DFT spectrum to resolve these oscillations. Due to the Fourier Uncertainty Principle, the runtime should be at least ~1/frequency resolution. Here, we found that a larger runtime of 2/frequency resolution was sufficient to converge to the desired accuracy. Technically, what we are doing is spectral density estimation of the periodogram by an ensemble average with a rectangular window, but many variations on this general idea exist. (3) The material properties for Ag are imported from the materials library. (4) At the end of the run, the flux spectra from all iterations are saved to a file using NumPy's uncompressed .npz format. This data is used to plot the results in post processing. (5) For Method 1, independent random numbers can be used for the white-noise dipole emitters in the trial runs for the two cases of a flat and textured surface, since all that matters is that they average to the same power spectrum.

Results for Methods 1 and 2 for the two cases of a flat and textured surface are generated using the following shell script:

#!/bin/bash

# Method 1: flat surface
python stochastic_emitter.py -method 1 -res 50 -nf 500 -nd 10 -nr 500

# Method 1: textured surface
python stochastic_emitter.py -method 1 -res 50 -nf 500 -nd 10 -nr 500 -textured

# Method 2: flat surface
python stochastic_emitter.py -method 2 -res 50 -nf 500 -nd 10

# Method 2: textured surface
python stochastic_emitter.py -method 2 -res 50 -nf 500 -nd 10 -textured

Afterwards, the four NumPy files produced by each run are used to plot the normalized flux for each method.

import numpy as np
import matplotlib.pyplot as plt

method1_freqs = method1_f0['freqs']
method1_f0_mean = np.mean(method1_f0['fluxes'],axis=1)
method1_f1_mean = np.mean(method1_f1['fluxes'],axis=1)

method2_freqs = method2_f0['freqs']
method2_f0_mean = np.mean(method2_f0['fluxes'],axis=1)
method2_f1_mean = np.mean(method2_f1['fluxes'],axis=1)

plt.semilogy(method1_freqs,method1_f1_mean/method1_f0_mean,'b-',label='Method 1')
plt.semilogy(method2_freqs,method2_f1_mean/method2_f0_mean,'r-',label='Method 2')
plt.xlabel('frequency')
plt.ylabel('normalized flux')
plt.legend()
plt.show()

Results for Method 1 for three different numbers of trials/iterations (10, 50, and 500) are shown in the following three figures. Each trial/iteration involves two runs: one each for the flat and textured surface. As the number of trials/iterations is increased, the "noisiness" in the plot is gradually reduced. However, the total runtime increases significantly.

The next figure shows a comparison of the normalized radiated flux for Method 1 (500 trials) and 2 (20 runs; 10 runs each for the flat and textured surface). The results show good agreement over the entire bandwidth spectrum. The Method 1 results (labeled "Monte Carlo") required almost four days of compute time using an Intel Xeon processor with two single-threaded cores at 3.8 GHz whereas the Method 2 results (labeled "Deterministic") were obtained in 24 minutes. In general, deterministic approaches tend to be more efficient than brute-force Monte Carlo.

One situation in which you may need to perform brute-force Monte Carlo simulations is that of nonlinear or time-varying media, because the equivalence between random and deterministic simulations above relied on linearity and time-invariance. However, in such media one also cannot directly employ white-noise sources, but you must instead input the noise with the correct spectrum for your desired emission process. For example, to model thermal emission in a nonlinear medium one must have a noise spectrum consistent with the fluctuation-dissipation theorem, which can be achieved using the NoisyLorentzianSusceptibility feature in Meep.

An Efficient Approach for Large Numbers of Dipole Emitters

For cases involving a large number $N$ of spatially incoherent dipole emitters, a more computationally efficient approach than $N$ single-dipole simulations (Method 2) is $M$ separate simulations involving a line source with a different set of basis functions (Method 3). $M$ is typically independent of the resolution and $\ll N$. (For Method 2, the number of point dipoles $N$ comprising the same line source increases with resolution.) The mathematical details of Method 3 are described in Physical Review A, vol. 80, 012119, (2010). The source amplitude function of the $m$th run in the ensemble is defined by a cosine Fourier series:

where $c_m = 1$ if $m=0$ and $c_m=2$ otherwise, $L$ is the length of the line source.

As a demonstration of Method 3 and to compare with Method 2, a similar 2d LED-like periodic structure with a 1d light-emitting layer is used. (The geometric parameters are slightly different than the first example comparing Method 1 and 2 in order to produce a different flux spectrum.) As before, results for the radiated flux of the textured surface are normalized using the flat surface so that the two methods can be directly compared without any additional post processing.

The simulation script is in examples/stochastic_emitter_line.py.

import meep as mp
from meep.materials import Ag
import numpy as np

import argparse
parser = argparse.ArgumentParser()
parser.add_argument('-nf', type=int, default=500, help='number of frequencies')
parser.add_argument('-nsrc', type=int, default=15, help='number of line sources with cosine Fourier series amplitude function (method 3)')
parser.add_argument('-textured', action='store_true', default=False, help='flat (default) or textured surface')
help='type of method: (2) single dipole with 1 run per dipole or (3) line source with cosine Fourier series amplitude function')
args = parser.parse_args()

resolution = args.res

dpml = 1.0
dair = 0.9
hrod = 0.6
wrod = 0.8
dsub = 5.4
dAg = 0.4

sx = 1.5
sy = dpml+dair+hrod+dsub+dAg

cell_size = mp.Vector3(sx,sy)

pml_layers = [mp.PML(direction=mp.Y,
thickness=dpml,
side=mp.High)]

fcen = 1.0
df = 0.2
nfreq = args.nf
nsrc = args.nsrc
ndipole = int(sx*resolution)
run_time = 2*nfreq/df

geometry = [mp.Block(material=mp.Medium(index=3.45),
center=mp.Vector3(0,0.5*sy-dpml-dair-hrod-0.5*dsub),
size=mp.Vector3(mp.inf,dsub,mp.inf)),
mp.Block(material=Ag,
center=mp.Vector3(0,-0.5*sy+0.5*dAg),
size=mp.Vector3(mp.inf,dAg,mp.inf))]

if args.textured:
geometry.append(mp.Block(material=mp.Medium(index=3.45),
center=mp.Vector3(0,0.5*sy-dpml-dair-0.5*hrod),
size=mp.Vector3(wrod,hrod,mp.inf)))

def src_amp_func(n):
def _src_amp_func(p):
if n == 0:
return 1/np.sqrt(sx)
else:
return np.sqrt(2/sx) * np.cos(n*np.pi*(p.x+0.5*sx)/sx)
return _src_amp_func

def compute_flux(m,n):
if m == 2:
sources = [mp.Source(mp.GaussianSource(fcen,fwidth=df),
component=mp.Ez,
center=mp.Vector3(sx*(-0.5+n/ndipole),-0.5*sy+dAg+0.5*dsub))]
else:
sources = [mp.Source(mp.GaussianSource(fcen,fwidth=df),
component=mp.Ez,
center=mp.Vector3(0,-0.5*sy+dAg+0.5*dsub),
size=mp.Vector3(sx,0),
amp_func=src_amp_func(n))]

sim = mp.Simulation(cell_size=cell_size,
resolution=resolution,
k_point=mp.Vector3(),
boundary_layers=pml_layers,
geometry=geometry,
sources=sources)

mp.FluxRegion(center=mp.Vector3(0,0.5*sy-dpml),size=mp.Vector3(sx)))

sim.run(until=run_time)

flux = mp.get_fluxes(flux_mon)
freqs = mp.get_flux_freqs(flux_mon)

return freqs, flux

if args.method == 2:
fluxes = np.zeros((nfreq,ndipole))
for d in range(ndipole):
freqs, fluxes[:,d] = compute_flux(2,d)
else:
fluxes = np.zeros((nfreq,nsrc))
for d in range(nsrc):
freqs, fluxes[:,d] = compute_flux(3,d)

if mp.am_master():
with open('method{}_{}_res{}_nfreq{}_{}{}.npz'.format(args.method,
"textured" if args.textured else "flat",
resolution,
nfreq,
"ndipole" if args.method == 2 else "nsrc",
ndipole if args.method == 2 else nsrc),'wb') as f:
np.savez(f,freqs=freqs,fluxes=fluxes)

There are three items to note in this script. (1) The line source spans the entire length of the cell in the $x$ direction (i.e., $L$ is sx). (2) The number of point dipoles in Method 2 is sx*resolution, one per pixel. (3) The source amplitude function in Method 3 is specified by the amp_func property of the Source object. In the case of a Fourier cosine series as conventionally written, $\cos (m\pi x)/L$ is defined over the interval $x=[0,L]$ such that $x=0$ corresponds to the edge of the source, not the center. Since the source region in this example is defined in $[-L/2,+L/2]$, the amplitude function must shift its $x$ coordinate argument by $+L/2$ or 0.5*sx.

Method 3 requires a convergence check in which $M$ (nsrc in the script) is repeatedly doubled until the change in the results are within a desired tolerance of e.g., < 1%. For this example, $M=15$ was found to be sufficient. Note that because a line source with a cosine amplitude function in homogeneous media is analogous to generating a planewave at a discrete angle, at each frequency $\omega$ there exists a cutoff $M$ beyond which there are no propagating planewaves. The cutoff $M$ can be computed analytically using the grating equation: $\sqrt{\omega^2n^2 - \left(k_x+\frac{M\pi}{L}\right)^2} = 0$, where $n$ is the refractive index of the source medium and $k_x$ is the Bloch-periodic wavevector in the $x$ direction. In this example, $\omega=2\pi$ (pulse center frequency), $L=1.5$, $k_x=0$, and $n=3.45$ for which the largest propagating $M$ is 10. For $M > 10$ the cosine source produces evanescent waves in the material, but these waves scatter into other Fourier components (including propagating waves) once they hit the grating. Thus, the source still produces propagating waves, but the amplitude of the propagating waves (and hence the contribution to the power) decreases exponentially for $M > 10$ as the coupling between the source and the grating decreases. This effect is demonstrated in the figure below which is a semilog plot of the $L_2$ norm of the error in the normalized flux of the cosine source relative to the "correct" result computed using 75 point dipoles (Method 2) as a function of the number of terms in the cosine source. The error decreases exponentially for $M > 10$ until $M=12$, at which point it becomes limited by discretization error (both the point-dipole and cosine-source methods have discretization errors at finite resolution, but are discretized slightly differently); as resolution is increased, this minimum difference will go to zero. Note that this analysis is only applicable to periodic structures where the line source extends the entire width of the cell with periodic boundary conditions. A finite length source in a non-periodic cell has no such cutoff and thus will typically require a large number of cosine terms for convergence.

Results for Methods 2 and 3 for the two cases of a flat and textured surface are generated using the following shell script:

#!/bin/bash

# Method 2: flat surface
python stochastic_emitter_line.py -method 2 -res 50 -nf 500

# Method 2: textured surface
python stochastic_emitter_line.py -method 2 -res 50 -nf 500 -textured

# Method 3: flat surface
python stochastic_emitter_line.py -method 3 -res 50 -nf 500 -nsrc 15

# Method 3: textured surface
python stochastic_emitter_line.py -method 3 -res 50 -nf 500 -nsrc 15 -textured

Afterwards, the four NumPy files produced by each run are used to plot the normalized flux for each method.

import numpy as np
import matplotlib.pyplot as plt

method2_freqs = method2_f0['freqs']
method2_f0_mean = np.mean(method2_f0['fluxes'],axis=1)
method2_f1_mean = np.mean(method2_f1['fluxes'],axis=1)

method3_freqs = method3_f0['freqs']
method3_f0_mean = np.mean(method3_f0['fluxes'],axis=1)
method3_f1_mean = np.mean(method3_f1['fluxes'],axis=1)

plt.semilogy(method2_freqs,method2_f1_mean/method2_f0_mean,'b-',label='Method 2')
plt.semilogy(method3_freqs,method3_f1_mean/method3_f0_mean,'r-',label='Method 3')
plt.xlabel('frequency')
plt.ylabel('normalized flux')
plt.legend()
plt.show()

Results for Method 2 and 3 are shown in the following figure. The agreement is good but there is a significant difference in the runtime. Method 2 (labeled "Point Source") involves $N=75$ simulations each for the flat and textured surface for a total of 150 runs which required 10.6 hours. Method 3 (labeled "Line Source") involves $M=15$ simulations for the flat/textured surface for a total of 30 runs which required just 2.0 hours.

Note regarding convergence properties of Method 2: In this demonstration, the number of point dipoles used in Method 2 is one per pixel. However, because this example is a unit-cell calculation involving periodic boundaries, the number of point dipoles (equivalent to the number of simulations) that are actually necessary to obtain results with sufficient accuracy can be significantly reduced. For smooth periodic functions, it is well known that a trapezoidal rule converges quite fast — generally even faster than the cosine-series expansion (used in Method 3) and comparable to a cosine+sine Fourier series (not shown). See these tutorial notes for the mathematical details. In this example, placing one dipole at every fifth pixel for a total of 15 rather than 75 simulations produces nearly equivalent results for the flux spectrum. More generally, an alternative approach for Method 2 would be to sample a set of dipole points and repeatedly double the sampling density until it converges — and in periodic cases, this could have similar or even better efficiency than the cosine-sine series approach if the right density of points is used. Sampling every grid point is usually not necessary. However, this approach to Method 2 has two major limitations: (1) it probably won't converge as quickly for non-periodic cases (where a trapezoidal rule becomes much less accurate, unlike Method 3), and (2) repeatedly doubling the number of sampling points might overshoot the minimum number of points by more than if the number of cosine-series terms in Method 3 is increased by one at a time, especially if one is not shooting for extremely high accuracies.

Note regarding polarization: The previous demonstrations involved a single-polarization source. For random polarizations, three separate simulations (for $\mathcal{J}_x$, $\mathcal{J}_y$, and $\mathcal{J}_z$) are required regardless of the type of source: point (Method 2) or line (Method 3). Since the different polarizations are uncorrelated, the results (i.e., the flux spectrum) from each set of single-polarization simulations (which, in general, will have different convergence properties) are then summed in post processing. If the emitters involve anisotropic polarizability then the polarizations are correlated. However, in this case choosing the polarizations to correspond to the three principal axes will again make them uncorrelated.

Method 3 can be extended to 3d using a planar source layer ($L_x \times L_y$) involving the product of two cosine series:

where $c_m = c_n = 1$ if $m=n=0$ and $c_m = c_n = 2$ otherwise, $L_x$ and $L_y$ are the lengths of the planar source in the $x$ and $y$ directions. This requires two nested loops to sum over $m$ and $n$ separately. For example, if $M=N=10$, there are a total of 100 simulations. A 3d emission volume would require the product of 3 cosine series leading to e.g. 1000 simulations if there are 10 different sources in each dimension.

The previous examples involved emission in the normal direction. To investigate emission in all directions for a periodic surface requires integrating over $k_x$ and $k_y$ (the Bloch wavevectors of the unit cell; specified using k_point). Each ($k_x,k_y$) corresponds to a different emission direction. However, if you have designed your periodic emitter to emit mostly vertically (via a resonance at $k_x=k_y=0$), then the emission drops off rapidly at other $k$ points. Thus, you can probably estimate the angular width of the resonance by sampling only a few $k$ points near (0,0).

Exploiting Reciprocity

Finally, for cases which involve computing the emission from a collection of $N$ random dipoles in a single direction, one can exploit reciprocity to obtain the same result using a single simulation rather than Monte Carlo sampling of a large number of runs simply by swapping the sources and monitors and running the simulation backwards. This approach is analogous to computing thermal radiation from a blackbody via Kirchhoff's law for thermal radiation which states that emissivity and absorptivity are equivalent under thermal equilibrium. However, the approach demonstrated in this tutorial, which was originally developed for computing the extraction efficiency of LEDs in Optics Express, Vol. 18, pp. 24522-35 (2010), does not require lossy materials.

The reciprocal calculation involves two parts: (1) for the simulation (with the setup shown in the figure below), send an input planewave in the opposite direction of the output mode of the forward calculation and monitor the DFT fields at the same location as the sources in the forward calculation, and (2) in post processing, evaluate an inner product of the DFT fields using a correlation operator of the random currents. Since the currents are uncorrelated in space and consist only of electric currents (because, in practice, they are generated by quantum wells), the correlation operator is a diagonal matrix. Also, since these are 2d simulations with $E_z$ polarization and involve an isotropic dielectric medium, the inner product of the fields is therefore just a sum of $|E_z|^2$ from the DFT monitors (equation 10 of the reference). In 3d, we would need to perform two separate calculations for the $\mathcal{S}$ and $\mathcal{P}$ polarizations and sum the results. (For more general cases, including multiple output channels, spatially correlated emitters, anisotropy, and other complications, see Section 2.3 of arXiv:2111.13046.)

In this example, we will demonstrate that the broadband emission spectrum in the normal direction ($+y$) in air from a collection of $N=10$ point dipoles in the LED-like structure (same as the original example in this series) computed using 10 single-dipole simulations ("forward" runs) can be computed using a single reciprocal calculation involving a planewave at normal incidence ($-y$) in air and a DFT line monitor inside the LED-like structure ("backward" run). (In the forward calculation, we had to discretize the sources into $N$ point sources manually, but a line monitor is discretized for us automatically; in either case, the goal is to approach a continuous spatial distribution as the resolution increases.)

The simulation script is in examples/stochastic_emitter_reciprocity.py.

from typing import List
import numpy as np
import matplotlib.pyplot as plt
import meep as mp
from meep.materials import Ag

resolution = 200  # pixels/μm

nfreq = 100  # number of frequencies
ndipole = 10 # number of point dipoles in forward simulation

fcen = 1.0  # center frequency of Gaussian source/monitors
df = 0.2    # frequency bandwidth of source/monitors

dpml = 1.0  # PML thickness
dair = 2.0  # air padding thickness
hrod = 0.7  # grating height
wrod = 0.5  # graing width
dsub = 5.0  # substrate thickness
dAg = 0.5   # Ag back reflecter thickness

sx = 1.1
sy = dpml + dair + hrod + dsub + dAg

cell_size = mp.Vector3(sx, sy)

pml_layers = [mp.PML(direction=mp.Y, thickness=dpml, side=mp.High)]

def substrate_geometry(is_textured: bool):
"""Returns the geometry of the LED-like structure.

Args:
is_textured: whether the substrate is textured or not.
"""
geometry = [
mp.Block(
material=mp.Medium(index=3.45),
center=mp.Vector3(0, 0.5 * sy - dpml - dair - hrod - 0.5 * dsub),
size=mp.Vector3(mp.inf, dsub, mp.inf),
),
mp.Block(
material=Ag,
center=mp.Vector3(0, -0.5 * sy + 0.5 * dAg),
size=mp.Vector3(mp.inf, dAg, mp.inf),
),
]

if is_textured:
geometry.append(
mp.Block(
material=mp.Medium(index=3.45),
center=mp.Vector3(0, 0.5 * sy - dpml - dair - 0.5 * hrod),
size=mp.Vector3(wrod, hrod, mp.inf),
)
)

return geometry

def forward(n: int, rt: int, is_textured: bool) -> [List, np.ndarray]:
"""Computes the Poynting flux in the +y direction in air
given a point dipole source positioned somewhere along a
line in the middle of the high-index substrate.

Args:
n: n'th position along a line of equally spaced dipoles.
rt: runtime of simulation after the source has turned off
in units of nfreq/df.
is_textured: whether the substrate is textured or not.

Returns:
The frequency and Poynting flux spectra.
"""
sources = [
mp.Source(
mp.GaussianSource(fcen, fwidth=df),
component=mp.Ez,
center=mp.Vector3(
sx * (-0.5 + n / ndipole),
-0.5 * sy + dAg + 0.5 * dsub,
),
)
]

geometry = substrate_geometry(is_textured)

sim = mp.Simulation(
cell_size=cell_size,
resolution=resolution,
k_point=mp.Vector3(),
boundary_layers=pml_layers,
geometry=geometry,
sources=sources,
)

fcen,
df,
nfreq,
mp.FluxRegion(
center=mp.Vector3(0, 0.5 * sy - dpml), size=mp.Vector3(sx)),
)

run_time = rt * nfreq / df
sim.run(until_after_sources=run_time)

res = sim.get_eigenmode_coefficients(flux_mon, [1], eig_parity=mp.ODD_Z)

flux = np.abs(res.alpha[0, :, 0]) ** 2
freqs = mp.get_flux_freqs(flux_mon)

return freqs, flux

def backward(rt: int, is_textured: bool) -> [List, np.ndarray]:
"""Computes the Poynting flux spectrum of the dipole emission using
an overlap integral of the DFT fields from a line monitor in the
high-index substrate given a planewave source in air propagating
in the -y direction.

Args:
rt: runtime of simulation after the source has turned off
in units of nfreq/df.
is_textured: whether the substrate is textured or not.

Returns:
The frequency and Poynting flux spectra.
"""
sources = [
mp.Source(
mp.GaussianSource(fcen, fwidth=df),
component=mp.Ez,
center=mp.Vector3(0, 0.5 * sy - dpml),
size=mp.Vector3(sx, 0),
)
]

geometry = substrate_geometry(is_textured)

sim = mp.Simulation(
cell_size=cell_size,
resolution=resolution,
k_point=mp.Vector3(),
boundary_layers=pml_layers,
geometry=geometry,
sources=sources,
)

[mp.Ez],
fcen,
df,
nfreq,
center=mp.Vector3(
0,
-0.5*sy+dAg+0.5*dsub
),
size=mp.Vector3(sx),
)

run_time = rt * nfreq / df
sim.run(until_after_sources=run_time)

freqs = mp.get_flux_freqs(dft_mon)

abs_flux = np.zeros(nfreq)
for nf in range(nfreq):
dft_ez = sim.get_dft_array(dft_mon, mp.Ez, nf)
abs_flux[nf] = np.sum(np.abs(dft_ez)**2)

return freqs, abs_flux

if __name__ == "__main__":
fwd_flat_flux = np.zeros((nfreq, ndipole))
fwd_text_flux = np.zeros((nfreq, ndipole))
for d in range(ndipole):
fwd_freqs, fwd_flat_flux[:, d] = forward(d, 2, False)
_, fwd_text_flux[:, d] = forward(d, 4, True)

fwd_norm_flux = (np.mean(fwd_text_flux, axis=1) /
np.mean(fwd_flat_flux, axis=1))

bwd_freqs, bwd_flat_flux = backward(2, False)
_, bwd_text_flux = backward(4, True)
bwd_norm_flux = bwd_text_flux / bwd_flat_flux

plt.figure()
plt.semilogy(fwd_freqs, fwd_norm_flux, "b-", label="forward")
plt.semilogy(bwd_freqs, bwd_norm_flux, "r-", label="backward")
plt.xlabel("frequency")
plt.ylabel("normalized flux")
plt.legend()

if mp.am_master():
plt.savefig(
"forward_vs_backward_flux_spectrum.png",
bbox_inches="tight",
dpi=150,
)

There are four items to note in this script. (1) Since the exact equivalence between the forward and reciprocal calculations only applies to the continuum model, demonstrating agreement using a discretized model typically requires going to fine grid resolutions. (In principle, there is an exact discrete version of reciprocity, but that would require more precise attention to how the sources and monitors are discretized.) In this example, we found that a resolution of 200 pixels/μm produced sufficient agreement in the flux spectrum between the forward and reciprocal calculations (a high resolution is required to accurately resolve sharp resonant effects). (2) Measuring the flux in the normal direction in the forward calculation requires mode decomposition rather than a Poynting flux monitor. (3) To compare the results of the forward and reciprocal calculations, similar to the previous examples, the flux spectrum of the grating structure must be normalized using the flux spectrum of a flat structure. (4) A planewave propagating in air in the $+y$ direction would have a wavevector $\vec{k} = \omega(0,1,0)$ with $|\vec{k}|=\omega$. However, the 2d unit cell is periodic only in the $x$ direction. There are PMLs in the $y$ direction which means $k_y$ is actually irrelevant. We could have defined the correct non-zero k_point of the Simulation object but that would have unnecessarily involved complex fields which doubles the memory consumption and the number of floating-point operations during the time stepping. As such, we used a k_point of zero. This should not affect the results since the boundary conditions in the $y$ direction are irrelevant assuming the PML is sufficiently thick to absorb all incoming waves. (For emission in a homogeneous medium with index $n$ at angle $\theta$ from the $+y$ axis, the k_point would be $n\omega(\sin(\theta),\cos(\theta))$ where $\omega$ is the source/monitor frequency. Note that this is valid for a single frequency.)

A plot of the results from the forward and backward simulations is shown below. There is good agreeement across the entire frequency bandwidth. However, there is a significant difference in computational efficiency: the forward calculation had a runtime which was more than three times that of the backward calculation.