From 253521faa135ce0dfebb71aac60c9575f0e8eea6 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Fri, 24 Nov 2023 20:51:58 +0000 Subject: [PATCH 01/10] tutorial for extraction efficiency of a collection of dipoles in a disc in cylindrical coordinates --- .../Near_to_Far_Field_Spectra.md | 5 + python/examples/disc_extraction_efficiency.py | 327 ++++++++++++++++++ 2 files changed, 332 insertions(+) create mode 100644 python/examples/disc_extraction_efficiency.py diff --git a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md index 020da8e39..8c470692d 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -656,6 +656,11 @@ if __name__ == "__main__": ) ``` +### Extraction Efficiency of a Collection of Dipoles in a Disc + +The simulation script is in [examples/disc_extraction_efficiency.py](https://github.com/NanoComp/meep/blob/master/python/examples/disc_extraction_efficiency.py). + + Focusing Properties of a Metasurface Lens ----------------------------------------- diff --git a/python/examples/disc_extraction_efficiency.py b/python/examples/disc_extraction_efficiency.py new file mode 100644 index 000000000..5f7979886 --- /dev/null +++ b/python/examples/disc_extraction_efficiency.py @@ -0,0 +1,327 @@ +"""Computes the extraction efficiency of a collection of dipoles in a disc. + +tutorial reference: https://meep.readthedocs.io/en/latest/Python_Tutorials/Near_to_Far_Field_Spectra/#extraction-efficiency-of-a-disc-in-cylindrical-coordinates +""" + +import math +from typing import Tuple + +import matplotlib +import meep as mp +import numpy as np + +matplotlib.use("agg") +import matplotlib.pyplot as plt + + +N_DISC = 2.4 # refractive index of disc +DISC_RADIUS_UM = 1.2 # radius of disc +WAVELENGTH_UM = 1.0 # wavelength (in vacuum) + +# radius of quarter circle for computing flux in far field +FARFIELD_RADIUS_UM = 1000 * WAVELENGTH_UM + + +def plot_radiation_pattern_polar(theta_rad: np.ndarray, radial_flux: np.ndarray): + """Plots the radiation pattern in polar coordinates. + + Args: + theta_rad: angles of the radiation pattern. The angles increase clockwise + with zero at the pole (+z direction) and π/2 at the equator (+r + direction). + radial_flux: radial flux of the far fields in polar coordinates. + """ + fig, ax = plt.subplots(subplot_kw={"projection": "polar"}, figsize=(6, 6)) + ax.plot( + theta_rad, + radial_flux, + "b-", + ) + ax.set_theta_direction(-1) + ax.set_theta_offset(0.5 * math.pi) + ax.set_thetalim(0, 0.5 * math.pi) + ax.grid(True) + ax.set_rlabel_position(22) + ax.set_ylabel("radial flux (a.u.)") + ax.set_title("radiation pattern in polar coordinates") + + if mp.am_master(): + fig.savefig( + "disc_radiation_pattern_polar.png", + dpi=150, + bbox_inches="tight", + ) + + +def plot_radiation_pattern_3d(theta_rad: np.ndarray, radial_flux: np.ndarray): + """Plots the radiation pattern in 3d Cartesian coordinates. + + Args: + theta_rad: angles of the radiation pattern. + radial_flux: radial flux of the far fields in polar coordinates. + """ + phis = np.linspace(0, 2 * np.pi, num_angles) + + xs = np.zeros((len(theta_rad), len(phis))) + ys = np.zeros((len(theta_rad), len(phis))) + zs = np.zeros((len(theta_rad), len(phis))) + + for i, theta in enumerate(theta_rad): + for j, phi in enumerate(phis): + xs[i, j] = radial_flux[i] * np.sin(theta) * np.cos(phi) + ys[i, j] = radial_flux[i] * np.sin(theta) * np.sin(phi) + zs[i, j] = radial_flux[i] * np.cos(theta) + + fig, ax = plt.subplots(subplot_kw={"projection": "3d"}, figsize=(6, 6)) + ax.plot_surface(xs, ys, zs, cmap="inferno") + ax.set_title("radiation pattern in 3d") + ax.set_box_aspect((np.amax(xs), np.amax(ys), np.amax(zs))) + ax.set_zlabel("radial flux (a.u.)") + ax.set(xticklabels=[], yticklabels=[]) + + if mp.am_master(): + fig.savefig( + "disc_radiation_pattern_3d.png", + dpi=150, + bbox_inches="tight", + ) + + +def radiation_pattern( + theta_rad: np.ndarray, sim: mp.Simulation, n2f_mon: mp.DftNear2Far +) -> np.ndarray: + """Computes the radiation pattern from the near fields. + + Args: + theta_rad: angles of the radiation pattern. + sim: a `Simulation` object. + n2f_mon: a `DftNear2Far` object returned by `Simulation.add_near2far`. + """ + e_field = np.zeros((theta_rad.shape[0], 3), dtype=np.complex128) + h_field = np.zeros((theta_rad.shape[0], 3), dtype=np.complex128) + for n in range(num_angles): + far_field = sim.get_farfield( + n2f_mon, + mp.Vector3( + FARFIELD_RADIUS_UM * math.sin(theta_rad[n]), + 0, + FARFIELD_RADIUS_UM * math.cos(theta_rad[n]), + ), + ) + e_field[n, :] = [np.conj(far_field[j]) for j in range(3)] + h_field[n, :] = [far_field[j + 3] for j in range(3)] + + flux_r = np.real(e_field[:, 1] * h_field[:, 2] - e_field[:, 2] * h_field[:, 1]) + flux_z = np.real(e_field[:, 0] * h_field[:, 1] - e_field[:, 1] * h_field[:, 0]) + flux_rz = np.sqrt(np.square(flux_r) + np.square(flux_z)) + + return flux_rz + + +def radiation_pattern_flux(theta_rad: np.ndarray, radial_flux: np.ndarray) -> float: + """Computes the total flux from the radiation pattern. + + Based on integrating the radiation pattern over solid angles spanned by + polar angles in the range of [0, π/2]. + + Args: + theta_rad: angles of the radiation pattern. + radial_flux: radial flux of the far fields in polar coordinates. + """ + dphi = 2 * math.pi + dtheta_rad = theta_rad[1] - theta_rad[0] + + total_flux = ( + np.sum(radial_flux * np.sin(theta_rad)) + * FARFIELD_RADIUS_UM**2 + * dtheta_rad + * dphi + ) + + return total_flux + + +def dipole_in_disc( + t_disc_um: float, h_disc: float, rpos: float, m: int, theta_rad: np.ndarray +) -> Tuple[float, np.ndarray]: + """Computes the total flux and radiation pattern of a dipole in a disc. + + Args: + t_disc_um: thickness of disc. + h_disc: height of dipole above ground plane as fraction of t_disc_um. + rpos: radial position of dipole. + m: angular φ dependence of the fields exp(imφ). + theta_rad: angles of the radiation pattern. + + Returns: + A 2-tuple of the total flux and the radiation pattern. + """ + resolution = 50 # pixels/μm + + t_pml_um = 0.5 # thickness of PML + t_air_um = 1.0 # thickness of air padding above disc + length_r_um = 5.0 # length of cell in r + + frequency = 1 / WAVELENGTH_UM # center frequency of source/monitor + + # field decay threshold for runtime termination criteria + decay_tol = 1e-6 + + size_r = length_r_um + t_pml_um + size_z = t_disc_um + t_air_um + t_pml_um + cell_size = mp.Vector3(size_r, 0, size_z) + + boundary_layers = [ + mp.PML(t_pml_um, direction=mp.R), + mp.PML(t_pml_um, direction=mp.Z, side=mp.High), + ] + + # An Er source at r=0 needs to be slighty offset. + # https://github.com/NanoComp/meep/issues/2704 + if rpos == 0: + rpos = 1.5 / resolution + + src_cmpt = mp.Er + src_pt = mp.Vector3(rpos, 0, -0.5 * size_z + h_disc * t_disc_um) + sources = [ + mp.Source( + src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), + component=src_cmpt, + center=src_pt, + ) + ] + + geometry = [ + mp.Block( + material=mp.Medium(index=N_DISC), + center=mp.Vector3(0.5 * DISC_RADIUS_UM, 0, -0.5 * size_z + 0.5 * t_disc_um), + size=mp.Vector3(DISC_RADIUS_UM, mp.inf, t_disc_um), + ) + ] + + sim = mp.Simulation( + resolution=resolution, + cell_size=cell_size, + dimensions=mp.CYLINDRICAL, + m=m, + boundary_layers=boundary_layers, + sources=sources, + geometry=geometry, + ) + + n2f_mon = sim.add_near2far( + frequency, + 0, + 1, + mp.FluxRegion( + center=mp.Vector3(0.5 * length_r_um, 0, 0.5 * size_z - t_pml_um), + size=mp.Vector3(length_r_um, 0, 0), + ), + mp.FluxRegion( + center=mp.Vector3( + length_r_um, 0, 0.5 * size_z - t_pml_um - 0.5 * (t_air_um + t_disc_um) + ), + size=mp.Vector3(0, 0, t_air_um + t_disc_um), + ), + ) + + sim.run( + mp.dft_ldos(frequency, 0, 1), + until_after_sources=mp.stop_when_fields_decayed( + 50, + src_cmpt, + src_pt, + decay_tol, + ), + ) + + delta_vol = 2 * np.pi * rpos / (resolution**2) + dipole_flux = -np.real(sim.ldos_Fdata[0] * np.conj(sim.ldos_Jdata[0])) * delta_vol + + dipole_radiation_pattern = radiation_pattern(theta_rad, sim, n2f_mon) + + return dipole_flux, dipole_radiation_pattern + + +if __name__ == "__main__": + disc_thickness = 0.7 * WAVELENGTH_UM / N_DISC + dipole_height = 0.5 + num_dipoles = 11 + dipole_rpos = np.linspace(0, DISC_RADIUS_UM, num_dipoles) + + delta_rpos = dipole_rpos[2] - dipole_rpos[1] + + # number of angular grid points in [0, π/2] + num_angles = 100 + + # grid of polar angles for computing radiated flux in far field + theta_rad = np.linspace(0, 0.5 * math.pi, num_angles) + + # r = 0 source requires a single simulation with m = ±1. + m = -1 + dipole_flux, dipole_radiation_pattern = dipole_in_disc( + disc_thickness, dipole_height, dipole_rpos[0], m, theta_rad + ) + + flux_total = dipole_flux * dipole_rpos[0] * delta_rpos + radiation_pattern_total = dipole_radiation_pattern * dipole_rpos[0] * delta_rpos + + print( + f"dipole:, {dipole_rpos[0]:.4f}, " + f"{radiation_pattern_flux(theta_rad, dipole_radiation_pattern):.6f}" + ) + + # r > 0 source requires Fourier-series expansion of φ. + flux_tol = 1e-3 # threshold flux to determine when to truncate expansion + for rpos in dipole_rpos[1:]: + dipole_flux_total = 0 + dipole_radiation_pattern_total = np.zeros((num_angles,)) + dipole_radiation_pattern_flux_max = 0 + m = 0 + while True: + dipole_flux, dipole_radiation_pattern = dipole_in_disc( + disc_thickness, dipole_height, rpos, m, theta_rad + ) + dipole_flux_total += dipole_flux if m == 0 else 2 * dipole_flux + dipole_radiation_pattern_total += ( + dipole_radiation_pattern if m == 0 else 2 * dipole_radiation_pattern + ) + + dipole_radiation_pattern_flux = radiation_pattern_flux( + theta_rad, dipole_radiation_pattern + ) + if dipole_radiation_pattern_flux > dipole_radiation_pattern_flux_max: + dipole_radiation_pattern_flux_max = dipole_radiation_pattern_flux + + if m > 0 and ( + (dipole_radiation_pattern_flux / dipole_radiation_pattern_flux_max) + < flux_tol + ): + break + + print( + f"dipole-m:, {rpos:.4f}, {m}, " f"{dipole_radiation_pattern_flux:.6f}" + ) + m += 1 + + scale_factor = 1 / (2 * (rpos / dipole_rpos[0]) ** 2) + flux_total += dipole_flux_total * scale_factor * rpos * delta_rpos + radiation_pattern_total += ( + dipole_radiation_pattern_total * scale_factor * rpos * delta_rpos + ) + + dipole_radiation_pattern_total_flux = radiation_pattern_flux( + theta_rad, dipole_radiation_pattern_total + ) + print( + f"dipole:, {rpos:.4f}, {m}, " f"{dipole_radiation_pattern_total_flux:.6f}" + ) + + radiation_pattern_total_flux = radiation_pattern_flux( + theta_rad, radiation_pattern_total + ) + extraction_efficiency = radiation_pattern_total_flux / flux_total + print(f"exteff:, {extraction_efficiency:.6f}") + + plot_radiation_pattern_polar(radiation_pattern_total * FARFIELD_RADIUS_UM**2) + plot_radiation_pattern_3d(radiation_pattern_total * FARFIELD_RADIUS_UM**2) From e59ca3afeb7fc610cf9426a2ab5b70e3fe0d98d8 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Wed, 29 Nov 2023 22:50:52 -0800 Subject: [PATCH 02/10] add text and figure to tutorial --- .../Near_to_Far_Field_Spectra.md | 10 ++++++++++ .../images/disc_dipoles_radiation_pattern.png | Bin 0 -> 205930 bytes 2 files changed, 10 insertions(+) create mode 100644 doc/docs/images/disc_dipoles_radiation_pattern.png diff --git a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md index 8c470692d..eaf9576c1 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -658,6 +658,16 @@ if __name__ == "__main__": ### Extraction Efficiency of a Collection of Dipoles in a Disc +[Tutorial/Radiation Pattern of a Disc in Cylindrical Coordinates](Near_to_Far_Field_Spectra.md#radiation-pattern-of-a-disc-in-cylindrical-coordinates) demonstrated the procedure for computing the radiation pattern of a *single* dipole (actually a "ring" current source) with a fixed angular dependence $exp(im\phi)$. Separately, [Tutorial/Nonaxisymmetric Dipole Sources](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) described the method for modeling a point dipole at $r > 0$ in cylindrical coordinates using a Fourier-series expansion of the fields in $\phi$. These two results can be combined to compute the radiation pattern for a point dipole anywhere in the cylindrical cell. Computing the extraction efficiency of a light-emitting diode (LED), however, involves a collection of spatially incoherent dipole emitters. [Tutorial/Stochastic Dipole Emission in Light Emitting Diodes](Custom_Source.md#stochastic-dipole-emission-in-light-emitting-diodes) described a method for computing the emission of an LED using a series of single-dipole simulations and averaging the emission profiles in post processing. The example used a 2D simulation involving a 1D binary grating. This tutorial demonstrates how this approach can be extended to cylindrical coordinates for structures with rotational symmetry. + +The example uses the same setup as the [previous tutorial](#radiation-pattern-of-a-disc-in-cylindrical-coordinates) involving a dielectric disc above a lossless-reflector ground plane. The dipoles are arranged with equal spacing $\Delta r$ on a line extending from $r = 0$ to $r = R$ where $R$ is the radius of the disc. The height of the dipoles ($z$ coordinate) is fixed. We compute the radiation pattern $P(r,\theta)$ for a dipole at $r$ using a Fourier-series expansion in $\phi$. The *total* radiation pattern $P(\theta)$ for $N$ dipoles is the average of the per-dipole results: + +$$P(\theta) = \int_0^R P(r,\theta) s(r) 2\pi rdr = \sum_{n=0}^{N-1} P(r_n,\theta) s(r_n) 2\pi r_n \Delta r$$, + +where $s(r)$ is a weighting function necessary for ensuring equal contribution from all dipoles relative to the dipole at $r = 0$. Note: an $E_r$ dipole at $r = 0$ must be placed at $r_0 = 1.5\Delta r$ due to an [interpolation bug](https://github.com/NanoComp/meep/issues/2704). $s(r)$ can be determined empirically by computing the emission in vacuum for a set of dipoles at different radial positions. The emission profiles of all dipoles in vacuum must be identical which yields: $s(r) = \frac{1}{2(r/r_0)^2}$. + +![](../images/disc_dipoles_radiation_pattern.png#center) + The simulation script is in [examples/disc_extraction_efficiency.py](https://github.com/NanoComp/meep/blob/master/python/examples/disc_extraction_efficiency.py). diff --git a/doc/docs/images/disc_dipoles_radiation_pattern.png b/doc/docs/images/disc_dipoles_radiation_pattern.png new file mode 100644 index 0000000000000000000000000000000000000000..4bd20deb9271bf7da7fb5559858b04746127ec6c GIT binary patch literal 205930 zcmZ^LcRbc@|2}1<5}B#2q>>b6BqKtUG7_>fGO}f67TJU(nGGZf$tGkc2}xF$*^*uM z{2ia}@AK#Hd3o-;TU^)YJkR$y-p6sASBTohv)gIeY01dQwks>iYmkv`9wj5&L`6e| z?;O7Z zEsVvoUUPN*5#Q6(l5(=uiDL@`1Dk2MP{ZUKR&uY-&k;uWbHefKm&9%fHT?SvHq$PL zsE_~m$8ZtGJ!!}P_ZKa9?lk`UuW|buw5N41iTUAotUrn?S>Q$f{YJu;$;toE?@~~Y z{_m%Bj#tY{l#dANm)x2g>pGLsK~5H~xbMr$EB;mM%USA0s`BEF!xWRA5EbZLDc1xSz&rW2%el2X%|7KT6NXYQ(wUq_a<_AYE 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zfStpZ=I(d^Kl!1&#5s71W1-$trdK)ER1$pa`)iGCy^?VW?V!Hz;hxC_- z5E~0E$qgto1mzq4{iG_`EI*)E0B}D&)JmdRW4^B=3p8JX5yqh+8^zE+=*KnFk<)Nl zT>TtD&6Y30<^OT%L&YOOO!fcI|E-KRVDrHL`U9I&x86PbZ$w=_{~nj0?ERnG|380< zLPId$|Mepet_}M4$d`Y|m1P{#_wD}Q|2K3;>rVcEeuMf`{B!f!&A-k|My>^ bXK7VN!|=EG$EkrU@LzH=il@_0UiSPy-?B&0 literal 0 HcmV?d00001 From 7d9987ca562dfa6cf182c53bbd9d4e93af94faf2 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Thu, 30 Nov 2023 15:38:08 +0000 Subject: [PATCH 03/10] add more details of calculation and include the script at the end of the tutorial --- .../Near_to_Far_Field_Spectra.md | 333 +++++++++++++++++- 1 file changed, 330 insertions(+), 3 deletions(-) diff --git a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md index eaf9576c1..72f6b19e8 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -658,19 +658,346 @@ if __name__ == "__main__": ### Extraction Efficiency of a Collection of Dipoles in a Disc -[Tutorial/Radiation Pattern of a Disc in Cylindrical Coordinates](Near_to_Far_Field_Spectra.md#radiation-pattern-of-a-disc-in-cylindrical-coordinates) demonstrated the procedure for computing the radiation pattern of a *single* dipole (actually a "ring" current source) with a fixed angular dependence $exp(im\phi)$. Separately, [Tutorial/Nonaxisymmetric Dipole Sources](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) described the method for modeling a point dipole at $r > 0$ in cylindrical coordinates using a Fourier-series expansion of the fields in $\phi$. These two results can be combined to compute the radiation pattern for a point dipole anywhere in the cylindrical cell. Computing the extraction efficiency of a light-emitting diode (LED), however, involves a collection of spatially incoherent dipole emitters. [Tutorial/Stochastic Dipole Emission in Light Emitting Diodes](Custom_Source.md#stochastic-dipole-emission-in-light-emitting-diodes) described a method for computing the emission of an LED using a series of single-dipole simulations and averaging the emission profiles in post processing. The example used a 2D simulation involving a 1D binary grating. This tutorial demonstrates how this approach can be extended to cylindrical coordinates for structures with rotational symmetry. +[Tutorial/Radiation Pattern of a Disc in Cylindrical Coordinates](Near_to_Far_Field_Spectra.md#radiation-pattern-of-a-disc-in-cylindrical-coordinates) demonstrated the procedure for computing the radiation pattern of a *single* dipole (actually a "ring" current source with angular dependence $exp(im\phi)$). [Tutorial/Nonaxisymmetric Dipole Sources](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) described the method for modeling a point dipole at $r > 0$ in cylindrical coordinates using a Fourier-series expansion of the fields in $\phi$. [Tutorial/Extraction Efficiency of a Light-Emitting Diode](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) described the procedure for computing the extraction efficiency of a dipole at $r = 0$. These three results can be combined to compute the extraction efficiency for a point dipole anywhere in the cylindrical cell. Computing the extraction efficiency of an actual light-emitting diode (LED), however, involves a collection of spatially incoherent dipole emitters. [Tutorial/Stochastic Dipole Emission in Light Emitting Diodes](Custom_Source.md#stochastic-dipole-emission-in-light-emitting-diodes) described a method for computing the emission of a collection of dipoles using a series of single-dipole calculations and averaging the emission profiles in post processing. The example used a 2D simulation involving a 1D binary grating (or photonic crystal). This tutorial demonstrates how this approach for modeling spatially incoherent dipoles can be extended to cylindrical coordinates for structures with rotational symmetry. -The example uses the same setup as the [previous tutorial](#radiation-pattern-of-a-disc-in-cylindrical-coordinates) involving a dielectric disc above a lossless-reflector ground plane. The dipoles are arranged with equal spacing $\Delta r$ on a line extending from $r = 0$ to $r = R$ where $R$ is the radius of the disc. The height of the dipoles ($z$ coordinate) is fixed. We compute the radiation pattern $P(r,\theta)$ for a dipole at $r$ using a Fourier-series expansion in $\phi$. The *total* radiation pattern $P(\theta)$ for $N$ dipoles is the average of the per-dipole results: +The example uses the same setup as the [previous tutorial](#radiation-pattern-of-a-disc-in-cylindrical-coordinates) involving a dielectric disc above a lossless-reflector ground plane. The dipoles are arranged on a line extending from $r = 0$ to $r = R$ where $R$ is the disc radius. The height of the dipoles ($z$ coordinate) within the disc is fixed. The radiation pattern $P(r,\theta)$ for a dipole at $r$ is computed using a Fourier-series expansion in $\phi$. The *total* radiation pattern $P(\theta)$ for $N$ dipoles with equal spacing $\Delta r$ is the average of the per-dipole results: $$P(\theta) = \int_0^R P(r,\theta) s(r) 2\pi rdr = \sum_{n=0}^{N-1} P(r_n,\theta) s(r_n) 2\pi r_n \Delta r$$, -where $s(r)$ is a weighting function necessary for ensuring equal contribution from all dipoles relative to the dipole at $r = 0$. Note: an $E_r$ dipole at $r = 0$ must be placed at $r_0 = 1.5\Delta r$ due to an [interpolation bug](https://github.com/NanoComp/meep/issues/2704). $s(r)$ can be determined empirically by computing the emission in vacuum for a set of dipoles at different radial positions. The emission profiles of all dipoles in vacuum must be identical which yields: $s(r) = \frac{1}{2(r/r_0)^2}$. +where $s(r)$ is a weighting function necessary for ensuring equal contribution from all dipoles relative to the dipole at $r = 0$. Note: an $E_r$ dipole at $r = 0$ must be placed at $r_0 = 1.5\Delta r$ due to an [interpolation bug](https://github.com/NanoComp/meep/issues/2704). $s(r)$ can be determined empirically by computing the emission in vacuum for a set of dipoles at different radial positions. The emission profile of a dipole in vacuum is a constant indepnedent of its position. This criteria is used to obtain: $s(r) = \frac{1}{2(r/r_0)^2}$. This weighting function is also used to average the flux emitted by each dipole (obtained using using the LDoS feature). This quantity is the denominator in the expression for the extraction efficiency. + +This figure shows the radiation pattern from $N=11$ dipoles with $\lambda$ of 1.0 $\mu$m in the middle of a disc of radius 1.2 $\mu$m, height 0.29 $\mu$m, and refractive index 2.4. ![](../images/disc_dipoles_radiation_pattern.png#center) The simulation script is in [examples/disc_extraction_efficiency.py](https://github.com/NanoComp/meep/blob/master/python/examples/disc_extraction_efficiency.py). +```py +import math +from typing import Tuple + +import matplotlib +import meep as mp +import numpy as np + +matplotlib.use("agg") +import matplotlib.pyplot as plt + + +N_DISC = 2.4 # refractive index of disc +DISC_RADIUS_UM = 1.2 # radius of disc +WAVELENGTH_UM = 1.0 # wavelength (in vacuum) + +# radius of quarter circle for computing flux in far field +FARFIELD_RADIUS_UM = 1000 * WAVELENGTH_UM + + +def plot_radiation_pattern_polar(theta_rad: np.ndarray, radial_flux: np.ndarray): + """Plots the radiation pattern in polar coordinates. + + Args: + theta_rad: angles of the radiation pattern. The angles increase clockwise + with zero at the pole (+z direction) and π/2 at the equator (+r + direction). + radial_flux: radial flux of the far fields in polar coordinates. + """ + fig, ax = plt.subplots(subplot_kw={"projection": "polar"}, figsize=(6, 6)) + ax.plot( + theta_rad, + radial_flux, + "b-", + ) + ax.set_theta_direction(-1) + ax.set_theta_offset(0.5 * math.pi) + ax.set_thetalim(0, 0.5 * math.pi) + ax.grid(True) + ax.set_rlabel_position(22) + ax.set_ylabel("radial flux (a.u.)") + ax.set_title("radiation pattern in polar coordinates") + + if mp.am_master(): + fig.savefig( + "disc_radiation_pattern_polar.png", + dpi=150, + bbox_inches="tight", + ) + + +def plot_radiation_pattern_3d(theta_rad: np.ndarray, radial_flux: np.ndarray): + """Plots the radiation pattern in 3d Cartesian coordinates. + + Args: + theta_rad: angles of the radiation pattern. + radial_flux: radial flux of the far fields in polar coordinates. + """ + phis = np.linspace(0, 2 * np.pi, num_angles) + + xs = np.zeros((len(theta_rad), len(phis))) + ys = np.zeros((len(theta_rad), len(phis))) + zs = np.zeros((len(theta_rad), len(phis))) + + for i, theta in enumerate(theta_rad): + for j, phi in enumerate(phis): + xs[i, j] = radial_flux[i] * np.sin(theta) * np.cos(phi) + ys[i, j] = radial_flux[i] * np.sin(theta) * np.sin(phi) + zs[i, j] = radial_flux[i] * np.cos(theta) + + fig, ax = plt.subplots(subplot_kw={"projection": "3d"}, figsize=(6, 6)) + ax.plot_surface(xs, ys, zs, cmap="inferno") + ax.set_title("radiation pattern in 3d") + ax.set_box_aspect((np.amax(xs), np.amax(ys), np.amax(zs))) + ax.set_zlabel("radial flux (a.u.)") + ax.set(xticklabels=[], yticklabels=[]) + + if mp.am_master(): + fig.savefig( + "disc_radiation_pattern_3d.png", + dpi=150, + bbox_inches="tight", + ) + + +def radiation_pattern( + theta_rad: np.ndarray, sim: mp.Simulation, n2f_mon: mp.DftNear2Far +) -> np.ndarray: + """Computes the radiation pattern from the near fields. + + Args: + theta_rad: angles of the radiation pattern. + sim: a `Simulation` object. + n2f_mon: a `DftNear2Far` object returned by `Simulation.add_near2far`. + """ + e_field = np.zeros((theta_rad.shape[0], 3), dtype=np.complex128) + h_field = np.zeros((theta_rad.shape[0], 3), dtype=np.complex128) + for n in range(num_angles): + far_field = sim.get_farfield( + n2f_mon, + mp.Vector3( + FARFIELD_RADIUS_UM * math.sin(theta_rad[n]), + 0, + FARFIELD_RADIUS_UM * math.cos(theta_rad[n]), + ), + ) + e_field[n, :] = [np.conj(far_field[j]) for j in range(3)] + h_field[n, :] = [far_field[j + 3] for j in range(3)] + + flux_r = np.real(e_field[:, 1] * h_field[:, 2] - e_field[:, 2] * h_field[:, 1]) + flux_z = np.real(e_field[:, 0] * h_field[:, 1] - e_field[:, 1] * h_field[:, 0]) + flux_rz = np.sqrt(np.square(flux_r) + np.square(flux_z)) + + return flux_rz + + +def radiation_pattern_flux(theta_rad: np.ndarray, radial_flux: np.ndarray) -> float: + """Computes the total flux from the radiation pattern. + + Based on integrating the radiation pattern over solid angles spanned by + polar angles in the range of [0, π/2]. + + Args: + theta_rad: angles of the radiation pattern. + radial_flux: radial flux of the far fields in polar coordinates. + """ + dphi = 2 * math.pi + dtheta_rad = theta_rad[1] - theta_rad[0] + + total_flux = ( + np.sum(radial_flux * np.sin(theta_rad)) + * FARFIELD_RADIUS_UM**2 + * dtheta_rad + * dphi + ) + + return total_flux + + +def dipole_in_disc( + t_disc_um: float, h_disc: float, rpos: float, m: int, theta_rad: np.ndarray +) -> Tuple[float, np.ndarray]: + """Computes the total flux and radiation pattern of a dipole in a disc. + + Args: + t_disc_um: thickness of disc. + h_disc: height of dipole above ground plane as fraction of t_disc_um. + rpos: radial position of dipole. + m: angular φ dependence of the fields exp(imφ). + theta_rad: angles of the radiation pattern. + + Returns: + A 2-tuple of the total flux and the radiation pattern. + """ + resolution = 50 # pixels/μm + + t_pml_um = 0.5 # thickness of PML + t_air_um = 1.0 # thickness of air padding above disc + length_r_um = 5.0 # length of cell in r + + frequency = 1 / WAVELENGTH_UM # center frequency of source/monitor + + # field decay threshold for runtime termination criteria + decay_tol = 1e-6 + + size_r = length_r_um + t_pml_um + size_z = t_disc_um + t_air_um + t_pml_um + cell_size = mp.Vector3(size_r, 0, size_z) + + boundary_layers = [ + mp.PML(t_pml_um, direction=mp.R), + mp.PML(t_pml_um, direction=mp.Z, side=mp.High), + ] + + # An Er source at r=0 needs to be slighty offset. + # https://github.com/NanoComp/meep/issues/2704 + if rpos == 0: + rpos = 1.5 / resolution + + src_cmpt = mp.Er + src_pt = mp.Vector3(rpos, 0, -0.5 * size_z + h_disc * t_disc_um) + sources = [ + mp.Source( + src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), + component=src_cmpt, + center=src_pt, + ) + ] + + geometry = [ + mp.Block( + material=mp.Medium(index=N_DISC), + center=mp.Vector3(0.5 * DISC_RADIUS_UM, 0, -0.5 * size_z + 0.5 * t_disc_um), + size=mp.Vector3(DISC_RADIUS_UM, mp.inf, t_disc_um), + ) + ] + + sim = mp.Simulation( + resolution=resolution, + cell_size=cell_size, + dimensions=mp.CYLINDRICAL, + m=m, + boundary_layers=boundary_layers, + sources=sources, + geometry=geometry, + ) + + n2f_mon = sim.add_near2far( + frequency, + 0, + 1, + mp.FluxRegion( + center=mp.Vector3(0.5 * length_r_um, 0, 0.5 * size_z - t_pml_um), + size=mp.Vector3(length_r_um, 0, 0), + ), + mp.FluxRegion( + center=mp.Vector3( + length_r_um, 0, 0.5 * size_z - t_pml_um - 0.5 * (t_air_um + t_disc_um) + ), + size=mp.Vector3(0, 0, t_air_um + t_disc_um), + ), + ) + + sim.run( + mp.dft_ldos(frequency, 0, 1), + until_after_sources=mp.stop_when_fields_decayed( + 50, + src_cmpt, + src_pt, + decay_tol, + ), + ) + + delta_vol = 2 * np.pi * rpos / (resolution**2) + dipole_flux = -np.real(sim.ldos_Fdata[0] * np.conj(sim.ldos_Jdata[0])) * delta_vol + + dipole_radiation_pattern = radiation_pattern(theta_rad, sim, n2f_mon) + + return dipole_flux, dipole_radiation_pattern + + +if __name__ == "__main__": + disc_thickness = 0.7 * WAVELENGTH_UM / N_DISC + dipole_height = 0.5 + num_dipoles = 11 + dipole_rpos = np.linspace(0, DISC_RADIUS_UM, num_dipoles) + + delta_rpos = dipole_rpos[2] - dipole_rpos[1] + + # number of angular grid points in [0, π/2] + num_angles = 100 + + # grid of polar angles for computing radiated flux in far field + theta_rad = np.linspace(0, 0.5 * math.pi, num_angles) + + # r = 0 source requires a single simulation with m = ±1. + m = -1 + dipole_flux, dipole_radiation_pattern = dipole_in_disc( + disc_thickness, dipole_height, dipole_rpos[0], m, theta_rad + ) + + flux_total = dipole_flux * dipole_rpos[0] * delta_rpos + radiation_pattern_total = dipole_radiation_pattern * dipole_rpos[0] * delta_rpos + + print( + f"dipole:, {dipole_rpos[0]:.4f}, " + f"{radiation_pattern_flux(theta_rad, dipole_radiation_pattern):.6f}" + ) + + # r > 0 source requires Fourier-series expansion of φ. + flux_tol = 1e-3 # threshold flux to determine when to truncate expansion + for rpos in dipole_rpos[1:]: + dipole_flux_total = 0 + dipole_radiation_pattern_total = np.zeros((num_angles,)) + dipole_radiation_pattern_flux_max = 0 + m = 0 + while True: + dipole_flux, dipole_radiation_pattern = dipole_in_disc( + disc_thickness, dipole_height, rpos, m, theta_rad + ) + dipole_flux_total += dipole_flux if m == 0 else 2 * dipole_flux + dipole_radiation_pattern_total += ( + dipole_radiation_pattern if m == 0 else 2 * dipole_radiation_pattern + ) + + dipole_radiation_pattern_flux = radiation_pattern_flux( + theta_rad, dipole_radiation_pattern + ) + if dipole_radiation_pattern_flux > dipole_radiation_pattern_flux_max: + dipole_radiation_pattern_flux_max = dipole_radiation_pattern_flux + + if m > 0 and ( + (dipole_radiation_pattern_flux / dipole_radiation_pattern_flux_max) + < flux_tol + ): + break + + print( + f"dipole-m:, {rpos:.4f}, {m}, " f"{dipole_radiation_pattern_flux:.6f}" + ) + m += 1 + + scale_factor = 1 / (2 * (rpos / dipole_rpos[0]) ** 2) + flux_total += dipole_flux_total * scale_factor * rpos * delta_rpos + radiation_pattern_total += ( + dipole_radiation_pattern_total * scale_factor * rpos * delta_rpos + ) + + dipole_radiation_pattern_total_flux = radiation_pattern_flux( + theta_rad, dipole_radiation_pattern_total + ) + print( + f"dipole:, {rpos:.4f}, {m}, " f"{dipole_radiation_pattern_total_flux:.6f}" + ) + + radiation_pattern_total_flux = radiation_pattern_flux( + theta_rad, radiation_pattern_total + ) + extraction_efficiency = radiation_pattern_total_flux / flux_total + print(f"exteff:, {extraction_efficiency:.6f}") + + plot_radiation_pattern_polar(radiation_pattern_total * FARFIELD_RADIUS_UM**2) + plot_radiation_pattern_3d(radiation_pattern_total * FARFIELD_RADIUS_UM**2) +``` + Focusing Properties of a Metasurface Lens ----------------------------------------- From b84f589e41a6534f412b8d0aa909e4edd163bc6a Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Wed, 6 Dec 2023 23:23:12 +0000 Subject: [PATCH 04/10] fixes --- doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md index 72f6b19e8..af8a4cfa2 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -658,15 +658,15 @@ if __name__ == "__main__": ### Extraction Efficiency of a Collection of Dipoles in a Disc -[Tutorial/Radiation Pattern of a Disc in Cylindrical Coordinates](Near_to_Far_Field_Spectra.md#radiation-pattern-of-a-disc-in-cylindrical-coordinates) demonstrated the procedure for computing the radiation pattern of a *single* dipole (actually a "ring" current source with angular dependence $exp(im\phi)$). [Tutorial/Nonaxisymmetric Dipole Sources](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) described the method for modeling a point dipole at $r > 0$ in cylindrical coordinates using a Fourier-series expansion of the fields in $\phi$. [Tutorial/Extraction Efficiency of a Light-Emitting Diode](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) described the procedure for computing the extraction efficiency of a dipole at $r = 0$. These three results can be combined to compute the extraction efficiency for a point dipole anywhere in the cylindrical cell. Computing the extraction efficiency of an actual light-emitting diode (LED), however, involves a collection of spatially incoherent dipole emitters. [Tutorial/Stochastic Dipole Emission in Light Emitting Diodes](Custom_Source.md#stochastic-dipole-emission-in-light-emitting-diodes) described a method for computing the emission of a collection of dipoles using a series of single-dipole calculations and averaging the emission profiles in post processing. The example used a 2D simulation involving a 1D binary grating (or photonic crystal). This tutorial demonstrates how this approach for modeling spatially incoherent dipoles can be extended to cylindrical coordinates for structures with rotational symmetry. +[Tutorial/Radiation Pattern of a Disc in Cylindrical Coordinates](Near_to_Far_Field_Spectra.md#radiation-pattern-of-a-disc-in-cylindrical-coordinates) demonstrated the procedure for computing the radiation pattern of a *single* dipole (actually a "ring" current source with angular dependence $exp(im\phi)$). [Tutorial/Nonaxisymmetric Dipole Sources](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) described the method for modeling a point dipole at $r > 0$ in cylindrical coordinates using a Fourier-series expansion of the fields in $\phi$. [Tutorial/Extraction Efficiency of a Light-Emitting Diode](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) described the procedure for computing the extraction efficiency of a dipole at $r = 0$. These three results can be combined to compute the extraction efficiency for a point dipole *anywhere* in the cylindrical cell. Computing the extraction efficiency of an actual light-emitting diode (LED), however, involves a collection of spatially incoherent dipole emitters. [Tutorial/Stochastic Dipole Emission in Light Emitting Diodes](Custom_Source.md#stochastic-dipole-emission-in-light-emitting-diodes) described a method for computing the emission of a collection of dipoles using a series of single-dipole calculations and averaging the emission profiles in post processing. The example used a 2D simulation involving a 1D binary grating (or photonic crystal). This tutorial demonstrates how this approach for modeling spatially incoherent dipoles can be extended to cylindrical coordinates for structures with rotational symmetry. The example uses the same setup as the [previous tutorial](#radiation-pattern-of-a-disc-in-cylindrical-coordinates) involving a dielectric disc above a lossless-reflector ground plane. The dipoles are arranged on a line extending from $r = 0$ to $r = R$ where $R$ is the disc radius. The height of the dipoles ($z$ coordinate) within the disc is fixed. The radiation pattern $P(r,\theta)$ for a dipole at $r$ is computed using a Fourier-series expansion in $\phi$. The *total* radiation pattern $P(\theta)$ for $N$ dipoles with equal spacing $\Delta r$ is the average of the per-dipole results: -$$P(\theta) = \int_0^R P(r,\theta) s(r) 2\pi rdr = \sum_{n=0}^{N-1} P(r_n,\theta) s(r_n) 2\pi r_n \Delta r$$, +$$P(\theta) = \frac{1}{R} \int_0^R P(r,\theta) s(r) 2\pi rdr = \frac{1}{N} \sum_{n=0}^{N-1} P(r_n,\theta) s(r_n) 2\pi r_n \Delta r$$, -where $s(r)$ is a weighting function necessary for ensuring equal contribution from all dipoles relative to the dipole at $r = 0$. Note: an $E_r$ dipole at $r = 0$ must be placed at $r_0 = 1.5\Delta r$ due to an [interpolation bug](https://github.com/NanoComp/meep/issues/2704). $s(r)$ can be determined empirically by computing the emission in vacuum for a set of dipoles at different radial positions. The emission profile of a dipole in vacuum is a constant indepnedent of its position. This criteria is used to obtain: $s(r) = \frac{1}{2(r/r_0)^2}$. This weighting function is also used to average the flux emitted by each dipole (obtained using using the LDoS feature). This quantity is the denominator in the expression for the extraction efficiency. +where $s(r)$ is a weighting function necessary for ensuring equal contribution from all dipoles relative to the dipole at $r = 0$. Note: an $E_r$ dipole at $r = 0$ must be placed at $r_0 = 1.5\Delta r$ due to an [interpolation bug](https://github.com/NanoComp/meep/issues/2704). $s(r)$ can be determined empirically by computing the emission in vacuum for a set of dipoles at different radial positions. The emission profile of a dipole in vacuum is a constant independent of its position. This criteria is used to obtain: $s(r) = \frac{1}{2(r/r_0)^2}$. This weighting function is also used to average the flux emitted by each dipole (obtained using using the LDOS feature). This quantity is the denominator in the expression for the extraction efficiency. -This figure shows the radiation pattern from $N=11$ dipoles with $\lambda$ of 1.0 $\mu$m in the middle of a disc of radius 1.2 $\mu$m, height 0.29 $\mu$m, and refractive index 2.4. +This figure shows the radiation pattern from $N=11$ dipoles with $\lambda$ of 1.0 $\mu$m in the middle of a disc of height 0.29 $\mu$m, radius 1.2 $\mu$m, and refractive index 2.4. ![](../images/disc_dipoles_radiation_pattern.png#center) From 527b0ff941d5b9b23d15ce3da2cf25c76f2f5f37 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Fri, 8 Dec 2023 05:13:57 +0000 Subject: [PATCH 05/10] fix bugs in script --- .../Near_to_Far_Field_Spectra.md | 20 ++++++++++--------- python/examples/disc_extraction_efficiency.py | 20 ++++++++++--------- 2 files changed, 22 insertions(+), 18 deletions(-) diff --git a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md index af8a4cfa2..f4c7ed213 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -692,6 +692,8 @@ WAVELENGTH_UM = 1.0 # wavelength (in vacuum) # radius of quarter circle for computing flux in far field FARFIELD_RADIUS_UM = 1000 * WAVELENGTH_UM +RESOLUTION = 50 # pixels/μm + def plot_radiation_pattern_polar(theta_rad: np.ndarray, radial_flux: np.ndarray): """Plots the radiation pattern in polar coordinates. @@ -827,8 +829,6 @@ def dipole_in_disc( Returns: A 2-tuple of the total flux and the radiation pattern. """ - resolution = 50 # pixels/μm - t_pml_um = 0.5 # thickness of PML t_air_um = 1.0 # thickness of air padding above disc length_r_um = 5.0 # length of cell in r @@ -847,11 +847,6 @@ def dipole_in_disc( mp.PML(t_pml_um, direction=mp.Z, side=mp.High), ] - # An Er source at r=0 needs to be slighty offset. - # https://github.com/NanoComp/meep/issues/2704 - if rpos == 0: - rpos = 1.5 / resolution - src_cmpt = mp.Er src_pt = mp.Vector3(rpos, 0, -0.5 * size_z + h_disc * t_disc_um) sources = [ @@ -871,7 +866,7 @@ def dipole_in_disc( ] sim = mp.Simulation( - resolution=resolution, + resolution=RESOLUTION, cell_size=cell_size, dimensions=mp.CYLINDRICAL, m=m, @@ -906,7 +901,7 @@ def dipole_in_disc( ), ) - delta_vol = 2 * np.pi * rpos / (resolution**2) + delta_vol = 2 * np.pi * rpos / (RESOLUTION**2) dipole_flux = -np.real(sim.ldos_Fdata[0] * np.conj(sim.ldos_Jdata[0])) * delta_vol dipole_radiation_pattern = radiation_pattern(theta_rad, sim, n2f_mon) @@ -920,6 +915,10 @@ if __name__ == "__main__": num_dipoles = 11 dipole_rpos = np.linspace(0, DISC_RADIUS_UM, num_dipoles) + # An Er source at r=0 needs to be slighty offset. + # https://github.com/NanoComp/meep/issues/2704 + dipole_rpos[0] = 1.5 / RESOLUTION + delta_rpos = dipole_rpos[2] - dipole_rpos[1] # number of angular grid points in [0, π/2] @@ -988,6 +987,9 @@ if __name__ == "__main__": f"dipole:, {rpos:.4f}, {m}, " f"{dipole_radiation_pattern_total_flux:.6f}" ) + flux_total /= num_dipoles + radiation_pattern_total /= num_dipoles + radiation_pattern_total_flux = radiation_pattern_flux( theta_rad, radiation_pattern_total ) diff --git a/python/examples/disc_extraction_efficiency.py b/python/examples/disc_extraction_efficiency.py index 5f7979886..b6e5446d4 100644 --- a/python/examples/disc_extraction_efficiency.py +++ b/python/examples/disc_extraction_efficiency.py @@ -21,6 +21,8 @@ # radius of quarter circle for computing flux in far field FARFIELD_RADIUS_UM = 1000 * WAVELENGTH_UM +RESOLUTION = 50 # pixels/μm + def plot_radiation_pattern_polar(theta_rad: np.ndarray, radial_flux: np.ndarray): """Plots the radiation pattern in polar coordinates. @@ -156,8 +158,6 @@ def dipole_in_disc( Returns: A 2-tuple of the total flux and the radiation pattern. """ - resolution = 50 # pixels/μm - t_pml_um = 0.5 # thickness of PML t_air_um = 1.0 # thickness of air padding above disc length_r_um = 5.0 # length of cell in r @@ -176,11 +176,6 @@ def dipole_in_disc( mp.PML(t_pml_um, direction=mp.Z, side=mp.High), ] - # An Er source at r=0 needs to be slighty offset. - # https://github.com/NanoComp/meep/issues/2704 - if rpos == 0: - rpos = 1.5 / resolution - src_cmpt = mp.Er src_pt = mp.Vector3(rpos, 0, -0.5 * size_z + h_disc * t_disc_um) sources = [ @@ -200,7 +195,7 @@ def dipole_in_disc( ] sim = mp.Simulation( - resolution=resolution, + resolution=RESOLUTION, cell_size=cell_size, dimensions=mp.CYLINDRICAL, m=m, @@ -235,7 +230,7 @@ def dipole_in_disc( ), ) - delta_vol = 2 * np.pi * rpos / (resolution**2) + delta_vol = 2 * np.pi * rpos / (RESOLUTION**2) dipole_flux = -np.real(sim.ldos_Fdata[0] * np.conj(sim.ldos_Jdata[0])) * delta_vol dipole_radiation_pattern = radiation_pattern(theta_rad, sim, n2f_mon) @@ -249,6 +244,10 @@ def dipole_in_disc( num_dipoles = 11 dipole_rpos = np.linspace(0, DISC_RADIUS_UM, num_dipoles) + # An Er source at r=0 needs to be slighty offset. + # https://github.com/NanoComp/meep/issues/2704 + dipole_rpos[0] = 1.5 / RESOLUTION + delta_rpos = dipole_rpos[2] - dipole_rpos[1] # number of angular grid points in [0, π/2] @@ -317,6 +316,9 @@ def dipole_in_disc( f"dipole:, {rpos:.4f}, {m}, " f"{dipole_radiation_pattern_total_flux:.6f}" ) + flux_total /= num_dipoles + radiation_pattern_total /= num_dipoles + radiation_pattern_total_flux = radiation_pattern_flux( theta_rad, radiation_pattern_total ) From 62f3402c3245f94bbc6e985afd4f5c368ea9b8fd Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Wed, 17 Jan 2024 22:03:55 -0800 Subject: [PATCH 06/10] update scaling factor for dipoles at different positions and various improvements to simulation script --- .../Near_to_Far_Field_Spectra.md | 223 ++++++++---------- python/examples/disc_extraction_efficiency.py | 223 +++++++++--------- 2 files changed, 209 insertions(+), 237 deletions(-) diff --git a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md index f4c7ed213..c34916868 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -664,7 +664,7 @@ The example uses the same setup as the [previous tutorial](#radiation-pattern-of $$P(\theta) = \frac{1}{R} \int_0^R P(r,\theta) s(r) 2\pi rdr = \frac{1}{N} \sum_{n=0}^{N-1} P(r_n,\theta) s(r_n) 2\pi r_n \Delta r$$, -where $s(r)$ is a weighting function necessary for ensuring equal contribution from all dipoles relative to the dipole at $r = 0$. Note: an $E_r$ dipole at $r = 0$ must be placed at $r_0 = 1.5\Delta r$ due to an [interpolation bug](https://github.com/NanoComp/meep/issues/2704). $s(r)$ can be determined empirically by computing the emission in vacuum for a set of dipoles at different radial positions. The emission profile of a dipole in vacuum is a constant independent of its position. This criteria is used to obtain: $s(r) = \frac{1}{2(r/r_0)^2}$. This weighting function is also used to average the flux emitted by each dipole (obtained using using the LDOS feature). This quantity is the denominator in the expression for the extraction efficiency. +where $s(r)$ is a weighting function necessary for ensuring equal contribution from all dipoles relative to the dipole at $r = 0$. Note: an $E_r$ dipole at $r = 0$ must be placed at $r_0 = 1.5\Delta r$ due to an [interpolation bug](https://github.com/NanoComp/meep/issues/2704). $s(r)$ can be determined empirically by computing the emission in vacuum for a set of dipoles at different radial positions. The emission profile of a dipole in vacuum is a constant independent of its position. This criteria is used to obtain: $s(r) = (r_0/r)^2$. This weighting function is also used to average the flux emitted by each dipole (obtained using using the LDOS feature). This quantity is the denominator in the expression for the extraction efficiency. This figure shows the radiation pattern from $N=11$ dipoles with $\lambda$ of 1.0 $\mu$m in the middle of a disc of height 0.29 $\mu$m, radius 1.2 $\mu$m, and refractive index 2.4. @@ -677,36 +677,30 @@ The simulation script is in [examples/disc_extraction_efficiency.py](https://git import math from typing import Tuple -import matplotlib +import matplotlib.pyplot as plt import meep as mp import numpy as np -matplotlib.use("agg") -import matplotlib.pyplot as plt - - -N_DISC = 2.4 # refractive index of disc -DISC_RADIUS_UM = 1.2 # radius of disc -WAVELENGTH_UM = 1.0 # wavelength (in vacuum) -# radius of quarter circle for computing flux in far field -FARFIELD_RADIUS_UM = 1000 * WAVELENGTH_UM - -RESOLUTION = 50 # pixels/μm +RESOLUTION_UM = 50 +N_DISC = 2.4 +DISC_RADIUS_UM = 1.2 +WAVELENGTH_UM = 1.0 +NUM_FARFIELD_PTS = 200 +FARFIELD_ANGLES = np.linspace(0, 0.5 * math.pi, NUM_FARFIELD_PTS) +FARFIELD_RADIUS_UM = 1e6 * WAVELENGTH_UM +NUM_DIPOLES = 11 -def plot_radiation_pattern_polar(theta_rad: np.ndarray, radial_flux: np.ndarray): +def plot_radiation_pattern_polar(radial_flux: np.ndarray): """Plots the radiation pattern in polar coordinates. Args: - theta_rad: angles of the radiation pattern. The angles increase clockwise - with zero at the pole (+z direction) and π/2 at the equator (+r - direction). radial_flux: radial flux of the far fields in polar coordinates. """ fig, ax = plt.subplots(subplot_kw={"projection": "polar"}, figsize=(6, 6)) ax.plot( - theta_rad, + FARFIELD_ANGLES, radial_flux, "b-", ) @@ -726,20 +720,19 @@ def plot_radiation_pattern_polar(theta_rad: np.ndarray, radial_flux: np.ndarray) ) -def plot_radiation_pattern_3d(theta_rad: np.ndarray, radial_flux: np.ndarray): +def plot_radiation_pattern_3d(radial_flux: np.ndarray): """Plots the radiation pattern in 3d Cartesian coordinates. Args: - theta_rad: angles of the radiation pattern. radial_flux: radial flux of the far fields in polar coordinates. """ - phis = np.linspace(0, 2 * np.pi, num_angles) + phis = np.linspace(0, 2 * np.pi, NUM_FARFIELD_PTS) - xs = np.zeros((len(theta_rad), len(phis))) - ys = np.zeros((len(theta_rad), len(phis))) - zs = np.zeros((len(theta_rad), len(phis))) + xs = np.zeros((NUM_FARFIELD_PTS, NUM_FARFIELD_PTS)) + ys = np.zeros((NUM_FARFIELD_PTS, NUM_FARFIELD_PTS)) + zs = np.zeros((NUM_FARFIELD_PTS, NUM_FARFIELD_PTS)) - for i, theta in enumerate(theta_rad): + for i, theta in enumerate(FARFIELD_ANGLES): for j, phi in enumerate(phis): xs[i, j] = radial_flux[i] * np.sin(theta) * np.cos(phi) ys[i, j] = radial_flux[i] * np.sin(theta) * np.sin(phi) @@ -760,54 +753,55 @@ def plot_radiation_pattern_3d(theta_rad: np.ndarray, radial_flux: np.ndarray): ) -def radiation_pattern( - theta_rad: np.ndarray, sim: mp.Simulation, n2f_mon: mp.DftNear2Far -) -> np.ndarray: +def radiation_pattern(sim: mp.Simulation, n2f_mon: mp.DftNear2Far) -> np.ndarray: """Computes the radiation pattern from the near fields. Args: - theta_rad: angles of the radiation pattern. sim: a `Simulation` object. n2f_mon: a `DftNear2Far` object returned by `Simulation.add_near2far`. + + Returns: + The radiation pattern (radial flux at each angle) as a 1d array. """ - e_field = np.zeros((theta_rad.shape[0], 3), dtype=np.complex128) - h_field = np.zeros((theta_rad.shape[0], 3), dtype=np.complex128) - for n in range(num_angles): + e_field = np.zeros((NUM_FARFIELD_PTS, 3), dtype=np.complex128) + h_field = np.zeros((NUM_FARFIELD_PTS, 3), dtype=np.complex128) + for n in range(NUM_FARFIELD_PTS): far_field = sim.get_farfield( n2f_mon, mp.Vector3( - FARFIELD_RADIUS_UM * math.sin(theta_rad[n]), + FARFIELD_RADIUS_UM * math.sin(FARFIELD_ANGLES[n]), 0, - FARFIELD_RADIUS_UM * math.cos(theta_rad[n]), + FARFIELD_RADIUS_UM * math.cos(FARFIELD_ANGLES[n]), ), ) - e_field[n, :] = [np.conj(far_field[j]) for j in range(3)] + e_field[n, :] = [far_field[j] for j in range(3)] h_field[n, :] = [far_field[j + 3] for j in range(3)] - flux_r = np.real(e_field[:, 1] * h_field[:, 2] - e_field[:, 2] * h_field[:, 1]) - flux_z = np.real(e_field[:, 0] * h_field[:, 1] - e_field[:, 1] * h_field[:, 0]) - flux_rz = np.sqrt(np.square(flux_r) + np.square(flux_z)) + flux_x = np.real(np.conj(e_field[:, 1]) * h_field[:, 2] - + np.conj(e_field[:, 2]) * h_field[:, 1]) + flux_z = np.real(np.conj(e_field[:, 0]) * h_field[:, 1] - + np.conj(e_field[:, 1]) * h_field[:, 0]) + flux_r = np.sqrt(np.square(flux_x) + np.square(flux_z)) - return flux_rz + return flux_r -def radiation_pattern_flux(theta_rad: np.ndarray, radial_flux: np.ndarray) -> float: +def radiation_pattern_flux(radial_flux: np.ndarray) -> float: """Computes the total flux from the radiation pattern. Based on integrating the radiation pattern over solid angles spanned by polar angles in the range of [0, π/2]. Args: - theta_rad: angles of the radiation pattern. radial_flux: radial flux of the far fields in polar coordinates. """ dphi = 2 * math.pi - dtheta_rad = theta_rad[1] - theta_rad[0] + dtheta = FARFIELD_ANGLES[1] - FARFIELD_ANGLES[0] total_flux = ( - np.sum(radial_flux * np.sin(theta_rad)) + np.sum(radial_flux * np.sin(FARFIELD_ANGLES)) * FARFIELD_RADIUS_UM**2 - * dtheta_rad + * dtheta * dphi ) @@ -815,40 +809,39 @@ def radiation_pattern_flux(theta_rad: np.ndarray, radial_flux: np.ndarray) -> fl def dipole_in_disc( - t_disc_um: float, h_disc: float, rpos: float, m: int, theta_rad: np.ndarray + disc_um: float, zpos: float, rpos_um: float, m: int ) -> Tuple[float, np.ndarray]: """Computes the total flux and radiation pattern of a dipole in a disc. Args: - t_disc_um: thickness of disc. - h_disc: height of dipole above ground plane as fraction of t_disc_um. - rpos: radial position of dipole. + disc_um: thickness of disc. + zpos: height of dipole above ground plane as fraction of disc_um. + rpos_um: radial position of dipole. m: angular φ dependence of the fields exp(imφ). - theta_rad: angles of the radiation pattern. Returns: A 2-tuple of the total flux and the radiation pattern. """ - t_pml_um = 0.5 # thickness of PML - t_air_um = 1.0 # thickness of air padding above disc - length_r_um = 5.0 # length of cell in r + pml_um = 1.0 # thickness of PML + padding_um = 1.0 # thickness of air padding above disc + r_um = 5.0 # length of cell in r frequency = 1 / WAVELENGTH_UM # center frequency of source/monitor # field decay threshold for runtime termination criteria - decay_tol = 1e-6 + field_decay_threshold = 1e-6 - size_r = length_r_um + t_pml_um - size_z = t_disc_um + t_air_um + t_pml_um + size_r = r_um + pml_um + size_z = disc_um + padding_um + pml_um cell_size = mp.Vector3(size_r, 0, size_z) boundary_layers = [ - mp.PML(t_pml_um, direction=mp.R), - mp.PML(t_pml_um, direction=mp.Z, side=mp.High), + mp.PML(pml_um, direction=mp.R), + mp.PML(pml_um, direction=mp.Z, side=mp.High), ] src_cmpt = mp.Er - src_pt = mp.Vector3(rpos, 0, -0.5 * size_z + h_disc * t_disc_um) + src_pt = mp.Vector3(rpos_um, 0, -0.5 * size_z + zpos * disc_um) sources = [ mp.Source( src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), @@ -860,13 +853,15 @@ def dipole_in_disc( geometry = [ mp.Block( material=mp.Medium(index=N_DISC), - center=mp.Vector3(0.5 * DISC_RADIUS_UM, 0, -0.5 * size_z + 0.5 * t_disc_um), - size=mp.Vector3(DISC_RADIUS_UM, mp.inf, t_disc_um), + center=mp.Vector3( + 0.5 * DISC_RADIUS_UM, 0, -0.5 * size_z + 0.5 * disc_um + ), + size=mp.Vector3(DISC_RADIUS_UM, mp.inf, disc_um), ) ] sim = mp.Simulation( - resolution=RESOLUTION, + resolution=RESOLUTION_UM, cell_size=cell_size, dimensions=mp.CYLINDRICAL, m=m, @@ -880,14 +875,16 @@ def dipole_in_disc( 0, 1, mp.FluxRegion( - center=mp.Vector3(0.5 * length_r_um, 0, 0.5 * size_z - t_pml_um), - size=mp.Vector3(length_r_um, 0, 0), + center=mp.Vector3(0.5 * r_um, 0, 0.5 * size_z - pml_um), + size=mp.Vector3(r_um, 0, 0), ), mp.FluxRegion( center=mp.Vector3( - length_r_um, 0, 0.5 * size_z - t_pml_um - 0.5 * (t_air_um + t_disc_um) + r_um, + 0, + 0.5 * size_z - pml_um - 0.5 * (padding_um + disc_um) ), - size=mp.Vector3(0, 0, t_air_um + t_disc_um), + size=mp.Vector3(0, 0, padding_um + disc_um), ), ) @@ -897,14 +894,15 @@ def dipole_in_disc( 50, src_cmpt, src_pt, - decay_tol, + field_decay_threshold, ), ) - delta_vol = 2 * np.pi * rpos / (RESOLUTION**2) - dipole_flux = -np.real(sim.ldos_Fdata[0] * np.conj(sim.ldos_Jdata[0])) * delta_vol + delta_vol = 2 * np.pi * rpos_um / (RESOLUTION_UM**2) + dipole_flux = (-np.real(sim.ldos_Fdata[0] * np.conj(sim.ldos_Jdata[0])) * + delta_vol) - dipole_radiation_pattern = radiation_pattern(theta_rad, sim, n2f_mon) + dipole_radiation_pattern = radiation_pattern(sim, n2f_mon) return dipole_flux, dipole_radiation_pattern @@ -912,92 +910,79 @@ def dipole_in_disc( if __name__ == "__main__": disc_thickness = 0.7 * WAVELENGTH_UM / N_DISC dipole_height = 0.5 - num_dipoles = 11 - dipole_rpos = np.linspace(0, DISC_RADIUS_UM, num_dipoles) + dipole_rpos_um = np.linspace(0, DISC_RADIUS_UM, NUM_DIPOLES) - # An Er source at r=0 needs to be slighty offset. + # An Er source at r = 0 needs to be slighty offset. # https://github.com/NanoComp/meep/issues/2704 - dipole_rpos[0] = 1.5 / RESOLUTION - - delta_rpos = dipole_rpos[2] - dipole_rpos[1] + dipole_rpos_um[0] = 1.5 / RESOLUTION_UM - # number of angular grid points in [0, π/2] - num_angles = 100 + delta_rpos_um = dipole_rpos_um[2] - dipole_rpos_um[1] - # grid of polar angles for computing radiated flux in far field - theta_rad = np.linspace(0, 0.5 * math.pi, num_angles) - - # r = 0 source requires a single simulation with m = ±1. + # Er source at r = 0 requires a single simulation with m = ±1. m = -1 dipole_flux, dipole_radiation_pattern = dipole_in_disc( - disc_thickness, dipole_height, dipole_rpos[0], m, theta_rad + disc_thickness, dipole_height, dipole_rpos_um[0], m, ) - flux_total = dipole_flux * dipole_rpos[0] * delta_rpos - radiation_pattern_total = dipole_radiation_pattern * dipole_rpos[0] * delta_rpos + flux_total = dipole_flux * dipole_rpos_um[0] * delta_rpos_um + radiation_pattern_total = dipole_radiation_pattern * dipole_rpos_um[0] * delta_rpos_um print( - f"dipole:, {dipole_rpos[0]:.4f}, " - f"{radiation_pattern_flux(theta_rad, dipole_radiation_pattern):.6f}" + f"dipole:, {dipole_rpos_um[0]:.4f}, " + f"{radiation_pattern_flux(dipole_radiation_pattern):.6f}" ) - # r > 0 source requires Fourier-series expansion of φ. - flux_tol = 1e-3 # threshold flux to determine when to truncate expansion - for rpos in dipole_rpos[1:]: + # Er source at r > 0 requires Fourier-series expansion of φ. + + # Threshold flux to determine when to truncate expansion. + flux_decay_threshold = 1e-2 + + for rpos_um in dipole_rpos_um[1:]: dipole_flux_total = 0 - dipole_radiation_pattern_total = np.zeros((num_angles,)) + dipole_radiation_pattern_total = np.zeros((NUM_FARFIELD_PTS,)) dipole_radiation_pattern_flux_max = 0 m = 0 while True: dipole_flux, dipole_radiation_pattern = dipole_in_disc( - disc_thickness, dipole_height, rpos, m, theta_rad - ) - dipole_flux_total += dipole_flux if m == 0 else 2 * dipole_flux - dipole_radiation_pattern_total += ( - dipole_radiation_pattern if m == 0 else 2 * dipole_radiation_pattern + disc_thickness, dipole_height, rpos_um, m ) + dipole_flux_total += dipole_flux * (2 if m == 0 else 1) + dipole_radiation_pattern_total += (dipole_radiation_pattern * + (2 if m == 0 else 1)) dipole_radiation_pattern_flux = radiation_pattern_flux( - theta_rad, dipole_radiation_pattern + dipole_radiation_pattern ) - if dipole_radiation_pattern_flux > dipole_radiation_pattern_flux_max: + if (dipole_radiation_pattern_flux > + dipole_radiation_pattern_flux_max): dipole_radiation_pattern_flux_max = dipole_radiation_pattern_flux - if m > 0 and ( - (dipole_radiation_pattern_flux / dipole_radiation_pattern_flux_max) - < flux_tol - ): + if (m > 0 and + (dipole_radiation_pattern_flux / + dipole_radiation_pattern_flux_max) < flux_decay_threshold): break print( - f"dipole-m:, {rpos:.4f}, {m}, " f"{dipole_radiation_pattern_flux:.6f}" + f"dipole:, {rpos_um:.4f}, {m}, {dipole_radiation_pattern_flux:.6f}" ) m += 1 - scale_factor = 1 / (2 * (rpos / dipole_rpos[0]) ** 2) - flux_total += dipole_flux_total * scale_factor * rpos * delta_rpos + scale_factor = (dipole_rpos_um[0] / rpos_um)**2 + flux_total += dipole_flux_total * scale_factor * rpos_um * delta_rpos_um radiation_pattern_total += ( - dipole_radiation_pattern_total * scale_factor * rpos * delta_rpos + dipole_radiation_pattern_total * scale_factor * rpos_um * delta_rpos_um ) - dipole_radiation_pattern_total_flux = radiation_pattern_flux( - theta_rad, dipole_radiation_pattern_total - ) - print( - f"dipole:, {rpos:.4f}, {m}, " f"{dipole_radiation_pattern_total_flux:.6f}" - ) - flux_total /= num_dipoles - radiation_pattern_total /= num_dipoles - - radiation_pattern_total_flux = radiation_pattern_flux( - theta_rad, radiation_pattern_total - ) + flux_total /= NUM_DIPOLES + radiation_pattern_total /= NUM_DIPOLES + radiation_pattern_total_flux = radiation_pattern_flux(radiation_pattern_total) extraction_efficiency = radiation_pattern_total_flux / flux_total print(f"exteff:, {extraction_efficiency:.6f}") - plot_radiation_pattern_polar(radiation_pattern_total * FARFIELD_RADIUS_UM**2) - plot_radiation_pattern_3d(radiation_pattern_total * FARFIELD_RADIUS_UM**2) + radiation_pattern_scaled = radiation_pattern_total * FARFIELD_RADIUS_UM**2 + plot_radiation_pattern_polar(radiation_pattern_scaled) + plot_radiation_pattern_3d(radiation_pattern_scaled) ``` Focusing Properties of a Metasurface Lens diff --git a/python/examples/disc_extraction_efficiency.py b/python/examples/disc_extraction_efficiency.py index b6e5446d4..5dde21108 100644 --- a/python/examples/disc_extraction_efficiency.py +++ b/python/examples/disc_extraction_efficiency.py @@ -1,41 +1,36 @@ """Computes the extraction efficiency of a collection of dipoles in a disc. -tutorial reference: https://meep.readthedocs.io/en/latest/Python_Tutorials/Near_to_Far_Field_Spectra/#extraction-efficiency-of-a-disc-in-cylindrical-coordinates +tutorial reference: +https://meep.readthedocs.io/en/latest/Python_Tutorials/Near_to_Far_Field_Spectra/#extraction-efficiency-of-a-disc-in-cylindrical-coordinates """ import math from typing import Tuple -import matplotlib +import matplotlib.pyplot as plt import meep as mp import numpy as np -matplotlib.use("agg") -import matplotlib.pyplot as plt - - -N_DISC = 2.4 # refractive index of disc -DISC_RADIUS_UM = 1.2 # radius of disc -WAVELENGTH_UM = 1.0 # wavelength (in vacuum) - -# radius of quarter circle for computing flux in far field -FARFIELD_RADIUS_UM = 1000 * WAVELENGTH_UM -RESOLUTION = 50 # pixels/μm +RESOLUTION_UM = 50 +N_DISC = 2.4 +DISC_RADIUS_UM = 1.2 +WAVELENGTH_UM = 1.0 +NUM_FARFIELD_PTS = 200 +FARFIELD_ANGLES = np.linspace(0, 0.5 * math.pi, NUM_FARFIELD_PTS) +FARFIELD_RADIUS_UM = 1e6 * WAVELENGTH_UM +NUM_DIPOLES = 11 -def plot_radiation_pattern_polar(theta_rad: np.ndarray, radial_flux: np.ndarray): +def plot_radiation_pattern_polar(radial_flux: np.ndarray): """Plots the radiation pattern in polar coordinates. Args: - theta_rad: angles of the radiation pattern. The angles increase clockwise - with zero at the pole (+z direction) and π/2 at the equator (+r - direction). radial_flux: radial flux of the far fields in polar coordinates. """ fig, ax = plt.subplots(subplot_kw={"projection": "polar"}, figsize=(6, 6)) ax.plot( - theta_rad, + FARFIELD_ANGLES, radial_flux, "b-", ) @@ -55,20 +50,19 @@ def plot_radiation_pattern_polar(theta_rad: np.ndarray, radial_flux: np.ndarray) ) -def plot_radiation_pattern_3d(theta_rad: np.ndarray, radial_flux: np.ndarray): +def plot_radiation_pattern_3d(radial_flux: np.ndarray): """Plots the radiation pattern in 3d Cartesian coordinates. Args: - theta_rad: angles of the radiation pattern. radial_flux: radial flux of the far fields in polar coordinates. """ - phis = np.linspace(0, 2 * np.pi, num_angles) + phis = np.linspace(0, 2 * np.pi, NUM_FARFIELD_PTS) - xs = np.zeros((len(theta_rad), len(phis))) - ys = np.zeros((len(theta_rad), len(phis))) - zs = np.zeros((len(theta_rad), len(phis))) + xs = np.zeros((NUM_FARFIELD_PTS, NUM_FARFIELD_PTS)) + ys = np.zeros((NUM_FARFIELD_PTS, NUM_FARFIELD_PTS)) + zs = np.zeros((NUM_FARFIELD_PTS, NUM_FARFIELD_PTS)) - for i, theta in enumerate(theta_rad): + for i, theta in enumerate(FARFIELD_ANGLES): for j, phi in enumerate(phis): xs[i, j] = radial_flux[i] * np.sin(theta) * np.cos(phi) ys[i, j] = radial_flux[i] * np.sin(theta) * np.sin(phi) @@ -89,54 +83,57 @@ def plot_radiation_pattern_3d(theta_rad: np.ndarray, radial_flux: np.ndarray): ) -def radiation_pattern( - theta_rad: np.ndarray, sim: mp.Simulation, n2f_mon: mp.DftNear2Far -) -> np.ndarray: +def radiation_pattern(sim: mp.Simulation, n2f_mon: mp.DftNear2Far) -> np.ndarray: """Computes the radiation pattern from the near fields. Args: - theta_rad: angles of the radiation pattern. sim: a `Simulation` object. n2f_mon: a `DftNear2Far` object returned by `Simulation.add_near2far`. + + Returns: + The radiation pattern (radial flux at each angle) as a 1d array. """ - e_field = np.zeros((theta_rad.shape[0], 3), dtype=np.complex128) - h_field = np.zeros((theta_rad.shape[0], 3), dtype=np.complex128) - for n in range(num_angles): + e_field = np.zeros((NUM_FARFIELD_PTS, 3), dtype=np.complex128) + h_field = np.zeros((NUM_FARFIELD_PTS, 3), dtype=np.complex128) + for n in range(NUM_FARFIELD_PTS): far_field = sim.get_farfield( n2f_mon, mp.Vector3( - FARFIELD_RADIUS_UM * math.sin(theta_rad[n]), + FARFIELD_RADIUS_UM * math.sin(FARFIELD_ANGLES[n]), 0, - FARFIELD_RADIUS_UM * math.cos(theta_rad[n]), + FARFIELD_RADIUS_UM * math.cos(FARFIELD_ANGLES[n]), ), ) - e_field[n, :] = [np.conj(far_field[j]) for j in range(3)] + e_field[n, :] = [far_field[j] for j in range(3)] h_field[n, :] = [far_field[j + 3] for j in range(3)] - flux_r = np.real(e_field[:, 1] * h_field[:, 2] - e_field[:, 2] * h_field[:, 1]) - flux_z = np.real(e_field[:, 0] * h_field[:, 1] - e_field[:, 1] * h_field[:, 0]) - flux_rz = np.sqrt(np.square(flux_r) + np.square(flux_z)) + flux_x = np.real( + np.conj(e_field[:, 1]) * h_field[:, 2] - np.conj(e_field[:, 2]) * h_field[:, 1] + ) + flux_z = np.real( + np.conj(e_field[:, 0]) * h_field[:, 1] - np.conj(e_field[:, 1]) * h_field[:, 0] + ) + flux_r = np.sqrt(np.square(flux_x) + np.square(flux_z)) - return flux_rz + return flux_r -def radiation_pattern_flux(theta_rad: np.ndarray, radial_flux: np.ndarray) -> float: +def radiation_pattern_flux(radial_flux: np.ndarray) -> float: """Computes the total flux from the radiation pattern. Based on integrating the radiation pattern over solid angles spanned by polar angles in the range of [0, π/2]. Args: - theta_rad: angles of the radiation pattern. radial_flux: radial flux of the far fields in polar coordinates. """ dphi = 2 * math.pi - dtheta_rad = theta_rad[1] - theta_rad[0] + dtheta = FARFIELD_ANGLES[1] - FARFIELD_ANGLES[0] total_flux = ( - np.sum(radial_flux * np.sin(theta_rad)) + np.sum(radial_flux * np.sin(FARFIELD_ANGLES)) * FARFIELD_RADIUS_UM**2 - * dtheta_rad + * dtheta * dphi ) @@ -144,40 +141,39 @@ def radiation_pattern_flux(theta_rad: np.ndarray, radial_flux: np.ndarray) -> fl def dipole_in_disc( - t_disc_um: float, h_disc: float, rpos: float, m: int, theta_rad: np.ndarray + disc_um: float, zpos: float, rpos_um: float, m: int ) -> Tuple[float, np.ndarray]: """Computes the total flux and radiation pattern of a dipole in a disc. Args: - t_disc_um: thickness of disc. - h_disc: height of dipole above ground plane as fraction of t_disc_um. - rpos: radial position of dipole. + disc_um: thickness of disc. + zpos: height of dipole above ground plane as fraction of disc_um. + rpos_um: radial position of dipole. m: angular φ dependence of the fields exp(imφ). - theta_rad: angles of the radiation pattern. Returns: A 2-tuple of the total flux and the radiation pattern. """ - t_pml_um = 0.5 # thickness of PML - t_air_um = 1.0 # thickness of air padding above disc - length_r_um = 5.0 # length of cell in r + pml_um = 1.0 # thickness of PML + padding_um = 1.0 # thickness of air padding above disc + r_um = 5.0 # length of cell in r frequency = 1 / WAVELENGTH_UM # center frequency of source/monitor # field decay threshold for runtime termination criteria - decay_tol = 1e-6 + field_decay_threshold = 1e-6 - size_r = length_r_um + t_pml_um - size_z = t_disc_um + t_air_um + t_pml_um + size_r = r_um + pml_um + size_z = disc_um + padding_um + pml_um cell_size = mp.Vector3(size_r, 0, size_z) boundary_layers = [ - mp.PML(t_pml_um, direction=mp.R), - mp.PML(t_pml_um, direction=mp.Z, side=mp.High), + mp.PML(pml_um, direction=mp.R), + mp.PML(pml_um, direction=mp.Z, side=mp.High), ] src_cmpt = mp.Er - src_pt = mp.Vector3(rpos, 0, -0.5 * size_z + h_disc * t_disc_um) + src_pt = mp.Vector3(rpos_um, 0, -0.5 * size_z + zpos * disc_um) sources = [ mp.Source( src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), @@ -189,13 +185,13 @@ def dipole_in_disc( geometry = [ mp.Block( material=mp.Medium(index=N_DISC), - center=mp.Vector3(0.5 * DISC_RADIUS_UM, 0, -0.5 * size_z + 0.5 * t_disc_um), - size=mp.Vector3(DISC_RADIUS_UM, mp.inf, t_disc_um), + center=mp.Vector3(0.5 * DISC_RADIUS_UM, 0, -0.5 * size_z + 0.5 * disc_um), + size=mp.Vector3(DISC_RADIUS_UM, mp.inf, disc_um), ) ] sim = mp.Simulation( - resolution=RESOLUTION, + resolution=RESOLUTION_UM, cell_size=cell_size, dimensions=mp.CYLINDRICAL, m=m, @@ -209,14 +205,14 @@ def dipole_in_disc( 0, 1, mp.FluxRegion( - center=mp.Vector3(0.5 * length_r_um, 0, 0.5 * size_z - t_pml_um), - size=mp.Vector3(length_r_um, 0, 0), + center=mp.Vector3(0.5 * r_um, 0, 0.5 * size_z - pml_um), + size=mp.Vector3(r_um, 0, 0), ), mp.FluxRegion( center=mp.Vector3( - length_r_um, 0, 0.5 * size_z - t_pml_um - 0.5 * (t_air_um + t_disc_um) + r_um, 0, 0.5 * size_z - pml_um - 0.5 * (padding_um + disc_um) ), - size=mp.Vector3(0, 0, t_air_um + t_disc_um), + size=mp.Vector3(0, 0, padding_um + disc_um), ), ) @@ -226,14 +222,14 @@ def dipole_in_disc( 50, src_cmpt, src_pt, - decay_tol, + field_decay_threshold, ), ) - delta_vol = 2 * np.pi * rpos / (RESOLUTION**2) + delta_vol = 2 * np.pi * rpos_um / (RESOLUTION_UM**2) dipole_flux = -np.real(sim.ldos_Fdata[0] * np.conj(sim.ldos_Jdata[0])) * delta_vol - dipole_radiation_pattern = radiation_pattern(theta_rad, sim, n2f_mon) + dipole_radiation_pattern = radiation_pattern(sim, n2f_mon) return dipole_flux, dipole_radiation_pattern @@ -241,89 +237,80 @@ def dipole_in_disc( if __name__ == "__main__": disc_thickness = 0.7 * WAVELENGTH_UM / N_DISC dipole_height = 0.5 - num_dipoles = 11 - dipole_rpos = np.linspace(0, DISC_RADIUS_UM, num_dipoles) + dipole_rpos_um = np.linspace(0, DISC_RADIUS_UM, NUM_DIPOLES) - # An Er source at r=0 needs to be slighty offset. + # An Er source at r = 0 needs to be slighty offset. # https://github.com/NanoComp/meep/issues/2704 - dipole_rpos[0] = 1.5 / RESOLUTION - - delta_rpos = dipole_rpos[2] - dipole_rpos[1] + dipole_rpos_um[0] = 1.5 / RESOLUTION_UM - # number of angular grid points in [0, π/2] - num_angles = 100 + delta_rpos_um = dipole_rpos_um[2] - dipole_rpos_um[1] - # grid of polar angles for computing radiated flux in far field - theta_rad = np.linspace(0, 0.5 * math.pi, num_angles) - - # r = 0 source requires a single simulation with m = ±1. + # Er source at r = 0 requires a single simulation with m = ±1. m = -1 dipole_flux, dipole_radiation_pattern = dipole_in_disc( - disc_thickness, dipole_height, dipole_rpos[0], m, theta_rad + disc_thickness, + dipole_height, + dipole_rpos_um[0], + m, ) - flux_total = dipole_flux * dipole_rpos[0] * delta_rpos - radiation_pattern_total = dipole_radiation_pattern * dipole_rpos[0] * delta_rpos + flux_total = dipole_flux * dipole_rpos_um[0] * delta_rpos_um + radiation_pattern_total = ( + dipole_radiation_pattern * dipole_rpos_um[0] * delta_rpos_um + ) print( - f"dipole:, {dipole_rpos[0]:.4f}, " - f"{radiation_pattern_flux(theta_rad, dipole_radiation_pattern):.6f}" + f"dipole:, {dipole_rpos_um[0]:.4f}, " + f"{radiation_pattern_flux(dipole_radiation_pattern):.6f}" ) - # r > 0 source requires Fourier-series expansion of φ. - flux_tol = 1e-3 # threshold flux to determine when to truncate expansion - for rpos in dipole_rpos[1:]: + # Er source at r > 0 requires Fourier-series expansion of φ. + + # Threshold flux to determine when to truncate expansion. + flux_decay_threshold = 1e-2 + + for rpos_um in dipole_rpos_um[1:]: dipole_flux_total = 0 - dipole_radiation_pattern_total = np.zeros((num_angles,)) + dipole_radiation_pattern_total = np.zeros((NUM_FARFIELD_PTS,)) dipole_radiation_pattern_flux_max = 0 m = 0 while True: dipole_flux, dipole_radiation_pattern = dipole_in_disc( - disc_thickness, dipole_height, rpos, m, theta_rad + disc_thickness, dipole_height, rpos_um, m ) - dipole_flux_total += dipole_flux if m == 0 else 2 * dipole_flux - dipole_radiation_pattern_total += ( - dipole_radiation_pattern if m == 0 else 2 * dipole_radiation_pattern + dipole_flux_total += dipole_flux * (2 if m == 0 else 1) + dipole_radiation_pattern_total += dipole_radiation_pattern * ( + 2 if m == 0 else 1 ) dipole_radiation_pattern_flux = radiation_pattern_flux( - theta_rad, dipole_radiation_pattern + dipole_radiation_pattern ) if dipole_radiation_pattern_flux > dipole_radiation_pattern_flux_max: dipole_radiation_pattern_flux_max = dipole_radiation_pattern_flux - if m > 0 and ( - (dipole_radiation_pattern_flux / dipole_radiation_pattern_flux_max) - < flux_tol + if ( + m > 0 + and (dipole_radiation_pattern_flux / dipole_radiation_pattern_flux_max) + < flux_decay_threshold ): break - print( - f"dipole-m:, {rpos:.4f}, {m}, " f"{dipole_radiation_pattern_flux:.6f}" - ) + print(f"dipole:, {rpos_um:.4f}, {m}, {dipole_radiation_pattern_flux:.6f}") m += 1 - scale_factor = 1 / (2 * (rpos / dipole_rpos[0]) ** 2) - flux_total += dipole_flux_total * scale_factor * rpos * delta_rpos + scale_factor = (dipole_rpos_um[0] / rpos_um) ** 2 + flux_total += dipole_flux_total * scale_factor * rpos_um * delta_rpos_um radiation_pattern_total += ( - dipole_radiation_pattern_total * scale_factor * rpos * delta_rpos + dipole_radiation_pattern_total * scale_factor * rpos_um * delta_rpos_um ) - dipole_radiation_pattern_total_flux = radiation_pattern_flux( - theta_rad, dipole_radiation_pattern_total - ) - print( - f"dipole:, {rpos:.4f}, {m}, " f"{dipole_radiation_pattern_total_flux:.6f}" - ) - - flux_total /= num_dipoles - radiation_pattern_total /= num_dipoles - - radiation_pattern_total_flux = radiation_pattern_flux( - theta_rad, radiation_pattern_total - ) + flux_total /= NUM_DIPOLES + radiation_pattern_total /= NUM_DIPOLES + radiation_pattern_total_flux = radiation_pattern_flux(radiation_pattern_total) extraction_efficiency = radiation_pattern_total_flux / flux_total print(f"exteff:, {extraction_efficiency:.6f}") - plot_radiation_pattern_polar(radiation_pattern_total * FARFIELD_RADIUS_UM**2) - plot_radiation_pattern_3d(radiation_pattern_total * FARFIELD_RADIUS_UM**2) + radiation_pattern_scaled = radiation_pattern_total * FARFIELD_RADIUS_UM**2 + plot_radiation_pattern_polar(radiation_pattern_scaled) + plot_radiation_pattern_3d(radiation_pattern_scaled) From e20690ca9c60a2650604c0ee4948b35c821449e4 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Thu, 18 Jan 2024 11:45:30 -0800 Subject: [PATCH 07/10] switch runtime termination criteria to stop_when_dft_decayed due to cavity modes with small decay rates for certain dipole configurations --- .../Near_to_Far_Field_Spectra.md | 101 +++++++++--------- python/examples/disc_extraction_efficiency.py | 67 ++++++------ 2 files changed, 83 insertions(+), 85 deletions(-) diff --git a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md index c34916868..f5efaf3d8 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -683,14 +683,16 @@ import numpy as np RESOLUTION_UM = 50 +WAVELENGTH_UM = 1.0 N_DISC = 2.4 DISC_RADIUS_UM = 1.2 -WAVELENGTH_UM = 1.0 +DISC_THICKNESS_UM = 0.7 * WAVELENGTH_UM / N_DISC NUM_FARFIELD_PTS = 200 -FARFIELD_ANGLES = np.linspace(0, 0.5 * math.pi, NUM_FARFIELD_PTS) FARFIELD_RADIUS_UM = 1e6 * WAVELENGTH_UM NUM_DIPOLES = 11 +farfield_angles = np.linspace(0, 0.5 * math.pi, NUM_FARFIELD_PTS) + def plot_radiation_pattern_polar(radial_flux: np.ndarray): """Plots the radiation pattern in polar coordinates. @@ -700,7 +702,7 @@ def plot_radiation_pattern_polar(radial_flux: np.ndarray): """ fig, ax = plt.subplots(subplot_kw={"projection": "polar"}, figsize=(6, 6)) ax.plot( - FARFIELD_ANGLES, + farfield_angles, radial_flux, "b-", ) @@ -732,7 +734,7 @@ def plot_radiation_pattern_3d(radial_flux: np.ndarray): ys = np.zeros((NUM_FARFIELD_PTS, NUM_FARFIELD_PTS)) zs = np.zeros((NUM_FARFIELD_PTS, NUM_FARFIELD_PTS)) - for i, theta in enumerate(FARFIELD_ANGLES): + for i, theta in enumerate(farfield_angles): for j, phi in enumerate(phis): xs[i, j] = radial_flux[i] * np.sin(theta) * np.cos(phi) ys[i, j] = radial_flux[i] * np.sin(theta) * np.sin(phi) @@ -769,18 +771,20 @@ def radiation_pattern(sim: mp.Simulation, n2f_mon: mp.DftNear2Far) -> np.ndarray far_field = sim.get_farfield( n2f_mon, mp.Vector3( - FARFIELD_RADIUS_UM * math.sin(FARFIELD_ANGLES[n]), + FARFIELD_RADIUS_UM * math.sin(farfield_angles[n]), 0, - FARFIELD_RADIUS_UM * math.cos(FARFIELD_ANGLES[n]), + FARFIELD_RADIUS_UM * math.cos(farfield_angles[n]), ), ) e_field[n, :] = [far_field[j] for j in range(3)] h_field[n, :] = [far_field[j + 3] for j in range(3)] - flux_x = np.real(np.conj(e_field[:, 1]) * h_field[:, 2] - - np.conj(e_field[:, 2]) * h_field[:, 1]) - flux_z = np.real(np.conj(e_field[:, 0]) * h_field[:, 1] - - np.conj(e_field[:, 1]) * h_field[:, 0]) + flux_x = np.real( + np.conj(e_field[:, 1]) * h_field[:, 2] - np.conj(e_field[:, 2]) * h_field[:, 1] + ) + flux_z = np.real( + np.conj(e_field[:, 0]) * h_field[:, 1] - np.conj(e_field[:, 1]) * h_field[:, 0] + ) flux_r = np.sqrt(np.square(flux_x) + np.square(flux_z)) return flux_r @@ -789,17 +793,17 @@ def radiation_pattern(sim: mp.Simulation, n2f_mon: mp.DftNear2Far) -> np.ndarray def radiation_pattern_flux(radial_flux: np.ndarray) -> float: """Computes the total flux from the radiation pattern. - Based on integrating the radiation pattern over solid angles spanned by - polar angles in the range of [0, π/2]. + Based on integrating the radiation pattern over solid angles + spanned by polar angles in the range of [0, π/2]. Args: radial_flux: radial flux of the far fields in polar coordinates. """ dphi = 2 * math.pi - dtheta = FARFIELD_ANGLES[1] - FARFIELD_ANGLES[0] + dtheta = farfield_angles[1] - farfield_angles[0] total_flux = ( - np.sum(radial_flux * np.sin(FARFIELD_ANGLES)) + np.sum(radial_flux * np.sin(farfield_angles)) * FARFIELD_RADIUS_UM**2 * dtheta * dphi @@ -809,13 +813,12 @@ def radiation_pattern_flux(radial_flux: np.ndarray) -> float: def dipole_in_disc( - disc_um: float, zpos: float, rpos_um: float, m: int + zpos: float, rpos_um: float, m: int ) -> Tuple[float, np.ndarray]: """Computes the total flux and radiation pattern of a dipole in a disc. Args: - disc_um: thickness of disc. - zpos: height of dipole above ground plane as fraction of disc_um. + zpos: height of dipole above ground plane as fraction of disc thickness. rpos_um: radial position of dipole. m: angular φ dependence of the fields exp(imφ). @@ -828,11 +831,11 @@ def dipole_in_disc( frequency = 1 / WAVELENGTH_UM # center frequency of source/monitor - # field decay threshold for runtime termination criteria - field_decay_threshold = 1e-6 + # runtime termination criteria + dft_decay_threshold = 1e-4 size_r = r_um + pml_um - size_z = disc_um + padding_um + pml_um + size_z = DISC_THICKNESS_UM + padding_um + pml_um cell_size = mp.Vector3(size_r, 0, size_z) boundary_layers = [ @@ -841,7 +844,7 @@ def dipole_in_disc( ] src_cmpt = mp.Er - src_pt = mp.Vector3(rpos_um, 0, -0.5 * size_z + zpos * disc_um) + src_pt = mp.Vector3(rpos_um, 0, -0.5 * size_z + zpos * DISC_THICKNESS_UM) sources = [ mp.Source( src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), @@ -853,10 +856,8 @@ def dipole_in_disc( geometry = [ mp.Block( material=mp.Medium(index=N_DISC), - center=mp.Vector3( - 0.5 * DISC_RADIUS_UM, 0, -0.5 * size_z + 0.5 * disc_um - ), - size=mp.Vector3(DISC_RADIUS_UM, mp.inf, disc_um), + center=mp.Vector3(0.5 * DISC_RADIUS_UM, 0, -0.5 * size_z + 0.5 * DISC_THICKNESS_UM), + size=mp.Vector3(DISC_RADIUS_UM, mp.inf, DISC_THICKNESS_UM), ) ] @@ -880,27 +881,21 @@ def dipole_in_disc( ), mp.FluxRegion( center=mp.Vector3( - r_um, - 0, - 0.5 * size_z - pml_um - 0.5 * (padding_um + disc_um) + r_um, 0, 0.5 * size_z - pml_um - 0.5 * (padding_um + DISC_THICKNESS_UM) ), - size=mp.Vector3(0, 0, padding_um + disc_um), + size=mp.Vector3(0, 0, padding_um + DISC_THICKNESS_UM), ), ) sim.run( mp.dft_ldos(frequency, 0, 1), - until_after_sources=mp.stop_when_fields_decayed( - 50, - src_cmpt, - src_pt, - field_decay_threshold, + until_after_sources=mp.stop_when_dft_decayed( + tol=dft_decay_threshold, ), ) delta_vol = 2 * np.pi * rpos_um / (RESOLUTION_UM**2) - dipole_flux = (-np.real(sim.ldos_Fdata[0] * np.conj(sim.ldos_Jdata[0])) * - delta_vol) + dipole_flux = -np.real(sim.ldos_Fdata[0] * np.conj(sim.ldos_Jdata[0])) * delta_vol dipole_radiation_pattern = radiation_pattern(sim, n2f_mon) @@ -908,7 +903,6 @@ def dipole_in_disc( if __name__ == "__main__": - disc_thickness = 0.7 * WAVELENGTH_UM / N_DISC dipole_height = 0.5 dipole_rpos_um = np.linspace(0, DISC_RADIUS_UM, NUM_DIPOLES) @@ -921,11 +915,15 @@ if __name__ == "__main__": # Er source at r = 0 requires a single simulation with m = ±1. m = -1 dipole_flux, dipole_radiation_pattern = dipole_in_disc( - disc_thickness, dipole_height, dipole_rpos_um[0], m, + dipole_height, + dipole_rpos_um[0], + m, ) flux_total = dipole_flux * dipole_rpos_um[0] * delta_rpos_um - radiation_pattern_total = dipole_radiation_pattern * dipole_rpos_um[0] * delta_rpos_um + radiation_pattern_total = ( + dipole_radiation_pattern * dipole_rpos_um[0] * delta_rpos_um + ) print( f"dipole:, {dipole_rpos_um[0]:.4f}, " @@ -944,36 +942,37 @@ if __name__ == "__main__": m = 0 while True: dipole_flux, dipole_radiation_pattern = dipole_in_disc( - disc_thickness, dipole_height, rpos_um, m + dipole_height, rpos_um, m ) dipole_flux_total += dipole_flux * (2 if m == 0 else 1) - dipole_radiation_pattern_total += (dipole_radiation_pattern * - (2 if m == 0 else 1)) + dipole_radiation_pattern_total += dipole_radiation_pattern * ( + 2 if m == 0 else 1 + ) dipole_radiation_pattern_flux = radiation_pattern_flux( dipole_radiation_pattern ) - if (dipole_radiation_pattern_flux > - dipole_radiation_pattern_flux_max): + print( + f"dipole:, {rpos_um:.4f}, {m}, " + f"{dipole_radiation_pattern_flux:.6f}" + ) + + if dipole_radiation_pattern_flux > dipole_radiation_pattern_flux_max: dipole_radiation_pattern_flux_max = dipole_radiation_pattern_flux if (m > 0 and (dipole_radiation_pattern_flux / dipole_radiation_pattern_flux_max) < flux_decay_threshold): break + else: + m += 1 - print( - f"dipole:, {rpos_um:.4f}, {m}, {dipole_radiation_pattern_flux:.6f}" - ) - m += 1 - - scale_factor = (dipole_rpos_um[0] / rpos_um)**2 + scale_factor = (dipole_rpos_um[0] / rpos_um) ** 2 flux_total += dipole_flux_total * scale_factor * rpos_um * delta_rpos_um radiation_pattern_total += ( dipole_radiation_pattern_total * scale_factor * rpos_um * delta_rpos_um ) - flux_total /= NUM_DIPOLES radiation_pattern_total /= NUM_DIPOLES radiation_pattern_total_flux = radiation_pattern_flux(radiation_pattern_total) diff --git a/python/examples/disc_extraction_efficiency.py b/python/examples/disc_extraction_efficiency.py index 5dde21108..7dbbd0aa3 100644 --- a/python/examples/disc_extraction_efficiency.py +++ b/python/examples/disc_extraction_efficiency.py @@ -13,14 +13,16 @@ RESOLUTION_UM = 50 +WAVELENGTH_UM = 1.0 N_DISC = 2.4 DISC_RADIUS_UM = 1.2 -WAVELENGTH_UM = 1.0 +DISC_THICKNESS_UM = 0.7 * WAVELENGTH_UM / N_DISC NUM_FARFIELD_PTS = 200 -FARFIELD_ANGLES = np.linspace(0, 0.5 * math.pi, NUM_FARFIELD_PTS) FARFIELD_RADIUS_UM = 1e6 * WAVELENGTH_UM NUM_DIPOLES = 11 +farfield_angles = np.linspace(0, 0.5 * math.pi, NUM_FARFIELD_PTS) + def plot_radiation_pattern_polar(radial_flux: np.ndarray): """Plots the radiation pattern in polar coordinates. @@ -30,7 +32,7 @@ def plot_radiation_pattern_polar(radial_flux: np.ndarray): """ fig, ax = plt.subplots(subplot_kw={"projection": "polar"}, figsize=(6, 6)) ax.plot( - FARFIELD_ANGLES, + farfield_angles, radial_flux, "b-", ) @@ -62,7 +64,7 @@ def plot_radiation_pattern_3d(radial_flux: np.ndarray): ys = np.zeros((NUM_FARFIELD_PTS, NUM_FARFIELD_PTS)) zs = np.zeros((NUM_FARFIELD_PTS, NUM_FARFIELD_PTS)) - for i, theta in enumerate(FARFIELD_ANGLES): + for i, theta in enumerate(farfield_angles): for j, phi in enumerate(phis): xs[i, j] = radial_flux[i] * np.sin(theta) * np.cos(phi) ys[i, j] = radial_flux[i] * np.sin(theta) * np.sin(phi) @@ -99,9 +101,9 @@ def radiation_pattern(sim: mp.Simulation, n2f_mon: mp.DftNear2Far) -> np.ndarray far_field = sim.get_farfield( n2f_mon, mp.Vector3( - FARFIELD_RADIUS_UM * math.sin(FARFIELD_ANGLES[n]), + FARFIELD_RADIUS_UM * math.sin(farfield_angles[n]), 0, - FARFIELD_RADIUS_UM * math.cos(FARFIELD_ANGLES[n]), + FARFIELD_RADIUS_UM * math.cos(farfield_angles[n]), ), ) e_field[n, :] = [far_field[j] for j in range(3)] @@ -121,17 +123,17 @@ def radiation_pattern(sim: mp.Simulation, n2f_mon: mp.DftNear2Far) -> np.ndarray def radiation_pattern_flux(radial_flux: np.ndarray) -> float: """Computes the total flux from the radiation pattern. - Based on integrating the radiation pattern over solid angles spanned by - polar angles in the range of [0, π/2]. + Based on integrating the radiation pattern over solid angles + spanned by polar angles in the range of [0, π/2]. Args: radial_flux: radial flux of the far fields in polar coordinates. """ dphi = 2 * math.pi - dtheta = FARFIELD_ANGLES[1] - FARFIELD_ANGLES[0] + dtheta = farfield_angles[1] - farfield_angles[0] total_flux = ( - np.sum(radial_flux * np.sin(FARFIELD_ANGLES)) + np.sum(radial_flux * np.sin(farfield_angles)) * FARFIELD_RADIUS_UM**2 * dtheta * dphi @@ -140,14 +142,11 @@ def radiation_pattern_flux(radial_flux: np.ndarray) -> float: return total_flux -def dipole_in_disc( - disc_um: float, zpos: float, rpos_um: float, m: int -) -> Tuple[float, np.ndarray]: +def dipole_in_disc(zpos: float, rpos_um: float, m: int) -> Tuple[float, np.ndarray]: """Computes the total flux and radiation pattern of a dipole in a disc. Args: - disc_um: thickness of disc. - zpos: height of dipole above ground plane as fraction of disc_um. + zpos: height of dipole above ground plane as fraction of disc thickness. rpos_um: radial position of dipole. m: angular φ dependence of the fields exp(imφ). @@ -160,11 +159,11 @@ def dipole_in_disc( frequency = 1 / WAVELENGTH_UM # center frequency of source/monitor - # field decay threshold for runtime termination criteria - field_decay_threshold = 1e-6 + # runtime termination criteria + dft_decay_threshold = 1e-4 size_r = r_um + pml_um - size_z = disc_um + padding_um + pml_um + size_z = DISC_THICKNESS_UM + padding_um + pml_um cell_size = mp.Vector3(size_r, 0, size_z) boundary_layers = [ @@ -173,7 +172,7 @@ def dipole_in_disc( ] src_cmpt = mp.Er - src_pt = mp.Vector3(rpos_um, 0, -0.5 * size_z + zpos * disc_um) + src_pt = mp.Vector3(rpos_um, 0, -0.5 * size_z + zpos * DISC_THICKNESS_UM) sources = [ mp.Source( src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), @@ -185,8 +184,10 @@ def dipole_in_disc( geometry = [ mp.Block( material=mp.Medium(index=N_DISC), - center=mp.Vector3(0.5 * DISC_RADIUS_UM, 0, -0.5 * size_z + 0.5 * disc_um), - size=mp.Vector3(DISC_RADIUS_UM, mp.inf, disc_um), + center=mp.Vector3( + 0.5 * DISC_RADIUS_UM, 0, -0.5 * size_z + 0.5 * DISC_THICKNESS_UM + ), + size=mp.Vector3(DISC_RADIUS_UM, mp.inf, DISC_THICKNESS_UM), ) ] @@ -210,19 +211,16 @@ def dipole_in_disc( ), mp.FluxRegion( center=mp.Vector3( - r_um, 0, 0.5 * size_z - pml_um - 0.5 * (padding_um + disc_um) + r_um, 0, 0.5 * size_z - pml_um - 0.5 * (padding_um + DISC_THICKNESS_UM) ), - size=mp.Vector3(0, 0, padding_um + disc_um), + size=mp.Vector3(0, 0, padding_um + DISC_THICKNESS_UM), ), ) sim.run( mp.dft_ldos(frequency, 0, 1), - until_after_sources=mp.stop_when_fields_decayed( - 50, - src_cmpt, - src_pt, - field_decay_threshold, + until_after_sources=mp.stop_when_dft_decayed( + tol=dft_decay_threshold, ), ) @@ -235,7 +233,6 @@ def dipole_in_disc( if __name__ == "__main__": - disc_thickness = 0.7 * WAVELENGTH_UM / N_DISC dipole_height = 0.5 dipole_rpos_um = np.linspace(0, DISC_RADIUS_UM, NUM_DIPOLES) @@ -248,7 +245,6 @@ def dipole_in_disc( # Er source at r = 0 requires a single simulation with m = ±1. m = -1 dipole_flux, dipole_radiation_pattern = dipole_in_disc( - disc_thickness, dipole_height, dipole_rpos_um[0], m, @@ -276,7 +272,7 @@ def dipole_in_disc( m = 0 while True: dipole_flux, dipole_radiation_pattern = dipole_in_disc( - disc_thickness, dipole_height, rpos_um, m + dipole_height, rpos_um, m ) dipole_flux_total += dipole_flux * (2 if m == 0 else 1) dipole_radiation_pattern_total += dipole_radiation_pattern * ( @@ -286,6 +282,10 @@ def dipole_in_disc( dipole_radiation_pattern_flux = radiation_pattern_flux( dipole_radiation_pattern ) + print( + f"dipole:, {rpos_um:.4f}, {m}, " f"{dipole_radiation_pattern_flux:.6f}" + ) + if dipole_radiation_pattern_flux > dipole_radiation_pattern_flux_max: dipole_radiation_pattern_flux_max = dipole_radiation_pattern_flux @@ -295,9 +295,8 @@ def dipole_in_disc( < flux_decay_threshold ): break - - print(f"dipole:, {rpos_um:.4f}, {m}, {dipole_radiation_pattern_flux:.6f}") - m += 1 + else: + m += 1 scale_factor = (dipole_rpos_um[0] / rpos_um) ** 2 flux_total += dipole_flux_total * scale_factor * rpos_um * delta_rpos_um From 098ff7ace54304ad6899143f93651a6372a5908a Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Fri, 19 Jan 2024 10:33:49 -0800 Subject: [PATCH 08/10] add force_complex_fields=True to Simulation constructor and update results --- .../Near_to_Far_Field_Spectra.md | 46 ++++++++++-------- .../images/disc_dipoles_radiation_pattern.png | Bin 205930 -> 183460 bytes python/examples/disc_extraction_efficiency.py | 22 ++++++--- 3 files changed, 40 insertions(+), 28 deletions(-) diff --git a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md index f5efaf3d8..d01fc0dc0 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -658,15 +658,15 @@ if __name__ == "__main__": ### Extraction Efficiency of a Collection of Dipoles in a Disc -[Tutorial/Radiation Pattern of a Disc in Cylindrical Coordinates](Near_to_Far_Field_Spectra.md#radiation-pattern-of-a-disc-in-cylindrical-coordinates) demonstrated the procedure for computing the radiation pattern of a *single* dipole (actually a "ring" current source with angular dependence $exp(im\phi)$). [Tutorial/Nonaxisymmetric Dipole Sources](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) described the method for modeling a point dipole at $r > 0$ in cylindrical coordinates using a Fourier-series expansion of the fields in $\phi$. [Tutorial/Extraction Efficiency of a Light-Emitting Diode](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) described the procedure for computing the extraction efficiency of a dipole at $r = 0$. These three results can be combined to compute the extraction efficiency for a point dipole *anywhere* in the cylindrical cell. Computing the extraction efficiency of an actual light-emitting diode (LED), however, involves a collection of spatially incoherent dipole emitters. [Tutorial/Stochastic Dipole Emission in Light Emitting Diodes](Custom_Source.md#stochastic-dipole-emission-in-light-emitting-diodes) described a method for computing the emission of a collection of dipoles using a series of single-dipole calculations and averaging the emission profiles in post processing. The example used a 2D simulation involving a 1D binary grating (or photonic crystal). This tutorial demonstrates how this approach for modeling spatially incoherent dipoles can be extended to cylindrical coordinates for structures with rotational symmetry. +[Tutorial/Radiation Pattern of a Disc in Cylindrical Coordinates](Near_to_Far_Field_Spectra.md#radiation-pattern-of-a-disc-in-cylindrical-coordinates) demonstrated the procedure for computing the radiation pattern of a *single* dipole (actually a "ring" current source with angular dependence $exp(im\phi)$). [Tutorial/Nonaxisymmetric Dipole Sources](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) described the method for modeling a point dipole at $r > 0$ in cylindrical coordinates using a Fourier-series expansion of the fields in $\phi$. [Tutorial/Extraction Efficiency of a Light-Emitting Diode](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) described the procedure for computing the extraction efficiency of a dipole at $r = 0$. These three demonstrations can be combined to compute the extraction efficiency for a point dipole *anywhere* in the cylindrical cell. Computing the extraction efficiency of an actual light-emitting diode (LED), however, involves a collection of spatially incoherent dipole emitters. [Tutorial/Stochastic Dipole Emission in Light Emitting Diodes](Custom_Source.md#stochastic-dipole-emission-in-light-emitting-diodes) described a method for computing the emission of a collection of dipoles using a series of single-dipole calculations and then averaging the emission profiles in post processing. The example used a 2D simulation involving a 1D binary grating (or photonic crystal). This tutorial demonstrates how this approach for modeling spatially incoherent dipoles can be extended to cylindrical coordinates for structures with rotational symmetry. The example uses the same setup as the [previous tutorial](#radiation-pattern-of-a-disc-in-cylindrical-coordinates) involving a dielectric disc above a lossless-reflector ground plane. The dipoles are arranged on a line extending from $r = 0$ to $r = R$ where $R$ is the disc radius. The height of the dipoles ($z$ coordinate) within the disc is fixed. The radiation pattern $P(r,\theta)$ for a dipole at $r$ is computed using a Fourier-series expansion in $\phi$. The *total* radiation pattern $P(\theta)$ for $N$ dipoles with equal spacing $\Delta r$ is the average of the per-dipole results: $$P(\theta) = \frac{1}{R} \int_0^R P(r,\theta) s(r) 2\pi rdr = \frac{1}{N} \sum_{n=0}^{N-1} P(r_n,\theta) s(r_n) 2\pi r_n \Delta r$$, -where $s(r)$ is a weighting function necessary for ensuring equal contribution from all dipoles relative to the dipole at $r = 0$. Note: an $E_r$ dipole at $r = 0$ must be placed at $r_0 = 1.5\Delta r$ due to an [interpolation bug](https://github.com/NanoComp/meep/issues/2704). $s(r)$ can be determined empirically by computing the emission in vacuum for a set of dipoles at different radial positions. The emission profile of a dipole in vacuum is a constant independent of its position. This criteria is used to obtain: $s(r) = (r_0/r)^2$. This weighting function is also used to average the flux emitted by each dipole (obtained using using the LDOS feature). This quantity is the denominator in the expression for the extraction efficiency. +where $s(r)$ is a weighting function necessary for ensuring equal contribution from all dipoles relative to the dipole at $r = 0$. Note: an $E_r$ dipole at $r = 0$ must be placed at $r_0 = 1.5\Delta r$ due to an [interpolation bug](https://github.com/NanoComp/meep/issues/2704). $s(r)$ can be determined empirically by computing the emission in vacuum for a set of dipoles at different radial positions. The emission profile of a dipole in vacuum is a constant independent of its position. This criteria is used to obtain $s(r) = 0.5(r_0/r)^2$. This weighting function is also used to average the flux emitted by each dipole (obtained using using the LDOS feature). This quantity is the denominator in the expression for the extraction efficiency. -This figure shows the radiation pattern from $N=11$ dipoles with $\lambda$ of 1.0 $\mu$m in the middle of a disc of height 0.29 $\mu$m, radius 1.2 $\mu$m, and refractive index 2.4. +This figure shows the radiation pattern from $N=11$ dipoles with $\lambda$ of 1.0 $\mu$m in the middle of a disc of height 0.29 $\mu$m, radius 1.2 $\mu$m, and refractive index 2.4. The extraction efficiency for this setup is 0.933517. The runtime is about two hours using two Intel 4.2 GHz Xeon cores. ![](../images/disc_dipoles_radiation_pattern.png#center) @@ -812,9 +812,7 @@ def radiation_pattern_flux(radial_flux: np.ndarray) -> float: return total_flux -def dipole_in_disc( - zpos: float, rpos_um: float, m: int -) -> Tuple[float, np.ndarray]: +def dipole_in_disc(zpos: float, rpos_um: float, m: int) -> Tuple[float, np.ndarray]: """Computes the total flux and radiation pattern of a dipole in a disc. Args: @@ -827,7 +825,7 @@ def dipole_in_disc( """ pml_um = 1.0 # thickness of PML padding_um = 1.0 # thickness of air padding above disc - r_um = 5.0 # length of cell in r + r_um = 4.0 # length of cell in r frequency = 1 / WAVELENGTH_UM # center frequency of source/monitor @@ -847,7 +845,7 @@ def dipole_in_disc( src_pt = mp.Vector3(rpos_um, 0, -0.5 * size_z + zpos * DISC_THICKNESS_UM) sources = [ mp.Source( - src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), + src=mp.GaussianSource(frequency, fwidth=0.05 * frequency), component=src_cmpt, center=src_pt, ) @@ -856,7 +854,9 @@ def dipole_in_disc( geometry = [ mp.Block( material=mp.Medium(index=N_DISC), - center=mp.Vector3(0.5 * DISC_RADIUS_UM, 0, -0.5 * size_z + 0.5 * DISC_THICKNESS_UM), + center=mp.Vector3( + 0.5 * DISC_RADIUS_UM, 0, -0.5 * size_z + 0.5 * DISC_THICKNESS_UM + ), size=mp.Vector3(DISC_RADIUS_UM, mp.inf, DISC_THICKNESS_UM), ) ] @@ -869,6 +869,7 @@ def dipole_in_disc( boundary_layers=boundary_layers, sources=sources, geometry=geometry, + force_complex_fields=True ) n2f_mon = sim.add_near2far( @@ -937,40 +938,45 @@ if __name__ == "__main__": for rpos_um in dipole_rpos_um[1:]: dipole_flux_total = 0 - dipole_radiation_pattern_total = np.zeros((NUM_FARFIELD_PTS,)) + dipole_radiation_pattern_total = np.zeros(NUM_FARFIELD_PTS) dipole_radiation_pattern_flux_max = 0 m = 0 while True: dipole_flux, dipole_radiation_pattern = dipole_in_disc( dipole_height, rpos_um, m ) - dipole_flux_total += dipole_flux * (2 if m == 0 else 1) + dipole_flux_total += dipole_flux * (1 if m == 0 else 2) dipole_radiation_pattern_total += dipole_radiation_pattern * ( - 2 if m == 0 else 1 + 1 if m == 0 else 2 ) dipole_radiation_pattern_flux = radiation_pattern_flux( dipole_radiation_pattern ) print( - f"dipole:, {rpos_um:.4f}, {m}, " - f"{dipole_radiation_pattern_flux:.6f}" + f"dipole:, {rpos_um:.4f}, {m}, " f"{dipole_radiation_pattern_flux:.6f}" ) if dipole_radiation_pattern_flux > dipole_radiation_pattern_flux_max: dipole_radiation_pattern_flux_max = dipole_radiation_pattern_flux - if (m > 0 and - (dipole_radiation_pattern_flux / - dipole_radiation_pattern_flux_max) < flux_decay_threshold): + if ( + m > 0 + and (dipole_radiation_pattern_flux / dipole_radiation_pattern_flux_max) + < flux_decay_threshold + ): break else: m += 1 - scale_factor = (dipole_rpos_um[0] / rpos_um) ** 2 - flux_total += dipole_flux_total * scale_factor * rpos_um * delta_rpos_um + dipole_position_scale_factor = 0.5 * (dipole_rpos_um[0] / rpos_um) ** 2 + flux_total += ( + dipole_flux_total * dipole_position_scale_factor * + rpos_um * delta_rpos_um + ) radiation_pattern_total += ( - dipole_radiation_pattern_total * scale_factor * rpos_um * delta_rpos_um + dipole_radiation_pattern_total * dipole_position_scale_factor * + rpos_um * delta_rpos_um ) flux_total /= NUM_DIPOLES diff --git a/doc/docs/images/disc_dipoles_radiation_pattern.png b/doc/docs/images/disc_dipoles_radiation_pattern.png index 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zfStpZ=I(d^Kl!1&#5s71W1-$trdK)ER1$pa`)iGCy^?VW?V!Hz;hxC_- z5E~0E$qgto1mzq4{iG_`EI*)E0B}D&)JmdRW4^B=3p8JX5yqh+8^zE+=*KnFk<)Nl zT>TtD&6Y30<^OT%L&YOOO!fcI|E-KRVDrHL`U9I&x86PbZ$w=_{~nj0?ERnG|380< zLPId$|Mepet_}M4$d`Y|m1P{#_wD}Q|2K3;>rVcEeuMf`{B!f!&A-k|My>^ bXK7VN!|=EG$EkrU@LzH=il@_0UiSPy-?B&0 diff --git a/python/examples/disc_extraction_efficiency.py b/python/examples/disc_extraction_efficiency.py index 7dbbd0aa3..f85ab8bfd 100644 --- a/python/examples/disc_extraction_efficiency.py +++ b/python/examples/disc_extraction_efficiency.py @@ -155,7 +155,7 @@ def dipole_in_disc(zpos: float, rpos_um: float, m: int) -> Tuple[float, np.ndarr """ pml_um = 1.0 # thickness of PML padding_um = 1.0 # thickness of air padding above disc - r_um = 5.0 # length of cell in r + r_um = 4.0 # length of cell in r frequency = 1 / WAVELENGTH_UM # center frequency of source/monitor @@ -175,7 +175,7 @@ def dipole_in_disc(zpos: float, rpos_um: float, m: int) -> Tuple[float, np.ndarr src_pt = mp.Vector3(rpos_um, 0, -0.5 * size_z + zpos * DISC_THICKNESS_UM) sources = [ mp.Source( - src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), + src=mp.GaussianSource(frequency, fwidth=0.05 * frequency), component=src_cmpt, center=src_pt, ) @@ -199,6 +199,7 @@ def dipole_in_disc(zpos: float, rpos_um: float, m: int) -> Tuple[float, np.ndarr boundary_layers=boundary_layers, sources=sources, geometry=geometry, + force_complex_fields=True, ) n2f_mon = sim.add_near2far( @@ -267,16 +268,16 @@ def dipole_in_disc(zpos: float, rpos_um: float, m: int) -> Tuple[float, np.ndarr for rpos_um in dipole_rpos_um[1:]: dipole_flux_total = 0 - dipole_radiation_pattern_total = np.zeros((NUM_FARFIELD_PTS,)) + dipole_radiation_pattern_total = np.zeros(NUM_FARFIELD_PTS) dipole_radiation_pattern_flux_max = 0 m = 0 while True: dipole_flux, dipole_radiation_pattern = dipole_in_disc( dipole_height, rpos_um, m ) - dipole_flux_total += dipole_flux * (2 if m == 0 else 1) + dipole_flux_total += dipole_flux * (1 if m == 0 else 2) dipole_radiation_pattern_total += dipole_radiation_pattern * ( - 2 if m == 0 else 1 + 1 if m == 0 else 2 ) dipole_radiation_pattern_flux = radiation_pattern_flux( @@ -298,10 +299,15 @@ def dipole_in_disc(zpos: float, rpos_um: float, m: int) -> Tuple[float, np.ndarr else: m += 1 - scale_factor = (dipole_rpos_um[0] / rpos_um) ** 2 - flux_total += dipole_flux_total * scale_factor * rpos_um * delta_rpos_um + dipole_position_scale_factor = 0.5 * (dipole_rpos_um[0] / rpos_um) ** 2 + flux_total += ( + dipole_flux_total * dipole_position_scale_factor * rpos_um * delta_rpos_um + ) radiation_pattern_total += ( - dipole_radiation_pattern_total * scale_factor * rpos_um * delta_rpos_um + dipole_radiation_pattern_total + * dipole_position_scale_factor + * rpos_um + * delta_rpos_um ) flux_total /= NUM_DIPOLES From 6b3e09e8d4befa97674ee4087c9900f36ce2bd6a Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Thu, 25 Jan 2024 12:41:26 -0800 Subject: [PATCH 09/10] Update doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md Co-authored-by: Steven G. Johnson --- doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md index d01fc0dc0..4eb93a922 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -664,7 +664,7 @@ The example uses the same setup as the [previous tutorial](#radiation-pattern-of $$P(\theta) = \frac{1}{R} \int_0^R P(r,\theta) s(r) 2\pi rdr = \frac{1}{N} \sum_{n=0}^{N-1} P(r_n,\theta) s(r_n) 2\pi r_n \Delta r$$, -where $s(r)$ is a weighting function necessary for ensuring equal contribution from all dipoles relative to the dipole at $r = 0$. Note: an $E_r$ dipole at $r = 0$ must be placed at $r_0 = 1.5\Delta r$ due to an [interpolation bug](https://github.com/NanoComp/meep/issues/2704). $s(r)$ can be determined empirically by computing the emission in vacuum for a set of dipoles at different radial positions. The emission profile of a dipole in vacuum is a constant independent of its position. This criteria is used to obtain $s(r) = 0.5(r_0/r)^2$. This weighting function is also used to average the flux emitted by each dipole (obtained using using the LDOS feature). This quantity is the denominator in the expression for the extraction efficiency. +where $s(r)$ is a weighting function necessary for ensuring equal contribution from all dipoles relative to the dipole at $r = 0$. Note: an $E_r$ dipole at $r = 0$ must be placed at $r_0 = 1.5\Delta r$ due to an [interpolation bug](https://github.com/NanoComp/meep/issues/2704). $s(r)$ can be determined empirically by computing the emission in vacuum for a set of dipoles at different radial positions. The emission profile of a dipole in vacuum is a constant independent of its position. This criteria is used to obtain $s(r) = 0.5(r_0/r)^2$. (A $1/r^2$ dependence is expected because a cylindrical delta function should really include a $1/r$ factor in order to integrate to 1 with $\int r \, dr$, but Meep currently does not include this $1/r$ in sources that have zero radial width, and power goes like the square of the current amplitude—therefore, we must include an additional $1/r^2$ factor to obtain the correct relative power for dipoles at different radii.) This weighting function is also used to average the flux emitted by each dipole (obtained using using the LDOS feature). This quantity is the denominator in the expression for the extraction efficiency. This figure shows the radiation pattern from $N=11$ dipoles with $\lambda$ of 1.0 $\mu$m in the middle of a disc of height 0.29 $\mu$m, radius 1.2 $\mu$m, and refractive index 2.4. The extraction efficiency for this setup is 0.933517. The runtime is about two hours using two Intel 4.2 GHz Xeon cores. From 81c13d463e4e50d40093d2a98c17582857c3746a Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Thu, 25 Jan 2024 12:47:42 -0800 Subject: [PATCH 10/10] Update doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md Co-authored-by: Steven G. Johnson --- doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md index 4eb93a922..c34139330 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -660,9 +660,9 @@ if __name__ == "__main__": [Tutorial/Radiation Pattern of a Disc in Cylindrical Coordinates](Near_to_Far_Field_Spectra.md#radiation-pattern-of-a-disc-in-cylindrical-coordinates) demonstrated the procedure for computing the radiation pattern of a *single* dipole (actually a "ring" current source with angular dependence $exp(im\phi)$). [Tutorial/Nonaxisymmetric Dipole Sources](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) described the method for modeling a point dipole at $r > 0$ in cylindrical coordinates using a Fourier-series expansion of the fields in $\phi$. [Tutorial/Extraction Efficiency of a Light-Emitting Diode](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) described the procedure for computing the extraction efficiency of a dipole at $r = 0$. These three demonstrations can be combined to compute the extraction efficiency for a point dipole *anywhere* in the cylindrical cell. Computing the extraction efficiency of an actual light-emitting diode (LED), however, involves a collection of spatially incoherent dipole emitters. [Tutorial/Stochastic Dipole Emission in Light Emitting Diodes](Custom_Source.md#stochastic-dipole-emission-in-light-emitting-diodes) described a method for computing the emission of a collection of dipoles using a series of single-dipole calculations and then averaging the emission profiles in post processing. The example used a 2D simulation involving a 1D binary grating (or photonic crystal). This tutorial demonstrates how this approach for modeling spatially incoherent dipoles can be extended to cylindrical coordinates for structures with rotational symmetry. -The example uses the same setup as the [previous tutorial](#radiation-pattern-of-a-disc-in-cylindrical-coordinates) involving a dielectric disc above a lossless-reflector ground plane. The dipoles are arranged on a line extending from $r = 0$ to $r = R$ where $R$ is the disc radius. The height of the dipoles ($z$ coordinate) within the disc is fixed. The radiation pattern $P(r,\theta)$ for a dipole at $r$ is computed using a Fourier-series expansion in $\phi$. The *total* radiation pattern $P(\theta)$ for $N$ dipoles with equal spacing $\Delta r$ is the average of the per-dipole results: +The example uses the same setup as the [previous tutorial](#radiation-pattern-of-a-disc-in-cylindrical-coordinates) involving a dielectric disc above a lossless-reflector ground plane. The dipoles are arranged on a line extending from $r = 0$ to $r = R$ where $R$ is the disc radius. The height of the dipoles ($z$ coordinate) within the disc is fixed. The radiation pattern $P(r,\theta)$ for a dipole at $r$ is computed using a Fourier-series expansion in $\phi$. The *total* radiation pattern $P(\theta)$ for an ensemble of incoherent dipoles is just the integral of the individual dipole powers, which we can approximate by a sum:: -$$P(\theta) = \frac{1}{R} \int_0^R P(r,\theta) s(r) 2\pi rdr = \frac{1}{N} \sum_{n=0}^{N-1} P(r_n,\theta) s(r_n) 2\pi r_n \Delta r$$, +$$P(\theta) \approx \int_0^R P(r,\theta) s(r) 2\pi rdr = \sum_{n=0}^{N-1} P(r_n,\theta) s(r_n) 2\pi r_n \Delta r$$, where $s(r)$ is a weighting function necessary for ensuring equal contribution from all dipoles relative to the dipole at $r = 0$. Note: an $E_r$ dipole at $r = 0$ must be placed at $r_0 = 1.5\Delta r$ due to an [interpolation bug](https://github.com/NanoComp/meep/issues/2704). $s(r)$ can be determined empirically by computing the emission in vacuum for a set of dipoles at different radial positions. The emission profile of a dipole in vacuum is a constant independent of its position. This criteria is used to obtain $s(r) = 0.5(r_0/r)^2$. (A $1/r^2$ dependence is expected because a cylindrical delta function should really include a $1/r$ factor in order to integrate to 1 with $\int r \, dr$, but Meep currently does not include this $1/r$ in sources that have zero radial width, and power goes like the square of the current amplitude—therefore, we must include an additional $1/r^2$ factor to obtain the correct relative power for dipoles at different radii.) This weighting function is also used to average the flux emitted by each dipole (obtained using using the LDOS feature). This quantity is the denominator in the expression for the extraction efficiency.