From 2b36f7e47e75e8337891570c385bfbec5f3e296e Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Thu, 26 Dec 2024 11:33:07 -0800 Subject: [PATCH 01/15] add scripts for computing the radiation pattern of an axisymmetric and nonaxisymmetric dipole in cylindrical coordinates --- .../dipole_in_vacuum_cyl_axisymmetric.py | 305 ++++++++++++++++++ .../dipole_in_vacuum_cyl_nonaxisymmetric.py | 278 ++++++++++++++++ 2 files changed, 583 insertions(+) create mode 100644 python/examples/dipole_in_vacuum_cyl_axisymmetric.py create mode 100644 python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py diff --git a/python/examples/dipole_in_vacuum_cyl_axisymmetric.py b/python/examples/dipole_in_vacuum_cyl_axisymmetric.py new file mode 100644 index 000000000..54d9be6d0 --- /dev/null +++ b/python/examples/dipole_in_vacuum_cyl_axisymmetric.py @@ -0,0 +1,305 @@ +"""Radiation pattern of an axisymmetric dipole in cylindrical coordinates.""" + +import argparse +import math +from typing import Tuple + +import matplotlib.pyplot as plt +import meep as mp +import numpy as np + + +RESOLUTION_UM = 50 +WAVELENGTH_UM = 1.0 +PML_UM = 1.0 +FARFIELD_RADIUS_UM = 1e6 * WAVELENGTH_UM +NUM_FARFIELD_PTS = 50 +POWER_DECAY_THRESHOLD = 1e-4 +DEBUG_OUTPUT = False + +frequency = 1 / WAVELENGTH_UM +polar_rad = np.linspace(0, 0.5 * math.pi, NUM_FARFIELD_PTS) + + +def plot_radiation_pattern(dipole_pol: int, radial_flux: np.ndarray): + """Plots the radiation pattern in polar coordinates. + + The angles increase clockwise with zero in the +z direction (the "pole") + and π/2 in the +r direction (the "equator"). + + Args: + dipole_pol: the polarization the electric dipole. Either x or z. + radial_flux: the radial flux in polar coordinates. + """ + normalized_radial_flux = radial_flux / np.max(radial_flux) + + if dipole_pol == mp.X: + dipole_radial_flux = np.square(np.cos(polar_rad)) + dipole_radial_flux_label = "$\cos^2θ$" + dipole_name = "$E_x$" + elif dipole_pol == mp.Z: + dipole_radial_flux = np.square(np.sin(polar_rad)) + dipole_radial_flux_label = "$\sin^2θ$" + dipole_name = "$E_z$" + + fig, ax = plt.subplots(subplot_kw={"projection": "polar"}, figsize=(6, 6)) + ax.plot(polar_rad, normalized_radial_flux, "b-", label="Meep") + ax.plot(polar_rad, dipole_radial_flux, "r-", label=dipole_radial_flux_label) + ax.legend() + ax.set_theta_direction(-1) + ax.set_theta_offset(0.5 * math.pi) + ax.set_thetalim(0, 0.5 * math.pi) + ax.set_rmax(1) + ax.set_rticks([0, 0.5, 1]) + ax.grid(True) + ax.set_rlabel_position(22) + ax.set_ylabel("radial flux (a.u.)") + ax.set_title("radiation pattern of an axisymmetric " + dipole_name + + " dipole") + + if mp.am_master(): + fig.savefig( + "dipole_radiation_pattern_axisymmetric.png", + dpi=100, + bbox_inches="tight", + ) + + relative_error = ( + np.linalg.norm(normalized_radial_flux - dipole_radial_flux) / + np.linalg.norm(dipole_radial_flux) + ) + print(f"relative error in radiation pattern:, {relative_error}") + + +def radiation_pattern(e_field: np.ndarray, h_field: np.ndarray) -> np.ndarray: + """Computes the radiation pattern from the far fields. + + Args: + e_field, h_field: the electric (Er, Ep, Ez) and magnetic (Hr, Hp, Hz) + far fields, respectively. + + Returns: + The radial Poynting flux as a 1D array. One element for each point on + the circumference of a quarter circle with angular range of + [0, π/2] rad. 0 radians is the +z direction (the "pole") and π/2 is + the +r direction (the "equator"). + """ + flux_x = 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_r = np.sqrt(np.square(flux_x) + np.square(flux_z)) + + return flux_r + + +def get_farfields( + sim: mp.Simulation, + n2f_mon: mp.DftNear2Far +) -> Tuple[np.ndarray, np.ndarray]: + """Computes the far fields from the near fields. + + Args: + sim: a `Simulation` object. + n2f_mon: a `DftNear2Far` object returned by `Simulation.add_near2far`. + + Returns: + The electric (Er, Ep, Ez) and magnetic (Hr, Hp, Hz) far fields. One row + with six columns for the fields for each point on the circumference of + a quarter circle with angular range of [0, π/2] rad. 0 radians is the + +z direction (the "pole") and π/2 is the +r direction (the "equator"). + """ + 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(polar_rad[n]), + 0, + FARFIELD_RADIUS_UM * math.cos(polar_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)] + + return e_field, h_field + + +def dipole_in_vacuum( + dipole_pol: int, + m: int +) -> Tuple[np.ndarray, np.ndarray]: + """Computes the far fields of an axisymmetric point source. + + Args: + dipole_pol: the polarization of the electric dipole. Either x or z. + m: angular φ dependence of the fields exp(imφ). + + Returns: + A 2-tuple containing the electric and magnetic far fields as 1D arrays. + """ + sr = 2.0 + sz = 4.0 + cell_size = mp.Vector3(sr + PML_UM, 0, sz + 2 * PML_UM) + + boundary_layers = [mp.PML(thickness=PML_UM)] + + if dipole_pol == mp.X: + # An Er source at r = 0 needs to be slighty offset due to a bug. + # https://github.com/NanoComp/meep/issues/2704 + dipole_pos_r = 1.5 / RESOLUTION_UM + sources = [ + mp.Source( + src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), + component=mp.Er, + center=mp.Vector3(dipole_pos_r, 0, 0) + ), + mp.Source( + src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), + component=mp.Ep, + center=mp.Vector3(dipole_pos_r, 0, 0), + amplitude=1j if m == 1 else -1j + ) + ] + elif dipole_pol == mp.Z: + dipole_pos_r = 0 + sources = [ + mp.Source( + src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), + component=mp.Ez, + center=mp.Vector3(dipole_pos_r, 0, 0) + ) + ] + + sim = mp.Simulation( + resolution=RESOLUTION_UM, + cell_size=cell_size, + dimensions=mp.CYLINDRICAL, + m=m, + boundary_layers=boundary_layers, + sources=sources, + force_complex_fields=True + ) + + nearfields_monitor = sim.add_near2far( + frequency, + 0, + 1, + mp.FluxRegion( + center=mp.Vector3(0.5 * sr, 0, 0.5 * sz), + size=mp.Vector3(sr, 0, 0) + ), + mp.FluxRegion( + center=mp.Vector3(sr, 0, 0), + size=mp.Vector3(0, 0, sz) + ), + mp.FluxRegion( + center=mp.Vector3(0.5 * sr, 0, -0.5 * sz), + size=mp.Vector3(sr, 0, 0), + weight=-1.0 + ) + ) + + sim.run( + until_after_sources=mp.stop_when_fields_decayed( + 20.0, + mp.Er if dipole_pol == mp.X else mp.Ez, + mp.Vector3(dipole_pos_r, 0, 0), + 1e-6 + ) + ) + + if DEBUG_OUTPUT: + fig, ax = plt.subplots() + sim.plot2D(ax=ax, show_monitors=True) + if mp.am_master(): + fig.savefig( + "dipole_in_vacuum_cyl_layout.png", + dpi=150, + bbox_inches="tight" + ) + + e_field, h_field = get_farfields(sim, nearfields_monitor) + + return e_field, h_field + + +def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: + """Computes the flux from the far fields. + + Args: + e_field, h_field: the electric (Er, Ep, Ez) and magnetic (Hr, Hp, Hz) + far fields, respectively. + + Returns: + The Poynting flux obtained from the far fields. + """ + dphi = 2 * math.pi + dtheta = 0.5 * math.pi / (NUM_FARFIELD_PTS - 1) + dipole_radiation_pattern = radiation_pattern(e_field, h_field) + flux = ( + np.sum(dipole_radiation_pattern * np.sin(polar_rad)) * + FARFIELD_RADIUS_UM**2 * + dtheta * + dphi + ) + + return flux + + +if __name__ == "__main__": + parser = argparse.ArgumentParser() + parser.add_argument( + "dipole_pol", + type=str, + choices=["x", "z"], + help="polarization of the electric dipole (x or z)" + ) + args = parser.parse_args() + + e_field_total = np.zeros((NUM_FARFIELD_PTS, 3), dtype=np.complex128) + h_field_total = np.zeros((NUM_FARFIELD_PTS, 3), dtype=np.complex128) + + if args.dipole_pol == "x": + # A linearly polarized dipole can be formed from the superposition + # of two left- and right-circularly polarized dipoles. + + e_field, h_field = dipole_in_vacuum(mp.X, +1) + e_field_total += 0.5 * e_field + h_field_total += 0.5 * h_field + + e_field, h_field = dipole_in_vacuum(mp.X, -1) + e_field_total += 0.5 * e_field + h_field_total += 0.5 * h_field + + elif args.dipole_pol == "z": + e_field, h_field = dipole_in_vacuum(mp.Z, 0) + e_field_total += e_field + h_field_total += h_field + + dipole_radiation_pattern = radiation_pattern( + e_field_total, h_field_total + ) + dipole_radiation_pattern_scaled = ( + dipole_radiation_pattern * FARFIELD_RADIUS_UM**2 + ) + plot_radiation_pattern( + mp.X if args.dipole_pol == "x" else mp.Z, + dipole_radiation_pattern_scaled + ) + + if mp.am_master(): + np.savez( + "dipole_farfields_axisymmetric.npz", + FARFIELD_RADIUS_UM=FARFIELD_RADIUS_UM, + PML_UM=PML_UM, + POWER_DECAY_THRESHOLD=POWER_DECAY_THRESHOLD, + RESOLUTION_UM=RESOLUTION_UM, + WAVELENGTH_UM=WAVELENGTH_UM, + dipole_pol=args.dipole_pol, + dipole_radiation_pattern=dipole_radiation_pattern, + e_field_total=e_field_total, + h_field_total=h_field_total, + polar_rad=polar_rad, + ) diff --git a/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py b/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py new file mode 100644 index 000000000..816c862a9 --- /dev/null +++ b/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py @@ -0,0 +1,278 @@ +"""Radiation pattern of a nonaxisymmetric dipole in cylindrical coordinates.""" + +import argparse +import math +from typing import Tuple + +import matplotlib.pyplot as plt +import meep as mp +import numpy as np + + +RESOLUTION_UM = 50 +WAVELENGTH_UM = 1.0 +PML_UM = 1.0 +FARFIELD_RADIUS_UM = 1e6 * WAVELENGTH_UM +NUM_FARFIELD_PTS = 50 +POWER_DECAY_THRESHOLD = 1e-4 +DEBUG_OUTPUT = True + +frequency = 1 / WAVELENGTH_UM +polar_rad = np.linspace(0, 0.5 * math.pi, NUM_FARFIELD_PTS) + + +def plot_radiation_pattern(radial_flux: np.ndarray): + """Plots the radiation pattern in polar coordinates. + + The angles increase clockwise with zero in the +z direction (the "pole") + and π/2 in the +r direction (the "equator"). + + Args: + radial_flux: the radial flux in polar coordinates. + """ + normalized_radial_flux = radial_flux / np.max(radial_flux) + dipole_radial_flux = np.square(np.cos(polar_rad)) + + fig, ax = plt.subplots(subplot_kw={"projection": "polar"}, figsize=(6, 6)) + ax.plot(polar_rad, normalized_radial_flux, "b-", label="Meep") + ax.plot(polar_rad, dipole_radial_flux, "r-", label="$\cos^2θ$") + ax.legend() + ax.set_theta_direction(-1) + ax.set_theta_offset(0.5 * math.pi) + ax.set_thetalim(0, 0.5 * math.pi) + ax.set_rmax(1) + ax.set_rticks([0, 0.5, 1]) + ax.grid(True) + ax.set_rlabel_position(22) + ax.set_ylabel("radial flux (a.u.)") + ax.set_title("radiation pattern of a nonaxisymmetric $E_r$ dipole") + + if mp.am_master(): + fig.savefig( + "dipole_radiation_pattern_nonaxisymmetric.png", + dpi=100, + bbox_inches="tight", + ) + + relative_error = ( + np.linalg.norm(normalized_radial_flux - dipole_radial_flux) / + np.linalg.norm(dipole_radial_flux) + ) + print(f"relative error in radiation pattern:, {relative_error}") + + +def radiation_pattern(e_field: np.ndarray, h_field: np.ndarray) -> np.ndarray: + """Computes the radiation pattern from the far fields. + + Args: + e_field, h_field: the electric (Er, Ep, Ez) and magnetic (Hr, Hp, Hz) + far fields, respectively. + + Returns: + The radial Poynting flux as a 1D array. One element for each point on + the circumference of a quarter circle with angular range of + [0, π/2] rad. 0 radians is the +z direction (the "pole") and π/2 is + the +r direction (the "equator"). + """ + flux_x = 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_r = np.sqrt(np.square(flux_x) + np.square(flux_z)) + + return flux_r + + +def get_farfields( + sim: mp.Simulation, + n2f_mon: mp.DftNear2Far +) -> Tuple[np.ndarray, np.ndarray]: + """Computes the far fields from the near fields. + + Args: + sim: a `Simulation` object. + n2f_mon: a `DftNear2Far` object returned by `Simulation.add_near2far`. + + Returns: + The electric (Er, Ep, Ez) and magnetic (Hr, Hp, Hz) far fields. One row + with six columns for the fields for each point on the circumference of + a quarter circle with angular range of [0, π/2] rad. 0 radians is the + +z direction (the "pole") and π/2 is the +r direction (the "equator"). + """ + 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(polar_rad[n]), + 0, + FARFIELD_RADIUS_UM * math.cos(polar_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)] + + return e_field, h_field + + +def dipole_in_vacuum( + dipole_pos_r: mp.Vector3, + m: int +) -> Tuple[np.ndarray, np.ndarray]: + """Computes the far fields of a nonaxisymmetric point source. + + Args: + dipole_pos_r: the radial position of the dipole. + m: angular φ dependence of the fields exp(imφ). + + Returns: + A 2-tuple containing the electric and magnetic far fields as 1D arrays. + """ + sr = 2.0 + sz = 4.0 + cell_size = mp.Vector3(sr + PML_UM, 0, sz + 2 * PML_UM) + + boundary_layers = [mp.PML(thickness=PML_UM)] + + sources = [ + mp.Source( + src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), + component=mp.Er, + center=mp.Vector3(dipole_pos_r, 0, 0) + ) + ] + + sim = mp.Simulation( + resolution=RESOLUTION_UM, + cell_size=cell_size, + dimensions=mp.CYLINDRICAL, + m=m, + boundary_layers=boundary_layers, + sources=sources, + force_complex_fields=True + ) + + nearfields_monitor = sim.add_near2far( + frequency, + 0, + 1, + mp.FluxRegion( + center=mp.Vector3(0.5 * sr, 0, 0.5 * sz), + size=mp.Vector3(sr, 0, 0) + ), + mp.FluxRegion( + center=mp.Vector3(sr, 0, 0), + size=mp.Vector3(0, 0, sz) + ), + mp.FluxRegion( + center=mp.Vector3(0.5 * sr, 0, -0.5 * sz), + size=mp.Vector3(sr, 0, 0), + weight=-1.0 + ) + ) + + sim.run( + until_after_sources=mp.stop_when_fields_decayed( + 20.0, mp.Er, mp.Vector3(dipole_pos_r, 0, 0), 1e-6 + ) + ) + + if DEBUG_OUTPUT: + fig, ax = plt.subplots() + sim.plot2D(ax=ax, show_monitors=True) + if mp.am_master(): + fig.savefig( + "dipole_in_vacuum_cyl_layout.png", + dpi=150, + bbox_inches="tight" + ) + + e_field, h_field = get_farfields(sim, nearfields_monitor) + + return e_field, h_field + + +def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: + """Computes the flux from the far fields. + + Args: + e_field, h_field: the electric (Er, Ep, Ez) and magnetic (Hr, Hp, Hz) + far fields, respectively. + + Returns: + The Poynting flux obtained from the far fields. + """ + dphi = 2 * math.pi + dtheta = 0.5 * math.pi / (NUM_FARFIELD_PTS - 1) + dipole_radiation_pattern = radiation_pattern(e_field, h_field) + flux = ( + np.sum(dipole_radiation_pattern * np.sin(polar_rad)) * + FARFIELD_RADIUS_UM**2 * + dtheta * + dphi + ) + + return flux + + +if __name__ == "__main__": + parser = argparse.ArgumentParser() + parser.add_argument( + "dipole_pos_r", + type=float, + help="radial position of the dipole" + ) + args = parser.parse_args() + + # Fourier series expansion of the fields from a ring current source + # used to generate a point dipole localized in the azimuthal direction. + + e_field_total = np.zeros((NUM_FARFIELD_PTS, 3), dtype=np.complex128) + h_field_total = np.zeros((NUM_FARFIELD_PTS, 3), dtype=np.complex128) + flux_max = 0 + m = 0 + while True: + e_field, h_field = dipole_in_vacuum(args.dipole_pos_r, m) + e_field_total += e_field + h_field_total += h_field + + if m > 0: + e_field, h_field = dipole_in_vacuum(args.dipole_pos_r, -m) + e_field_total += e_field + h_field_total += h_field + + flux = flux_from_farfields(e_field, h_field) + if flux > flux_max: + flux_max = flux + power_decay = flux / flux_max + print(f"power_decay:, {m}, {flux}, {flux_max}, {power_decay}") + + if m > 0 and power_decay < POWER_DECAY_THRESHOLD: + break + else: + m += 1 + + dipole_radiation_pattern = radiation_pattern( + e_field_total, h_field_total + ) + dipole_radiation_pattern_scaled = ( + dipole_radiation_pattern * FARFIELD_RADIUS_UM**2 + ) + plot_radiation_pattern(dipole_radiation_pattern_scaled) + + if mp.am_master(): + np.savez( + "dipole_farfields_nonaxisymmetric.npz", + FARFIELD_RADIUS_UM=FARFIELD_RADIUS_UM, + PML_UM=PML_UM, + POWER_DECAY_THRESHOLD=POWER_DECAY_THRESHOLD, + RESOLUTION_UM=RESOLUTION_UM, + WAVELENGTH_UM=WAVELENGTH_UM, + dipole_pos_r=args.dipole_pos_r, + dipole_radiation_pattern=dipole_radiation_pattern, + e_field_total=e_field_total, + h_field_total=h_field_total, + m=m, + polar_rad=polar_rad, + ) From ae5cc48a6888d1b50f26ef63ff57769e0b1e9ad5 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Thu, 26 Dec 2024 13:20:58 -0800 Subject: [PATCH 02/15] add figures --- .../Near_to_Far_Field_Spectra.md | 12 +++++++++++- .../radiation_pattern_axisymmetric_dipole.png | Bin 0 -> 116839 bytes ...adiation_pattern_nonaxisymmetric_dipole.png | Bin 0 -> 115740 bytes .../dipole_in_vacuum_cyl_axisymmetric.py | 2 +- .../dipole_in_vacuum_cyl_nonaxisymmetric.py | 2 +- 5 files changed, 13 insertions(+), 3 deletions(-) create mode 100644 doc/docs/images/radiation_pattern_axisymmetric_dipole.png create mode 100644 doc/docs/images/radiation_pattern_nonaxisymmetric_dipole.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 49c795138..302956a21 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -189,7 +189,6 @@ plt.show() ![](../images/Source_radiation_pattern.png#center) - ### Antenna above a Perfect Electric Conductor Ground Plane As a second example, we compute the radiation pattern of an antenna positioned a given height $h$ above a perfect-electric conductor (PEC) ground plane. Depending on the wavelength and height of the antenna, self-interference effects due to reflections from the ground plane will produce well-defined lobes in the radiation pattern. The challenge in setting up this calculation is that because the ground plane is infinitely extended, it is not possible to enclose the antenna by a near-field surface. A non-closed near-field surface unfortunately gives rise to truncation errors which is described in more detail in the [section below](#truncation-errors-from-a-non-closed-near-field-surface). @@ -1702,3 +1701,14 @@ In this example, to demonstrate agreement between the far fields and DFT fields, When these two conditions are not met as in the example below involving a small `dpad` and large `d2`, the error from the finite truncation and numerical dispersion can be large and therefore result in a significant mismatch between the far fields computed using the near-to-far field transformation versus the actual DFT fields at the same location. ![Comparison of the far fields from the near-to-far field transformation dominated by errors and the DFT fields at the same location for a holey-waveguide cavity.](../images/farfields_vs_DFTfields_holeycavity_mismatch.png#center) + +Radiation Pattern of an Antenna in Cylindrical Coordinates +---------------------------------------------------------- + +In an earlier section, we showed how to compute the [radiation pattern of an antennna](#radiation-pattern-of-an-antenna) using 2D Cartesian coordinates. The same calculation can also be performed using cylindrical coordinates. + +The simulation scripts are in [examples/dipole_in_vacuum_cyl_axisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_axisymmetric.py) and [examples/dipole_in_vacuum_cyl_nonaxisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py). + +![](../images/radiation_pattern_axisymmetric_dipole.png#center) + +![](../images/radiation_pattern_nonaxisymmetric_dipole.png#center) diff --git a/doc/docs/images/radiation_pattern_axisymmetric_dipole.png b/doc/docs/images/radiation_pattern_axisymmetric_dipole.png new file mode 100644 index 0000000000000000000000000000000000000000..9193b1ce08e1fac04d92bc4550267ba4a1ae205f GIT binary patch literal 116839 zcmcG0cU;eJ`}VsX6&XcK8bWAkX-^HMAqowt(9+h>)X<=kky%6(X%V5Nq*O#AB~2-j z_Fm6%@w@Nm^*nz)|2$sz_vQY6U-kLCuj@Lm^E}SuIL>#B@opWqrQAy?iel5#)iR|h 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plt.subplots(subplot_kw={"projection": "polar"}, figsize=(6, 6)) ax.plot(polar_rad, normalized_radial_flux, "b-", label="Meep") - ax.plot(polar_rad, dipole_radial_flux, "r-", label=dipole_radial_flux_label) + ax.plot(polar_rad, dipole_radial_flux, "r--", label=dipole_radial_flux_label) ax.legend() ax.set_theta_direction(-1) ax.set_theta_offset(0.5 * math.pi) diff --git a/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py b/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py index 816c862a9..07f1ac0cf 100644 --- a/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py +++ b/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py @@ -35,7 +35,7 @@ def plot_radiation_pattern(radial_flux: np.ndarray): fig, ax = plt.subplots(subplot_kw={"projection": "polar"}, figsize=(6, 6)) ax.plot(polar_rad, normalized_radial_flux, "b-", label="Meep") - ax.plot(polar_rad, dipole_radial_flux, "r-", label="$\cos^2θ$") + ax.plot(polar_rad, dipole_radial_flux, "r--", label="$\cos^2θ$") ax.legend() ax.set_theta_direction(-1) ax.set_theta_offset(0.5 * math.pi) From 9286237ff92dc54e9f21aa6e282fbb6e765ba07e Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Thu, 26 Dec 2024 16:11:18 -0800 Subject: [PATCH 03/15] add text for tutorial --- doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md | 8 +++++++- python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py | 2 +- 2 files changed, 8 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 302956a21..94dcd8df7 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -1705,7 +1705,13 @@ When these two conditions are not met as in the example below involving a small Radiation Pattern of an Antenna in Cylindrical Coordinates ---------------------------------------------------------- -In an earlier section, we showed how to compute the [radiation pattern of an antennna](#radiation-pattern-of-an-antenna) using 2D Cartesian coordinates. The same calculation can also be performed using cylindrical coordinates. +In an earlier section, we showed how to compute the [radiation pattern of an antennna](#radiation-pattern-of-an-antenna) with **linear** polarization using 2D Cartesian coordinates. The same calculation can also be performed using [cylindrical coordinates](Exploiting_Symmetry.md#cylindrical-symmetry). We will demonstrate this for two cases in which the dipole is (1) axisymmetric (i.e., at $r = 0$) or (2) nonaxisymmetric (at $r > 0$). + +For (1), an $E_z$ dipole is positioned exactly at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of two left- and right-circularly polarized dipoles as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This involves two simulations. + +For (2), an $E_x$ dipole positioned anywhere at $r > 0$ requires a Fourier-series expansion of the fields from an $E_r$ ring current source with azimuthal dependence $\exp(im\phi)$. + +The figures below show the radiation pattern for each of these cases compared with the analytic result. The simulation and analytic results show good agreement. The simulation scripts are in [examples/dipole_in_vacuum_cyl_axisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_axisymmetric.py) and [examples/dipole_in_vacuum_cyl_nonaxisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py). diff --git a/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py b/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py index 07f1ac0cf..6c344d5e6 100644 --- a/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py +++ b/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py @@ -15,7 +15,7 @@ FARFIELD_RADIUS_UM = 1e6 * WAVELENGTH_UM NUM_FARFIELD_PTS = 50 POWER_DECAY_THRESHOLD = 1e-4 -DEBUG_OUTPUT = True +DEBUG_OUTPUT = False frequency = 1 / WAVELENGTH_UM polar_rad = np.linspace(0, 0.5 * math.pi, NUM_FARFIELD_PTS) From e9c92e17a9803432b15cbc7f93e7a95949d99138 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Thu, 26 Dec 2024 16:58:23 -0800 Subject: [PATCH 04/15] fixes and tweaks to text --- doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md | 9 +++++---- 1 file changed, 5 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 94dcd8df7..0d26ce106 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -189,6 +189,7 @@ plt.show() ![](../images/Source_radiation_pattern.png#center) + ### Antenna above a Perfect Electric Conductor Ground Plane As a second example, we compute the radiation pattern of an antenna positioned a given height $h$ above a perfect-electric conductor (PEC) ground plane. Depending on the wavelength and height of the antenna, self-interference effects due to reflections from the ground plane will produce well-defined lobes in the radiation pattern. The challenge in setting up this calculation is that because the ground plane is infinitely extended, it is not possible to enclose the antenna by a near-field surface. A non-closed near-field surface unfortunately gives rise to truncation errors which is described in more detail in the [section below](#truncation-errors-from-a-non-closed-near-field-surface). @@ -1705,13 +1706,13 @@ When these two conditions are not met as in the example below involving a small Radiation Pattern of an Antenna in Cylindrical Coordinates ---------------------------------------------------------- -In an earlier section, we showed how to compute the [radiation pattern of an antennna](#radiation-pattern-of-an-antenna) with **linear** polarization using 2D Cartesian coordinates. The same calculation can also be performed using [cylindrical coordinates](Exploiting_Symmetry.md#cylindrical-symmetry). We will demonstrate this for two cases in which the dipole is (1) axisymmetric (i.e., at $r = 0$) or (2) nonaxisymmetric (at $r > 0$). +In an earlier section, we showed how to compute the [radiation pattern of an antennna](#radiation-pattern-of-an-antenna) in vacuum with **linear** polarization using 2D Cartesian coordinates. The same calculation can also be performed using [cylindrical coordinates](Exploiting_Symmetry.md#cylindrical-symmetry). We will demonstrate this for two cases in which the dipole is (1) axisymmetric (i.e., at $r = 0$) or (2) nonaxisymmetric (i.e., at $r > 0$). In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). -For (1), an $E_z$ dipole is positioned exactly at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of two left- and right-circularly polarized dipoles as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This involves two simulations. +For (1), an $E_z$ dipole is positioned exactly at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This involves *two* simulations. Note that computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because in the latter a circularly polarized source emits the same power as a linearly polarized source. -For (2), an $E_x$ dipole positioned anywhere at $r > 0$ requires a Fourier-series expansion of the fields from an $E_r$ ring current source with azimuthal dependence $\exp(im\phi)$. +For (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $\exp(im\phi)$. The $\pm m$ fields are computed separately (even though they are complex conjugates) and combined as part of the calculation of the total fields of the expansion. -The figures below show the radiation pattern for each of these cases compared with the analytic result. The simulation and analytic results show good agreement. +The figures show the radiation pattern for each of these cases obtained using the scripts. The simulation and analytic results show good agreement. The simulation scripts are in [examples/dipole_in_vacuum_cyl_axisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_axisymmetric.py) and [examples/dipole_in_vacuum_cyl_nonaxisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py). From 2f8b532e4859d40425baa5c6bdaf1d18eb64bcd9 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Fri, 27 Dec 2024 21:08:10 -0800 Subject: [PATCH 05/15] simplify the nonaxisymmetric computation by obtaining the -m fields directly from the +m fields using the analytic expression --- .../Near_to_Far_Field_Spectra.md | 10 +- ...diation_pattern_nonaxisymmetric_dipole.png | Bin 115740 -> 66676 bytes .../dipole_in_vacuum_cyl_axisymmetric.py | 92 ++++++++---------- .../dipole_in_vacuum_cyl_nonaxisymmetric.py | 89 +++++++---------- 4 files changed, 80 insertions(+), 111 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 0d26ce106..faa80d63f 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -2,7 +2,7 @@ # Near to Far Field Spectra --- -The [near-to-far field transformation](../Python_User_Interface.md#near-to-far-field-spectra) feature in Cartesian (2D/3D) and [cylindrical](../Cylindrical_Coordinates.md) coordinates is demonstrated using six different examples. Generally, there are three steps involved in this type of calculation. First, the "near" surface(s) is defined as a set of surfaces capturing *all* outgoing radiation in *free space* in the desired direction(s). Second, the simulation is run using a pulsed source (or alternatively, a CW source via the [frequency-domain solver](../Python_User_Interface.md#frequency-domain-solver)) to allow Meep to accumulate the DFT fields on the near surface(s). Third, Meep computes the "far" fields at any desired points with the option to save the far fields to an HDF5 file. +The [near-to-far field transformation](../Python_User_Interface.md#near-to-far-field-spectra) feature in Cartesian (2D/3D) and [cylindrical](Cylindrical_Coordinates.md) coordinates is demonstrated using six different examples. Generally, there are three steps involved in this type of calculation. First, the "near" surface(s) is defined as a set of surfaces capturing *all* outgoing radiation in *free space* in the desired direction(s). Second, the simulation is run using a pulsed source (or alternatively, a CW source via the [frequency-domain solver](../Python_User_Interface.md#frequency-domain-solver)) to allow Meep to accumulate the DFT fields on the near surface(s). Third, Meep computes the "far" fields at any desired points with the option to save the far fields to an HDF5 file. [TOC] @@ -1706,13 +1706,13 @@ When these two conditions are not met as in the example below involving a small Radiation Pattern of an Antenna in Cylindrical Coordinates ---------------------------------------------------------- -In an earlier section, we showed how to compute the [radiation pattern of an antennna](#radiation-pattern-of-an-antenna) in vacuum with **linear** polarization using 2D Cartesian coordinates. The same calculation can also be performed using [cylindrical coordinates](Exploiting_Symmetry.md#cylindrical-symmetry). We will demonstrate this for two cases in which the dipole is (1) axisymmetric (i.e., at $r = 0$) or (2) nonaxisymmetric (i.e., at $r > 0$). In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). +Earlier we showed how to compute the [radiation pattern of an antennna in vacuum with linear polarization](#radiation-pattern-of-an-antenna) using a simulation in 2D Cartesian coordinates. The same calculation can also be performed using [cylindrical coordinates](../Exploiting_Symmetry.md#cylindrical-symmetry). We will demonstrate this for two different cases in which the dipole is (1) axisymmetric (i.e., at $r = 0$) or (2) nonaxisymmetric (i.e., at $r > 0$). In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). -For (1), an $E_z$ dipole is positioned exactly at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This involves *two* simulations. Note that computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because in the latter a circularly polarized source emits the same power as a linearly polarized source. +For (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations. Note that computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because the latter calculation involves a circularly polarized source which emits exactly half the power as a linearly polarized source even though their radiation patterns are different: $\frac{1}{2}(1 + \cos^2\theta)$ vs. $\cos^2\theta$. -For (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $\exp(im\phi)$. The $\pm m$ fields are computed separately (even though they are complex conjugates) and combined as part of the calculation of the total fields of the expansion. +For (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $\exp(im\phi)$. The $m = -1$ fields can be obtained directly from the $m = +1$ fields using the formulas $(E_r, E_\phi, E_z)_m = (E_r, -E_\phi, E_z)_{-m}$ and $(H_r, H_\phi, H_z)_m = (-H_r, H_\phi, -H_z)_{-m}$. These formulas can be used to simplify the expressions for the Fourier series expansion of the fields at $\phi = 0$: $\vec{E}_{tot}(\theta) = \vec{E}_{m=0}(\theta) + 2\sum_{m=1}^M (E_r, 0, E_z)_m$ and $\vec{H}_{tot}(\theta) = \vec{H}_{m=0}(\theta) + 2\sum_{m=1}^M (0, H_\phi, 0)_m$. -The figures show the radiation pattern for each of these cases obtained using the scripts. The simulation and analytic results show good agreement. +The radiation pattern for each of these cases obtained using the corresponding scripts is shown below. The simulation results show agreement with the analytic formulas. 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cylindrical coordinates. + +Tutorial Reference: + +https://meep.readthedocs.io/en/latest/Python_Tutorials/Near_to_Far_Field_Spectra/#radiation-pattern-of-an-antenna-in-cylindrical-coordinates +""" import argparse import math @@ -35,11 +40,11 @@ def plot_radiation_pattern(dipole_pol: int, radial_flux: np.ndarray): if dipole_pol == mp.X: dipole_radial_flux = np.square(np.cos(polar_rad)) - dipole_radial_flux_label = "$\cos^2θ$" + dipole_radial_flux_label = r"$\cos^2θ$" dipole_name = "$E_x$" elif dipole_pol == mp.Z: dipole_radial_flux = np.square(np.sin(polar_rad)) - dipole_radial_flux_label = "$\sin^2θ$" + dipole_radial_flux_label = r"$\sin^2θ$" dipole_name = "$E_z$" fig, ax = plt.subplots(subplot_kw={"projection": "polar"}, figsize=(6, 6)) @@ -54,20 +59,18 @@ def plot_radiation_pattern(dipole_pol: int, radial_flux: np.ndarray): ax.grid(True) ax.set_rlabel_position(22) ax.set_ylabel("radial flux (a.u.)") - ax.set_title("radiation pattern of an axisymmetric " + dipole_name + - " dipole") + ax.set_title("radiation pattern of an axisymmetric " + dipole_name + " dipole") if mp.am_master(): fig.savefig( "dipole_radiation_pattern_axisymmetric.png", - dpi=100, + dpi=150, bbox_inches="tight", ) - relative_error = ( - np.linalg.norm(normalized_radial_flux - dipole_radial_flux) / - np.linalg.norm(dipole_radial_flux) - ) + relative_error = np.linalg.norm( + normalized_radial_flux - dipole_radial_flux + ) / np.linalg.norm(dipole_radial_flux) print(f"relative error in radiation pattern:, {relative_error}") @@ -84,18 +87,15 @@ def radiation_pattern(e_field: np.ndarray, h_field: np.ndarray) -> np.ndarray: [0, π/2] rad. 0 radians is the +z direction (the "pole") and π/2 is the +r direction (the "equator"). """ - flux_x = 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_x = 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_r = np.sqrt(np.square(flux_x) + np.square(flux_z)) return flux_r def get_farfields( - sim: mp.Simulation, - n2f_mon: mp.DftNear2Far + sim: mp.Simulation, n2f_mon: mp.DftNear2Far ) -> Tuple[np.ndarray, np.ndarray]: """Computes the far fields from the near fields. @@ -117,8 +117,8 @@ def get_farfields( mp.Vector3( FARFIELD_RADIUS_UM * math.sin(polar_rad[n]), 0, - FARFIELD_RADIUS_UM * math.cos(polar_rad[n]) - ) + FARFIELD_RADIUS_UM * math.cos(polar_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)] @@ -126,10 +126,7 @@ def get_farfields( return e_field, h_field -def dipole_in_vacuum( - dipole_pol: int, - m: int -) -> Tuple[np.ndarray, np.ndarray]: +def dipole_in_vacuum(dipole_pol: int, m: int) -> Tuple[np.ndarray, np.ndarray]: """Computes the far fields of an axisymmetric point source. Args: @@ -153,14 +150,14 @@ def dipole_in_vacuum( mp.Source( src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), component=mp.Er, - center=mp.Vector3(dipole_pos_r, 0, 0) + center=mp.Vector3(dipole_pos_r, 0, 0), ), mp.Source( src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), component=mp.Ep, center=mp.Vector3(dipole_pos_r, 0, 0), - amplitude=1j if m == 1 else -1j - ) + amplitude=1j if m == 1 else -1j, + ), ] elif dipole_pol == mp.Z: dipole_pos_r = 0 @@ -168,7 +165,7 @@ def dipole_in_vacuum( mp.Source( src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), component=mp.Ez, - center=mp.Vector3(dipole_pos_r, 0, 0) + center=mp.Vector3(dipole_pos_r, 0, 0), ) ] @@ -179,7 +176,7 @@ def dipole_in_vacuum( m=m, boundary_layers=boundary_layers, sources=sources, - force_complex_fields=True + force_complex_fields=True, ) nearfields_monitor = sim.add_near2far( @@ -187,18 +184,14 @@ def dipole_in_vacuum( 0, 1, mp.FluxRegion( - center=mp.Vector3(0.5 * sr, 0, 0.5 * sz), - size=mp.Vector3(sr, 0, 0) - ), - mp.FluxRegion( - center=mp.Vector3(sr, 0, 0), - size=mp.Vector3(0, 0, sz) + center=mp.Vector3(0.5 * sr, 0, 0.5 * sz), size=mp.Vector3(sr, 0, 0) ), + mp.FluxRegion(center=mp.Vector3(sr, 0, 0), size=mp.Vector3(0, 0, sz)), mp.FluxRegion( center=mp.Vector3(0.5 * sr, 0, -0.5 * sz), size=mp.Vector3(sr, 0, 0), - weight=-1.0 - ) + weight=-1.0, + ), ) sim.run( @@ -206,7 +199,7 @@ def dipole_in_vacuum( 20.0, mp.Er if dipole_pol == mp.X else mp.Ez, mp.Vector3(dipole_pos_r, 0, 0), - 1e-6 + 1e-6, ) ) @@ -214,11 +207,7 @@ def dipole_in_vacuum( fig, ax = plt.subplots() sim.plot2D(ax=ax, show_monitors=True) if mp.am_master(): - fig.savefig( - "dipole_in_vacuum_cyl_layout.png", - dpi=150, - bbox_inches="tight" - ) + fig.savefig("dipole_in_vacuum_cyl_layout.png", dpi=150, bbox_inches="tight") e_field, h_field = get_farfields(sim, nearfields_monitor) @@ -239,10 +228,10 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: dtheta = 0.5 * math.pi / (NUM_FARFIELD_PTS - 1) dipole_radiation_pattern = radiation_pattern(e_field, h_field) flux = ( - np.sum(dipole_radiation_pattern * np.sin(polar_rad)) * - FARFIELD_RADIUS_UM**2 * - dtheta * - dphi + np.sum(dipole_radiation_pattern * np.sin(polar_rad)) + * FARFIELD_RADIUS_UM**2 + * dtheta + * dphi ) return flux @@ -254,7 +243,7 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: "dipole_pol", type=str, choices=["x", "z"], - help="polarization of the electric dipole (x or z)" + help="polarization of the electric dipole (x or z)", ) args = parser.parse_args() @@ -278,15 +267,10 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: e_field_total += e_field h_field_total += h_field - dipole_radiation_pattern = radiation_pattern( - e_field_total, h_field_total - ) - dipole_radiation_pattern_scaled = ( - dipole_radiation_pattern * FARFIELD_RADIUS_UM**2 - ) + dipole_radiation_pattern = radiation_pattern(e_field_total, h_field_total) + dipole_radiation_pattern_scaled = dipole_radiation_pattern * FARFIELD_RADIUS_UM**2 plot_radiation_pattern( - mp.X if args.dipole_pol == "x" else mp.Z, - dipole_radiation_pattern_scaled + mp.X if args.dipole_pol == "x" else mp.Z, dipole_radiation_pattern_scaled ) if mp.am_master(): diff --git a/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py b/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py index 6c344d5e6..1ef97157d 100644 --- a/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py +++ b/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py @@ -1,4 +1,9 @@ -"""Radiation pattern of a nonaxisymmetric dipole in cylindrical coordinates.""" +"""Radiation pattern of a nonaxisymmetric dipole in cylindrical coordinates. + +Tutorial Reference: + +https://meep.readthedocs.io/en/latest/Python_Tutorials/Near_to_Far_Field_Spectra/#radiation-pattern-of-an-antenna-in-cylindrical-coordinates +""" import argparse import math @@ -35,7 +40,7 @@ def plot_radiation_pattern(radial_flux: np.ndarray): fig, ax = plt.subplots(subplot_kw={"projection": "polar"}, figsize=(6, 6)) ax.plot(polar_rad, normalized_radial_flux, "b-", label="Meep") - ax.plot(polar_rad, dipole_radial_flux, "r--", label="$\cos^2θ$") + ax.plot(polar_rad, dipole_radial_flux, "r--", label=r"$\cos^2θ$") ax.legend() ax.set_theta_direction(-1) ax.set_theta_offset(0.5 * math.pi) @@ -45,19 +50,18 @@ def plot_radiation_pattern(radial_flux: np.ndarray): ax.grid(True) ax.set_rlabel_position(22) ax.set_ylabel("radial flux (a.u.)") - ax.set_title("radiation pattern of a nonaxisymmetric $E_r$ dipole") + ax.set_title("radiation pattern of a nonaxisymmetric $E_x$ dipole") if mp.am_master(): fig.savefig( "dipole_radiation_pattern_nonaxisymmetric.png", - dpi=100, + dpi=150, bbox_inches="tight", ) - relative_error = ( - np.linalg.norm(normalized_radial_flux - dipole_radial_flux) / - np.linalg.norm(dipole_radial_flux) - ) + relative_error = np.linalg.norm( + normalized_radial_flux - dipole_radial_flux + ) / np.linalg.norm(dipole_radial_flux) print(f"relative error in radiation pattern:, {relative_error}") @@ -74,18 +78,15 @@ def radiation_pattern(e_field: np.ndarray, h_field: np.ndarray) -> np.ndarray: [0, π/2] rad. 0 radians is the +z direction (the "pole") and π/2 is the +r direction (the "equator"). """ - flux_x = 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_x = 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_r = np.sqrt(np.square(flux_x) + np.square(flux_z)) return flux_r def get_farfields( - sim: mp.Simulation, - n2f_mon: mp.DftNear2Far + sim: mp.Simulation, n2f_mon: mp.DftNear2Far ) -> Tuple[np.ndarray, np.ndarray]: """Computes the far fields from the near fields. @@ -107,8 +108,8 @@ def get_farfields( mp.Vector3( FARFIELD_RADIUS_UM * math.sin(polar_rad[n]), 0, - FARFIELD_RADIUS_UM * math.cos(polar_rad[n]) - ) + FARFIELD_RADIUS_UM * math.cos(polar_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)] @@ -116,10 +117,7 @@ def get_farfields( return e_field, h_field -def dipole_in_vacuum( - dipole_pos_r: mp.Vector3, - m: int -) -> Tuple[np.ndarray, np.ndarray]: +def dipole_in_vacuum(dipole_pos_r: mp.Vector3, m: int) -> Tuple[np.ndarray, np.ndarray]: """Computes the far fields of a nonaxisymmetric point source. Args: @@ -139,7 +137,7 @@ def dipole_in_vacuum( mp.Source( src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), component=mp.Er, - center=mp.Vector3(dipole_pos_r, 0, 0) + center=mp.Vector3(dipole_pos_r, 0, 0), ) ] @@ -150,7 +148,7 @@ def dipole_in_vacuum( m=m, boundary_layers=boundary_layers, sources=sources, - force_complex_fields=True + force_complex_fields=True, ) nearfields_monitor = sim.add_near2far( @@ -158,18 +156,14 @@ def dipole_in_vacuum( 0, 1, mp.FluxRegion( - center=mp.Vector3(0.5 * sr, 0, 0.5 * sz), - size=mp.Vector3(sr, 0, 0) - ), - mp.FluxRegion( - center=mp.Vector3(sr, 0, 0), - size=mp.Vector3(0, 0, sz) + center=mp.Vector3(0.5 * sr, 0, 0.5 * sz), size=mp.Vector3(sr, 0, 0) ), + mp.FluxRegion(center=mp.Vector3(sr, 0, 0), size=mp.Vector3(0, 0, sz)), mp.FluxRegion( center=mp.Vector3(0.5 * sr, 0, -0.5 * sz), size=mp.Vector3(sr, 0, 0), - weight=-1.0 - ) + weight=-1.0, + ), ) sim.run( @@ -182,11 +176,7 @@ def dipole_in_vacuum( fig, ax = plt.subplots() sim.plot2D(ax=ax, show_monitors=True) if mp.am_master(): - fig.savefig( - "dipole_in_vacuum_cyl_layout.png", - dpi=150, - bbox_inches="tight" - ) + fig.savefig("dipole_in_vacuum_cyl_layout.png", dpi=150, bbox_inches="tight") e_field, h_field = get_farfields(sim, nearfields_monitor) @@ -207,10 +197,10 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: dtheta = 0.5 * math.pi / (NUM_FARFIELD_PTS - 1) dipole_radiation_pattern = radiation_pattern(e_field, h_field) flux = ( - np.sum(dipole_radiation_pattern * np.sin(polar_rad)) * - FARFIELD_RADIUS_UM**2 * - dtheta * - dphi + np.sum(dipole_radiation_pattern * np.sin(polar_rad)) + * FARFIELD_RADIUS_UM**2 + * dtheta + * dphi ) return flux @@ -219,9 +209,7 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: if __name__ == "__main__": parser = argparse.ArgumentParser() parser.add_argument( - "dipole_pos_r", - type=float, - help="radial position of the dipole" + "dipole_pos_r", type=float, help="radial position of the dipole" ) args = parser.parse_args() @@ -234,13 +222,14 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: m = 0 while True: e_field, h_field = dipole_in_vacuum(args.dipole_pos_r, m) - e_field_total += e_field - h_field_total += h_field - if m > 0: - e_field, h_field = dipole_in_vacuum(args.dipole_pos_r, -m) + if m == 0: e_field_total += e_field h_field_total += h_field + else: + e_field_total[:, 0] += 2 * e_field[:, 0] + h_field_total[:, 1] += 2 * h_field[:, 1] + e_field_total[:, 2] += 2 * e_field[:, 2] flux = flux_from_farfields(e_field, h_field) if flux > flux_max: @@ -253,12 +242,8 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: else: m += 1 - dipole_radiation_pattern = radiation_pattern( - e_field_total, h_field_total - ) - dipole_radiation_pattern_scaled = ( - dipole_radiation_pattern * FARFIELD_RADIUS_UM**2 - ) + dipole_radiation_pattern = radiation_pattern(e_field_total, h_field_total) + dipole_radiation_pattern_scaled = dipole_radiation_pattern * FARFIELD_RADIUS_UM**2 plot_radiation_pattern(dipole_radiation_pattern_scaled) if mp.am_master(): From 8ba215777a61dcc5a50b0b9bef33af5070d750ab Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Sat, 28 Dec 2024 20:32:08 -0800 Subject: [PATCH 06/15] add example of a nonaxisymmetric dipole with Ey polarization --- .../Near_to_Far_Field_Spectra.md | 8 +-- .../radiation_pattern_axisymmetric_dipole.png | Bin 116839 -> 190355 bytes ...diation_pattern_nonaxisymmetric_dipole.png | Bin 66676 -> 189142 bytes .../dipole_in_vacuum_cyl_axisymmetric.py | 42 ++++++----- .../dipole_in_vacuum_cyl_nonaxisymmetric.py | 66 +++++++++++++----- 5 files changed, 75 insertions(+), 41 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 faa80d63f..7f447f64a 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -1706,13 +1706,13 @@ When these two conditions are not met as in the example below involving a small Radiation Pattern of an Antenna in Cylindrical Coordinates ---------------------------------------------------------- -Earlier we showed how to compute the [radiation pattern of an antennna in vacuum with linear polarization](#radiation-pattern-of-an-antenna) using a simulation in 2D Cartesian coordinates. The same calculation can also be performed using [cylindrical coordinates](../Exploiting_Symmetry.md#cylindrical-symmetry). We will demonstrate this for two different cases in which the dipole is (1) axisymmetric (i.e., at $r = 0$) or (2) nonaxisymmetric (i.e., at $r > 0$). In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). +Earlier we showed how to compute the [radiation pattern of an antenna with linear polarization in vacuum](#radiation-pattern-of-an-antenna) using a simulation in 2D Cartesian coordinates. The same calculation can also be performed using [cylindrical coordinates](../Exploiting_Symmetry.md#cylindrical-symmetry). We will compute the radiation pattern for two different cases in which the dipole is (1) axisymmetric (i.e., at $r = 0$) or (2) nonaxisymmetric (i.e., at $r > 0$). The radiation pattern will be validated using the analytic result from antenna theory. In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). Note that the radiation pattern for an $\hat{x}$ or $\hat{y}$ polarized dipole is *not* rotationally invariant about the $z$ axis. This means that the radiation pattern depends on the choice of $\phi$. -For (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations. Note that computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because the latter calculation involves a circularly polarized source which emits exactly half the power as a linearly polarized source even though their radiation patterns are different: $\frac{1}{2}(1 + \cos^2\theta)$ vs. $\cos^2\theta$. +For (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations. The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because the latter calculation involves a circularly polarized source which emits exactly half the power as a linearly polarized source even though their radiation patterns are different: $\frac{1}{2}(1 + \cos^2\theta)$ vs. $\cos^2\theta$. -For (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $\exp(im\phi)$. The $m = -1$ fields can be obtained directly from the $m = +1$ fields using the formulas $(E_r, E_\phi, E_z)_m = (E_r, -E_\phi, E_z)_{-m}$ and $(H_r, H_\phi, H_z)_m = (-H_r, H_\phi, -H_z)_{-m}$. These formulas can be used to simplify the expressions for the Fourier series expansion of the fields at $\phi = 0$: $\vec{E}_{tot}(\theta) = \vec{E}_{m=0}(\theta) + 2\sum_{m=1}^M (E_r, 0, E_z)_m$ and $\vec{H}_{tot}(\theta) = \vec{H}_{m=0}(\theta) + 2\sum_{m=1}^M (0, H_\phi, 0)_m$. +For (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $\exp(im\phi)$. The $m = -1$ fields can be obtained directly from the $m = +1$ fields using the formulas $(E_r, E_\phi, E_z)_m = (E_r, -E_\phi, E_z)_{-m}$ and $(H_r, H_\phi, H_z)_m = (-H_r, H_\phi, -H_z)_{-m}$. These formulas can be used to simplify the expressions for the Fourier-series expansion of the fields at $\phi = 0$: $\vec{E}_{tot}(\theta) = (E_r, 0, E_z)_{m=0} + 2\sum_{m=1}^M (E_r, 0, E_z)_m$ and $\vec{H}_{tot}(\theta) = (0, H_\phi, 0)_{m=0} + 2\sum_{m=1}^M (0, H_\phi, 0)_m$. An $E_y$ (or equivalently an $E_\phi$) dipole involves flipping the sign of the $-m$ fields resulting in: $\vec{E}_{tot}(\theta) = (0, E_\phi, 0)_{m=0} + 2\sum_{m=1}^M (0, E_\phi, 0)_m$ and $\vec{H}_{tot}(\theta) = (H_r, 0, H_z)_{m=0} + 2\sum_{m=1}^M (H_r, 0, H_z)_m$. We will compute the radiation pattern for $E_x$ and $E_y$. -The radiation pattern for each of these cases obtained using the corresponding scripts is shown below. The simulation results show agreement with the analytic formulas. +The radiation pattern for each case is shown below. The simulation results agree with the analytic formulas. The simulation scripts are in [examples/dipole_in_vacuum_cyl_axisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_axisymmetric.py) and [examples/dipole_in_vacuum_cyl_nonaxisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py). diff --git a/doc/docs/images/radiation_pattern_axisymmetric_dipole.png b/doc/docs/images/radiation_pattern_axisymmetric_dipole.png index 9193b1ce08e1fac04d92bc4550267ba4a1ae205f..d17706e351c3f61fab4b0e4ff7fd18bf7de4cd91 100644 GIT binary patch literal 190355 zcmb@ucU;eX|33U7vKksR6-i0a)E-6}GzbX|ZE0zjrj?c^?Qv2{q%^d&D5X+SsI+%! z4-KvRcyoQP?{)p|`*Gj@+#V0tb)F}E#_RPQ$MZOj=iB$Jnj+DdMilitP*h%y6Um+ZI z?>aa5{Kv1=)c42a|M%Ag{ku*ZtC3K!k^ zWR*;(gPY&WEVSRPHZmCh@y79Cxk09Y!_N!MVpahU9&F(X+ZI+`5qjXv7;}Wo$cmQp zNM3RFO6=RD1ErM*?qBtg)f_8!=W%;9WEwcLvT|mq>iwBg%ioQ^J!WPX!V?U$87?v^ zg|9C51*t@f`2HGgZj*X+f;+Lx^4tpLi4!LxB^?eK*F5T38cN7>5_dVo!Er}9jO$#f 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""" - flux_x = 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_x = np.real( + e_field[:, 1] * np.conj(h_field[:, 2]) - e_field[:, 2] * np.conj(h_field[:, 1]) + ) + flux_z = np.real( + e_field[:, 0] * np.conj(h_field[:, 1]) - e_field[:, 1] * np.conj(h_field[:, 0]) + ) flux_r = np.sqrt(np.square(flux_x) + np.square(flux_z)) return flux_r @@ -120,13 +126,13 @@ def get_farfields( FARFIELD_RADIUS_UM * math.cos(polar_rad[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)] return e_field, h_field -def dipole_in_vacuum(dipole_pol: int, m: int) -> Tuple[np.ndarray, np.ndarray]: +def dipole_in_vacuum(dipole_pol: str, m: int) -> Tuple[np.ndarray, np.ndarray]: """Computes the far fields of an axisymmetric point source. Args: @@ -142,7 +148,7 @@ def dipole_in_vacuum(dipole_pol: int, m: int) -> Tuple[np.ndarray, np.ndarray]: boundary_layers = [mp.PML(thickness=PML_UM)] - if dipole_pol == mp.X: + if dipole_pol == "x": # An Er source at r = 0 needs to be slighty offset due to a bug. # https://github.com/NanoComp/meep/issues/2704 dipole_pos_r = 1.5 / RESOLUTION_UM @@ -159,7 +165,7 @@ def dipole_in_vacuum(dipole_pol: int, m: int) -> Tuple[np.ndarray, np.ndarray]: amplitude=1j if m == 1 else -1j, ), ] - elif dipole_pol == mp.Z: + else: dipole_pos_r = 0 sources = [ mp.Source( @@ -197,7 +203,7 @@ def dipole_in_vacuum(dipole_pol: int, m: int) -> Tuple[np.ndarray, np.ndarray]: sim.run( until_after_sources=mp.stop_when_fields_decayed( 20.0, - mp.Er if dipole_pol == mp.X else mp.Ez, + mp.Er if dipole_pol == "x" else mp.Ez, mp.Vector3(dipole_pos_r, 0, 0), 1e-6, ) @@ -251,27 +257,25 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: h_field_total = np.zeros((NUM_FARFIELD_PTS, 3), dtype=np.complex128) if args.dipole_pol == "x": - # A linearly polarized dipole can be formed from the superposition + # An x-polarized dipole can be formed from the superposition # of two left- and right-circularly polarized dipoles. - e_field, h_field = dipole_in_vacuum(mp.X, +1) + e_field, h_field = dipole_in_vacuum("x", +1) e_field_total += 0.5 * e_field h_field_total += 0.5 * h_field - e_field, h_field = dipole_in_vacuum(mp.X, -1) + e_field, h_field = dipole_in_vacuum("x", -1) e_field_total += 0.5 * e_field h_field_total += 0.5 * h_field - elif args.dipole_pol == "z": - e_field, h_field = dipole_in_vacuum(mp.Z, 0) + else: + e_field, h_field = dipole_in_vacuum("z", 0) e_field_total += e_field h_field_total += h_field dipole_radiation_pattern = radiation_pattern(e_field_total, h_field_total) dipole_radiation_pattern_scaled = dipole_radiation_pattern * FARFIELD_RADIUS_UM**2 - plot_radiation_pattern( - mp.X if args.dipole_pol == "x" else mp.Z, dipole_radiation_pattern_scaled - ) + plot_radiation_pattern(args.dipole_pol, dipole_radiation_pattern_scaled) if mp.am_master(): np.savez( diff --git a/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py b/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py index 1ef97157d..aa94e2f7e 100644 --- a/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py +++ b/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py @@ -26,21 +26,29 @@ polar_rad = np.linspace(0, 0.5 * math.pi, NUM_FARFIELD_PTS) -def plot_radiation_pattern(radial_flux: np.ndarray): +def plot_radiation_pattern(dipole_pol: str, radial_flux: np.ndarray): """Plots the radiation pattern in polar coordinates. The angles increase clockwise with zero in the +z direction (the "pole") and π/2 in the +r direction (the "equator"). Args: + dipole_pol: the dipole polarization. radial_flux: the radial flux in polar coordinates. """ normalized_radial_flux = radial_flux / np.max(radial_flux) - dipole_radial_flux = np.square(np.cos(polar_rad)) + if dipole_pol == "x": + dipole_radial_flux = np.square(np.cos(polar_rad)) + dipole_radial_flux_label = r"$\cos^2θ$" + dipole_name = "$E_x$" + else: + dipole_radial_flux = np.ones(NUM_FARFIELD_PTS) + dipole_radial_flux_label = "constant (1.0)" + dipole_name = "$E_y$" fig, ax = plt.subplots(subplot_kw={"projection": "polar"}, figsize=(6, 6)) ax.plot(polar_rad, normalized_radial_flux, "b-", label="Meep") - ax.plot(polar_rad, dipole_radial_flux, "r--", label=r"$\cos^2θ$") + ax.plot(polar_rad, dipole_radial_flux, "r--", label=dipole_radial_flux_label) ax.legend() ax.set_theta_direction(-1) ax.set_theta_offset(0.5 * math.pi) @@ -50,7 +58,9 @@ def plot_radiation_pattern(radial_flux: np.ndarray): ax.grid(True) ax.set_rlabel_position(22) ax.set_ylabel("radial flux (a.u.)") - ax.set_title("radiation pattern of a nonaxisymmetric $E_x$ dipole") + ax.set_title( + "radiation pattern (φ = 0) of a nonaxisymmetric " f"{dipole_name} dipole" + ) if mp.am_master(): fig.savefig( @@ -78,8 +88,12 @@ def radiation_pattern(e_field: np.ndarray, h_field: np.ndarray) -> np.ndarray: [0, π/2] rad. 0 radians is the +z direction (the "pole") and π/2 is the +r direction (the "equator"). """ - flux_x = 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_x = np.real( + e_field[:, 1] * np.conj(h_field[:, 2]) - e_field[:, 2] * np.conj(h_field[:, 1]) + ) + flux_z = np.real( + e_field[:, 0] * np.conj(h_field[:, 1]) - e_field[:, 1] * np.conj(h_field[:, 0]) + ) flux_r = np.sqrt(np.square(flux_x) + np.square(flux_z)) return flux_r @@ -111,16 +125,19 @@ def get_farfields( FARFIELD_RADIUS_UM * math.cos(polar_rad[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)] return e_field, h_field -def dipole_in_vacuum(dipole_pos_r: mp.Vector3, m: int) -> Tuple[np.ndarray, np.ndarray]: +def dipole_in_vacuum( + dipole_pol: str, dipole_pos_r: mp.Vector3, m: int +) -> Tuple[np.ndarray, np.ndarray]: """Computes the far fields of a nonaxisymmetric point source. Args: + dipole_pol: the dipole polarization. dipole_pos_r: the radial position of the dipole. m: angular φ dependence of the fields exp(imφ). @@ -133,10 +150,12 @@ def dipole_in_vacuum(dipole_pos_r: mp.Vector3, m: int) -> Tuple[np.ndarray, np.n boundary_layers = [mp.PML(thickness=PML_UM)] + src_cmpt = mp.Er if dipole_pol == "x" else mp.Ep + sources = [ mp.Source( src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), - component=mp.Er, + component=src_cmpt, center=mp.Vector3(dipole_pos_r, 0, 0), ) ] @@ -168,7 +187,7 @@ def dipole_in_vacuum(dipole_pos_r: mp.Vector3, m: int) -> Tuple[np.ndarray, np.n sim.run( until_after_sources=mp.stop_when_fields_decayed( - 20.0, mp.Er, mp.Vector3(dipole_pos_r, 0, 0), 1e-6 + 20.0, src_cmpt, mp.Vector3(dipole_pos_r, 0, 0), 1e-6 ) ) @@ -208,11 +227,20 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: if __name__ == "__main__": parser = argparse.ArgumentParser() + parser.add_argument( + "dipole_pol", + type=str, + choices=["x", "y"], + help="polarization of the electric dipole (x or y)", + ) parser.add_argument( "dipole_pos_r", type=float, help="radial position of the dipole" ) args = parser.parse_args() + if args.dipole_pos_r == 0: + raise ValueError(f"dipole_pos_r must be nonzero.") + # Fourier series expansion of the fields from a ring current source # used to generate a point dipole localized in the azimuthal direction. @@ -221,15 +249,16 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: flux_max = 0 m = 0 while True: - e_field, h_field = dipole_in_vacuum(args.dipole_pos_r, m) + e_field, h_field = dipole_in_vacuum(args.dipole_pol, args.dipole_pos_r, m) - if m == 0: - e_field_total += e_field - h_field_total += h_field + if args.dipole_pol == "x": + e_field_total[:, 0] += e_field[:, 0] * (1 if m == 0 else 2) + h_field_total[:, 1] += h_field[:, 1] * (1 if m == 0 else 2) + e_field_total[:, 2] += e_field[:, 2] * (1 if m == 0 else 2) else: - e_field_total[:, 0] += 2 * e_field[:, 0] - h_field_total[:, 1] += 2 * h_field[:, 1] - e_field_total[:, 2] += 2 * e_field[:, 2] + h_field_total[:, 0] += h_field[:, 0] * (1 if m == 0 else 2) + e_field_total[:, 1] += e_field[:, 1] * (1 if m == 0 else 2) + h_field_total[:, 2] += h_field[:, 2] * (1 if m == 0 else 2) flux = flux_from_farfields(e_field, h_field) if flux > flux_max: @@ -244,7 +273,7 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: dipole_radiation_pattern = radiation_pattern(e_field_total, h_field_total) dipole_radiation_pattern_scaled = dipole_radiation_pattern * FARFIELD_RADIUS_UM**2 - plot_radiation_pattern(dipole_radiation_pattern_scaled) + plot_radiation_pattern(args.dipole_pol, dipole_radiation_pattern_scaled) if mp.am_master(): np.savez( @@ -254,6 +283,7 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: POWER_DECAY_THRESHOLD=POWER_DECAY_THRESHOLD, RESOLUTION_UM=RESOLUTION_UM, WAVELENGTH_UM=WAVELENGTH_UM, + dipole_pol=args.dipole_pol, dipole_pos_r=args.dipole_pos_r, dipole_radiation_pattern=dipole_radiation_pattern, e_field_total=e_field_total, From eaffa7cb06f6f7c78768e8ebd7ac9e6e2215731f Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Wed, 1 Jan 2025 15:10:49 -0800 Subject: [PATCH 07/15] =?UTF-8?q?make=20=CF=95=20a=20parameter=20with=20de?= =?UTF-8?q?fault=20value=20of=20zero?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../Near_to_Far_Field_Spectra.md | 10 +++--- .../radiation_pattern_axisymmetric_dipole.png | Bin 190355 -> 190404 bytes .../dipole_in_vacuum_cyl_axisymmetric.py | 29 +++++++-------- .../dipole_in_vacuum_cyl_nonaxisymmetric.py | 33 ++++++++---------- 4 files changed, 32 insertions(+), 40 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 7f447f64a..abbf6ead8 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -1706,15 +1706,15 @@ When these two conditions are not met as in the example below involving a small Radiation Pattern of an Antenna in Cylindrical Coordinates ---------------------------------------------------------- -Earlier we showed how to compute the [radiation pattern of an antenna with linear polarization in vacuum](#radiation-pattern-of-an-antenna) using a simulation in 2D Cartesian coordinates. The same calculation can also be performed using [cylindrical coordinates](../Exploiting_Symmetry.md#cylindrical-symmetry). We will compute the radiation pattern for two different cases in which the dipole is (1) axisymmetric (i.e., at $r = 0$) or (2) nonaxisymmetric (i.e., at $r > 0$). The radiation pattern will be validated using the analytic result from antenna theory. In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). Note that the radiation pattern for an $\hat{x}$ or $\hat{y}$ polarized dipole is *not* rotationally invariant about the $z$ axis. This means that the radiation pattern depends on the choice of $\phi$. +Earlier we showed how to compute the [radiation pattern of an antenna with linear polarization in vacuum](#radiation-pattern-of-an-antenna) using a simulation in 2D Cartesian coordinates. The same calculation can also be performed using [cylindrical coordinates](../Exploiting_Symmetry.md#cylindrical-symmetry). We will compute the radiation pattern for two different cases in which the dipole is (1) axisymmetric (i.e., at $r = 0$) or (2) nonaxisymmetric (i.e., at $r > 0$). The radiation pattern is validated using the analytic result from antenna theory. -For (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations. The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because the latter calculation involves a circularly polarized source which emits exactly half the power as a linearly polarized source even though their radiation patterns are different: $\frac{1}{2}(1 + \cos^2\theta)$ vs. $\cos^2\theta$. +In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). Note that the radiation pattern for an $\hat{x}$ or $\hat{y}$ polarized dipole is *not* rotationally invariant about the $z$ axis. This means that the radiation pattern depends on the choice of $\phi$. (This is different than the computation of the [extraction efficiency for nonaxisymmetric dipoles](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) for which the radiation pattern is rotationally invariant because the dipoles are arranged along the circumference of a circle.) -For (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $\exp(im\phi)$. The $m = -1$ fields can be obtained directly from the $m = +1$ fields using the formulas $(E_r, E_\phi, E_z)_m = (E_r, -E_\phi, E_z)_{-m}$ and $(H_r, H_\phi, H_z)_m = (-H_r, H_\phi, -H_z)_{-m}$. These formulas can be used to simplify the expressions for the Fourier-series expansion of the fields at $\phi = 0$: $\vec{E}_{tot}(\theta) = (E_r, 0, E_z)_{m=0} + 2\sum_{m=1}^M (E_r, 0, E_z)_m$ and $\vec{H}_{tot}(\theta) = (0, H_\phi, 0)_{m=0} + 2\sum_{m=1}^M (0, H_\phi, 0)_m$. An $E_y$ (or equivalently an $E_\phi$) dipole involves flipping the sign of the $-m$ fields resulting in: $\vec{E}_{tot}(\theta) = (0, E_\phi, 0)_{m=0} + 2\sum_{m=1}^M (0, E_\phi, 0)_m$ and $\vec{H}_{tot}(\theta) = (H_r, 0, H_z)_{m=0} + 2\sum_{m=1}^M (H_r, 0, H_z)_m$. We will compute the radiation pattern for $E_x$ and $E_y$. +For (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles ($E_r \pm iE_\phi$) as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations. The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because the latter calculation involves a circularly polarized source which emits exactly half the power as a linearly polarized source even though their radiation patterns are different: $\frac{1}{2}(1 + \cos^2\theta)$ vs. $\cos^2\theta$. -The radiation pattern for each case is shown below. The simulation results agree with the analytic formulas. +For (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $\exp(im\phi)$. The $m = -1$ fields can be obtained directly from the $m = +1$ fields using the formulas $(E_r, E_\phi, E_z)_m = (E_r, -E_\phi, E_z)_{-m}$ and $(H_r, H_\phi, H_z)_m = (-H_r, H_\phi, -H_z)_{-m}$. These formulas can be used to simplify the expressions for the Fourier-series expansion of the fields at $\phi = 0$: $\vec{E}_{tot}(\theta) = (E_r, 0, E_z)_{m=0} + 2\sum_{m=1}^M (E_r, 0, E_z)_m$ and $\vec{H}_{tot}(\theta) = (0, H_\phi, 0)_{m=0} + 2\sum_{m=1}^M (0, H_\phi, 0)_m$. An $E_y$ (or equivalently an $E_\phi$) dipole involves flipping the sign of the $-m$ fields resulting in: $\vec{E}_{tot}(\theta) = (0, E_\phi, 0)_{m=0} + 2\sum_{m=1}^M (0, E_\phi, 0)_m$ and $\vec{H}_{tot}(\theta) = (H_r, 0, H_z)_{m=0} + 2\sum_{m=1}^M (H_r, 0, H_z)_m$. We will compute the radiation pattern for $E_x$ and $E_y$ dipoles at $r = 0.1$ μm. -The simulation scripts are in [examples/dipole_in_vacuum_cyl_axisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_axisymmetric.py) and [examples/dipole_in_vacuum_cyl_nonaxisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py). +The radiation pattern for each case is shown below. The simulation results agree with the analytic formulas. 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z$+cj&+5uSzxU4OZ`%j2nATjqhe*l44xDmJ1ubxYP5NSPlydC3{;a^%;4e)MO9| zVCy|(rdQsbEbp^LWA45F>@R)454!$UsbdRU_Qp`8^4ji2d1~Ya3zO3Y=$BI$B8;+moA@~73c&QUkPOa5k%0$vXJYN zCp!-kJOc6OsCL+g5C@7__2>`FS@8tc-6V5!pOaDI#IF{q^dt(b&Y9xa8N^QZsNnhN zPR9xsI>Pv+(V9?A+vLdchrbaZ8xog~_0Hi84a$4K4)yzisjKT*cVe%#Ve!jI}~Nm+`#@_+y6wM2r0;2&FUe&Bbl z?&sa0-03bhMt)V$;Mv_M5@xg#ot8d)ppNzUn3HCY~c@a z2*oy0XHn80nI3Mdd4^sltZRzJJoS;aF`t@N}N&e0aY}&PHh)+ zFD?Z-aQx@F=;aEKz3*glKaNo@MKn`xcU*}|9X+>m$K@tPFh(d)0Vsf?IR;NM-^Bq9 znn&?ipuFQ?YVMzF!qknu#>oC~q$6Yq*g}b&u$bDo1||hAKo}j|Bqm0Y@(Wyktz_i_ zWFK_3Kfr#0`c{Bu9KRk8V%SG-iWR@ZVVx5hQ9j}6ev5-(N1eYMQSO=YoEvuV)5T;O z{^*^=znI%C-J(J`8Q_J4#MvoV&Z=gX(?LY~r$%Eb+H15Z7Blio3njD%h=BG*Q-(yP zka5G9K(&C_5|IxNH{QLM5xh9@Y9OkJ30!`=0nLu19`#<$C4L$x+Iq)pgo%MPx1vFF zP>wQrH|S~PF ziO4!54V3HqT+$dLZePr%Xq|N8^FCBIJ>C3J-@&IZxpuY+28 K`_nWnul_$jb3bnY diff --git a/python/examples/dipole_in_vacuum_cyl_axisymmetric.py b/python/examples/dipole_in_vacuum_cyl_axisymmetric.py index b6336d760..dc9256905 100644 --- a/python/examples/dipole_in_vacuum_cyl_axisymmetric.py +++ b/python/examples/dipole_in_vacuum_cyl_axisymmetric.py @@ -6,6 +6,7 @@ """ import argparse +import cmath import math from typing import Tuple @@ -19,8 +20,8 @@ PML_UM = 1.0 FARFIELD_RADIUS_UM = 1e6 * WAVELENGTH_UM NUM_FARFIELD_PTS = 50 +AZIMUTHAL_RAD = 0 POWER_DECAY_THRESHOLD = 1e-4 -DEBUG_OUTPUT = False frequency = 1 / WAVELENGTH_UM polar_rad = np.linspace(0, 0.5 * math.pi, NUM_FARFIELD_PTS) @@ -103,7 +104,7 @@ def radiation_pattern(e_field: np.ndarray, h_field: np.ndarray) -> np.ndarray: def get_farfields( sim: mp.Simulation, n2f_mon: mp.DftNear2Far ) -> Tuple[np.ndarray, np.ndarray]: - """Computes the far fields from the near fields. + """Computes the far fields from the near fields for φ = 0 (rz plane). Args: sim: a `Simulation` object. @@ -111,9 +112,10 @@ def get_farfields( Returns: The electric (Er, Ep, Ez) and magnetic (Hr, Hp, Hz) far fields. One row - with six columns for the fields for each point on the circumference of - a quarter circle with angular range of [0, π/2] rad. 0 radians is the - +z direction (the "pole") and π/2 is the +r direction (the "equator"). + for each point on the circumference of a quarter circle with angular + range of [0, π/2] rad. Each row has six columns for the fields. + 0 radians is the +z direction (the "pole") and π/2 is the +r direction + (the "equator"). """ e_field = np.zeros((NUM_FARFIELD_PTS, 3), dtype=np.complex128) h_field = np.zeros((NUM_FARFIELD_PTS, 3), dtype=np.complex128) @@ -209,12 +211,6 @@ def dipole_in_vacuum(dipole_pol: str, m: int) -> Tuple[np.ndarray, np.ndarray]: ) ) - if DEBUG_OUTPUT: - fig, ax = plt.subplots() - sim.plot2D(ax=ax, show_monitors=True) - if mp.am_master(): - fig.savefig("dipole_in_vacuum_cyl_layout.png", dpi=150, bbox_inches="tight") - e_field, h_field = get_farfields(sim, nearfields_monitor) return e_field, h_field @@ -258,15 +254,15 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: if args.dipole_pol == "x": # An x-polarized dipole can be formed from the superposition - # of two left- and right-circularly polarized dipoles. + # of left- and right-circularly polarized dipoles. e_field, h_field = dipole_in_vacuum("x", +1) - e_field_total += 0.5 * e_field - h_field_total += 0.5 * h_field + e_field_total += 0.5 * e_field * cmath.exp(1j * AZIMUTHAL_RAD) + h_field_total += 0.5 * h_field * cmath.exp(1j * AZIMUTHAL_RAD) e_field, h_field = dipole_in_vacuum("x", -1) - e_field_total += 0.5 * e_field - h_field_total += 0.5 * h_field + e_field_total += 0.5 * e_field * cmath.exp(-1j * AZIMUTHAL_RAD) + h_field_total += 0.5 * h_field * cmath.exp(-1j * AZIMUTHAL_RAD) else: e_field, h_field = dipole_in_vacuum("z", 0) @@ -280,6 +276,7 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: if mp.am_master(): np.savez( "dipole_farfields_axisymmetric.npz", + AZIMUTHAL_RAD=AZIMUTHAL_RAD, FARFIELD_RADIUS_UM=FARFIELD_RADIUS_UM, PML_UM=PML_UM, POWER_DECAY_THRESHOLD=POWER_DECAY_THRESHOLD, diff --git a/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py b/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py index aa94e2f7e..f86562b60 100644 --- a/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py +++ b/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py @@ -6,6 +6,7 @@ """ import argparse +import cmath import math from typing import Tuple @@ -19,8 +20,8 @@ PML_UM = 1.0 FARFIELD_RADIUS_UM = 1e6 * WAVELENGTH_UM NUM_FARFIELD_PTS = 50 +AZIMUTHAL_RAD = 0 POWER_DECAY_THRESHOLD = 1e-4 -DEBUG_OUTPUT = False frequency = 1 / WAVELENGTH_UM polar_rad = np.linspace(0, 0.5 * math.pi, NUM_FARFIELD_PTS) @@ -102,7 +103,7 @@ def radiation_pattern(e_field: np.ndarray, h_field: np.ndarray) -> np.ndarray: def get_farfields( sim: mp.Simulation, n2f_mon: mp.DftNear2Far ) -> Tuple[np.ndarray, np.ndarray]: - """Computes the far fields from the near fields. + """Computes the far fields from the near fields for φ = 0 (rz plane). Args: sim: a `Simulation` object. @@ -110,9 +111,10 @@ def get_farfields( Returns: The electric (Er, Ep, Ez) and magnetic (Hr, Hp, Hz) far fields. One row - with six columns for the fields for each point on the circumference of - a quarter circle with angular range of [0, π/2] rad. 0 radians is the - +z direction (the "pole") and π/2 is the +r direction (the "equator"). + for each point on the circumference of a quarter circle with angular + range of [0, π/2] rad. Each row has six columns for the fields. + 0 radians is the +z direction (the "pole") and π/2 is the +r direction + (the "equator"). """ e_field = np.zeros((NUM_FARFIELD_PTS, 3), dtype=np.complex128) h_field = np.zeros((NUM_FARFIELD_PTS, 3), dtype=np.complex128) @@ -191,12 +193,6 @@ def dipole_in_vacuum( ) ) - if DEBUG_OUTPUT: - fig, ax = plt.subplots() - sim.plot2D(ax=ax, show_monitors=True) - if mp.am_master(): - fig.savefig("dipole_in_vacuum_cyl_layout.png", dpi=150, bbox_inches="tight") - e_field, h_field = get_farfields(sim, nearfields_monitor) return e_field, h_field @@ -250,15 +246,13 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: m = 0 while True: e_field, h_field = dipole_in_vacuum(args.dipole_pol, args.dipole_pos_r, m) + e_field_total += e_field * cmath.exp(1j * m * AZIMUTHAL_RAD) + h_field_total += h_field * cmath.exp(1j * m * AZIMUTHAL_RAD) - if args.dipole_pol == "x": - e_field_total[:, 0] += e_field[:, 0] * (1 if m == 0 else 2) - h_field_total[:, 1] += h_field[:, 1] * (1 if m == 0 else 2) - e_field_total[:, 2] += e_field[:, 2] * (1 if m == 0 else 2) - else: - h_field_total[:, 0] += h_field[:, 0] * (1 if m == 0 else 2) - e_field_total[:, 1] += e_field[:, 1] * (1 if m == 0 else 2) - h_field_total[:, 2] += h_field[:, 2] * (1 if m == 0 else 2) + if m > 0: + e_field, h_field = dipole_in_vacuum(args.dipole_pol, args.dipole_pos_r, -m) + e_field_total += e_field * cmath.exp(-1j * m * AZIMUTHAL_RAD) + h_field_total += h_field * cmath.exp(-1j * m * AZIMUTHAL_RAD) flux = flux_from_farfields(e_field, h_field) if flux > flux_max: @@ -278,6 +272,7 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: if mp.am_master(): np.savez( "dipole_farfields_nonaxisymmetric.npz", + AZIMUTHAL_RAD=AZIMUTHAL_RAD, FARFIELD_RADIUS_UM=FARFIELD_RADIUS_UM, PML_UM=PML_UM, POWER_DECAY_THRESHOLD=POWER_DECAY_THRESHOLD, From bedac65aa44c8cd1a28dbd2ffcd16ff80fbdc752 Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Fri, 3 Jan 2025 15:48:57 -0500 Subject: [PATCH 08/15] Update doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md --- 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 abbf6ead8..7be3acc26 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -1710,7 +1710,7 @@ Earlier we showed how to compute the [radiation pattern of an antenna with linea In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). Note that the radiation pattern for an $\hat{x}$ or $\hat{y}$ polarized dipole is *not* rotationally invariant about the $z$ axis. This means that the radiation pattern depends on the choice of $\phi$. (This is different than the computation of the [extraction efficiency for nonaxisymmetric dipoles](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) for which the radiation pattern is rotationally invariant because the dipoles are arranged along the circumference of a circle.) -For (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles ($E_r \pm iE_\phi$) as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations. The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because the latter calculation involves a circularly polarized source which emits exactly half the power as a linearly polarized source even though their radiation patterns are different: $\frac{1}{2}(1 + \cos^2\theta)$ vs. $\cos^2\theta$. +For (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles ($E_r \pm iE_\phi$) as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations. The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because extraction efficiency involves integration over $\phi$, in which case the $m = \pm 1$ cross terms cancel and you simply get identical integrated powers from $m = \pm 1$. For (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $\exp(im\phi)$. The $m = -1$ fields can be obtained directly from the $m = +1$ fields using the formulas $(E_r, E_\phi, E_z)_m = (E_r, -E_\phi, E_z)_{-m}$ and $(H_r, H_\phi, H_z)_m = (-H_r, H_\phi, -H_z)_{-m}$. These formulas can be used to simplify the expressions for the Fourier-series expansion of the fields at $\phi = 0$: $\vec{E}_{tot}(\theta) = (E_r, 0, E_z)_{m=0} + 2\sum_{m=1}^M (E_r, 0, E_z)_m$ and $\vec{H}_{tot}(\theta) = (0, H_\phi, 0)_{m=0} + 2\sum_{m=1}^M (0, H_\phi, 0)_m$. An $E_y$ (or equivalently an $E_\phi$) dipole involves flipping the sign of the $-m$ fields resulting in: $\vec{E}_{tot}(\theta) = (0, E_\phi, 0)_{m=0} + 2\sum_{m=1}^M (0, E_\phi, 0)_m$ and $\vec{H}_{tot}(\theta) = (H_r, 0, H_z)_{m=0} + 2\sum_{m=1}^M (H_r, 0, H_z)_m$. We will compute the radiation pattern for $E_x$ and $E_y$ dipoles at $r = 0.1$ μm. From 6167ba43222adce6d8638db95cbd75aff696bc04 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Sat, 4 Jan 2025 15:34:57 -0800 Subject: [PATCH 09/15] use two sources and DFT field monitors to simplify calculation of nonaxisymmetric dipoles --- .../Near_to_Far_Field_Spectra.md | 6 +-- .../dipole_in_vacuum_cyl_axisymmetric.py | 4 +- .../dipole_in_vacuum_cyl_nonaxisymmetric.py | 53 +++++++++++++------ 3 files changed, 41 insertions(+), 22 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 7be3acc26..0f30d018d 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -1708,11 +1708,11 @@ Radiation Pattern of an Antenna in Cylindrical Coordinates Earlier we showed how to compute the [radiation pattern of an antenna with linear polarization in vacuum](#radiation-pattern-of-an-antenna) using a simulation in 2D Cartesian coordinates. The same calculation can also be performed using [cylindrical coordinates](../Exploiting_Symmetry.md#cylindrical-symmetry). We will compute the radiation pattern for two different cases in which the dipole is (1) axisymmetric (i.e., at $r = 0$) or (2) nonaxisymmetric (i.e., at $r > 0$). The radiation pattern is validated using the analytic result from antenna theory. -In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). Note that the radiation pattern for an $\hat{x}$ or $\hat{y}$ polarized dipole is *not* rotationally invariant about the $z$ axis. This means that the radiation pattern depends on the choice of $\phi$. (This is different than the computation of the [extraction efficiency for nonaxisymmetric dipoles](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) for which the radiation pattern is rotationally invariant because the dipoles are arranged along the circumference of a circle.) +In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). Note that the radiation pattern for an $\hat{x}$ or $\hat{y}$ polarized dipole is *nonaxisymmetric*. This means that the radiation pattern depends on the choice of $\phi$. (This is different than the computation of the [extraction efficiency for nonaxisymmetric dipoles](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) which involves an integration over $\phi$ resulting in a radiation pattern that is axisymmetric.) -For (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles ($E_r \pm iE_\phi$) as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations. The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because extraction efficiency involves integration over $\phi$, in which case the $m = \pm 1$ cross terms cancel and you simply get identical integrated powers from $m = \pm 1$. +For (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles ($E_r \pm iE_\phi$) as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations. The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because extraction efficiency involves integration over $\phi$, in which case the $m = \pm 1$ cross terms cancel and you simply get identical integrated powers from $m = \pm 1$. -For (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $\exp(im\phi)$. The $m = -1$ fields can be obtained directly from the $m = +1$ fields using the formulas $(E_r, E_\phi, E_z)_m = (E_r, -E_\phi, E_z)_{-m}$ and $(H_r, H_\phi, H_z)_m = (-H_r, H_\phi, -H_z)_{-m}$. These formulas can be used to simplify the expressions for the Fourier-series expansion of the fields at $\phi = 0$: $\vec{E}_{tot}(\theta) = (E_r, 0, E_z)_{m=0} + 2\sum_{m=1}^M (E_r, 0, E_z)_m$ and $\vec{H}_{tot}(\theta) = (0, H_\phi, 0)_{m=0} + 2\sum_{m=1}^M (0, H_\phi, 0)_m$. An $E_y$ (or equivalently an $E_\phi$) dipole involves flipping the sign of the $-m$ fields resulting in: $\vec{E}_{tot}(\theta) = (0, E_\phi, 0)_{m=0} + 2\sum_{m=1}^M (0, E_\phi, 0)_m$ and $\vec{H}_{tot}(\theta) = (H_r, 0, H_z)_{m=0} + 2\sum_{m=1}^M (H_r, 0, H_z)_m$. We will compute the radiation pattern for $E_x$ and $E_y$ dipoles at $r = 0.1$ μm. +For (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $e^{im\phi}$. The Fourier-series expansion involves nonnegative $m$ values because the $-m$ fields can be obtained directly from the $+m$ fields using the relation $\vec{E}_{m}(r, z)e^{im\phi-i\omega t} = \left[\vec{E}_{-m}(r, z)e^{-im\phi+i\omega t} \right]^*$. There is a similar relation for $\vec{H}_{m}$. This involves taking the $\vec{E}_{m}$ fields at $-\omega$ (not $\omega$!) and conjugating them to get the $\vec{E}_{-m}$ fields at $\omega$. Implementing this in a simulation involves two aspects: (1) two sources with frequencies $\omega$ and $-\omega$ and (2) two DFT field monitors at $\omega$ and $-\omega$. (1) is necessary because it ensures the source current $\mathcal{J}$ is purely real and therefore does not change under a complex conjugation. We will compute the radiation pattern for $E_x$ and $E_y$ dipoles at $r = 0.1$ μm. The radiation pattern for each case is shown below. The simulation results agree with the analytic formulas. The simulation scripts are in [examples/dipole_in_vacuum_cyl_axisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_axisymmetric.py) and [examples/dipole_in_vacuum_cyl_nonaxisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py). diff --git a/python/examples/dipole_in_vacuum_cyl_axisymmetric.py b/python/examples/dipole_in_vacuum_cyl_axisymmetric.py index dc9256905..16d2e3cf6 100644 --- a/python/examples/dipole_in_vacuum_cyl_axisymmetric.py +++ b/python/examples/dipole_in_vacuum_cyl_axisymmetric.py @@ -60,9 +60,7 @@ def plot_radiation_pattern(dipole_pol: str, radial_flux: np.ndarray): ax.grid(True) ax.set_rlabel_position(22) ax.set_ylabel("radial flux (a.u.)") - ax.set_title( - "radiation pattern (φ = 0) of an axisymmetric " f"{dipole_name} dipole" - ) + ax.set_title("radiation pattern (φ = 0) of an axisymmetric {dipole_name} dipole") if mp.am_master(): fig.savefig( diff --git a/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py b/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py index f86562b60..079a77f30 100644 --- a/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py +++ b/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py @@ -59,9 +59,7 @@ def plot_radiation_pattern(dipole_pol: str, radial_flux: np.ndarray): ax.grid(True) ax.set_rlabel_position(22) ax.set_ylabel("radial flux (a.u.)") - ax.set_title( - "radiation pattern (φ = 0) of a nonaxisymmetric " f"{dipole_name} dipole" - ) + ax.set_title("radiation pattern (φ = 0) of a nonaxisymmetric {dipole_name} dipole") if mp.am_master(): fig.savefig( @@ -135,7 +133,7 @@ def get_farfields( def dipole_in_vacuum( dipole_pol: str, dipole_pos_r: mp.Vector3, m: int -) -> Tuple[np.ndarray, np.ndarray]: +) -> Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray]: """Computes the far fields of a nonaxisymmetric point source. Args: @@ -144,7 +142,8 @@ def dipole_in_vacuum( m: angular φ dependence of the fields exp(imφ). Returns: - A 2-tuple containing the electric and magnetic far fields as 1D arrays. + A 4-tuple containing the electric and magnetic far fields at positive + and negative frequencies, respectively, as 1D arrays. """ sr = 2.0 sz = 4.0 @@ -159,7 +158,12 @@ def dipole_in_vacuum( src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), component=src_cmpt, center=mp.Vector3(dipole_pos_r, 0, 0), - ) + ), + mp.Source( + src=mp.GaussianSource(-frequency, fwidth=0.1 * frequency), + component=src_cmpt, + center=mp.Vector3(dipole_pos_r, 0, 0), + ), ] sim = mp.Simulation( @@ -172,7 +176,7 @@ def dipole_in_vacuum( force_complex_fields=True, ) - nearfields_monitor = sim.add_near2far( + nearfields_monitor_plus = sim.add_near2far( frequency, 0, 1, @@ -187,15 +191,31 @@ def dipole_in_vacuum( ), ) + nearfields_monitor_minus = sim.add_near2far( + -frequency, + 0, + 1, + mp.FluxRegion( + center=mp.Vector3(0.5 * sr, 0, 0.5 * sz), size=mp.Vector3(sr, 0, 0) + ), + mp.FluxRegion(center=mp.Vector3(sr, 0, 0), size=mp.Vector3(0, 0, sz)), + mp.FluxRegion( + center=mp.Vector3(0.5 * sr, 0, -0.5 * sz), + size=mp.Vector3(sr, 0, 0), + weight=-1.0, + ), + ) + sim.run( until_after_sources=mp.stop_when_fields_decayed( 20.0, src_cmpt, mp.Vector3(dipole_pos_r, 0, 0), 1e-6 ) ) - e_field, h_field = get_farfields(sim, nearfields_monitor) + e_field_plus, h_field_plus = get_farfields(sim, nearfields_monitor_plus) + e_field_minus, h_field_minus = get_farfields(sim, nearfields_monitor_minus) - return e_field, h_field + return e_field_plus, h_field_plus, e_field_minus, h_field_minus def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: @@ -245,16 +265,17 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: flux_max = 0 m = 0 while True: - e_field, h_field = dipole_in_vacuum(args.dipole_pol, args.dipole_pos_r, m) - e_field_total += e_field * cmath.exp(1j * m * AZIMUTHAL_RAD) - h_field_total += h_field * cmath.exp(1j * m * AZIMUTHAL_RAD) + (e_field_plus, h_field_plus, e_field_minus, h_field_minus) = dipole_in_vacuum( + args.dipole_pol, args.dipole_pos_r, m + ) + e_field_total += e_field_plus * cmath.exp(1j * m * AZIMUTHAL_RAD) + h_field_total += h_field_plus * cmath.exp(1j * m * AZIMUTHAL_RAD) if m > 0: - e_field, h_field = dipole_in_vacuum(args.dipole_pol, args.dipole_pos_r, -m) - e_field_total += e_field * cmath.exp(-1j * m * AZIMUTHAL_RAD) - h_field_total += h_field * cmath.exp(-1j * m * AZIMUTHAL_RAD) + e_field_total += np.conj(e_field_minus) * cmath.exp(-1j * m * AZIMUTHAL_RAD) + h_field_total += np.conj(h_field_minus) * cmath.exp(-1j * m * AZIMUTHAL_RAD) - flux = flux_from_farfields(e_field, h_field) + flux = flux_from_farfields(e_field_plus, h_field_plus) if flux > flux_max: flux_max = flux power_decay = flux / flux_max From 2f151b0ef293e60f26d3c48fc94fed6cccd8d06e Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Thu, 9 Jan 2025 16:02:20 -0500 Subject: [PATCH 10/15] Update doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md --- 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 0f30d018d..366a24d15 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -1710,7 +1710,7 @@ Earlier we showed how to compute the [radiation pattern of an antenna with linea In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). Note that the radiation pattern for an $\hat{x}$ or $\hat{y}$ polarized dipole is *nonaxisymmetric*. This means that the radiation pattern depends on the choice of $\phi$. (This is different than the computation of the [extraction efficiency for nonaxisymmetric dipoles](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) which involves an integration over $\phi$ resulting in a radiation pattern that is axisymmetric.) -For (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles ($E_r \pm iE_\phi$) as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations. The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because extraction efficiency involves integration over $\phi$, in which case the $m = \pm 1$ cross terms cancel and you simply get identical integrated powers from $m = \pm 1$. +For (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles ($E_r \pm iE_\phi$) as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations at $m = \pm 1$ (although it is possible to combine these into a single $m=+1$ simulation at $\pm \omega$ as described below). The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because extraction efficiency involves integration over $\phi$, in which case the $m = \pm 1$ cross terms cancel and you simply get identical integrated powers from $m = \pm 1$. For (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $e^{im\phi}$. The Fourier-series expansion involves nonnegative $m$ values because the $-m$ fields can be obtained directly from the $+m$ fields using the relation $\vec{E}_{m}(r, z)e^{im\phi-i\omega t} = \left[\vec{E}_{-m}(r, z)e^{-im\phi+i\omega t} \right]^*$. There is a similar relation for $\vec{H}_{m}$. This involves taking the $\vec{E}_{m}$ fields at $-\omega$ (not $\omega$!) and conjugating them to get the $\vec{E}_{-m}$ fields at $\omega$. Implementing this in a simulation involves two aspects: (1) two sources with frequencies $\omega$ and $-\omega$ and (2) two DFT field monitors at $\omega$ and $-\omega$. (1) is necessary because it ensures the source current $\mathcal{J}$ is purely real and therefore does not change under a complex conjugation. We will compute the radiation pattern for $E_x$ and $E_y$ dipoles at $r = 0.1$ μm. From 920e16e82d0ef22f21a38d33ffb6c7fa2d286f6c Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Thu, 9 Jan 2025 16:07:50 -0500 Subject: [PATCH 11/15] Update doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md --- 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 366a24d15..7b22b7f1e 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -1706,7 +1706,7 @@ When these two conditions are not met as in the example below involving a small Radiation Pattern of an Antenna in Cylindrical Coordinates ---------------------------------------------------------- -Earlier we showed how to compute the [radiation pattern of an antenna with linear polarization in vacuum](#radiation-pattern-of-an-antenna) using a simulation in 2D Cartesian coordinates. The same calculation can also be performed using [cylindrical coordinates](../Exploiting_Symmetry.md#cylindrical-symmetry). We will compute the radiation pattern for two different cases in which the dipole is (1) axisymmetric (i.e., at $r = 0$) or (2) nonaxisymmetric (i.e., at $r > 0$). The radiation pattern is validated using the analytic result from antenna theory. +Earlier we showed how to compute the [radiation pattern of an antenna with linear polarization in vacuum](#radiation-pattern-of-an-antenna) using a simulation in 2D Cartesian coordinates. The same calculation can also be performed using [cylindrical coordinates](../Exploiting_Symmetry.md#cylindrical-symmetry). We will compute the radiation pattern for two different cases in which the dipole is (1) on-axis (i.e., at $r = 0$) or (2) off-axis (i.e., at $r > 0$). The radiation pattern is validated using the analytic result from antenna theory in vacuum—of course, in a homogeneous medium, the on- and off-axis cases are equivalent to observers at infinity. In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). Note that the radiation pattern for an $\hat{x}$ or $\hat{y}$ polarized dipole is *nonaxisymmetric*. This means that the radiation pattern depends on the choice of $\phi$. (This is different than the computation of the [extraction efficiency for nonaxisymmetric dipoles](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) which involves an integration over $\phi$ resulting in a radiation pattern that is axisymmetric.) From 05ab70896bd11db781a64f8993cf34aa3ea018f0 Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Thu, 9 Jan 2025 16:08:53 -0500 Subject: [PATCH 12/15] Update doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md --- 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 7b22b7f1e..da53432bb 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -1708,7 +1708,7 @@ Radiation Pattern of an Antenna in Cylindrical Coordinates Earlier we showed how to compute the [radiation pattern of an antenna with linear polarization in vacuum](#radiation-pattern-of-an-antenna) using a simulation in 2D Cartesian coordinates. The same calculation can also be performed using [cylindrical coordinates](../Exploiting_Symmetry.md#cylindrical-symmetry). We will compute the radiation pattern for two different cases in which the dipole is (1) on-axis (i.e., at $r = 0$) or (2) off-axis (i.e., at $r > 0$). The radiation pattern is validated using the analytic result from antenna theory in vacuum—of course, in a homogeneous medium, the on- and off-axis cases are equivalent to observers at infinity. -In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). Note that the radiation pattern for an $\hat{x}$ or $\hat{y}$ polarized dipole is *nonaxisymmetric*. This means that the radiation pattern depends on the choice of $\phi$. (This is different than the computation of the [extraction efficiency for nonaxisymmetric dipoles](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) which involves an integration over $\phi$ resulting in a radiation pattern that is axisymmetric.) +In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). Note that the radiation pattern for an $\hat{x}$ or $\hat{y}$ polarized dipole is *nonaxisymmetric* even if it is on-axis. This means that the radiation pattern depends on the choice of $\phi$. (This is different than the computation of the [extraction efficiency for nonaxisymmetric dipoles](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) which involves an integration over $\phi$ resulting in a radiation pattern that is axisymmetric.) For (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles ($E_r \pm iE_\phi$) as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations at $m = \pm 1$ (although it is possible to combine these into a single $m=+1$ simulation at $\pm \omega$ as described below). The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because extraction efficiency involves integration over $\phi$, in which case the $m = \pm 1$ cross terms cancel and you simply get identical integrated powers from $m = \pm 1$. From e6320802a065efd439bd08afb1c82ec67291da18 Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Thu, 9 Jan 2025 16:09:36 -0500 Subject: [PATCH 13/15] Update doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md --- 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 da53432bb..45c3ba211 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -1710,7 +1710,7 @@ Earlier we showed how to compute the [radiation pattern of an antenna with linea In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). Note that the radiation pattern for an $\hat{x}$ or $\hat{y}$ polarized dipole is *nonaxisymmetric* even if it is on-axis. This means that the radiation pattern depends on the choice of $\phi$. (This is different than the computation of the [extraction efficiency for nonaxisymmetric dipoles](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) which involves an integration over $\phi$ resulting in a radiation pattern that is axisymmetric.) -For (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles ($E_r \pm iE_\phi$) as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations at $m = \pm 1$ (although it is possible to combine these into a single $m=+1$ simulation at $\pm \omega$ as described below). The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because extraction efficiency involves integration over $\phi$, in which case the $m = \pm 1$ cross terms cancel and you simply get identical integrated powers from $m = \pm 1$. +For on-axis (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles ($E_r \pm iE_\phi$) as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations at $m = \pm 1$ (although it is possible to combine these into a single $m=+1$ simulation at $\pm \omega$ as described below). The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because extraction efficiency involves integration over $\phi$, in which case the $m = \pm 1$ cross terms cancel and you simply get identical integrated powers from $m = \pm 1$. For (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $e^{im\phi}$. The Fourier-series expansion involves nonnegative $m$ values because the $-m$ fields can be obtained directly from the $+m$ fields using the relation $\vec{E}_{m}(r, z)e^{im\phi-i\omega t} = \left[\vec{E}_{-m}(r, z)e^{-im\phi+i\omega t} \right]^*$. There is a similar relation for $\vec{H}_{m}$. This involves taking the $\vec{E}_{m}$ fields at $-\omega$ (not $\omega$!) and conjugating them to get the $\vec{E}_{-m}$ fields at $\omega$. Implementing this in a simulation involves two aspects: (1) two sources with frequencies $\omega$ and $-\omega$ and (2) two DFT field monitors at $\omega$ and $-\omega$. (1) is necessary because it ensures the source current $\mathcal{J}$ is purely real and therefore does not change under a complex conjugation. We will compute the radiation pattern for $E_x$ and $E_y$ dipoles at $r = 0.1$ μm. From 8d5ebda17608a918bf258458c861623aa7a3c533 Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Thu, 9 Jan 2025 16:10:26 -0500 Subject: [PATCH 14/15] Update doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md --- 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 45c3ba211..142c8674e 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -1712,7 +1712,7 @@ In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz For on-axis (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles ($E_r \pm iE_\phi$) as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations at $m = \pm 1$ (although it is possible to combine these into a single $m=+1$ simulation at $\pm \omega$ as described below). The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because extraction efficiency involves integration over $\phi$, in which case the $m = \pm 1$ cross terms cancel and you simply get identical integrated powers from $m = \pm 1$. -For (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $e^{im\phi}$. The Fourier-series expansion involves nonnegative $m$ values because the $-m$ fields can be obtained directly from the $+m$ fields using the relation $\vec{E}_{m}(r, z)e^{im\phi-i\omega t} = \left[\vec{E}_{-m}(r, z)e^{-im\phi+i\omega t} \right]^*$. There is a similar relation for $\vec{H}_{m}$. This involves taking the $\vec{E}_{m}$ fields at $-\omega$ (not $\omega$!) and conjugating them to get the $\vec{E}_{-m}$ fields at $\omega$. Implementing this in a simulation involves two aspects: (1) two sources with frequencies $\omega$ and $-\omega$ and (2) two DFT field monitors at $\omega$ and $-\omega$. (1) is necessary because it ensures the source current $\mathcal{J}$ is purely real and therefore does not change under a complex conjugation. We will compute the radiation pattern for $E_x$ and $E_y$ dipoles at $r = 0.1$ μm. +For off-axis (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $e^{im\phi}$. As an optimization, our Fourier-series expansion involves only nonnegative $m$ values because the $-m$ fields can be obtained directly from the $+m$ fields using the relation $\vec{E}_{m}(r, z)e^{im\phi-i\omega t} = \left[\vec{E}_{-m}(r, z)e^{-im\phi+i\omega t} \right]^*$. There is a similar relation for $\vec{H}_{m}$. This involves taking the $\vec{E}_{m}$ fields at $-\omega$ (not $\omega$!) and conjugating them to get the $\vec{E}_{-m}$ fields at $\omega$. Implementing this in a simulation involves two aspects: (1) two sources with frequencies $\omega$ and $-\omega$ and (2) two DFT field monitors at $\omega$ and $-\omega$. (1) is necessary because it ensures the source current $\mathcal{J}$ is purely real and therefore does not change under a complex conjugation. We will compute the radiation pattern for $E_x$ and $E_y$ dipoles at $r = 0.1$ μm. The radiation pattern for each case is shown below. The simulation results agree with the analytic formulas. The simulation scripts are in [examples/dipole_in_vacuum_cyl_axisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_axisymmetric.py) and [examples/dipole_in_vacuum_cyl_nonaxisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py). From 4fd9c083b7d98694801fd8b59b717ec7d130e9a1 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Thu, 9 Jan 2025 20:54:17 -0800 Subject: [PATCH 15/15] rename nonaxisymmetric, axisymmetric to off-axis, on-axis in script filename, within the script, and in plot titles --- .../Near_to_Far_Field_Spectra.md | 317 +++++++++++++++++- .../dipole_radiation_pattern_off_axis.png | Bin 0 -> 187844 bytes .../dipole_radiation_pattern_on_axis.png | Bin 0 -> 189308 bytes .../radiation_pattern_axisymmetric_dipole.png | Bin 190404 -> 0 bytes ...diation_pattern_nonaxisymmetric_dipole.png | Bin 189142 -> 0 bytes ...ic.py => dipole_in_vacuum_cyl_off_axis.py} | 10 +- ...ric.py => dipole_in_vacuum_cyl_on_axis.py} | 10 +- 7 files changed, 321 insertions(+), 16 deletions(-) create mode 100644 doc/docs/images/dipole_radiation_pattern_off_axis.png create mode 100644 doc/docs/images/dipole_radiation_pattern_on_axis.png delete mode 100644 doc/docs/images/radiation_pattern_axisymmetric_dipole.png delete mode 100644 doc/docs/images/radiation_pattern_nonaxisymmetric_dipole.png rename python/examples/{dipole_in_vacuum_cyl_nonaxisymmetric.py => dipole_in_vacuum_cyl_off_axis.py} (96%) rename python/examples/{dipole_in_vacuum_cyl_axisymmetric.py => dipole_in_vacuum_cyl_on_axis.py} (96%) 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 142c8674e..2a3e14a7d 100644 --- a/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md +++ b/doc/docs/Python_Tutorials/Near_to_Far_Field_Spectra.md @@ -1708,14 +1708,319 @@ Radiation Pattern of an Antenna in Cylindrical Coordinates Earlier we showed how to compute the [radiation pattern of an antenna with linear polarization in vacuum](#radiation-pattern-of-an-antenna) using a simulation in 2D Cartesian coordinates. The same calculation can also be performed using [cylindrical coordinates](../Exploiting_Symmetry.md#cylindrical-symmetry). We will compute the radiation pattern for two different cases in which the dipole is (1) on-axis (i.e., at $r = 0$) or (2) off-axis (i.e., at $r > 0$). The radiation pattern is validated using the analytic result from antenna theory in vacuum—of course, in a homogeneous medium, the on- and off-axis cases are equivalent to observers at infinity. -In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). Note that the radiation pattern for an $\hat{x}$ or $\hat{y}$ polarized dipole is *nonaxisymmetric* even if it is on-axis. This means that the radiation pattern depends on the choice of $\phi$. (This is different than the computation of the [extraction efficiency for nonaxisymmetric dipoles](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) which involves an integration over $\phi$ resulting in a radiation pattern that is axisymmetric.) +In this example, the radiation pattern is computed for $\phi = 0$ (i.e., the $rz$ or $xz$ plane). Note that the radiation pattern for an $\hat{x}$ or $\hat{y}$ polarized dipole is *nonaxisymmetric* even if it is on-axis. This means that the radiation pattern depends on the choice of $\phi$. (This is different than the computation of the [extraction efficiency for off-axis dipoles](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) which involves an integration over $\phi$ resulting in a radiation pattern that is axisymmetric.) -For on-axis (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles ($E_r \pm iE_\phi$) as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations at $m = \pm 1$ (although it is possible to combine these into a single $m=+1$ simulation at $\pm \omega$ as described below). The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because extraction efficiency involves integration over $\phi$, in which case the $m = \pm 1$ cross terms cancel and you simply get identical integrated powers from $m = \pm 1$. +For on-axis (1), there are two dipole configurations: $E_x$ and $E_z$. An $E_z$ dipole is positioned at $r = 0$ with $m = 0$. This involves a single simulation. An $E_x$ dipole at $r = 0$, however, involves the superposition of left- and right-circularly polarized dipoles ($E_r \pm iE_\phi$) as described in [Tutorial/Scattering Cross Section of a Finite Dielectric Cylinder](Cylindrical_Coordinates.md#scattering-cross-section-of-a-finite-dielectric-cylinder). This requires *two* simulations at $m = \pm 1$ (although it is possible to combine these into a single $m=+1$ simulation at $\pm \omega$ frequencies as described below). The computation of the radiation pattern of an $E_x$ dipole at $r = 0$ is different from the [computation of its extraction efficiency](Local_Density_of_States.md#extraction-efficiency-of-a-light-emitting-diode-led) which involves a *single* $E_r$ source with either $m = +1$ or $m = -1$. This is because extraction efficiency involves integration over $\phi$, in which case the $m = \pm 1$ cross terms cancel and you simply get identical integrated powers from $m = \pm 1$. -For off-axis (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $e^{im\phi}$. As an optimization, our Fourier-series expansion involves only nonnegative $m$ values because the $-m$ fields can be obtained directly from the $+m$ fields using the relation $\vec{E}_{m}(r, z)e^{im\phi-i\omega t} = \left[\vec{E}_{-m}(r, z)e^{-im\phi+i\omega t} \right]^*$. There is a similar relation for $\vec{H}_{m}$. This involves taking the $\vec{E}_{m}$ fields at $-\omega$ (not $\omega$!) and conjugating them to get the $\vec{E}_{-m}$ fields at $\omega$. Implementing this in a simulation involves two aspects: (1) two sources with frequencies $\omega$ and $-\omega$ and (2) two DFT field monitors at $\omega$ and $-\omega$. (1) is necessary because it ensures the source current $\mathcal{J}$ is purely real and therefore does not change under a complex conjugation. We will compute the radiation pattern for $E_x$ and $E_y$ dipoles at $r = 0.1$ μm. +For off-axis (2), an $E_x$ (or equivalently an $E_r$) dipole positioned at $r > 0$ requires a [Fourier-series expansion of the fields](Cylindrical_Coordinates.md#nonaxisymmetric-dipole-sources) from an $E_r$ "ring" current source with azimuthal dependence $e^{im\phi}$. As an optimization, our Fourier-series expansion involves only nonnegative $m$ values because the $-m$ fields can be obtained directly from the $+m$ fields using the relation $\vec{E}_{m}(r, z)e^{im\phi-i\omega t} = \left[\vec{E}_{-m}(r, z)e^{-im\phi+i\omega t} \right]^*$. There is a similar relation for $\vec{H}_{m}$. This involves taking the $\vec{E}_{m}$ fields at $-\omega$ (not $+\omega$!) and conjugating them to get the $\vec{E}_{-m}$ fields at $+\omega$. Implementing this in a simulation involves two aspects: (1) two sources with frequencies $+\omega$ and $-\omega$ and (2) two DFT field monitors at $+\omega$ and $-\omega$. (1) is necessary because it ensures the source current $\mathcal{J}$ is purely real and therefore does not change under a complex conjugation. We will compute the radiation pattern using this approach for $E_x$ and $E_y$ dipoles at $r = 0.1$ μm. -The radiation pattern for each case is shown below. The simulation results agree with the analytic formulas. The simulation scripts are in [examples/dipole_in_vacuum_cyl_axisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_axisymmetric.py) and [examples/dipole_in_vacuum_cyl_nonaxisymmetric.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_nonaxisymmetric.py). +The radiation pattern for each case is shown below. The simulation results agree with the analytic formulas. The simulation scripts are in [examples/dipole_in_vacuum_cyl_on_axis.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_on_axis.py) and [examples/dipole_in_vacuum_cyl_off_axis.py](https://github.com/NanoComp/meep/blob/master/python/examples/dipole_in_vacuum_cyl_off_axis.py). -![](../images/radiation_pattern_axisymmetric_dipole.png#center) +![](../images/dipole_radiation_pattern_on_axis.png#center) -![](../images/radiation_pattern_nonaxisymmetric_dipole.png#center) +![](../images/dipole_radiation_pattern_off_axis.png#center) + +```py +import argparse +import cmath +import math +from typing import Tuple + +import matplotlib.pyplot as plt +import meep as mp +import numpy as np + + +RESOLUTION_UM = 50 +WAVELENGTH_UM = 1.0 +PML_UM = 1.0 +FARFIELD_RADIUS_UM = 1e6 * WAVELENGTH_UM +NUM_FARFIELD_PTS = 50 +AZIMUTHAL_RAD = 0 +POWER_DECAY_THRESHOLD = 1e-4 + +frequency = 1 / WAVELENGTH_UM +polar_rad = np.linspace(0, 0.5 * math.pi, NUM_FARFIELD_PTS) + + +def plot_radiation_pattern(dipole_pol: str, radial_flux: np.ndarray): + """Plots the radiation pattern in polar coordinates. + + The angles increase clockwise with zero in the +z direction (the "pole") + and π/2 in the +r direction (the "equator"). + + Args: + dipole_pol: the dipole polarization. + radial_flux: the radial flux in polar coordinates. + """ + normalized_radial_flux = radial_flux / np.max(radial_flux) + if dipole_pol == "x": + dipole_radial_flux = np.square(np.cos(polar_rad)) + dipole_radial_flux_label = r"$\cos^2θ$" + dipole_name = "$E_x$" + else: + dipole_radial_flux = np.ones(NUM_FARFIELD_PTS) + dipole_radial_flux_label = "constant (1.0)" + dipole_name = "$E_y$" + + fig, ax = plt.subplots(subplot_kw={"projection": "polar"}, figsize=(6, 6)) + ax.plot(polar_rad, normalized_radial_flux, "b-", label="Meep") + ax.plot(polar_rad, dipole_radial_flux, "r--", label=dipole_radial_flux_label) + ax.legend() + ax.set_theta_direction(-1) + ax.set_theta_offset(0.5 * math.pi) + ax.set_thetalim(0, 0.5 * math.pi) + ax.set_rmax(1) + ax.set_rticks([0, 0.5, 1]) + ax.grid(True) + ax.set_rlabel_position(22) + ax.set_ylabel("radial flux (a.u.)") + ax.set_title(f"radiation pattern (φ = 0) of an off-axis {dipole_name} dipole") + + if mp.am_master(): + fig.savefig( + "dipole_radiation_pattern_off_axis.png", + dpi=150, + bbox_inches="tight", + ) + + relative_error = np.linalg.norm( + normalized_radial_flux - dipole_radial_flux + ) / np.linalg.norm(dipole_radial_flux) + print(f"relative error in radiation pattern:, {relative_error}") + + +def radiation_pattern(e_field: np.ndarray, h_field: np.ndarray) -> np.ndarray: + """Computes the radiation pattern from the far fields. + + Args: + e_field, h_field: the electric (Er, Ep, Ez) and magnetic (Hr, Hp, Hz) + far fields, respectively. + + Returns: + The radial Poynting flux as a 1D array. One element for each point on + the circumference of a quarter circle with angular range of + [0, π/2] rad. 0 radians is the +z direction (the "pole") and π/2 is + the +r direction (the "equator"). + """ + flux_x = np.real( + e_field[:, 1] * np.conj(h_field[:, 2]) - e_field[:, 2] * np.conj(h_field[:, 1]) + ) + flux_z = np.real( + e_field[:, 0] * np.conj(h_field[:, 1]) - e_field[:, 1] * np.conj(h_field[:, 0]) + ) + flux_r = np.sqrt(np.square(flux_x) + np.square(flux_z)) + + return flux_r + + +def get_farfields( + sim: mp.Simulation, n2f_mon: mp.DftNear2Far +) -> Tuple[np.ndarray, np.ndarray]: + """Computes the far fields from the near fields for φ = 0 (rz plane). + + Args: + sim: a `Simulation` object. + n2f_mon: a `DftNear2Far` object returned by `Simulation.add_near2far`. + + Returns: + The electric (Er, Ep, Ez) and magnetic (Hr, Hp, Hz) far fields. One row + for each point on the circumference of a quarter circle with angular + range of [0, π/2] rad. Each row has six columns for the fields. + 0 radians is the +z direction (the "pole") and π/2 is the +r direction + (the "equator"). + """ + 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(polar_rad[n]), + 0, + FARFIELD_RADIUS_UM * math.cos(polar_rad[n]), + ), + ) + e_field[n, :] = [far_field[j] for j in range(3)] + h_field[n, :] = [far_field[j + 3] for j in range(3)] + + return e_field, h_field + + +def dipole_in_vacuum( + dipole_pol: str, dipole_pos_r: mp.Vector3, m: int +) -> Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray]: + """Computes the far fields of an off-axis point source. + + Args: + dipole_pol: the dipole polarization. + dipole_pos_r: the radial position of the dipole. + m: angular φ dependence of the fields exp(imφ). + + Returns: + A 4-tuple containing the electric and magnetic far fields at positive + and negative frequencies, respectively, as 1D arrays. + """ + sr = 2.0 + sz = 4.0 + cell_size = mp.Vector3(sr + PML_UM, 0, sz + 2 * PML_UM) + + boundary_layers = [mp.PML(thickness=PML_UM)] + + src_cmpt = mp.Er if dipole_pol == "x" else mp.Ep + + sources = [ + mp.Source( + src=mp.GaussianSource(frequency, fwidth=0.1 * frequency), + component=src_cmpt, + center=mp.Vector3(dipole_pos_r, 0, 0), + ), + mp.Source( + src=mp.GaussianSource(-frequency, fwidth=0.1 * frequency), + component=src_cmpt, + center=mp.Vector3(dipole_pos_r, 0, 0), + ), + ] + + sim = mp.Simulation( + resolution=RESOLUTION_UM, + cell_size=cell_size, + dimensions=mp.CYLINDRICAL, + m=m, + boundary_layers=boundary_layers, + sources=sources, + force_complex_fields=True, + ) + + nearfields_monitor_plus = sim.add_near2far( + frequency, + 0, + 1, + mp.FluxRegion( + center=mp.Vector3(0.5 * sr, 0, 0.5 * sz), size=mp.Vector3(sr, 0, 0) + ), + mp.FluxRegion(center=mp.Vector3(sr, 0, 0), size=mp.Vector3(0, 0, sz)), + mp.FluxRegion( + center=mp.Vector3(0.5 * sr, 0, -0.5 * sz), + size=mp.Vector3(sr, 0, 0), + weight=-1.0, + ), + ) + + nearfields_monitor_minus = sim.add_near2far( + -frequency, + 0, + 1, + mp.FluxRegion( + center=mp.Vector3(0.5 * sr, 0, 0.5 * sz), size=mp.Vector3(sr, 0, 0) + ), + mp.FluxRegion(center=mp.Vector3(sr, 0, 0), size=mp.Vector3(0, 0, sz)), + mp.FluxRegion( + center=mp.Vector3(0.5 * sr, 0, -0.5 * sz), + size=mp.Vector3(sr, 0, 0), + weight=-1.0, + ), + ) + + sim.run( + until_after_sources=mp.stop_when_fields_decayed( + 20.0, src_cmpt, mp.Vector3(dipole_pos_r, 0, 0), 1e-6 + ) + ) + + e_field_plus, h_field_plus = get_farfields(sim, nearfields_monitor_plus) + e_field_minus, h_field_minus = get_farfields(sim, nearfields_monitor_minus) + + return e_field_plus, h_field_plus, e_field_minus, h_field_minus + + +def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: + """Computes the flux from the far fields. + + Args: + e_field, h_field: the electric (Er, Ep, Ez) and magnetic (Hr, Hp, Hz) + far fields, respectively. + + Returns: + The Poynting flux obtained from the far fields. + """ + dphi = 2 * math.pi + dtheta = 0.5 * math.pi / (NUM_FARFIELD_PTS - 1) + dipole_radiation_pattern = radiation_pattern(e_field, h_field) + flux = ( + np.sum(dipole_radiation_pattern * np.sin(polar_rad)) + * FARFIELD_RADIUS_UM**2 + * dtheta + * dphi + ) + + return flux + + +if __name__ == "__main__": + parser = argparse.ArgumentParser() + parser.add_argument( + "dipole_pol", + type=str, + choices=["x", "y"], + help="polarization of the electric dipole (x or y)", + ) + parser.add_argument( + "dipole_pos_r", type=float, help="radial position of the dipole" + ) + args = parser.parse_args() + + if args.dipole_pos_r == 0: + raise ValueError(f"dipole_pos_r must be nonzero.") + + # Fourier series expansion of the fields from a ring current source + # used to generate a point dipole localized in the azimuthal direction. + + e_field_total = np.zeros((NUM_FARFIELD_PTS, 3), dtype=np.complex128) + h_field_total = np.zeros((NUM_FARFIELD_PTS, 3), dtype=np.complex128) + flux_max = 0 + m = 0 + while True: + (e_field_plus, h_field_plus, e_field_minus, h_field_minus) = dipole_in_vacuum( + args.dipole_pol, args.dipole_pos_r, m + ) + e_field_total += e_field_plus * cmath.exp(1j * m * AZIMUTHAL_RAD) + h_field_total += h_field_plus * cmath.exp(1j * m * AZIMUTHAL_RAD) + + if m > 0: + e_field_total += np.conj(e_field_minus) * cmath.exp(-1j * m * AZIMUTHAL_RAD) + h_field_total += np.conj(h_field_minus) * cmath.exp(-1j * m * AZIMUTHAL_RAD) + + flux = flux_from_farfields(e_field_plus, h_field_plus) + if flux > flux_max: + flux_max = flux + power_decay = flux / flux_max + print(f"power_decay:, {m}, {flux}, {flux_max}, {power_decay}") + + if m > 0 and power_decay < POWER_DECAY_THRESHOLD: + break + else: + m += 1 + + dipole_radiation_pattern = radiation_pattern(e_field_total, h_field_total) + dipole_radiation_pattern_scaled = dipole_radiation_pattern * FARFIELD_RADIUS_UM**2 + plot_radiation_pattern(args.dipole_pol, dipole_radiation_pattern_scaled) + + if mp.am_master(): + np.savez( + "dipole_farfields_off_axis.npz", + AZIMUTHAL_RAD=AZIMUTHAL_RAD, + FARFIELD_RADIUS_UM=FARFIELD_RADIUS_UM, + PML_UM=PML_UM, + POWER_DECAY_THRESHOLD=POWER_DECAY_THRESHOLD, + RESOLUTION_UM=RESOLUTION_UM, + WAVELENGTH_UM=WAVELENGTH_UM, + dipole_pol=args.dipole_pol, + dipole_pos_r=args.dipole_pos_r, + dipole_radiation_pattern=dipole_radiation_pattern, + e_field_total=e_field_total, + h_field_total=h_field_total, + m=m, + polar_rad=polar_rad, + ) +``` diff --git a/doc/docs/images/dipole_radiation_pattern_off_axis.png b/doc/docs/images/dipole_radiation_pattern_off_axis.png new file mode 100644 index 0000000000000000000000000000000000000000..9eaa8c0c3057d081a58a735283a17e07b1bfef07 GIT binary patch literal 187844 zcmcG$by!vF+AlmsK|n;7bPFo2APo`xfmW9M~YZQw3D)MzsKk=Ia>KqCsDRy7UA#P>d-hp6X==0`! zx~T+>_f`OFdTd$HqKf$+HPdzn{v8nY{PopR1V3bTp}D 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+++ b/python/examples/dipole_in_vacuum_cyl_off_axis.py @@ -1,4 +1,4 @@ -"""Radiation pattern of a nonaxisymmetric dipole in cylindrical coordinates. +"""Radiation pattern of an off-axis dipole in cylindrical coordinates. Tutorial Reference: @@ -59,11 +59,11 @@ def plot_radiation_pattern(dipole_pol: str, radial_flux: np.ndarray): ax.grid(True) ax.set_rlabel_position(22) ax.set_ylabel("radial flux (a.u.)") - ax.set_title("radiation pattern (φ = 0) of a nonaxisymmetric {dipole_name} dipole") + ax.set_title(f"radiation pattern (φ = 0) of an off-axis {dipole_name} dipole") if mp.am_master(): fig.savefig( - "dipole_radiation_pattern_nonaxisymmetric.png", + "dipole_radiation_pattern_off_axis.png", dpi=150, bbox_inches="tight", ) @@ -134,7 +134,7 @@ def get_farfields( def dipole_in_vacuum( dipole_pol: str, dipole_pos_r: mp.Vector3, m: int ) -> Tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray]: - """Computes the far fields of a nonaxisymmetric point source. + """Computes the far fields of an off-axis point source. Args: dipole_pol: the dipole polarization. @@ -292,7 +292,7 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: if mp.am_master(): np.savez( - "dipole_farfields_nonaxisymmetric.npz", + "dipole_farfields_off_axis.npz", AZIMUTHAL_RAD=AZIMUTHAL_RAD, FARFIELD_RADIUS_UM=FARFIELD_RADIUS_UM, PML_UM=PML_UM, diff --git a/python/examples/dipole_in_vacuum_cyl_axisymmetric.py b/python/examples/dipole_in_vacuum_cyl_on_axis.py similarity index 96% rename from python/examples/dipole_in_vacuum_cyl_axisymmetric.py rename to python/examples/dipole_in_vacuum_cyl_on_axis.py index 16d2e3cf6..3f7f17e6b 100644 --- a/python/examples/dipole_in_vacuum_cyl_axisymmetric.py +++ b/python/examples/dipole_in_vacuum_cyl_on_axis.py @@ -1,4 +1,4 @@ -"""Radiation pattern of an axisymmetric dipole in cylindrical coordinates. +"""Radiation pattern of an on-axis dipole in cylindrical coordinates. Tutorial Reference: @@ -60,11 +60,11 @@ def plot_radiation_pattern(dipole_pol: str, radial_flux: np.ndarray): ax.grid(True) ax.set_rlabel_position(22) ax.set_ylabel("radial flux (a.u.)") - ax.set_title("radiation pattern (φ = 0) of an axisymmetric {dipole_name} dipole") + ax.set_title("radiation pattern (φ = 0) of an on-axis {dipole_name} dipole") if mp.am_master(): fig.savefig( - "dipole_radiation_pattern_axisymmetric.png", + "dipole_radiation_pattern_on_axis.png", dpi=150, bbox_inches="tight", ) @@ -133,7 +133,7 @@ def get_farfields( def dipole_in_vacuum(dipole_pol: str, m: int) -> Tuple[np.ndarray, np.ndarray]: - """Computes the far fields of an axisymmetric point source. + """Computes the far fields of an on-axis point source. Args: dipole_pol: the polarization of the electric dipole. Either x or z. @@ -273,7 +273,7 @@ def flux_from_farfields(e_field: np.ndarray, h_field: np.ndarray) -> float: if mp.am_master(): np.savez( - "dipole_farfields_axisymmetric.npz", + "dipole_farfields_on_axis.npz", AZIMUTHAL_RAD=AZIMUTHAL_RAD, FARFIELD_RADIUS_UM=FARFIELD_RADIUS_UM, PML_UM=PML_UM,