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"""Computes the extraction efficiency in 3D and cylindrical coordinates.
Verifies that the extraction efficiency of a point dipole in a dielectric
layer above a lossless metallic ground plane computed in two different
coordinate systems agree.
"""
import matplotlib.pyplot as plt
import meep as mp
import numpy as np
resolution = 80 # pixels/μm
dpml = 0.5 # thickness of PML
dair = 1.0 # thickness of air padding
L = 6.0 # length of non-PML region
n = 2.4 # refractive index of surrounding medium
wvl = 1.0 # wavelength (in vacuum)
fcen = 1 / wvl # center frequency of source/monitor
# runtime termination criteria
tol = 1e-8
def extraction_eff_cyl(dmat: float, h: float) -> float:
"""Computes the extraction efficiency in cylindrical coordinates.
Args:
dmat: thickness of dielectric layer.
h: height of dipole above ground plane as fraction of dmat.
Returns:
The extraction efficiency of the dipole within the dielecric layer.
"""
sr = L + dpml
sz = dmat + dair + dpml
cell_size = mp.Vector3(sr, 0, sz)
boundary_layers = [
mp.PML(dpml, direction=mp.R),
mp.PML(dpml, direction=mp.Z, side=mp.High),
]
src_cmpt = mp.Er
# Because (1) Er is not defined at r=0 on the Yee grid, and (2) there
# seems to be a bug in the interpolation of an Er point source at r=0,
# the source is placed at r=~Δr (just outside the first voxel).
# This incurs a small error which decreases linearly with resolution.
# Ref: https://github.com/NanoComp/meep/issues/2704
src_pt = mp.Vector3(1.5 / resolution, 0, -0.5 * sz + h * dmat)
sources = [
mp.Source(
src=mp.GaussianSource(fcen, fwidth=0.1 * fcen),
component=src_cmpt,
center=src_pt,
)
]
geometry = [
mp.Block(
material=mp.Medium(index=n),
center=mp.Vector3(0, 0, -0.5 * sz + 0.5 * dmat),
size=mp.Vector3(mp.inf, mp.inf, dmat),
)
]
sim = mp.Simulation(
resolution=resolution,
cell_size=cell_size,
dimensions=mp.CYLINDRICAL,
m=-1,
boundary_layers=boundary_layers,
sources=sources,
geometry=geometry,
)
flux_air = sim.add_flux(
fcen,
0,
1,
mp.FluxRegion(
center=mp.Vector3(0.5 * L, 0, 0.5 * sz - dpml),
size=mp.Vector3(L, 0, 0),
),
mp.FluxRegion(
center=mp.Vector3(L, 0, 0.5 * sz - dpml - 0.5 * dair),
size=mp.Vector3(0, 0, dair),
),
)
sim.run(
mp.dft_ldos(fcen, 0, 1),
until_after_sources=mp.stop_when_fields_decayed(20, src_cmpt, src_pt, tol),
)
out_flux = mp.get_fluxes(flux_air)[0]
if src_pt.x == 0:
dV = np.pi / (resolution**3)
else:
dV = 2 * np.pi * src_pt.x / (resolution**2)
total_flux = -np.real(sim.ldos_Fdata[0] * np.conj(sim.ldos_Jdata[0])) * dV
ext_eff = out_flux / total_flux
print(f"extraction efficiency (cyl):, " f"{dmat:.4f}, {h:.4f}, {ext_eff:.6f}")
return ext_eff
def extraction_eff_3D(dmat: float, h: float) -> float:
"""Computes the extraction efficiency in 3D Cartesian coordinates.
Args:
dmat: thickness of dielectric layer.
h: height of dipole above ground plane as fraction of dmat.
Returns:
The extraction efficiency of the dipole within the dielecric layer.
"""
sxy = L + 2 * dpml
sz = dmat + dair + dpml
cell_size = mp.Vector3(sxy, sxy, sz)
symmetries = [
mp.Mirror(direction=mp.X, phase=-1),
mp.Mirror(direction=mp.Y),
]
boundary_layers = [
mp.PML(dpml, direction=mp.X),
mp.PML(dpml, direction=mp.Y),
mp.PML(dpml, direction=mp.Z, side=mp.High),
]
src_cmpt = mp.Ex
src_pt = mp.Vector3(0, 0, -0.5 * sz + h * dmat)
sources = [
mp.Source(
src=mp.GaussianSource(fcen, fwidth=0.1 * fcen),
component=src_cmpt,
center=src_pt,
)
]
geometry = [
mp.Block(
material=mp.Medium(index=n),
center=mp.Vector3(0, 0, -0.5 * sz + 0.5 * dmat),
size=mp.Vector3(mp.inf, mp.inf, dmat),
)
]
sim = mp.Simulation(
resolution=resolution,
cell_size=cell_size,
boundary_layers=boundary_layers,
sources=sources,
geometry=geometry,
symmetries=symmetries,
)
flux_air = sim.add_flux(
fcen,
0,
1,
mp.FluxRegion(
center=mp.Vector3(0, 0, 0.5 * sz - dpml),
size=mp.Vector3(L, L, 0),
),
mp.FluxRegion(
center=mp.Vector3(0.5 * L, 0, 0.5 * sz - dpml - 0.5 * dair),
size=mp.Vector3(0, L, dair),
),
mp.FluxRegion(
center=mp.Vector3(-0.5 * L, 0, 0.5 * sz - dpml - 0.5 * dair),
size=mp.Vector3(0, L, dair),
weight=-1.0,
),
mp.FluxRegion(
center=mp.Vector3(0, 0.5 * L, 0.5 * sz - dpml - 0.5 * dair),
size=mp.Vector3(L, 0, dair),
),
mp.FluxRegion(
center=mp.Vector3(0, -0.5 * L, 0.5 * sz - dpml - 0.5 * dair),
size=mp.Vector3(L, 0, dair),
weight=-1.0,
),
)
sim.run(
mp.dft_ldos(fcen, 0, 1),
until_after_sources=mp.stop_when_fields_decayed(20, src_cmpt, src_pt, tol),
)
out_flux = mp.get_fluxes(flux_air)[0]
dV = 1 / (resolution**3)
total_flux = -np.real(sim.ldos_Fdata[0] * np.conj(sim.ldos_Jdata[0])) * dV
ext_eff = out_flux / total_flux
print(f"extraction efficiency (3D):, {dmat:.4f}, {h:.4f}, {ext_eff:.6f}")
return ext_eff
if __name__ == "__main__":
layer_thickness = 0.7 * wvl / n
dipole_height = np.linspace(0.1, 0.9, 21)
exteff_cyl = np.zeros(len(dipole_height))
exteff_3D = np.zeros(len(dipole_height))
for j in range(len(dipole_height)):
exteff_cyl[j] = extraction_eff_cyl(layer_thickness, dipole_height[j])
exteff_3D[j] = extraction_eff_3D(layer_thickness, dipole_height[j])
plt.plot(dipole_height, exteff_cyl, "bo-", label="cylindrical")
plt.plot(dipole_height, exteff_3D, "ro-", label="3D Cartesian")
plt.xlabel("height of dipole above ground plane (fraction of layer thickness)")
plt.ylabel("extraction efficiency")
plt.legend()
if mp.am_master():
plt.savefig("extraction_eff_vs_dipole_height.png", dpi=150, bbox_inches="tight")
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