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# Computes the Purcell enhancement factor of a parallel dipole in a planar
# dielectric cavity with lossless metallic walls. The result is computed in
# cylindrical and 3D coordinates and compared with the analytic theory from:
# I. Abram et al., IEEE J. Quantum Electronics, Vol. 34, pp. 71-76 (1998).
import matplotlib
import numpy as np
import meep as mp
matplotlib.use("agg")
import matplotlib.pyplot as plt
# important note:
# Meep may round the cell dimensions to an integer number
# of pixels which could modify the cavity structure.
resolution = 70 # pixels/μm
dpml = 0.5 # thickness of PML
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
def bulk_ldos_cyl():
sr = L + dpml
sz = L + 2 * dpml
cell_size = mp.Vector3(sr, 0, sz)
pml_layers = [mp.PML(dpml)]
sources = [
mp.Source(
src=mp.GaussianSource(fcen, fwidth=0.2 * fcen),
component=mp.Er,
center=mp.Vector3(),
)
]
sim = mp.Simulation(
resolution=resolution,
cell_size=cell_size,
boundary_layers=pml_layers,
sources=sources,
dimensions=mp.CYLINDRICAL,
m=-1,
default_material=mp.Medium(index=n),
)
sim.run(
mp.dft_ldos(fcen, 0, 1),
until_after_sources=mp.stop_when_fields_decayed(20, mp.Er, mp.Vector3(), 1e-6),
)
return sim.ldos_data[0]
def cavity_ldos_cyl(sz):
sr = L + dpml
cell_size = mp.Vector3(sr, 0, sz)
pml_layers = [mp.PML(dpml, direction=mp.R)]
sources = [
mp.Source(
src=mp.GaussianSource(fcen, fwidth=0.2 * fcen),
component=mp.Er,
center=mp.Vector3(),
)
]
sim = mp.Simulation(
resolution=resolution,
cell_size=cell_size,
boundary_layers=pml_layers,
sources=sources,
dimensions=mp.CYLINDRICAL,
m=-1,
default_material=mp.Medium(index=n),
)
sim.run(
mp.dft_ldos(fcen, 0, 1),
until_after_sources=mp.stop_when_fields_decayed(20, mp.Er, mp.Vector3(), 1e-6),
)
return sim.ldos_data[0]
def bulk_ldos_3D():
s = L + 2 * dpml
cell_size = mp.Vector3(s, s, s)
pml_layers = [mp.PML(dpml)]
sources = [
mp.Source(
src=mp.GaussianSource(fcen, fwidth=0.2 * fcen),
component=mp.Ex,
center=mp.Vector3(),
)
]
symmetries = [
mp.Mirror(direction=mp.X, phase=-1),
mp.Mirror(direction=mp.Y),
mp.Mirror(direction=mp.Z),
]
sim = mp.Simulation(
resolution=resolution,
cell_size=cell_size,
boundary_layers=pml_layers,
sources=sources,
symmetries=symmetries,
default_material=mp.Medium(index=n),
)
sim.run(
mp.dft_ldos(fcen, 0, 1),
until_after_sources=mp.stop_when_fields_decayed(20, mp.Ex, mp.Vector3(), 1e-6),
)
return sim.ldos_data[0]
def cavity_ldos_3D(sz):
sxy = L + 2 * dpml
cell_size = mp.Vector3(sxy, sxy, sz)
boundary_layers = [mp.PML(dpml, direction=mp.X), mp.PML(dpml, direction=mp.Y)]
sources = [
mp.Source(
src=mp.GaussianSource(fcen, fwidth=0.2 * fcen),
component=mp.Ex,
center=mp.Vector3(),
)
]
symmetries = [
mp.Mirror(direction=mp.X, phase=-1),
mp.Mirror(direction=mp.Y),
mp.Mirror(direction=mp.Z),
]
sim = mp.Simulation(
resolution=resolution,
cell_size=cell_size,
boundary_layers=boundary_layers,
sources=sources,
symmetries=symmetries,
default_material=mp.Medium(index=n),
)
sim.run(
mp.dft_ldos(fcen, 0, 1),
until_after_sources=mp.stop_when_fields_decayed(20, mp.Ex, mp.Vector3(), 1e-6),
)
return sim.ldos_data[0]
if __name__ == "__main__":
ldos_bulk_cyl = bulk_ldos_cyl()
ldos_bulk_3D = bulk_ldos_3D()
# units of wavelength in medium
cavity_thickness = np.arange(0.50, 2.55, 0.05)
gap = cavity_thickness * wvl / n
ldos_cavity_cyl = np.zeros(len(cavity_thickness))
ldos_cavity_3D = np.zeros(len(cavity_thickness))
for idx, g in enumerate(gap):
ldos_cavity_cyl[idx] = cavity_ldos_cyl(g)
ldos_cavity_3D[idx] = cavity_ldos_3D(g)
print(
"purcell-enh:, {:.3f}, "
"{:.6f} (cyl.), {:.6f} (3D)".format(
cavity_thickness[idx],
ldos_cavity_cyl[idx] / ldos_bulk_cyl,
ldos_cavity_3D[idx] / ldos_bulk_3D,
)
)
# Purcell enhancement factor (relative to bulk medium)
pe_meep_cyl = ldos_cavity_cyl / ldos_bulk_cyl
pe_meep_3D = ldos_cavity_3D / ldos_bulk_3D
# equation 7 of reference
pe_theory = 3 * np.fix(cavity_thickness + 0.5) / (4 * cavity_thickness) + (
4 * np.power(np.fix(cavity_thickness + 0.5), 3) - np.fix(cavity_thickness + 0.5)
) / (16 * np.power(cavity_thickness, 3))
if mp.am_master():
plt.plot(cavity_thickness, pe_meep_3D, "b-", label="Meep (3D)")
plt.plot(cavity_thickness, pe_meep_cyl, "r-", label="Meep (cylin.)")
plt.plot(cavity_thickness, pe_theory, "g-", label="theory")
plt.plot(cavity_thickness, np.ones(len(cavity_thickness)), "k--")
plt.xlabel(r"cavity thickness, $nL/\lambda$")
plt.ylabel("Purcell enhancement factor")
plt.title(
"planar point dipole at λ=1.0 μm in a planar cavity\n"
"with n=2.4 and lossless metallic walls"
)
plt.axis([0.5, 2.5, 0.4, 3.1])
plt.legend()
plt.savefig("cavity_purcell_factor_vs_thickness.png", bbox_inches="tight")
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