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"""
Verifies that the magnitude and phase of the reflection coefficient of a
total internal reflected (TIR) mode of a flat interface of two lossless
materials given an incident planewave at oblique incidence computed using
the mode-decomposition feature matches the Fresnel equations.
ref: https://meep.readthedocs.io/en/latest/Python_Tutorials/Mode_Decomposition/#phase-of-a-total-internal-reflected-mode
"""
import cmath
from enum import Enum
import math
import meep as mp
import numpy as np
Polarization = Enum("Polarization", "S P")
# refractive indices of materials 1 and 2
n1 = 1.5
n2 = 1.0
def refl_coeff_meep(pol: Polarization, theta: float, L: float) -> complex:
"""Returns the complex reflection coefficient of a TIR mode computed
using mode decomposition.
Args:
pol: polarization of the incident planewave (S or P).
theta: angle of the incident planewave (radians).
L: position of the mode monitor relative to the flat interface.
"""
if theta < math.asin(n2 / n1):
raise ValueError(
f"incident angle of {math.degrees(theta):.2f}° is "
f"not a total internal reflected mode."
)
resolution = 50.0
# cell size is arbitrary
sx = 7.0
sy = 3.0
dpml = 2.0
cell_size = mp.Vector3(sx + 2 * dpml, sy, 0)
pml_layers = [mp.PML(dpml, direction=mp.X)]
fcen = 1.0 # source/monitor frequency
df = 0.1 * fcen
# k (in source medium) with correct length
# plane of incidence is xy
k = mp.Vector3(n1 * fcen, 0, 0).rotate(mp.Vector3(0, 0, 1), theta)
# planewave amplitude function (for line source)
def pw_amp(k, x0):
def _pw_amp(x):
return cmath.exp(1j * 2 * math.pi * k.dot(x + x0))
return _pw_amp
src_pt = mp.Vector3(-0.5 * sx, 0, 0)
if pol.name == "S":
src_cmpt = mp.Ez
eig_parity = mp.ODD_Z
elif pol.name == "P":
src_cmpt = mp.Hz
eig_parity = mp.EVEN_Z
else:
raise ValueError("pol must be S or P, only.")
sources = [
mp.Source(
mp.GaussianSource(fcen, fwidth=df),
component=src_cmpt,
center=src_pt,
size=mp.Vector3(0, cell_size.y, 0),
amp_func=pw_amp(k, src_pt),
),
]
sim = mp.Simulation(
resolution=resolution,
cell_size=cell_size,
default_material=mp.Medium(index=n1),
boundary_layers=pml_layers,
k_point=k,
sources=sources,
)
# DFT monitor for incident fields
mode_mon = sim.add_mode_monitor(
fcen,
0,
1,
mp.FluxRegion(
center=mp.Vector3(-L, 0, 0),
size=mp.Vector3(0, cell_size.y, 0),
),
)
sim.run(
until_after_sources=mp.stop_when_fields_decayed(
50,
src_cmpt,
mp.Vector3(-L, 0, 0),
1e-6,
),
)
res = sim.get_eigenmode_coefficients(
mode_mon,
bands=[1],
eig_parity=eig_parity,
kpoint_func=lambda *not_used: k,
direction=mp.NO_DIRECTION,
)
input_mode_coeff = res.alpha[0, 0, 0]
input_flux_data = sim.get_flux_data(mode_mon)
sim.reset_meep()
geometry = [
mp.Block(
material=mp.Medium(index=n1),
center=mp.Vector3(-0.25 * (sx + 2 * dpml), 0, 0),
size=mp.Vector3(0.5 * (sx + 2 * dpml), mp.inf, mp.inf),
),
mp.Block(
material=mp.Medium(index=n2),
center=mp.Vector3(0.25 * (sx + 2 * dpml), 0, 0),
size=mp.Vector3(0.5 * (sx + 2 * dpml), mp.inf, mp.inf),
),
]
sim = mp.Simulation(
resolution=resolution,
cell_size=cell_size,
boundary_layers=pml_layers,
k_point=k,
sources=sources,
geometry=geometry,
)
# DFT monitor for reflected fields
mode_mon = sim.add_mode_monitor(
fcen,
0,
1,
mp.FluxRegion(
center=mp.Vector3(-L, 0, 0),
size=mp.Vector3(0, cell_size.y, 0),
),
)
sim.load_minus_flux_data(mode_mon, input_flux_data)
sim.run(
until_after_sources=mp.stop_when_fields_decayed(
50,
src_cmpt,
mp.Vector3(-L, 0, 0),
1e-6,
),
)
res = sim.get_eigenmode_coefficients(
mode_mon,
bands=[1],
eig_parity=eig_parity,
kpoint_func=lambda *not_used: k,
direction=mp.NO_DIRECTION,
)
# mode coefficient of reflected planewave
refl_mode_coeff = res.alpha[0, 0, 1]
# reflection coefficient
refl_coeff = refl_mode_coeff / input_mode_coeff
# apply phase correction factor
refl_coeff /= cmath.exp(1j * k.x * 2 * math.pi * 2 * L)
return refl_coeff
def refl_coeff_Fresnel(pol: Polarization, theta: float) -> complex:
"""Returns the complex reflection coefficient of a TIR mode computed
using the Fresnel equations.
Args:
pol: polarization of the incident planewave (S or P).
theta: angle of the incident planewave (radians).
"""
if pol.name == "S":
refl_coeff = (
math.cos(theta) - ((n2 / n1) ** 2 - math.sin(theta) ** 2) ** 0.5
) / (math.cos(theta) + ((n2 / n1) ** 2 - math.sin(theta) ** 2) ** 0.5)
else:
refl_coeff = (
-((n2 / n1) ** 2) * math.cos(theta)
+ ((n2 / n1) ** 2 - math.sin(theta) ** 2) ** 0.5
) / (
(n2 / n1) ** 2 * math.cos(theta)
+ ((n2 / n1) ** 2 - math.sin(theta) ** 2) ** 0.5
)
return refl_coeff
if __name__ == "__main__":
# angle of incident planewave (degrees)
thetas = [54.3, 48.5]
# position of mode monitor relative to flat interface
Ls = [0.4, 1.2]
# polarization of incident planewave
pols = [Polarization.S, Polarization.P]
for pol, theta, L in zip(pols, thetas, Ls):
theta_rad = np.radians(theta)
R_meep = refl_coeff_meep(pol, theta_rad, L)
R_fres = refl_coeff_Fresnel(pol, theta_rad)
complex_to_str = lambda cnum: f"{cnum.real:.5f}{cnum.imag:+.5f}j"
print(
f"refl-coeff:, {pol.name}, {theta}, "
f"{complex_to_str(R_meep)} (Meep), "
f"{complex_to_str(R_fres)} (Fresnel)"
)
mag_meep = abs(R_meep)
mag_fres = abs(R_fres)
err_mag = abs(mag_meep - mag_fres) / mag_fres
print(
f"magnitude:, {mag_meep:.5f} (Meep), {mag_fres:.5f} (Fresnel), "
f"{err_mag:.5f} (error)"
)
phase_meep = cmath.phase(R_meep)
phase_fres = cmath.phase(R_fres)
err_phase = abs(phase_meep - phase_fres) / abs(phase_fres)
print(
f"phase:, {phase_meep:.5f} (Meep), {phase_fres:.5f} (Fresnel), "
f"{err_phase:.5f} (error)"
)
|
