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"""
1d simulation of a plane wave propagating through a Kerr medium
and generating the third-harmonic frequency component.
ref: https://meep.readthedocs.io/en/latest/Python_Tutorials/Third_Harmonic_Generation/
"""
from typing import Tuple, List, Union
import numpy as np
import matplotlib
matplotlib.use("agg")
import matplotlib.pyplot as plt
import meep as mp
def third_harmonic_generation(
k: float, amp: float = 1.0, nfreq: int = 10, flux_spectrum: bool = True
) -> Union[Tuple[List[float], List[float]], Tuple[float, float]]:
"""Computes the transmission spectrum of a plane wave propagating
through a Kerr medium.
Args:
k: strength of Kerr susceptibility.
amp: amplitude of the incident planewave.
nfreq: number of frequencies in flux spectrum.
flux_spectrum: compute the flux spectrum over broad bandwidth (True) or
just the two harmonic frequencies ω and 3ω (False).
Returns:
The frequencies and transmitted flux over the flux spectrum or
the transmitted flux at the harmonic frequencies ω and 3ω.
"""
sz = 100 # size of cell in z direction
fcen = 1 / 3.0 # center frequency of source
df = fcen / 20.0 # frequency width of source
dpml = 1.0 # PML thickness
# We'll use an explicitly 1d simulation. Setting dimensions=1 will actually
# result in faster execution than just using two no-size dimensions. However,
# in this case Meep requires us to use E in the x direction (and H in y),
# and our one no-size dimension must be z.
dimensions = 1
cell = mp.Vector3(0, 0, sz)
pml_layers = [mp.PML(dpml)]
resolution = 25
# to put the same material in all space, we can just set the default material
# and pass it to the Simulation constructor
default_material = mp.Medium(index=1, chi3=k)
sources = [
mp.Source(
mp.GaussianSource(fcen, fwidth=df),
component=mp.Ex,
center=mp.Vector3(0, 0, -0.5 * sz + dpml),
amplitude=amp,
)
]
# frequency range for flux calculation
fmin = fcen / 2.0
fmax = fcen * 4
sim = mp.Simulation(
cell_size=cell,
sources=sources,
boundary_layers=pml_layers,
default_material=default_material,
resolution=resolution,
dimensions=dimensions,
)
mon_pt = mp.Vector3(0, 0, 0.5 * sz - dpml - 0.5)
if flux_spectrum:
trans = sim.add_flux(
0.5 * (fmin + fmax),
fmax - fmin,
nfreq,
mp.FluxRegion(mon_pt),
)
else:
trans1 = sim.add_flux(fcen, 0, 1, mp.FluxRegion(mon_pt))
trans3 = sim.add_flux(3 * fcen, 0, 1, mp.FluxRegion(mon_pt))
sim.run(until_after_sources=mp.stop_when_fields_decayed(50, mp.Ex, mon_pt, 1e-6))
if flux_spectrum:
freqs = mp.get_flux_freqs(trans)
trans_flux = mp.get_fluxes(trans)
return freqs, trans_flux
else:
print(
f"harmonics:, {k}, {amp}, {mp.get_fluxes(trans1)[0]}, "
f"{mp.get_fluxes(trans3)[0]}"
)
return mp.get_fluxes(trans1)[0], mp.get_fluxes(trans3)[0]
if __name__ == "__main__":
# Part 1: plot transmitted power spectrum for several values of χ(3).
nfreq = 400
logk = range(-3, 1)
tflux = np.zeros((nfreq, len(logk)))
for i, lk in enumerate(logk):
freqs, tflux[:, i] = third_harmonic_generation(
10**lk, nfreq=nfreq, flux_spectrum=True
)
fig, ax = plt.subplots()
ax.semilogy(freqs, tflux[:, 0], "bo-", label=r"$\chi^{(3)}$=0.001")
ax.semilogy(freqs, tflux[:, 1], "ro-", label=r"$\chi^{(3)}$=0.01")
ax.semilogy(freqs, tflux[:, 2], "go-", label=r"$\chi^{(3)}$=0.1")
ax.semilogy(freqs, tflux[:, 3], "co-", label=r"$\chi^{(3)}$=1")
ax.set_xlabel("frequency")
ax.set_ylabel("transmitted power (a.u.)")
ax.set_xlim(0.2, 1.2)
ax.set_ylim(1e-15, 1e2)
ax.legend()
ax.grid(True)
fig.savefig(
"transmitted_power_vs_frequency_vary_chi3.png", dpi=150, bbox_inches="tight"
)
# Part 2: plot transmittance vs. χ(3) for frequencies ω and 3ω.
logk = np.arange(-6.0, 0.2, 0.2)
first_order = np.zeros(len(logk))
third_order = np.zeros(len(logk))
for i, lk in enumerate(logk):
first_order[i], third_order[i] = third_harmonic_generation(
10**lk, flux_spectrum=False
)
input_flux = first_order[0]
fig, ax = plt.subplots()
ax.loglog(10**logk, first_order / input_flux, "ro-", label=r"$\omega$")
ax.loglog(10**logk, third_order / input_flux, "bo-", label=r"$3\omega$")
ax.loglog(10**logk, (10**logk) ** 2, "k-", label="quadratic line")
ax.set_xlabel(r"$\chi^{(3)}$")
ax.set_ylabel("transmission / incident power")
ax.legend()
ax.grid(True, "both")
fig.savefig("transmittance_vs_chi3.png", dpi=150, bbox_inches="tight")
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