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"""
A module for fitting the complex refractive index profile over a broad
bandwidth to a sum of Lorentzian polarizability terms using gradient-based
optimization via NLopt (nlopt.readthedocs.io). The fitting parameters are
then used to define a `Medium` object.
"""
from typing import Tuple
import matplotlib
import meep as mp
import nlopt
import numpy as np
matplotlib.use("agg")
import matplotlib.pyplot as plt
def lorentzfunc(p: np.ndarray, x: np.ndarray) -> np.ndarray:
"""
Returns the complex ε profile given a set of Lorentzian parameters p
(σ_0, ω_0, γ_0, σ_1, ω_1, γ_1, ...) for a set of frequencies x.
"""
N = len(p) // 3
y = np.zeros(len(x))
for n in range(N):
A_n = p[3 * n + 0]
x_n = p[3 * n + 1]
g_n = p[3 * n + 2]
y = y + A_n / (np.square(x_n) - np.square(x) - 1j * x * g_n)
return y
def lorentzerr(p: np.ndarray, x: np.ndarray, y: np.ndarray, grad: np.ndarray) -> float:
"""
Returns the error (or residual or loss) as the L2 norm
of the difference of ε(p,x) and y over a set of frequencies x as
well as the gradient of this error with respect to each Lorentzian
polarizability parameter in p and saving the result in grad.
"""
N = len(p) // 3
yp = lorentzfunc(p, x)
val = np.sum(np.square(abs(y - yp)))
for n in range(N):
A_n = p[3 * n + 0]
x_n = p[3 * n + 1]
g_n = p[3 * n + 2]
d = 1 / (np.square(x_n) - np.square(x) - 1j * x * g_n)
if grad.size > 0:
grad[3 * n + 0] = 2 * np.real(np.dot(np.conj(yp - y), d))
grad[3 * n + 1] = (
-4 * x_n * A_n * np.real(np.dot(np.conj(yp - y), np.square(d)))
)
grad[3 * n + 2] = (
-2 * A_n * np.imag(np.dot(np.conj(yp - y), x * np.square(d)))
)
return val
def lorentzfit(
p0: np.ndarray,
x: np.ndarray,
y: np.ndarray,
alg=nlopt.LD_LBFGS,
tol: float = 1e-25,
maxeval: float = 10000,
) -> Tuple[np.ndarray, float]:
"""
Returns the optimal Lorentzian polarizability parameters and error
which minimize the error in ε(p0,x) relative to y for an initial
set of Lorentzian polarizability parameters p0 over a set of
frequencies x using the NLopt algorithm alg for a relative
tolerance tol and a maximum number of iterations maxeval.
"""
opt = nlopt.opt(alg, len(p0))
opt.set_ftol_rel(tol)
opt.set_maxeval(maxeval)
opt.set_lower_bounds(np.zeros(len(p0)))
opt.set_upper_bounds(float("inf") * np.ones(len(p0)))
opt.set_min_objective(lambda p, grad: lorentzerr(p, x, y, grad))
local_opt = nlopt.opt(nlopt.LD_LBFGS, len(p0))
local_opt.set_ftol_rel(1e-10)
local_opt.set_xtol_rel(1e-8)
opt.set_local_optimizer(local_opt)
popt = opt.optimize(p0)
minf = opt.last_optimum_value()
return popt, minf
if __name__ == "__main__":
# Import the complex refractive index profile from a CSV file.
# The file format is three comma-separated columns:
# wavelength (nm), real(n), imag(n).
mydata = np.genfromtxt("mymaterial.csv", delimiter=",")
n = mydata[:, 1] + 1j * mydata[:, 2]
# Fitting parameter: the instantaneous (infinite frequency) dielectric.
# Should be > 1.0 for stability and chosen such that
# np.amin(np.real(eps)) is ~1.0. eps is defined below.
eps_inf = 1.1
eps = np.square(n) - eps_inf
# Fit only the data in the wavelength range of [wl_min, wl_max].
wl = mydata[:, 0]
wl_min = 399 # minimum wavelength (units of nm)
wl_max = 701 # maximum wavelength (units of nm)
start_idx = np.where(wl > wl_min)
idx_start = start_idx[0][0]
end_idx = np.where(wl < wl_max)
idx_end = end_idx[0][-1] + 1
# The fitting function is ε(f) where f is the frequency, rather than ε(λ).
# Note: an equally spaced grid of wavelengths results in the larger
# wavelengths having a finer frequency grid than smaller ones.
# This feature may impact the accuracy of the fit.
freqs = 1000 / wl # units of 1/μm
freqs_reduced = freqs[idx_start:idx_end]
wl_reduced = wl[idx_start:idx_end]
eps_reduced = eps[idx_start:idx_end]
# Fitting parameter: number of Lorentzian terms to use in the fit
num_lorentzians = 2
# Number of times to repeat local optimization with random initial values.
num_repeat = 30
ps = np.zeros((num_repeat, 3 * num_lorentzians))
mins = np.zeros(num_repeat)
for m in range(num_repeat):
# Initial values for the Lorentzian polarizability terms. Each term
# consists of three parameters (σ, ω, γ) and is chosen randomly.
# Note: for the case of no absorption, γ should be set to zero.
p_rand = [10 ** (np.random.random()) for _ in range(3 * num_lorentzians)]
ps[m, :], mins[m] = lorentzfit(
p_rand, freqs_reduced, eps_reduced, nlopt.LD_MMA, 1e-25, 50000
)
ps_str = "( " + ", ".join(f"{prm:.4f}" for prm in ps[m, :]) + " )"
print(f"iteration:, {m:3d}, ps_str, {mins[m]:.6f}")
# Find the best performing set of parameters.
idx_opt = np.where(np.min(mins) == mins)[0][0]
popt_str = "( " + ", ".join(f"{prm:.4f}" for prm in ps[idx_opt]) + " )"
print(f"optimal:, {popt_str}, {mins[idx_opt]:.6f}")
# Define a `Medium` class object using the optimal fitting parameters.
E_susceptibilities = []
for n in range(num_lorentzians):
mymaterial_freq = ps[idx_opt][3 * n + 1]
mymaterial_gamma = ps[idx_opt][3 * n + 2]
if mymaterial_freq == 0:
mymaterial_sigma = ps[idx_opt][3 * n + 0]
E_susceptibilities.append(
mp.DrudeSusceptibility(
frequency=1.0, gamma=mymaterial_gamma, sigma=mymaterial_sigma
)
)
else:
mymaterial_sigma = ps[idx_opt][3 * n + 0] / mymaterial_freq**2
E_susceptibilities.append(
mp.LorentzianSusceptibility(
frequency=mymaterial_freq,
gamma=mymaterial_gamma,
sigma=mymaterial_sigma,
)
)
mymaterial = mp.Medium(epsilon=eps_inf, E_susceptibilities=E_susceptibilities)
# Plot the fit and the actual data for comparison.
mymaterial_eps = [mymaterial.epsilon(f)[0][0] for f in freqs_reduced]
fig, ax = plt.subplots(ncols=2)
ax[0].plot(wl_reduced, np.real(eps_reduced) + eps_inf, "bo-", label="actual")
ax[0].plot(wl_reduced, np.real(mymaterial_eps), "ro-", label="fit")
ax[0].set_xlabel("wavelength (nm)")
ax[0].set_ylabel(r"real($\epsilon$)")
ax[0].legend()
ax[1].plot(wl_reduced, np.imag(eps_reduced), "bo-", label="actual")
ax[1].plot(wl_reduced, np.imag(mymaterial_eps), "ro-", label="fit")
ax[1].set_xlabel("wavelength (nm)")
ax[1].set_ylabel(r"imag($\epsilon$)")
ax[1].legend()
fig.suptitle(
f"Comparison of Actual Material Data and Fit\n"
f"using Drude-Lorentzian Susceptibility"
)
fig.subplots_adjust(wspace=0.3)
fig.savefig("eps_fit_sample.png", dpi=150, bbox_inches="tight")
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