1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
|
# fit complex refractive index profile over broad bandwidth imported from
# a file to a sum of Lorentzian polarizability terms using gradient-based
# optimization via NLopt (nlopt.readthedocs.io)
import matplotlib
import nlopt
import numpy as np
matplotlib.use("agg")
import matplotlib.pyplot as plt
import meep as mp
def lorentzfunc(p, x):
"""Return 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, x, y, grad):
"""Return the error (or residual or loss funnction) 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, x, y, alg=nlopt.LD_LBFGS, tol=1e-25, maxeval=10000):
"""Return 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
# format of input data file is three comma-separated columns:
# wavelength (nm), real(n), complex(n)
mydata = np.genfromtxt("mymaterial.csv", delimiter=",")
n = mydata[:, 1] + 1j * mydata[:, 2]
eps_inf = 1.1 # should be >= 1.0 for stability and chosen such that np.amin(np.real(eps)) is ~1.0
eps = np.square(n) - eps_inf
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
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]
num_lorentzians = 2
num_repeat = (
30 # number of times to repeat local optimization with random initial values
)
ps = np.zeros((num_repeat, 3 * num_lorentzians))
mins = np.zeros(num_repeat)
for m in range(num_repeat):
# specify Lorentzian polarizability terms each consisting of three parameters (σ_0, ω_0, γ_0)
# with random initial values
# note: for the case of no absorption, set γ (every third parameter) 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
)
print(f"iteration{m}:, {ps[m,:]}, {mins[m]}")
# find the best performing set of parameters
idx_opt = np.where(np.min(mins) == mins)
print(f"optimal:, {ps[idx_opt]}, {mins[idx_opt]}")
# define a Meep `Medium` class object using the optimal fitting parameters
E_susceptibilities = []
for n in range(num_lorentzians):
mymaterial_freq = ps[idx_opt][0][3 * n + 1]
mymaterial_gamma = ps[idx_opt][0][3 * n + 2]
if mymaterial_freq == 0:
mymaterial_sigma = ps[idx_opt][0][3 * n + 0]
E_susceptibilities.append(
mp.DrudeSusceptibility(
frequency=1.0, gamma=mymaterial_gamma, sigma=mymaterial_sigma
)
)
else:
mymaterial_sigma = ps[idx_opt][0][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]
plt.subplot(1, 2, 1)
plt.plot(wl_reduced, np.real(eps_reduced) + eps_inf, "bo-", label="actual")
plt.plot(wl_reduced, np.real(mymaterial_eps), "ro-", label="fit")
plt.xlabel("wavelength (nm)")
plt.ylabel(r"real($\epsilon$)")
plt.legend()
plt.subplot(1, 2, 2)
plt.plot(wl_reduced, np.imag(eps_reduced), "bo-", label="actual")
plt.plot(wl_reduced, np.imag(mymaterial_eps), "ro-", label="fit")
plt.xlabel("wavelength (nm)")
plt.ylabel(r"imag($\epsilon$)")
plt.legend()
plt.suptitle(
"Comparison of Actual Material Data and Fit using Drude-Lorentzian Susceptibility"
)
plt.subplots_adjust(wspace=0.3)
plt.savefig("eps_fit_sample.png", dpi=150, bbox_inches="tight")
|
