Modified Debye model parameters of metals applicable
for broadband calculations
Hongfeng Gai, Jia Wang, and Qian Tian
The finite-difference time-domain method can provide broadband results if the excitation source is a
pulse. This demands that the parameters of modeled materials have to be applicable over broad frequency
bands. We optimize the modified Debye model parameters for gold, silver, copper, platinum, and aluminum using a large-scale nonlinear optimization algorithm. The complex relative permittivities calculated
using the optimized parameters agree well with experimental values over broad frequency bands. The
associated root-mean-square deviations are 0.49%, 3.52%, 4.13%, 1.64%, and 0.66%, respectively. We also
provide an example of broadband calculations. The obtained broadband results are verified by a series of
steady-state calculations. © 2007 Optical Society of America
OCIS codes: 120.4530, 160.3900, 310.0310.
1. Introduction
The finite-difference time-domain (FDTD) method
is one of the most popular numerical methods for
solving electromagnetic problems with arbitrary
geometries2– 4 and inhomogeneous materials. One of
the major advantages of FDTD is that broadband
results can be obtained with the FDTD algorithm
run only once, taking a pulse as the excitation
source.5 This will dramatically save calculation
time compared with a series of steady-state calculations. However, the parameters of modeled materials have to be applicable over broad frequency
bands in order to perform broadband calculations.
Here we focus our attention on real metals, which
are dispersive materials. Because the constitutive
parameters (␧, ␮, ␴, and ␶) must be specified as
constants in FDTD simulations,5 the modified Debye model (MDM)5 is usually used to describe the
frequency-dependent behavior of metals. It is very
difficult to optimize the MDM parameters of a metal
so that they are applicable over a broad frequency
band. Krug et al. have tried to get gold parameters in
the near-infrared range.6 But their results deviate
1
H. Gai (gaihf99@mails.tsinghua.edu.cn), J. Wang, and Q. Tian
are with the State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments,
Tsinghua University, Beijing 100084, China.
Received 14 November 2006; accepted 8 January 2007; posted 10
January 2007 (Doc. ID 77012); published 3 April 2007.
0003-6935/07/122229-05$15.00/0
© 2007 Optical Society of America
seriously from the experimental values.7 Jin et al.
have recently determined gold parameters applicable
in the wavelength range 550–950 nm.8 Nevertheless,
the parameters for other types of metals have not
been reported to our knowledge. Because of the lack
of appropriate parameters, researchers usually take
a perfect electric conductor to construct their simulation models,9,10 where the effect of real metals cannot be considered. The MDM parameters of metals
that are applicable over broad frequency bands are
urgently desired.
In this paper, we optimize MDM parameters for
gold, silver, copper, platinum, and aluminum, using a
large-scale nonlinear optimization algorithm. At the
end, we provide an example on calculating the broadband power throughputs of a circular aperture. The
broadband results are verified by a series of steadystate calculations. This study is important because it
reports the MDM parameters of metals that are applicable over broad frequency bands.
2. Optimization Method and Results
The frequency-dependent permittivity function of the
MDM is5
␧ˆ 共␻兲 ⫽ ␧⬁ ⫹
␧s ⫺ ␧⬁
␴
⫽ ␧⬘ ⫺ i␧ ⬙,
⫹
1 ⫹ i␻␶ i␻␧0
(1)
where ␧ˆ is the complex relative permittivity, ␻ is the
angular frequency, ␧⬁ is the infinite-frequency relative permittivity, ␧s is the zero-frequency relative
permittivity (static relative permittivity), i is the
imaginary unit, ␶ is the relaxation time, ␴ is the
20 April 2007 兾 Vol. 46, No. 12 兾 APPLIED OPTICS
2229
Fig. 1. (Color online) Comparison between our results and the experimental determined complex relative permittivities7 of (a) gold,
(b) silver, (c) copper, (d) platinum, and (e) aluminum. The squares are the real parts of the experimental values; the diamonds are
the imaginary parts. The solid curves are our results.
conductivity, ␧0 is the permittivity of free space, and
␧⬘ and ␧⬙ are the real and imaginary parts of the
complex relative permittivity. It can be seen from Eq.
(1) that the MDM can be described by ␧⬁, ␧s, ␶, and ␴.
These parameters are not independent. The relationship between them can be derived by comparing Eq.
(1) with Eq. (2), which is the frequency-dependent
permittivity function of the Drude model:5
␧ˆ 共␻兲 ⫽ 1 ⫹
2230
2
␻p2
共␻p兾␯c兲2 共␻p 兾␯c兲
⫽1⫺
⫹
,
␻
␻共i␯c ⫺ ␻兲
i␻
1⫹i
␯c
APPLIED OPTICS 兾 Vol. 46, No. 12 兾 20 April 2007
(2)
where ␻p is the radian plasma frequency, and ␯c is
the collision frequency. Both Eq. (1) and Eq. (2) are
taken from Ref. 5. The derived relation is illustrated by Eq. (3):
␴␶ ⫽ ␧0共␧⬁ ⫺ ␧s兲.
(3)
Therefore there are actually three independent parameters. The other one can be indirectly obtained
using Eq. (3). Now the function [Eq. (1)] is known. Its
results, i.e., the complex relative permittivities, can
be found from Ref. 7. The next step is to optimize the
parameters to best fit the function to the results. We
Table 1. Modified Debye Model Parameters of Metals Applicable over Broad Frequency Bands
Materials
␧s
␧⬁
␴
S兾m
␶
s
Wavelength Ranges
nm
rms
Deviations
Gold
Silver
Copper
Platinum
Aluminum
⫺15789
⫺9530.5
⫺6672.7
⫺30.005
⫺656.21
11.575
3.8344
12.076
5.3741
1.8614
1.6062 ⫻ 107
1.1486 ⫻ 107
1.0513 ⫻ 107
9.5505 ⫻ 105
5.4455 ⫻ 106
8.71 ⫻ 10⫺15
7.35 ⫻ 10⫺15
5.63 ⫻ 10⫺15
3.28 ⫻ 10⫺16
1.07 ⫻ 10⫺15
700–1200
450–1200
550–850
400–1200
200–700
0.49%
3.52%
4.13%
1.64%
0.66%
choose ␧⬁, ␧s, and ␶ as the parameters to be optimized.
A nonlinear optimization method should be used because the function itself is nonlinear.
Numerical methods for nonlinear optimization
problems are iterative in nature. One of the iterative
methods is the trust-region method.11 This method is
robust when the initial values are far from the true
values. It can also handle the case when the Jacobian
matrix is singular. Resorting to the optimization toolbox of MATLAB, we developed a program to optimize the
chosen parameters. The core of the program is the
large-scale algorithm,12,13 which is a subspace trustregion method.14,15 The program is based on the leastsquares method. It starts at a set of initial values, and
finds optimal solutions to minimize f共␧⬁, ␧s, ␶兲:
min f共␧⬁, ␧s, ␶兲 ⫽
1
2
兺j 㛳␧ˆ j共␧⬁,
␧s, ␶兲 ⫺ 共␧j⬘ ⫺ i␧j⬙兲㛳 22. (4)
our results show a rms deviation of only 0.61%. In the
wavelength range 550–950 nm, Jin’s results8 show a
rms deviation of 6.52%, while our results show a rms
deviation of 5.76%. Therefore it seems that our results are better than theirs.
We optimize the parameters for five different metals: gold, silver, copper, platinum, and aluminum.
The complex relative permittivities calculated using
our parameters agree well with experimental values7
over broad frequency bands. The associated rms deviations are 0.49%, 3.52%, 4.13%, 1.64%, and 0.66%,
respectively. Table 1 summarizes the obtained parameters. The comparison between our results and
the experimental values7 is shown in Figs. 1(a)–1(e).
The good agreement implies that the large-scale algorithm may be a versatile tool for optimizing the
MDM parameters of metals, and these parameters
are applicable over broad frequency bands.
3. Example on Broadband Calculations
Bound constraints are specified as ␧⬁ ⬎ 1, ␧s ⬍ 0, and
␶ ⬎ 0. An iteration of the program has the following
form. First, a two-dimensional subspace is determined
by the preconditioned conjugated gradient method.
Second, an approximate model is constructed in the
subspace. Third, the approximate model is solved
within the trust region, giving a solution that is called
the trial step. Fourth, the previous iterative solution is
updated if the trial step reduces the objective function.
Otherwise, the previous iterative solution remains unchanged, and the trust region is shrunk. Then the trial
step computation is repeated until the obtained solution satisfies the given termination tolerance.
Taking the parameters provided in Ref. 6 as the
initial values, we successfully obtain the parameters
for gold: ␧⬁ ⫽ 11.575, ␧s ⫽ ⫺15789, ␶ ⫽ 8.71 ⫻ 10⫺15 s,
and ␴ ⫽ 1.6062 ⫻ 107 S兾m. The accuracy of the optimized parameters is indicated by the number of the
stated digits. These parameters are applicable over
a wavelength range 700–1200 nm, larger than the
wavelength range 700–1000 nm presented in Ref. 6.
In order to validate the parameters, the complex relative permittivities of gold are calculated from Eq. (1)
using the optimized parameters. Then they are compared with the experimental values provided in Ref.
7. The comparison results are shown in Fig. 1(a). The
associated rms deviation is calculated to be only
0.49%. So the optimized gold parameters are valid.
We also compare our results with other researchers.
In the wavelength range 700–1000 nm, Krug’s results6 show a rms deviation as large as 161%, while
In this section, we provide an example of broadband
calculations using the optimized parameters. The
broadband power throughputs of a circular aperture
are obtained with the FDTD algorithm run only once.
The power throughput is defined as the ratio of the
total transmitted electric field intensity to the total
incident electric field intensity over the aperture area.
We choose commercial software XFDTD V6.2 for the
FDTD calculations. This software has been validated
by our previous paper.16
Figure 2 shows the simulation model. The aperture
diameter d is 100 nm, and the silver film thickness h
is 50 nm. The aperture is centered in a simulation
volume of 1000 nm ⫻ 1000 nm ⫻ 1250 nm. The simulation volume is divided into 5 nm ⫻ 5 nm ⫻ 5 nm
Fig. 2. (Color online) FDTD simulation model: a circular aperture
in a silver film.
20 April 2007 兾 Vol. 46, No. 12 兾 APPLIED OPTICS
2231
Yee cells.1 The Liao absorbing boundary condition3
is applied to all sides of the simulation volume. The
excitation source is a modulated Gaussian pulse17
polarized along the x direction. This type of pulse can
prevent low-frequency components from exciting nonradiating modes that are not what we want. The pulse
width is adjusted17 to cover the wavelength range
450–1200 nm. The time step is 8.66625 ⫻ 10⫺18 s,
which is determined according to the stability criteria
of the FDTD algorithm.5
The pulse–probe technique18 is employed to calculate the broadband power throughputs. The electric
field response is probed at a plane 10 nm behind the
exit plane of the aperture. Fourier transforms are
applied to both the probe field and the incident field.
Then the broadband power throughputs are obtained
by normalizing the probe field spectrum to the incident field spectrum. The FDTD algorithm runs only
once during these processes. The curve of Fig. 3
shows the obtained broadband power throughputs. It
can be seen that the power throughput of the circular
aperture decreases when the incident wavelength is
increased. This is in accordance with the Bethe–
Bouwkamp model.19,20
For validating the obtained power throughputs, we
implement a series of steady-state calculations using
a time harmonic wave as the excitation source. Instead of providing broadband results, a steady-state
calculation can only give solutions at a single wavelength of interest. So the power throughputs are actually obtained wavelength by wavelength. In our study,
the incident wavelength is changed from 450 nm to
1200 nm in 50 nm steps. The power throughput at
each wavelength is obtained by normalizing the probe
field intensity to the incident field intensity. The
squares of Fig. 3 show the obtained power throughputs. It can be calculated that these discrete power
throughputs deviate from the continuous ones with a
rms deviation of 7.43%. Therefore the broadband
power throughputs are verified by the steady-state calculations. However, there are mismatches between the
results for short wavelengths. This is because the cal-
culated complex relative permittivities deviate from
the experimental values more seriously for short wavelengths than for long wavelengths.
Although both these methods can be used to calculate the power throughputs, the former method is
more efficient if broadband results are required. The
FDTD algorithm is only run once in the former
method instead of several times in the latter method.
This is particularly important because of the extensive calculation time needed by the FDTD algorithm.
Besides this, the latter method can only provide discrete results, instead of continuous results provided
by the former method. So the latter method cannot be
used for broadband calculations.
4. Conclusions
A large-scale nonlinear optimization algorithm is
used to optimize the MDM parameters for gold, silver, copper, platinum, and aluminum. The complex
relative permittivities calculated using the optimized
parameters agree well with experimental values7
over broad frequency bands. The associated rms deviations are 0.49%, 3.52%, 4.13%, 1.64%, and 0.66%,
respectively. Broadband calculations can be performed since the parameters have been obtained. A
lot of calculation time can be saved because the FDTD
algorithm needs to be run only once. An example on
this issue is provided. The obtained broadband results are verified by a series of steady-state calculations.
The large-scale algorithm is robust when the initial
values are far from the true values. It can also handle
the case when the Jacobian matrix is singular. This
algorithm is suitable for nonlinear optimization problems, such as what has been presented in this paper.
The limitation of this algorithm is that the function to
be minimized must be continuous. Besides this, the
nonlinear system of equations cannot be underdetermined; that is, the number of equations must be at
least as many as the number of parameters to be
optimized.
The research is sponsored by the National Natural
Science Foundation of China (grant 60678028) and
the National High Technology Research and Development Program of China (grant 2003AA311132).
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Fig. 3. (Color online) Comparison between the power throughputs calculated using two types of excitation sources: a pulse and
a time harmonic wave. The solid curve is the broadband results
obtained with the pulse excitation. The squares are the discrete
results obtained with the time harmonic wave excitation. The inset
shows the incident pulse field.
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APPLIED OPTICS 兾 Vol. 46, No. 12 兾 20 April 2007
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