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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 alumi-

num 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

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

geometries

2–4

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.

This will dramatically save calculation

time compared with a series of steady-state calcu-

lations. However, the parameters of modeled mate-

rials 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,

the modiﬁed De-

bye model (MDM)

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.

But their results deviate

seriously from the experimental values.

Jin et al.

have recently determined gold parameters applicable

in the wavelength range 550–950 nm.

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 simu-

lation models,

9,10

where the effect of real metals can-

not 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 broad-

band power throughputs of a circular aperture. The

broadband results are veriﬁed by a series of steady-

state calculations. This study is important because it

reports the MDM parameters of metals that are ap-

plicable over broad frequency bands.

2. Optimization Method and Results

The frequency-dependent permittivity function of the

MDM is

␧ˆ

共

␻

兲

⫽␧

⬁

⫹

␧

⫺␧

⬁

1 ⫹ i␻␶

⫹

␴

i␻␧

⫽␧⬘ ⫺ i␧ ⬙, (1)

where ␧ˆ is the complex relative permittivity, ␻ is the

angular frequency, ␧

⬁

is the inﬁnite-frequency rela-

tive permittivity, ␧

is the zero-frequency relative

permittivity (static relative permittivity), i is the

imaginary unit, ␶ is the relaxation time, ␴ is the

H. Gai (gaihf99@mails.tsinghua.edu.cn), J. Wang, and Q. Tian

are with the State Key Laboratory of Precision Measurement Tech-

nology 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

20 April 2007 兾 Vol. 46, No. 12 兾 APPLIED OPTICS 2229

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