Abstract-A general perfectly matched layer (PML) formulation is presented for lossy anisotropic media with arbitrary permittivity, permeability, and conductivity tensors. This PML is then used in a pseudospectral time-domain (PSTD) algorithm for solutions of electromagnetic fields in lossy anisotropic media. Numerical results verify the efficiency of the PML absorbing boundary condition and the PSTD algorithm

Senior Member, IEEE Qing Huo Liu

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48 IEEE MICROWAVE AND GUIDED WAVE LETTERS, VOL. 9, NO. 2, FEBRUARY 1999
PML and PSTD Algorithm for
Arbitrary Lossy Anisotropic Media
Qing Huo Liu, Senior Member, IEEE
Abstract—A general perfectly matched layer (PML) formu-
lation is presented for lossy anisotropic media with arbitrary
permittivity, permeability, and conductivity tensors. This PML
is then used in a pseudospectral time-domain (PSTD) algorithm
for solutions of electromagnetic fields in lossy anisotropic media.
Numerical results verify the efficiency of the PML absorbing
boundary condition and the PSTD algorithm.
Index Terms—Anisotropic media, pseudospectral time-domain
(PSTD) method, time-domain analysis.
I. INTRODUCTION
BERENGER’S perfectly matched layer (PML) [1] hasbecome one of the best absorbing boundary conditions
(ABC’s) for finite-difference time-domain (FDTD) and finite-
element methods. It has been used in many applications
involving isotropic and conductive media and in diagonally
anisotropic media [2]–[4]. Recently, the extension of the PML
formulations to more general anisotropic media has received
considerable attention [5]–[7].
For the time-domain solutions in anisotropic media,
one convenient approach (the material-independent PML
(MIPML) [6]) for the PML formulation is to develop
the absorber for vectors and , recognizing that these
vectors are more natural than and for anisotropic
media. The formulation with and can become very
difficult [5]. An elegant idea of the analytic continuation of
partial differential equations to a complex variable spatial
domain [8], [9] provides a straightforward way to formulate
PML for anisotropic media. Recently, this method has been
used to formulate the frequency-domain closed-form PML
constitutive tensors for bianisotropic and dispersive media
[7]. The formulation in [7] is Maxwellian and is particularly
advantageous for finite-element methods in the frequency
domain.
The objective of this work is twofold: 1) we extend the
method of analytic continuation [9] to formulate the PML
for lossy anisotropic media with arbitrary permittivity, per-
meability, and conductivity tensors, , and and 2) we
develop a pseudospectral time-domain (PSTD) algorithm for
field solutions in these arbitrary anisotropic media. In the
conventional FDTD method for anisotropic media, it becomes
necessary to interpolate field components over the Yee grid
Manuscript received June 22, 1998. This work was supported by the
US EPA under PECASE Grant CR-825-225-010 and by the National Science
Foundation under CAREER Grant ECS-9702195.
The author is with the Klipsch School of Electrical and Computer Engi-
neering, New Mexico State University, Las Cruces, NM 88003 USA.
Publisher Item Identifier S 1051-8207(99)02651-3.
since all components are needed at any field location [10].
This requirement is avoided in the PSTD algorithm [11] since
it uses a centered grid. Three-dimensional (3-D) numerical
results for media with large anisotropy confirm the efficiency
of the PML and the PSTD algorithm.
II. FORMULATION
A. PML for a Lossy Anisotropic Medium
Consider an inhomogeneous lossy anisotropic medium with
space-dependent permittivity tensor , permeability tensor
, and conductivity tensor . The frequency-domain
Maxwell’s curl equations are
(1)
where and are the imposed electric and magnetic current
densities, respectively, and the constitutive relations are
(2)
To formulate the PML equations using the analytic contin-
uation [9], the coordinate variable( ) is replaced
by a complex coordinate variablesuch that
(3)
where time convention is used, and the real vari-
ables and are the PML scaling and attenuation coeffi-
cients, respectively. With the help of (3), the second equation
of (1) can be rewritten as
(4)
If we split the field into six scalar components
( but )
(5)
(4) can be simplified by using (3) as
(6)
where is the split electric current source. A similar split
equation can be obtained for the first equation of (1).
1051–8207/99$10.00 1999 IEEE
LIU: PML AND PSTD ALGORITHM 49
The time-domain equations can be easily obtained from
these split equations by Fourier transform:
(7)
(8)
These two equations are combined with the time-domain
constitutive relations
(9)
to obtain the solutions for , , , and . Note that and
are the total fields and do not require a splitting. The FDTD
and pseudospectral time-domain (PSTD) methods can then be
utilized to solve (7)–(9).
B. The PSTD Algorithm
Let us first consider the time integration of partial differ-
ential equations (7) and (8) together with (9). If and
are sampled at , and if and are sampled at
where is an integer, using the central time
differencing scheme in (7) and (8) yields
(10)
(11)
where , , and
. These two equations give the explicit time-stepping
formulas for updating the split fields from the previous time
step. The total fields are then obtained from the constitutive
relations (9) which, also after central time differencing, relate
and to and
(12)
(13)
where and are the total fields defined as in (5).
The difference between the FDTD and PSTD methods
lies in the approximation of the spatial derivative operator
. In the FDTD method with a conventional staggered Yee
grid, because of the nondiagonal nature of, , and , the
interpolation of fields is required, as detailed by [10]. This
interpolation decreases the FDTD accuracy, as the cell size
( ) is effectively doubled. With a centered grid where all
field components are located at the cell center, the PSTD
algorithm [11] avoids this interpolation. Furthermore, the
PSTD algorithm provides a much higher order of accuracy
for spatial derivatives, as described in [11].
In the PSTD algorithm, the spatial derivative is approx-
imated by the fast Fourier transform (FFT) as
(14)
(a)
(b)
Fig. 1. (a) Comparison of the 3-D PSTD result for a 32 32  32 case
with a large reference case forx- andy-line receivers and (b) local reflection
near the PML boundaries.
where and denote the forward and inverse FFT’s, and
is the Fourier variable in direction. The representation in
(14) is exact up to the Nyquist sampling rate, i.e., two cells
per minimum wavelength. Therefore, there is no degradation
of accuracy compared to the isotropic case.
III. N UMERICAL RESULTS
Three-dimensional PSTD results are shown here for media
with large anisotropy. The source has a Blackman–Harris
window time function with a center frequency of
MHz, and the discretization is m,
ps. The first example shows the PML performance.
It compares a 32 32 32 case with a much larger reference
case to obtain the reflection from the PML absorbing boundary.
The homogeneous lossy anisotropic medium has the following
material properties:
S/m
50 IEEE MICROWAVE AND GUIDED WAVE LETTERS, VOL. 9, NO. 2, FEBRUARY 1999
Fig. 2. Propagation of EM waves along thez direction in an electrically
anisotropic medium. The dashed lines show the arrival times of waves
if they would propagate with the principal velocitiesvx = 1=
p
01,
vy = 1=
p
02, and vz = 1=
p
03, respectively. For clarity, thejth
waveform is multiplied by0:25j wherej = 1;    ; 18.
Ten PML cells with a linear profile ( , )
are used in each edge. The source is an-oriented electric
dipole located at , and two receiver arrays are:
1) the line receivers at locations and 2) the
line receivers at locations where .
The comparison of the results with the reference results at
in Fig. 1(a) displays excellent agreement between
the two results in the non-PML region of interest. The local
reflection in Fig. 1(b) shows that the maximum reflection is
below 60 dB.
In the second example we study the propagation of EM
waves in an electrically anisotropic medium with the same
tensor as in the last example, but isotropic and .
An electric dipole in direction is located at in
a grid of 128 128 128, and an array of receivers are
located along the direction at where
. Fig. 2 shows the received component.
Clearly, two types of waves are excited which propagate in
direction with the apparent velocities of
and , where are the
principal (diagonal) components of thetensor.
IV. CONCLUSIONS
The PML formulation for a lossy anisotropic medium with
arbitrary permittivity, permeability, and conductivity tensors
are presented using the idea of analytic continuation. A pseu-
dospectral time-domain algorithm is developed to improve the
accuracy of time-domain solutions of Maxwell’s equations in
arbitrary anisotropic media and to avoid the interpolation of
field components in the FDTD methods. The PML and PSTD
formulation can also be easily extended to lossy bianisotropic
media.
REFERENCES
[1] J.-P. Berenger, “A perfectly matched layer for the absorption of elec-
tromagnetic waves,”J. Comput. Phys.,vol. 114, pp. 185–200, 1994.
[2] Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly
matched anisotropic absorber for use as an absorbing boundary con-
dition,” IEEE Trans. Antennas Propagat.,vol. 43, pp. 1460–1463, 1995.
[3] S. D. Gedney, “An anisotropic PML absorbing media for the FDTD
simulation of fields in lossy and dispersive media,”Electromagn.,vol.
16, pp. 399–415, 1996.
[4] L. Zhao and A. C. Cangellaris, “GT-PML: Generalized theory of per-
fectly matched layers and its application to the reflectionless truncation
of finite-difference time-domain grids,”IEEE Trans. Microwave Theory
Tech.,vol. 44, pp. 2555–2563, 1996.
[5] S. G. Garcia, I. Villo Perez, R. Gomez Martin, and B. Garcia Olmedo,
“Extension of Berenger’s absorbing boundary conditions to match
dielectric anisotropic media,”IEEE Microwave Guided Wave Lett.,vol.
7, pp. 302–304, Sept. 1997.
[6] A. P. Zhao, J. Juntunen, and A. V. Raisanen, “Generalized material-
independent PML absorbers for the FDTD simulation of electromagnetic
waves in arbitrary anisotropic dielectric and magnetic media,”IEEE
Microwave Guided Wave Lett.,vol. 8, pp. 52–54, Feb. 1998.
[7] F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive
tensors to match arbitrary bianisotropic and dispersive linear media,”
IEEE Microwave Guided Wave Lett.,vol. 8, pp. 223–225, June 1998.
[8] W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from
modified Maxwell’s equations with stretched coordinates,”Microwave
Opt. Technol. Lett.,vol. 7, pp. 599–604, 1994.
[9] W. C. Chew, J. M. Jin, and E. Michelssen, “Complex coordinate
system as a generalized absorbing boundary condition,”Microwave Opt.
Technol. Lett.,vol. 15, pp. 363–369, 1997.
[10] J. Schneider and S. Hudson, “The finite-difference time-domain method
applied to anisotropic material,”IEEE Trans. Antennas Propagat.,vol.
41, pp. 994–999, July 1993.
[11] Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only
two cells per wavelength,”Microwave Opt. Technol. Lett.,vol. 15, pp.
158–165, 1997.


PML and PSTD Algorithm for Arbitrary Lossy Anisotropic Media

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PML and PSTD Algorithm for Arbitrary Lossy Anisotropic Media

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