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Accurate evaluation of Green’s functions
in a layered medium by SDP-FLAM
SONG Zhe†, ZHOU HouXing, HU Jun & HONG Wei
State Key Lab of Millimeter Waves, School of Information Technology and Engineering, Southeast University, Nanjing 210096, China
Based on local Taylor expansions on the complex plane, a method for fast locating all modes (FLAM) of
spectral-domain Green’s Functions in a planar layered medium is developed in this paper. SDP-FLAM,
a combination of FLAM with the steepest descent path algorithm (SDP), is employed to accurately eval-
uate the spatial-domain Green’s functions in a layered medium. According to the theory of complex
analysis, the relationship among the poles, branch points and Riemann sheets is also analyzed rigor-
ously. To inverse the Green’s functions from spectral to spatial domain, SDP-FLAM method and discrete
complex image method (DCIM) are applied to the non-near field region and the near filed region, respec-
tively. The significant advantage of SDP-FLAM lies in its capability of calculating Green’s functions in
a layered medium of moderate thickness with loss or without loss. Some numerical examples are pre-
sented to validate SDP-FLAM method.
Green’s functions, layered medium, fast all modes method, surface wave poles, leaky wave poles, branch cut, Riemann sheet
With the development in microwave and millime-
ter wave integrated circuit design and VLSI tech-
nology, multi-functionality, high-performance and
sub-miniaturization will be the advancing trends in
the near future. Meanwhile, with the progress in
microwave monolithic integrated circuit (MMIC)
and low temperature co-fired ceramic (LTCC)
techniques and with the popularity in the concep-
tions of system on chip (SoC) and system in pack-
age (SiP), more and more attention has been paid
to the rigorous, accurate and fast modeling and
simulation method of layered circuits.
The spatial-domain Green’s functions in a lay-
ered medium are vital parts in full wave analy-
ses of complicated microstrip structures using the
method of moment (MoM). Based on the geom-
etry features of a layered medium, the spectral-
domain Green’s functions can be expressed in
closed forms, and hence can be inversed to spatial
domain through the well-known Sommerfeld inte-
grals (SI)[1], whose integrands are Bessel/Hankel
functions. Both pole point and branch point singu-
larities of the spectral-domain Green’s functions as
well as the high oscillation of Bessel/Hankel func-
tions result in slow attenuation and strong oscilla-
tion of the Sommerfeld integrals on their paths so
that the direct calculations of the Sommerfeld inte-
grals are quite time-consuming, which is a bottle-
Received February 26, 2008; accepted May 14, 2008; published online March 25, 2009
doi: 10.1007/s11432-009-0021-0
†Corresponding author (email: zhesong@emfield.org)
Supported by the National Natural Science Foundation of China (Grant No. 60621002), and the State Key Development Program for Basic
Research of China (Grant No. 2009CB320200)
Citation: Song Z, Zhou H X, Hu J, et al. Accurate evaluation of Green’s functions in a layered medium by SDP-FLAM. Sci China Ser F-Inf
Sci, 2009, 52(5): 867–875, doi: 10.1007/s11432-009-0021-0

neck in modeling and analyzing the planar layered
medium structures based on the method of mo-
ment.
The steepest descent path (SDP)[2,3] method
is the foremost fast algorithm for calculating the
Sommerfeld integrals. The main idea of SDP is
to select such a deformed integral path on com-
plex plane that the convergence of the integration
can be accelerated. However, because of the re-
striction in a numerical computer, accurate calcu-
lation of complex variable Bessel/Hankel functions
is still difficult. In 1984, Lindell[4] suggested the
accurate complex image method. During 1988 and
1991, Fang et al.[5,6] proposed the discrete com-
plex image method (DCIM), which laid the founda-
tion for semi-numerical and semi-analytical meth-
ods for inversing the spectral-domain Green func-
tions to spatial domain. DCIM[7,8] decomposes
the spectral-domain Green’s functions into three
parts, namely quasi-static term, surface wave term,
and complex image term, among which the last is
obtained with the function fitting technique. In
1995, Sakar, et al.[9] used the generalized pencil of
function (GPOF) to do fitting in the complex im-
age part, which is more robust and reliable than
the previous Prony[5] method. In 1996, Aksun et
al.[10,11] developed the original DCIM into the two-
level version, which can enhance the accuracy and
adaptability simultaneously without pole extrac-
tions. In 1997, Michalsky and Mosig[12] systemati-
cally derived the integral equation formulations of
Green’s functions in a multilayered medium and
also provided the mixed-potential integral equa-
tion (MPIE). In 1999, Cui and Chew[13] obtained
the further acceleration of SDP method based on
the symmetrical characteristic of the Green’s func-
tions. From 2000 up to now, the use of DCIM
to evaluate the spatial-domain Green’s functions
in MPIE becomes more and more popular for full
wave analysis of multilayered medium structures
based on MoM. In this aspect, Ling and Jin et
al.[14−17], Li and Liu et al.[18−20], Boix, Medina and
Horno et al.[21], Simsek and Liu et al.[22,23] made
their contributions. However, most researches
have been focusing on thin layer structures. Be-
side the works mentioned above, there are also
some methods from different points of view. In
1998, Hsieh and Kuo[24] introduced the fast Han-
kel transform (FHT) method into this area. In
1999, Song and Hong et al.[25] used the Chebyshev
pseudo spectrum expansion combined with Krylov
subspace order-reduction techniques to calculate
spatial-domain Green’s functions. In 2006, Kourk-
oulos and Cangellaris[26] proposed an accurate ap-
proximation of spectral domain Green’s functions
as a set of spherical and cylindrical waves based on
vector fitting (VECTFIT). Meanwhile, many re-
searchers, including King and Sandler et al.[27,28],
Zhang et al.[29], Li et al.[30] did a lot of work in this
area. The significant advantage of DCIM is that it
can get the spatial-domain Green’s functions with-
out any numerical calculation. However, DCIM is
not suit for a moderately thick layered medium, es-
pecially the far field region. Moreover, its precision
in the far field region could not be improved even if
the number of complex images is further increased.
In 2003, Teo et al.[31] presents a method for pole
extraction. In 2006, Neve and Paknys[32] suggested
using the Newton-Raphson algorithm to find poles.
Then, in 2007, Tsang, Ong and Wu et al.[33−35] pro-
posed a fast all modes (FAM) method combined
with numerical modified steepest path (NMSP) to
calculate Sommerfeld integrals. This method could
deal with layered media with a moderate thickness,
and its significant advantage is that it can find all
modes without any contour integration in a proper
range with the origin as the center on the com-
plex plane, namely surface wave modes, leaky wave
modes and improper modes. However, this method
could not ensure the validity in a large range.
Figure 1 Model of single layered medium with source (HED)
and field points locate on boundary.
In this paper, based on local Taylor expansions
on the complex plane, a method for fast locating
868 SONG Zhe et al. Sci China Ser F-Inf Sci | May 2009 | vol. 52 | no. 5 | 867-875