5.5. NUMERICAL RESULTS 75
5.5 NUMERICAL RESULTS
In this section, numerical examples are provided to prove the validity of the proposed algorithm.
A test plan is summariz ed in Table 5.1, which cov ers different multi-layer categories described in
Section 5.2 with different types of plane wave incidence (normal and oblique). In all the test cases, the
results of the cascading technique are compared with the FDTD simulation of the entire structure.
The FDTD code was developed in MATLAB and run on a computer with an Intel Core 2 CPU
6700, 2.66 GHz with 2 GB RAM.
Table 5.1: Test plan of multi-layer periodic structure analysis code
Test Case
Number
Periodicity / Gap
size
Incidence Cross-
polarization
Higher
harmonics
1 Dielectric slab Normal / Oblique No No
2 1:1 / Large Normal No No
3 1:1 / Small Normal No Yes
4 1:1 / Large Oblique Yes No
5 1:1 / Small Oblique Yes Yes
6 n:m / Large Normal No No
7 n:m / Small Normal No Yes
8 n:m / Large Oblique Yes No
5.5.1 TEST CASE 1 (INFINITE DIELECTRIC SLAB)
Due to the homogeneity of the dielectric slab,it is considered a good veriﬁcation case.In addition,the
results can be compared with the analytical solution.The code is used to analyze an inﬁnite dielectric
slab with thickness h = 9.375 mm and relative permittivity ε
r
= 2.56. The slab is illuminated by TM
z
and TE
z
plane waves, respectively. The plane wave is incident normally (k
x
= k
y
=0m
1
) and
obliquely (k
x
= 104.8 m
1
, k
y
=0m
1
for min frequency of 5 GHz).
In this test case, a dielectric slab with half the thickness of the original slab was simulated,
and the cascading technique was used to simulate the original dielectric slab. As shown in Fig. 5.11,
(a) the dielectric slab is analyzed analy tically with different excitation polarizations and angles of
incidence; (b) half the dielectric slab is analyzed using the FDTD/PBC technique and the scattering
parameters are extracted as previously described; and (c) the cascading technique is used to get
the scattering parameters of the whole dielectric slab from the scattering parameters of half of the
original slab. The slab is excited using a cosine-modulated Gaussian pulse centered at 10 GHz with
a 20 GHz bandwidth. The FDTD grid cell size is x = y = z = 0.3125 mm, and the slab is
represented by 5x5 cells. In the FDTD code, 2,500 time steps and a 0.9 reduction factor of CFL
time step are used. The CPML is used as the absorbing boundaries at the top and the bottom of the
computational domain. The results are compared with analytical results in Figs. 5.12, 5.13, and 5.14.
It should also be noticed that due to the homogeneity of the dielectric slab, the harmonics effects
76 5. MULTILAYERED PERIODIC STRUCTURES
(a) (b) (c)
Figure 5.11: Dielectric slab simulation using cascading technique.
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Frequency [GHz]
Magnitude
Γ
FDTD Casc
T FDTD Casc
Γ
Anal yti cal Enti r e
T Analyti cal Enti re
0 2 4 6 8 10 12 14 16 18 20
-200
-150
-100
-50
0
50
100
150
200
Frequency [GHz]
Phase [deg]
Γ
FDTD Casc
T F DTD Cas c
Γ
Anal yti c al Enti r e
T Analyti cal Enti re
(a) (b)
Figure 5.12: Reﬂection and transmission coefﬁcients of inﬁnite dielectric slab with normal incidence,
(a) Magnitude, (b) Phase.
do not exist even for a very small gap (zero gap), and only the dominant mode is considered in the
analysis.
From Figs. 5.12, 5.13, and 5.14, it should be noticed that the cascading technique is very
accurate in calculating the S-parameters of the entire structure. In addition, good agreement can be
noticed between the proposed technique and the analytic al solution for both magnitude and phase
of the reﬂection and transmission coefﬁcients, with both oblique and normal incidence TE
z
and
TM
z
cases.

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