4.4. NUMERICAL RESULTS 51
E
Update H, D, E
Middle Region
E
DD
EE
CPML (Update H,E)
CPML (Update H,E)
Figure 4.2: The four different regions of the new FDTD/DPBC computational domain.
lutions, the FDTD method with conventional PBC, and Ansoft high frequenc y structural simulator
(HFSS), which is based on the finite element method (FEM) [ 43].
4.4.1 AN INFINITE WATER SLAB
The algorithm is first used to analyze an infinite water slab with thickness h = 6 mm. The slab is
illuminated by TM
z
and TE
z
plane waves in two different simulations. The geometry of the slab is
shown in Fig. 4.3. The parameters of water permittivity are obtained from [40]asε
s1
= 81, ε
s2
=
1.8, ε
=1.8, τ
1
= 9.4 × 10
12
and τ
2
= 0. The permittivity of water versus frequency is shown in
Fig. 4.4. The FDTD grid cell size is x = y = z = 0.125 mm, and the slab is represented by 2
× 2 cells (due to the homogeneity of the infinite slab it could be considered as a per iodic structure
with any periodicit y). In the FDTD code 10,000 time steps and a 0.9 reduction factor of CFL time
step are used. The CPML was used for the absorbing boundaries at the top and the bottom of
the computational domain. The slab is excited using a cosine-modulated Gaussian pulse centered
at 10 GHz with 20 GHz bandwidth for the normal incidence case (k
x
=0m
1
), and it is excited
using a cosine-modulated Gaussian pulse centered at 12.75 GHz with 14.5 GHz bandwidth for
the oblique incidence case (k
x
=104.8 m
1
for minimum frequency of 5 GHz) [44]. The results are
compared with those obtained from the analytic al formulations.
From Figs. 4.5 and 4.6 good agreement between analytical solutions and results generated by
the new FDTD/DPBC algorithm can be noticed for both TM
z
and TE
z
cases (normal and oblique
incidence). The computational time is equal to 1.17 minutes for each FDTD simulation.
52 4. DISPERSIVE PERIODIC STRUCTURES
Figure 4.3: Geometry of the simulated infinite water slab (from [44]©IEEE).
0 5 10 15 20
30
40
50
60
70
80
90
Fre
q
uenc
y
[GHz]
Magnitude
ε
r
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Frequency [GHz]
Magnitude
Loss Tangent
(a) (b)
Figure 4.4: Water dispersive property versus frequency, (a) Relative permittivity, (b) L oss tangent.
4.4. NUMERICAL RESULTS 53
ε
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Frequency [GHz]
Reflection Coefficients Magnitude
FDTD
Analytical
Figure 4.5: Reflection coefficient for infinite water slab of thickness 6 mm under normal incidence
(k
x
= 0m
1
) (from [44]©IEEE).
ε
8 10 12 14 16 18 205.5
0
0.2
0.4
0.6
0.8
1
Frequency [GHz]
Reflection Coefficients Magnitude
FDTD TE
z
Analytical TE
z
FDTD TM
z
Analytical TM
z
Figure 4.6: Reflection coefficients for infinite water slab of thickness 6 mm TM
z
and TE
z
oblique
incidence (k
x
= 104.8 m
1
) (from [44]©IEEE).

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