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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