5.5. NUMERICAL RESULTS 77
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 Cas c
Γ
Analytical Enti re
T Anal yt i c al Enti r e
8 10 12 14 16
-200
-150
-100
-50
0
50
100
150
200
Frequency [GHz]
Phase [deg]
Γ
FDTD Casc
T FDTD Cas c
Γ
Analytical Enti re
T Analyti cal Enti re
(a) (b)
Γ
Γ
Γ
Γ
Figure 5.13: Reflection and transmission coefficients of infinite dielectric slab with oblique incidence
k
x
= 104.8 m
1
TE
z
, (a) Magnitude, (b) Phase.
Γ
Γ
Γ
Γ
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
Analyti cal Enti re
T Anal yt i c al Ent i re
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 Casc
Γ
Analyti cal Enti re
T Anal yti cal Enti re
(a) (b)
Figure 5.14: Reflection and transmission coefficients of infinite dielectric slab with oblique incidence
k
x
= 104.8 m
1
TM
z
, (a) Magnitude, (b) Phase.
5.5.2 TEST CASE 2 (1:1 CASE, NORMAL INCIDENCE AND LARGE GAP)
In this test case, the multi-layer geometry consists of two identical FSS structures consisting of
dipole elements (1:1 case) separated by an air gap of width d. The dipole length is 12 mm and width
is 3 mm. The per iodicity is 15 mm in both x- and y-directions. The substrate has a thickness of
6 mm and relative permittivity ε
r
= 2.2, as shown in Fig. 5.15. The structure is illuminated by a TE
z
78 5. MULTILAYERED PERIODIC STRUCTURES
Figure 5.15: Two identical dipole FSS geometry (all dimensions are in mm).
normally incident plane wave (with polarization along y-axis). The frequency range of interest is
0-16 GHz. The FDTD grid cell size is x = y = z = 0.5 mm and 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 first step is to determine the distance d after
which the level of all the harmonics are less than 40 dB relative to the corresponding magnitude
of the incident field components. Using the gap determination procedure:
1) The two layers are identical; analyzing the harmonics of one layer is enough. The reflection
and transmission harmonics must be calculated.
2) The frequency range of interest as specified by the problem is 0-16 GHz (as shown in Fig. 5.9,
at the highest frequency the effect of harmonics is maximum).
3) k
i
x
and k
i
y
are equal to zero (normal incidence). Determine the cut-off frequencies for the first
eight harmonics as follows:
M
0,1
,M
0,1
f
0,1
cut-off
= f
0,1
cut-off
= 20GHz
M
1,0
,M
1,0
f
1,0
cut-off
= f
1,0
cut-off
= 20GHz
M
1,1
,M
1,1
f
1,1
cut-off
= f
1,1
cut-off
= 28.3GHz
M
1,1
,M
1,1
f
1,1
cut-off
= f
1,1
cut-off
= 28.3GHz
4) Use the harmonic analysis to calculate the magnitude coefficient of the eight harmonics and
plot the behavior of these harmonics versus frequency, as shown in Figs. 5.16 and 5.17.
5.5. NUMERICAL RESULTS 79
M-1,-1 M-1,1 M-1,0 M0,-1 M0,0 M0,1 M1,0 M1,1 M1,-1
-60
-50
-40
-30
-20
-10
0
Harmonics in x- and y- directions
|E
ty
/E
i
| [dB]
0 5 10 15 20
-60
-50
-40
-30
-20
-10
0
d [mm]
|E
m
/E
i
|[dB]
M
0,1
& M
0,-1
M
1,0
& M
-1,0
M
-1-,1
& M
-1,1
& M
1,1
& M
1,-1
(a) (b)
Figure 5.16: The eight transmitted harmonics at 16 GHz: (a) Magnitude compared to incident electric
field, (b) Decaying relative magnitude versus gap distance.
M-1,-1 M-1,1 M-1,0 M0,-1 M0,0 M0,1 M1,0 M1,1 M1,-1
-60
-50
-40
-30
-20
-10
0
Harmonics in x and y directions
|E
r
/E
i
| [dB]
0 5 10 15 20
-60
-50
-40
-30
-20
-10
0
d [mm]
|E
m
/E
i
|[dB]
M
0,1
&M
0,-1
M
1,0
&M
-1,0
M
-1,-1
&M
-1,1
&M
1,1
&M
1,-1
(a) (b)
Figure 5.17: The eight reflected harmonics at 16 GHz: (a) Magnitude compared to incident electric
field, (b) Decaying relative magnitude versus gap distance.
As can be noticed from Figs. 5.16 and 5.17, almost 95% of the dominant mode will be
transmitted.In addition,a distance d = 15.5 mm between the two layers for this range of frequencies is
considered enough to neglect all the higher harmonics effects (the magnitude of all higher harmonics
are less than 40 dB compared to the incident field magnitude).To validate the cascading technique

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