Thore Magath and Andriy E. Serebryannikov, "Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs," J. Opt. Soc. Am. A 22, 2405-2418 (2005)
A fast coupled-integral-equation (CIE) technique is developed to compute the plane-TE-wave scattering by a wide class of periodic 2D inhomogeneous structures with curvilinear boundaries, which includes finite-thickness relief and rod gratings made of homogeneous material as special cases. The CIEs in the spectral domain are derived from the standard volume electric field integral equation. The kernel of the CIEs is of Picard type and offers therefore the possibility of deriving recursions, which allow the computation of the convolution integrals occurring in the CIEs with linear amounts of arithmetic complexity and memory. To utilize this advantage, the CIEs are solved iteratively. We apply the biconjugate gradient stabilized method. To make the iterative solution process faster, an efficient preconditioning operator (PO) is proposed that is based on a formal analytical inversion of the CIEs. The application of the PO also takes only linear complexity and memory. Numerical studies are carried out to demonstrate the potential and flexibility of the CIE technique proposed. Though the best efficiency and accuracy are observed at either low permittivity contrast or high conductivity, the technique can be used in a wide range of variation of material parameters of the structures including when they contain components made of both dielectrics with high permittivity and typical metals.
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MMM, RFM, IM, and CIE denote multilayer modal, Rayleigh–Fourier, and integral methods, and our CIE technique, respectively.
I corresponds to , ; II to , ; III to , ; and IV to , , .
Table 2
Effect of , Q, and N on the DE Values for a Lossless Sinusoidal Gratinga
Q
N
PB
J
Diffraction Efficiency
400
50
300
668
0.2539
0.0041
0.3200
200
20
195
68
0.2535
0.0042
0.3192
100
10
240
18
0.2532
0.0042
0.3157
2.0
200
5
12
6.9
0.0239
0.8766
—
2.0
100
5
12
3.4
0.0240
0.8767
—
2.0
25
5
12
1.9
0.0244
0.8775
—
, .
Table 3
Effect of , Q, and N on the DE Values for a Lossless Sinusoidal Grating with High Permittivity Contrasta
Q
N
PB
J
Diffraction Efficiency
3.8
500
30
118
132
3.8
200
20
138
47
3.8
100
10
105
10
2.0
500
20
48
108
2.0
300
15
48
36
2.0
50
10
48
3.8
, .
Table 4
Effect of , Q, and N on the DE Values for a Metallic Sinusoidal Gratinga
Q
N
J
Diffraction Efficiency
3.8
500
50
38
268
0.2658
0.3042
0.0248
0.3249
3.8
400
40
43
147
0.2658
0.3042
0.0247
0.3249
3.8
200
20
37
24
0.2657
0.3037
0.0248
0.3243
2.0
500
50
26
236
—
0.3461
0.5715
—
2.0
300
40
25
78
—
0.3460
0.5715
—
2.0
100
10
23
5.7
—
0.3458
0.5703
—
, .
Table 5
Effect of , Q, and N on the DE Values for Deep Sinusoidal Gratings Made of Various Materialsa
Q
N
PB
J
Diffraction Efficiency
500
20
65
117
200
20
63
29
100
10
73
8.3
400
60
195
638
300
50
192
327
700
20
37
188
300
50
37
113
100
40
38
21
600
50
—
24
316
—
—
500
40
—
24
180
—
—
200
40
—
21
40
—
—
.
Table 6
Effect of Q and N on the DE Values for an Inhomogeneous Slab with Curvilinear Boundaries and an Intermediate Permittivity Contrasta
Q
N
PB
J
Diffraction Efficiency
500
50
110
425
400
40
97
212
200
20
92
37
.
Table 7
Effect of , Q, and N on the DE Values for Inhomogeneous Slabs with Curvilinear Boundaries and High Permittivity Contrasta
Q
N
PB
J
Diffraction Efficiency
Inhomogeneous Slab With Curvilinear Boundaries,
700
50
152
860
—
500
40
180
432
—
600
20
115
203
—
300
40
165
203
—
Inhomogeneous Slab With Curvilinear Boundaries,
700
50
190
1012
800
40
170
832
700
30
200
530
Inhomogeneous Slab With Curvilinear Boundaries And Metallic Insert,
600
50
—
117
779
500
50
—
105
512
500
30
—
110
326
.
Tables (7)
Table 1
Comparison of DE Values Calculated Using Different Methods
MMM, RFM, IM, and CIE denote multilayer modal, Rayleigh–Fourier, and integral methods, and our CIE technique, respectively.
I corresponds to , ; II to , ; III to , ; and IV to , , .
Table 2
Effect of , Q, and N on the DE Values for a Lossless Sinusoidal Gratinga
Q
N
PB
J
Diffraction Efficiency
400
50
300
668
0.2539
0.0041
0.3200
200
20
195
68
0.2535
0.0042
0.3192
100
10
240
18
0.2532
0.0042
0.3157
2.0
200
5
12
6.9
0.0239
0.8766
—
2.0
100
5
12
3.4
0.0240
0.8767
—
2.0
25
5
12
1.9
0.0244
0.8775
—
, .
Table 3
Effect of , Q, and N on the DE Values for a Lossless Sinusoidal Grating with High Permittivity Contrasta
Q
N
PB
J
Diffraction Efficiency
3.8
500
30
118
132
3.8
200
20
138
47
3.8
100
10
105
10
2.0
500
20
48
108
2.0
300
15
48
36
2.0
50
10
48
3.8
, .
Table 4
Effect of , Q, and N on the DE Values for a Metallic Sinusoidal Gratinga
Q
N
J
Diffraction Efficiency
3.8
500
50
38
268
0.2658
0.3042
0.0248
0.3249
3.8
400
40
43
147
0.2658
0.3042
0.0247
0.3249
3.8
200
20
37
24
0.2657
0.3037
0.0248
0.3243
2.0
500
50
26
236
—
0.3461
0.5715
—
2.0
300
40
25
78
—
0.3460
0.5715
—
2.0
100
10
23
5.7
—
0.3458
0.5703
—
, .
Table 5
Effect of , Q, and N on the DE Values for Deep Sinusoidal Gratings Made of Various Materialsa
Q
N
PB
J
Diffraction Efficiency
500
20
65
117
200
20
63
29
100
10
73
8.3
400
60
195
638
300
50
192
327
700
20
37
188
300
50
37
113
100
40
38
21
600
50
—
24
316
—
—
500
40
—
24
180
—
—
200
40
—
21
40
—
—
.
Table 6
Effect of Q and N on the DE Values for an Inhomogeneous Slab with Curvilinear Boundaries and an Intermediate Permittivity Contrasta
Q
N
PB
J
Diffraction Efficiency
500
50
110
425
400
40
97
212
200
20
92
37
.
Table 7
Effect of , Q, and N on the DE Values for Inhomogeneous Slabs with Curvilinear Boundaries and High Permittivity Contrasta
Q
N
PB
J
Diffraction Efficiency
Inhomogeneous Slab With Curvilinear Boundaries,
700
50
152
860
—
500
40
180
432
—
600
20
115
203
—
300
40
165
203
—
Inhomogeneous Slab With Curvilinear Boundaries,
700
50
190
1012
800
40
170
832
700
30
200
530
Inhomogeneous Slab With Curvilinear Boundaries And Metallic Insert,