Oscar P. Bruno and Michael C. Haslam, "Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences," J. Opt. Soc. Am. A 26, 658-668 (2009)
We present a superalgebraically convergent integral equation algorithm for evaluation of TE and TM electromagnetic scattering by smooth perfectly conducting periodic surfaces . For grating-diffraction problems in the resonance regime (heights and periods up to a few wavelengths) the proposed algorithm produces solutions with full double-precision accuracy in single-processor computing times of the order of a few seconds. The algorithm can also produce, in reasonable computing times, highly accurate solutions for very challenging problems, such as (a) a problem of diffraction by a grating for which the peak-to-trough distance equals 40 times its period that, in turn, equals 20 times the wavelength; and (b) a high-frequency problem with very small incidence, up to 0.01° from glancing. The algorithm is based on the concurrent use of Floquet and Chebyshev expansions together with certain integration weights that are computed accurately by means of an asymptotic expansion as the number of integration points tends to infinity.
You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
Parameters Defining Interpolants of the Function a
1
25
5.0731
25
200
3.5681
200
1000
3.2753
1000
2000
3.2126
2000
4000
3.1820
4000
6000
3.1684
6000
8000
3.1591
For we use the linear relationship , which leads to slight overestimates of the truncation index .
Table 3
Results Provided by Our Method for the TE Test Case Considered in Table 3 of [16]a
N
M
Error
Error
4
6
8
10
Our results for the TM case exhibit nearly identical behavior. Our algorithm’s parameters are , , , for all the cases considered in this table. The reference solution was computed with the same parameters listed in each row of the table and . The execution times required by our code for these tests are very small; for the case , for example, the execution time was .
Table 4
Code Parameters, Computation Times, and Resulting Accuracies for Various Problems in the Classical Resonance Regime, TE Problem (TM Results Are Nearly Identical)a
N
M
Error
N
M
Error
N
M
Error
0.5
15
0.67
20
0.74
30
1.02
1.0
25
0.78
25
1.32
45
2.30
1.5
35
1.82
35
1.64
55
2.45
2.0
65
3.21
65
2.84
70
5.96
The code execution time is measured in seconds. The reference solution was computed with the same parameters listed in each row of the table and . In all examples presented here, the value was used for the evaluation of the Green’s function; see Appendix A.
Table 5
Code Performance for Various Surface Heights a With Parameters and Resulting Accuracy and Energy Balance for Both Cases of TE and TM Scattering from Surface Profile with and a
N
M
Error (TE)
(TE)
Error (TM)
(TM)
4
550
700
10
8
1050
1200
64
12
1600
1700
236
16
2100
2200
392
20
2600
2700
570
In each case there are 40 propagating modes. The code execution time is measured in minutes. In all examples presented here we use to compute the Green’s function; see Appendix A.
Table 6
Code Performance for Grazing Incidence Problem: Incidence Angles θ (Measured in Degrees from Horizontal) with Parameters and Resulting Accuracies for Both Cases of TE and TM Scattering from Surface Profile with and a
θ (deg)
N
M
Error (TE)
(TE)
Error (TM)
(TM)
1.00
800
50
0.10
800
50
0.01
800
72
In each case there are 200 propagating modes. The code execution time is measured in minutes. The reference solution was computed with the same parameters listed in each row of the table and . For cases and 0.10° we use , while for we use ; see Appendix A.
Table 7
Code Performance for Various Incidence Wavelengths λ with Parameters and Resulting Accuracies for Both Cases of TE and TM Scattering from Surface Profile with and a
N
M
Error (TE)
(TE)
Error (TM)
(TM)
0.100
100
200
0.25
0.050
200
300
1.3
0.010
750
900
45
0.005
1500
1600
221
The number of propagating modes in the above examples are 20, 40, 200, and 400. This is a challenging problem in the resonance (and approaching high frequency) regime. The code execution time is measured in minutes. In all examples presented here we use to compute the Green’s function; see Appendix A.
Table 8
Convergence Results for a Problem of Scattering by the Composite Surface Depicted in Fig. 2a
N
M
Error (TE)
(TE)
Error (TM)
(TM)
150
62
175
74
200
85
225
98
Incidence data , ; 40 propagating modes. The code execution time is measured in seconds. The reference solution was computed using . In all examples presented in this table the value was used for the evaluation of the periodic Green’s function; see Appendix A.
Table 9
Comparison of the Spatial Collocation and Spectral Testing Methods, in a TM Problem with Scattering Surface Defined in Eq. (49) with , , (TE Results Are Nearly Identical)a
N
Spatial Collocation
Spectral Testing
Error
Error
60
65
70
75
80
85
90
95
Parameters: , . The reference solution in each case was obtained by using . Similar performance improvements were observed in a wide range of cases, and no cases were found in which spatial collocation resulted in better performance than its spectral counterpart.
Parameters Defining Interpolants of the Function a
1
25
5.0731
25
200
3.5681
200
1000
3.2753
1000
2000
3.2126
2000
4000
3.1820
4000
6000
3.1684
6000
8000
3.1591
For we use the linear relationship , which leads to slight overestimates of the truncation index .
Table 3
Results Provided by Our Method for the TE Test Case Considered in Table 3 of [16]a
N
M
Error
Error
4
6
8
10
Our results for the TM case exhibit nearly identical behavior. Our algorithm’s parameters are , , , for all the cases considered in this table. The reference solution was computed with the same parameters listed in each row of the table and . The execution times required by our code for these tests are very small; for the case , for example, the execution time was .
Table 4
Code Parameters, Computation Times, and Resulting Accuracies for Various Problems in the Classical Resonance Regime, TE Problem (TM Results Are Nearly Identical)a
N
M
Error
N
M
Error
N
M
Error
0.5
15
0.67
20
0.74
30
1.02
1.0
25
0.78
25
1.32
45
2.30
1.5
35
1.82
35
1.64
55
2.45
2.0
65
3.21
65
2.84
70
5.96
The code execution time is measured in seconds. The reference solution was computed with the same parameters listed in each row of the table and . In all examples presented here, the value was used for the evaluation of the Green’s function; see Appendix A.
Table 5
Code Performance for Various Surface Heights a With Parameters and Resulting Accuracy and Energy Balance for Both Cases of TE and TM Scattering from Surface Profile with and a
N
M
Error (TE)
(TE)
Error (TM)
(TM)
4
550
700
10
8
1050
1200
64
12
1600
1700
236
16
2100
2200
392
20
2600
2700
570
In each case there are 40 propagating modes. The code execution time is measured in minutes. In all examples presented here we use to compute the Green’s function; see Appendix A.
Table 6
Code Performance for Grazing Incidence Problem: Incidence Angles θ (Measured in Degrees from Horizontal) with Parameters and Resulting Accuracies for Both Cases of TE and TM Scattering from Surface Profile with and a
θ (deg)
N
M
Error (TE)
(TE)
Error (TM)
(TM)
1.00
800
50
0.10
800
50
0.01
800
72
In each case there are 200 propagating modes. The code execution time is measured in minutes. The reference solution was computed with the same parameters listed in each row of the table and . For cases and 0.10° we use , while for we use ; see Appendix A.
Table 7
Code Performance for Various Incidence Wavelengths λ with Parameters and Resulting Accuracies for Both Cases of TE and TM Scattering from Surface Profile with and a
N
M
Error (TE)
(TE)
Error (TM)
(TM)
0.100
100
200
0.25
0.050
200
300
1.3
0.010
750
900
45
0.005
1500
1600
221
The number of propagating modes in the above examples are 20, 40, 200, and 400. This is a challenging problem in the resonance (and approaching high frequency) regime. The code execution time is measured in minutes. In all examples presented here we use to compute the Green’s function; see Appendix A.
Table 8
Convergence Results for a Problem of Scattering by the Composite Surface Depicted in Fig. 2a
N
M
Error (TE)
(TE)
Error (TM)
(TM)
150
62
175
74
200
85
225
98
Incidence data , ; 40 propagating modes. The code execution time is measured in seconds. The reference solution was computed using . In all examples presented in this table the value was used for the evaluation of the periodic Green’s function; see Appendix A.
Table 9
Comparison of the Spatial Collocation and Spectral Testing Methods, in a TM Problem with Scattering Surface Defined in Eq. (49) with , , (TE Results Are Nearly Identical)a
N
Spatial Collocation
Spectral Testing
Error
Error
60
65
70
75
80
85
90
95
Parameters: , . The reference solution in each case was obtained by using . Similar performance improvements were observed in a wide range of cases, and no cases were found in which spatial collocation resulted in better performance than its spectral counterpart.