The Brekhovskikh variant of the Kirchhoff approximation (KA), as it applies to the description of the scattered field at large distances from a perfectly conducting sinusoidal grating (of amplitude h and period d) exposed to a plane wave of wavelength λ and incidence angle θi, is evaluated by means of energy-balance error calculations and comparisons with quasi-rigorous reference results obtained from the extensive form of the Rayleigh theory (RT). The RT is theoretically exact for all d/λ provided that h/d ≲ 0.072 and can be made to yield accurate numerical solutions at all θi and for all polarizations roughly for h/d < 0.39 − 0.0177 (d/λ) when d/λ ≤ 15. The KA is found to yield sufficiently accurate solutions at normal incidence only for roughly h/d < 0.011 (d/λ). Furthermore, the quality of the KA diminishes rapidly with increasing angle of incidence, as is shown by the fact that unacceptable errors occur for h/d ≳ 0.1 above θi ≃ 6°, for h/d ≳ 0.05 above θi ≃ 37°, and for h/d ≳ 0.016 above θi ≃ 63° at the period/wavelength ratio of 10. At higher frequencies (d/λ ≃ 36) there is some improvement in that, for example, the error becomes excessive for h/d ≳ 0.1 only above θi ≃ 25.5°. The KA proves to be unsafe at any frequency and incident angle as soon as h/d exceeds 0.13. This threshold is shown to be related to the onset of multiple scattering in the sense of geometrical optics.
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Plane Wave Incident at θi = 80° on a Sinusoidal Gratinga
n
ξnRAY,E
ξnRAY,H
ξnBRE
ζnE
ζnH
ηnE
ηnH
0
0.9235
0.8126
0.9410
1.89
15.80
0.0175
0.1284
−1
0.0711
0.1726
0.1224
72.15
29.08
0.0513
0.0502
−2
0.0051
0.0142
0.0092
80.39
35.21
0.0041
0.0050
−3
0.0002
0.0007
0.0004
100.00
42.86
0.0002
0.0003
ξ
1.0000
1.0000
1.0731
ζ
0.0731
η
0.0731
0.1839
kh = 1, d/λ = 10 (i.e., h/d = 0.0159). Grating efficiencies (ξn) and total energy (ξ) resulting from the RT (ξnRAY,H, ξnRAY,H) and from the Brekhovskikh KA (ξnBRE). Also included are the per-cent relative error per grating order of the KA (ζnE, ζnH), the absolute error per grating order of the KA (ηnE, ηnH), the energy-balance error of the KA (δ), and the total absolute error of the KA (ηE, ηH). The efficiencies of the remaining nonevanescent orders(n = −4, −5, …, −9) are all inferior to 5 × 10−5.
Table 2
Plane Wave Incident at θi = 80° on a Sinusoidal Gratinga
n
ξnRAY,E
ξnRAY,H
ξnBRE
ζnE
ζnH
ηnE
ηnH
0
0.5556
0.1101
0.5572
0.29
406.09
0.0016
0.4471
−1
0.2196
0.3669
0.4472
103.64
21.89
0.2276
0.0803
−2
0.1390
0.3164
0.3026
117.70
4.36
0.1636
0.0138
−3
0.0619
0.1502
0.1294
109.05
13.85
0.0675
0.0208
−4
0.0191
0.0455
0.0371
94.24
18.46
0.0180
0.0084
−5
0.0042
0.0094
0.0075
78.57
20.21
0.0033
0.0019
−6
0.0007
0.0014
0.0011
57.14
21.43
0.0004
0.0003
−7
0.0001
0.0001
0.0001
0.00
0.00
0.0000
0.0000
ξ
1.0000
1.0000
1.4823
ζ
0.4823
η
0.4820
0.5726
kh = 3, d/λ = 10. The efficiencies (ξn) of the remaining nonevanescent grating orders (n = −8, −9, …, −19) are all inferior to 5 × 10−5.
Table 3
Plane Wave Incident at θi = 45° on a Sinusoidal Gratinga
n
ξnRAY,E
ξnRAY,H
ξnBRE
ζnE
ζnH
ηnE
ηnH
0
0.0658
0.0352
0.0021
96.81
94.03
0.0637
0.0331
−1
0.1950
0.0092
0.0603
69.08
555.43
0.1347
0.0511
+1
0.0020
0.1289
0.0402
1910.00
68.81
0.0382
0.0887
−2
0.1421
0.0111
0.0492
65.38
343.24
0.0929
0.0381
+2
0.0309
0.1359
0.1114
260.52
18.03
0.0805
0.0245
−3
0.0145
0.0239
0.0001
99.31
99.58
0.0144
0.0238
−4
0.0394
0.1037
0.0586
48.73
43.49
0.0192
0.0451
−5
0.0576
0.0998
0.0726
26.04
27.25
0.0150
0.0272
−6
0.0029
0.0094
0.0048
65.52
48.94
0.0019
0.0046
−7
0.0468
0.0373
0.0425
9.19
13.94
0.0043
0.0052
−8
0.1465
0.1418
0.1447
1.23
2.05
0.0018
0.0029
−9
0.1495
0.1519
0.1509
0.94
0.66
0.0014
0.0010
−10
0.0787
0.0818
0.0803
2.03
1.83
0.0016
0.0015
−11
0.0238
0.0251
0.0245
2.94
2.39
0.0007
0.0006
−12
0.0042
0.0044
0.0043
2.38
2.27
0.0001
0.0001
−13
0.0004
0.0004
0.0004
0.00
0.00
0.0000
0.0000
ξ
1.0000
1.0000
0.8469
ζ
0.1531
η
0.4707
0.3475
kh = 6, d/λ = 10 (i.e., h/d = 0.095493). All the remaining nonevanescent grating (n = −14, −15, …, −17) efficiencies are inferior to 5 × 10−5.
Table 4
Plane Wave Incident at θi = 45° on a Sinusoidal Gratinga
n
ξnRAY,E
ξnRAY,H
ξnBRE
ζnE
ζnH
ηnE
ηnH
0
0.0227
0.0438
0.0227
0.00
48.17
0.0000
0.0211
−1
0.0208
0.0930
0.0424
103.85
54.41
0.0216
0.0506
+1
0.0143
0.0069
0.0058
59.44
15.94
0.0085
0.0011
−2
0.0037
0.0840
0.0256
591.89
69.52
0.0219
0.0584
+2
0.0115
0.0017
0.0070
39.13
311.76
0.0045
0.0053
−3
0.0205
0.0141
0.0000
100.00
100.00
0.0205
0.0141
−4
0.0708
0.0112
0.0320
54.80
185.71
0.0388
0.0208
−5
0.0415
0.0245
0.0381
8.19
55.51
0.0034
0.0136
−6
0.0024
0.0012
0.0000
100.00
100.00
0.0024
0.0012
−7
0.0695
0.0715
0.0464
33.24
35.10
0.0231
0.0251
−8
0.0491
0.0855
0.0378
23.01
55.79
0.0113
0.0477
−9
0.0358
0.0017
0.0068
81.01
300.00
0.0290
0.0051
−10
0.1348
0.0583
0.0898
33.38
54.03
0.0450
0.0315
−11
0.0356
0.0172
0.0256
28.09
48.84
0.0100
0.0084
−12
0.0500
0.0606
0.0537
7.40
11.39
0.0037
0.0069
−13
0.2313
0.2184
0.2339
1.12
7.10
0.0026
0.0155
−14
0.1584
0.1765
0.1607
1.45
8.95
0.0023
0.0158
−15
0.0274
0.0253
0.0283
3.28
11.86
0.0009
0.0030
−16
0.0009
0.0015
0.0008
11.11
46.67
0.0001
0.0007
−17
0.0000
0.0000
0.0000
0.00
0.00
0.0000
0.0000
ξ
1.0008
0.9971
0.8574
ζ
0.1426
η
0.2496
0.3459
kh = 10, d/λ = 10 (i.e., h/d = 0.159155).
Table 5
Geometrical-Optics Multiple Scattering (
) and Overshadowing (
) Thresholds for a Sinusoidal Grating Illuminated by a Plane Wave at θi Incidencea
θi
0°
10°
20°
30°
40°
50°
60°
70°
80°
0.8349
0.8113
0.7845
0.7532
0.7157
0.6687
0.6066
0.5176
0.3705
(
)
0.1329
0.1291
0.1249
0.1199
0.1139
0.1064
0.0965
0.0824
0.0590
(
)
∞
0.9028
0.4372
0.2756
0.1897
0.1335
0.0919
0.0579
0.0281
Multiple scattering occurs for h/d > (
), whereas overshadowing occurs for h/d >
.
Table 6
Plane Wave Normally Incident on a Sinusoidal Gratinga
Domains of applicability of the KA (positive slope, //-slashed region) and the RT (negative slope, \\-slashed region).
Table 7
Plane Wave Incidence at Angle θi on a Sinusoidal Grating when d/λ = 10a
Domains of applicability of the KA and the RT (as defined in Table 6).
Table 8
Plane Wave Incident at Angle θi on a Sinusoidal Grating when d/λ = 36.18a
Domains of applicability of the KA and the RT (as defined in Table 6).
Tables (8)
Table 1
Plane Wave Incident at θi = 80° on a Sinusoidal Gratinga
n
ξnRAY,E
ξnRAY,H
ξnBRE
ζnE
ζnH
ηnE
ηnH
0
0.9235
0.8126
0.9410
1.89
15.80
0.0175
0.1284
−1
0.0711
0.1726
0.1224
72.15
29.08
0.0513
0.0502
−2
0.0051
0.0142
0.0092
80.39
35.21
0.0041
0.0050
−3
0.0002
0.0007
0.0004
100.00
42.86
0.0002
0.0003
ξ
1.0000
1.0000
1.0731
ζ
0.0731
η
0.0731
0.1839
kh = 1, d/λ = 10 (i.e., h/d = 0.0159). Grating efficiencies (ξn) and total energy (ξ) resulting from the RT (ξnRAY,H, ξnRAY,H) and from the Brekhovskikh KA (ξnBRE). Also included are the per-cent relative error per grating order of the KA (ζnE, ζnH), the absolute error per grating order of the KA (ηnE, ηnH), the energy-balance error of the KA (δ), and the total absolute error of the KA (ηE, ηH). The efficiencies of the remaining nonevanescent orders(n = −4, −5, …, −9) are all inferior to 5 × 10−5.
Table 2
Plane Wave Incident at θi = 80° on a Sinusoidal Gratinga
n
ξnRAY,E
ξnRAY,H
ξnBRE
ζnE
ζnH
ηnE
ηnH
0
0.5556
0.1101
0.5572
0.29
406.09
0.0016
0.4471
−1
0.2196
0.3669
0.4472
103.64
21.89
0.2276
0.0803
−2
0.1390
0.3164
0.3026
117.70
4.36
0.1636
0.0138
−3
0.0619
0.1502
0.1294
109.05
13.85
0.0675
0.0208
−4
0.0191
0.0455
0.0371
94.24
18.46
0.0180
0.0084
−5
0.0042
0.0094
0.0075
78.57
20.21
0.0033
0.0019
−6
0.0007
0.0014
0.0011
57.14
21.43
0.0004
0.0003
−7
0.0001
0.0001
0.0001
0.00
0.00
0.0000
0.0000
ξ
1.0000
1.0000
1.4823
ζ
0.4823
η
0.4820
0.5726
kh = 3, d/λ = 10. The efficiencies (ξn) of the remaining nonevanescent grating orders (n = −8, −9, …, −19) are all inferior to 5 × 10−5.
Table 3
Plane Wave Incident at θi = 45° on a Sinusoidal Gratinga
n
ξnRAY,E
ξnRAY,H
ξnBRE
ζnE
ζnH
ηnE
ηnH
0
0.0658
0.0352
0.0021
96.81
94.03
0.0637
0.0331
−1
0.1950
0.0092
0.0603
69.08
555.43
0.1347
0.0511
+1
0.0020
0.1289
0.0402
1910.00
68.81
0.0382
0.0887
−2
0.1421
0.0111
0.0492
65.38
343.24
0.0929
0.0381
+2
0.0309
0.1359
0.1114
260.52
18.03
0.0805
0.0245
−3
0.0145
0.0239
0.0001
99.31
99.58
0.0144
0.0238
−4
0.0394
0.1037
0.0586
48.73
43.49
0.0192
0.0451
−5
0.0576
0.0998
0.0726
26.04
27.25
0.0150
0.0272
−6
0.0029
0.0094
0.0048
65.52
48.94
0.0019
0.0046
−7
0.0468
0.0373
0.0425
9.19
13.94
0.0043
0.0052
−8
0.1465
0.1418
0.1447
1.23
2.05
0.0018
0.0029
−9
0.1495
0.1519
0.1509
0.94
0.66
0.0014
0.0010
−10
0.0787
0.0818
0.0803
2.03
1.83
0.0016
0.0015
−11
0.0238
0.0251
0.0245
2.94
2.39
0.0007
0.0006
−12
0.0042
0.0044
0.0043
2.38
2.27
0.0001
0.0001
−13
0.0004
0.0004
0.0004
0.00
0.00
0.0000
0.0000
ξ
1.0000
1.0000
0.8469
ζ
0.1531
η
0.4707
0.3475
kh = 6, d/λ = 10 (i.e., h/d = 0.095493). All the remaining nonevanescent grating (n = −14, −15, …, −17) efficiencies are inferior to 5 × 10−5.
Table 4
Plane Wave Incident at θi = 45° on a Sinusoidal Gratinga
n
ξnRAY,E
ξnRAY,H
ξnBRE
ζnE
ζnH
ηnE
ηnH
0
0.0227
0.0438
0.0227
0.00
48.17
0.0000
0.0211
−1
0.0208
0.0930
0.0424
103.85
54.41
0.0216
0.0506
+1
0.0143
0.0069
0.0058
59.44
15.94
0.0085
0.0011
−2
0.0037
0.0840
0.0256
591.89
69.52
0.0219
0.0584
+2
0.0115
0.0017
0.0070
39.13
311.76
0.0045
0.0053
−3
0.0205
0.0141
0.0000
100.00
100.00
0.0205
0.0141
−4
0.0708
0.0112
0.0320
54.80
185.71
0.0388
0.0208
−5
0.0415
0.0245
0.0381
8.19
55.51
0.0034
0.0136
−6
0.0024
0.0012
0.0000
100.00
100.00
0.0024
0.0012
−7
0.0695
0.0715
0.0464
33.24
35.10
0.0231
0.0251
−8
0.0491
0.0855
0.0378
23.01
55.79
0.0113
0.0477
−9
0.0358
0.0017
0.0068
81.01
300.00
0.0290
0.0051
−10
0.1348
0.0583
0.0898
33.38
54.03
0.0450
0.0315
−11
0.0356
0.0172
0.0256
28.09
48.84
0.0100
0.0084
−12
0.0500
0.0606
0.0537
7.40
11.39
0.0037
0.0069
−13
0.2313
0.2184
0.2339
1.12
7.10
0.0026
0.0155
−14
0.1584
0.1765
0.1607
1.45
8.95
0.0023
0.0158
−15
0.0274
0.0253
0.0283
3.28
11.86
0.0009
0.0030
−16
0.0009
0.0015
0.0008
11.11
46.67
0.0001
0.0007
−17
0.0000
0.0000
0.0000
0.00
0.00
0.0000
0.0000
ξ
1.0008
0.9971
0.8574
ζ
0.1426
η
0.2496
0.3459
kh = 10, d/λ = 10 (i.e., h/d = 0.159155).
Table 5
Geometrical-Optics Multiple Scattering (
) and Overshadowing (
) Thresholds for a Sinusoidal Grating Illuminated by a Plane Wave at θi Incidencea
θi
0°
10°
20°
30°
40°
50°
60°
70°
80°
0.8349
0.8113
0.7845
0.7532
0.7157
0.6687
0.6066
0.5176
0.3705
(
)
0.1329
0.1291
0.1249
0.1199
0.1139
0.1064
0.0965
0.0824
0.0590
(
)
∞
0.9028
0.4372
0.2756
0.1897
0.1335
0.0919
0.0579
0.0281
Multiple scattering occurs for h/d > (
), whereas overshadowing occurs for h/d >
.
Table 6
Plane Wave Normally Incident on a Sinusoidal Gratinga
Domains of applicability of the KA (positive slope, //-slashed region) and the RT (negative slope, \\-slashed region).
Table 7
Plane Wave Incidence at Angle θi on a Sinusoidal Grating when d/λ = 10a
Domains of applicability of the KA and the RT (as defined in Table 6).
Table 8
Plane Wave Incident at Angle θi on a Sinusoidal Grating when d/λ = 36.18a
Domains of applicability of the KA and the RT (as defined in Table 6).