Laboratoire d’Optique Électromagnétique, Unité de Recherche Associée, Centre National de la Recherche Scientifique 843, Faculté des Sciences de Saint-Jérôme, Case 262, Université d’Aix-Marseille III, Avenue Escadrille Normandie-Niemen, 13397 Marseille Cedex 13, France
A method is described for computing the electromagnetic field that is diffracted by a grating in TE polarization. It uses a Green function that takes into account the air-substrate interface, and it is based on the solution of a two-dimensional integral equation. It is shown that this particular Green’s function reduces the integration domain to a bump emerging from the substrate and reduces dramatically the computation requirements for large-period gratings. Numerical examples are given.
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Comparison of Diffraction Efficienciesa Calculated by the Eigenmode Methodb and by the Step-by-Step Methodc with Values Calculated in the Present Studyd
Order
Angle of Diffraction
Reflected Efficiencies
Present Study
Eigenmode Computation
Step-by-Step Computation
0
0.000
0.2725
0.2680288
0.2680287
−1
39.257
0.0117
0.0122915
0.0122915
Transmitted Efficiencies
Present Study
Eigenmode Computation
Step-by-Step Computation
0
0.000
0.1083
0.1030529
0.1030528
1
9.384
0.1315
0.1256182
0.1256183
2
19.032
0.0296
0.0309360
0.0309360
3
29.285
0.0459
0.0495929
0.0495928
4
40.708
0.0484
0.0516321
0.0516321
5
54.612
0.0336
0.0354155
0.0354155
6
78.043
0.0085
0.0089727
0.0089727
θi = 0°, λ = 0.6328 μm, d = 1 μm, lamellar profile with dimensions b ×y0 = 0.2 × 0.4 μm; ν1 = ν2 = 3.881. Ref. 4. The number of terms in the Fourier series is nF = 19. Ref. 2. nF = 19.
A 20 × 40 grid was used.
Table 2
Comparison of Diffraction Efficienciesa Calculated by Moharam and Gaylord’s Methodb with Values Calculated in the Present Study
Order
Angle of Diffraction
Reflected Efficiencies
Present Study
Eigenmode Computation
0
0.000
0.1971
0.1931
−1
39.257
0.0688
0.0690
Transmitted Efficiencies
Present Study
Eigenmode Computation
0
0.000
0.1491
0.1592
1
9.384
0.1504
0.1485
2
19.032
0.0303
0.0288
3
29.285
0.0323
0.0313
4
40.708
0.0264
0.0264
5
54.612
0.0152
0.0158
6
78.043
0.0033
0.0036
Same parameters as in Table 1, but the height of the grating is doubled: y0 = 0.8 μm. Ref. 4.
Table 3
Comparison of Diffraction Efficienciesa Calculated by the Step-by-Step Integration Methodb and by Maystre’s Integral Methodc with Values Calculated in the Present Studyd
Order
Angle of Diffraction
Reflected Efficiencies
Present Study
Step-by-Step Computation
Maystre’s Method
0
0.000
0.1520
0.1481
0.1492
Transmitted Efficiencies
Present Study
Step-by-Step Computation
Maystre’s Method
0
0.000
0.7937316
0.7974795
0.8000
1
19.032
0.0271077
0.0271560
0.02723
2
40.708
0.0000300
0.0000319
0.0000423
3
78.043
0.0000012
0.0000015
0.0000027
θi = 0°, λ = 0.6328 μm, d = 0.5 μm, triangular profile with base b = 0.5 μm and height y0 = 0.2 μm; ν1 = 3.881, ν2 = 1.46. Ref. 2. Ref. 10.
A grid of 474 cells was used.
Tables (3)
Table 1
Comparison of Diffraction Efficienciesa Calculated by the Eigenmode Methodb and by the Step-by-Step Methodc with Values Calculated in the Present Studyd
Order
Angle of Diffraction
Reflected Efficiencies
Present Study
Eigenmode Computation
Step-by-Step Computation
0
0.000
0.2725
0.2680288
0.2680287
−1
39.257
0.0117
0.0122915
0.0122915
Transmitted Efficiencies
Present Study
Eigenmode Computation
Step-by-Step Computation
0
0.000
0.1083
0.1030529
0.1030528
1
9.384
0.1315
0.1256182
0.1256183
2
19.032
0.0296
0.0309360
0.0309360
3
29.285
0.0459
0.0495929
0.0495928
4
40.708
0.0484
0.0516321
0.0516321
5
54.612
0.0336
0.0354155
0.0354155
6
78.043
0.0085
0.0089727
0.0089727
θi = 0°, λ = 0.6328 μm, d = 1 μm, lamellar profile with dimensions b ×y0 = 0.2 × 0.4 μm; ν1 = ν2 = 3.881. Ref. 4. The number of terms in the Fourier series is nF = 19. Ref. 2. nF = 19.
A 20 × 40 grid was used.
Table 2
Comparison of Diffraction Efficienciesa Calculated by Moharam and Gaylord’s Methodb with Values Calculated in the Present Study
Order
Angle of Diffraction
Reflected Efficiencies
Present Study
Eigenmode Computation
0
0.000
0.1971
0.1931
−1
39.257
0.0688
0.0690
Transmitted Efficiencies
Present Study
Eigenmode Computation
0
0.000
0.1491
0.1592
1
9.384
0.1504
0.1485
2
19.032
0.0303
0.0288
3
29.285
0.0323
0.0313
4
40.708
0.0264
0.0264
5
54.612
0.0152
0.0158
6
78.043
0.0033
0.0036
Same parameters as in Table 1, but the height of the grating is doubled: y0 = 0.8 μm. Ref. 4.
Table 3
Comparison of Diffraction Efficienciesa Calculated by the Step-by-Step Integration Methodb and by Maystre’s Integral Methodc with Values Calculated in the Present Studyd
Order
Angle of Diffraction
Reflected Efficiencies
Present Study
Step-by-Step Computation
Maystre’s Method
0
0.000
0.1520
0.1481
0.1492
Transmitted Efficiencies
Present Study
Step-by-Step Computation
Maystre’s Method
0
0.000
0.7937316
0.7974795
0.8000
1
19.032
0.0271077
0.0271560
0.02723
2
40.708
0.0000300
0.0000319
0.0000423
3
78.043
0.0000012
0.0000015
0.0000027
θi = 0°, λ = 0.6328 μm, d = 0.5 μm, triangular profile with base b = 0.5 μm and height y0 = 0.2 μm; ν1 = 3.881, ν2 = 1.46. Ref. 2. Ref. 10.
A grid of 474 cells was used.