Diffraction of electromagnetic waves by dielectric crossed gratings: a three-dimensional Rayleigh–Fourier solution

V. Bagnoud; S. Mainguy

doi:10.1364/JOSAA.16.001277

Journal of the Optical Society of America A
Vol. 16,
Issue 6,
pp. 1277-1285
(1999)
•https://doi.org/10.1364/JOSAA.16.001277

Diffraction of electromagnetic waves by dielectric crossed gratings: a three-dimensional Rayleigh–Fourier solution

V. Bagnoud and S. Mainguy

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V. Bagnoud and S. Mainguy, "Diffraction of electromagnetic waves by dielectric crossed gratings: a three-dimensional Rayleigh–Fourier solution," J. Opt. Soc. Am. A 16, 1277-1285 (1999)

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Abstract

We present a solution based on the Rayleigh hypothesis. The calculation leads to a matrix system that can be computed to yield the efficiencies of a two-dimensional dielectric grating. The results are compared with previously reported numerical results obtained both with exact theories and with perturbation methods. We show that our method can treat rougher surfaces than can classic perturbation methods.

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Order	Transmitted and Reflected Efficiencies
Order	Method of Ref. 4	Method of Ref. 6	Method of Ref. 5	Method of Ref. 12	RFS
$(- 1,$ $- 1) T$	0.00092	0.00081	0.00097	0.00086	0.00085
$(- 1,$ $0) R$	0.00254	0.00259	0.00272	0.00249	0.00243
$(- 1,$ $0) T$	0.00303	0.00321	0.00324	0.00294	0.00282
$(0,$ $- 1) T$	0.00704	0.00683	0.00695	0.00677	0.00640
$(0,$ $0) R$	0.01984	0.01948	0.01904	0.01963	0.01970
$(0,$ $0) T$	0.96219	0.96430	0.96243	0.96448	0.96584
$(1,$ $0) T$	0.00299	0.00276	0.00299	0.00282	0.00269
E	0.9987	0.99998	0.9983	0.99989	1.00074

	Method of Ref. 16				RFS ( $P = 8,$ $N = 91$ )
$h / D$	$(- 1,$ $0) R$	$(0,$ $- 1) R$	$(0,$ $0) R$	$1 - E$	$(- 1,$ $0) R$	$(0,$ $- 1) R$	$(0,$ $0) R$	$1 - E$
0.1	$3.27739 \times 10^{- 3}$	$3.72936 \times 10^{- 3}$	$9.62746 \times 10^{- 2}$	$3.9 \times 10^{- 14}$	$3.29383 \times 10^{- 3}$	$3.74909 \times 10^{- 3}$	$9.6201 \times 10^{- 2}$	$4.66 \times 10^{- 15}$
0.2	$1.04428 \times 10^{- 2}$	$1.18396 \times 10^{- 2}$	$6.17586 \times 10^{- 2}$	$1.62 \times 10^{- 10}$	$1.04336 \times 10^{- 2}$	$1.18332 \times 10^{- 2}$	$6.18082 \times 10^{- 2}$	$6.88 \times 10^{- 15}$
0.3	$1.5987 \times 10^{- 2}$	$1.82072 \times 10^{- 2}$	$2.77715 \times 10^{- 2}$	$1.68 \times 10^{- 8}$	$1.59927 \times 10^{- 2}$	$1.82209 \times 10^{- 2}$	$2.77324 \times 10^{- 2}$	$7.55 \times 10^{- 12}$
0.4	$1.62416 \times 10^{- 2}$	$1.9071 \times 10^{- 2}$	$7.76777 \times 10^{- 3}$	$3.6 \times 10^{- 7}$	$1.6249 \times 10^{- 2}$	$1.90873 \times 10^{- 2}$	$7.79654 \times 10^{- 3}$	$7.44 \times 10^{- 10}$
0.5	$1.17478 \times 10^{- 2}$	$1.50842 \times 10^{- 2}$	$1.97568 \times 10^{- 3}$	$2.8 \times 10^{- 6}$	$1.17447 \times 10^{- 2}$	$1.50945 \times 10^{- 2}$	$1.97286 \times 10^{- 2}$	$2.15 \times 10^{- 8}$
0.6	$5.95639 \times 10^{- 3}$	$9.39993 \times 10^{- 3}$	$2.96950 \times 10^{- 3}$	$8.2 \times 10^{- 6}$	$5.94102 \times 10^{- 3}$	$9.39239 \times 10^{- 3}$	$2.97211 \times 10^{- 3}$	$2.33 \times 10^{- 6}$
0.7	$1.99328 \times 10^{- 3}$	$4.6860 \times 10^{- 3}$	$4.12517 \times 10^{- 3}$	$- 1 \times 10^{- 5}$	$1.99065 \times 10^{- 3}$	$4.68494 \times 10^{- 3}$	$4.12694 \times 10^{- 3}$	$5.3 \times 10^{- 5}$
0.8	$6.8640 \times 10^{- 4}$	$1.81897 \times 10^{- 3}$	$3.40891 \times 10^{- 3}$	$- 1.9 \times 10^{- 4}$	$6.74655 \times 10^{- 4}$	$1.78964 \times 10^{- 3}$	$3.39347 \times 10^{- 3}$	$4.17 \times 10^{- 4}$
0.9	$1.1486 \times 10^{- 3}$	$4.73788 \times 10^{- 4}$	$1.8427 \times 10^{- 3}$	$- 8.4 \times 10^{- 4}$	$1.28816 \times 10^{- 3}$	$3.3649 \times 10^{- 4}$	$2.08597 \times 10^{- 3}$	$3.73 \times 10^{- 4}$
1.0	$2.1132 \times 10^{- 3}$	$3.7408 \times 10^{- 5}$	$6.4855 \times 10^{- 4}$	$- 2.4 \times 10^{- 3}$	$3.68175 \times 10^{- 5}$	$2.96237 \times 10^{- 5}$	$1.71293 \times 10^{- 3}$	$5.15 \times 10^{- 2}$

	Method of Ref. 16				RFS ( $P = 8,$ $N = 91$ )
$h / D$	$(- 1, 0) R$	$(0, - 1) R$	$(0,0) R$	$1 - E$	$(- 1,0) R$	$(0, - 1) R$	$(0,0) R$	$1 - E$
0.1	$1.881 \times 10^{- 2}$	$5.961 \times 10^{- 2}$	$8.42996 \times 10^{- 1}$	$4 \times 10^{- 9}$	$1.88225 \times 10^{- 2}$	$5.96898 \times 10^{- 2}$	$8.42975 \times 10^{- 1}$	$2.3 \times 10^{- 16}$
0.2	$6.3551 \times 10^{- 2}$	$1.92968 \times 10^{- 1}$	$4.86961 \times 10^{- 1}$	$- 1 \times 10^{- 8}$	$6.35919 \times 10^{- 2}$	$1.92964 \times 10^{- 1}$	$4.86888 \times 10^{- 1}$	$3.3 \times 10^{- 16}$
0.3	$1.10711 \times 10^{- 11}$	$3.08565 \times 10^{- 1}$	$1.61448 \times 10^{- 1}$	$1 \times 10^{- 16}$	$1.10777 \times 10^{- 1}$	$3.08554 \times 10^{- 1}$	$1.61337 \times 10^{- 1}$	$7.2 \times 10^{- 13}$
0.4	$1.39785 \times 10^{- 1}$	$3.42546 \times 10^{- 1}$	$3.35336 \times 10^{- 2}$	$1.2 \times 10^{- 8}$	$1.39848 \times 10^{- 1}$	$3.42510 \times 10^{- 1}$	$3.52834 \times 10^{- 2}$	$1.1 \times 10^{- 10}$
0.5	$0.134627 \times 10^{- 1}$	$2.83651 \times 10^{- 1}$	$1.63443 \times 10^{- 1}$	$1.8 \times 10^{- 8}$	$1.34640 \times 10^{- 1}$	$2.83567 \times 10^{- 1}$	$1.63588 \times 10^{- 1}$	$1.8 \times 10^{- 8}$
0.6	$8.9611 \times 10^{- 2}$	$1.6838 \times 10^{- 1}$	$4.84015 \times 10^{- 1}$	$2.2 \times 10^{- 7}$	$8.95569 \times 10^{- 2}$	$1.68281 \times 10^{- 1}$	$4.84328 \times 10^{- 1}$	$2.6 \times 10^{- 6}$
0.7	$3.6319 \times 10^{- 2}$	$6.8512 \times 10^{- 2}$	$7.90337 \times 10^{- 1}$	$8.9 \times 10^{- 7}$	$3.62971 \times 10^{- 2}$	$6.85115 \times 10^{- 2}$	$7.90493 \times 10^{- 1}$	$1.1 \times 10^{- 4}$
0.8	$3.5332 \times 10^{- 2}$	$3.3868 \times 10^{- 2}$	$8.61599 \times 10^{- 1}$	$8.9 \times 10^{- 7}$	$3.55558 \times 10^{- 2}$	$3.40413 \times 10^{- 2}$	$8.61411 \times 10^{- 1}$	$6 \times 10^{- 4}$
0.9	$9.7902 \times 10^{- 2}$	$3.9987 \times 10^{- 2}$	$7.24224 \times 10^{- 1}$	$- 1.7 \times 10^{- 6}$	$1.05611 \times 10^{- 1}$	$4.29679 \times 10^{- 2}$	$6.96417 \times 10^{- 1}$	$6.5 \times 10^{- 3}$
1.0	$1.83175 \times 10^{- 1}$	$4.4119 \times 10^{- 2}$	$5.44826 \times 10^{- 1}$	$- 2.34 \times 10^{- 7}$	$6.95374 \times 10^{- 2}$	$2.86982 \times 10^{- 1}$	$2.90415 \times 10^{- 1}$	$3.5 \times 10^{- 3}$

Order	Transmitted and Reflected Efficiencies
Order	Method of Ref. 4	Method of Ref. 6	Method of Ref. 5	Method of Ref. 12	RFS
$(- 1,$ $- 1) T$	0.00092	0.00081	0.00097	0.00086	0.00085
$(- 1,$ $0) R$	0.00254	0.00259	0.00272	0.00249	0.00243
$(- 1,$ $0) T$	0.00303	0.00321	0.00324	0.00294	0.00282
$(0,$ $- 1) T$	0.00704	0.00683	0.00695	0.00677	0.00640
$(0,$ $0) R$	0.01984	0.01948	0.01904	0.01963	0.01970
$(0,$ $0) T$	0.96219	0.96430	0.96243	0.96448	0.96584
$(1,$ $0) T$	0.00299	0.00276	0.00299	0.00282	0.00269
E	0.9987	0.99998	0.9983	0.99989	1.00074

	Method of Ref. 16				RFS ( $P = 8,$ $N = 91$ )
$h / D$	$(- 1,$ $0) R$	$(0,$ $- 1) R$	$(0,$ $0) R$	$1 - E$	$(- 1,$ $0) R$	$(0,$ $- 1) R$	$(0,$ $0) R$	$1 - E$
0.1	$3.27739 \times 10^{- 3}$	$3.72936 \times 10^{- 3}$	$9.62746 \times 10^{- 2}$	$3.9 \times 10^{- 14}$	$3.29383 \times 10^{- 3}$	$3.74909 \times 10^{- 3}$	$9.6201 \times 10^{- 2}$	$4.66 \times 10^{- 15}$
0.2	$1.04428 \times 10^{- 2}$	$1.18396 \times 10^{- 2}$	$6.17586 \times 10^{- 2}$	$1.62 \times 10^{- 10}$	$1.04336 \times 10^{- 2}$	$1.18332 \times 10^{- 2}$	$6.18082 \times 10^{- 2}$	$6.88 \times 10^{- 15}$
0.3	$1.5987 \times 10^{- 2}$	$1.82072 \times 10^{- 2}$	$2.77715 \times 10^{- 2}$	$1.68 \times 10^{- 8}$	$1.59927 \times 10^{- 2}$	$1.82209 \times 10^{- 2}$	$2.77324 \times 10^{- 2}$	$7.55 \times 10^{- 12}$
0.4	$1.62416 \times 10^{- 2}$	$1.9071 \times 10^{- 2}$	$7.76777 \times 10^{- 3}$	$3.6 \times 10^{- 7}$	$1.6249 \times 10^{- 2}$	$1.90873 \times 10^{- 2}$	$7.79654 \times 10^{- 3}$	$7.44 \times 10^{- 10}$
0.5	$1.17478 \times 10^{- 2}$	$1.50842 \times 10^{- 2}$	$1.97568 \times 10^{- 3}$	$2.8 \times 10^{- 6}$	$1.17447 \times 10^{- 2}$	$1.50945 \times 10^{- 2}$	$1.97286 \times 10^{- 2}$	$2.15 \times 10^{- 8}$
0.6	$5.95639 \times 10^{- 3}$	$9.39993 \times 10^{- 3}$	$2.96950 \times 10^{- 3}$	$8.2 \times 10^{- 6}$	$5.94102 \times 10^{- 3}$	$9.39239 \times 10^{- 3}$	$2.97211 \times 10^{- 3}$	$2.33 \times 10^{- 6}$
0.7	$1.99328 \times 10^{- 3}$	$4.6860 \times 10^{- 3}$	$4.12517 \times 10^{- 3}$	$- 1 \times 10^{- 5}$	$1.99065 \times 10^{- 3}$	$4.68494 \times 10^{- 3}$	$4.12694 \times 10^{- 3}$	$5.3 \times 10^{- 5}$
0.8	$6.8640 \times 10^{- 4}$	$1.81897 \times 10^{- 3}$	$3.40891 \times 10^{- 3}$	$- 1.9 \times 10^{- 4}$	$6.74655 \times 10^{- 4}$	$1.78964 \times 10^{- 3}$	$3.39347 \times 10^{- 3}$	$4.17 \times 10^{- 4}$
0.9	$1.1486 \times 10^{- 3}$	$4.73788 \times 10^{- 4}$	$1.8427 \times 10^{- 3}$	$- 8.4 \times 10^{- 4}$	$1.28816 \times 10^{- 3}$	$3.3649 \times 10^{- 4}$	$2.08597 \times 10^{- 3}$	$3.73 \times 10^{- 4}$
1.0	$2.1132 \times 10^{- 3}$	$3.7408 \times 10^{- 5}$	$6.4855 \times 10^{- 4}$	$- 2.4 \times 10^{- 3}$	$3.68175 \times 10^{- 5}$	$2.96237 \times 10^{- 5}$	$1.71293 \times 10^{- 3}$	$5.15 \times 10^{- 2}$

Abstract

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Equations (60)

Journal of the Optical Society of America A