Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences

Oscar P. Bruno; Michael C. Haslam

doi:10.1364/JOSAA.26.000658

Journal of the Optical Society of America A
Vol. 26,
Issue 3,
pp. 658-668
(2009)
•https://doi.org/10.1364/JOSAA.26.000658

Efficient high-order evaluation of scattering by periodic surfaces: deep gratings, high frequencies, and glancing incidences

Oscar P. Bruno and Michael C. Haslam

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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)

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Abstract

We present a superalgebraically convergent integral equation algorithm for evaluation of TE and TM electromagnetic scattering by smooth perfectly conducting periodic surfaces $z = f (x)$ . 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.

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Equations (60)

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$l$	$α_{2 l}$
1	0
2	3
3	0
4	63
5	1320
6	49,203
7	2,653,560
8	196,707,423
9	19,194,804,720
10	2,385,684,870,723
11	367,985,503,366,800
12	68,980,888,889,771,103

$z_{i}$	$z_{i + 1}$	$a_{0}^{(i)}$	$a_{1}^{(i)}$	$a_{2}^{(i)}$
1	25	$1.9349 \times 10^{1}$	5.0731	$- 3.4114 \times 10^{- 2}$
25	200	$3.8393 \times 10^{1}$	3.5681	$- 8.1905 \times 10^{- 4}$
200	1000	$6.8173 \times 10^{1}$	3.2753	$- 4.8158 \times 10^{- 5}$
1000	2000	$9.4061 \times 10^{1}$	3.2126	$- 1.0606 \times 10^{- 5}$
2000	4000	$1.2388 \times 10^{2}$	3.1820	$- 2.6515 \times 10^{- 6}$
4000	6000	$1.5245 \times 10^{2}$	3.1684	$- 9.9068 \times 10^{- 7}$
6000	8000	$1.8265 \times 10^{2}$	3.1591	$- 2.7473 \times 10^{- 7}$

N	M	$θ = π ∕ 2$		$θ = π ∕ 6$
N	M	$∣ 1 - E ∣$	Error	$∣ 1 - E ∣$	Error
4	$2^{6}$	$1.26 \times 10^{- 4}$	$9.56 \times 10^{- 4}$	$1.67 \times 10^{- 4}$	$6.66 \times 10^{- 4}$
6	$2^{6}$	$1.26 \times 10^{- 7}$	$3.14 \times 10^{- 7}$	$1.39 \times 10^{- 7}$	$1.03 \times 10^{- 7}$
8	$2^{6}$	$8.59 \times 10^{- 10}$	$1.45 \times 10^{- 10}$	$1.43 \times 10^{- 10}$	$1.78 \times 10^{- 10}$
10	$2^{6}$	$1.32 \times 10^{- 13}$	$1.28 \times 10^{- 13}$	$2.11 \times 10^{- 13}$	$1.36 \times 10^{- 13}$

$a ∕ P$	$P ∕ λ = 1.0$				$P ∕ λ = 2.0$				$P ∕ λ = 4.0$
$a ∕ P$	N	M	Error	$t_{exe}$	N	M	Error	$t_{exe}$	N	M	Error	$t_{exe}$
0.5	15	$2^{7}$	$1.3 \times 10^{- 14}$	0.67	20	$2^{7}$	$1.0 \times 10^{- 14}$	0.74	30	$2^{7}$	$1.0 \times 10^{- 14}$	1.02
1.0	25	$2^{7}$	$3.9 \times 10^{- 12}$	0.78	25	$2^{8}$	$2.2 \times 10^{- 13}$	1.32	45	$2^{8}$	$1.8 \times 10^{- 14}$	2.30
1.5	35	$2^{8}$	$4.4 \times 10^{- 14}$	1.82	35	$2^{8}$	$3.0 \times 10^{- 14}$	1.64	55	$2^{8}$	$2.1 \times 10^{- 12}$	2.45
2.0	65	$2^{8}$	$1.0 \times 10^{- 14}$	3.21	65	$2^{8}$	$1.0 \times 10^{- 14}$	2.84	70	$2^{9}$	$5.2 \times 10^{- 13}$	5.96

$a ∕ P$	N	M	$N^{*}$	Error (TE)	$∣ 1 - E ∣$ (TE)	Error (TM)	$∣ 1 - E ∣$ (TM)	$t_{exe}$
4	550	$2^{11}$	700	$7.35 \times 10^{- 13}$	$5.73 \times 10^{- 12}$	$7.58 \times 10^{- 13}$	$1.09 \times 10^{- 12}$	10
8	1050	$2^{12}$	1200	$6.14 \times 10^{- 12}$	$1.48 \times 10^{- 11}$	$7.52 \times 10^{- 11}$	$5.19 \times 10^{- 11}$	64
12	1600	$2^{13}$	1700	$1.49 \times 10^{- 11}$	$4.00 \times 10^{- 12}$	$3.75 \times 10^{- 11}$	$8.28 \times 10^{- 11}$	236
16	2100	$2^{13}$	2200	$3.59 \times 10^{- 11}$	$8.72 \times 10^{- 11}$	$5.74 \times 10^{- 11}$	$2.83 \times 10^{- 11}$	392
20	2600	$2^{13}$	2700	$9.80 \times 10^{- 11}$	$3.51 \times 10^{- 11}$	$2.05 \times 10^{- 10}$	$1.55 \times 10^{- 10}$	570

θ (deg)	N	M	$M_{v}$	Error (TE)	$∣ 1 - E ∣$ (TE)	Error (TM)	$∣ 1 - E ∣$ (TM)	$t_{exe}$
1.00	800	$2^{12}$	$2^{10} + 1$	$3.00 \times 10^{- 14}$	$2.84 \times 10^{- 14}$	$2.61 \times 10^{- 14}$	$2.38 \times 10^{- 14}$	50
0.10	800	$2^{12}$	$2^{10} + 1$	$3.09 \times 10^{- 15}$	$2.32 \times 10^{- 13}$	$4.80 \times 10^{- 14}$	$2.35 \times 10^{- 13}$	50
0.01	800	$2^{12}$	$2^{12} + 1$	$2.47 \times 10^{- 13}$	$7.58 \times 10^{- 14}$	$1.21 \times 10^{- 12}$	$1.45 \times 10^{- 12}$	72

$λ ∕ P$	N	M	$N^{*}$	Error (TE)	$∣ 1 - E ∣$ (TE)	Error (TM)	$∣ 1 - E ∣$ (TM)	$t_{exe}$
0.100	100	$2^{9}$	200	$1.40 \times 10^{- 14}$	$2.22 \times 10^{- 16}$	$1.30 \times 10^{- 14}$	$1.77 \times 10^{- 14}$	0.25
0.050	200	$2^{10}$	300	$2.40 \times 10^{- 14}$	$1.95 \times 10^{- 14}$	$2.32 \times 10^{- 14}$	$5.22 \times 10^{- 14}$	1.3
0.010	750	$2^{12}$	900	$2.90 \times 10^{- 14}$	$7.69 \times 10^{- 14}$	$4.85 \times 10^{- 14}$	$1.60 \times 10^{- 13}$	45
0.005	1500	$2^{13}$	1600	$6.99 \times 10^{- 14}$	$1.43 \times 10^{- 13}$	$7.59 \times 10^{- 14}$	$5.27 \times 10^{- 13}$	221

N	M	Error (TE)	$∣ 1 - E ∣$ (TE)	Error (TM)	$∣ 1 - E ∣$ (TM)	$t_{exe}$
150	$2^{10}$	$2.14 \times 10^{- 4}$	$1.03 \times 10^{- 3}$	$1.11 \times 10^{- 1}$	$8.67 \times 10^{- 1}$	62
175	$2^{10}$	$3.13 \times 10^{- 5}$	$4.94 \times 10^{- 5}$	$1.90 \times 10^{- 4}$	$2.80 \times 10^{- 5}$	74
200	$2^{10}$	$2.67 \times 10^{- 9}$	$3.35 \times 10^{- 9}$	$7.13 \times 10^{- 9}$	$1.66 \times 10^{- 8}$	85
225	$2^{10}$	$1.30 \times 10^{- 14}$	$2.89 \times 10^{- 14}$	$2.06 \times 10^{- 14}$	$1.42 \times 10^{- 14}$	98

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Journal of the Optical Society of America A