Reflection by a grating: Rayleigh methods: comments

A. Wirgin

doi:10.1364/JOSA.72.000963

Journal of the Optical Society of America
Vol. 72,
Issue 7,
pp. 963-969
(1982)
•https://doi.org/10.1364/JOSA.72.000963

Reflection by a grating: Rayleigh methods: comments

A. Wirgin

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A. Wirgin, "Reflection by a grating: Rayleigh methods: comments," J. Opt. Soc. Am. 72, 963-969 (1982)

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Abstract

It is shown, contrary to the assertions of van den Berg,1 that (1) the validity of the Rayleigh hypothesis has not been confirmed numerically, (2) the Rayleigh–Fourier (RF) method gives useful far-field results for sinusoidal gratings with height/period ratios far exceeding 0.3, (3) the Rayleigh–Yasuura method is not generally convergent in a numerical sense and is much less useful than the RF method, (4) the integrated-square error of the boundary condition is not a more reliable check of a numerical solution than the energy-balance error.

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	E Polarization				H Polarization

	ξ₀	ξ_±1	ξ_±2	ξ	ξ₀	ξ_±1	ξ_±2	ξ
RY
N = 3	0.3436	0.0052	0.2338	0.8216	0.0020	0.0563	0.4048	0.9243
4	0.3756	0.0066	0.2557	0.9003	0.0020	0.0554	0.4055	0.9238
5	0.3947	0.0083	0.2658	0.9429	0.0018	0.0560	0.4041	0.9218
6	0.4060	0.0095	0.2709	0.9667	0.0013	0.0575	0.4042	0.9245
7	0.4127	0.0103	0.2735	0.9803	0.0008	0.0594	0.4058	0.9312
8	0.4167	0.0108	0.2750	0.9882	0.0004	0.0614	0.4084	0.9400
9	0.4191	0.0111	0.2758	0.9928	0.0002	0.0632	0.4114	0.9495
10	0.4205	0.0113	0.2762	0.9956	0.0001	0.0648	0.4144	0.9586
RF
N = 3	0.4083	0.0130	0.2894	1.0131	0.0012	0.0616	0.4139	0.9521
4	0.4380	0.0133	0.2809	1.0263	0.0006	0.0723	0.4198	0.9847
5	0.4251	0.0117	0.2772	1.0027	0.0006	0.0725	0.4284	1.0026
6	0.4226	0.0116	0.2768	0.9994	0.0004	0.0709	0.4293	1.0008
7	0.4228	0.0116	0.2769	0.9999	0.0004	0.0708	0.4290	1.0000
8	0.4229	0.0116	0.2769	1.0000	0.0004	0.0709	0.4289	1.0000
9	0.4229	0.0116	0.2769	1.0000	0.0004	0.0709	0.4289	1.0000
10	0.4229	0.0116	0.2769	1.0000	0.0004	0.0709	0.4289	1.0000
VB	0.4223	0.0115	0.2773	1.0000	0.0004	0.0707	0.4291	1.0000
PB	0.4180	0.0143	0.2767	0.9999	0.0004	0.0756	0.4234	0.9984

	E Polarization				H Polarization

	ξ₀	ξ_±1	ξ_±2	ξ	ξ₀	ξ_±1	ξ_±2	ξ
RY
N = 7	0.2355	0.1687	0.1318	0.8366	0.4230	0.1047	0.1006	0.8335
8	0.2590	0.1720	0.1388	0.8805	0.4536	0.0985	0.0939	0.8383
9	0.2769	0.1735	0.1440	0.9119	0.4816	0.0939	0.0882	0.8458
10	0.2905	0.1740	0.1480	0.9344	0.5073	0.0907	0.0833	0.8552
11	0.3008	0.1740	0.1510	0.9508	0.5307	0.0883	0.0792	0.8657
12	0.3088	0.1737	0.1534	0.9630	0.5522	0.0865	0.0757	0.8768
RF
N = 7	0.3362	0.1698	0.1616	0.9989	0.7075	0.0840	0.0620	0.9996
8	0.3370	0.1700	0.1616	1.0003	0.7113	0.0832	0.0611	1.0000
9	0.3368	0.1700	0.1616	1.0001	0.7116	0.0831	0.0611	1.0000
10	0.3368	0.1700	0.1616	1.0000	0.7114	0.0832	0.0611	1.0000
11	0.3368	0.1700	0.1616	1.0000	0.7114	0.0832	0.0611	1.0000
12	0.3368	0.1700	0.1616	1.0000	0.7114	0.0832	0.0611	1.0000
VB	0.3352	0.1698	0.1611	0.9971	0.7078	0.0831	0.0612	0.9964
PB	0.3259	0.1693	0.1693	1.0029	0.7150	0.0834	0.0610	1.0037

	E Polarization				H Polarization

	ξ₀	ξ_±1	ξ_±2	ξ	ξ₀	ξ_±1	ξ_±2	ξ
RY
N = 8	0.3848	0.0575	0.0551	0.6102	0.0976	0.2057	0.0460	0.6009
9	0.4001	0.0629	0.0702	0.6664	0.1148	0.2366	0.0524	0.6929
10	0.4094	0.0666	0.0849	0.7124	0.1285	0.2615	0.0572	0.7659
11	0.4143	0.0691	0.0988	0.7501	0.1391	0.2812	0.0605	0.8227
12	0.4161	0.0710	0.1116	0.7812	0.1474	0.2967	0.0628	0.8664
13	0.4159	0.0724	0.1233	0.8072	0.1539	0.3089	0.0643	0.9002
RF
N = 8	0.3296	0.0900	0.2289	0.9674	0.1684	0.3745	0.0526	1.0227
9	0.3440	0.0940	0.2311	0.9941	0.1650	0.3717	0.0529	1.0142
10	0.3476	0.0954	0.2323	1.0031	0.1574	0.3684	0.0516	0.9976
11	0.3460	0.0951	0.2323	1.0001	0.1577	0.3689	0.0515	0.9986
12	0.3456	0.0949	0.2322	0.9998	0.1585	0.3692	0.0516	1.0002
13	0.3457	0.0950	0.2322	1.0000	0.1585	0.3691	0.0517	1.0001
VB	0.3443	0.0944	0.2322	0.9975	0.1568	0.3672	0.0518	0.9949
PB	0.3481	0.0970	0.2394	1.0209	0.1576	0.3759	0.0526	1.0147

	E Polarization						H Polarization

	ξ₀	ξ₋₁	ξ₁	ξ₋₂	ξ₂	ξ	ξ₀	ξ₋₁	ξ₁	ξ₋₂	ξ₂	ξ
RY
N = 12	0.1950	0.1389	0.1389	0.0246	0.0246	0.5221	0.4234	0.0167	0.0167	0.0069	0.0069	0.4704
13	0.1882	0.1363	0.1363	0.0297	0.0297	0.5202	0.4234	0.0161	0.0161	0.0070	0.0070	0.4695
14	0.1818	0.1345	0.1345	0.0347	0.0347	0.5203	0.4233	0.0156	0.0156	0.0070	0.0070	0.4686
15	0.1761	0.1336	0.1336	0.0396	0.0396	0.5226	0.4231	0.0150	0.0150	0.0071	0.0071	0.4675
16	0.1714	0.1331	0.1334	0.0444	0.0444	0.5267	0.4229	0.0145	0.0146	0.0072	0.0072	0.4464
17	0.1680	0.1330	0.1338	0.0485	0.0486	0.5318	0.4228	0.0143	0.0144	0.0072	0.0072	0.4659
18	0.1718	0.1402	0.1266	0.0442	0.0434	0.5262	0.4226	0.0135	0.0146	0.0076	0.0070	0.4652
RF
N = 12	0.3656	0.1345	0.1345	0.1730	0.1730	0.9806	0.7099	0.0382	0.0382	0.1277	0.1277	1.0417
13	0.3724	0.1381	0.1381	0.1733	0.1733	0.9953	0.7050	0.0312	0.0312	0.1237	0.1237	1.0150
14	0.3756	0.1398	0.1398	0.1734	0.1734	1.0020	0.6935	0.0273	0.0273	0.1238	0.1238	0.9956
15	0.3747	0.1395	0.1394	0.1733	0.1732	1.0001	0.6942	0.0279	0.0279	0.1242	0.1241	0.9983
16	0.3754	0.1395	0.1386	0.1727	0.1724	0.9986	0.6974	0.0287	0.0283	0.1237	0.1234	1.0016
17	0.3937	0.1376	0.1391	0.1699	0.1671	1.0075	0.7122	0.0313	0.0261	0.1191	0.1188	1.0075
18	0.1363	0.8023	0.4509	0.0223	0.0665	1.4783	0.7081	0.0250	0.0155	0.1412	0.0996	0.9894

	Re(A_±3⁺)	Re(A_±5⁺)	Re(A_±7⁺)	Re(A_±9⁺)
	Im(A_±3⁺)	Im(A_±5⁺)	Im(A_±7⁺)	Im (A_±9⁺)
RY
N = 5	0.13 × 10¹	0.84 × 10⁰
	−0.18 × 10¹	−0.11 × 10¹
6	0.17 × 10¹	0.31 × 10¹
	−0.23 × 10¹	−0.39 × 10¹
7	0.20 × 10¹	0.67 × 10¹	0.17 × 10¹
	−0.27 × 10¹	−0.83 × 10¹	−0.20 × 10¹
3	0.22 × 10¹	0.12 × 10²	0.86 × 10¹
	−0.30 × 10¹	−0.14 × 10²	−0.99 × 10¹
9	0.23 × 10¹	0.17 × 10²	0.25 × 10²	0.35 × 10¹
	−0.32 × 10¹	−0.20 × 10²	−0.27 × 10²	−0.38 × 10¹
10	0.25 × 10¹	0.23 × 10²	0.52 × 10²	0.22 × 10²
	−0.33 × 10¹	−0.26 × 10²	−0.56 × 10²	−0.24 × 10²
RF
N = 5	0.24 × 10¹	0.42 × 10¹
	−0.33 × 10¹	−0.50 × 10¹
6	0.27 × 10¹	0.15 × 10²
	−0.36 × 10¹	−0.17 × 10²
7	0.28 × 10¹	0.30 × 10²	0.22 × 10²
	−0.37 × 10¹	−0.32 × 10²	−0.23 × 10²
8	0.28 × 10¹	0.42 × 10²	0.10 × 10³
	−0.37 × 10¹	−0.44 × 10²	−0.10 × 10³
9	0.28 × 10¹	0.50 × 10²	0.25 × 10³	0.11 ×10³
	−0.37 × 10¹	−0.51 × 10²	−0.24 × 10³	−0.10 × 10³
10	0.28 × 10¹	0.53 × 10²	0.44 × 10³	0.63 × 10³
	−0.37 × 10¹	−0.55 × 10²	−0.41 × 10³	−0.58 × 10³

Abstract

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Tables (5)

Equations (2)

Journal of the Optical Society of America