Wavelet element method for lamellar gratings

Zhangyi Liu; Jiu Hui Wu; Li Shen

doi:10.1364/JOSAA.30.001021

Journal of the Optical Society of America A
Vol. 30,
Issue 5,
pp. 1021-1029
(2013)
•https://doi.org/10.1364/JOSAA.30.001021

Wavelet element method for lamellar gratings

Zhangyi Liu, Jiu Hui Wu, and Li Shen

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Zhangyi Liu, Jiu Hui Wu, and Li Shen, "Wavelet element method for lamellar gratings," J. Opt. Soc. Am. A 30, 1021-1029 (2013)

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Abstract

A wavelet element method is developed for analyzing lamellar diffraction gratings or grating stacks. The eigenmodes of the grating layers are accurately calculated by this method, and then the diffraction efficiencies of the gratings are calculated by the S-matrix algorithm. The method proposed in this paper consists in mapping each homogeneous layer to a wavelet element, and then matching them according to the boundary conditions between the layers. By this method the boundary conditions are satisfied rigorously and the Gibbs phenomenon in the Fourier modal method (FMM) can be avoided. The method performs better than the standard FMM for gratings involving metals. It can also be applied to analyze other discontinuous structures.

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$N_{f}$	FMM	$N$	$M$	WEM(B3.5)	$N$	$M$	WEM(B5.5)
57	0.8456	56	50	0.84864	53	50	0.84835
81	0.8468	81	50	0.82935	81	50	0.84793
121	0.8475	120	50	0.84846	116	50	0.84841
141	0.8477	140	50	0.84841	140	50	0.84844
161	0.8478	160	50	0.84844	160	60	0.84835
249	0.8481	248	69	0.84846	244	85	0.84848
321	0.8482	321	72	0.84835	321	85	0.84846
371	0.8482	370	78	0.84851	370	87	0.84845
441	0.8483	440	79	0.84845	440	105	0.84847
505	0.8483	504	117	0.84848	500	151	0.84848

$N_{f}$	FMM	$N$	$M$	WEM(B3.5)	$N$	$M$	WEM(B5.5)
57	0.735908	56	45	0.736985	53	45	0.734821
81	0.734852	8	45	0.731422	81	45	0.734765
121	0.734452	120	50	0.734337	116	50	0.734284
141	0.734388	140	54	0.734332	140	50	0.734284
161	0.734352	160	58	0.734343	160	50	0.734283
249	0.734299	248	90	0.734280	244	95	0.734279
321	0.734288	321	90	0.734280	321	100	0.734279
371	0.734285	370	90	0.734279	370	98	0.734279
441	0.734282	440	120	0.734279	440	110	0.734279
505	0.734281	504	150	0.734279	500	170	0.734279

$N_{f}$	FMM	$N$	$M$	WEM(B3.5)	$N$	$M$	WEM(B5.5)
57	0.549665	56	50	0.55035	53	52	0.55009
81	0.550025	8	50	0.55046	81	51	0.54991
121	0.550153	120	52	0.55032	116	51	0.55039
141	0.550173	140	51	0.55041	140	51	0.55036
161	0.550184	160	50	0.55029	160	51	0.55038
249	0.550199	248	73	0.55026	244	89	0.55024
321	0.550202	321	71	0.55029	321	93	0.55024
371	0.550203	370	97	0.55024	370	93	0.55023
441	0.550204	440	104	0.54994	440	119	0.55024
505	0.550204	504	127	0.55023	500	165	0.55021

$N_{f}$	FMM	$N$	$M$	WEM(B3.5)	$N$	$M$	WEM(B5.5)
57	0.165678	56	45	0.165633	53	45	0.165649
81	0.165656	8	45	0.165543	81	45	0.165662
121	0.165647	120	50	0.165643	116	50	0.165643
141	0.165646	140	54	0.165643	140	50	0.165643
161	0.165645	160	58	0.165644	160	50	0.165642
249	0.165644	248	90	0.165644	244	95	0.165644
321	0.165644	321	90	0.165644	321	100	0.165644

$N_{f}$	FMM	$N$	$M$	WEM(B3.5)	$N$	$M$	WEM(B5.5)
57	0.8894	56	50	0.89293	53	53	0.89306
81	0.8935	81	50	0.86845	81	50	0.89229
121	0.8666	120	50	0.89294	116	50	0.89291
141	0.8938	140	50	0.89290	140	50	0.89293
161	0.8943	160	50	0.89294	160	50	0.89299
249	0.8929	248	69	0.89294	244	85	0.89296
321	0.8928	321	72	0.89285	321	85	0.89295
371	0.8930	370	78	0.89300	370	87	0.89295
441	0.8919	440	79	0.89294	440	87	0.89302
505	0.8925	504	117	0.89298	500	157	0.89298

$N_{f}$	FMM	$N$	$M$	WEM(B3.5)	$N$	$M$	WEM(B5.5)
57	0.85108	56	45	0.853143	53	45	0.849917
81	0.84972	8	45	0.845036	81	45	0.849907
121	0.84921	120	50	0.849082	116	50	0.848993
141	0.84913	140	54	0.849076	140	50	0.848992
161	0.84908	160	58	0.849095	160	50	0.848990
249	0.84901	248	90	0.848987	244	95	0.848986
321	0.84900	321	90	0.848988	321	100	0.848986
371	0.84899	370	90	0.848986	370	98	0.848986

	MMGE	PSMM(f)	WEM
Example 1 (TM)	16	162	244
Example 1 (TE)	20	42	116
Example 3 (TM)	12	182	160
Example 3 (TE)	20	62	116

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Tables (7)

Equations (47)

Journal of the Optical Society of America A