Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions

Guozhen Yang; Benyuan Gu; Xin Tan; Ming-Pin Chang; Bizhen Dong; Okan K. Ersoy

doi:10.1364/JOSAA.11.001632

Journal of the Optical Society of America A
Vol. 11,
Issue 5,
pp. 1632-1640
(1994)
•https://doi.org/10.1364/JOSAA.11.001632

Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions

Guozhen Yang, Benyuan Gu, Xin Tan, Ming-Pin Chang, Bizhen Dong, and Okan K. Ersoy

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Guozhen Yang, Benyuan Gu, Xin Tan, Ming-Pin Chang, Bizhen Dong, and Okan K. Ersoy, "Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions," J. Opt. Soc. Am. A 11, 1632-1640 (1994)

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Abstract

A new iterative optimization method based on the Yang–Gu (Y-G) algorithm for the design of diffractive phase elements (DPE’s) that incorporate several optical functions is presented. The Y-G algorithm has been previously used to deal with various amplitude-phase retrieval problems in linear optical transform systems involving a monochromatic wave. We extend the general theory to the case of linear imaging systems with the illumination consisting of a number of components at different wavelengths. Based on a rigorous mathematical derivation, an iterative algorithm is developed to integrate several functions into one DPE. In the first set of examples numerical simulations are carried out in the design of several one-dimensional DPE’s capable of both demultiplexing different wavelength components and focusing each component wave. The second example illustrates the design of a two-dimensional DPE that performs as a demultiplexer in one direction and an array generator in the other direction.

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Position for 1-D Sequence l	Sequence Wavelength g	Desired r₂	Common-Phase Case r₂	Actual-Phase Case r₂
1	1	0.5	0.4845	0.4975
	2	0.0	0.0515	0.0515
	3	0.0	0.0467	0.0270
	4	0.0	0.0470	0.0461
2	1	0.0	0.0556	0.0412
	2	0.5	0.4993	0.4993
	3	0.0	0.0870	0.1185
	4	0.0	0.0127	0.0628
3	1	0.0	0.0076	0.1193
	2	0.0	0.0225	0.0225
	3	0.5	0.5189	0.5202
	4	0.0	0.0688	0.0237
4	1	0.0	0.0241	0.1148
	2	0.0	0.0442	0.0442
	3	0.0	0.0790	0.0639
	4	0.5	0.4634	0.4414

Position for 1-D Sequence l	Sequence Wavelength g	Desired r₂	Common-Phase Case r₂	Actual-Phase Case r₂
1	1	0.3536	0.3336	0.3168
	2	0.0	0.0204	0.0297
	3	0.0	0.0080	0.0073
	4	0.0	0.0157	0.0157
	5	0.0	0.0223	0.0298
	6	0.0	0.0211	0.0145
	7	0.0	0.0334	0.0701
	8	0.0	0.0250	0.0625
2	1	0.0	0.0321	0.0630
	2	0.3536	0.3549	0.3466
	3	0.0	0.0047	0.0090
	4	0.0	0.0086	0.0157
	5	0.0	0.0202	0.0027
	6	0.0	0.0240	0.0410
	7	0.0	0.0103	0.0226
	8	0.0	0.0152	0.0213
3	1	0.0	0.0223	0.0040
	2	0.0	0.0175	0.0334
	3	0.3536	0.3569	0.3369
	4	0.0	0.0286	0.0286
	5	0.0	0.0205	0.0258
	6	0.0	0.0064	0.0045
	7	0.0	0.0118	0.0247
	8	0.0	0.0143	0.0246
4	1	0.0	0.0278	0.0582
	2	0.0	0.0354	0.0581
	3	0.0	0.0420	0.0591
	4	0.3536	0.3415	0.3415
	5	0.0	0.0237	0.0258
	6	0.0	0.0187	0.0182
	7	0.0	0.0240	0.0141
	8	0.0	0.0079	0.0487
5	1	0.0	0.0199	0.0057
	2	0.0	0.0195	0.0295
	3	0.0	0.0219	0.0467
	4	0.0	0.0309	0.0309
	5	0.3536	0.3472	0.3465
	6	0.0	0.0127	0.0173
	7	0.0	0.0184	0.0154
	8	0.0	0.0055	0.0160
6	1	0.0	0.0146	0.0811
	2	0.0	0.0264	0.0321
	3	0.0	0.0362	0.0333
	4	0.0	0.0163	0.0163
	5	0.0	0.0233	0.0123
	6	0.3536	0.3589	0.3525
	7	0.0	0.0175	0.0731
	8	0.0	0.0080	0.0200
7	1	0.0	0.0341	0.0798
	2	0.0	0.0077	0.0336
	3	0.0	0.0054	0.0113
	4	0.0	0.0182	0.0182
	5	0.0	0.0100	0.0187
	6	0.0	0.0037	0.0264
	7	0.3536	0.3377	0.3262
	8	0.0	0.0168	0.0594
8	1	0.0	0.0128	0.0342
	2	0.0	0.0171	0.0230
	3	0.0	0.0211	0.0252
	4	0.0	0.0268	0.0268
	5	0.0	0.0280	0.0354
	6	0.0	0.0592	0.0907
	7	0.0	0.0222	0.0383
	8	0.3536	0.3567	0.3061

b		%		w		%
At 4 wavelengths
b₁		2.22		ω₁		11.77
b₂		7.90		ω₂		2.05
b₃		5.66		ω₃		6.97
b₄		9.87		ω₄		3.40
At 8 wavelengths
b₁		11.08		ω₁		21.44
b₂		5.63		ω₂		7.44
b₃		3.47		ω₃		6.77
b₄		11.85		ω₄		2.92
b₅		4.05		ω₅		3.39
b₆		11.97		ω₆		9.22
b₇		11.77		ω₇		12.48
b₈		14.93		ω₈		12.32
At 16 wavelengths
b₁	9.56	b₉	10.91	ω₁	19.91	ω₉	3.84
b₂	7.44	b₁₀	15.27	ω₂	33.09	ω₁₀	5.56
b₃	10.08	b₁₁	13.28	ω₃	21.74	ω₁₁	8.19
b₄	10.00	b₁₂	21.33	ω₄	19.14	ω₁₂	7.14
b₅	9.39	b₁₃	15.68	ω₅	14.79	ω₁₃	7.89
b₆	9.91	b₁₄	11.88	ω₆	7.62	ω₁₄	10.64
b₇	10.58	b₁₅	15.57	ω₇	2.80	ω₁₅	17.42
b₈	12.56	b₁₆	20.56	ω₈	3.40	ω₁₆	19.70

Position for 2-D Sequence l, J	Sequence Wavelength g	Desired r₂	Common-Phase Case r₂	Actual-Phase Case
1, 1	1	0.3536	0.3500	0.4031
1, 1	2	0.0	0.0212	0.0212
1, 1	3	0.0	0.0099	0.0394
1, 1	4	0.0	0.0169	0.0356
1, 2	1	0.0	0.0448	0.0476
1, 2	2	0.3536	0.3610	0.3609
1, 2	3	0.0	0.0407	0.0322
1, 2	4	0.0	0.0200	0.0442
1, 3	1	0.0	0.0268	0.0397
1, 3	2	0.0	0.0203	0.0203
1, 3	3	0.3536	0.3318	0.3112
1, 3	4	0.0	0.0447	0.0624
1, 4	1	0.0	0.0317	0.0531
1, 4	2	0.0	0.0162	0.0162
1, 4	3	0.0	0.0204	0.0218
1, 4	4	0.3536	0.3557	0.3049
2, 1	1	0.0	0.0080	0.0370
2, 1	2	0.0	0.0173	0.0173
2, 1	3	0.0	0.0123	0.0232
2, 1	4	0.0	0.0063	0.0044
2, 2	1	0.0	0.0172	0.0442
2, 2	2	0.0	0.0168	0.0168
2, 2	3	0.0	0.0231	0.0280
2, 2	4	0.0	0.0110	0.0375
2, 3	1	0.0	0.0097	0.0356
2, 3	2	0.0	0.0145	0.0145
2, 3	3	0.0	0.0105	0.0179
2, 3	4	0.0	0.0184	0.0259
2, 4	1	0.0	0.0071	0.0238
2, 4	2	0.0	0.0060	0.0060
2, 4	3	0.0	0.0131	0.0267
2, 4	4	0.0	0.0086	0.0356
3, 1	1	0.3536	0.3470	0.3010
3, 1	2	0.0	0.0293	0,0293
3, 1	3	0.0	0.0186	0.0252
3, 1	4	0.0	0.0094	0.0356
3, 2	1	0.0	0.0029	0.0141
3, 2	2	0.3536	0.3434	0.3434
3, 2	3	0.0	0.0278	0.0306
3, 2	4	0.0	0.0315	0.0541
3, 3	1	0.0	0.0252	0.0515
3, 3	2	0.0	0.0247	0.0248
3, 3	3	0.3536	0.3504	0.3750
3, 3	4	0.0	0.0082	0.0220
3, 4	1	0.0	0.0102	0.0792
3, 4	2	0.0	0.0109	0.0110
3, 4	3	0.0	0.0324	0.0247
3, 4	4	0.3536	0.3582	0.3711
4, 1	1	0.0	0.0080	0.0408
4, 1	2	0.0	0.0173	0.0166
4, 1	3	0.0	0.0124	0.0196
4, 1	4	0.0	0.0066	0.0410
4, 2	1	0.0	0.0170	0.0472
4, 2	2	0.0	0.0166	0.0166
4, 2	3	0.0	0.0230	0.0196
4, 2	4	0.0	0.0109	0.0515
4, 3	1	0.0	0.0098	0.0488
4, 3	2	0.0	0.0147	0.0147
4, 3	3	0.0	0.0103	0.0179
4, 3	4	0.0	0.0182	0.0285
4, 4	1	0.0	0.0070	0.0515
4, 4	2	0.0	0.0060	0.0061
4, 4	3	0.0	0.0131	0.0273
4, 4	4	0.0	0.0085	0.0379

ω	%	Bright Spots		Dark Spots

		b	%	b	%
ω₁	11.03	b₁	2.06	b₅	0.22
ω₂	1.80	b₂	3.90	b₆	0.41
ω₃	5.13	b₃	5.83	b₇	0.48
ω₄	9.74	b₄	3.67	b₈	0.38
		b₉	3.28	b₁₃	0.26
		b₁₀	4.88	b₁₄	0.55
		b₁₁	1.48	b₁₅	0.38
		b₁₂	5.49	b₁₆	0.44

Abstract

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Figures (4)

Tables (5)

Equations (36)

Journal of the Optical Society of America A