Refinement of optical multilayer systems with different optimization
procedures

J. A. Dobrowolski; R. A. Kemp

doi:10.1364/AO.29.002876

Applied Optics
Vol. 29,
Issue 19,
pp. 2876-2893
(1990)
•https://doi.org/10.1364/AO.29.002876

Refinement of optical multilayer systems with different optimization procedures

J. A. Dobrowolski and R. A. Kemp

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J. A. Dobrowolski and R. A. Kemp, "Refinement of optical multilayer systems with different optimization procedures," Appl. Opt. 29, 2876-2893 (1990)

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Abstract

Ten different optimization methods, representing both local and global minimum seeking algorithms, were applied to the solution of three different optical thin film design problems. Because all methods were incorporated in the same thin film design program, and because a routine was invoked that reads CPU time, the relative efficiencies of the various methods could be compared directly.

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1	2	3	4	5	6	7	8

A Adaptive Random Search	ARS	Yes	Yes	No	D	0.107	0.221
B Damped Least Squares	DLSQ	No	Yes	Yes	A	0.058	0.091
C Modified Gradient	GRAD	Yes	Yes	Yes	A	0.134	0.188
D Golden Section	GOLD	Yes	No	No	N.A.	0.086	0.166
E Hooke and Jeeves Pattern Search	H&J	Yes	Yes	No	A, B	0.054	0.094
F Basic Powell’s Conjugate Search	BPCS	Yes	Yes	No	C	0.062	0.163
G Rosenbrock’s Rotating Coordinates	ROSE	Yes	Yes	No	B	0.074	0.178
H Generalized Simulated Annealing	GSA	Yes	Yes	No	D	0.105	0.376
I Monte Carlo Simulated Annealing	MCSA	Yes	Yes	No	D	0.099	0.198
J Revised Nelder-Mead Simplex	SIMP	Yes	Yes	No	B	0.114	0.309

REFINEMENT METHOD		INITIAL DESIGN	ADAPT. RANDOM SEARCH	DAMPED LEAST SQUARES	MODIFIED GRADIENT	GOLDEN SECTION	HOOKE & JEEVES
REFINEMENT METHOD		INITIAL DESIGN	ADAPT. RANDOM SEARCH	DAMPED LEAST SQUARES	MODIFIED GRADIENT	GOLDEN SECTION	1	2

Layer No.	n	nt	nt	nt	nt	nt	nt	nt

Substrate	4.00	–	–	–	–	–	–
1	2.20	0.7510	0.4488	0.7973	0.3741	0.6674	0.8559	0.6758
2	4.20	0.7675	1.7578	1.1112	1.2269	0.4708	0.6605	0.5250
3	2.20	0.7300	0.9105	1.2924	0.5267	4.3018	2.7639	3.8084
4	4.20	0.7825	1.2941	2.9719	0.7861	0.9299	0.9282	1.0343
5	2.20	0.7300	0.8149	0.0529	0.7785	0.7231	0.4068	0.1727
6	4.20	0.7675	0.1166	1.9708	0.4304	0.8621	3.5005	1.8640
7	2.20	0.7150	0.6653	3.6648	0.9247	0.7506	1.5573	0.6478
8	4.20	0.7675	0.3715	0.7752	0.4751	0.7071	0.1450	1.4454
9	2.20	1.1000	1.0293	0.5898	1.1019	1.3093	0.9582	1.5884
10	4.20	0.1450	0.2359	2.3699	0.1344	0.0752	0.3116	0.0711
11	2.20	1.1000	1.1103	5.3107	1.1644	1.1103	1.5535	0.3944
12	4.20	0.3050	0.5959	0.9999	0.4095	0.3506	0.6626	0.3255
13	2.20	1.6450	1.1917	5.6406	6.7621	1.4035	0.3108	0.6974
14	4.20	0.3050	0.8995	1.2672	0.4653	0.1266	0.7181	0.7485
15	2.20	1.1000	1.4521	2.8033	6.2371	1.2219	5.6707	0.4721
16	4.20	0.1450	0.0007		0.8820	0.0008	0.8949	2.5070
17	2.20	4.3250	4.5410		3.0558	4.3241	5.7373	5.2899
18	4.20	0.5825	0.7072			0.5414	1.2211	1.0021
19	2.20	6.0125	5.9344			6.0125	2.8317	5.6327
20	4.20	1.3875	1.0692			1.0051		1.2697
21	2.20	2.7975	2.9262			2.9544		2.7994
air	1.00	–	–	–	–	–	–	–

∑(nt)		26.93	28.07	31.62	25.73	29.85	31.69	32.39

Merit Function		10.58	1.72	1.13	2.81	2.11	1.21	1.13

REFINEMENT METHOD		POWELL CONJUG. SEARCH	ROSENBROCK		GENERALIZED SIMUL. ANNEAL.		MONTE CARLO S. A.	SIMPLEX
REFINEMENT METHOD		POWELL CONJUG. SEARCH	1	2	1	2	MONTE CARLO S. A.	SIMPLEX

Layer No.	n	nt	nt	nt	nt	nt	nt	nt

Substrate	4.00	–	–	–	–	–	–	–
1	2.20	0.7833	0.4909	0.4138	0.0445	0.1686	0.3311	0.5327
2	4.20	1.0276	1.2856	1.3415	5.0341	6.3377	1.6947	1.0354
3	2.20	1.8699	0.8690	0.9133	0.0593	0.5383	0.3801	1.2109
4	4.20	1.2699	0.7792	0.8318	1.3728	2.2559	1.1612	0.6607
5	2.20	0.8460	0.3689	0.4219	0.4407	0.1535	0.7390	1.1008
6	4.20	1.8154	0.8837	0.9140	2.3597	3.0119	1.1626	1.6114
7	2.20	0.8947	1.0810	1.1038	0.2054	4.7514	1.5199	0.7245
8	4.20	0.3564	0.8653	0.9413	2.5716	4.7285	0.8931	1.4327
9	2.20	1.4245	2.7359	2.9083	1.0084	0.1436	0.1666	1.6635
10	4.20	0.3716	0.6486	0.6485	0.1863	4.8876	0.8152	0.0315
11	2.20	1.1115	1.5662	1.5485	0.7948	5.1950	0.5167	1.3713
12	4.20	1.5134	1.1278	1.2149	0.3723	0.9591	2.0732	0.5193
13	2.20	0.1945	5.9433	5.9371	1.6989	5.0185	0.1146	1.5119
14	4.20	2.2223	0.7820	0.8198	5.0590	3.5764	0.2309	1.2954
15	2.20	5.2448	5.8353	5.8331	0.5112	2.0298	1.3067	1.2258
16	4.20	0.9073	1.1214	1.1440	4.9453		0.0524	0.0009
17	2.20	5.6949	2.8750	2.8785	5.0891		4.1416	4.6132
18	4.20	1.2007			0.8796		0.8481	0.8781
19	2.20	2.8381			4.9614		5.8146	5.7835
20	4.20				3.5863		1.1410	1.1834
21	2.20				2.0117		2.9159	2.8592
air	1.00	–	–	–	–	–	–	–

∑(nt)		31.59	29.26	29.81	43.19	43.76	28.02	31.25

Merit Function		1.27	1.60	1.46	1.77	1.18	1.62	1.28

REFINEMENT METHOD	INITIAL DESIGN		ADAPT. RANDOM SEARCH		DAMPED LEAST SQUARES		MODIFIED GRADIENT				GOLDEN SECTION
REFINEMENT METHOD	INITIAL DESIGN		ADAPT. RANDOM SEARCH		DAMPED LEAST SQUARES		1		2		GOLDEN SECTION

Layer	nt	n	nt	n	nt	n	nt	n	nt	n	nt	n

Substrate	–	1.520	–	1.520	–	1.520	–	1.520	–	1.520	–	1.520
1	0.1375	2.150	0.2256	2.155	0.1321	1.491	0.1322	1.380	0.1387	1.350	0.2658	1.655
2	0.1375	1.550	0.1877	1.444	0.1435	1.546	0.1410	1.350	0.1408	1.540	0.2674	1.380
3	0.1375	2.150	0.2186	2.273	0.1376	1.865	0.1081	1.553	0.2832	2.350	0.1026	1.973
4	0.1375	1.550	0.1327	1.517	0.1702	2.261	0.1900	1.675	0.1421	1.350	0.0090	1.350
5	0.1375	2.150	0.2507	2.250	0.1286	1.973	0.1191	1.367	0.1409	2.350	0.1362	2.137
6	0.1375	1.550	0.0985	1.647	0.1920	1.390	0.1523	1.350			0.2432	1.588
7	0.1375	2.150	0.0952	2.169	0.0891	1.395	0.1103	2.001			0.2283	2.278
8	0.1375	1.550	0.1386	1.512	0.1410	2.187	0.1780	2.307			0.2365	1.394
9	0.1375	2.150	0.1489	2.240	0.1417	2.350	0.1481	1.350			0.1500	2.320
10	0.1375	1.550	0.1589	1.525	0.1424	1.350	0.1271	2.350			0.1167	1.360
11	0.1375	2.150	0.0352	2.158	0.1410	2.350					0.1211	1.996
air	–	1.000	–	1.000	–	1.000	–	1.000	–	1.000	–	1.000

∑(nt)	1.51		1.69		1.56		1.41		0.85		1.88

Merit Function	36.62		2.57		0.28		1.47		2.91		0.97

REFINEMENT METHOD	HOOKE & JEEVES		POWELL’S CONJUGATE SEARCH		ROSENBROCK		GENERALIZED SIMULATED ANNEALING		MONTE CARLO SIMULATED ANNEALING		SIMPLEX

Layer	nt	n	nt	n	nt	n	nt	n	nt	n	nt	n

Substrate	–	1.520	–	1.520	–	1.520	–	1.520	–	1.520	–	1.520
1	0.1198	1.725	0.2474	1.578	0.1268	1.667	0.1039	2.096	0.3127	1.972	0.0322	1.996
2	0.2364	2.053	0.2344	1.448	0.1976	1.891	0.0997	1.539	0.0692	1.388	0.2318	1.410
3	0.1466	1.856	0.0786	1.624	0.3378	1.927	0.1004	2.269	0.0494	2.264	0.1829	1.482
4	0.2587	1.679	0.3272	1.737	0.0394	1.647	0.1195	1.460	0.4153	1.646	0.2385	2.279
5	0.2338	2.319	0.2422	2.133	0.2579	2.128	0.4069	2.305	0.0304	2.265	0.0268	1.620
6	0.2074	1.350	0.2325	1.471	0.2069	1.390	0.1901	1.429	0.1536	1.470	0.1598	2.245
7	0.2168	2.350	0.2150	2.270	0.2066	2.350	0.1632	2.135	0.2502	2.348	0.0880	1.403
8	0.0970	1.350	0.2230	1.366	0.1465	1.350	0.1940	1.440	0.1822	1.440	0.1509	2.350
9	0.1310	2.303	0.1108	2.350	0.1115	2.322	0.1183	2.069	0.1348	2.323	0.1611	1.477
10	0.1218	1.499	0.1183	1.350	0.1420	1.525	0.1021	1.606	0.1081	1.462	0.2477	2.299
11	0.1070	1.592	0.1187	1.899	0.0768	1.816	0.0003	2.177	0.4101	2.059
air	–	1.000	–	1.000	–	1.000	–	1.000	–	1.000	–	1.000

∑(nt)	1.88		2.15		1.85		1.60		2.12		1.52

Merit Function	0.37		0.59		0.50		3.69		3.08		3.81

REFINEMENT METHOD		INITIAL DESIGN	ADAPT. RANDOM SEARCH	DAMPED LEAST SQUARES	MODIFIED GRADIENT	GOLDEN SECTION	HOOKE & JEEVES

Layer	n	t	t	t	t	t	t

Substrate	Ni	–	–	–	–	–	–
1	4.00	0.6000	0.6871	0.2839	0.6748	0.1483	0.2957
2	Ni	0.0300	0.0183	0.0145	0.0156	0.0121	0.0208
3	4.00	0.6000	0.3564	0.7611	0.7204	0.1596	0.7706
4	Ni	0.0300	0.0098	0.0054	0.0075	0.0070	0.0150
5	4.00	0.6000	0.7121	0.7409	0.7656	0.1612	0.2968
6	Ni	0.0300	0.0068	0.0048	0.0057	0.0147	0.0099
7	4.00	0.6000	0.7273	0.7125	0.7246	0.2674	0.7361
8	Ni	0.0300	0.0064	0.0054	0.0056	0.0102	0.0060
9	2.35	0.6000	0.3714	0.3655	0.3659	0.3665	0.3676
air	1.00	–	–	–	–	–	–

∑(t)		3.12	2.90	2.89	3.29	1.15	2.52

Merit Function		77.50	6.82	4.59	7.50	2.54	2.86

REFINEMENT METHOD		INITIAL DESIGN	POWELL CONJUG. SEARCH	ROSENBROCK	GENERAL. SIMUL. ANNEAL.	MONTE CARLO S. A.	SIMPLEX

Layer	n	t	t	t	t	t	t

Substrate	Ni	–	–	–	–	–	–
1	4.00	0.6000	0.6969	0.2846	1.0880	0.5331	0.2636
2	Ni	0.0300	0.0295	0.0145	0.5206	0.1280	0.0033
3	4.00	0.6000	0.3011	0.7589	0.7936	0.7010	0.7851
4	Ni	0.0300	0.0114	0.0058	0.2533	0.0280	0.0106
5	4.00	0.6000	0.7658	0.7419	0.3113	0.3009	0.7411
6	Ni	0.0300	0.0055	0.0050	0.0267	0.0115	0.0076
7	4.00	0.6000	0.7188	0.7133	0.3008	0.7436	0.7228
8	Ni	0.0300	0.0057	0.0056	0.0120	0.0063	0.0064
9	2.35	0.6000	0.3671	0.3647	0.3832	0.3661	0.3682
air	1.00	–	–	–	–	–	–

∑(t)		3.12	2.90	2.89	3.69	2.82	2.91

Merit Function		77.50	3.93	4.56	3.77	4.68	9.07

PROBLEM	B		A		A		A

SYSTEM	A - DSLQ + H&J		B - MCSA + DSLQ		C - DSLQ + H&J		D - DSLQ

Layer	nt	n	nt	n	nt	n	nt	n

Substrate	–	1.5200	–	4.00	–	4.00	–	4.00
1	0.1299	1.4644	0.5552	2.20	0.5021	2.20	0.4607	2.20
2	0.1603	1.4163	1.3897	4.20	1.0940	4.20	0.7000	4.20
3	0.1375	1.6043	0.9552	2.20	2.0346	2.20	1.0067	2.20
4	0.1540	2.0459	5.4688	4.20	0.6214	4.20	0.0999	4.20
5	0.1286	2.0891	2.3601	2.20	1.3537	2.20	3.3174	2.20
6	0.1735	1.6078	0.4590	4.20	3.3368	4.20	1.0824	4.20
7	0.1057	1.5368	1.3621	2.20	0.5572	2.20	0.6214	2.20
8	0.1632	2.2468	3.3825	4.20	0.9641	4.20	1.7496	4.20
9	0.1215	2.3500	0.3787	2.20	2.9986	2.20	0.8576	2.20
10	0.1430	1.3500	1.6333	4.20	0.8611	4.20	0.7991	4.20
11	0.1425	2.3500	0.2579	2.20	1.5230	2.20	1.0683	2.20
12			4.1197	4.20	0.8934	4.20	4.7904	4.20
13			5.2659	2.20	2.6623	2.20	3.0578	2.20
14			0.9635	4.20	1.3413	4.20	0.6164	4.20
15			5.6640	2.20	0.4930	2.20	0.6831	2.20
16			1.2585	4.20	2.5477	4.20	2.8713	4.20
17			2.8099	2.20	3.1304	2.20	5.1887	2.20
18					0.3902	4.20	1.0345	4.20
19					1.0866	2.20	5.5899	2.20
20					2.4895	4.20	1.2970	4.20
21					2.3987	2.20	2.7811	2.20
air	–	1.0000	–	1.00	–	1.00	–	1.00

∑(nt)	1.56		38.28		33.28		39.67

Merit Function	0.58		1.08		0.64		1.00

Abstract

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Figures (7)

Tables (5)

Equations (20)

Applied Optics