Application of linear programming to antireflection coating design

Philip Baumeister; Robert Moore; Kenneth Walsh

doi:10.1364/JOSA.67.001039

Journal of the Optical Society of America
Vol. 67,
Issue 8,
pp. 1039-1045
(1977)
•https://doi.org/10.1364/JOSA.67.001039

Application of linear programming to antireflection coating design

Philip Baumeister, Robert Moore, and Kenneth Walsh

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Philip Baumeister, Robert Moore, and Kenneth Walsh, "Application of linear programming to antireflection coating design," J. Opt. Soc. Am. 67, 1039-1045 (1977)

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Abstract

The antireflection coatings consist of layers whose optical thicknesses are quater waves or multiples thereof. Linear programming is used to minimize the reflectance and at the same time restrain the refractive indices to values that are attainable. Examples are given of coatings for visual optics and for higher refractive index substrates such as lithium niobate.

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Equations (17)

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Coefficient	2	3
a₀	sinh u₀	sinh u₀
a₁	sinh (u₀ − 2u₁)	sinh (u₀ − 2u₁)
	+ sinh (u₀ − 2u₂)	+ sinh (u₀ − 2u₂)
		+sinh (u₀ − 2u₃)
a₂	sinh (u₀ − 2u₁+2u₂)	sinh (u₀ − 2u₁ + 2u₂)
		+ sinh (u₀ − 2u₁ + 2u₃)
		+ sinh (u₀ − 2u₂ + 2u₃)
a₃		sinh (u₀ − 2u₁ + 2u₂ − 2u₃)

b_mi
l	m	1	2	3	4	5	6	7	8
1	1	−2
2	1	−2	−2
2	2	−2	2
3	1	−2	−2	−2
3	2	−4	0	4
3	3	−2	2	−2
4	1	−2	−2	−2	−2
4	2	−6	−2	2	6
4	3	−6	2	2	−6
4	4	−2	2	−2	2
5	1	−2	−2	−2	−2	−2
5	2	−8	−4	0	4	8
5	3	−12	0	4	0	−12
5	4	−8	4	0	−4	8
5	5	−2	2	−2	2	−2
6	1	−2	−2	−2	−2	−2	−2
6	2	−10	−6	−2	2	6	10
6	3	−20	−4	4	4	−4	−20
6	4	−20	4	4	−4	−4	20
6	5	−10	6	−2	−2	6	−10
6	6	−2	2	−2	2	−2	2
7	1	−2	−2	−2	−2	−2	−2	−2
7	2	−12	−8	−4	0	4	8	12
7	8	−30	−10	2	6	2	−10	−30
7	4	−40	0	8	0	−8	0	40
7	5	−30	10	2	−6	2	10	−30
7	6	−12	8	−4	0	4	−8	12
7	7	−2	2	−2	2	−2	2	−2
8	1	−2	−2	−2	−2	−2	−2	−2	−2
8	2	−14	−10	−6	−2	2	6	10	14
8	3	−42	−18	−2	6	6	−2	−18	−42
8	4	−70	−10	10	6	−6	−10	10	70
8	5	−70	10	10	−6	−6	10	10	−70
8	6	−42	18	−2	−6	6	2	−18	42
8	7	−14	10	−6	2	2	−6	10	−14
8	8	−2	2	−2	2	−2	2	−2	2

Design	Condition R = 0	Condition for quasimaximally flat	Example of comment
QH	$n_{1} = n_{0}^{1 / 2} n_{s}^{1 / 2}$	$n_{2} = n_{0}^{- 1 / 4} n_{s}^{5 / 4}$	1.00/1.38/2.23/1.90
QQH	$n_{2} = n_{0}^{- 1 / 2} n_{1} n_{s}^{1 / 2}$	$n_{3} = n_{0}^{3 / 4} n_{0}^{- 1} n_{s}^{5 / 4}$	1.00/1.35/1. 86/1.65/1.90
QQQ	$n_{3} = n_{0}^{1 / 2} n_{2} n_{s}^{- 1} n_{s}^{1 / 2}$	$n_{1} = n_{0}^{7 / 8} n_{0}^{1 / 8}, n_{2} = n_{0}^{1 / 2} n_{s}^{1 / 2}$	no constraints
QQQ	$n_{3} = n_{0}^{1 / 2} n_{s}^{- 1} n_{2} n_{s}^{1 / 2}$	$n_{2} = n_{0}^{- 5 / 4} n_{1}^{2} n_{s}^{1 / 4}$	1.00/1.38/2.11/1.89/1.52
QHQ	$n_{3} = n_{0}^{- 1 / 2} n_{1} n_{s}^{1 / 2}$	$n_{2} = n_{0}^{- 7 / 4} n_{1}^{3} n_{s}^{- 1 / 4}$	1.00/1.38/2.37/1.70/1.52
QQQQ	$n_{3} = n_{0}^{1 / 2} n_{s}^{- 1} n_{2} n_{4} n_{s}^{- 1 / 2}$	$n_{1} = n_{0}^{α} n_{s}^{1 - α}, n_{2} = n_{0}^{γ} n_{s}^{1 - γ}, n_{4} = n_{0} n_{1}^{- 1} n_{s}$	no constraints
QQQQ	$n_{3} = n_{0}^{1 / 2} n_{s}^{- 1} n_{2} n_{4} n_{s}^{- 1 / 2}$	$n_{4} = n_{0}^{5 / 4} n_{1}^{- 2} n_{2} n_{s}^{3 / 4}$	1.00/1.38/2.30/2.24/1.65/1.52
QHQQ	$n_{4} = n_{0}^{1 / 2} n_{s}^{- 1} n_{3} n_{s}^{1 / 2}$	$n_{3} = n_{0}^{- 9 / 4} n_{1}^{4} n_{2}^{- 1} n_{s}^{1 / 4}$	1.00/1.38/2.30/1.75/1.56/1.52

Fig.	Design	n_s	n_i	λ₀/λ_i	W_i	B
1*	QQH	1.90	1.35 L
			1.86
			1.65
1	QH	1.90	1.38 L
			2.23
2*	Q⁵	1.52	1.38 L
			2.15	0.75	1
			2.25 U	0.87	1	1.86
			1.66	1.00	1
			1.41
2	Q⁵	1.52	1.38 L	0.72	1	2.0
			2.06	0.87	1.5
			2.25 U	1.00	1
			1.74
			1.45
3*	Q⁴	1.80	1.38 L	0.65	1	1.78
			2.07	0.75	1
			2.25 U	0.85	2
			1.93	1.00	1
3	Q⁵	1.80	1.38 L	0.65	1.5	1.91
			2.05	0.75	1
			2.25 U	0.85	1.5
			1.89	1.00	1
			1.74
4	Q⁷ = Q⁶,	1.80	1.38 L	0.65	1	1.80
			2.17	0.75	1
			2.25 U	0.85	2
			1.57	1.00	1.5
			1.38 L
			1.57
5*	QQ	2.25	1.38 L
			1.63
5	Q³	2.25	1.38 L	0.55	1
			1.72	0.67	1.25
			2.15 U	0.78	1
6*	Q⁴	2.25	1.38 L	0.55	1
			1.53	0.67	1.25
			2.15 U	0.78	1
			1.97
6	QQHQ	2.25	1.38 L	0.55	1
			1.46	0.67	1.7
			1.38 L	0.78	1
			1.68
7	Q⁶	2.25	1.38 L	0.55	1
			1.85	0.67	2.5
			2.15 U	0.78	1
			1.70
			1.65
			1.85
8	QQQHQ	2.25	1.38 L	0.55	1
			1.79	0.67	2.5
			2.03	0.78	1
			1.56
			1.79
9	QQQH	2.25	1.38 L	0.55	1
			1.59	0.67	10
			2.15 U	0.75	1
			1.59
			1.74

Coefficient	2	3
a₀	sinh u₀	sinh u₀
a₁	sinh (u₀ − 2u₁)	sinh (u₀ − 2u₁)
	+ sinh (u₀ − 2u₂)	+ sinh (u₀ − 2u₂)
		+sinh (u₀ − 2u₃)
a₂	sinh (u₀ − 2u₁+2u₂)	sinh (u₀ − 2u₁ + 2u₂)
		+ sinh (u₀ − 2u₁ + 2u₃)
		+ sinh (u₀ − 2u₂ + 2u₃)
a₃		sinh (u₀ − 2u₁ + 2u₂ − 2u₃)

Abstract

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Figures (9)

Tables (4)

Equations (17)

Journal of the Optical Society of America