Design of three-layer antireflection coatings: a generalized approach

C. L. Nagendra; G. K. M. Thutupalli

doi:10.1364/AO.27.002320

Applied Optics
Vol. 27,
Issue 11,
pp. 2320-2333
(1988)
•https://doi.org/10.1364/AO.27.002320

Design of three-layer antireflection coatings: a generalized approach

C. L. Nagendra and G. K. M. Thutupalli

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C. L. Nagendra and G. K. M. Thutupalli, "Design of three-layer antireflection coatings: a generalized approach," Appl. Opt. 27, 2320-2333 (1988)

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Abstract

A generalized theoretical formulation for multilayer antireflection coatings (ARCs) has been established, which has scope for considering both zero and nonzero reflectance at one or more than one wavelength. Starting with this, a new design method for three-layer ARCs making use of input reflectance parameters at two wavelengths (R₁ and R₂ at λ₁ and λ₂) has been developed. It allows for determination of all the unknown design parameters explicitly without any a priori conditions on them. The scope of the method has been extended by incorporating optimization of the resultant solutions through an optimizing parameter, integrated reflection loss R* to obtain efficient designs with respect to spectral bandwidth, residual reflection loss, etc. The resulting hybrid methodology is found to be generalized in that it can be used to design a wide range of ARC systems, broad band to V-type, irrespective of the nature of the substrate (absorbing or nonabsorbing) and spectral region of use, with adaptability to simpler/complex ARC systems and other thin film systems such as dichroics/beam splitters and achromatic reflectors.

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	Methodology		Scope of the method

Sl. Design method	Principle/scheme of evaluation	Design parameters for evaluation	Merits	Demerits
Graphical:
(1) Graphical/Argand diagram method (Vasicek, Turner, Muchmore)	Vector diagram analysis of reflection coefficients	For known integral thickness ratios, refractive indices are evaluated or for known refractive indices, thicknesses subjected to integral ratio conditions are evaluated.		Constraints on both refractive indices and thicknesses. Method of evaluation is cumbersome and less efficient. Approximate reflection coefficient relations are used.
Analytical:
(2) Classical quarterwave method (Hass, Thun, Thelen)	Establishing ref. ind. relations using zero refl. condition, thicknesses set to m λ/4 (m = 1,2,3)	Determination of the refractive index of the layers for given n_s and n₀	Simple design method	Often mismatch of the theoretical refractive index values with those of practical values arises. Limited scope for developing efficient designs (restricted to zero reflection-based designs and to λ/4 system)
(3) Electrical filter design method (Young, Kucirkova)	Application of microwave network theory for optical admittance calculations	Evaluation of refractive indices for quarterwave thickness configurations		Design analysis is complex. Mismatch of theoretical refractive-index values with material-index values arises. Limited to only quarterwave systems
(4) Method of effective interfaces (Thetford, Baer)	Application of Fabry-Perot etalon theory to ARCs	Evaluation of thicknesses for given refractive-index configuration	Baer’s method offers analytical equations for evaluation of thicknesses	Thetford’s scheme is approximate and a trial and error approach. In Baer’s method, designs with zero base reflection can only be possible. Existence of dual set of solutions. Refractive-index limitations. Only two out of three thicknesses can be evaluated.
(5) Exact synthesis method I (Kard, Mouchart)	Establishing closed form equations from zero reflection condition to determine the thicknesses	Evaluation of thicknesses for given refractive-index configuration	Simple method of evaluation	Ambiguity of dual set of solutions. Only two out of the three thicknesses can be evaluated. Possible to obtain only zero reflection based designs. Existence of ref index limitations (Kard’s scheme applicable to only symmetrical systems)
(6) Exact synthesis method II—NTa (Nagendra, Thutupalli)	Formulation of suitable nonlinear equation for three-layer ARC from zero reflectance condition	Solving the nonlinear equation through Newton’s method to determine the thicknesses for a given combination of refractive indices	Method of evaluation is straightforward. No ambiguity of dual solutions	Similar to Mouchart’s method
(7) Exact synthesis method III—NTb (Nagendra, Viswanathan, Thutupalli)	Establishing a theoretical formulation applicable for both zero and nonzero base reflectance and evolution of suitable nonlinear equation for three-layer ARCs	Direct synthesis of the nonlinear equation to determine the thicknesses for a given combination of refractive indices	Scope for both zero and nonzero reflectance-based designs and optimization through IRL R. Flexibility over selection of materials. Provision for in situ* correction.	Only two out of the three thicknesses can be evaluated exactly. Applicable to ARCs on transparent substrates
Numerical:
(8) Linear programming method (Baumiester et al).	Linear programming optimization to ARCs	Refractive-index optimization for quarterwave configurations	Refractive-index values can be bounded and realizable values can be achieved	Method involves complex programming. Applicable to quarterwave systems only
(9) Refractive-index optimization method (Furman, Stolina)	Lagrange’s method of optimization applied to ARCs	For known thickness values, refractive indices are optimized.		Involves complex optimization routine. Often optimized values should be rounded off to the nearest material values. Thicknesses are fixed a priori.
(10) Method of least squares (Heavens, Liddel, Appa Rao)	Optimization of design variables using least-squares merit function	Optimization of thicknesses for given refractive indices	Scope for generating efficient designs	Mathematical analysis is complex. Needs extensive computation and a good starting design.
(11) Evolutionary design method (Dobrowolski)	Merit function optimization	Both refractive-index and thickness optimization	Does not need a starting design. Flexibility over choice of Merit function. Realizable index values can be achieved. Efficient designs can be generated.	Involves complex mathematical analysis and computation

Sl.	Refractive-index configuration	Normalized refractive-index ratio N = n₂n₃/(n₁n_s)	Upper and lower wavelength limits		$R_{G M}^{*} \times 10^{3}$	Design wavelength (λ₁)
Sl.	Refractive-index configuration	Normalized refractive-index ratio N = n₂n₃/(n₁n_s)	λ_U	λ_L	$R_{G M}^{*} \times 10^{3}$	Calculated value	Set value
(1)	1.5/4.0/1.5/4.0	1.000	18.10	10.80	7.30	14.40	15.0
(2)	1.5/4.0/1.5/3.45	1.159	18.00	10.75	6.60	14.65	15.0
(3)	1.44/4.0/1.44/2.56	1.563	12.68	7.75	3.53	10.70	10.60
(4)	1.44/4.0/1.44/4.0	1.000	12.50	7.20	2.75	9.900	10.60
(5)	1.44/2.3/2.7/2.56	1.685	12.50	7.10	3.25	10.489	10.60
(6)	1.38/2.56/1.38/2.56	1.000	4.65	2.35	6.48	3.50	3.8
(7)	1.38/2.30/3.45/4.0	1.440	6.20	2.75	12.20	4.786	5.0
(8)	1.38/2.10/1.88/1.52	1.882	0.720	0.420	25.57	0.616	0.6
(9)	1.38/1.45/2.10/1.80	1.226	0.710	0.530	19.05	0.629	0.6
(10)	1.38/1.54/1.75/1.62	1.204	0.7	0.475	10.68	0.598	0.6

	Methodology		Scope of the method

Sl. Design method	Principle/scheme of evaluation	Design parameters for evaluation	Merits	Demerits
Graphical:
(1) Graphical/Argand diagram method (Vasicek, Turner, Muchmore)	Vector diagram analysis of reflection coefficients	For known integral thickness ratios, refractive indices are evaluated or for known refractive indices, thicknesses subjected to integral ratio conditions are evaluated.		Constraints on both refractive indices and thicknesses. Method of evaluation is cumbersome and less efficient. Approximate reflection coefficient relations are used.
Analytical:
(2) Classical quarterwave method (Hass, Thun, Thelen)	Establishing ref. ind. relations using zero refl. condition, thicknesses set to m λ/4 (m = 1,2,3)	Determination of the refractive index of the layers for given n_s and n₀	Simple design method	Often mismatch of the theoretical refractive index values with those of practical values arises. Limited scope for developing efficient designs (restricted to zero reflection-based designs and to λ/4 system)
(3) Electrical filter design method (Young, Kucirkova)	Application of microwave network theory for optical admittance calculations	Evaluation of refractive indices for quarterwave thickness configurations		Design analysis is complex. Mismatch of theoretical refractive-index values with material-index values arises. Limited to only quarterwave systems
(4) Method of effective interfaces (Thetford, Baer)	Application of Fabry-Perot etalon theory to ARCs	Evaluation of thicknesses for given refractive-index configuration	Baer’s method offers analytical equations for evaluation of thicknesses	Thetford’s scheme is approximate and a trial and error approach. In Baer’s method, designs with zero base reflection can only be possible. Existence of dual set of solutions. Refractive-index limitations. Only two out of three thicknesses can be evaluated.
(5) Exact synthesis method I (Kard, Mouchart)	Establishing closed form equations from zero reflection condition to determine the thicknesses	Evaluation of thicknesses for given refractive-index configuration	Simple method of evaluation	Ambiguity of dual set of solutions. Only two out of the three thicknesses can be evaluated. Possible to obtain only zero reflection based designs. Existence of ref index limitations (Kard’s scheme applicable to only symmetrical systems)
(6) Exact synthesis method II—NTa (Nagendra, Thutupalli)	Formulation of suitable nonlinear equation for three-layer ARC from zero reflectance condition	Solving the nonlinear equation through Newton’s method to determine the thicknesses for a given combination of refractive indices	Method of evaluation is straightforward. No ambiguity of dual solutions	Similar to Mouchart’s method
(7) Exact synthesis method III—NTb (Nagendra, Viswanathan, Thutupalli)	Establishing a theoretical formulation applicable for both zero and nonzero base reflectance and evolution of suitable nonlinear equation for three-layer ARCs	Direct synthesis of the nonlinear equation to determine the thicknesses for a given combination of refractive indices	Scope for both zero and nonzero reflectance-based designs and optimization through IRL R. Flexibility over selection of materials. Provision for in situ* correction.	Only two out of the three thicknesses can be evaluated exactly. Applicable to ARCs on transparent substrates
Numerical:
(8) Linear programming method (Baumiester et al).	Linear programming optimization to ARCs	Refractive-index optimization for quarterwave configurations	Refractive-index values can be bounded and realizable values can be achieved	Method involves complex programming. Applicable to quarterwave systems only
(9) Refractive-index optimization method (Furman, Stolina)	Lagrange’s method of optimization applied to ARCs	For known thickness values, refractive indices are optimized.		Involves complex optimization routine. Often optimized values should be rounded off to the nearest material values. Thicknesses are fixed a priori.
(10) Method of least squares (Heavens, Liddel, Appa Rao)	Optimization of design variables using least-squares merit function	Optimization of thicknesses for given refractive indices	Scope for generating efficient designs	Mathematical analysis is complex. Needs extensive computation and a good starting design.
(11) Evolutionary design method (Dobrowolski)	Merit function optimization	Both refractive-index and thickness optimization	Does not need a starting design. Flexibility over choice of Merit function. Realizable index values can be achieved. Efficient designs can be generated.	Involves complex mathematical analysis and computation

Sl.	Refractive-index configuration	Normalized refractive-index ratio N = n₂n₃/(n₁n_s)	Upper and lower wavelength limits		$R_{G M}^{*} \times 10^{3}$	Design wavelength (λ₁)
Sl.	Refractive-index configuration	Normalized refractive-index ratio N = n₂n₃/(n₁n_s)	λ_U	λ_L	$R_{G M}^{*} \times 10^{3}$	Calculated value	Set value
(1)	1.5/4.0/1.5/4.0	1.000	18.10	10.80	7.30	14.40	15.0
(2)	1.5/4.0/1.5/3.45	1.159	18.00	10.75	6.60	14.65	15.0
(3)	1.44/4.0/1.44/2.56	1.563	12.68	7.75	3.53	10.70	10.60
(4)	1.44/4.0/1.44/4.0	1.000	12.50	7.20	2.75	9.900	10.60
(5)	1.44/2.3/2.7/2.56	1.685	12.50	7.10	3.25	10.489	10.60
(6)	1.38/2.56/1.38/2.56	1.000	4.65	2.35	6.48	3.50	3.8
(7)	1.38/2.30/3.45/4.0	1.440	6.20	2.75	12.20	4.786	5.0
(8)	1.38/2.10/1.88/1.52	1.882	0.720	0.420	25.57	0.616	0.6
(9)	1.38/1.45/2.10/1.80	1.226	0.710	0.530	19.05	0.629	0.6
(10)	1.38/1.54/1.75/1.62	1.204	0.7	0.475	10.68	0.598	0.6

Abstract

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Figures (13)

Tables (2)

Equations (28)

Applied Optics