Abstract

Starting with the A B C B A multilayer design of Thelen, a computer-aided method is presented to replace the three-material by a two-material design. The method, which utilizes an optimized equivalent layer design, is illustrated by the two commonly used coating materials SiO2 and TiO2.

© 1987 Optical Society of America

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References

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  1. A. Thelen, “Multilayer Filters with Wide Transmission Bands,” J. Opt. Soc. Am. 53, 1266 (1963).
    [CrossRef]
  2. A. Thelen, “Wide-Band Multilayer Filters, II,” J. Opt. Soc. Am. 63, 65 (1973).
    [CrossRef]
  3. M. C. Ohmer, “Design of Three-Layer Equivalent Films,” J. Opt. Soc. Am. 68, 137 (1978).
    [CrossRef]
  4. J. A. Nelder, R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7, 308 (1965).
    [CrossRef]

1978 (1)

1973 (1)

1965 (1)

J. A. Nelder, R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7, 308 (1965).
[CrossRef]

1963 (1)

Mead, R.

J. A. Nelder, R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7, 308 (1965).
[CrossRef]

Nelder, J. A.

J. A. Nelder, R. Mead, “A Simplex Method for Function Minimization,” Comput. J. 7, 308 (1965).
[CrossRef]

Ohmer, M. C.

Thelen, A.

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Figures (11)

Fig. 1
Fig. 1

Calculated transmittance of a quarterwave stack of SiO2/TiO2 at a design wavelength of 1500 nm. Design: S/(0.5L 1H0.5L)10/M, with nLdL = nHdH = (1/4) 1500 nm, nS = 1.52, nL = 1.46, nH = 2.3, nM = 1.0.

Fig. 2
Fig. 2

Calculated transmittance of the design S/(A B C B A)10/M, with nAdA = nBdB = nCdC = (1/4) 600 nm, nS = 1.52, nA = 1.46, nB = 1.94, nC = 2.3, nM = 1.0.

Fig. 3
Fig. 3

Calculated transmittances of the building blocks: dotted line, original design S/A B C B A/M; solid line, equivalent layer design 1 S/1.277 A 0.415 C 0.277 A 1.000 C 0.277 A 0.415 C 1.277 A/M; dash–dot line, equivalent layer design 1 after refinement S/1.572 A 0.415 C 0.302 A 0.933 C 0.303 A 0.408 C 1.277 A/M, with nAdA = nBdB = nCdC = (1/4) 600 nm, nA = 1.46, nB = 1.94, nC = 2.3, nS = 1.52, nM = 1.0.

Fig. 4
Fig. 4

Calculated transmittances of the building blocks: dotted line, original design S/A B C B A/M; solid line, equivalent layer design 2 S/1.000 A 0.368 C 0.240 A 1.737 C 0.240 A 0.368 C 1.000 A/M; dash–dot line, equivalent layer design 2 after refinement S/1.035 A 0.348 C 0.233 A 1.640 C 0.235 A 0.345 C 1.085 A/M, with nAdA = nBdB = nCdC = (1/4) 600 nm, nA = 1.46, nB = 1.94, nC = 2.3, nS = 1.52, nM = 1.0.

Fig. 5
Fig. 5

Calculated transmittances of the building blocks: dotted line, original design S/A B C B A/M; solid line, equivalent layer system 3 S/1.277 A 0.415 C 0.277 A 1.368 C 0.240 A 0.368 C 1.000 A/ M; dash–dot line, equivalent layer design 3 after refinement S/1.348 A 0.377 C 0.288 A 1.480 C 0.235 A 0.368 C 1.112 A/M, with nAdA = nBdB = nCdC = (1/4) 600 nm, nA = 1.46, nB = 1.94, nC = 2.3, nS = 1.52, nM = 1.0.

Fig. 6
Fig. 6

Calculated transmittances of the building blocks: dotted line, original design S/A B C B A/M, solid line, equivalent layer system 4 S/1.000 A 0.368 C 0.240 A 1.368 C 0.277 A 0.415 C 1.277 A/ M; dash–dot line, equivalent layer design 4 after refinement S/1.038 A 0.358 C 0.235 A 1.382 C 0.203 A 0.402 C 1.210 A/M, with nAdA = nBdB = nCdC = (1/4) 600 nm, nA = 1.46, nB = 1.94, nC = 2.3, nS = 1.52, nM = 1.0.

Fig. 7
Fig. 7

Calculated filter curves with building block 1 before and after refinement: solid line, unrefined system S/(1.277 A 0.415 C 0.277 A 1.000 C 0.277 A 0.415 C 1.277 A)10/M, dashed line, refined system S/(1.572 A 0.415 C 0.302 A 0.933 C 0.303 A 0.408 C 1.277 A)10/ M, with nAdA = nCdC = (1/4) 600 nm, nA = 1.46, nC = 2.3, nS = 1.52, nM = 1.0.

Fig. 8
Fig. 8

Calculated filter curves with building block 2 before and after refinement: solid line, unrefined system S/(1.000 A 0.368 C 0.240 A 1.737 C 0.240 A 0.368 C 1.000 A)10/M; dashed line, refined system S/(1.035 A 0.348 C 0.233 A 1.640 C 0.235 A 0.345 C 1.085 A)10/ M, with nAdA = nCdC = (1/4) 600 nm, nA = 1.46, nC = 2.3, nS = 1.52, nM = 1.0.

Fig. 9
Fig. 9

Calculated filter curves with building block 3 before and after refinement: solid line, unrefined system S/(1.277 A 0.415 C 0.277 A 1.368 C 0.240 A 0.368 C 1.000 A)10/M; dashed line, refined system S/(1.348 A 0.377 C 0.228 A 1.480 C 0.235 A 0.368 C 1.112 A)10/ M, with nAdA = nCdC = (1/4) 600 nm, nA = 1.46, nC = 2.3, nS = 1.52, nM = 1.0.

Fig. 10
Fig. 10

Calculated filter curves with building block 4 before and after refinement: solid line, unrefined system S/(1.000 A 0.368 C 0.240 A 1.368 C 0.277 A 0.415 C 1.277 A)10/M; dashed line, refined system S/(1.038 A 0.358 C 0.235 A 1.382 C 0.203 A 0.402 C 1.210 A)10/ M, with nAdA = nCdC = (1/4) 600 nm, nA = 1.46, nC = 2.3, nS = 1.52, nM = 1.0.

Fig. 11
Fig. 11

Calculated filter curves of the original Thelen design and the optimized equivalent layer system: solid line, original design S/ (A B C B A)10/M; dotted line, equivalent layer design with optimized building block 2 S/(1.035 A 0.348 C 0.233 A 1.640 C 0.235 A 0.345 C 1.085 C)10/M, with nAdA = nBdB = nCdC = (1/4) 600 nm, nA = 1.46, nB = 1.94, nC = 2.3, nS = 1.52, nM = 1.0.

Tables (1)

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Table I Optical Thickness of the Building Block Layers 1–4 Before and After Optimization by Automatic Refinement Using the Simplex Method; Design Wavelength is 600 nm

Equations (6)

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y 2 ( 1 + x ) ( 1 + x 2 ) y 2 x + x = 0 ,
Φ q = sin 1 ( n p / N N / n p n p / n q n q / n p sin μ ) , μ = 2 π N D / λ 0 , λ 0 is the design wavelength ,
Φ p = 0.5 [ sin 1 ( c r ) tan 1 ( b a ) ] ,
a = 0.5 ( n q / n p + n p / n q ) sin Φ q b = cos Φ q , c = cos μ , r = a cos ( tan 1 b a ) .
p q p stack : 0.277  A 0.415  C 0.277  A ,
q p q stack : 0.368  C 0.240  A 0.368  C .

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