Abstract

New optical coating design algorithm with the equivalent layers theory is presented. The algorithm is based on the merit-function-constrained optimization in the accessible domain of equivalent phase thicknesses and equivalent refractive indices. It allows for creation of design coatings with sophisticated narrowband spectral characteristics.

© 2006 Optical Society of America

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References

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  1. N. Kaiser and H. K. Pulker, Optical Interference Coatings (Springer-Verlag, Berlin, 2003).
  2. A. V. Tikhonravov, M. K. Trubetskov, and G. DeBell, 'Application of the needle optimization technique to the design of optical coatings,' Appl. Opt. 35, 5493-5508 (1996).
    [CrossRef] [PubMed]
  3. B. T. Sullivan and J. A. Dobrowolski, 'Implementation of a numerical needle method for thin-film design,' Appl. Opt. 35, 5484-5492 (1996).
    [CrossRef] [PubMed]
  4. A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, 'Key role of the coating total optical thickness in solving design problems,' in Advances in Optical Film, C. Amra, N. Kaiser, and H. A. Macleod, eds., Proc. SPIE 5250, 312-321 (2003).
    [CrossRef]
  5. L. I. Epstein, 'The design of optical filters,' J. Opt. Soc. Am. 42, 806-810 (1952).
    [CrossRef]
  6. A. Thelen, 'Equivalent layers in multilayer filters,' J. Opt. Soc. Am. 56, 1533-1538 (1966).
    [CrossRef]
  7. M. C. Ohmer, 'Design of three-layer equivalent films,' J. Opt. Soc. Am. 68, 137-139 (1978).
    [CrossRef]
  8. H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, New York, 1986).
    [CrossRef]
  9. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1988).
  10. A. Herpin, 'Calcul du pouvoir reflecteur d'un systeme stratifie quelconque,' C. R. Acad. Sci. 225, 182-183 (1947).
  11. J. A. Dobrowolski and S. H. C. Piotrowski, 'Refractive index as a variable in the numerical design of optical thin film systems,' Appl. Opt. 21, 1502-1511 (1982).
    [CrossRef] [PubMed]
  12. J. A. Dobrowolski, 'Comparison of the Fourier transform and flip-flop thin-film synthesis methods,' Appl. Opt. 25, 1966-1972 (1986).
    [CrossRef] [PubMed]
  13. J. Kruschwitz, 'Software tools speed optical thin-film design,' Laser Focus World 39, 157-166 (2003).
  14. S. Furman and A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Edition Frontieres, Gif-sur-Yvette, 1992).
  15. C. J. van der Laan and H. J. Frankena, 'Equivalent layers: another way to look at them,' Appl. Opt. 34, 681-687 (1995).
    [CrossRef] [PubMed]
  16. D. M. Himmelblau, Applied Nonlinear Programming (McGraw-Hill, New York, 1972).
  17. M. Tilsch, C. A. Hulse, K. D. Hendrix, and R. B. Sargent, 'Design and demonstration of a thin-film based gain equalization filter for C-band EDFAs,' in Proceedings of the National Fiber Optics Engineering Conference (NFOEC), D. Thorp, J. Petitt, D. Klemisch, F. Kapron, and J. Varachi, eds. (Telcordia Technologies, Chicago, 1999), Vol. II, pp. 390-395.
  18. A. Thelen, A. V. Tikhonravov, M. K. Trubetskov, M. Tilsch, and U. Brauneck, 'Topical Meeting on Optical Interference Coatings (OIC '2001): design contest results,' Appl. Opt. 41, 3022-3038 (2002).
    [CrossRef] [PubMed]

2003 (2)

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, 'Key role of the coating total optical thickness in solving design problems,' in Advances in Optical Film, C. Amra, N. Kaiser, and H. A. Macleod, eds., Proc. SPIE 5250, 312-321 (2003).
[CrossRef]

J. Kruschwitz, 'Software tools speed optical thin-film design,' Laser Focus World 39, 157-166 (2003).

2002 (1)

1996 (2)

1995 (1)

1986 (1)

1982 (1)

1978 (1)

1966 (1)

1952 (1)

1947 (1)

A. Herpin, 'Calcul du pouvoir reflecteur d'un systeme stratifie quelconque,' C. R. Acad. Sci. 225, 182-183 (1947).

Amotchkina, T. V.

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, 'Key role of the coating total optical thickness in solving design problems,' in Advances in Optical Film, C. Amra, N. Kaiser, and H. A. Macleod, eds., Proc. SPIE 5250, 312-321 (2003).
[CrossRef]

Brauneck, U.

DeBell, G.

Dobrowolski, J. A.

Epstein, L. I.

Frankena, H. J.

Furman, S.

S. Furman and A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Edition Frontieres, Gif-sur-Yvette, 1992).

Hendrix, K. D.

M. Tilsch, C. A. Hulse, K. D. Hendrix, and R. B. Sargent, 'Design and demonstration of a thin-film based gain equalization filter for C-band EDFAs,' in Proceedings of the National Fiber Optics Engineering Conference (NFOEC), D. Thorp, J. Petitt, D. Klemisch, F. Kapron, and J. Varachi, eds. (Telcordia Technologies, Chicago, 1999), Vol. II, pp. 390-395.

Herpin, A.

A. Herpin, 'Calcul du pouvoir reflecteur d'un systeme stratifie quelconque,' C. R. Acad. Sci. 225, 182-183 (1947).

Himmelblau, D. M.

D. M. Himmelblau, Applied Nonlinear Programming (McGraw-Hill, New York, 1972).

Hulse, C. A.

M. Tilsch, C. A. Hulse, K. D. Hendrix, and R. B. Sargent, 'Design and demonstration of a thin-film based gain equalization filter for C-band EDFAs,' in Proceedings of the National Fiber Optics Engineering Conference (NFOEC), D. Thorp, J. Petitt, D. Klemisch, F. Kapron, and J. Varachi, eds. (Telcordia Technologies, Chicago, 1999), Vol. II, pp. 390-395.

Kaiser, N.

N. Kaiser and H. K. Pulker, Optical Interference Coatings (Springer-Verlag, Berlin, 2003).

Kokarev, M. A.

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, 'Key role of the coating total optical thickness in solving design problems,' in Advances in Optical Film, C. Amra, N. Kaiser, and H. A. Macleod, eds., Proc. SPIE 5250, 312-321 (2003).
[CrossRef]

Kruschwitz, J.

J. Kruschwitz, 'Software tools speed optical thin-film design,' Laser Focus World 39, 157-166 (2003).

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, New York, 1986).
[CrossRef]

Ohmer, M. C.

Piotrowski, S. H. C.

Pulker, H. K.

N. Kaiser and H. K. Pulker, Optical Interference Coatings (Springer-Verlag, Berlin, 2003).

Sargent, R. B.

M. Tilsch, C. A. Hulse, K. D. Hendrix, and R. B. Sargent, 'Design and demonstration of a thin-film based gain equalization filter for C-band EDFAs,' in Proceedings of the National Fiber Optics Engineering Conference (NFOEC), D. Thorp, J. Petitt, D. Klemisch, F. Kapron, and J. Varachi, eds. (Telcordia Technologies, Chicago, 1999), Vol. II, pp. 390-395.

Sullivan, B. T.

Thelen, A.

Tikhonravov, A. V.

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, 'Key role of the coating total optical thickness in solving design problems,' in Advances in Optical Film, C. Amra, N. Kaiser, and H. A. Macleod, eds., Proc. SPIE 5250, 312-321 (2003).
[CrossRef]

A. Thelen, A. V. Tikhonravov, M. K. Trubetskov, M. Tilsch, and U. Brauneck, 'Topical Meeting on Optical Interference Coatings (OIC '2001): design contest results,' Appl. Opt. 41, 3022-3038 (2002).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and G. DeBell, 'Application of the needle optimization technique to the design of optical coatings,' Appl. Opt. 35, 5493-5508 (1996).
[CrossRef] [PubMed]

S. Furman and A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Edition Frontieres, Gif-sur-Yvette, 1992).

Tilsch, M.

A. Thelen, A. V. Tikhonravov, M. K. Trubetskov, M. Tilsch, and U. Brauneck, 'Topical Meeting on Optical Interference Coatings (OIC '2001): design contest results,' Appl. Opt. 41, 3022-3038 (2002).
[CrossRef] [PubMed]

M. Tilsch, C. A. Hulse, K. D. Hendrix, and R. B. Sargent, 'Design and demonstration of a thin-film based gain equalization filter for C-band EDFAs,' in Proceedings of the National Fiber Optics Engineering Conference (NFOEC), D. Thorp, J. Petitt, D. Klemisch, F. Kapron, and J. Varachi, eds. (Telcordia Technologies, Chicago, 1999), Vol. II, pp. 390-395.

Trubetskov, M. K.

van der Laan, C. J.

Appl. Opt. (6)

C. R. Acad. Sci. (1)

A. Herpin, 'Calcul du pouvoir reflecteur d'un systeme stratifie quelconque,' C. R. Acad. Sci. 225, 182-183 (1947).

J. Opt. Soc. Am. (3)

Laser Focus World (1)

J. Kruschwitz, 'Software tools speed optical thin-film design,' Laser Focus World 39, 157-166 (2003).

Proc. SPIE (1)

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, 'Key role of the coating total optical thickness in solving design problems,' in Advances in Optical Film, C. Amra, N. Kaiser, and H. A. Macleod, eds., Proc. SPIE 5250, 312-321 (2003).
[CrossRef]

Other (6)

H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, New York, 1986).
[CrossRef]

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1988).

N. Kaiser and H. K. Pulker, Optical Interference Coatings (Springer-Verlag, Berlin, 2003).

S. Furman and A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Edition Frontieres, Gif-sur-Yvette, 1992).

D. M. Himmelblau, Applied Nonlinear Programming (McGraw-Hill, New York, 1972).

M. Tilsch, C. A. Hulse, K. D. Hendrix, and R. B. Sargent, 'Design and demonstration of a thin-film based gain equalization filter for C-band EDFAs,' in Proceedings of the National Fiber Optics Engineering Conference (NFOEC), D. Thorp, J. Petitt, D. Klemisch, F. Kapron, and J. Varachi, eds. (Telcordia Technologies, Chicago, 1999), Vol. II, pp. 390-395.

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Figures (7)

Fig. 1
Fig. 1

Symmetrical combination of three homogeneous nonabsorbing layers: n 1 and n 2 are layer refractive indices, d 1 and 2 d 2 are layer physical thicknesses, φ 1 and 2 φ 2 are layer phase thicknesses.

Fig. 2
Fig. 2

Curves of constant u values in the admissible domain of phase thicknesses φ 1 , φ 2 calculated with Eq. (8) for five values of the parameter u: u = 0 , u = ± 1 / 2 , u = ± 1 . Bold black curves correspond to the boundaries of the admissible domain. Layer refractive indices are n 1 = 2.35 , n 2 = 1.45 .

Fig. 3
Fig. 3

Curves of constant v values in the admissible domain of phase thicknesses φ 1 , φ 2 calculated with Eq. (9). Bold curves indicate the boundaries of the admissible domain. The admissible domain is subdivided into 24 regions marked by the single numbers (5, 6, 7, 8) and by double numbers (1.1 and so on) (see the text for details). Layer refractive indices are n 1 = 2.35 and n 2 = 1.45 .

Fig. 4
Fig. 4

Accessible domain is marked as a gray area: vertical stripes marked by letters C 1 , C 2 , C 3 , C 4 and divided by vertical dashed lines contain the images of the regions marked by letters A 1 , A 2 , A 3 , A 4 in Fig. 2. Accessible domain is subdivided into 14 regions marked by Roman numerals for the discussion of mapping properties (see the text for details). Layer refractive indices are n 1 = 2.35 and n 2 = 1.45 .

Fig. 5
Fig. 5

Schematic of the design algorithm.

Fig. 6
Fig. 6

Replacing an optimized prototype design by its two-component original: d 1 ( 1 ) , d 2 ( 1 ) , , d 1 ( m ) , d 2 ( m ) are thicknesses of layers of three-layer combinations representing m equivalent layers with parameters ( Φ 1 , N 1 ) , , ( Φ m , N m ) .

Fig. 7
Fig. 7

Solution of the OIC'2001 design contest problem: target reflectance (crosses), reflectance of the optimized prototype design (dashed curve), reflectance of the two-component original of the prototype design (dotted curve), reflectance of the refined two-component design (solid curve).

Equations (50)

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M = [ cos 2 φ 1 cos 2 φ 2 - p sin 2 φ 1 sin 2 φ 2 i n 1 ( sin 2 φ 1  cos  2 φ 2 + p  cos  2 φ 1 sin 2 φ 2 + q sin 2 φ 2 ) i n 1 ( sin 2 φ 1  cos  2 φ 2 + p  cos  2 φ 1  sin  2 φ 2 - q  sin  2 φ 2 cos 2 φ 1 cos 2 φ 2 - p  sin  2 φ 1 sin 2 φ 2 ] ,
p = 1 2 ( n 1 n 2 + n 2 n 1 ) , q = 1 2 ( n 1 n 2 - n 2 n 1 ) ,
φ 1 = 2 π λ ( n 1 d 1 ) , φ 2 = 2 π λ ( n 2 d 2 )
M e = [ cos Φ i N sin Φ i N sin Φ cos Φ ] .
cos Φ = cos 2 φ 1 cos 2 φ 2 - p  sin  2 φ 1  sin  2 φ 2 ,
N = n 1 sin 2 φ 1 cos 2 φ 2 + p cos 2 φ 1  sin  2 φ 2 - q  sin  2 φ 2 sin 2 φ 1  cos  2 φ 2 + p  cos  2 φ 1  sin  2 φ 2 + q  sin  2 φ 2 .
| cos 2 φ 1  cos  2 φ 2 - p  sin  2 φ 1  sin  2 φ 2 | 1 ,
sin 2 φ 1  cos  2 φ 2 + p  cos  2 φ 1  sin  2 φ 2 - q  sin  2 φ 2 sin 2 φ 1  cos  2 φ 2 + p  cos  2 φ 1  sin  2 φ 2 + q  sin  2 φ 2 0.
cos 2 φ 1  cos  2 φ 2 - p  sin  2 φ 1  sin  2 φ 2 = u ,
n 1 sin 2 φ 1 cos 2 φ 2 + p cos 2 φ 1  sin  2 φ 2 - q  sin  2 φ 2 sin 2 φ 1  cos  2 φ 2 + p  cos  2 φ 1  sin  2 φ 2 + q  sin  2 φ 2 = v ,
N u ( Φ ) = { n 1 , if   Φ [ 0 , π ] [ 3 π , 4 π ] n 1 p 2 - cos 2 Φ + q p 2 - cos 2 Φ - q , if   Φ [ π , 3 π ] ,
N l ( Φ ) = { n 2 , if   Φ [ 0 , π / 2 ] [ π , 3 π / 2 ] [ 5 π / 2 , 3 π ] [ 7 π / 2 , 4 π ] , n 1 p 2 - cos 2 Φ - q p 2 - cos 2 Φ + q , if   Φ [ π / 2 , π ] [ 3 π / 2 , 5 π / 2 ] [ 3 π , 3 π / 2 ] , n 1 > n 2 .
Y = { Φ 1 , N 1 , , Φ m , N m } .
F e ( Y ) = 1 L j = 1 L [ R ( Y , λ j ) - R ˜ ( λ j ) Δ R j ] 2 ,
P ( Y ) = i = 1 m [ ( max { N i - N u ( Φ i ) , 0 } ) 2 + ( max { N l ( Φ i ) - N i , 0 } ) 2 ] ,
F ( X ) = 1 L j = 1 L [ R ( X , λ j ) - R ˜ ( λ j ) Δ R j ] 2
a cos 2 φ 2 + b sin 2 φ 2 = c ,
a = a ( φ 1 ) = cos 2 φ 1 cos 2 2 φ 1 + p 2 sin 2 φ 1 ,
b = b ( φ 1 ) = p sin 2 φ 1 cos 2 2 φ 1 + p 2 sin 2 φ 1 ,
c = c ( φ 1 ) = u cos 2 2 φ 1 + p 2 sin 2 φ 1 .
( φ 2 ) 1 = arccos c - arccos a 2 + π , ( φ 2 ) 2 = ( φ 2 ) 1 - π 2 , ( φ 2 ) 3 = - arccos c - arccos a 2 + π , ( φ 2 ) 4 = ( φ 2 ) 3 - π 2 , for   φ 1 [ 0 , π 2 ] , ( φ 2 ) 5 = - arccos ( - c ) - arccos ( - a ) 2 + π , ( φ 2 ) 6 = ( φ 2 ) 5 + π 2 , ( φ 2 ) 7 = arccos ( - c ) - arccos ( - a ) 2 , ( φ 2 ) 8 = ( φ 2 ) 7 + π 2 , for   φ 1 [ π 2 , π ] .
a cos 2 φ 2 + b sin 2 φ 2 = 0 ,
a = a ( φ 1 ) = ( n 1 2 - v 2 ) sin 2 φ 1 sin 2 2 φ 1 ( n 1 2 - v 2 ) 2 + [ p cos 2 φ 1 ( n 1 2 - v 2 ) - q ( n 1 2 + v 2 ) ] 2 ,
b = b ( φ 1 ) = p ( n 1 2 - v 2 ) cos 2 φ 1 - q ( n 1 2 + v 2 ) sin 2 2 φ 1 ( n 1 2 - v 2 ) 2 + [ p cos 2 φ 1 ( n 1 2 - v 2 ) - q ( n 1 2 + v 2 ) ] 2
a = a ( φ 1 ) = sin 2 φ 1 sin 2 2 φ 1 + [ p cos 2 φ 1 - q ) ] 2 ,
b = b ( φ 1 ) = p cos 2 φ 1 + q sin 2 2 φ 1 + [ p cos 2 φ 1 - q ) ] 2
f 1 ( φ 1 ) = f 1 [ a ( φ 1 ) ] = arccos a 2 + π 4 ,
f 2 ( φ 1 ) = f 2 [ a ( φ 1 ) ] = - arccos a 2 + π 4 .
φ 2 ( 1.1 ) = f 1 ( φ 1 ) , φ 2 ( 2.1 ) = f 1 ( φ 1 ) + π 2 ,
φ 1 [ 0 , Ψ v ] , φ 2 ( 1.2 ) = f 2 ( φ 1 ) ,
φ 2 ( 2.2 ) = f 2 ( φ 1 ) + π 2 , φ 1 [ Ψ v , π 2 ] ,
φ 2 ( 3.1 ) = f 2 ( φ 1 ) + π 2 , φ 2 ( 4.1 ) = f 2 ( φ 1 ) + π ,
φ 1 [ π 2 , π - Ψ ] , φ 2 ( 3.2 ) = - f 2 ( φ 1 ) ,
φ 2 ( 4.2 ) = f 1 ( φ 1 ) , φ 1 [ π - Ψ v , π ] , for   0 v n 2 ,
φ 2 ( 5 ) = f 2 ( φ 1 ) , φ 2 ( 6 ) = f 2 ( φ 1 ) + π 2 ,
φ 1 [ 0 , π 2 ] , φ 2 ( 7 ) = f 2 ( φ 1 ) + π 2 ,
φ 2 ( 8 ) = f 2 ( φ 1 ) + π , φ 1 [ π 2 , π ] , for   n 2 < v n 1
φ 2 ( 9.1 ) = f 2 ( φ 1 ) + π 2 , φ 2 ( 10.1 ) = f 2 ( φ 1 ) + π ,
φ 1 [ 0 , π 2 ] , φ 2 ( 9.2 ) = f 2 ( φ 1 ) ,
φ 2 ( 10.2 ) = f 2 ( φ 1 ) + π 2 , φ 1 [ π 2 , π ] ,
for   n 1 v n 1 2 n 2 ;
φ 2 ( 11.1 ) = f 2 ( φ 1 ) + π 2 , φ 2 ( 12.1 ) = f 2 ( φ 1 ) + π ,
φ 1 [ 0 , Ψ v ] , φ 2 ( 11.2 ) = - f 2 ( φ 1 ) ,
φ 2 ( 12.2 ) = f 1 ( φ 1 ) , φ 1 [ Ψ v , π 2 ] ,
φ 2 ( 13.1 ) = f 1 ( φ 1 ) , φ 2 ( 14.1 ) = f 1 ( φ 1 ) + π 2 ,
φ 1 [ π 2 , π - Ψ v ] , φ 2 ( 13.2 ) = f 2 ( φ 1 ) ,
φ 2 ( 14.2 ) = f 2 ( φ 1 ) + π 2 , φ 1 [ π - Ψ v , π ]
for   n 1 2 n 2 < v + .
Ψ v = 1 2 arccos q ( n 1 2 + v 2 ) p ( n 1 2 - v 2 )   for   0 v < + ,
Ψ v = 1 2 arccos ( - q p )   for   v = + .

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