Abstract

Although in theory the Fourier transform method is valid only for small rejections, in practice it can be modified for the synthesis of high rejection filters with minimum transmittances as low as 10−4. Two new spectral functions are proposed for use in the Fourier transforms. An empirical procedure which is much faster than refinement is described for optimization of the spectral performance. The method and optimization are illustrated numerically for several different spectral shapes.

© 1989 Optical Society of America

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References

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  1. E. Berkcan, G. H. Cohen, “A New CAD Technique for (Narrow Band) Inhomogeneous Thin Films Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 518, 98–104 (1984).
  2. J. A. Dobrowolski, “Subtractive Method of Optical Thin-Film Interference Filter Design,” Appl. Opt. 12, 1885–1893 (1973).
    [CrossRef] [PubMed]
  3. A. Thelen, “Design of Optical Minus Filters,” J. Opt. Soc. Am. 61, 365–369 (1971).
    [CrossRef]
  4. L. Young, “Multilayer Interference Coatings with Narrow Stop Bands,” Appl. Opt. 6, 297–315 (1967).
    [CrossRef] [PubMed]
  5. P. Baumeister, “Theory of Rejection Filters with Ultranarrow Bandwidths,” J. Opt. Soc. Am. 71, 604–608 (1981).
    [CrossRef]
  6. P. Baumeister, “Simulation of a Rugate Filter via a Stepped-Index Dielectric Multilayer,” Appl. Opt. 25, 2644–2645 (1986).
    [CrossRef] [PubMed]
  7. R. Jacobsson, “Optical Properties of Periodically Stratified Media with Continuously Varying Refractive Index,” Ark. Fys. 31, No. 14, 191–200 (1966).
  8. W. E. Johnson, R. L. Crane, “An Overview of Rugate Filter Technology,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 118–121.
  9. W. H. Southwell, “Rugate Side Lobe Suppression Using Quintics and Rugated Matching Layers,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 146–149.
  10. W. H. Southwell, “Rugate Index Profile which Suppresses all Harmonic Stop Bands,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 142–145.
  11. E. Delano, “Fourier Synthesis of Multilayer Filters,” J. Opt. Soc. Am. 57, 1529–1553 (1967).
    [CrossRef]
  12. L. Sossi, “A Method for the Synthesis of Multilayer Dielectric Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, No. 3, 229–237 (1974) (Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6).
  13. L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filters,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, No. 2, 171–176 (1976) (see Ref. 12 for translation).
  14. L. Sossi, “Synthesis of Dielectric Interference Filters with Narrow Reflection Bands,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 28, No. 3, 213–220 (1979) (see Ref. 12 for translation).
  15. J. A. Dobrowolski, D. Lowe, “Optical Thin Film Synthesis Program Based on the Use of Fourier Transforms,” Appl. Opt. 17, 3039–3050 (1978).
    [CrossRef] [PubMed]
  16. W. H. Southwell, “Use of Gradient Index for Spectral Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 464, 110–114 (1984).
  17. G. Boivin, D. St. Germain, “Synthesis of Gradient-Index Profiles Corresponding to Spectral Reflectance Derived by Inverse Fourier Transform Method,” Appl. Opt. 26, 4209–4213 (1987).
    [CrossRef] [PubMed]
  18. D. Soulaire, “Synthèse des filtres spectraux multidiélectriques. Etude d’une méthode analytique par transformation de Fourier, Thèse de Doctorat,” Ecole Nationale supérieure de physique de Marseille (1987).
  19. O. Arnon, P. Baumeister, “Electric Field Distribution and the Reduction of Laser Damage in Multilayers,” Appl. Opt. 19, 1853–1855 (1980).
    [CrossRef] [PubMed]
  20. P. G. Verly, J. A. Dobrowolski, W. J. Wild, R. J. Burton, “Design of High Rejection Filters by the Inverse Fourier Transform Method,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 391–394.

1987 (1)

1986 (1)

1984 (2)

E. Berkcan, G. H. Cohen, “A New CAD Technique for (Narrow Band) Inhomogeneous Thin Films Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 518, 98–104 (1984).

W. H. Southwell, “Use of Gradient Index for Spectral Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 464, 110–114 (1984).

1981 (1)

1980 (1)

1979 (1)

L. Sossi, “Synthesis of Dielectric Interference Filters with Narrow Reflection Bands,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 28, No. 3, 213–220 (1979) (see Ref. 12 for translation).

1978 (1)

1976 (1)

L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filters,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, No. 2, 171–176 (1976) (see Ref. 12 for translation).

1974 (1)

L. Sossi, “A Method for the Synthesis of Multilayer Dielectric Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, No. 3, 229–237 (1974) (Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6).

1973 (1)

1971 (1)

1967 (2)

1966 (1)

R. Jacobsson, “Optical Properties of Periodically Stratified Media with Continuously Varying Refractive Index,” Ark. Fys. 31, No. 14, 191–200 (1966).

Arnon, O.

Baumeister, P.

Berkcan, E.

E. Berkcan, G. H. Cohen, “A New CAD Technique for (Narrow Band) Inhomogeneous Thin Films Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 518, 98–104 (1984).

Boivin, G.

Burton, R. J.

P. G. Verly, J. A. Dobrowolski, W. J. Wild, R. J. Burton, “Design of High Rejection Filters by the Inverse Fourier Transform Method,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 391–394.

Cohen, G. H.

E. Berkcan, G. H. Cohen, “A New CAD Technique for (Narrow Band) Inhomogeneous Thin Films Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 518, 98–104 (1984).

Crane, R. L.

W. E. Johnson, R. L. Crane, “An Overview of Rugate Filter Technology,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 118–121.

Delano, E.

Dobrowolski, J. A.

J. A. Dobrowolski, D. Lowe, “Optical Thin Film Synthesis Program Based on the Use of Fourier Transforms,” Appl. Opt. 17, 3039–3050 (1978).
[CrossRef] [PubMed]

J. A. Dobrowolski, “Subtractive Method of Optical Thin-Film Interference Filter Design,” Appl. Opt. 12, 1885–1893 (1973).
[CrossRef] [PubMed]

P. G. Verly, J. A. Dobrowolski, W. J. Wild, R. J. Burton, “Design of High Rejection Filters by the Inverse Fourier Transform Method,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 391–394.

Germain, D. St.

Jacobsson, R.

R. Jacobsson, “Optical Properties of Periodically Stratified Media with Continuously Varying Refractive Index,” Ark. Fys. 31, No. 14, 191–200 (1966).

Johnson, W. E.

W. E. Johnson, R. L. Crane, “An Overview of Rugate Filter Technology,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 118–121.

Lowe, D.

Sossi, L.

L. Sossi, “Synthesis of Dielectric Interference Filters with Narrow Reflection Bands,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 28, No. 3, 213–220 (1979) (see Ref. 12 for translation).

L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filters,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, No. 2, 171–176 (1976) (see Ref. 12 for translation).

L. Sossi, “A Method for the Synthesis of Multilayer Dielectric Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, No. 3, 229–237 (1974) (Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6).

Soulaire, D.

D. Soulaire, “Synthèse des filtres spectraux multidiélectriques. Etude d’une méthode analytique par transformation de Fourier, Thèse de Doctorat,” Ecole Nationale supérieure de physique de Marseille (1987).

Southwell, W. H.

W. H. Southwell, “Use of Gradient Index for Spectral Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 464, 110–114 (1984).

W. H. Southwell, “Rugate Index Profile which Suppresses all Harmonic Stop Bands,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 142–145.

W. H. Southwell, “Rugate Side Lobe Suppression Using Quintics and Rugated Matching Layers,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 146–149.

Thelen, A.

Verly, P. G.

P. G. Verly, J. A. Dobrowolski, W. J. Wild, R. J. Burton, “Design of High Rejection Filters by the Inverse Fourier Transform Method,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 391–394.

Wild, W. J.

P. G. Verly, J. A. Dobrowolski, W. J. Wild, R. J. Burton, “Design of High Rejection Filters by the Inverse Fourier Transform Method,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 391–394.

Young, L.

Appl. Opt. (6)

Ark. Fys. (1)

R. Jacobsson, “Optical Properties of Periodically Stratified Media with Continuously Varying Refractive Index,” Ark. Fys. 31, No. 14, 191–200 (1966).

Eesti NSV Tead. Akad. Toim. Fuus. Mat. (3)

L. Sossi, “A Method for the Synthesis of Multilayer Dielectric Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, No. 3, 229–237 (1974) (Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6).

L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filters,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, No. 2, 171–176 (1976) (see Ref. 12 for translation).

L. Sossi, “Synthesis of Dielectric Interference Filters with Narrow Reflection Bands,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 28, No. 3, 213–220 (1979) (see Ref. 12 for translation).

J. Opt. Soc. Am. (3)

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

W. H. Southwell, “Use of Gradient Index for Spectral Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 464, 110–114 (1984).

E. Berkcan, G. H. Cohen, “A New CAD Technique for (Narrow Band) Inhomogeneous Thin Films Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 518, 98–104 (1984).

Other (5)

D. Soulaire, “Synthèse des filtres spectraux multidiélectriques. Etude d’une méthode analytique par transformation de Fourier, Thèse de Doctorat,” Ecole Nationale supérieure de physique de Marseille (1987).

P. G. Verly, J. A. Dobrowolski, W. J. Wild, R. J. Burton, “Design of High Rejection Filters by the Inverse Fourier Transform Method,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 391–394.

W. E. Johnson, R. L. Crane, “An Overview of Rugate Filter Technology,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 118–121.

W. H. Southwell, “Rugate Side Lobe Suppression Using Quintics and Rugated Matching Layers,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 146–149.

W. H. Southwell, “Rugate Index Profile which Suppresses all Harmonic Stop Bands,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 142–145.

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Figures (7)

Fig. 1
Fig. 1

Geometry of an inhomogeneous layer.

Fig. 2
Fig. 2

Effects of the parameters used for the optimization. Each figure is described in the text, and the values of the parameters are listed in Table I.

Fig. 3
Fig. 3

Variation of the weight of the spectral function Q4 vs the desired minimum transmission for two different discretizations: curve 1, 1/16 wave layers; curve 2, 1/4 wave layers. (These layer thicknesses are approximate; see note e in Table I).

Fig. 4
Fig. 4

Synthesis of high rejection filters with Gaussian transmittance curves.

Fig. 5
Fig. 5

Synthesis of high rejection filters with different halfwidths.

Fig. 6
Fig. 6

Synthesis of medium and high rejection filters with square transmittance curves.

Fig. 7
Fig. 7

Electric fields in high rejection filters.

Tables (1)

Tables Icon

Table I Parameters Used in the Synthesis of the Rejection Filters

Equations (20)

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1 t ( σ ) = G ( x , σ ) exp ( j π σ x ) ,
x ( z ) = 2 0 z n ( z ) d z
G ( x , σ ) = 1 + 0 x d Q ( x 1 , σ ) 0 x 1 d Q + ( x 2 , σ ) + + ,
d Q ± ( x , σ ) = 1 2 n d n d x exp ( ± j 2 π σ x ) d x .
Q + ( σ ) = [ Q ( σ ) ] * Q ( σ ) ,
Q ( σ ) = 1 2 n d n d x exp ( j 2 π σ x ) d x .
ln n ( x ) n 0 = j π Q ( σ ) σ exp ( j 2 π σ x ) d σ .
1 T = 1 + | Q | 2 + O ( Q 4 ) ,
T = 1 | Q | 2 + O ( Q 4 ) .
| Q 1 | = 1 2 ( 1 T T ) ;
| Q 2 | = 1 T ;
| Q 3 | = 1 T 1 .
S ( T ) = S ( 1 ) + ( S T ) T = 1 ( T 1 ) + 1 2 ( 2 S T 2 ) T = 1 ( T 1 ) 2 + .
| Q | = S ( T ) .
S = 0 ;
S T = 1 .
| Q 4 | = w 1 T + ( 1 w ) 1 T 1 .
| Q 5 | = ln ( γ + γ 2 1 ) ,
γ = 1 + 1 4 ( 1 T T ) .
( log T log T min ) new = ( log T log T min ) old ,

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