Abstract

Several errors inherent to the Fourier transform method for optical thin film synthesis, including the inaccuracy of the spectral functions Q˜(σ) used in the Fourier transforms, are compensated numerically by using successive approximations. We show that the complex phase of Q˜(σ) is a key parameter which can be exploited to reduce significantly the thickness of the synthesized films and to control the shape of the refractive index profiles without affecting the spectral performance. This method is compared to other well established thin film design techniques.

© 1990 Optical Society of America

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References

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  1. R. Jacobsson, “Light Reflection from Films of Continuously Varying Refractive Index,” in Progress in Optics, Vol. 5 (North-Holland, Amsterdam, 1966).
    [CrossRef]
  2. R. W. Bertram, M. F. Ouellette, P. Y. Tse, “New Approach to Design and Production of Inhomogeneous Optical Coatings,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 150–153.
  3. W. H. Southwell, “Gradient-Index Antireflection Coatings,” Opt. Lett. 8, 584–586 (1983).
    [CrossRef] [PubMed]
  4. 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.
  5. M. Zukic, K. H. Guenther, “Optical Coatings with Graded Index Layers for High Power Laser Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 895, 271–277 (1988).
  6. L. Sossi, “A Method for the Synthesis of Multilayer Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat.23, 229–237 (1974) (Translation Services of the Canada Institute for Technical and Scientific Information, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6).
  7. L. Sossi, “On the Synthesis of Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 26, 28–36 (1977) (see Ref. 6 for translation).
  8. E. Delano, “Fourier Synthesis of Multilayer Filters,” J. Opt. Soc. Am. 57, 1529–1533 (1967).
    [CrossRef]
  9. J. A. Dobrowolski, D. G. Lowe, “Optical Thin Film Synthesis Program Based on the Use of Fourier Transforms,” Appl. Opt. 17, 3039–3050 (1978).
    [CrossRef] [PubMed]
  10. P. G. Verly, J. A. Dobrowolski, W. J. Wild, R. L. Burton, “Synthesis of High Rejection Filters with the Fourier Transform Method,” Appl. Opt. 28, 2864–2875 (1989).
    [CrossRef] [PubMed]
  11. B. G. Bovard, “Derivation of a Matrix Describing a Rugate Dielectric Thin Film,” Appl. Opt. 27, 1998–1905 (1988).
    [CrossRef] [PubMed]
  12. B. G. Bovard, “Fourier Transform Technique Applied to Quarterwave Optical Coatings,” Appl. Opt. 27, 3062–3063 (1988).
    [CrossRef] [PubMed]
  13. J. A. Dobrowolski, “Computer Design of Optical Coatings,” Thin Solid Films 163, 97–110 (1988).
    [CrossRef]
  14. R. F. Potter, “Optical Properties of Lamelliform Materials,” Proc. Soc. Photo-Opt. Instrum. Eng. 276, 204–213 (1981).
  15. J. A. Dobrowolski, “Subtractive Method of Optical Thin-Film Interference Filter Design,” Appl. Opt. 12, 1885–1893 (1973).
    [CrossRef] [PubMed]

1989 (1)

1988 (4)

B. G. Bovard, “Fourier Transform Technique Applied to Quarterwave Optical Coatings,” Appl. Opt. 27, 3062–3063 (1988).
[CrossRef] [PubMed]

M. Zukic, K. H. Guenther, “Optical Coatings with Graded Index Layers for High Power Laser Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 895, 271–277 (1988).

J. A. Dobrowolski, “Computer Design of Optical Coatings,” Thin Solid Films 163, 97–110 (1988).
[CrossRef]

B. G. Bovard, “Derivation of a Matrix Describing a Rugate Dielectric Thin Film,” Appl. Opt. 27, 1998–1905 (1988).
[CrossRef] [PubMed]

1983 (1)

1981 (1)

R. F. Potter, “Optical Properties of Lamelliform Materials,” Proc. Soc. Photo-Opt. Instrum. Eng. 276, 204–213 (1981).

1978 (1)

1977 (1)

L. Sossi, “On the Synthesis of Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 26, 28–36 (1977) (see Ref. 6 for translation).

1973 (1)

1967 (1)

Bertram, R. W.

R. W. Bertram, M. F. Ouellette, P. Y. Tse, “New Approach to Design and Production of Inhomogeneous Optical Coatings,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 150–153.

Bovard, B. G.

Burton, R. L.

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.

Guenther, K. H.

M. Zukic, K. H. Guenther, “Optical Coatings with Graded Index Layers for High Power Laser Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 895, 271–277 (1988).

Jacobsson, R.

R. Jacobsson, “Light Reflection from Films of Continuously Varying Refractive Index,” in Progress in Optics, Vol. 5 (North-Holland, Amsterdam, 1966).
[CrossRef]

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. G.

Ouellette, M. F.

R. W. Bertram, M. F. Ouellette, P. Y. Tse, “New Approach to Design and Production of Inhomogeneous Optical Coatings,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 150–153.

Potter, R. F.

R. F. Potter, “Optical Properties of Lamelliform Materials,” Proc. Soc. Photo-Opt. Instrum. Eng. 276, 204–213 (1981).

Sossi, L.

L. Sossi, “On the Synthesis of Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 26, 28–36 (1977) (see Ref. 6 for translation).

L. Sossi, “A Method for the Synthesis of Multilayer Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat.23, 229–237 (1974) (Translation Services of the Canada Institute for Technical and Scientific Information, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6).

Southwell, W. H.

Tse, P. Y.

R. W. Bertram, M. F. Ouellette, P. Y. Tse, “New Approach to Design and Production of Inhomogeneous Optical Coatings,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 150–153.

Verly, P. G.

Wild, W. J.

Zukic, M.

M. Zukic, K. H. Guenther, “Optical Coatings with Graded Index Layers for High Power Laser Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 895, 271–277 (1988).

Appl. Opt. (5)

Eesti NSV Tead. Akad. Toim. Fuus. Mat. (1)

L. Sossi, “On the Synthesis of Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 26, 28–36 (1977) (see Ref. 6 for translation).

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

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

R. F. Potter, “Optical Properties of Lamelliform Materials,” Proc. Soc. Photo-Opt. Instrum. Eng. 276, 204–213 (1981).

M. Zukic, K. H. Guenther, “Optical Coatings with Graded Index Layers for High Power Laser Applications,” Proc. Soc. Photo-Opt. Instrum. Eng. 895, 271–277 (1988).

Thin Solid Films (1)

J. A. Dobrowolski, “Computer Design of Optical Coatings,” Thin Solid Films 163, 97–110 (1988).
[CrossRef]

Other (4)

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.

L. Sossi, “A Method for the Synthesis of Multilayer Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat.23, 229–237 (1974) (Translation Services of the Canada Institute for Technical and Scientific Information, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6).

R. Jacobsson, “Light Reflection from Films of Continuously Varying Refractive Index,” in Progress in Optics, Vol. 5 (North-Holland, Amsterdam, 1966).
[CrossRef]

R. W. Bertram, M. F. Ouellette, P. Y. Tse, “New Approach to Design and Production of Inhomogeneous Optical Coatings,” in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, Washington, DC, 1988), pp. 150–153.

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Figures (7)

Fig. 1
Fig. 1

Variation of several Q-functions Qi(T) against transmittance. Q4(T,w) is represented by a family of intermediate curves depending on the value of w. For example, Q4(T,1) ≡ Q2(T), Q4(T,0.85) ≈ Q5(T), Q4(T,0) ≡ Q3(T).

Fig. 2
Fig. 2

Flow of the calculations pertaining to the correction process.

Fig. 3
Fig. 3

Synthesis examples showing the importance of the initial Q-phase.

Fig. 4
Fig. 4

Variation of the value of the merit function with time for Figs. 3(C) and 5(A)–(D) (see text).

Fig. 5
Fig. 5

Synthesis examples with full determination of the Q-phase.

Fig. 6
Fig. 6

Comparison of the correction process with refinement.

Fig. 7
Fig. 7

Variation of the values of the merit function with time for Figs. 6(A)–(D) (see text).

Tables (1)

Tables Icon

Table I Parameters Used In the Numerical Examples

Equations (30)

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n ( x ) = n 0 exp [ j π - Q ˜ ( σ ) σ exp ( - j 2 π σ x ) d σ ]
Q ˜ ( σ ) = Q [ T ( σ ) ] exp [ j ϕ ( σ ) ] ,
x = 2 0 z n ( u ) d u ,
Q ˜ ( σ ) = r ( σ ) t ( σ ) .
Q 2 ( T ) = 1 - T ,
Q 3 ( T ) = R T = 1 T - 1 ,
Q 4 ( T , w ) = w Q 2 ( T ) + ( 1 - w ) Q 3 ( T ) ,
Q 5 ( T ) = ln ( γ + γ 2 - 1 ) ,
γ = 1 + 1 4 ( 1 T - T ) .
Q i + 1 ( σ ) = Q i ( σ ) + Δ Q i ( σ )             i = 1 , 2 , 3 , ,
Δ Q i ( σ ) = Q [ T D ( σ ) ] - Q [ T i ( σ ) ] ,
Q 1 ( σ ) = Q [ T D ( σ ) ] .
Q ˜ i + 1 ( σ ) = Q ˜ i ( σ ) + s i Δ Q ˜ i ( σ )             i = 1 , 2 , 3 , ,
Δ Q ˜ i ( σ ) = Δ Q i ( σ ) exp [ j ψ i ( σ ) ] ,
Δ Q i ( σ ) = Q [ T D ( σ ) ] - Q [ T i ( σ ) ] ,
Q 1 ( σ ) = Q [ T D ( σ ) ] exp [ j ϕ 1 ( σ ) ] .
M = { 1 N i = 1 N [ T ( σ i ) - T D ( σ i ) δ T ( σ i ) ] 2 } 1 / 2 ,
n i + 1 ( x ) = n i ( x ) Δ n i ( x )             i = 1 , 2 , 3 , ,
Δ n i ( x ) = exp [ j s i π - Δ Q ˜ i ( σ ) σ exp ( - j 2 π σ x ) d σ ] ,
ψ i ( σ ) = ϕ i ( σ )             i = 1 , 2 , 3 ,
ϕ 1 ( σ ) = arg [ r ( σ ) t ( σ ) ] ,
ψ i ( σ ) = arg [ r ( σ ) t ( σ ) ] i             i = 1 , 2 , 3 ,
Q ˜ ( σ ) σ = 2 a - Q ˜ ( u ) u sinc [ 2 a ( σ - u ) ] d u ,
ϕ m = ϕ ( σ m )             m = 1 , 2 , 3 , ,
Q m + ( σ ) = [ [ Q ( σ ) ] 2 + [ Δ Q m ( σ ) ] 2 + 2 Q ( σ ) Δ Q m ( σ ) sin [ ϕ ( σ ) - ϕ m ] ] 1 / 2 ,
Δ Q m ( σ ) = 2 a σ ϕ σ Q ( σ m ) σ m sinc [ 2 a ( σ - σ m ) ] ,
M Q = ( 1 N i = 1 N { Q ( σ i ) - Q [ T D ( σ i ) ] δ Q ( σ i ) } 2 ) 1 / 2 ,
( M Q ) m = M Q ϕ m = 1 ϕ [ M Q ( ϕ m + ϕ ) - M Q ( ϕ m ) ]
ϕ m , new = ϕ m , old - s ( M Q ) m ,
n ¯ = ½ ( n a + n b ) ,

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