Abstract

An empirical procedure based on optical-density-bandwidth products was recently proposed for thickness estimation of dielectric thin film reflectors. A parallel is established with new results derived from the Fourier transform thin film synthesis technique. Two Fourier-transform approaches are proposed and justified by numerical examples.

© 2006 Optical Society of America

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References

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  1. H. A. Macleod, Thin-Film Optical Filters (Macmillan, 1986).
    [CrossRef]
  2. P. W. Baumeister, Optical Coating Technology (SPIE Optical Engineering Press, 2004).
    [CrossRef]
  3. R. R. Willey, "Estimating the number of layers required and other properties of blocker and dichroic optical thin films," Appl. Opt. 35, 4982-4986 (1996).
    [CrossRef] [PubMed]
  4. R. R. Willey, "Predicting achievable design performance of broadband antireflection coatings," Appl. Opt. 32, 5447-5451 (1993).
    [CrossRef] [PubMed]
  5. J. A. Dobrowolski, "Subtractive method of optical thin-film interference filter design," Appl. Opt. 12, 1885-1893 (1973).
    [CrossRef] [PubMed]
  6. J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, "Optimal single-band normal-incidence antireflection coatings," Appl. Opt. 35, 644-658 (1996).
    [CrossRef] [PubMed]
  7. E. Delano, "Fourier synthesis of multilayer filters," J. Opt. Soc. Am. 57, 1529-1553 (1967).
    [CrossRef]
  8. L. Sossi, "A method for the synthesis of multilayer dielectric interference coatings," Eesti NSV Tead. Akad. Toim. Fuus. , Mat. 23, 229-237 (1974). An English translation is available from the Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6.
  9. J. A. Dobrowolski and D. 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, and R. L. Burton, "Synthesis of high rejection filters with the Fourier transform method," Appl. Opt. 28, 2864-2875 (1989).
    [CrossRef] [PubMed]
  11. P. G. Verly and J. A. Dobrowolski, "Iterative correction process for optical thin film synthesis with the Fourier transform method," Appl. Opt. 29, 3672-3684 (1990).
    [CrossRef] [PubMed]
  12. P. G. Verly, "Design of inhomogeneous and quasi-inhomogeneous optical coatings at the NRC," in Inhomogeneous and Quasi-Inhomogenous Optical Coatings, J.A.Dobrowolski and P.G.Verly, eds., Proc. SPIE 2046, 36-45 (1993).
  13. P. G. Verly, J. A. Dobrowolski, and R. R. Willey, "Fourier transform method for the design of wideband antireflection coatings," Appl. Opt. 31, 3836-3846 (1992).
    [CrossRef] [PubMed]
  14. B. G. Bovard, "Derivation of a matrix describing a rugate dielectric thin film," Appl. Opt. 27, 1998-2005 (1988).
    [CrossRef] [PubMed]
  15. H. Fabricius, "Gradient-index filters: conversion into a two-index solution by taking into account dispersion," Appl. Opt. 31, 5216-5220 (1992).
    [CrossRef] [PubMed]
  16. P. V. Bulkin, P. L. Swart, and B. M. Lacquet, "Fourier-transform design and electron cyclotron resonance plasma-enhanced deposition of lossy graded-index optical coatings," Appl. Opt. 35, 4413-4419 (1996).
    [CrossRef] [PubMed]
  17. P. G. Verly, A. V. Tikhonravov, and A. D. Poezd, "Multiple solutions to the synthesis of graded index optical coatings," in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J.A.Dobrowolski and P.G.Verly, eds., Proc. SPIE 2046, 9-16 (1993).
  18. P. G. Verly, "Fourier transform approach for the estimation of optical thin film thickness," in Technical Digest of Topical Meeting on Optical Interference Coatings (Optical Society of America, 2001), pp. TuA9 1-3.
  19. Equation (5) is more general than stated in Ref. 18; an equal proportion of the high and low refractive index is not necessary.

1996 (3)

1993 (1)

1992 (2)

1990 (1)

1989 (1)

1988 (1)

1978 (1)

1974 (1)

L. Sossi, "A method for the synthesis of multilayer dielectric interference coatings," Eesti NSV Tead. Akad. Toim. Fuus. , Mat. 23, 229-237 (1974). An English translation is available from the Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6.

1973 (1)

1967 (1)

Baumeister, P. W.

P. W. Baumeister, Optical Coating Technology (SPIE Optical Engineering Press, 2004).
[CrossRef]

Bovard, B. G.

Bulkin, P. V.

Burton, R. L.

Delano, E.

Dobrowolski, J. A.

Fabricius, H.

Lacquet, B. M.

Lowe, D.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Macmillan, 1986).
[CrossRef]

Poezd, A. D.

P. G. Verly, A. V. Tikhonravov, and A. D. Poezd, "Multiple solutions to the synthesis of graded index optical coatings," in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J.A.Dobrowolski and P.G.Verly, eds., Proc. SPIE 2046, 9-16 (1993).

Sossi, L.

L. Sossi, "A method for the synthesis of multilayer dielectric interference coatings," Eesti NSV Tead. Akad. Toim. Fuus. , Mat. 23, 229-237 (1974). An English translation is available from the Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6.

Sullivan, B. T.

Swart, P. L.

Tikhonravov, A. V.

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, "Optimal single-band normal-incidence antireflection coatings," Appl. Opt. 35, 644-658 (1996).
[CrossRef] [PubMed]

P. G. Verly, A. V. Tikhonravov, and A. D. Poezd, "Multiple solutions to the synthesis of graded index optical coatings," in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J.A.Dobrowolski and P.G.Verly, eds., Proc. SPIE 2046, 9-16 (1993).

Trubetskov, M. K.

Verly, P. G.

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, "Optimal single-band normal-incidence antireflection coatings," Appl. Opt. 35, 644-658 (1996).
[CrossRef] [PubMed]

P. G. Verly, J. A. Dobrowolski, and R. R. Willey, "Fourier transform method for the design of wideband antireflection coatings," Appl. Opt. 31, 3836-3846 (1992).
[CrossRef] [PubMed]

P. G. Verly and J. A. Dobrowolski, "Iterative correction process for optical thin film synthesis with the Fourier transform method," Appl. Opt. 29, 3672-3684 (1990).
[CrossRef] [PubMed]

P. G. Verly, J. A. Dobrowolski, W. J. Wild, and R. L. Burton, "Synthesis of high rejection filters with the Fourier transform method," Appl. Opt. 28, 2864-2875 (1989).
[CrossRef] [PubMed]

P. G. Verly, "Fourier transform approach for the estimation of optical thin film thickness," in Technical Digest of Topical Meeting on Optical Interference Coatings (Optical Society of America, 2001), pp. TuA9 1-3.

P. G. Verly, "Design of inhomogeneous and quasi-inhomogeneous optical coatings at the NRC," in Inhomogeneous and Quasi-Inhomogenous Optical Coatings, J.A.Dobrowolski and P.G.Verly, eds., Proc. SPIE 2046, 36-45 (1993).

P. G. Verly, A. V. Tikhonravov, and A. D. Poezd, "Multiple solutions to the synthesis of graded index optical coatings," in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J.A.Dobrowolski and P.G.Verly, eds., Proc. SPIE 2046, 9-16 (1993).

Wild, W. J.

Willey, R. R.

Appl. Opt. (11)

R. R. Willey, "Estimating the number of layers required and other properties of blocker and dichroic optical thin films," Appl. Opt. 35, 4982-4986 (1996).
[CrossRef] [PubMed]

R. R. Willey, "Predicting achievable design performance of broadband antireflection coatings," Appl. Opt. 32, 5447-5451 (1993).
[CrossRef] [PubMed]

J. A. Dobrowolski, "Subtractive method of optical thin-film interference filter design," Appl. Opt. 12, 1885-1893 (1973).
[CrossRef] [PubMed]

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, "Optimal single-band normal-incidence antireflection coatings," Appl. Opt. 35, 644-658 (1996).
[CrossRef] [PubMed]

J. A. Dobrowolski and D. Lowe, "Optical thin film synthesis program based on the use of Fourier transforms," Appl. Opt. 17, 3039-3050 (1978).
[CrossRef] [PubMed]

P. G. Verly, J. A. Dobrowolski, W. J. Wild, and R. L. Burton, "Synthesis of high rejection filters with the Fourier transform method," Appl. Opt. 28, 2864-2875 (1989).
[CrossRef] [PubMed]

P. G. Verly and J. A. Dobrowolski, "Iterative correction process for optical thin film synthesis with the Fourier transform method," Appl. Opt. 29, 3672-3684 (1990).
[CrossRef] [PubMed]

P. G. Verly, J. A. Dobrowolski, and R. R. Willey, "Fourier transform method for the design of wideband antireflection coatings," Appl. Opt. 31, 3836-3846 (1992).
[CrossRef] [PubMed]

B. G. Bovard, "Derivation of a matrix describing a rugate dielectric thin film," Appl. Opt. 27, 1998-2005 (1988).
[CrossRef] [PubMed]

H. Fabricius, "Gradient-index filters: conversion into a two-index solution by taking into account dispersion," Appl. Opt. 31, 5216-5220 (1992).
[CrossRef] [PubMed]

P. V. Bulkin, P. L. Swart, and B. M. Lacquet, "Fourier-transform design and electron cyclotron resonance plasma-enhanced deposition of lossy graded-index optical coatings," Appl. Opt. 35, 4413-4419 (1996).
[CrossRef] [PubMed]

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

L. Sossi, "A method for the synthesis of multilayer dielectric interference coatings," Eesti NSV Tead. Akad. Toim. Fuus. , Mat. 23, 229-237 (1974). An English translation is available from the Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6.

J. Opt. Soc. Am. (1)

Other (6)

P. G. Verly, "Design of inhomogeneous and quasi-inhomogeneous optical coatings at the NRC," in Inhomogeneous and Quasi-Inhomogenous Optical Coatings, J.A.Dobrowolski and P.G.Verly, eds., Proc. SPIE 2046, 36-45 (1993).

H. A. Macleod, Thin-Film Optical Filters (Macmillan, 1986).
[CrossRef]

P. W. Baumeister, Optical Coating Technology (SPIE Optical Engineering Press, 2004).
[CrossRef]

P. G. Verly, A. V. Tikhonravov, and A. D. Poezd, "Multiple solutions to the synthesis of graded index optical coatings," in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J.A.Dobrowolski and P.G.Verly, eds., Proc. SPIE 2046, 9-16 (1993).

P. G. Verly, "Fourier transform approach for the estimation of optical thin film thickness," in Technical Digest of Topical Meeting on Optical Interference Coatings (Optical Society of America, 2001), pp. TuA9 1-3.

Equation (5) is more general than stated in Ref. 18; an equal proportion of the high and low refractive index is not necessary.

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Figures (6)

Fig. 1
Fig. 1

Unconstrained Fourier transform design nC (x) used for estimating the optical thickness of a two-material broadband reflector by means of Eq. (8). The design target is represented by a horizontal straight line at T = −20 dB. See text for details.

Fig. 2
Fig. 2

Two-material broadband reflector of thickness estimated by integration of the refractive index profile in Fig. 1.

Fig. 3
Fig. 3

Unconstrained Fourier transform design of a linear reflector. The spectral target is composed of two linear segments in the region 1 ≤ σ ≤ 2.5 μm−1.

Fig. 4
Fig. 4

Refined two-material multilayer of thickness estimated by integration of the refractive index profile shown in Fig. 3.

Fig. 5
Fig. 5

Reflecting filter thickness variation with index range. Solid curve: arbitrary reflectance target (fixed). Dashed curve: QW stack of constant peak reflectance. See text for details.

Fig. 6
Fig. 6

Broadband reflector designs obtained for two different refractive index ranges. The target is the same as in Fig. 2. Solid curve: nH = 2.0. Dashed curve: nH = 2.5. nL = 1.45 in both cases.

Equations (12)

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N σ MIN σ MAX OD σ d σ 2   ln ( n H n L ) sin 1 ( n H n L n H + n L ) ,
ln [ n ( x ) n 0 ] FT i π Q ˜ ( T , σ ) σ
Q ˜ ( T , σ ) = Q ( T ) e i ϕ ( σ )
ln 2 [ n ( x ) n 0 ] d x = 1 π 2 [ Q ( T ) σ ] 2 d σ .
ln 2 [ n ( x ) n 0 ] d x 1 2 ( Σ n t ) ln 2 [ n H n L ] .
Q ( T ) = | Q ˜ ( T , σ ) | = ln ( 1 T ) = ln ( 10 ) OD .
Σ n t 0.93 0 OD σ 2  d σ ln 2 [ n H n L ] .
Σ n t 2 ln 2 [ n C ( x ) n 0 ] d x ln 2 [ n H n L ] ,
1 + 1 3 2 + 1 5 2 + = π 2 8 = 1.23 ,
ln [ n ( x ) n 0 ] = 2 p m = l F Q ( m p ) , l p x l + 1 p ,
( n H n L ) n t = const .
( n H n L ) n t = const.,

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