Abstract

A Fourier transform (FT) approach based on the evaluation of optical-density-bandwidth products in the spectral region of interest was recently proposed for the thickness estimation of reflecting thin-film dielectric filters. For simplicity, the initial discussion was limited to a particular type of immersed coating. The theory is generalized to more realistic filter configurations and confirmed by numerical examples. It is shown that good results are possible although the problem is more complex from a FT point of view.

© 2007 Optical Society of America

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  1. E. Delano, "Fourier synthesis of multilayer filters," J. Opt. Soc. Am. 57, 1529-1553 (1967).
    [CrossRef]
  2. 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 KIA 0R6, Canada.
  3. 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]
  4. 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]
  5. P. G. Verly, "Design of inhomogeneous and quasi-inhomogeneous optical coatings at the NRC," in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski and P. G. Verly, eds., Proc. SPIE 2046, 36-45 (1993).
    [CrossRef]
  6. P. G. Verly, "Fourier transform approach for the estimation of optical thin film thickness," in Technical Digest, Topical Meeting on Optical Interference Coatings (Optical Society of America, 2001), pp. 1-3, paper TuA9.
  7. P. G. Verly, "Fourier transform approach for thickness estimation of reflecting interference filters," Appl. Opt. 45, 636-641 (2006).
    [CrossRef]
  8. 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]
  9. B. G. Bovard, "Derivation of a matrix describing a rugate dielectric thin film," Appl. Opt. 27, 1998-2005 (1988).
    [CrossRef] [PubMed]
  10. B. G. Bovard, "Fourier transform technique applied to quarterwave optical coatings," Appl. Opt. 27, 3062-3063 (1988).
    [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, A. V. Tikhonravov, and A. D. Poezd, "Multiple solutions to the synthesis of graded index optical coatings," in Inhomogeneous and Quasi-Inhomogeneous Coatings, J. A. Dobrowolski and P. G. Verly, eds., Proc. SPIE 2046, 9-16 (1993).
    [CrossRef]
  13. A. Dinca, V. Lupei, and M. P. Dinca, "Interferential filter design with continuously variable refractive index," in ROMOPTO '97: Fifth Conference on Optics, V. I. Vlad and D. C. Dumitras, eds., Proc. SPIE 3405, 1173-1176 (1998).
    [CrossRef]
  14. P. G. Verly, "Fourier transform technique with frequency filtering for optical thin-film design," Appl. Opt. 34, 688-694 (1995).
    [CrossRef] [PubMed]
  15. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).
  16. P. G. Verly, "Optical coating synthesis by simultaneous refractive-index and thickness refinement of inhomogeneous films," Appl. Opt. 37, 7327-7333 (1998).
    [CrossRef]
  17. J. A. Dobrowolski, S. Browning, M. Jacobson, and M. Nadal, "Topical Meeting on Optical Interference Coatings (OIC '2001): manufacturing problem," Appl. Opt. 41, 3039-3052 (2002).
    [CrossRef] [PubMed]
  18. P. G. Verly, "Fourier transform estimation of reflecting thin film thickness," in Advances in Optical Thin Films II, edited by C. Amra, N. Kaiser, and H. A. Macleod, eds., Proc. SPIE 5963, 596306 (2005).
    [CrossRef]

2006 (1)

P. G. Verly, "Fourier transform approach for thickness estimation of reflecting interference filters," Appl. Opt. 45, 636-641 (2006).
[CrossRef]

2005 (1)

P. G. Verly, "Fourier transform estimation of reflecting thin film thickness," in Advances in Optical Thin Films II, edited by C. Amra, N. Kaiser, and H. A. Macleod, eds., Proc. SPIE 5963, 596306 (2005).
[CrossRef]

2002 (1)

2001 (1)

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

1998 (2)

A. Dinca, V. Lupei, and M. P. Dinca, "Interferential filter design with continuously variable refractive index," in ROMOPTO '97: Fifth Conference on Optics, V. I. Vlad and D. C. Dumitras, eds., Proc. SPIE 3405, 1173-1176 (1998).
[CrossRef]

P. G. Verly, "Optical coating synthesis by simultaneous refractive-index and thickness refinement of inhomogeneous films," Appl. Opt. 37, 7327-7333 (1998).
[CrossRef]

1996 (1)

1995 (1)

1993 (2)

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 Coatings, J. A. Dobrowolski and P. G. Verly, eds., Proc. SPIE 2046, 9-16 (1993).
[CrossRef]

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

1990 (1)

1989 (1)

1988 (2)

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 KIA 0R6, Canada.

1967 (1)

Bovard, B. G.

Browning, S.

Burton, R. L.

Delano, E.

Dinca, A.

A. Dinca, V. Lupei, and M. P. Dinca, "Interferential filter design with continuously variable refractive index," in ROMOPTO '97: Fifth Conference on Optics, V. I. Vlad and D. C. Dumitras, eds., Proc. SPIE 3405, 1173-1176 (1998).
[CrossRef]

Dinca, M. P.

A. Dinca, V. Lupei, and M. P. Dinca, "Interferential filter design with continuously variable refractive index," in ROMOPTO '97: Fifth Conference on Optics, V. I. Vlad and D. C. Dumitras, eds., Proc. SPIE 3405, 1173-1176 (1998).
[CrossRef]

Dobrowolski, J. A.

Jacobson, M.

Lowe, D.

Lupei, V.

A. Dinca, V. Lupei, and M. P. Dinca, "Interferential filter design with continuously variable refractive index," in ROMOPTO '97: Fifth Conference on Optics, V. I. Vlad and D. C. Dumitras, eds., Proc. SPIE 3405, 1173-1176 (1998).
[CrossRef]

Nadal, M.

Palik, E. D.

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

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 Coatings, J. A. Dobrowolski and P. G. Verly, eds., Proc. SPIE 2046, 9-16 (1993).
[CrossRef]

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 KIA 0R6, Canada.

Tikhonravov, A. V.

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 Coatings, J. A. Dobrowolski and P. G. Verly, eds., Proc. SPIE 2046, 9-16 (1993).
[CrossRef]

Verly, P. G.

P. G. Verly, "Fourier transform approach for thickness estimation of reflecting interference filters," Appl. Opt. 45, 636-641 (2006).
[CrossRef]

P. G. Verly, "Fourier transform estimation of reflecting thin film thickness," in Advances in Optical Thin Films II, edited by C. Amra, N. Kaiser, and H. A. Macleod, eds., Proc. SPIE 5963, 596306 (2005).
[CrossRef]

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

P. G. Verly, "Optical coating synthesis by simultaneous refractive-index and thickness refinement of inhomogeneous films," Appl. Opt. 37, 7327-7333 (1998).
[CrossRef]

P. G. Verly, "Fourier transform technique with frequency filtering for optical thin-film design," Appl. Opt. 34, 688-694 (1995).
[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 Coatings, J. A. Dobrowolski and P. G. Verly, eds., Proc. SPIE 2046, 9-16 (1993).
[CrossRef]

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

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]

Wild, W. J.

Willey, R. R.

Appl. Opt. (10)

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]

B. G. Bovard, "Derivation of a matrix describing a rugate dielectric thin film," Appl. Opt. 27, 1998-2005 (1988).
[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, "Optical coating synthesis by simultaneous refractive-index and thickness refinement of inhomogeneous films," Appl. Opt. 37, 7327-7333 (1998).
[CrossRef]

P. G. Verly, "Fourier transform technique with frequency filtering for optical thin-film design," Appl. Opt. 34, 688-694 (1995).
[CrossRef] [PubMed]

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]

J. A. Dobrowolski, S. Browning, M. Jacobson, and M. Nadal, "Topical Meeting on Optical Interference Coatings (OIC '2001): manufacturing problem," Appl. Opt. 41, 3039-3052 (2002).
[CrossRef] [PubMed]

P. G. Verly, "Fourier transform approach for thickness estimation of reflecting interference filters," Appl. Opt. 45, 636-641 (2006).
[CrossRef]

B. G. Bovard, "Fourier transform technique applied to quarterwave optical coatings," Appl. Opt. 27, 3062-3063 (1988).
[CrossRef] [PubMed]

Eesti NSV Tead. Akad. Toim. Fuus. Mat. (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 KIA 0R6, Canada.

J. Opt. Soc. Am. (1)

Proc. SPIE (4)

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 Coatings, J. A. Dobrowolski and P. G. Verly, eds., Proc. SPIE 2046, 9-16 (1993).
[CrossRef]

A. Dinca, V. Lupei, and M. P. Dinca, "Interferential filter design with continuously variable refractive index," in ROMOPTO '97: Fifth Conference on Optics, V. I. Vlad and D. C. Dumitras, eds., Proc. SPIE 3405, 1173-1176 (1998).
[CrossRef]

P. G. Verly, "Fourier transform estimation of reflecting thin film thickness," in Advances in Optical Thin Films II, edited by C. Amra, N. Kaiser, and H. A. Macleod, eds., Proc. SPIE 5963, 596306 (2005).
[CrossRef]

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

Other (2)

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

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

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Figures (10)

Fig. 1
Fig. 1

Plot of Q functions Q 7 and Q 8 [Eq. (3)].

Fig. 2
Fig. 2

Subdivision of a refractive index profile n(x) in a homogeneous support layer n A ( x ) and a refractive index modulation n B ( x ) . See text for details.

Fig. 3
Fig. 3

Design target apodization. Curve 1: original stepped target. Curve 2: sinc 2 function in Eq. (13). Curve 3: effective target obtained by the convolution of the stepped target with the sinc function in Eq. (12). Thick black curve: apodized target used in the calculations. See text for details.

Fig. 4
Fig. 4

Semi-inhomogeneous refractive index profile n(x) and its support layer n A ( x ) (solid and dashed curves, respectively). The transmittances are T and T A . The spectral range plotted in (A) shows gradually decreasing harmonics in the region where the right-hand side of approximation (10) has its main contribution.

Fig. 5
Fig. 5

Seven-layer QW-stack reflector and its support layer. See text for details.

Fig. 6
Fig. 6

Seven-layer QW stack immersed in a medium of midrefractive index n O .

Fig. 7
Fig. 7

Preliminary design used for the thickness estimation. The thick curve represents the spectral target. See text for details.

Fig. 8
Fig. 8

Preliminary design used for the thickness estimation of a wideband reflector in the infrared. The thick black curve represents the spectral target. Thin black curves: filter and its transmittance. Black dashed curves: support layer and its transmittance. Gray dashed curves: target apodization.

Fig. 9
Fig. 9

Reflecting filter of thickness estimated from the missing ODBWP in Fig. 8. The thick curve represents the spectral target. See text for details.

Fig. 10
Fig. 10

Comparison of an immersed and nonimmersed reflector of essentially the same transmittance in the region of the design target (thick black curve).

Equations (16)

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ln 2 ( n ( x ) n o ) d x = 2 π 2 0 ( Q ( T ) σ ) 2 d σ ,
n o = n H n L .
Q 7 = | Q ˜ 7 | = ln ( 1 T ) = ln ( 10 ) OD ,
Q 8 = ln ( 1 T 1 T 1 ) .
ln 2 ( n ( x ) n o ) d x 1 2 ( Σ n t ) ln 2 ( n H n L ) + C ,
1 2 ( Σ n t ) ln 2 ( n H n L ) + C 2 π 2 0 ( Q ( T ) σ ) 2 d σ .
( Q ( T ) σ ) 2 d σ OD σ 2 d σ ,
1 2 ( Σ n t ) ln 2 ( n H n L ) + C ln 2 ( n C ( x ) n o ) d x .
n ( x ) = n A ( x ) n B ( x ) / n o ,
ln 2 ( n B ( x ) n o ) d x = a a ln 2 ( n ( x ) n o ) d x = 2 π 2 [ 0 ( Q ( T ) σ ) 2 d σ 0 ( Q ( T A ) σ ) 2 d σ ] ,
1 2 ( Σ n t ) ln 2 ( n H n L ) 2 π 2 [ 0 ( Q ( T ) σ ) 2 d σ 0 ( Q ( T A ) σ ) 2 d σ ] .
1 2 ( Σ n t ) ln 2 ( n H n L ) a a ln 2 ( n C ( x ) n o ) d x ,
Q ˜ eff ( T D ) = Q ˜ ( T D )  ∗  sinc ( 2 σ Σ n t ) ,
OD sinc 2 ( 2 Σ n t ∂σ ) ,   0 σ 1 2 Σ n t ,
1 + 1 3 2 + 1 5 2 + = π 2 8 = 1.23 ,
Σ n t = Σ n t C Thq ( T D ) Thq ( T C ) .

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