Abstract

An optimum phase is developed for the synthesis of rugate reflectors by a simple Fourier transform. This phase belongs to a complex function of the desired spectral characteristics and is usually a free param eter. In general, it receives much less attention than the function magnitude, which is not known exactly. The current work shows that phase shaping alone produces surprisingly good results and has other advantages in rugate filter synthesis. In addition, the operating mode of this design procedure is quite unusual and interesting in itself.

© 2011 Optical Society of America

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References

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  1. E. Delano, “Fourier synthesis of multilayer filters,” J. Opt. Soc. Am. 57, 1529–1532 (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, Canada KlA 0R6.
  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]
  4. H. Fabricius, “Gradient-index filters: conversion into a two-index solution by taking into account dispersion,” Appl. Opt. 31, 5216–5220 (1992).
    [CrossRef]
  5. 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]
  6. D. Poitras, S. Larouche, and L. Martinu, “Design and plasma deposition of dispersion-corrected multiband rugate filters,” Appl. Opt. 41, 5249–5255 (2002).
    [CrossRef]
  7. S. Larouche and L. Martinu, “Dispersion implementation in optical filter design by the Fourier transform method using correction factors,” Appl. Opt. 46, 7436–7441 (2007).
    [CrossRef]
  8. B. G. Bovard, “Fourier transform technique applied to quarterwave optical coatings,” Appl. Opt. 27, 3062–3063(1988).
    [CrossRef]
  9. 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]
  10. W. J. Wild, “Analytic improvement of Sossi’s Q function,” Appl. Opt. 28, 3272–3273 (1989).
    [CrossRef]
  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]
  12. P. G. Verly, “Fourier transform technique with refinement in the frequency domain for the synthesis of optical thin films,” Appl. Opt. 35, 5148–5154 (1996).
    [CrossRef]
  13. X. Cheng, B. Fan, J. A. Dobrowolski, L. Wang, and Z. Wang, “Gradient-index optical filter synthesis with controllable and predictable refractive index profiles,” Opt. Express 16, 2315–2321 (2008).
    [CrossRef]
  14. R. Szipocs and A. Kohazi-Kis, “Theory and design of chirped dielectric laser mirrors,” Appl. Phys. B 65, 115–135(1997).
    [CrossRef]
  15. P. G. Verly, A. V. Tikhonravov, and A. D. Poezd, “Multiple solutions to the synthesis of graded index optical coatings,” Proc. SPIE 2046, 9–16 (1993) .
  16. A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their applications,” Appl. Opt. 32, 5417–5426(1993).
    [CrossRef]
  17. J. Druessel, J. Grantham, and P. Haaland, “Optimal phase modulation for gradient-index optical filters,” Opt. Lett. 18, 1583–1585 (1993).
    [CrossRef]
  18. P. G. Verly, A. V. Tikhonravov, and M. K. Trubetskov, “Efficient refinement algorithm for the synthesis of inhomogeneous optical coatings,” Appl. Opt. 36, 1487–1495 (1997).
    [CrossRef]
  19. P. G. Verly, “Fourier transform technique with frequency filtering for optical thin-film design,” Appl. Opt. 34, 688–694(1995).
    [CrossRef]
  20. P. G. Verly, “Hybrid approach for rugate filter design,” Appl. Opt. 47, C172–C178 (2007).
    [CrossRef]
  21. W. H. Southwell, R. L. Hall, and W. J. Gunning III, “Using wavelets to design gradient-index interference coatings,” Proc. SPIE 2046, 46–59 (1993).
  22. P. G. Verly, “Design of inhomogeneous and quasi-inhomogeneous optical coatings at the NRC,” Proc. SPIE 2046, 36–45(1993).
  23. M. Hacker, G. Stobrawa, and T. Feurer, “Iterative Fourier transform algorithm for phase-only pulse shaping,” Opt. Express 9, 191–199 (2001).
    [CrossRef]
  24. P. G. Verly, “Optimum phase for thin film synthesis by Fourier transforms,” in Optical Interference Coatings on CD-ROM (The Optical Society of America, 2010), pp. TuB2 1-3.

2008 (1)

2007 (2)

2002 (1)

2001 (1)

1997 (2)

1996 (2)

1995 (1)

1993 (5)

W. H. Southwell, R. L. Hall, and W. J. Gunning III, “Using wavelets to design gradient-index interference coatings,” Proc. SPIE 2046, 46–59 (1993).

P. G. Verly, “Design of inhomogeneous and quasi-inhomogeneous optical coatings at the NRC,” 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,” Proc. SPIE 2046, 9–16 (1993) .

A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their applications,” Appl. Opt. 32, 5417–5426(1993).
[CrossRef]

J. Druessel, J. Grantham, and P. Haaland, “Optimal phase modulation for gradient-index optical filters,” Opt. Lett. 18, 1583–1585 (1993).
[CrossRef]

1992 (1)

1990 (1)

1989 (2)

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 KlA 0R6.

1967 (1)

Bovard, B. G.

Bulkin, P. V.

Burton, R. L.

Cheng, X.

Delano, E.

Dobrowolski, J. A.

Druessel, J.

Fabricius, H.

Fan, B.

Feurer, T.

Grantham, J.

Gunning, W. J.

W. H. Southwell, R. L. Hall, and W. J. Gunning III, “Using wavelets to design gradient-index interference coatings,” Proc. SPIE 2046, 46–59 (1993).

Haaland, P.

Hacker, M.

Hall, R. L.

W. H. Southwell, R. L. Hall, and W. J. Gunning III, “Using wavelets to design gradient-index interference coatings,” Proc. SPIE 2046, 46–59 (1993).

Kohazi-Kis, A.

R. Szipocs and A. Kohazi-Kis, “Theory and design of chirped dielectric laser mirrors,” Appl. Phys. B 65, 115–135(1997).
[CrossRef]

Lacquet, B. M.

Larouche, S.

Lowe, D.

Martinu, L.

Poezd, A. D.

P. G. Verly, A. V. Tikhonravov, and A. D. Poezd, “Multiple solutions to the synthesis of graded index optical coatings,” Proc. SPIE 2046, 9–16 (1993) .

Poitras, D.

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 KlA 0R6.

Southwell, W. H.

W. H. Southwell, R. L. Hall, and W. J. Gunning III, “Using wavelets to design gradient-index interference coatings,” Proc. SPIE 2046, 46–59 (1993).

Stobrawa, G.

Swart, P. L.

Szipocs, R.

R. Szipocs and A. Kohazi-Kis, “Theory and design of chirped dielectric laser mirrors,” Appl. Phys. B 65, 115–135(1997).
[CrossRef]

Tikhonravov, A. V.

Trubetskov, M. K.

Verly, P. G.

Wang, L.

Wang, Z.

Wild, W. J.

Appl. Opt. (14)

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]

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

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]

D. Poitras, S. Larouche, and L. Martinu, “Design and plasma deposition of dispersion-corrected multiband rugate filters,” Appl. Opt. 41, 5249–5255 (2002).
[CrossRef]

S. Larouche and L. Martinu, “Dispersion implementation in optical filter design by the Fourier transform method using correction factors,” Appl. Opt. 46, 7436–7441 (2007).
[CrossRef]

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

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]

W. J. Wild, “Analytic improvement of Sossi’s Q function,” Appl. Opt. 28, 3272–3273 (1989).
[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]

P. G. Verly, “Fourier transform technique with refinement in the frequency domain for the synthesis of optical thin films,” Appl. Opt. 35, 5148–5154 (1996).
[CrossRef]

P. G. Verly, A. V. Tikhonravov, and M. K. Trubetskov, “Efficient refinement algorithm for the synthesis of inhomogeneous optical coatings,” Appl. Opt. 36, 1487–1495 (1997).
[CrossRef]

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

P. G. Verly, “Hybrid approach for rugate filter design,” Appl. Opt. 47, C172–C178 (2007).
[CrossRef]

A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their applications,” Appl. Opt. 32, 5417–5426(1993).
[CrossRef]

Appl. Phys. B (1)

R. Szipocs and A. Kohazi-Kis, “Theory and design of chirped dielectric laser mirrors,” Appl. Phys. B 65, 115–135(1997).
[CrossRef]

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, Canada KlA 0R6.

J. Opt. Soc. Am. (1)

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (3)

P. G. Verly, A. V. Tikhonravov, and A. D. Poezd, “Multiple solutions to the synthesis of graded index optical coatings,” Proc. SPIE 2046, 9–16 (1993) .

W. H. Southwell, R. L. Hall, and W. J. Gunning III, “Using wavelets to design gradient-index interference coatings,” Proc. SPIE 2046, 46–59 (1993).

P. G. Verly, “Design of inhomogeneous and quasi-inhomogeneous optical coatings at the NRC,” Proc. SPIE 2046, 36–45(1993).

Other (1)

P. G. Verly, “Optimum phase for thin film synthesis by Fourier transforms,” in Optical Interference Coatings on CD-ROM (The Optical Society of America, 2010), pp. TuB2 1-3.

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Figures (7)

Fig. 1
Fig. 1

Fourier transform design of a rugate reflector obtained from Eqs. (1, 2, 3): thick gray line, reflectance target; dotted black line, SWIFT phase; solid black lines, calculated refractive-index profile and reflectance.

Fig. 2
Fig. 2

Result of the SWIFT phase optimization. The starting and final phases are represented by a dashed and a solid black line, respectively. The upper part of the figure is the transmittance in decibels [ 10 log ( T ) ] .

Fig. 3
Fig. 3

Previous design recalculated over a wider thickness range. The shaded regions are ignored in Fig. 2.

Fig. 4
Fig. 4

Fourier analysis of the design shown in Fig. 2. The solid and dashed black lines represent the refractive-index modulation corresponding to the 1st and 2nd reflectance bands, respectively, and their reflectance. The dashed index profile is essentially a wavelet.

Fig. 5
Fig. 5

Result obtained with a different form of the Q-function magnitude [Eq. (5)]: dashed lines, SWIFT phase and starting design; solid lines, result after phase optimization.

Fig. 6
Fig. 6

Fourier transform design of a rugate reflector on a semi-infinite glass substrate in air. The filter is assumed to be a mixture of Si O 2 and Nb 2 O 5 . Material dispersion is included.

Fig. 7
Fig. 7

Fourier transform design of a rugate filter corresponding to the silhouette of the Taj Mahal.

Equations (6)

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ln ( n ( x ) n 0 ) FT i π Q ˜ ( T , σ ) σ .
Q ˜ ( T , σ ) = Q ( T ) exp [ i ϕ ( σ ) ]
ϕ ( σ ) 2 π = x 1 x 0 0 | Q ( T ) ν | 2 d ν 0 σ 0 η | Q ( T ) ν | 2 d ν d η + x 0 σ .
Q ( T ) = ln ( 1 T 1 T 1 ) .
M / ϕ k = i M / n i n i / ϕ k i , k = 1 , 2 , ,
Q ( T ) = R T .

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