Abstract

The Fourier transform thin film synthesis method often results in solutions that call for indices that lie outside the range of values of the available materials. To make the resulting refractive index profiles always realizable in our meta-mode sputtering machine, a modified Fourier transform synthesis method is proposed with which the reflectance spectra can be accurately synthesized with controllable and predictable refractive index profiles. In our procedure, an optimal phase function is explored to yield acceptable refractive index profiles. Then the overall thickness is estimated using the Parseval theorem. Finally, several errors inherent to the Fourier transform method, including the imprecision of the spectral function, the truncation of the film and the apodization of the refractive index profiles, are compensated by successive corrections to the magnitude of the spectral function. An explicit iterative formula based on the derivative of the magnitude function is proposed for the compensation of the spectral mismatches. We show with a number of examples that, with the proposed method, it is possible to synthesize gradient-index optical filters with almost any desired spectral performance using experimentally realizable refractive indices.

© 2008 Optical Society of America

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Corrections

Zhanshan Wang, Xinbin Cheng, Bin Fan, George Dobrowolski, and Li Wang, "Gradient-index optical filter synthesis with controllable and predictable refractive index profiles: erratum," Opt. Express 16, 8902-8903 (2008)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-16-12-8902

References

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  1. R. Delano, "Fourier Synthesis of Multilayer Filters," J. Opt. Soc. Am. 57, 1529-1533 (1967).
    [CrossRef]
  2. L. Sossi and P. Kard, "A Method for the Synthesis of Multilayer Interference Coatings," Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229-237 (1974). (An English translation of this paper is available from the Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council, Ottawa, Ontario K1A OS2, Canada.)
  3. J. A. Dobrowolski and D. G. 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. B. Bovard, "Rugate Filter Design: the Modified Fourier Transform Technique," Appl. Opt. 29,24-30 (1990).
    [CrossRef] [PubMed]
  6. H. Fabricius, "Gradient-index filters: designing filters with steep skirts, high reflection, and quintic matching layers," Appl. Opt. 31,5191-5196 (1992).
    [CrossRef] [PubMed]
  7. 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]
  8. C. C. Lee, C. J. Tang, and J. Y. Wu, "Rugate Filter Made with Composite Thin Films by Ion-beam Sputtering," Appl. Opt. 45,1333-1337 (2006).
    [CrossRef] [PubMed]
  9. D. Poitras, S. Larouche, and L. Martinu, "Design and Plasma Deposition of Dispersion-Corrected Multiband Rugate Filters," Appl. Opt. 41,5249-5255 (2002).
    [CrossRef] [PubMed]
  10. 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]
  11. P. G. Verly, "Fourier Transform Approach for Thickness Estimation of Reflecting Interference Filters," Appl. Opt. 32,5636-5641 (2006).
    [CrossRef]
  12. P. G. Verly, "Fourier Transform Approach for Thickness Estimation of Reflecting Interference Filters. 2. Generalized Theory," Appl. Opt. 46,76-83 (2007).
    [CrossRef]
  13. J. Druessel and J. Grantham, "Optimal Phase Modulation for Gradient-index Optical Filters," Opt. Lett. 18, 1583-1585 (1993).
    [CrossRef] [PubMed]
  14. A. V. Tikhonravov, "Some Aspect of Thin-film Optics and Their Applications," Appl. Opt. 32,5417-5426 (1993).
    [CrossRef] [PubMed]
  15. S. Guan and J. Chem, "General phase modulation method for stored waveform inverse Fourier transform excitation for Fourier transform ion cyclotron resonance mass spectrometry," J. Chem. Phys. 91, 775-777 (1989).
    [CrossRef]
  16. S. Guan and R. McIver, Jr., "Optimal phase modulation in stored wave form inverse Fourier transform excitation for Fourier transform mass spectrometry. I. Basic algorithm," J. Chem. Phys. 92, 5841-5846 (1990).
    [CrossRef]
  17. M. Hacker, G. Stobrawa, and T. Feurer, "Iterative Fourier transform algorithm for phase-only pulse shaping," Opt. Express 9, 191-199 (2001).
    [CrossRef] [PubMed]

2007 (1)

2006 (2)

C. C. Lee, C. J. Tang, and J. Y. Wu, "Rugate Filter Made with Composite Thin Films by Ion-beam Sputtering," Appl. Opt. 45,1333-1337 (2006).
[CrossRef] [PubMed]

P. G. Verly, "Fourier Transform Approach for Thickness Estimation of Reflecting Interference Filters," Appl. Opt. 32,5636-5641 (2006).
[CrossRef]

2002 (1)

2001 (1)

1996 (1)

1993 (2)

1992 (1)

1990 (3)

S. Guan and R. McIver, Jr., "Optimal phase modulation in stored wave form inverse Fourier transform excitation for Fourier transform mass spectrometry. I. Basic algorithm," J. Chem. Phys. 92, 5841-5846 (1990).
[CrossRef]

B. Bovard, "Rugate Filter Design: the Modified Fourier Transform Technique," Appl. Opt. 29,24-30 (1990).
[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]

1989 (2)

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]

S. Guan and J. Chem, "General phase modulation method for stored waveform inverse Fourier transform excitation for Fourier transform ion cyclotron resonance mass spectrometry," J. Chem. Phys. 91, 775-777 (1989).
[CrossRef]

1978 (1)

1967 (1)

Bovard, B.

Bulkin, P. V.

Burton, R. L.

Chem, J.

S. Guan and J. Chem, "General phase modulation method for stored waveform inverse Fourier transform excitation for Fourier transform ion cyclotron resonance mass spectrometry," J. Chem. Phys. 91, 775-777 (1989).
[CrossRef]

Delano, R.

Dobrowolski, J. A.

Druessel, J.

Fabricius, H.

Feurer, T.

Grantham, J.

Guan, S.

S. Guan and R. McIver, Jr., "Optimal phase modulation in stored wave form inverse Fourier transform excitation for Fourier transform mass spectrometry. I. Basic algorithm," J. Chem. Phys. 92, 5841-5846 (1990).
[CrossRef]

S. Guan and J. Chem, "General phase modulation method for stored waveform inverse Fourier transform excitation for Fourier transform ion cyclotron resonance mass spectrometry," J. Chem. Phys. 91, 775-777 (1989).
[CrossRef]

Hacker, M.

Lacquet, B. M.

Larouche, S.

Lee, C. C.

Lowe, D. G.

Martinu, L.

McIver, R.

S. Guan and R. McIver, Jr., "Optimal phase modulation in stored wave form inverse Fourier transform excitation for Fourier transform mass spectrometry. I. Basic algorithm," J. Chem. Phys. 92, 5841-5846 (1990).
[CrossRef]

Poitras, D.

Stobrawa, G.

Swart, P. L.

Tang, C. J.

Tikhonravov, A. V.

Verly, P. G.

Wild, W. J.

Wu, J. Y.

Appl. Opt. (11)

J. A. Dobrowolski and D. G. 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]

B. Bovard, "Rugate Filter Design: the Modified Fourier Transform Technique," Appl. Opt. 29,24-30 (1990).
[CrossRef] [PubMed]

H. Fabricius, "Gradient-index filters: designing filters with steep skirts, high reflection, and quintic matching layers," Appl. Opt. 31,5191-5196 (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]

C. C. Lee, C. J. Tang, and J. Y. Wu, "Rugate Filter Made with Composite Thin Films by Ion-beam Sputtering," Appl. Opt. 45,1333-1337 (2006).
[CrossRef] [PubMed]

D. Poitras, S. Larouche, and L. Martinu, "Design and Plasma Deposition of Dispersion-Corrected Multiband Rugate Filters," Appl. Opt. 41,5249-5255 (2002).
[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]

P. G. Verly, "Fourier Transform Approach for Thickness Estimation of Reflecting Interference Filters," Appl. Opt. 32,5636-5641 (2006).
[CrossRef]

P. G. Verly, "Fourier Transform Approach for Thickness Estimation of Reflecting Interference Filters. 2. Generalized Theory," Appl. Opt. 46,76-83 (2007).
[CrossRef]

A. V. Tikhonravov, "Some Aspect of Thin-film Optics and Their Applications," Appl. Opt. 32,5417-5426 (1993).
[CrossRef] [PubMed]

J. Chem. Phys. (2)

S. Guan and J. Chem, "General phase modulation method for stored waveform inverse Fourier transform excitation for Fourier transform ion cyclotron resonance mass spectrometry," J. Chem. Phys. 91, 775-777 (1989).
[CrossRef]

S. Guan and R. McIver, Jr., "Optimal phase modulation in stored wave form inverse Fourier transform excitation for Fourier transform mass spectrometry. I. Basic algorithm," J. Chem. Phys. 92, 5841-5846 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (1)

Opt. Lett. (1)

Other (1)

L. Sossi and P. Kard, "A Method for the Synthesis of Multilayer Interference Coatings," Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229-237 (1974). (An English translation of this paper is available from the Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council, Ottawa, Ontario K1A OS2, Canada.)

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Figures (3)

Fig. 1.
Fig. 1.

Refractive-index profiles and reflectance spectra for a 50% reflector calculated with (a) a zero phase, (b) a SWIFT-phase, (c) a SWIFT-phase combined with iterations. The red dash line is the design target and the black lines are the synthesized spectra.

Fig. 2.
Fig. 2.

Refractive index profiles and reflectance spectra for a house-like reflector calculated with (a) a zero phase, (b) a SWIFT-phase, (c) a SWIFT-phase combined with iterations. The red dash line is the design target and the black lines are synthesized spectra.

Fig. 3.
Fig. 3.

Refractive index profiles and reflectance spectra for a 50% reflector calculated with (a) a SWIFT-phase after apodization, (b) a SWIFT-phase after apodization followed by iterations. The red dash line is the design target and the black lines are synthesized spectra.

Equations (6)

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ln n ( x ) n 0 = i π Q ( σ ) σ · exp { i [ ϕ ( σ ) 2 π σ x ] } · d σ ,
ϕ ( σ ) = x 1 x 0 0 Q ( ξ ) ξ 2 d ξ 0 k 0 ξ Q ( η ) η 2 d η d ξ + x 0 σ + Φ 0 .
Q [ T ( σ ) ] = 0.5 ln [ T ( σ ) ] 1 2 + 0.5 ( 1 T ( σ ) T ( σ ) ) 1 2
Q i + 1 [ T ( σ ) ] = Q i [ T ( σ ) ] + dQ dT · Δ [ T ( σ ) ] i = 1,2,3 . . .
Δ T ( σ ) = T D ( σ ) T i ( σ )
Q 1 [ T ( σ ) ] = Q [ T D ( σ ) ]

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