Abstract

A Fourier analysis of existing and potentially complex refractive index profiles is often useful in the resolution of thin-film design problems. Applications of a frequency-filtering technique, which consists of the suppression of undesired spatial frequencies from the refractive index profiles, are described. A new, extremely simple antireflection coating synthesis method based on this concept is proposed.

© 1995 Optical Society of America

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References

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  1. W. H. Southwell, “Using apodization functions to reduce sidelobes in rugate filters,” Appl. Opt. 89, 5091–5094 (1989).
    [CrossRef]
  2. M. F. Ouellette, R. V. Lang, K. L. Yan, R. W. Bertram, R. S. Owles, “Experimental studies of inhomogeneous coatings for optical applications,” J. Vac. Sci. Technol. A 9, 1188–1193 (1989).
    [CrossRef]
  3. J. A. Dobrowolski, P. G. Verly, eds., Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, Proc. Soc. Photo-Opt. Instrum. Eng.2046 (1993).
  4. B. G. Bovard, “Rugate filter design: the modified Fourier transform technique,” Appl. Opt. 29, 24–30 (1990).
    [CrossRef] [PubMed]
  5. B. G. Bovard, “Rugate filter theory: an overview,” Appl. Opt. 33, 5427–5442 (1993).
    [CrossRef]
  6. W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 88–108 (1993).
  7. P. G. Verly, “Design of inhomogeneous and quasi-inhomogeneous optical coatings at the NRC,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 36–45 (1993).
  8. P. G. Verly, J. A. Dobrowolski, “Iterative correction process for optical thin film synthesis with the Fourier transform method,” Appl. Opt. 29, 3672–3684 (1990).
    [CrossRef] [PubMed]
  9. P. G. Verly, A. V. Tikhonravov, A. D. Poezd, “Multiple solutions to the synthesis of graded index optical coatings,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 9–16 (1993).
  10. P. G. Verly, J. A. Dobrowolski, R. R. Willey, “Fourier transform method for the design of wideband antireflection coatings,” Appl. Opt. 31, 3836–3846 (1992).
    [CrossRef] [PubMed]
  11. H. Fabricius, “Gradient-index filters: designing filters with steep skirts, high reflection, and quintic matching layers,” Appl. Opt. 31, 5191–5196 (1992).
    [CrossRef] [PubMed]
  12. W. H. Southwell, R. L. Hall, W. J. Gunning, “Using wavelets to design gradient-index interference coatings,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 46–59 (1993).
  13. J. A. Dobrowolski, “Subtractive method of optical thin film interference filter design,” Appl. Opt. 12, 1885–1893 (1973).
    [CrossRef] [PubMed]
  14. A. V. Tikhonravov, J. A. Dobrowolski, “Quasi-optimal synthesis for antireflection coatings: a new method,” Appl. Opt. 32, 4265–4275 (1993).
    [CrossRef] [PubMed]
  15. J. A. Dobrowolski, A. V. Tikhonravov, “A series of optimal or near-optimal solutions to an antireflection problem,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 62–68 (1993).
  16. P. G. Verly, “Optical thin film optimization by Fourier transform filtering techniques,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 152.

1993 (2)

1992 (2)

1990 (2)

1989 (2)

W. H. Southwell, “Using apodization functions to reduce sidelobes in rugate filters,” Appl. Opt. 89, 5091–5094 (1989).
[CrossRef]

M. F. Ouellette, R. V. Lang, K. L. Yan, R. W. Bertram, R. S. Owles, “Experimental studies of inhomogeneous coatings for optical applications,” J. Vac. Sci. Technol. A 9, 1188–1193 (1989).
[CrossRef]

1973 (1)

Bertram, R. W.

M. F. Ouellette, R. V. Lang, K. L. Yan, R. W. Bertram, R. S. Owles, “Experimental studies of inhomogeneous coatings for optical applications,” J. Vac. Sci. Technol. A 9, 1188–1193 (1989).
[CrossRef]

Bovard, B. G.

Crane, R. L.

W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 88–108 (1993).

Dobrowolski, J. A.

Fabricius, H.

Gunning, W. J.

W. H. Southwell, R. L. Hall, W. J. Gunning, “Using wavelets to design gradient-index interference coatings,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 46–59 (1993).

Hall, R. L.

W. H. Southwell, R. L. Hall, W. J. Gunning, “Using wavelets to design gradient-index interference coatings,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 46–59 (1993).

Johnson, W. E.

W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 88–108 (1993).

Lang, R. V.

M. F. Ouellette, R. V. Lang, K. L. Yan, R. W. Bertram, R. S. Owles, “Experimental studies of inhomogeneous coatings for optical applications,” J. Vac. Sci. Technol. A 9, 1188–1193 (1989).
[CrossRef]

Ouellette, M. F.

M. F. Ouellette, R. V. Lang, K. L. Yan, R. W. Bertram, R. S. Owles, “Experimental studies of inhomogeneous coatings for optical applications,” J. Vac. Sci. Technol. A 9, 1188–1193 (1989).
[CrossRef]

Owles, R. S.

M. F. Ouellette, R. V. Lang, K. L. Yan, R. W. Bertram, R. S. Owles, “Experimental studies of inhomogeneous coatings for optical applications,” J. Vac. Sci. Technol. A 9, 1188–1193 (1989).
[CrossRef]

Poezd, A. D.

P. G. Verly, A. V. Tikhonravov, A. D. Poezd, “Multiple solutions to the synthesis of graded index optical coatings,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 9–16 (1993).

Southwell, W. H.

W. H. Southwell, “Using apodization functions to reduce sidelobes in rugate filters,” Appl. Opt. 89, 5091–5094 (1989).
[CrossRef]

W. H. Southwell, R. L. Hall, W. J. Gunning, “Using wavelets to design gradient-index interference coatings,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 46–59 (1993).

Tikhonravov, A. V.

A. V. Tikhonravov, J. A. Dobrowolski, “Quasi-optimal synthesis for antireflection coatings: a new method,” Appl. Opt. 32, 4265–4275 (1993).
[CrossRef] [PubMed]

P. G. Verly, A. V. Tikhonravov, A. D. Poezd, “Multiple solutions to the synthesis of graded index optical coatings,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 9–16 (1993).

J. A. Dobrowolski, A. V. Tikhonravov, “A series of optimal or near-optimal solutions to an antireflection problem,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 62–68 (1993).

Verly, P. G.

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

P. G. Verly, 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 thin film optimization by Fourier transform filtering techniques,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 152.

P. G. Verly, A. V. Tikhonravov, A. D. Poezd, “Multiple solutions to the synthesis of graded index optical coatings,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 9–16 (1993).

P. G. Verly, “Design of inhomogeneous and quasi-inhomogeneous optical coatings at the NRC,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 36–45 (1993).

Willey, R. R.

Yan, K. L.

M. F. Ouellette, R. V. Lang, K. L. Yan, R. W. Bertram, R. S. Owles, “Experimental studies of inhomogeneous coatings for optical applications,” J. Vac. Sci. Technol. A 9, 1188–1193 (1989).
[CrossRef]

Appl. Opt. (8)

J. Vac. Sci. Technol. A (1)

M. F. Ouellette, R. V. Lang, K. L. Yan, R. W. Bertram, R. S. Owles, “Experimental studies of inhomogeneous coatings for optical applications,” J. Vac. Sci. Technol. A 9, 1188–1193 (1989).
[CrossRef]

Other (7)

J. A. Dobrowolski, P. G. Verly, eds., Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, Proc. Soc. Photo-Opt. Instrum. Eng.2046 (1993).

W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 88–108 (1993).

P. G. Verly, “Design of inhomogeneous and quasi-inhomogeneous optical coatings at the NRC,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 36–45 (1993).

P. G. Verly, A. V. Tikhonravov, A. D. Poezd, “Multiple solutions to the synthesis of graded index optical coatings,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 9–16 (1993).

W. H. Southwell, R. L. Hall, W. J. Gunning, “Using wavelets to design gradient-index interference coatings,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 46–59 (1993).

J. A. Dobrowolski, A. V. Tikhonravov, “A series of optimal or near-optimal solutions to an antireflection problem,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2046, 62–68 (1993).

P. G. Verly, “Optical thin film optimization by Fourier transform filtering techniques,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 152.

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Figures (6)

Fig. 1
Fig. 1

Notch filter designed with the FT method and subsequently smoothed by spatial frequency filtering (see text).

Fig. 2
Fig. 2

Apodized rugate filter matched into the external media, after Refs. 1 and 12.

Fig. 3
Fig. 3

Fourier domain analysis of the smoothed notch filter shown in Fig. 1B: A, wavelet subsystem; B, AR coating subsystem; C, combination of the two subsystems.

Fig. 4
Fig. 4

AR coating design by suppression of the high spatial frequencies σ ≥ 0.6 μm−1: A, starting design; B, C, results without and with design constraints, respectively.

Fig. 5
Fig. 5

AR coating design by suppression of the spatial frequencies in the AR band 0.6 ≤ σ ≤ 1.4 μm−1: A, B, results of single and multiple iterations, respectively.

Fig. 6
Fig. 6

Variation of the rms reflectance in Fig. 5 with the number of frequency-filtering iterations.

Equations (7)

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ln [ n ( x ) n 0 ] FT i π Q ˜ ( R , σ ) σ ,
Q ( σ ) | Q ˜ ( R , σ ) | R T for R 1 ,
n ( x ) = n A ( x ) · n B ( x ) ,
ln [ n ( x ) n 0 ] = ln [ n A ( x ) n 0 ] + ln [ n B ( x ) ] FT i πσ [ Q ˜ ( R A , σ ) + Q ˜ ( R B , σ ) ] .
n 0 = n m n s .
ln [ n A ( x ) n 0 ] FT 1 σ [ ( ln n s n m ) cos ( 2 πσ a ) ] ,
n A ( x ) = n ( x ) / n B ( x ) .

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