Abstract

Fourier transforms provide a basis for the design of gradient-index optical filters. A variety of techniques that differ in their treatment of the complex part or phase of the transformed refractive-index profile are reported. Here we describe a method of using the phase of the index profile as a variable to permit a closed-form, constrained optimization of rugate filters. Use of an optimal phase function in Fourier-based filter designs reduces the product of index contrast and thickness for desired reflectance spectra. The shape of the reflectance spectrum is recovered with greater fidelity by suppression of Gibbs oscillations and shifting of sidelobes into desired wavelength regions.

© 1993 Optical Society of America

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References

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  1. L. Sossi, P. Kard, Eestvi NSV Tead. Akad. Toim. Fuus. Mat. 17, 41 (1968). (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.)
  2. L. Sossi, Eestvi NSV Tead. Akad. Toim. Fuus. Mat. 23, 229 (1974).
  3. L. Sossi, Eestvi NSV Tead. Akad. Toim. Fuus. Mat. 25, 171 (1976).
  4. B. Bovard, Appl. Opt. 27, 3062 (1988).
    [CrossRef] [PubMed]
  5. B. Bovard, Barr Associates, Bedford, Mass. 01730 (personal communication, 1993).
  6. B. Bovard, Appl. Opt. 27, 1998 (1988).
    [CrossRef] [PubMed]
  7. B. Bovard, Appl. Opt. 29, 24 (1990).
    [CrossRef] [PubMed]
  8. J. A. Dobrowolski, D. Lowe, Appl. Opt. 17, 3039 (1978).
    [CrossRef] [PubMed]
  9. P. G. Verly, J. A. Dobrowolski, W. J. Wild, R. L. Burton, Appl. Opt. 28, 2864 (1989).
    [CrossRef] [PubMed]
  10. H. Fabricius, Appl. Opt. 31, 5191 (1992).
    [CrossRef] [PubMed]
  11. P. G. Verly, J. A. Dobrowolski, Appl. Opt. 29, 3672 (1990).
    [CrossRef] [PubMed]
  12. S. Guan, J. Chem. Phys. 91, 775 (1989).
    [CrossRef]
  13. S. Guan, R. McIver, J. Chem. Phys. 92, 5841 (1990).
    [CrossRef]
  14. K. Reihl, “Collisional detachment of negative ions using Fourier transform mass spectrometry,” Ph.D. dissertation (U.S. Air Force Institute of Technology, Dayton, Ohio, 1992).
  15. H. A. Mcleod, Thin-Film Optical Filters (Macmillan, New York, 1986), Chap. 2, pp. 11–48.
  16. Pro-Matlab version 3.5i, The Math Works, Inc., Natick, Mass.
  17. R. Jacobsson, in Physics of Thin Films, G. Haas, M. Francombe, R. Hoffman, eds. (Academic, New York, 1975), Vol. 8, pp. 51–97.

1992 (1)

1990 (3)

1989 (2)

1988 (2)

1978 (1)

1976 (1)

L. Sossi, Eestvi NSV Tead. Akad. Toim. Fuus. Mat. 25, 171 (1976).

1974 (1)

L. Sossi, Eestvi NSV Tead. Akad. Toim. Fuus. Mat. 23, 229 (1974).

1968 (1)

L. Sossi, P. Kard, Eestvi NSV Tead. Akad. Toim. Fuus. Mat. 17, 41 (1968). (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.)

Bovard, B.

Burton, R. L.

Dobrowolski, J. A.

Fabricius, H.

Guan, S.

S. Guan, R. McIver, J. Chem. Phys. 92, 5841 (1990).
[CrossRef]

S. Guan, J. Chem. Phys. 91, 775 (1989).
[CrossRef]

Jacobsson, R.

R. Jacobsson, in Physics of Thin Films, G. Haas, M. Francombe, R. Hoffman, eds. (Academic, New York, 1975), Vol. 8, pp. 51–97.

Kard, P.

L. Sossi, P. Kard, Eestvi NSV Tead. Akad. Toim. Fuus. Mat. 17, 41 (1968). (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.)

Lowe, D.

McIver, R.

S. Guan, R. McIver, J. Chem. Phys. 92, 5841 (1990).
[CrossRef]

Mcleod, H. A.

H. A. Mcleod, Thin-Film Optical Filters (Macmillan, New York, 1986), Chap. 2, pp. 11–48.

Reihl, K.

K. Reihl, “Collisional detachment of negative ions using Fourier transform mass spectrometry,” Ph.D. dissertation (U.S. Air Force Institute of Technology, Dayton, Ohio, 1992).

Sossi, L.

L. Sossi, Eestvi NSV Tead. Akad. Toim. Fuus. Mat. 25, 171 (1976).

L. Sossi, Eestvi NSV Tead. Akad. Toim. Fuus. Mat. 23, 229 (1974).

L. Sossi, P. Kard, Eestvi NSV Tead. Akad. Toim. Fuus. Mat. 17, 41 (1968). (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.)

Verly, P. G.

Wild, W. J.

Appl. Opt. (7)

Eestvi NSV Tead. Akad. Toim. Fuus. Mat. (3)

L. Sossi, P. Kard, Eestvi NSV Tead. Akad. Toim. Fuus. Mat. 17, 41 (1968). (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.)

L. Sossi, Eestvi NSV Tead. Akad. Toim. Fuus. Mat. 23, 229 (1974).

L. Sossi, Eestvi NSV Tead. Akad. Toim. Fuus. Mat. 25, 171 (1976).

J. Chem. Phys. (2)

S. Guan, J. Chem. Phys. 91, 775 (1989).
[CrossRef]

S. Guan, R. McIver, J. Chem. Phys. 92, 5841 (1990).
[CrossRef]

Other (5)

K. Reihl, “Collisional detachment of negative ions using Fourier transform mass spectrometry,” Ph.D. dissertation (U.S. Air Force Institute of Technology, Dayton, Ohio, 1992).

H. A. Mcleod, Thin-Film Optical Filters (Macmillan, New York, 1986), Chap. 2, pp. 11–48.

Pro-Matlab version 3.5i, The Math Works, Inc., Natick, Mass.

R. Jacobsson, in Physics of Thin Films, G. Haas, M. Francombe, R. Hoffman, eds. (Academic, New York, 1975), Vol. 8, pp. 51–97.

B. Bovard, Barr Associates, Bedford, Mass. 01730 (personal communication, 1993).

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Figures (2)

Fig. 1
Fig. 1

Refractive-index profile and reflectance spectra for a 90% reflector from 580 to 620 nm calculated with (a) Φ = 0, 20-μm filter, (b) optimal phase, 20-μm filter, (c) optimal phase, 30-μm filter.

Fig. 2
Fig. 2

Refractive-index profiles and reflectance spectra for a 40-μm 99% reflecting Ti:sapphire mirror calculated with (a) Φ = 0 and (b) optimal phase.

Equations (8)

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d n d x 1 2 n ( x ) exp ( ikx ) d x = Q ( k ) exp [ i Φ ( k ) ] ,
ln [ n ( x ) ] = i / π Q ( k ) k exp { i [ Φ ( k ) k x ] } d k ,
x = 2 0 z n ( u ) d u .
d x = c G ( k ) d k ,
x = c 0 k G ( ξ ) d ξ + x 0 .
d Φ ( k ) / d k = x .
Φ ( k ) = x 1 x 0 0 G ( ξ ) d ξ 0 k 0 ξ G ( η ) d η d ξ + x 0 k + Φ 0 .
Q ( k ) = 0.5 [ ln ( T ) ] 1 / 2 + 0.5 ( 1 T T ) 1 / 2 .

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