Abstract

In Sossi’s formulation of the Fourier transform method of optical multilayer design the refractive-index profile is derived for an inhomogeneous layer of infinite extent having the desired spectral transmittance. This layer is then approximated by a finite system of discrete homogeneous layers. Because it does not make any assumptions about the refractive indices, thicknesses, or number of layers, it is the most powerful analytical method proposed so far. The method has been programmed for a computer and combined with other numerical design procedures. With the program it is possible to design filters with almost any desired transmittance characteristics using realistic refractive indices.

© 1978 Optical Society of America

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References

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  1. J. A. Dobrowolski, Thin Films 34, 313 (1976).
    [CrossRef]
  2. H. Ikeda, Kogaku 3, 178,309 (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, Canada K1A 0S2.
  3. E. Delano, R. J. Pegis, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1969), Vol. 4, p. 67.
    [CrossRef]
  4. Z. Knittl, Optics of Thin Films (Wiley, London, 1976).
  5. E. Delano, Ph.D. thesis, U. Rochester (June1966).
  6. E. Delano, J. Opt. Soc. Am. 57, 1529 (1967).
    [CrossRef]
  7. L. Sossi, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229 (1974);an English translation is available, see Ref. 2.
  8. L. Sossi, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, 171 (1976);an English translation is available, see Ref. 2.
  9. L. Sossi, P. Kard, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 17, 41 (1968);an English translation is available, see Ref. 2.
  10. L. Sossi, P. Kard, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 21, 155 (1972).
  11. R. Jacobsson, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1966), Vol. 5, p. 247.
    [CrossRef]
  12. R. Jacobsson, in Physics of Thin Films, G. Hass, M. H. Francombe, R. W. Hoffman, Eds. (Academic, New York, 1975), Vol. 8, p. 51.
  13. A. Thelen, in Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, New York, 1969), Vol. 5, p. 47.
  14. R. M. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).
  15. J. A. Dobrowolski, Appl. Opt. 12, 1885 (1973).
    [CrossRef] [PubMed]
  16. J. A. Dobrowolski, Appl. Opt. 4, 937 (1965).
    [CrossRef]
  17. J. A. Dobrowolski, Appl. Opt. 9, 1396 (1970).
    [CrossRef] [PubMed]

1976 (2)

L. Sossi, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, 171 (1976);an English translation is available, see Ref. 2.

J. A. Dobrowolski, Thin Films 34, 313 (1976).
[CrossRef]

1974 (2)

H. Ikeda, Kogaku 3, 178,309 (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, Canada K1A 0S2.

L. Sossi, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229 (1974);an English translation is available, see Ref. 2.

1973 (1)

1972 (1)

L. Sossi, P. Kard, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 21, 155 (1972).

1970 (1)

1968 (1)

L. Sossi, P. Kard, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 17, 41 (1968);an English translation is available, see Ref. 2.

1967 (1)

1965 (1)

Bracewell, R. M.

R. M. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

Delano, E.

E. Delano, J. Opt. Soc. Am. 57, 1529 (1967).
[CrossRef]

E. Delano, Ph.D. thesis, U. Rochester (June1966).

E. Delano, R. J. Pegis, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1969), Vol. 4, p. 67.
[CrossRef]

Dobrowolski, J. A.

Ikeda, H.

H. Ikeda, Kogaku 3, 178,309 (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, Canada K1A 0S2.

Jacobsson, R.

R. Jacobsson, in Physics of Thin Films, G. Hass, M. H. Francombe, R. W. Hoffman, Eds. (Academic, New York, 1975), Vol. 8, p. 51.

R. Jacobsson, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1966), Vol. 5, p. 247.
[CrossRef]

Kard, P.

L. Sossi, P. Kard, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 21, 155 (1972).

L. Sossi, P. Kard, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 17, 41 (1968);an English translation is available, see Ref. 2.

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, London, 1976).

Pegis, R. J.

E. Delano, R. J. Pegis, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1969), Vol. 4, p. 67.
[CrossRef]

Sossi, L.

L. Sossi, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, 171 (1976);an English translation is available, see Ref. 2.

L. Sossi, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229 (1974);an English translation is available, see Ref. 2.

L. Sossi, P. Kard, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 21, 155 (1972).

L. Sossi, P. Kard, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 17, 41 (1968);an English translation is available, see Ref. 2.

Thelen, A.

A. Thelen, in Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, New York, 1969), Vol. 5, p. 47.

Appl. Opt. (3)

Eesti NSV Tead. Akad. Toim. Fuus. Mat. (4)

L. Sossi, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229 (1974);an English translation is available, see Ref. 2.

L. Sossi, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, 171 (1976);an English translation is available, see Ref. 2.

L. Sossi, P. Kard, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 17, 41 (1968);an English translation is available, see Ref. 2.

L. Sossi, P. Kard, Eesti NSV Tead. Akad. Toim. Fuus. Mat. 21, 155 (1972).

J. Opt. Soc. Am. (1)

Kogaku (1)

H. Ikeda, Kogaku 3, 178,309 (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, Canada K1A 0S2.

Thin Films (1)

J. A. Dobrowolski, Thin Films 34, 313 (1976).
[CrossRef]

Other (7)

E. Delano, R. J. Pegis, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1969), Vol. 4, p. 67.
[CrossRef]

Z. Knittl, Optics of Thin Films (Wiley, London, 1976).

E. Delano, Ph.D. thesis, U. Rochester (June1966).

R. Jacobsson, in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1966), Vol. 5, p. 247.
[CrossRef]

R. Jacobsson, in Physics of Thin Films, G. Hass, M. H. Francombe, R. W. Hoffman, Eds. (Academic, New York, 1975), Vol. 8, p. 51.

A. Thelen, in Physics of Thin Films, G. Hass, R. E. Thun, Eds. (Academic, New York, 1969), Vol. 5, p. 47.

R. M. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

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Figures (13)

Fig. 1
Fig. 1

Block diagram of the flow of calculations.

Fig. 2
Fig. 2

Refractive-index profiles for filters with different bandwidths and rejections.

Fig. 3
Fig. 3

Effect of higher wavenumber harmonics of the primary transmission curve on the refractive-index profile.

Fig. 4
Fig. 4

Homogeneous layer solutions obtained by the use of the replicating function with wavenumber displacements of 2 μm−1, 1.5 μm−1, and 1 μm−1, respectively.

Fig. 5
Fig. 5

Effect of lower wavenumber harmonics of the primary transmission curve on the refractive-index profile.

Fig. 6
Fig. 6

Effect of the phase factor on the refractive-index profile.

Fig. 7
Fig. 7

Use of the phase factor to control the refractive-index profile.

Fig. 8
Fig. 8

Illustration of refinement by successive approximations (A), (B) and of refractive-index steps that do not change the transmittance in a designated wavenumber range (C).

Fig. 9
Fig. 9

Effect of limiting the optical thickness of the inhomogeneous layer on the transmittance curve.

Fig. 10
Fig. 10

Effect of different approximations of the inhomogeneous layer on the transmittance curve.

Fig. 11
Fig. 11

y ¯ λ filter.

Fig. 12
Fig. 12

Comb filter.

Fig. 13
Fig. 13

Filter with a spectral transmittance curve that approximates the silhouette of the Parliament Buildings, Ottawa.

Equations (14)

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d n d x · 1 2 n · exp ( i k x ) · d x = Q ( k ) · exp [ i ϕ ( k ) ] = f ( k ) ,
x = 2 0 z n ( u ) · d u ,
n ( x ) = exp [ 2 π 0 Q ( k ) ( k ) sin [ ϕ ( k ) k x ] · d k } .
Q ( k ) = { 1 / 2 [ 1 T ( k ) T ( k ) ] } 1 / 2
( 1.35 × 2.35 / n L n H ) 1 / 2 ,
n ( x ) = n A ( x ) · n B ( x ) · n C ( x ) ,
T = 1 ( n s 1 n s + 1 ) 2 ,
n ( x ) = exp ( 1 π o p Q ( k ) · m = 0 m { sin [ ϕ ( k ) ( m p + k ) x ] ( m p + k ) } · d k ) .
n ( x ) = exp [ 4 p m = F ( m p ) ] ,
n ( x ) = n A ( x ) + n B ( x ) + n C ( x ) + ,
ϕ ( k ) = c k ,
ϕ ( k ) = c k sin ( h k ) ,
n 1 ( x ) = N ( x ) · n ( x ) ,
N ( x ) = exp ( 2 j = 1 m c j ) j exp { S i [ k L ( x x j ) ] S i [ k H ( x x j ) ] } .

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