Abstract

Analysis and synthesis operations for a stack of dielectric layers with equal optical thickness are described in terms of the Wiener–Khintchine theorem with variables reflectance R and distribution of optical paths f. The method yields an infinite sequence of refractive indices that converge to a substrate index. An iterative process is used to determine the minimum phase solution, and all practical solutions are constructed from it by a root-shifting procedure. Realizability and practicability conditions are discussed.

© 1978 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. J. Pegis, J. Opt. Soc. Am. 51, 1255 (1961).
    [CrossRef]
  2. P. G. Kard, Opt. Spectrosc. 16, 497 (1964).
  3. E. Delano, J. Opt. Soc. Am. 57, 107 (1967).
    [CrossRef]
  4. E. Delano, R. J. Pegis, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Chap. 7, p. 68.
  5. H. Pohlack, Jenaer Jahrbuch 181 (Zeiss, Jena, 1952).
  6. S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. P., Cambridge, England, 1969).
  7. E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974).
  8. A. V. Aho, J. E. Hopcroft, J. D. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, Mass., 1974), p. 256.
  9. B. P. Demidovich, I. A. Maron, Computational Mathematics (MIR, Moscow, 1976).

1967 (1)

1964 (1)

P. G. Kard, Opt. Spectrosc. 16, 497 (1964).

1961 (1)

Aho, A. V.

A. V. Aho, J. E. Hopcroft, J. D. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, Mass., 1974), p. 256.

Delano, E.

E. Delano, J. Opt. Soc. Am. 57, 107 (1967).
[CrossRef]

E. Delano, R. J. Pegis, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Chap. 7, p. 68.

Demidovich, B. P.

B. P. Demidovich, I. A. Maron, Computational Mathematics (MIR, Moscow, 1976).

Hecht, E.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974).

Hopcroft, J. E.

A. V. Aho, J. E. Hopcroft, J. D. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, Mass., 1974), p. 256.

Kard, P. G.

P. G. Kard, Opt. Spectrosc. 16, 497 (1964).

Lipson, H.

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. P., Cambridge, England, 1969).

Lipson, S. G.

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. P., Cambridge, England, 1969).

Maron, I. A.

B. P. Demidovich, I. A. Maron, Computational Mathematics (MIR, Moscow, 1976).

Pegis, R. J.

R. J. Pegis, J. Opt. Soc. Am. 51, 1255 (1961).
[CrossRef]

E. Delano, R. J. Pegis, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Chap. 7, p. 68.

Pohlack, H.

H. Pohlack, Jenaer Jahrbuch 181 (Zeiss, Jena, 1952).

Ullman, J. D.

A. V. Aho, J. E. Hopcroft, J. D. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, Mass., 1974), p. 256.

Zajac, A.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974).

J. Opt. Soc. Am. (2)

Opt. Spectrosc. (1)

P. G. Kard, Opt. Spectrosc. 16, 497 (1964).

Other (6)

E. Delano, R. J. Pegis, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Chap. 7, p. 68.

H. Pohlack, Jenaer Jahrbuch 181 (Zeiss, Jena, 1952).

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. P., Cambridge, England, 1969).

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974).

A. V. Aho, J. E. Hopcroft, J. D. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, Mass., 1974), p. 256.

B. P. Demidovich, I. A. Maron, Computational Mathematics (MIR, Moscow, 1976).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Beams reflected from a stack of dielectric films of equal optical thickness.

Fig. 2
Fig. 2

Synthesis of a 50% Gaussian reflector: (a) Reflectance data profile, RK = 0.50–0.46 exp(−16K2/N2); (b) and (c) distribution of optical paths f and refractive indices n for the minimum phase solution. (d) Roots of F = 0 plotted in the s plane for the minimum phase solution. All roots with |s| < 5 are plotted. All additional practical solutions can be constructed by moving roots from the shaded area into the circle of unit radius. (e) Real and imaginary parts of the amplitude F for two solutions that satisfy the reflectance profile.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

f = ( 0 , 0 , , 0 , f 0 , f 1 , , f N ) .
F = ( F - N + 1 , F - N + 2 , , F N - 1 , F N ) .
F K = x = - N + 1 N f x exp ( i π K x / N ) ;             K = - N + 1 , N .
f x = 1 2 N K = - N + 1 N F K exp ( - i π K x / N ) ;             x = - N + 1 , N .
R K = F K 2 .
R = FT ( f ) 2 ,             ( analysis ) ;
c ff = FT - 1 ( R ) ,             ( synthesis ) .
f 0 2 + f 1 2 + f 2 2 + f 3 2 + + f N 2 = [ FT - 1 ( R ) ] 0 , f 0 f 1 + f 1 f 2 + f 2 f 3 + + f N - 1 f N = [ FT - 1 ( R ) ] 1 , f 0 f 2 + f 1 f 3 + + f N - 2 f N = [ FT - 1 ( R ) ] 2 ,             f 0 f N = [ F T - 1 ( R ) ] N .
F = f 0 + f 1 s + f N s N = f N ( s - z 1 ) ( s - z N ) .
F = - z [ ( s - 1 / z ) F / ( s - z ) ] ,
1 < s f 0 ( 1 + n p ) / ( 1 - n p ) ,             ( real roots ) ; 1 < s [ f 0 ( 1 + n p ) / ( 1 - n p ) ] 1 / 2 ,             ( complex conjugate roots ) .

Metrics