Abstract

A brief description is given of a method for approximate synthesis of multilayer filters consisting of either homogeneous layers or inhomogeneous films. The method is based on the fact that the amplitude reflectance is the approximate Fourier transform of the function 12U(p)/U(p), where U(p) is the effective refractive index as a function of the optical thickness p. Two forms of the sampling theorem are applied to obtain explicit expressions for the film parameters in terms of the specified reflectivity at certain sampling values of the frequency. Numerical examples are included.

© 1967 Optical Society of America

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References

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  1. E. Delano, Ph. D. thesis, University of Rochester, June1966.
  2. O. S. Heavens, Optical Properties of Thin Solid Films (Dover Publications, Inc., New York, 1965), Sec. 4.10, p. 80.
  3. C. H. Greenewalt, W. Brandt, and D. D. Friel, J. Opt. Soc. Am. 50, 1005 (1960).
    [CrossRef]
  4. R. J. Pegis, J. Opt. Soc. Am. 51, 1255 (1961).
    [CrossRef]
  5. R. J. Pegis (Private communication, July1962).
  6. Z. Knittl, Appl. Opt. 6, 331, (1967).
    [CrossRef] [PubMed]
  7. H. Wolter, in Progress in Optics, Vol. I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), Ch. V, p. 157.
  8. D. A. Linden, Proc. IRE 47, 1219 (1959).
    [CrossRef]
  9. W. Weinstein, Vacuum 4, 3 (1954).
    [CrossRef]
  10. E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964), Appendix, p. 75.
  11. A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill Book Co., New York, 1962), Appendix 1, p. 269.
  12. S. Goldman, Information Theory (Prentice-Hall, Inc., New York, 1954), Secs. 2.1, 2.2; see also Appendix XII, p. 368.
  13. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill Book Co., New York, 1965), Ch. 10.
  14. C. Lanczos, Applied Analysis (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964) Ch. IV.

1967 (1)

1961 (1)

1960 (1)

1959 (1)

D. A. Linden, Proc. IRE 47, 1219 (1959).
[CrossRef]

1954 (1)

W. Weinstein, Vacuum 4, 3 (1954).
[CrossRef]

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill Book Co., New York, 1965), Ch. 10.

Brandt, W.

Delano, E.

E. Delano, Ph. D. thesis, University of Rochester, June1966.

Friel, D. D.

Goldman, S.

S. Goldman, Information Theory (Prentice-Hall, Inc., New York, 1954), Secs. 2.1, 2.2; see also Appendix XII, p. 368.

Greenewalt, C. H.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover Publications, Inc., New York, 1965), Sec. 4.10, p. 80.

Knittl, Z.

Lanczos, C.

C. Lanczos, Applied Analysis (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964) Ch. IV.

Linden, D. A.

D. A. Linden, Proc. IRE 47, 1219 (1959).
[CrossRef]

Linfoot, E. H.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964), Appendix, p. 75.

Papoulis, A.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill Book Co., New York, 1962), Appendix 1, p. 269.

Pegis, R. J.

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

R. J. Pegis (Private communication, July1962).

Weinstein, W.

W. Weinstein, Vacuum 4, 3 (1954).
[CrossRef]

Wolter, H.

H. Wolter, in Progress in Optics, Vol. I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), Ch. V, p. 157.

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Proc. IRE (1)

D. A. Linden, Proc. IRE 47, 1219 (1959).
[CrossRef]

Vacuum (1)

W. Weinstein, Vacuum 4, 3 (1954).
[CrossRef]

Other (9)

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964), Appendix, p. 75.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill Book Co., New York, 1962), Appendix 1, p. 269.

S. Goldman, Information Theory (Prentice-Hall, Inc., New York, 1954), Secs. 2.1, 2.2; see also Appendix XII, p. 368.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill Book Co., New York, 1965), Ch. 10.

C. Lanczos, Applied Analysis (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964) Ch. IV.

E. Delano, Ph. D. thesis, University of Rochester, June1966.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover Publications, Inc., New York, 1965), Sec. 4.10, p. 80.

R. J. Pegis (Private communication, July1962).

H. Wolter, in Progress in Optics, Vol. I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), Ch. V, p. 157.

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Figures (4)

Fig. 1
Fig. 1

Notation scheme.

Fig. 2
Fig. 2

Nine-layer, unsmoothed, high-reflectance filter. Solid curve: exact evaluation using refractive indices obtained from first-order synthesis. Dashed curve: first-order prediction based on sampling procedure.

Fig. 3
Fig. 3

Nine-layer, smoothed, high-reflectance filter. Solid curve: exact evaluation using modified refractive indices to eliminate secondary maxima. Dashed curve: same as solid curve in Fig. 2. Shows effect of smoothing in this case.

Fig. 4
Fig. 4

Inhomogeneous, high-reflectance filter. Solid curve: exact evaluation using inhomogeneous index obtained from first-order synthesis. Dashed curve: first-order prediction based on sampling procedure.

Equations (35)

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E j 1 = t j 1 1 ( e i g j E j + r j 1 e i g i E j + ) E j 1 + = t j 1 1 ( r j 1 e i g j E j + e i g i E j + ) ,
g j = 2 π λ 1 n j h j cos φ j ; u j = { n j cos φ j ( TE wave ) n j sec φ j ( TM wave ) ; r j 1 = ( u j 1 u j ) ( u j 1 + u j ) 1 ; t j 1 = 1 + r j 1 = 2 u j 1 ( u j 1 + u j ) 1 .
R = E 0 E 0 + ; T = 1 E 0 + ; R = | R | 2 ; T = u l + 1 u 0 | T | 2 ,
ρ = ( u 0 / u l + 1 ) 1 2 e i G E 0 ; τ = ( u 0 / u l + 1 ) 1 2 e i G E 0 + ,
R / T = | ρ | 2 ; 1 / T = | τ | 2 .
ρ = f 1 k = 0 l a k x 2 k ; τ = f 1 k = 0 l α k x 2 k ,
f = j = 0 l ( 1 r j 2 ) 1 2
P = j = 1 l n j h j cos φ j = λ ( 2 π ) 1 G = G / π μ
ρ ( μ ) = k = 0 l y k e 2 π i k μ / θ ;
τ ( μ ) = k = 0 l η k e 2 π i k μ / θ
ρ ( μ ) = 0 P F ( p ) e 2 π i μ p d p ;
F ( p ) = ρ ( μ ) e 2 π i μ p d μ ,
p = 0 z n ( z ) cos φ ( z ) d z
y k = r k = ( u k 1 u k ) ( u k 1 + u k ) 1 = 1 2 ( U / U ) d p ,
F ( p ) = 1 2 U ( p ) / U ( p )
U ( p ) = U ( 0 ) exp [ 2 0 p F ( p ) d p ] .
ρ ( μ ) = e π i μ P m = ( 1 ) m ρ ( m P 1 ) sinc ( μ P m ) ,
F ( p ) = P 1 rect ( p P 1 1 2 ) m = ρ ( m P 1 ) e 2 π i m p / P
rect p { 1 | p | < 1 2 0 | p | > 1 2 .
F ( p ) = P 1 rect ( p P 1 1 2 ) { A ( 0 ) cos ψ ( 0 ) + 2 m = 1 A ( m P 1 ) cos [ 2 π m p P 1 + ψ ( m P 1 ) ] } .
ρ ( μ ) = e π i μ P m = N N 1 e π i m l / l + 1 ρ ( m θ / l + 1 ) × sin { ( l + 1 ) π [ μ θ 1 m ( l + 1 ) 1 ] } ( l + 1 ) sin { π [ μ θ 1 m ( l + 1 ) 1 ] } ,
F ( p ) = k = 0 l y k δ ( p k θ 1 ) ,
y k = ( l + 1 ) 1 { A ( 0 ) cos ψ ( 0 ) + 2 m = 1 N A ( m θ / l + 1 ) × cos [ 2 π k m ( l + 1 ) 1 + ψ ( m θ / l + 1 ) ] }
= ( l + 1 ) 1 { A ( 0 ) cos ψ ( 0 ) + 2 m = 1 N A ( m θ / l + 1 ) × cos [ 2 π k m ( l + 1 ) 1 + ψ ( m θ / l + 1 ) ] } + ( 1 ) k A ( θ / 2 ) cos ψ ( θ / 2 ) }
u k + 1 = u k e 2 y k , k = 0 , 1 , , l .
ρ ˆ ( μ ) P rect μ P ρ ( μ ) ,
ρ ˆ ( μ ) = 0 P F ( p ) sinc ( p P 1 ) e 2 π i μ p d p ,
F ˆ F ( p ) = { F ( p ) + 1.696 0 F ( p ) [ 1 sinc ( p P 1 ) ] d p } × sinc ( p P 1 ) rect ( p P 1 1 2 ) .
y ˆ F k = ( y k + [ m = 0 l 1 sinc ( m l 1 ) 1 ] × { m = 0 l y m [ 1 sinc ( m l 1 ) ] } ) sinc ( k l 1 ) , k = 0 , 1 , , l
y k = ( 1 ) k 3 / 10 , k = 0 , 1 , , 9.
n 1 = n 3 = n 5 = n 7 = n 9 = 2.77 n 2 = n 4 = n 6 = n 8 = 1.52.
y ˆ 0 = 0.2744 y ˆ 1 = 0.3191 y ˆ 2 = 0.2373 y ˆ 3 = 0.2693 y ˆ 4 = 0.1935 y ˆ 5 = 0.1837 y ˆ 6 = 0.1135 y ˆ 7 = 0.0805 y ˆ 8 = 0.0335 y ˆ 9 = 0.
n 1 = 2.00 n 2 = 1.45 n 3 = 1.84 n 4 = 1.41 n 5 = 1.71 n 6 = 1.42 n 7 = 1.59 n 8 = 1.47 n 9 = 1.52.
k / P 1 = θ / 2 = 1 2 l / P .
F ( p ) = 4800 rect ( 800 p 1 2 ) cos 8000 π p N ( p ) = 1.52 exp ( 9600 0 p cos 8000 π p d p ) = 1.52 exp ( 0.382 sin 8000 π p ) , 0 p P 1 ,