Abstract

Multilayer optical filters are characterized by a signal flow graph. Standard feedback–network analysis techniques are modified to yield an iterative algorithm to compute the transmission of filters with any number of layers. The use of iteration results in an economical use of computer storage. The algorithm can take into account the frequency, polarization, and angle of incidence of the incident light wave as well as the index of refraction and thickness of each layer. Dispersion effects can be incorporated. Working programs have been developed for the IBM 1130 computer.

© 1970 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), Chap. 1.
  2. J. A. Dobrowolski, Appl. Opt. 4, 937 (1965).
    [CrossRef]
  3. R. J. Pegis, J. Opt. Soc. Amer. 51, 1255 (1961).
    [CrossRef]
  4. L. A. Catalan, J. Opt. Soc. Amer. 52, 437 (1962).
    [CrossRef]
  5. A. W. Crook, J. Opt. Soc. Amer. 38, 954 (1948).
    [CrossRef]
  6. L. Young, E. G. Cristal, IEEE Trans. Microwave Theory and Tech. 14, 75 (1966).
    [CrossRef]
  7. Ref. 1, footnote on p. 53.
  8. S. J. Mason, Proc. IRE 41, 1144 (1953).
    [CrossRef]
  9. S. J. Mason, Proc. IRE 44, 920 (1956).
    [CrossRef]

1966 (1)

L. Young, E. G. Cristal, IEEE Trans. Microwave Theory and Tech. 14, 75 (1966).
[CrossRef]

1965 (1)

1962 (1)

L. A. Catalan, J. Opt. Soc. Amer. 52, 437 (1962).
[CrossRef]

1961 (1)

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

1956 (1)

S. J. Mason, Proc. IRE 44, 920 (1956).
[CrossRef]

1953 (1)

S. J. Mason, Proc. IRE 41, 1144 (1953).
[CrossRef]

1948 (1)

A. W. Crook, J. Opt. Soc. Amer. 38, 954 (1948).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), Chap. 1.

Catalan, L. A.

L. A. Catalan, J. Opt. Soc. Amer. 52, 437 (1962).
[CrossRef]

Cristal, E. G.

L. Young, E. G. Cristal, IEEE Trans. Microwave Theory and Tech. 14, 75 (1966).
[CrossRef]

Crook, A. W.

A. W. Crook, J. Opt. Soc. Amer. 38, 954 (1948).
[CrossRef]

Dobrowolski, J. A.

Mason, S. J.

S. J. Mason, Proc. IRE 44, 920 (1956).
[CrossRef]

S. J. Mason, Proc. IRE 41, 1144 (1953).
[CrossRef]

Pegis, R. J.

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

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), Chap. 1.

Young, L.

L. Young, E. G. Cristal, IEEE Trans. Microwave Theory and Tech. 14, 75 (1966).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Microwave Theory and Tech. (1)

L. Young, E. G. Cristal, IEEE Trans. Microwave Theory and Tech. 14, 75 (1966).
[CrossRef]

J. Opt. Soc. Amer. (3)

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

L. A. Catalan, J. Opt. Soc. Amer. 52, 437 (1962).
[CrossRef]

A. W. Crook, J. Opt. Soc. Amer. 38, 954 (1948).
[CrossRef]

Proc. IRE (2)

S. J. Mason, Proc. IRE 41, 1144 (1953).
[CrossRef]

S. J. Mason, Proc. IRE 44, 920 (1956).
[CrossRef]

Other (2)

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), Chap. 1.

Ref. 1, footnote on p. 53.

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Figures (3)

Fig. 1
Fig. 1

A one-layer filter.

Fig. 2
Fig. 2

Signal flow graph for a one-layer filter.

Fig. 3
Fig. 3

Signal flow graph for a four-layer filter.

Equations (28)

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d 1 = exp [ i ( 2 π / λ 0 ) n 1 L 1 cos θ 1 ] ,
A 1 = t 01 d 1 t 12 .
G 1 = r 1 r 2 d 1 2 .
Δ 1 = 1 G 1 = 1 + r 1 r 2 d 1 2 .
H 1 = A 1 / Δ 1 = t 01 t 12 d 1 / ( 1 + r 1 r 2 d 1 2 ) .
T = | H 1 | 2 n 2 cos θ 2 n 0 cos θ 0 = t 01 2 t 12 2 | 1 + r 1 r 2 d 1 2 | 2 n 2 cos θ 2 n 0 cos θ 0 .
Δ M = Δ M 1 K = 0 M 1 G K Δ K .
Δ 0 = 1 ,
Δ 1 = 1 + r 1 r 2 d 1 2 .
A 4 = t 01 d 1 t 12 d 2 t 23 d 3 t 34 d 4 t 45 .
Δ 2 = Δ 1 2 G 0 Δ 0 2 G 1 Δ 1 .
2 G 0 = r 1 d 1 2 t 12 t 21 d 2 2 r 3 .
2 G 1 = r 2 d 2 2 r 3 .
Δ 2 = 1 + r 1 r 2 d 1 2 + r 1 d 1 2 t 12 t 21 d 2 2 r 3 + r 2 d 2 2 r 3 ( 1 + r 1 r 2 d 1 2 ) .
Δ 3 = Δ 2 3 G 0 Δ 0 3 G 1 Δ 1 3 G 2 Δ 2 .
3 G 0 = r 1 d 1 2 t 12 t 21 d 2 2 t 23 t 32 d 3 2 r 4 , 3 G 1 = r 2 d 2 2 t 23 t 32 d 3 2 r 4 , 3 G 2 = r 3 d 3 2 r 4 .
Δ 4 = Δ 3 4 G 0 Δ 0 4 G 1 Δ 1 4 G 2 Δ 2 4 G 3 Δ 3 .
4 G 0 = r 1 d 1 2 t 12 t 21 d 2 2 t 23 t 32 d 3 2 t 34 t 43 d 4 2 r 5 , 4 G 1 = r 2 d 2 2 t 23 t 32 d 3 2 t 34 t 43 d 4 2 r 5 ,
H 4 = A 4 / Δ 4 ,
T = ( | A 4 | 2 / | Δ 4 | 2 ) ( n 5 cos θ 5 / n 0 cos θ 0 ) .
M G K = G K t M 1 , M t M , M 1 d M 2 r M + 1 r M .
3 G 1 = r 2 d 2 2 t 23 t 32 d 3 2 r 4 .
4 G 1 = r 2 d 2 2 t 23 t 32 d 3 2 r 4 t 34 t 43 d 4 2 r 5 r 4 .
M G M 1 = r M d M 2 r M + 1 .
t K = ( t K 1 , K t K , K 1 ) 1 2 .
t 2 = ( t 12 t 21 ) 1 2 .
H = A / Δ .
T = | H | 2 = | A / Δ | 2 .

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