Abstract

A transmission matrix T is defined as a computational aid for determining refractive indices from the Fourier coefficients of the reflected amplitude in thin-film synthesis. It is shown that the matrix T acts as the term factor in geometric series expansions of the reflected and transmitted amplitudes for a stack of films of equal optical thickness.

© 1982 Optical Society of America

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References

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  1. I. J. Hodgkinson, “Fourier description of analysis and synthesis operations for a stack of thin films of equal optical thickness,” Opt. Lett. 3, 133–135 (1978).
    [CrossRef] [PubMed]
  2. R. J. Pegis, “An exact design method for multilayer dielectric films,” J. Opt. Soc. Am. 51, 1255–1264 (1961).
    [CrossRef]
  3. P. G. Kard, “On the synthesis of multilayer coatings,” Opt. Spectrosc. 16, 497–498 (1964).
  4. Sh. A. Furman, “The use of exact formulae for the synthesis of coatings with given spectral characteristics using layers of equal optical thickness,” Opt. Spectrosc. 22, 344–347 (1967).
  5. E. Delano and R. J. Pegis, “Methods of synthesis for dielectric multilayer filters,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. 7, Chap. 2.
    [CrossRef]
  6. A. V. Aho, J. E. Hopcroft, and J. D. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, Mass., 1974).
  7. M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1964).
  8. H. Pohlack, “Die Synthese optischer Interferenzschichtsysteme mit vorgegekenen Spektraleigenschaften,” Jenaer Jahrb.181–221 (1952).

1978 (1)

1967 (1)

Sh. A. Furman, “The use of exact formulae for the synthesis of coatings with given spectral characteristics using layers of equal optical thickness,” Opt. Spectrosc. 22, 344–347 (1967).

1964 (1)

P. G. Kard, “On the synthesis of multilayer coatings,” Opt. Spectrosc. 16, 497–498 (1964).

1961 (1)

1952 (1)

H. Pohlack, “Die Synthese optischer Interferenzschichtsysteme mit vorgegekenen Spektraleigenschaften,” Jenaer Jahrb.181–221 (1952).

Aho, A. V.

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

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1964).

Delano, E.

E. Delano and R. J. Pegis, “Methods of synthesis for dielectric multilayer filters,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. 7, Chap. 2.
[CrossRef]

Furman, Sh. A.

Sh. A. Furman, “The use of exact formulae for the synthesis of coatings with given spectral characteristics using layers of equal optical thickness,” Opt. Spectrosc. 22, 344–347 (1967).

Hodgkinson, I. J.

Hopcroft, J. E.

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

Kard, P. G.

P. G. Kard, “On the synthesis of multilayer coatings,” Opt. Spectrosc. 16, 497–498 (1964).

Pegis, R. J.

R. J. Pegis, “An exact design method for multilayer dielectric films,” J. Opt. Soc. Am. 51, 1255–1264 (1961).
[CrossRef]

E. Delano and R. J. Pegis, “Methods of synthesis for dielectric multilayer filters,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. 7, Chap. 2.
[CrossRef]

Pohlack, H.

H. Pohlack, “Die Synthese optischer Interferenzschichtsysteme mit vorgegekenen Spektraleigenschaften,” Jenaer Jahrb.181–221 (1952).

Ullman, J. D.

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

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1964).

J. Opt. Soc. Am. (1)

Jenaer Jahrb. (1)

H. Pohlack, “Die Synthese optischer Interferenzschichtsysteme mit vorgegekenen Spektraleigenschaften,” Jenaer Jahrb.181–221 (1952).

Opt. Lett. (1)

Opt. Spectrosc. (2)

P. G. Kard, “On the synthesis of multilayer coatings,” Opt. Spectrosc. 16, 497–498 (1964).

Sh. A. Furman, “The use of exact formulae for the synthesis of coatings with given spectral characteristics using layers of equal optical thickness,” Opt. Spectrosc. 22, 344–347 (1967).

Other (3)

E. Delano and R. J. Pegis, “Methods of synthesis for dielectric multilayer filters,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. 7, Chap. 2.
[CrossRef]

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

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1964).

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Figures (1)

Fig. 1
Fig. 1

Distribution of optical paths for light reflected (fi) and transmitted (gi) by a stack of films of equal optical thickness. The fi and gi are the Fourier coefficients of the reflected amplitude F and transmitted amplitude G, respectively. Paths with transmittance T21, T22, etc. occur repeatedly in the analysis of the structure.

Equations (25)

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F = f 0 + f 1 s + f 2 s 2 +
G = s N / 2 ( g 0 + g 1 s + g 2 s 2 + ) ,
r i , i + 1 = ( n i - n i + 1 ) / ( n i + n i + 1 ) , t i , i + 1 = 2 n i / ( n i + n i + 1 ) .
f 0 = r 01 = n 0 - n 1 n 0 + n 1 ,
f 1 = t 01 r 12 t 10 = 2 n 0 ( n 0 + n 1 ) ( n 1 - n 2 ) ( n 1 + n 2 ) 2 n 1 ( n 0 + n 1 ) ,
f 2 = t 01 t 12 r 23 t 21 t 10 + t 01 r 12 r 10 r 12 t 10 .
T = ( T 11 T 12 0 0 0 . . T 21 T 22 T 23 0 0 . . T 31 T 32 T 33 T 34 0 . . . . . . . . . ) .
T i j = r i , i + 1 r j , j - 1 l = j + 1 i t l , l - 1 , j < i = r i , i + 1 r i , i - 1 , j = i = t i , i + 1 , j = i + 1 = 0 , j > i + 1 ,
( x 1 , k + 1 , x 2 , k + 1 , ) = ( x 1 k , x 2 k , ) T .
f k = t 10 r 10 ( x 1 k T 11 + x 2 k T 21 + x k k T k 1 ) ,
n j = ( 2 t j - 1 , j - 1 ) n j - 1 ,             j = 1 , , N + 1
t 01 ( 1 , 0 , 0 , , 0 ) ,
( x 1 , k + 1 , x 2 , k + 1 ) = t 01 ( 1 , 0 , 0 ) T k .
f 0 = r 01 , f k = t 10 r 10 x 1 , k + 1 = t 01 t 10 r 10 ( T k ) 11 ,             k > 0 , g k = t N , N + 1 x N , N + k = t 01 t N , N + 1 ( T N + k - 1 ) 1 N .
Thus             F = f 0 + f 1 s + f 2 s 2 + = r 01 + t 01 t 10 r 10 ( s T + s 2 T 2 + ) 11 ,
F = [ ( - r 10 I + s T r 10 ) ( I - s T ) - 1 ] 11 ,
G = t 01 t N , N + 1 s N / 2 [ T N - 1 ( I - s T ) - 1 ] 1 N .
F = ( - r 10 + r 12 s ) ( 1 - r 10 r 12 s ) - 1
G = t 01 t 12 s 1 / 2 ( 1 - r 10 r 12 s ) - 1 .
G = t 01 t N , N + 1 s N / 2 s N - 1 [ ( I - s T ) - 1 - I - s T - s N - 2 T N - 2 ] 1 N
( T k ) 1 N = 0 ,             0 k N - 2 ,
[ ( I - s T ) - 1 ] 1 N = s N - 1 T 12 T N - 1 , N det ( I - s T ) .
G = s N / 2 t 01 T 12 T N - 1 , N t N , N + 1 det ( I - s T ) ,
F = - r 10 + ( T 11 r 10 + r 10 T 22 ) s - ( T 11 T 22 - T 12 T 21 ) r 10 s 2 1 - ( T 11 + T 22 ) s + ( T 11 T 22 - T 12 T 21 ) s 2 , G = s N / 2 t 01 T 12 t 23 1 - ( T 11 + T 22 ) s + ( T 11 T 22 - T 12 T 21 ) s 2 .
1 / G = 0