Abstract

The specified reflectance of a multilayer optical filter is formulated as a Fourier series. The coefficients are related via sample relations to the necessary refractive indices of the layers. The method is approximate but has the advantage of simplicity and efficiency, which makes it a useful design tool. The method is also useful as a companion to the exact methods of synthesis since it can provide the starting values for these exact methods. A versatile design example is solved to demonstrate the utility of the method. It is also shown that once a set of refractive indices is found, an infinite set of solutions can be found by simple scaling of the refractive indices.

© 1983 Optical Society of America

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References

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  1. See, for example, the excellent review article by E. Delano, R. J. Pegis, in Progress in Optics, Vol. 4, E. Wolf, Ed. (North-Holland, Amsterdam, 1969), p. 67.
    [CrossRef]
  2. E. Delano, J. Opt. Soc. Am. 57, 1529 (1967).
    [CrossRef]
  3. P. H. Berning, Theory and Calculations of Optical Thin Films in Physics of Thin Films, Vol. 1, G. Hass, Ed. (Academic, New York, 1963), p. 69.
  4. See, for example, CRC Standard Mathematical Tables, W. H. Peyer, Ed. (CRC Press, Cleveland, Ohio, 1979), p. 465.
  5. L. D. Faddeev, J. Math. Phys. 4, 72 (1963).
    [CrossRef]
  6. See, for example, the review article by P. K. Tien, Rev. Mod. Phys. 49, 361 (1977).
    [CrossRef]

1977 (1)

See, for example, the review article by P. K. Tien, Rev. Mod. Phys. 49, 361 (1977).
[CrossRef]

1967 (1)

1963 (1)

L. D. Faddeev, J. Math. Phys. 4, 72 (1963).
[CrossRef]

Berning, P. H.

P. H. Berning, Theory and Calculations of Optical Thin Films in Physics of Thin Films, Vol. 1, G. Hass, Ed. (Academic, New York, 1963), p. 69.

Delano, E.

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

See, for example, the excellent review article by E. Delano, R. J. Pegis, in Progress in Optics, Vol. 4, E. Wolf, Ed. (North-Holland, Amsterdam, 1969), p. 67.
[CrossRef]

Faddeev, L. D.

L. D. Faddeev, J. Math. Phys. 4, 72 (1963).
[CrossRef]

Pegis, R. J.

See, for example, the excellent review article by E. Delano, R. J. Pegis, in Progress in Optics, Vol. 4, E. Wolf, Ed. (North-Holland, Amsterdam, 1969), p. 67.
[CrossRef]

Tien, P. K.

See, for example, the review article by P. K. Tien, Rev. Mod. Phys. 49, 361 (1977).
[CrossRef]

J. Math. Phys. (1)

L. D. Faddeev, J. Math. Phys. 4, 72 (1963).
[CrossRef]

J. Opt. Soc. Am. (1)

Rev. Mod. Phys. (1)

See, for example, the review article by P. K. Tien, Rev. Mod. Phys. 49, 361 (1977).
[CrossRef]

Other (3)

P. H. Berning, Theory and Calculations of Optical Thin Films in Physics of Thin Films, Vol. 1, G. Hass, Ed. (Academic, New York, 1963), p. 69.

See, for example, CRC Standard Mathematical Tables, W. H. Peyer, Ed. (CRC Press, Cleveland, Ohio, 1979), p. 465.

See, for example, the excellent review article by E. Delano, R. J. Pegis, in Progress in Optics, Vol. 4, E. Wolf, Ed. (North-Holland, Amsterdam, 1969), p. 67.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Multilayer planar filter with (m + 1) layers. n0 is the refractive index of the superstrate, while nm corresponds to the refractive index of the substrate. rj is the total reflection at the boundary j.

Fig. 2
Fig. 2

Schematic representation of Eq. (3).

Fig. 3
Fig. 3

Design goal of a general bandpass (or band-rejection) filter. Note that the period is (2k0).

Fig. 4
Fig. 4

Performance of a filter synthesized by using four films (excluding the superstrate and the substrate). The parameters used are A = 0.5, W = 0.8, and n0 = 1.5. The dashed curve corresponds to the performance according to the exact relation of Eq. (1), while the crosses correspond to the predictions of Eq. (3).

Fig. 5
Fig. 5

Same as in Fig. 4 but for nine films.

Fig. 6
Fig. 6

Same as in Fig. 5 but for A = 0.8.

Fig. 7
Fig. 7

Same as in Fig. 5 but for A = 1.0.

Equations (26)

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r j - 1 = f j - 1 + r j exp ( - 2 i ϕ j ) 1 + f j - 1 r j exp ( - 2 i ϕ j ) .
r j - 1 = 1 f j - 1 + ( f j - 1 - 1 f j - 1 ) [ 1 + f j - 1 r j exp ( - 2 i ϕ j ) ] - 1 .
r j - 1 = f j - 1 + ( 1 - f j - 1 2 ) α r j - f j - 1 ( 1 - f j - 1 2 ) α 2 r j 2 + ,
r j - 1 = f j - 1 + β j - 1 α r j ,
r 0 = j = 0 M - 1 u j α j = j = 0 M - 1 u j exp [ - i ( 2 ϕ ) j ] ,
u j = β 0 β 1 , , β j - 1 f j .
r 0 ( k ) = j = - l l C j exp [ - i ( k / k 0 ) j ] ,
r 0 ( k ) = exp [ i ( k / k 0 ) / l ] j = 0 2 l C j - l exp [ - i ( k / k 0 ) j ] .
r 0 ( k ) = j = 0 2 l C j - l exp [ - i ( k / k 0 ) j ] .
u j = C j - l ,
n j h j = ½ k 0 ,
M = 2 l + 1.
f j = u j / p = 0 j - i ( 1 - f p 2 ) = ( u j u j - 1 ) ( f j - 1 1 - f j - 1 2 ) .
f j = n j - n j + 1 n j + n j + 1 ,
n j + 1 = n j ( 1 - f j 1 + f j ) .
r 0 ( k ) = A { ½ - 4 π 2 ( 1 - 2 a ) j = 1 , 3 , 5 l [ cos ( j π a ) j 2 ] cos ( j π k / k 0 ) } ,
a = W / 2 k .
u j = - 2 A cos [ ( j - l ) π a ] π 2 ( 1 - 2 a ) ( j - l ) 2 ;             j l ;
u l = A / 2.
u j = 0.
r 0 ( k ) = d 0 + j = 1 l d j cos ( j π k / k 0 ) ,
u l = d 0 ,
u j = ½ d j - l ;             j l .
f j = n j - n j + 1 n j + n j + 1 = n j δ - n j + 1 δ n j δ + n j + 1 δ , ϕ j = k ( n j δ ) ( h j / δ ) = k n j h j ,
r 0 = s + r 0 1 + s r 0 ,
r 0 r 0 + s .

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