Abstract

The objective of our synthesis technique is to find the refractive indices ni and the thicknesses ei of the layers in a nonabsorbing multilayer stack with prescribed properties. By introducing the constraint that each layer should have an optical thickness niei = ki λ0/4, where ki is an integer and λ0 a simple parameter as small as we wish, we have developed a calculation method for the optical properties, expressed as R/T, which is particularly advantageous. The form of the solution is a series of cosine terms. Given the reflectance R0 and the transmittance T0 of the filter to be synthesized, its R0/T0 curve is expanded in a Fourier series over a given frequency interval. The integers ki and the refractive indices of the layers are then adjusted until the coefficients of the cosine terms approximate those of the Fourier series of the filter being synthesized. The technique is demonstrated in the synthesis of a beamsplitter and a dichroic mirror.

© 1978 Optical Society of America

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References

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  1. H. Pohlack, Jenaer Jahrb. 5, 181–221(1952); A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960); Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).
  2. (a)E. Delano, J. Opt. Soc. Am. 57, 1529–1533(1967); (b)E. Delano and R. J. Pegis, in Progress in Optics, edited by E. Wolf (Wiley, New York, 1969), Vol. 7, p 67.
  3. P. H. Berning, in Physics of Thin Films, Vol. 1, edited by G. Hass (AcademicLondon, 1963); P. Baumeister, Optical Interference Coating Technology. Course given at the University of California, Los Angeles, June 27–July 1, 1977.
  4. P. Baumeister, Opt. Acta 8, 105–109(1961).
    [Crossref]
  5. J. A. Nelder and R. Mead, Comput. J. 7, 308–313(1965).
    [Crossref]
  6. E. Ritter in Physics of Thin Films, Vol. 8, edited by G. Hass (Academic, London, 1975); E. Ritter, in Laser Handbook, edited by F. T. Arrechi and E. O. Schulz-Dubois (North-Holland, Amsterdam, 1972).
  7. C. Dufour and A. Herpin, Opt. Acta 1, 1–8(1954); H. A. Macleod, Thin Film Optical Filters (Hilger, London, 1969).
    [Crossref]
  8. A. F. Turner and P. W. Baumeister, Appl. Opt. 5, 69–76(1966).
    [Crossref] [PubMed]
  9. O. S. Heavens and H. M. Liddell, Appl. Opt. 5, 373–376(1966).
    [Crossref] [PubMed]
  10. E. Pelletier, M. Klapish, and P. Giacomo, Nouv. Rev. Opt. Appl. 2, 247–259(1971); P. Baumeister, J. Opt. Soc. Am. 48, 955–958(1958); O. S. Heavens and H. M. Liddell, Opt. Acta 15, 129–138(1968).
    [Crossref]
  11. J. P. Borgogno and E. Pelletier, Thin Solid Films 34, 357–361(1976).
    [Crossref]
  12. E. Pelletier, P. Roche, and B. Vidal, Nouv. Rev. Opt. 7, 353–362(1976).
    [Crossref]

1976 (2)

J. P. Borgogno and E. Pelletier, Thin Solid Films 34, 357–361(1976).
[Crossref]

E. Pelletier, P. Roche, and B. Vidal, Nouv. Rev. Opt. 7, 353–362(1976).
[Crossref]

1971 (1)

E. Pelletier, M. Klapish, and P. Giacomo, Nouv. Rev. Opt. Appl. 2, 247–259(1971); P. Baumeister, J. Opt. Soc. Am. 48, 955–958(1958); O. S. Heavens and H. M. Liddell, Opt. Acta 15, 129–138(1968).
[Crossref]

1967 (1)

1966 (2)

1965 (1)

J. A. Nelder and R. Mead, Comput. J. 7, 308–313(1965).
[Crossref]

1961 (1)

P. Baumeister, Opt. Acta 8, 105–109(1961).
[Crossref]

1954 (1)

C. Dufour and A. Herpin, Opt. Acta 1, 1–8(1954); H. A. Macleod, Thin Film Optical Filters (Hilger, London, 1969).
[Crossref]

1952 (1)

H. Pohlack, Jenaer Jahrb. 5, 181–221(1952); A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960); Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

Baumeister, P.

P. Baumeister, Opt. Acta 8, 105–109(1961).
[Crossref]

Baumeister, P. W.

Berning, P. H.

P. H. Berning, in Physics of Thin Films, Vol. 1, edited by G. Hass (AcademicLondon, 1963); P. Baumeister, Optical Interference Coating Technology. Course given at the University of California, Los Angeles, June 27–July 1, 1977.

Borgogno, J. P.

J. P. Borgogno and E. Pelletier, Thin Solid Films 34, 357–361(1976).
[Crossref]

Delano, E.

Dufour, C.

C. Dufour and A. Herpin, Opt. Acta 1, 1–8(1954); H. A. Macleod, Thin Film Optical Filters (Hilger, London, 1969).
[Crossref]

Giacomo, P.

E. Pelletier, M. Klapish, and P. Giacomo, Nouv. Rev. Opt. Appl. 2, 247–259(1971); P. Baumeister, J. Opt. Soc. Am. 48, 955–958(1958); O. S. Heavens and H. M. Liddell, Opt. Acta 15, 129–138(1968).
[Crossref]

Heavens, O. S.

Herpin, A.

C. Dufour and A. Herpin, Opt. Acta 1, 1–8(1954); H. A. Macleod, Thin Film Optical Filters (Hilger, London, 1969).
[Crossref]

Klapish, M.

E. Pelletier, M. Klapish, and P. Giacomo, Nouv. Rev. Opt. Appl. 2, 247–259(1971); P. Baumeister, J. Opt. Soc. Am. 48, 955–958(1958); O. S. Heavens and H. M. Liddell, Opt. Acta 15, 129–138(1968).
[Crossref]

Liddell, H. M.

Mead, R.

J. A. Nelder and R. Mead, Comput. J. 7, 308–313(1965).
[Crossref]

Nelder, J. A.

J. A. Nelder and R. Mead, Comput. J. 7, 308–313(1965).
[Crossref]

Pelletier, E.

J. P. Borgogno and E. Pelletier, Thin Solid Films 34, 357–361(1976).
[Crossref]

E. Pelletier, P. Roche, and B. Vidal, Nouv. Rev. Opt. 7, 353–362(1976).
[Crossref]

E. Pelletier, M. Klapish, and P. Giacomo, Nouv. Rev. Opt. Appl. 2, 247–259(1971); P. Baumeister, J. Opt. Soc. Am. 48, 955–958(1958); O. S. Heavens and H. M. Liddell, Opt. Acta 15, 129–138(1968).
[Crossref]

Pohlack, H.

H. Pohlack, Jenaer Jahrb. 5, 181–221(1952); A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960); Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

Ritter, E.

E. Ritter in Physics of Thin Films, Vol. 8, edited by G. Hass (Academic, London, 1975); E. Ritter, in Laser Handbook, edited by F. T. Arrechi and E. O. Schulz-Dubois (North-Holland, Amsterdam, 1972).

Roche, P.

E. Pelletier, P. Roche, and B. Vidal, Nouv. Rev. Opt. 7, 353–362(1976).
[Crossref]

Turner, A. F.

Vidal, B.

E. Pelletier, P. Roche, and B. Vidal, Nouv. Rev. Opt. 7, 353–362(1976).
[Crossref]

Appl. Opt. (2)

Comput. J. (1)

J. A. Nelder and R. Mead, Comput. J. 7, 308–313(1965).
[Crossref]

J. Opt. Soc. Am. (1)

Jenaer Jahrb. (1)

H. Pohlack, Jenaer Jahrb. 5, 181–221(1952); A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960); Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

Nouv. Rev. Opt. (1)

E. Pelletier, P. Roche, and B. Vidal, Nouv. Rev. Opt. 7, 353–362(1976).
[Crossref]

Nouv. Rev. Opt. Appl. (1)

E. Pelletier, M. Klapish, and P. Giacomo, Nouv. Rev. Opt. Appl. 2, 247–259(1971); P. Baumeister, J. Opt. Soc. Am. 48, 955–958(1958); O. S. Heavens and H. M. Liddell, Opt. Acta 15, 129–138(1968).
[Crossref]

Opt. Acta (2)

P. Baumeister, Opt. Acta 8, 105–109(1961).
[Crossref]

C. Dufour and A. Herpin, Opt. Acta 1, 1–8(1954); H. A. Macleod, Thin Film Optical Filters (Hilger, London, 1969).
[Crossref]

Thin Solid Films (1)

J. P. Borgogno and E. Pelletier, Thin Solid Films 34, 357–361(1976).
[Crossref]

Other (2)

P. H. Berning, in Physics of Thin Films, Vol. 1, edited by G. Hass (AcademicLondon, 1963); P. Baumeister, Optical Interference Coating Technology. Course given at the University of California, Los Angeles, June 27–July 1, 1977.

E. Ritter in Physics of Thin Films, Vol. 8, edited by G. Hass (Academic, London, 1975); E. Ritter, in Laser Handbook, edited by F. T. Arrechi and E. O. Schulz-Dubois (North-Holland, Amsterdam, 1972).

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Figures (5)

FIG. 1
FIG. 1

Diagram of a multilayer illustrating the notation used in the text.

FIG. 2
FIG. 2

Flow diagram of computer program for calculating a multilayer system with prescribed optical properties. (Parameters are the thicknesses and indices of layers.)

FIG. 3
FIG. 3

Calculated reflectance as a function of wavelength of the approximate solutions for the beamsplitter at normal incidence consisting of 11 layers for which the values of kj listed from the substrate are 5 5 5 5 5 4 4 4 3 3 3. josa-68-7-964-i001, result obtained with material of indices alternating from 1.7 to 1.3. —, the same filter with successive indices of 1.9 1.5 1.9 1.3 1.7 1.3 1.7 1.3 1.7 1.3 1.9.

FIG. 4
FIG. 4

Calculated reflectance as a function of wavelength of the solution for a dichroic mirror for six wavelengths distributed through the visible and infrared. The mirror has eleven layers of TiO2 and MgF2 and the light is p polarized with angle of incidence 55°. The calculations take into account the dispersion of the indices of refraction of the materials.12

FIG. 5
FIG. 5

Calculated reflectance as a function of wavelength of an appropriate solution for a beamsplitter. The light is p polarized with angle of incidence of 55°. The layers have indices alternating from 2.35 to 1.35 with thicknesses given in the text.

Tables (2)

Tables Icon

TABLE I Design for a dichroic mirror which uses TiO2 and MgF2 on a substrate of index 1.52 (see text).

Tables Icon

TABLE II Design for a beamsplitter for p polarization at oblique incidence. Less than 1 min of computing time was required to establish this design.

Equations (30)

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( τ j - 1 ρ j - 1 ) = C j - 1 ( 1 r j - 1 e - i 4 π n j e j σ r j - 1 e - i 4 π n j e j σ ) ( τ j ρ j )
C j - 1 = n j - 1 + n j 2 ( n j - 1 n j ) 1 / 2             and             r j - 1 = n j - 1 - n j n j - 1 + n j .
n j e j = k j λ 0 / 4 ,
( τ j - 1 ρ j - 1 ) = C j - 1 ( 1 r j - 1 z k j r j - 1 z k j ) ( τ j ρ j )
( τ 0 ρ 0 ) = [ j = 1 P C j - 1 ( 1 r j - 1 z k j r j - 1 z k j ) ] ( 1 r P r P 1 ) ( 1 0 ) .
τ j = l = 1 l = m j α j , l z l             and             ρ j = l = 1 l = m j β j , l z l
( l = 1 l = m j - 1 α j - 1 , l z l l = 1 l = m j - 1 β j - 1 , l z l ) = C j - 1 ( 1 r j - 1 z k j r j - 1 z k j ) ( l = 1 l = m j α j , l z l l = 1 l = m j β j , l z l )
α j - 1 , l = C j - 1 ( α j , l + r j - 1 β j , l - k j ) , β j - 1 , l = C j - 1 ( r j - 1 α j , l + β j , l - k j ) ,
α j , l = 0 if l > m j , β j , l - k j = 0             if l < k j .
N = j = 1 P k j .
τ 0 = l = 1 N α 0 , l z l             and             ρ 0 = l = 1 N β 0 , l z l .
R / T = ½ a 0 + a 1 cos ( π λ 0 σ ) + a N cos ( N π λ 0 σ ) ,
a l = 2 i = 0 i = N - l β 0 , i β 0 , i + l ,
k j λ 0 / 4 = n j e j cos θ j ,
( R T ) s = ½ a 0 , s + l = 1 l = N a l , s cos ( l π λ 0 σ ) , ( R T ) p = ½ a 0 , p + l = 1 l = N a l , p cos ( l π λ 0 σ ) .
( R 0 / T 0 ) ( σ ) = ½ A 0 + A 1 cos ( π λ 0 σ ) + + A l cos ( l π λ 0 σ ) + .
f = m ( a m - A m ) 2 .
Φ ( σ ) = ½ φ 0 + φ 1 cos ( π λ 0 σ ) + + φ m cos ( m π λ 0 σ ) + .
( R 0 / T 0 ) Φ ( σ ) = ½ A 0 + A 1 cos ( π λ 0 σ ) + + A m cos ( m π λ 0 σ ) + .
( R / T ) Φ ( σ ) = ½ a 0 + a 1 cos ( π λ 0 σ ) + + a m cos ( m π λ 0 σ ) + ,
f = m ( a m - A m ) 2 .
( R ˆ / T ) ( x ) = ½ a 0 δ ( x ) + ½ a 1 [ δ ( x - λ 0 / 2 ) + δ ( x + λ 0 / 2 ) ] + + ½ a l [ δ ( x - l λ 0 / 2 ) + δ ( x + l λ 0 / 2 ) ] +
Φ ˆ ( x ) = ½ φ 0 δ ( x ) + ½ φ 1 [ δ ( x - λ 0 / 2 ) + δ ( x + λ 0 / 2 ) ] + + ½ φ l [ δ ( x - l λ 0 / 2 ) + δ ( x + l λ 0 / 2 ) ] + .
a m = 1 2 l = - N N a l φ l + m .
a - l = a l             and             φ - l = φ l .
Substrate ( H 1 L 1 ) ( H 2 L 2 ) ( H i L i ) air .
( 5 5 5 5 5 ) ( 4 4 4 ) ( 3 3 3 ) .
( 1.9 1.5 1.9 1.3 1.7 ) ( 1.3 1.7 1.3 ) ( 1.7 1.3 1.9 ) .
n j e j cos θ j = λ * / 4
n j e j cos θ j = 14 λ 0 / 4