Abstract

Inverse Fourier transform has been used to derive the gradient-index profiles of inhomogeneous films having spectral requirements. Two examples are given, and the corresponding experimental designs are presented. Results show a good agreement with the theory and evidences the reliability of the technology used to produce inhomogeneous media.

© 1987 Optical Society of America

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References

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  1. R. Jacobsson, “Inhomogeneous and Coevaporated Homogeneous Films for Optical Applications,” Phys. Thin Films 8, 51 (1975).
  2. P. W. Baumeister, “Utilization of Kard's Equations to Suppress the High Frequency Reflectance Bands of Periodic Multilayers,” Appl. Opt. 24, 2687 (1985).
    [CrossRef] [PubMed]
  3. W. H. Southwell, “Gradient-Index Antireflection Coatings,” Opt. Lett. 8, 584 (1983).
    [CrossRef] [PubMed]
  4. H. Sankur, W. H. Southwell, “Broadband Gradient—Index Antireflection Coating for ZnSe,” Appl. Opt. 23, 2770 (1984).
    [CrossRef] [PubMed]
  5. P. W. Baumeister, “Simulation of a Rugate Filter via a Stepped-Index Dielectric Multilayer,” Appl. Opt. 25, 2644 (1986).
    [CrossRef] [PubMed]
  6. J. A. Dobrowolski, D. G. Lowe, “Optical Thin Film Synthesis Program Based on the Use of the Fourier Transforms,” Appl. Opt. 17, 3039 (1978).
    [CrossRef] [PubMed]
  7. J. A. Dobrowolski, “Comparison of the Fourier Transform and Flip-Flop Thin-Film Synthesis Methods,” Appl. Opt. 25, 1966 (1986).
    [CrossRef] [PubMed]
  8. E. Delano, “Fourier Synthesis of Multilayer Filter,” V.O.S.A. 57, 1529 (1967).
    [CrossRef]
  9. L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filter,” Eesti Tead. Akad. Toim Fuus, Mat. 25, 171 (1976).
  10. P. Rouard, “Etude des propriétés optiques des lames métalliques très minces,” Ann. Phys. 7, 291 (1937).
  11. G. Boivin, “Automatic Variable Rate Evaporation Source for Thin Films Deposition,” Canadian Patent1,185,782,G. Boivin, U.S. Patent4579083.granted to Canadian Patent and Dev., Ltd., Ottawa, Canada.
  12. D. St.-Germain, “Méthode de fabrication et propriétés optiques de couches minces inhomogènes,” Ph.D. Thesis, U. Laval (1985).
  13. D. St.-Germain, G. Boivin, “Indices de réfraction des mélanges MgF2-PbCl2 and Na3AlF6-PbCl2,” Can. J. Phys. 64, 316 (1986).
    [CrossRef]
  14. R. Jacobsson, “Optical Properties or Periodically Stratified Media with Continuously Varying Refractive Index,” Ark. Fys. 31, 191 (1966).

1986 (3)

P. W. Baumeister, “Simulation of a Rugate Filter via a Stepped-Index Dielectric Multilayer,” Appl. Opt. 25, 2644 (1986).
[CrossRef] [PubMed]

D. St.-Germain, G. Boivin, “Indices de réfraction des mélanges MgF2-PbCl2 and Na3AlF6-PbCl2,” Can. J. Phys. 64, 316 (1986).
[CrossRef]

J. A. Dobrowolski, “Comparison of the Fourier Transform and Flip-Flop Thin-Film Synthesis Methods,” Appl. Opt. 25, 1966 (1986).
[CrossRef] [PubMed]

1985 (1)

1984 (1)

H. Sankur, W. H. Southwell, “Broadband Gradient—Index Antireflection Coating for ZnSe,” Appl. Opt. 23, 2770 (1984).
[CrossRef] [PubMed]

1983 (1)

1978 (1)

1976 (1)

L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filter,” Eesti Tead. Akad. Toim Fuus, Mat. 25, 171 (1976).

1975 (1)

R. Jacobsson, “Inhomogeneous and Coevaporated Homogeneous Films for Optical Applications,” Phys. Thin Films 8, 51 (1975).

1967 (1)

E. Delano, “Fourier Synthesis of Multilayer Filter,” V.O.S.A. 57, 1529 (1967).
[CrossRef]

1966 (1)

R. Jacobsson, “Optical Properties or Periodically Stratified Media with Continuously Varying Refractive Index,” Ark. Fys. 31, 191 (1966).

1937 (1)

P. Rouard, “Etude des propriétés optiques des lames métalliques très minces,” Ann. Phys. 7, 291 (1937).

Baumeister, P. W.

Boivin, G.

D. St.-Germain, G. Boivin, “Indices de réfraction des mélanges MgF2-PbCl2 and Na3AlF6-PbCl2,” Can. J. Phys. 64, 316 (1986).
[CrossRef]

G. Boivin, “Automatic Variable Rate Evaporation Source for Thin Films Deposition,” Canadian Patent1,185,782,G. Boivin, U.S. Patent4579083.granted to Canadian Patent and Dev., Ltd., Ottawa, Canada.

Delano, E.

E. Delano, “Fourier Synthesis of Multilayer Filter,” V.O.S.A. 57, 1529 (1967).
[CrossRef]

Dobrowolski, J. A.

Jacobsson, R.

R. Jacobsson, “Inhomogeneous and Coevaporated Homogeneous Films for Optical Applications,” Phys. Thin Films 8, 51 (1975).

R. Jacobsson, “Optical Properties or Periodically Stratified Media with Continuously Varying Refractive Index,” Ark. Fys. 31, 191 (1966).

Lowe, D. G.

Rouard, P.

P. Rouard, “Etude des propriétés optiques des lames métalliques très minces,” Ann. Phys. 7, 291 (1937).

Sankur, H.

H. Sankur, W. H. Southwell, “Broadband Gradient—Index Antireflection Coating for ZnSe,” Appl. Opt. 23, 2770 (1984).
[CrossRef] [PubMed]

Sossi, L.

L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filter,” Eesti Tead. Akad. Toim Fuus, Mat. 25, 171 (1976).

Southwell, W. H.

H. Sankur, W. H. Southwell, “Broadband Gradient—Index Antireflection Coating for ZnSe,” Appl. Opt. 23, 2770 (1984).
[CrossRef] [PubMed]

W. H. Southwell, “Gradient-Index Antireflection Coatings,” Opt. Lett. 8, 584 (1983).
[CrossRef] [PubMed]

St.-Germain, D.

D. St.-Germain, G. Boivin, “Indices de réfraction des mélanges MgF2-PbCl2 and Na3AlF6-PbCl2,” Can. J. Phys. 64, 316 (1986).
[CrossRef]

D. St.-Germain, “Méthode de fabrication et propriétés optiques de couches minces inhomogènes,” Ph.D. Thesis, U. Laval (1985).

Ann. Phys. (1)

P. Rouard, “Etude des propriétés optiques des lames métalliques très minces,” Ann. Phys. 7, 291 (1937).

Appl. Opt. (2)

H. Sankur, W. H. Southwell, “Broadband Gradient—Index Antireflection Coating for ZnSe,” Appl. Opt. 23, 2770 (1984).
[CrossRef] [PubMed]

P. W. Baumeister, “Simulation of a Rugate Filter via a Stepped-Index Dielectric Multilayer,” Appl. Opt. 25, 2644 (1986).
[CrossRef] [PubMed]

Appl. Opt. (3)

Ark. Fys. (1)

R. Jacobsson, “Optical Properties or Periodically Stratified Media with Continuously Varying Refractive Index,” Ark. Fys. 31, 191 (1966).

Can. J. Phys. (1)

D. St.-Germain, G. Boivin, “Indices de réfraction des mélanges MgF2-PbCl2 and Na3AlF6-PbCl2,” Can. J. Phys. 64, 316 (1986).
[CrossRef]

Eesti Tead. Akad. Toim Fuus, Mat. (1)

L. Sossi, “On the Theory of the Synthesis of Multilayer Dielectric Light Filter,” Eesti Tead. Akad. Toim Fuus, Mat. 25, 171 (1976).

Opt. Lett. (1)

Phys. Thin Films (1)

R. Jacobsson, “Inhomogeneous and Coevaporated Homogeneous Films for Optical Applications,” Phys. Thin Films 8, 51 (1975).

V.O.S.A. (1)

E. Delano, “Fourier Synthesis of Multilayer Filter,” V.O.S.A. 57, 1529 (1967).
[CrossRef]

Other (2)

G. Boivin, “Automatic Variable Rate Evaporation Source for Thin Films Deposition,” Canadian Patent1,185,782,G. Boivin, U.S. Patent4579083.granted to Canadian Patent and Dev., Ltd., Ottawa, Canada.

D. St.-Germain, “Méthode de fabrication et propriétés optiques de couches minces inhomogènes,” Ph.D. Thesis, U. Laval (1985).

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Figures (11)

Fig. 1
Fig. 1

Reflection on sublayers equivalent to a graded-index film.

Fig. 2
Fig. 2

(a) Spectral function Q(σ) equivalent to an approximate narrowband filter. (b) Derivative of the logarithm of the refractive index corresponding to the function shown in (a) as a function of the optical thickness γ.

Fig. 3
Fig. 3

(a) Refractive index for an approximate narrowband filter as a function of the mechanical thickness. (b) Corresponding reflectance calculated by the sublayers approximation and the Fourier transform.

Fig. 4
Fig. 4

(a) Refractive index for an approximate broadband filter as a function of the mechanical thickness. (b) Corresponding reflectance calculated by the sublayers approximation and Fourier transform.

Fig. 5
Fig. 5

Schematic diagram of the system used to evaporate films of variable composition.

Fig. 6
Fig. 6

Reference voltage for controlling the rate of deposition of lead chloride to produce a narrowband filter.

Fig. 7
Fig. 7

Rate of deposition as measured by the quartz crystal monitor during the deposition of lead chloride in the case of a narrowband filter.

Fig. 8
Fig. 8

Transmission spectrum of an experimental narrowband filter.

Fig. 9
Fig. 9

Reference voltage for controlling the rate of deposition of lead chloride to produce a broadband filter.

Fig. 10
Fig. 10

Rate of deposition as measured by the quartz crystal monitor during the deposition of lead chloride in the case of a broadband filter.

Fig. 11
Fig. 11

Transmission spectrum of an experimental broadband filter.

Equations (25)

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d F = n ( z ) n ( z + d z ) n ( z ) + n ( z + d z ) = 1 2 n ( z ) d n ( z ) .
F ( k ) = + n ( z ) 2 n ( z ) exp [ i 2 k 0 z n ( z ) d z ] d z .
γ = 2 0 z n ( z ) d z ,
1 2 n ( z ) d n ( z ) d z d z = 1 2 n ( γ ) d n ( γ ) d γ d γ ,
F ( σ ) = + 1 2 n ( γ ) d n ( γ ) d γ exp ( i k γ ) d γ ,
σ = 1 λ and k = 2 π λ .
F ( σ ) = Q ( σ ) exp [ i φ ( σ ) ] .
n ( γ ) = exp 0 2 + Q ( σ ) exp [ i ( φ k γ ) ] d σ d γ .
Q ( σ ) = ( R T ) 1 / 2 ,
Q ( σ ) = [ 1 2 ( 1 T T ) ] 1 / 2 ,
Q ( σ ) = A Δ γ 2 [ sin π Δ γ ( σ + σ 0 ) π Δ γ ( σ + σ 0 ) + sin π Δ γ ( σ σ 0 ) π Δ γ ( σ σ 0 ) ] ,
n ( γ ) 2 n ( γ ) = A П ( γ Δ γ ) cos 2 π σ 0 γ ,
П ( γ Δ γ ) = 1 for Δ γ 2 < γ < Δ γ 2 = 0 elsewhere ,
n ( γ ) = exp ( A π σ 0 sin 2 π σ 0 γ ) exp B .
B = 1 2 ln n H n L and A π σ 0 = 1 2 ln n H n L .
Z p = Z p 1 + δ γ 2 n ̅ ,
Q ( σ ) = A 2 Δ σ [ π ( σ + σ 0 Δ σ ) + П ( σ σ 0 Δ σ ) ] ,
n 2 n ( γ ) ( γ ) = A sin π Δ σ γ π Δ σ γ cos 2 π σ 0 γ
ln n ( γ ) = 2 A σ 2 Δ σ 0 d γ sin 2 π γ σ 2 2 π γ σ 2 2 A σ 1 Δ σ 0 d γ sin 2 π γ σ 1 2 π γ σ 2 .
y 1 = 2 π γ σ 1 y 2 = 2 π γ σ 2 ,
n ( γ ) = exp A π Δ σ [ S i ( y 2 ) S i ( y 1 ) ] exp B ,
B = 1 2 ln N M N m , A M π Δ σ = 1 2 In N M N m .
N s = ( N M N m ) 1 / 2 = N ( 0 ) .
H L + ( 1 L ) H = C D H L D + ( 1 L ) H ,
τ p = C τ c 1 C

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