Abstract

A new approach to inhomogeneous layer synthesis is proposed that differs from the classical Fourier transform methods. This approach, based on the wave-number discretization of the continuous Q function, is described together with the relationship between the smoothness of the inhomogeneous layer’s refractive-index profile, the complexity of the target reflectance specification, and the total optical thickness of the layer. An iterative procedure is introduced to overcome the inaccuracy connected with the approximate Q-function representation. Some control features of the Q-function phase are studied, and numerical examples are given.

© 1994 Optical Society of America

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References

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  1. R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, pp. 247–286.
    [CrossRef]
  2. W. E. Johnson, R. L. Crane, “An overview of rugate filter technology,” in Optical Interference Coatings, Vol. 6 of OSA 1988 Technical Digest Series (Optical Society of America, Washington, D.C., 1988), pp. 118–121.
  3. W. J. Gunning, R. L. Hall, F. J. Woodberry, W. H. Southwell, N. S. Gluck, “Codeposition of continuous composition rugate filters,” Appl. Opt. 28, 2945–2948 (1989).
    [CrossRef] [PubMed]
  4. E. Delano, “Fourier synthesis of multilayer filters,” J. Opt. Soc. Am. 57, 1529–1533 (1967).
    [CrossRef]
  5. L. Sossi, P. Kard, “On the theory of the reflection and transmission of light by thin inhomogeneous films,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 17, 41–48 (1968).
  6. L. Sossi, “A method for the synthesis of multilayer dielectric interference coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229–237 (1974).
  7. J. A. Dobrowolski, D. Lowe, “Optical thin-film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039–3050 (1978).
    [CrossRef] [PubMed]
  8. G. Boivin, D. St.- Germain, “Synthesis of gradient-index profiles corresponding to spectral reflectance derived by inverse Fourier transform,” Appl. Opt. 26, 4209–4213 (1987).
    [CrossRef] [PubMed]
  9. B. G. Bovard, “Rugate filter design: the modified Fourier transform technique,” Appl. Opt. 29, 24–30 (1990).
    [CrossRef] [PubMed]
  10. P. G. Verly, J. A. Dobrowolski, “Iterative correction process for optical thin film synthesis with the Fourier transform method,” Appl. Opt. 29, 3672–3684 (1990).
    [CrossRef] [PubMed]
  11. P. G. Verly, J. A. Dobrowolski, R. R. Willey, “Fourier transform method for the design of wideband antireflectance coatings,” Appl. Opt. 31, 3836–3846 (1992).
    [CrossRef] [PubMed]
  12. H. Fabricius, “Gradient-index filters: designing filters with steep skirts, high reflection, and quintic matching layers,” Appl. Opt. 31, 5191–5196 (1992).
    [CrossRef] [PubMed]
  13. W. H. Press, B. T. Flannery, S. A. Teukoksky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986), Chap. 12.
  14. A. V. Tikhonravov, J. A. Dobrowolski, “Quasi-optimal synthesis for antireflection coatings: a new method,” Appl. Opt. 32, 4265–4275 (1993).
    [CrossRef] [PubMed]
  15. W. H. Southwell, “Using apodization function to reduce sidelobes in rugate filters,” Appl. Opt. 28, 5091–5094 (1989).
    [CrossRef] [PubMed]
  16. A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their applications,” Appl. Opt. 32, 5417–5426 (1993).
    [CrossRef] [PubMed]
  17. A. V. Tikhonravov, “Amplitude-phase properties of the spectral coefficients of layered media,” USSR Comput. Math. Math. Phys. 25, 77–83 (1985).
    [CrossRef]

1993

1992

1990

1989

1987

1985

A. V. Tikhonravov, “Amplitude-phase properties of the spectral coefficients of layered media,” USSR Comput. Math. Math. Phys. 25, 77–83 (1985).
[CrossRef]

1978

1974

L. Sossi, “A method for the synthesis of multilayer dielectric interference coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229–237 (1974).

1968

L. Sossi, P. Kard, “On the theory of the reflection and transmission of light by thin inhomogeneous films,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 17, 41–48 (1968).

1967

Boivin, G.

Bovard, B. G.

Crane, R. L.

W. E. Johnson, R. L. Crane, “An overview of rugate filter technology,” in Optical Interference Coatings, Vol. 6 of OSA 1988 Technical Digest Series (Optical Society of America, Washington, D.C., 1988), pp. 118–121.

Delano, E.

Dobrowolski, J. A.

Fabricius, H.

Flannery, B. T.

W. H. Press, B. T. Flannery, S. A. Teukoksky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986), Chap. 12.

Germain, D. St.-

Gluck, N. S.

Gunning, W. J.

Hall, R. L.

Jacobsson, R.

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, pp. 247–286.
[CrossRef]

Johnson, W. E.

W. E. Johnson, R. L. Crane, “An overview of rugate filter technology,” in Optical Interference Coatings, Vol. 6 of OSA 1988 Technical Digest Series (Optical Society of America, Washington, D.C., 1988), pp. 118–121.

Kard, P.

L. Sossi, P. Kard, “On the theory of the reflection and transmission of light by thin inhomogeneous films,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 17, 41–48 (1968).

Lowe, D.

Press, W. H.

W. H. Press, B. T. Flannery, S. A. Teukoksky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986), Chap. 12.

Sossi, L.

L. Sossi, “A method for the synthesis of multilayer dielectric interference coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229–237 (1974).

L. Sossi, P. Kard, “On the theory of the reflection and transmission of light by thin inhomogeneous films,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 17, 41–48 (1968).

Southwell, W. H.

Teukoksky, S. A.

W. H. Press, B. T. Flannery, S. A. Teukoksky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986), Chap. 12.

Tikhonravov, A. V.

Verly, P. G.

Vetterling, W. T.

W. H. Press, B. T. Flannery, S. A. Teukoksky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986), Chap. 12.

Willey, R. R.

Woodberry, F. J.

Appl. Opt.

J. A. Dobrowolski, D. Lowe, “Optical thin-film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039–3050 (1978).
[CrossRef] [PubMed]

G. Boivin, D. St.- Germain, “Synthesis of gradient-index profiles corresponding to spectral reflectance derived by inverse Fourier transform,” Appl. Opt. 26, 4209–4213 (1987).
[CrossRef] [PubMed]

W. J. Gunning, R. L. Hall, F. J. Woodberry, W. H. Southwell, N. S. Gluck, “Codeposition of continuous composition rugate filters,” Appl. Opt. 28, 2945–2948 (1989).
[CrossRef] [PubMed]

W. H. Southwell, “Using apodization function to reduce sidelobes in rugate filters,” Appl. Opt. 28, 5091–5094 (1989).
[CrossRef] [PubMed]

B. G. Bovard, “Rugate filter design: the modified Fourier transform technique,” Appl. Opt. 29, 24–30 (1990).
[CrossRef] [PubMed]

P. G. Verly, J. A. Dobrowolski, “Iterative correction process for optical thin film synthesis with the Fourier transform method,” Appl. Opt. 29, 3672–3684 (1990).
[CrossRef] [PubMed]

P. G. Verly, J. A. Dobrowolski, R. R. Willey, “Fourier transform method for the design of wideband antireflectance coatings,” Appl. Opt. 31, 3836–3846 (1992).
[CrossRef] [PubMed]

H. Fabricius, “Gradient-index filters: designing filters with steep skirts, high reflection, and quintic matching layers,” Appl. Opt. 31, 5191–5196 (1992).
[CrossRef] [PubMed]

A. V. Tikhonravov, J. A. Dobrowolski, “Quasi-optimal synthesis for antireflection coatings: a new method,” Appl. Opt. 32, 4265–4275 (1993).
[CrossRef] [PubMed]

A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their applications,” Appl. Opt. 32, 5417–5426 (1993).
[CrossRef] [PubMed]

Eesti NSV Tead. Akad. Toim. Fuus. Mat.

L. Sossi, P. Kard, “On the theory of the reflection and transmission of light by thin inhomogeneous films,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 17, 41–48 (1968).

L. Sossi, “A method for the synthesis of multilayer dielectric interference coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229–237 (1974).

J. Opt. Soc. Am.

USSR Comput. Math. Math. Phys.

A. V. Tikhonravov, “Amplitude-phase properties of the spectral coefficients of layered media,” USSR Comput. Math. Math. Phys. 25, 77–83 (1985).
[CrossRef]

Other

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, pp. 247–286.
[CrossRef]

W. E. Johnson, R. L. Crane, “An overview of rugate filter technology,” in Optical Interference Coatings, Vol. 6 of OSA 1988 Technical Digest Series (Optical Society of America, Washington, D.C., 1988), pp. 118–121.

W. H. Press, B. T. Flannery, S. A. Teukoksky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986), Chap. 12.

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Figures (6)

Fig. 1
Fig. 1

Schematic of an inhomogeneous refractive-index profile n(x): x, optical thickness; x a , total optical thickness; n s , n a , index of the substrate and ambient n. Note that there are no discontinuities in the refractive index at the boundaries x = 0 and x = x a .

Fig. 2
Fig. 2

(a) Target and calculated reflectance for R = 20% and R = 70% rugate filters. Note that, whereas the calculated reflectance is close to the target reflectance for R = 20%, the calculated reflectance for the R = 70% filter is ≈ 15% greater than desired. (b) The inhomogeneous refractive-index profiles corresponding to the calculated reflectances in (a). Note that the two profiles are similar in shape with only their magnitude differing.

Fig. 3
Fig. 3

Modulus of Q(k) versus k for an inhomogeneous refractive-index profile (solid curve) and a multilayer (HL)2H (dashed curve) where n H = 2.35 and n L = 1.45 for all k (nondispersive). For k ≥ 10 the |Q(k)| values are negligible for the inhomogeneous layer, whereas this is not the case for the multilayer consisting of homogeneous layers. The values of |Q(k)| at higher wave numbers are related to the smoothness of the refractive-index profile.

Fig. 4
Fig. 4

Curves showing how the calculated reflectances (solid curves) differ from the target points (dots) as the target reflectance is increased. Note that the filter bandwidth is also increasing for higher target reflectances and is extremely wide for the R = 90% filter.

Fig. 5
Fig. 5

Effect of the scaling correction procedure on a filter designed for R = 70% for (a) x a = 10 μm and (b) x a = 20 μm. For both plots the calculated reflectance is shown for (1) no scaling, (2) after the first iteration of the scaling procedure, (3) after four iterations of the scaling procedure. Note that for the x a = 20-μm filter the transition region from low to high reflectance is nearly half of that for the 10-μm filter.

Fig. 6
Fig. 6

Effect of modifying the phase of Q(k) while keeping the target reflectance fixed for R = 50%. In all cases the total optical thickness is kept fixed at x a = 20 μm and a scaling procedure has been performed to achieve the specified target reflectances. The curves correspond to the calculated reflectance for (1) the zero phase across the region of interest, (2) the phase jump of 180° between 0.975 and 1.0 μm−1, (3) the smoother phase change of 0–20–160–180°, and (4) 0–30–150–180° over four wave-number points from 0.95 to 1.025 μm−1. Curves (1) and (2) in (b) are the refractive-index profiles corresponding to curves (1) and (2) in (a).

Equations (39)

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Q ( k ) = 0 x a p ( x ) exp [ i k ( 2 x - x a ) ] d x ,
p ( x ) = 1 2 n ( x ) n ( x ) .
h = x a / N .
x j = h ( j + ½ ) ,             j = 0 , 1 , , N - 1
p j = p ( x j ) .
Q ( k ) h j = 0 N - 1 p j exp [ i k ( 2 x j - x a ) ] = h exp [ i k ( h - x a ) ] j = 0 N - 1 p j exp ( i 2 j k x a / N ) .
f ( k ) = h - 1 exp [ - i k ( h - x a ) ] Q ( k ) ,
f ( k ) = j = 0 N - 1 p j exp ( 2 i j k x a / N ) .
k l = l ( π / x a ) ,             l = 0 , 1 , , N - 1.
f l = f ( k l ) = h - 1 exp [ i π l ( 1 - h / x a ) ] Q ( k l ) ,
f l = j = 0 N - 1 p j exp ( 2 π i l j / N ) ,             l = 0 , 1 , , N - 1.
p j = N - 1 l = 0 N - 1 f l exp ( - 2 π i l j / N ) ,             j = 0 , 1 , , N - 1.
p j = n ( x j ) / 2 n ( x j ) .
n ( x ) = n ( j h ) exp [ 2 p j ( x - j h ) ] .
f N - l = f l * ,             l = 1 , 2 , , N / 2 - 1.
Q ( k ) f 2 ( k ) = r ( k ) / τ ( k ) ,
Q ( k ) = Q ( k ) exp [ i φ ( k ) ] ,
Q ( k ) 2 = R ( k ) T ( k ) = R ( k ) 1 - R ( k ) .
n ( 0 ) = n s ,             n ( x a ) = n a .
k l = 0.1 π l ,             l = 0 , 1 , , N - 1.
k l = l ( π / x a ) ,             l = 0 , 1 , .
Δ k = π / x a .
Q ( 0 ) = n a - n s 2 ( n a n s ) 1 / 2 .
ν = k + i σ ,
R ( k ) = A [ n ( x ) ] .
n ( x ) = A - 1 [ R ( k ) ] .
n ( x ) B [ R ( k ) ] .
B A - 1 .
A [ n ( x ) ] = R 0 ( k ) .
n 0 ( x ) = B [ R 0 ( k ) ] .
R 1 ( k ) = A [ n 0 ( x ) ] = A B [ R 0 ( k ) ] .
A B [ s ( k ) R 0 ( k ) ] = A B R ˜ ( k ) = R 0 ( k ) ,
n ( x ) = B [ R ˜ ( k ) ] .
s ( k ) = R ˜ ( k ) R 0 ( k ) .
s ( k ) s 1 ( k ) = R 0 ( k ) R 1 ( k ) .
s i + 1 ( k ) = R ˜ i ( k ) R i + 1 ( k ) .
s ( k ) - s 1 ( k ) = ( A B ) - 1 [ R 0 ( k ) ] A B [ R 0 ( k ) ] - R 0 2 ( k ) R 0 ( k ) A B [ R 0 ( k ) ] .
A B I + E ,             ( A B ) - 1 I - E ,
s ( k ) - s 1 ( k ) = { E [ R 0 ( k ) ] } 2 R 0 ( k ) 2 + R 0 ( k ) E [ R 0 ( k ) ] .

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