Abstract

A new method for the design of antireflection coatings is described in which a linearly constrained quadratic programming optimization procedure is used to find an inhomogeneous layer solution that, for a given overall thickness of the layer, corresponds to the global optimum for the problem. The refractive-index profile of this solution is then approximated by a two-material multilayer system that serves as a starting design for further numerical refinement. We present the theoretical basis for this method and discuss the effect of the various approximnations made on the solution. Four different antireflection problems are solved to illustrate this new method.

© 1993 Optical Society of America

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References

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  1. I. V. Grebenshchikov, Prosvetlenie Optiki (Antireflection Coating of Optical Surfaces) (State Publishers of Technical and Theoretical Literature, Moscow, 1946).
  2. J. T. Cox, G. Hass, “Antireflection coatings for optical and infrared optical materials,” in Physics of Thin Films, G. Hass, R. E. Thun, eds. (Academic, New York, 1964), pp. 239–304.
  3. A. Mussett, A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics, E. Wolf, ed. (Pergamon, New York, 1970), pp. 203–237.
  4. P. G. Kard, Analysis and Synthesis of Multilayer Interference Coatings (Valgus, Tallin, Estonia, 1971) (in Russian).
  5. Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).
  6. H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, New York, 1986).
    [CrossRef]
  7. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1988).
  8. S. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992).
  9. L. Sossi, “A method for the synthesis of multilayer dielectric light filters,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 23, 229–237 (1974).
  10. L. Sossi, “On the theory of the synthesis of multilayer dielectric light filters,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, 171–176 (1976).
  11. 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]
  12. P. G. Verly, J. A. Dobrowolski, W. J. Wild, R. L. Burton, “Synthesis of high rejection filters with the Fourier transform method,” Appl. Opt. 28, 2864–2875 (1989).
    [CrossRef] [PubMed]
  13. P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).
  14. A. V. Tikhonravov, M. K. Trubetskov, “Thin film coating design using second order optimization methods,” in Thin Films for Optical Systems, K. Guenther, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1782 (to be published).
  15. J. A. Aguilera, J. Aguilera, P. Baumeister, A. Bloom, D. Coursen, J. A. Dobrowolski, F. T. Goldstein, D. E. Gustafson, R. A. Kemp, “Antireflection coatings for germanium IR optics: a comparison of numerical design methods,” Appl. Opt. 27, 2832–2840 (1988).
    [CrossRef] [PubMed]
  16. J. A. Dobrowolski, R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876–2893 (1990).
    [CrossRef] [PubMed]
  17. R. R. Willey, P. C. Verly, J. A. Dobrowolski, “Synthesis of wide band AR coatings with the Fourier transform method,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 36–44 (1990).
  18. P. G. Verly, J. A. Dobrowolski, R. R. Willey, “Fourier-transform method for the design of wideband antireflection coatings,” Appl. Opt. 31, 3836–3846 (1992).
    [CrossRef] [PubMed]
  19. J. A. Dobrowolski, R. A. Kemp, “Interface design methods for two-material optical multilayer coatings,” Appl. Opt. 31, 6747–6756 (1992).
    [CrossRef] [PubMed]
  20. R. F. Polter, “Germanium (Ge),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 465–478.
  21. I. Ohidal, K. Navratil, “Thorium fluoride (ThF4),” in Handbook of Optical Constants II, E. D. Palik, ed. (Academic, Boston, Mass., 1991), pp. 1049–1058.
  22. B. G. Bovard, “Rugate filter design: the modified Fourier transform technique,” Appl. Opt. 29, 24–30 (1990).
    [CrossRef] [PubMed]
  23. A. V. Tikhonravov, “On the optimality of thin film optical coating design,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 28–35 (1990).
  24. A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their applications,” Appl. Opt. (to be published).

1992 (2)

1990 (2)

1989 (1)

1988 (1)

1978 (1)

1976 (1)

L. Sossi, “On the theory of the synthesis of multilayer dielectric light filters,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, 171–176 (1976).

1974 (1)

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

Aguilera, J.

Aguilera, J. A.

Baumeister, P.

Bloom, A.

Bovard, B. G.

Burton, R. L.

Coursen, D.

Cox, J. T.

J. T. Cox, G. Hass, “Antireflection coatings for optical and infrared optical materials,” in Physics of Thin Films, G. Hass, R. E. Thun, eds. (Academic, New York, 1964), pp. 239–304.

Dobrowolski, J. A.

Furman, S.

S. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992).

Gill, P. E.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

Goldstein, F. T.

Grebenshchikov, I. V.

I. V. Grebenshchikov, Prosvetlenie Optiki (Antireflection Coating of Optical Surfaces) (State Publishers of Technical and Theoretical Literature, Moscow, 1946).

Gustafson, D. E.

Hass, G.

J. T. Cox, G. Hass, “Antireflection coatings for optical and infrared optical materials,” in Physics of Thin Films, G. Hass, R. E. Thun, eds. (Academic, New York, 1964), pp. 239–304.

Kard, P. G.

P. G. Kard, Analysis and Synthesis of Multilayer Interference Coatings (Valgus, Tallin, Estonia, 1971) (in Russian).

Kemp, R. A.

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

Lowe, D.

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, New York, 1986).
[CrossRef]

Murray, W.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

Mussett, A.

A. Mussett, A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics, E. Wolf, ed. (Pergamon, New York, 1970), pp. 203–237.

Navratil, K.

I. Ohidal, K. Navratil, “Thorium fluoride (ThF4),” in Handbook of Optical Constants II, E. D. Palik, ed. (Academic, Boston, Mass., 1991), pp. 1049–1058.

Ohidal, I.

I. Ohidal, K. Navratil, “Thorium fluoride (ThF4),” in Handbook of Optical Constants II, E. D. Palik, ed. (Academic, Boston, Mass., 1991), pp. 1049–1058.

Polter, R. F.

R. F. Polter, “Germanium (Ge),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 465–478.

Sossi, L.

L. Sossi, “On the theory of the synthesis of multilayer dielectric light filters,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, 171–176 (1976).

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

Thelen, A.

A. Mussett, A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics, E. Wolf, ed. (Pergamon, New York, 1970), pp. 203–237.

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1988).

Tikhonravov, A. V.

S. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992).

A. V. Tikhonravov, M. K. Trubetskov, “Thin film coating design using second order optimization methods,” in Thin Films for Optical Systems, K. Guenther, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1782 (to be published).

A. V. Tikhonravov, “On the optimality of thin film optical coating design,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 28–35 (1990).

A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their applications,” Appl. Opt. (to be published).

Trubetskov, M. K.

A. V. Tikhonravov, M. K. Trubetskov, “Thin film coating design using second order optimization methods,” in Thin Films for Optical Systems, K. Guenther, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1782 (to be published).

Verly, P. C.

R. R. Willey, P. C. Verly, J. A. Dobrowolski, “Synthesis of wide band AR coatings with the Fourier transform method,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 36–44 (1990).

Verly, P. G.

Wild, W. J.

Willey, R. R.

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

R. R. Willey, P. C. Verly, J. A. Dobrowolski, “Synthesis of wide band AR coatings with the Fourier transform method,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 36–44 (1990).

Wright, M. H.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

Appl. Opt. (7)

Eesti NSV Tead. Akad. Toim. Fuus. Mat. (2)

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

L. Sossi, “On the theory of the synthesis of multilayer dielectric light filters,” Eesti NSV Tead. Akad. Toim. Fuus. Mat. 25, 171–176 (1976).

Other (15)

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

A. V. Tikhonravov, M. K. Trubetskov, “Thin film coating design using second order optimization methods,” in Thin Films for Optical Systems, K. Guenther, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1782 (to be published).

R. R. Willey, P. C. Verly, J. A. Dobrowolski, “Synthesis of wide band AR coatings with the Fourier transform method,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 36–44 (1990).

R. F. Polter, “Germanium (Ge),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 465–478.

I. Ohidal, K. Navratil, “Thorium fluoride (ThF4),” in Handbook of Optical Constants II, E. D. Palik, ed. (Academic, Boston, Mass., 1991), pp. 1049–1058.

A. V. Tikhonravov, “On the optimality of thin film optical coating design,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 28–35 (1990).

A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their applications,” Appl. Opt. (to be published).

I. V. Grebenshchikov, Prosvetlenie Optiki (Antireflection Coating of Optical Surfaces) (State Publishers of Technical and Theoretical Literature, Moscow, 1946).

J. T. Cox, G. Hass, “Antireflection coatings for optical and infrared optical materials,” in Physics of Thin Films, G. Hass, R. E. Thun, eds. (Academic, New York, 1964), pp. 239–304.

A. Mussett, A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics, E. Wolf, ed. (Pergamon, New York, 1970), pp. 203–237.

P. G. Kard, Analysis and Synthesis of Multilayer Interference Coatings (Valgus, Tallin, Estonia, 1971) (in Russian).

Z. Knittl, Optics of Thin Films (Wiley, New York, 1976).

H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, New York, 1986).
[CrossRef]

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1988).

S. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992).

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Figures (8)

Fig. 1
Fig. 1

Schematic models of (a) an inhomogeneous layer and (b) a homogeneous multilayer system.

Fig. 2
Fig. 2

Block diagram of the calculations for the quadratic problem approximation method.

Fig. 3
Fig. 3

Transformation of the ηj values that result from the solution of the quadratic problem (A) into an inhomogeneous refractive-index profile (B) and the corresponding two-material equivalent multilayer design (C).

Fig. 4
Fig. 4

Antireflection problem over the 0.38 < λ < 0.78-μm spectral region for a substrate of refractive index 1.52, using coating materials with indices 1.45 and 2.2. Rows A, B, and C correspond to the quadratic program solution and its two-material equivalent before and after refinement.

Fig. 5
Fig. 5

Antireflection problem over the 0.4 < λ < 1.2-μm spectral region for a substrate of refractive index 1.52, using coating materials with indices 1.45 and 2.1 (see text). Rows A, B, and C correspond to the quadratic program solution and its two-material equivalent before and after refinement. The solutions in rows D and E are derived from that of row C by the elimination of 1 and 3 thin layers followed by further refinement.

Fig. 6
Fig. 6

Antireflection problem over the 7.7 < λ < 12.3-μm spectral region for a substrate of refractive index 4.00, using coating materials with indices 2.2 and 4.2 (see text). Rows A, B, and C correspond to the quadratic program solution and its two-material equivalent before and after refinement.

Fig. 7
Fig. 7

Antireflection problem over the 2.0 < λ < 14.0-μm spectral region for a substrate of refractive index 4.00, using coating materials with indices 1.45 and 4.0 (see text). Rows A, B, and C correspond to the quadratic program solution and its two-material equivalent before and after refinement. In the solution of row D the nondispersive refractive indices of solution C were replaced by dispersive optical constants.

Fig. 8
Fig. 8

Relationship between the quadratic and the homogeneous multilayer thin-film synthesis problems (see text).

Equations (41)

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X = { d 1 , d 2 d m } ,
x = 0 z n ( z ) d z .
x a = 0 z a n ( z ) d z .
Q ( k ) = 0 x a η ( x ) exp [ i k ( 2 x x a ) ] d x ,
η ( x ) = 1 2 n ( x ) n ( x ) .
Q ( k ) = h j = 1 N η ( ζ j ) exp [ i k ( 2 ζ j x a ) ] .
Q ( k ) = h exp [ i k ( h x a ) ] j = 1 N η j exp ( 2 i k j h ) .
F = k min k max | Q ( k ) | 2 d k .
F = { k j } R 2 ( k j ) ,
F = { k j } R ( k j ) ,
| Q ( k ) | 2 = R ( k ) / T ( k ) R ( k ) .
| Q ( k ) | 2 = j , l = 1 N η j η l exp [ 2 i k h ( l j ) ] .
| Q ( k ) | 2 = j , l = 1 N η j η l cos 2 k h ( l j ) .
F = j , l = 1 N a j l η j η l ,
a j l = h 2 ( l j ) [ sin 2 k max h ( l j ) sin 2 k min h ( l j ) ] , l j = h 2 ( k max k min ) , l = j .
n L n ( x ) n H ,
0 x η ( ζ ) d ζ = 1 2 0 x n ( ζ ) n ( ζ ) d ζ = 1 2 [ ln n ( x ) ] 0 x = 1 2 ln n ( x ) n s .
1 2 ln n L n s 0 x η ( ζ ) d ζ 1 2 ln n H n s .
0 x a η ( ζ ) d ζ = 1 2 ln n a n s .
1 2 ln n L n s 0 p h η ( ζ ) d ζ 1 2 ln n H n s , p = 1 , 2 , , N 1 ,
0 N h η ( ζ ) d ζ = 1 2 ln n a n s .
0 p h η ( ζ ) d ζ = h j = 1 p η j , p = 1 , 2 , , N
1 2 h ln n L n s j = 1 p η j 1 2 h ln n H n s , p = 1 , 2 , , N 1 ,
j = 1 N η j = 1 2 h ln n a n s .
k min = 2 π λ max , k max = 2 π λ min .
n ( x ) = n ( x j 1 ) exp [ 2 η j ( x x j 1 ) ] .
d u d z = i k υ , d υ d z = i k n 2 ( z ) u .
u ( 0 , k ) = 1 , υ ( 0 , k ) = n s .
r ( k ) = n a u ( z a , k ) υ ( z a , k ) n a u ( z a , k ) + υ ( z a , k ) , t ( k ) = 2 n a n s n a u ( z a , k ) + υ ( z a , k ) .
x = 0 z n ( z ) d z .
d u d x = i k υ n ( x ) , d υ d x = i k n ( x ) u .
f 1 ( x , k ) = 1 2 n s [ n ( x ) u ( x , k ) + υ ( x , k ) n ( x ) ] ,
f 2 ( x , k ) = 1 2 n s [ n ( x ) u ( x , k ) υ ( x , k ) n ( x ) ] .
r ( k ) = n ( x ) u ( x , k ) υ ( x , k ) n ( x ) u ( x , k ) + υ ( x , k ) , t ( k ) = 2 n ( x ) n s n ( x ) u ( x , k ) + υ ( x , k ) .
d f 1 d x = i k f 1 + η ( x ) f 2 , d f 2 d x = i k f 2 + η ( x ) f 1 .
η ( x ) = 1 2 n ( x ) n ( x ) ,
f 1 ( 0 , k ) = n ( 0 ) + n s 2 n ( 0 ) n s , f 2 ( 0 , k ) = n ( 0 ) n s 2 n ( 0 ) n s .
d f 1 d x = i k f 1 ,
f 1 ( x , k ) = exp ( i k x ) .
d f 2 d x = i k f 2 + η ( x ) exp ( i k x ) ,
f ( x a , k ) = Q ( k ) 0 x a η ( x ) exp [ i k ( 2 x x a ) ] d x ,

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