Abstract

Mathematical and computational evidence that strongly suggests that optimal solutions exist to single-band, normal-incidence antireflection coating problems is presented. It is shown that efficient synthesis and refinement techniques can quickly and accurately find such solutions. Several visible and infrared antireflection coating examples are presented to support this claim. Graphs that show the expected optimal performance for different representative substrates, refractive-index ratios, wavelength ranges, and overall optical thickness combinations are given. Typical designs exhibit a pronounced semiperiodic clustering of layers, which has also been observed in the past. Explanations of this phenomenon are proposed.

© 1996 Optical Society of America

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References

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  1. I. V. Grebenshchikov, L. G. Vlasov, B. S. Neporent, N. V. Suikovskaya, Prosvetleniye Optiki (Antireflection Coating of Optical Surfaces) (State Publishers of Technical and Theoretical Literature, Moscow–Leningrad, 1946).
  2. T. Sawaki, “Studies on anti-reflection films,” Research Rep. 315 (Osaka Industrial Research Institute, Osaka, Japan, 1960).
  3. A. Musset, A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics,E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 18, pp. 201–237.
    [CrossRef]
  4. L. I. Epstein, “The design of optical filters,” J. Opt. Soc. Am. 42, 806–810 (1952).
    [CrossRef]
  5. M. C. Ohmer, “Design of three-layer equivalent films,” J. Opt. Soc. Am. 68, 137–139 (1978).
    [CrossRef]
  6. W. H. Southwell, “Coating design using very thin high- and low-index layers,” Appl. Opt. 24, 457–460 (1985).
    [CrossRef] [PubMed]
  7. A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their applications,” Appl. Opt. 32, 5417–5426 (1993).
    [CrossRef] [PubMed]
  8. B. T. Sullivan, J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings. II. Experimental results—sputtering system,” Appl. Opt. 32, 2351–2360 (1993).
    [CrossRef] [PubMed]
  9. 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]
  10. M. J. Minot, “The angular reflectance of single-layer gradient refractive-index films,” J. Opt. Soc. Am. 67, 1046–1050 (1977).
    [CrossRef]
  11. E. Spiller, I. Haller, R. Feder, J. E. E. Baglin, W. N. Hammer, “Graded-index AR surfaces produced by ion implantation on plastic materials,” Appl. Opt. 19, 3022–3026 (1980).
    [CrossRef] [PubMed]
  12. W. H. Southwell, “Gradient-index antireflection coatings,” Opt. Lett. 8, 584–586 (1983).
    [CrossRef] [PubMed]
  13. L. Young, “Synthesis of multiple antireflection films over a prescribed frequency band,” J. Opt. Soc. Am. 51, 967–974 (1961).
    [CrossRef]
  14. R. R. Willey, “Broadband antireflection coating design performance estimation,” in Proceedings, 34th Annual Technical Conference of the Society of Vacuum Coaters (Society of Vacuum Coaters, Washington, D.C., 1991), pp. 205–209.
  15. R. R. Willey, “Predicting achievable design performance of broadband antireflection coatings,” Appl. Opt. 32, 5447–5451 (1993).
    [CrossRef] [PubMed]
  16. 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]
  17. A. V. Tikhonravov, J. A. Dobrowolski, “A new, quasi-optimal synthesis method for antireflection coatings,” Appl. Opt. 32, 4265–4275 (1993).
    [CrossRef] [PubMed]
  18. P. G. Verly, J. A. Dobrowolski, “Iterative correction process for optical thin film synthesis with Fourier transform method,” Appl. Opt. 29, 3672–3684 (1990).
    [CrossRef] [PubMed]
  19. P. G. Verly, “Fourier transform technique with frequency filtering for antireflection coating design,” in Optical Interference Coatings, F. Abeles, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2253, 161–167 (1994).
  20. P. G. Kard, Analysis and Synthesis of Multilayer Interference Coatings (Valrus, Tallin, 1971).
  21. Z. Knittl, Optics of Thin Films (Wiley, London, 1976).
  22. H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, New York, 1986).
    [CrossRef]
  23. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1988).
  24. S. A. Furman, A. V. Tikhonravov, Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992).
  25. A. V. Tikhonravov, M. K. Trubetskov, “Development of the needle optimization technique and new features of “Optilayer” design software,” in Optical Interference Coatings,F. Abeles, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2253, 10–20 (1994)
  26. B. T. Sullivan, J. A. Dobrowolski, “Implementation of a numerical needle method for thin film design,” Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 72–74.
  27. P. G. Verly, “Fourier transform technique with frequency filtering for optical thin-film design,” Appl. Opt. 34, 688–694 (1995)
    [CrossRef] [PubMed]
  28. J. A. Dobrowolski, “Completely automatic synthesis of optical thin film systems,” Appl. Opt. 4, 937–946 (1965).
    [CrossRef]
  29. 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]
  30. J. A. Dobrowolski, R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876–2893 (1990).
    [CrossRef] [PubMed]
  31. A. Premoli, M. L. Rastello, “Stochastic synthesis of multilayers,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 18–27 (1990).
  32. A. Premoli, M. L. Rastello, “Minimax refining of optical multilayer systems,” Appl. Opt. 31, 1597–1605 (1992).
    [CrossRef] [PubMed]
  33. M. L. Rastello, A. Premoli, “Continuation method for synthesizing antireflection coatings,” Appl. Opt. 31, 6741–6746 (1992).
    [CrossRef] [PubMed]
  34. J. A. Dobrowolski, R. A. Kemp, “Interface design method for two-material optical multilayer coatings,” Appl. Opt. 31, 6747–6756 (1992).
    [CrossRef] [PubMed]
  35. N. K. Sahoo, R. V. S. R. Apparao, “Modified complex method for constrained design and optimization of optical multilayer thin-film devices,” Appl. Phys. A 59, 317–326 (1994).
    [CrossRef]
  36. P. Baumeister, “Starting designs for the computer optimization of optical coatings,” Appl. Opt. 34, 4835–4843 (1995).
    [CrossRef] [PubMed]
  37. G. W. DeBell, “Antireflection coatings utilizing multiple half waves,” in Thin Film Technologies I, J. R. Jacobsson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.401, 127–137 (1983).
  38. J. A. Dobrowolski, J. R. Pekelsky, R. Pelletier, M. Ranger, B. T. Sullivan, A. J. Waldorf, “A practical magnetron sputtering system for the deposition of optical multilayer coatings,” Appl. Opt. 31, 3784–3789 (1992).
    [CrossRef] [PubMed]

1995

1994

N. K. Sahoo, R. V. S. R. Apparao, “Modified complex method for constrained design and optimization of optical multilayer thin-film devices,” Appl. Phys. A 59, 317–326 (1994).
[CrossRef]

1993

1992

1990

1988

1985

1983

1980

1978

1977

1965

1961

1952

Aguilera, J.

Aguilera, J. A.

Apparao, R. V. S. R.

N. K. Sahoo, R. V. S. R. Apparao, “Modified complex method for constrained design and optimization of optical multilayer thin-film devices,” Appl. Phys. A 59, 317–326 (1994).
[CrossRef]

Baglin, J. E. E.

Baumeister, P.

Bloom, A.

Coursen, D.

DeBell, G. W.

G. W. DeBell, “Antireflection coatings utilizing multiple half waves,” in Thin Film Technologies I, J. R. Jacobsson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.401, 127–137 (1983).

Dobrowolski, J. A.

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

B. T. Sullivan, J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings. II. Experimental results—sputtering system,” Appl. Opt. 32, 2351–2360 (1993).
[CrossRef] [PubMed]

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]

J. A. Dobrowolski, J. R. Pekelsky, R. Pelletier, M. Ranger, B. T. Sullivan, A. J. Waldorf, “A practical magnetron sputtering system for the deposition of optical multilayer coatings,” Appl. Opt. 31, 3784–3789 (1992).
[CrossRef] [PubMed]

J. A. Dobrowolski, R. A. Kemp, “Interface design method for two-material optical multilayer coatings,” Appl. Opt. 31, 6747–6756 (1992).
[CrossRef] [PubMed]

J. A. Dobrowolski, R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876–2893 (1990).
[CrossRef] [PubMed]

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

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]

J. A. Dobrowolski, “Completely automatic synthesis of optical thin film systems,” Appl. Opt. 4, 937–946 (1965).
[CrossRef]

B. T. Sullivan, J. A. Dobrowolski, “Implementation of a numerical needle method for thin film design,” Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 72–74.

Epstein, L. I.

Feder, R.

Furman, S. A.

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

Goldstein, F. T.

Grebenshchikov, I. V.

I. V. Grebenshchikov, L. G. Vlasov, B. S. Neporent, N. V. Suikovskaya, Prosvetleniye Optiki (Antireflection Coating of Optical Surfaces) (State Publishers of Technical and Theoretical Literature, Moscow–Leningrad, 1946).

Gustafson, D. E.

Haller, I.

Hammer, W. N.

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]

Kard, P. G.

P. G. Kard, Analysis and Synthesis of Multilayer Interference Coatings (Valrus, Tallin, 1971).

Kemp, R. A.

Knittl, Z.

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

Macleod, H. A.

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

Minot, M. J.

Musset, A.

A. Musset, A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics,E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 18, pp. 201–237.
[CrossRef]

Neporent, B. S.

I. V. Grebenshchikov, L. G. Vlasov, B. S. Neporent, N. V. Suikovskaya, Prosvetleniye Optiki (Antireflection Coating of Optical Surfaces) (State Publishers of Technical and Theoretical Literature, Moscow–Leningrad, 1946).

Ohmer, M. C.

Pekelsky, J. R.

Pelletier, R.

Premoli, A.

Ranger, M.

Rastello, M. L.

Sahoo, N. K.

N. K. Sahoo, R. V. S. R. Apparao, “Modified complex method for constrained design and optimization of optical multilayer thin-film devices,” Appl. Phys. A 59, 317–326 (1994).
[CrossRef]

Sawaki, T.

T. Sawaki, “Studies on anti-reflection films,” Research Rep. 315 (Osaka Industrial Research Institute, Osaka, Japan, 1960).

Southwell, W. H.

Spiller, E.

Suikovskaya, N. V.

I. V. Grebenshchikov, L. G. Vlasov, B. S. Neporent, N. V. Suikovskaya, Prosvetleniye Optiki (Antireflection Coating of Optical Surfaces) (State Publishers of Technical and Theoretical Literature, Moscow–Leningrad, 1946).

Sullivan, B. T.

Thelen, A.

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

A. Musset, A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics,E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 18, pp. 201–237.
[CrossRef]

Tikhonravov, A. V.

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

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

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

A. V. Tikhonravov, M. K. Trubetskov, “Development of the needle optimization technique and new features of “Optilayer” design software,” in Optical Interference Coatings,F. Abeles, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2253, 10–20 (1994)

Trubetskov, M. K.

A. V. Tikhonravov, M. K. Trubetskov, “Development of the needle optimization technique and new features of “Optilayer” design software,” in Optical Interference Coatings,F. Abeles, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2253, 10–20 (1994)

Verly, P. G.

Vlasov, L. G.

I. V. Grebenshchikov, L. G. Vlasov, B. S. Neporent, N. V. Suikovskaya, Prosvetleniye Optiki (Antireflection Coating of Optical Surfaces) (State Publishers of Technical and Theoretical Literature, Moscow–Leningrad, 1946).

Waldorf, A. J.

Willey, R. R.

R. R. Willey, “Predicting achievable design performance of broadband antireflection coatings,” Appl. Opt. 32, 5447–5451 (1993).
[CrossRef] [PubMed]

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, “Broadband antireflection coating design performance estimation,” in Proceedings, 34th Annual Technical Conference of the Society of Vacuum Coaters (Society of Vacuum Coaters, Washington, D.C., 1991), pp. 205–209.

Young, L.

Appl. Opt.

W. H. Southwell, “Coating design using very thin high- and low-index layers,” Appl. Opt. 24, 457–460 (1985).
[CrossRef] [PubMed]

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

B. T. Sullivan, J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings. II. Experimental results—sputtering system,” Appl. Opt. 32, 2351–2360 (1993).
[CrossRef] [PubMed]

R. R. Willey, “Predicting achievable design performance of broadband antireflection coatings,” Appl. Opt. 32, 5447–5451 (1993).
[CrossRef] [PubMed]

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]

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

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

E. Spiller, I. Haller, R. Feder, J. E. E. Baglin, W. N. Hammer, “Graded-index AR surfaces produced by ion implantation on plastic materials,” Appl. Opt. 19, 3022–3026 (1980).
[CrossRef] [PubMed]

P. G. Verly, “Fourier transform technique with frequency filtering for optical thin-film design,” Appl. Opt. 34, 688–694 (1995)
[CrossRef] [PubMed]

J. A. Dobrowolski, “Completely automatic synthesis of optical thin film systems,” Appl. Opt. 4, 937–946 (1965).
[CrossRef]

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]

J. A. Dobrowolski, R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876–2893 (1990).
[CrossRef] [PubMed]

A. Premoli, M. L. Rastello, “Minimax refining of optical multilayer systems,” Appl. Opt. 31, 1597–1605 (1992).
[CrossRef] [PubMed]

M. L. Rastello, A. Premoli, “Continuation method for synthesizing antireflection coatings,” Appl. Opt. 31, 6741–6746 (1992).
[CrossRef] [PubMed]

J. A. Dobrowolski, R. A. Kemp, “Interface design method for two-material optical multilayer coatings,” Appl. Opt. 31, 6747–6756 (1992).
[CrossRef] [PubMed]

P. Baumeister, “Starting designs for the computer optimization of optical coatings,” Appl. Opt. 34, 4835–4843 (1995).
[CrossRef] [PubMed]

J. A. Dobrowolski, J. R. Pekelsky, R. Pelletier, M. Ranger, B. T. Sullivan, A. J. Waldorf, “A practical magnetron sputtering system for the deposition of optical multilayer coatings,” Appl. Opt. 31, 3784–3789 (1992).
[CrossRef] [PubMed]

Appl. Phys. A

N. K. Sahoo, R. V. S. R. Apparao, “Modified complex method for constrained design and optimization of optical multilayer thin-film devices,” Appl. Phys. A 59, 317–326 (1994).
[CrossRef]

J. Opt. Soc. Am.

Opt. Lett.

Other

R. R. Willey, “Broadband antireflection coating design performance estimation,” in Proceedings, 34th Annual Technical Conference of the Society of Vacuum Coaters (Society of Vacuum Coaters, Washington, D.C., 1991), pp. 205–209.

A. Premoli, M. L. Rastello, “Stochastic synthesis of multilayers,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 18–27 (1990).

G. W. DeBell, “Antireflection coatings utilizing multiple half waves,” in Thin Film Technologies I, J. R. Jacobsson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.401, 127–137 (1983).

I. V. Grebenshchikov, L. G. Vlasov, B. S. Neporent, N. V. Suikovskaya, Prosvetleniye Optiki (Antireflection Coating of Optical Surfaces) (State Publishers of Technical and Theoretical Literature, Moscow–Leningrad, 1946).

T. Sawaki, “Studies on anti-reflection films,” Research Rep. 315 (Osaka Industrial Research Institute, Osaka, Japan, 1960).

A. Musset, A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics,E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 18, pp. 201–237.
[CrossRef]

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]

P. G. Verly, “Fourier transform technique with frequency filtering for antireflection coating design,” in Optical Interference Coatings, F. Abeles, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2253, 161–167 (1994).

P. G. Kard, Analysis and Synthesis of Multilayer Interference Coatings (Valrus, Tallin, 1971).

Z. Knittl, Optics of Thin Films (Wiley, London, 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. A. Furman, A. V. Tikhonravov, Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992).

A. V. Tikhonravov, M. K. Trubetskov, “Development of the needle optimization technique and new features of “Optilayer” design software,” in Optical Interference Coatings,F. Abeles, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2253, 10–20 (1994)

B. T. Sullivan, J. A. Dobrowolski, “Implementation of a numerical needle method for thin film design,” Optical Interference Coatings, Vol. 17 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 72–74.

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Figures (13)

Fig. 1
Fig. 1

Performance and refractive-index profiles of four different starting designs: A, multilayer approximation (dotted curves) of an inhomogeneous layer solution (solid curves) obtained by a quadratic programming method; B, C, a single-layer and an 18-layer quarter-wave stack; D, best 8-layer solution to the problem. After refinement and, in the case of D, the addition of layers at the glass interface, all calculations converged to the system shown in row E.

Fig. 2
Fig. 2

Optimal 20-layer AR coatings on glass for the 0.4–0.8-μm spectral region based on refractive-index pairs: A, 1.90 and 1.45;B, 1.45 and 2.10; C, 1.45 and 2.35; D, 1.38 and 2.35. E, as in C, except that the refractive index of the outermost layer is 1.38.

Fig. 3
Fig. 3

Optimal AR coatings on glass based on the refractive-index pair 1.38 and 2.35 for different overall thicknesses of the systems.

Fig. 4
Fig. 4

Optimal AR coatings on glass based on the refractive-index pair 1.45 and 2.35 for different overall thicknesses of the systems.

Fig. 5
Fig. 5

Lowest rms reflectances of a glass surface that can be achieved with AR coatings made of films of refractive indices 1.45 and 2.35 for different values of the ratio λ U L . The integers next to the calculated points correspond to the number of layers in the solutions. The spacing between the vertical lines is 0.625.

Fig. 6
Fig. 6

Lowest rms reflectances of a glass surface over a λ U L = 2.0 spectral range that can be achieved with AR coatings made of films of refractive indices 1.45 and 2.35. For comparison, the average reflectance values predicted by Willey are also shown.14,15

Fig. 7
Fig. 7

Optimal AR coatings on a nS = 4.0 substrate based on the refractive index pair 2.2 and 4.20 for different overall optical thicknesses of the systems.

Fig. 8
Fig. 8

Lowest calculated rms reflectances of a Ge surface that can be achieved with AR coatings made of films of refractive indices 2.2 and 4.2 for different values of the ratio λ U L . The number of layers in the solution is displayed next to calculated points. The spacing between the vertical lines is 0.598.

Fig. 9
Fig. 9

Optimal rms reflectances of a nS = 4.0 substrate over a λ U L = 1.6 spectral range that can be achieved with AR coatings made of films of refractive indices 2.2 and 4.2: A, optimal solutions obtained during this study. The integers represent the number of coincident points. B, Results obtained by others20,21,2532 for the same problem, including average reflectance values predicted by Willey.14,15

Fig. 10
Fig. 10

Reflectance of the one-to four-cluster optimal solutions of Figs. 3A–D are plotted in A–D for a larger wavelength region. As can be seen, reflectance peaks adjacent to the AR band are visible, and the amplitudes of these peaks increase as the number of clusters increases.

Fig. 11
Fig. 11

Demonstration that the reflectance peaks adjacent to the AR band determine the layer clustering evident in the refractive-index profile of optimal solutions. The dashed curves correspond to the optimal multilayer solution shown in Fig. 3C. A, The solid curve corresponds to a low-frequency rugate filter; B, the solid curve corresponds to a high-frequency rugate filter; C, the solid curve corresponds to the superposition of the previous two rugate filters, obtained when their respective refractive-index profiles are multiplied (see text).

Fig. 12
Fig. 12

Theoretical calculation of the performance of nonrefined AB N cluster systems with different values of N.

Fig. 13
Fig. 13

18-layer Nb2O5/SiO2 AR coating applied to one side of a quartz substrate: A, calculated and measured reflectance; B, refractive-index profile of the AR coating.

Equations (5)

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f = { 1 m j = 1 m [ R ( λ j ) Δ R j ] 2 } 1 / 2 ,
F = λ L λ U R ( λ ) 1 R ( λ ) d λ .
Λ 0 = λ 0 2 = 1 2 λ U ( 1 + Δ g ) .
Δ g = 2 π arcsin ( n H / n L 1 n H / n L + 1 ) .
Λ j = λ U , j λ U , i Λ i ,

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