Abstract

We deal with optimal two-material antireflection (AR) coatings for the visible and adjacent spectral regions. It has been shown before that, for a given set of input parameters (refractive indices of the substrate, ambient medium and high- and low-index coating materials, and for a given spectral width of the AR coating), such designs consist of one or more clusters of layers of approximately constant optical thickness and number of layers. We show that, through the analysis of many different optimal coatings, it is possible to derive two parameters for a simple empirical expression that relates the residual average reflectance in the AR region to the number of clusters. These parameters are given for all possible combinations of relative spectral bandwidth equal to 2, 3, and 4; low-index to ambient-medium index ratio equal to 1.38 and 1.45; and high-to-low index ratio equal to 1.4, 1.5, and 1.7. The agreement between the numerically and the empirically calculated values of residual average reflectance is excellent. From the information presented the optical thin-film designer can quickly calculate the required number of layers and the overall optical thickness of an AR coating having the desired achievable residual average reflectance.

© 2008 Optical Society of America

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References

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  1. N. Kaiser, "Optical coatings road-map," presented at the International Workshop on Optical Coatings, Ottawa, Canada, 11 May 2006.
  2. I. V. Grebenshchikov, L. G. Vlasov, B. S. Neporent, and N. V. Suikovskaya, Antireflection Coating of Optical Surfaces (State Publishers of Technical and Theoretical Literature, 1946) (in Russian).
  3. J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, "Optimal single-band normal-incidence antireflection coatings," Appl. Opt. 35, 644-658 (1996).
    [CrossRef] [PubMed]
  4. J. A. Dobrowolski, D. Poitras, P. Ma, H. Vakil, and M. Acree, "Toward perfect antireflection coatings: numerical investigation," Appl. Opt. 41, 3075-3083 (2002).
    [CrossRef] [PubMed]
  5. D. Poitras and J. A. Dobrowolski, "Toward perfect antireflection coatings. 2. Theory," Appl. Opt. 43, 1286-1295 (2004).
    [CrossRef] [PubMed]
  6. J. A. Dobrowolski, Y. Guo, T. Tiwald, P. Ma, and D. Poitras, "Toward perfect antireflection coatings. 3. Experimental results obtained with the use of Reststrahlen materials," Appl. Opt. 45, 1555-1562 (2006).
    [CrossRef] [PubMed]
  7. U. B.Schulz, U. B. Schallenberg, and N. Kaiser, "Symmetrical periods in antireflective coatings for plastic optics," Appl. Opt. 42, 1346-1351 (2003).
    [CrossRef] [PubMed]
  8. U. B. Schallenberg, "Antireflection design concepts with equivalent layers," Appl. Opt. 45, 1507-1514 (2006).
    [CrossRef] [PubMed]
  9. A. V. Tikhonravov, "Some theoretical aspects of thin film optics and their applications," Appl. Opt. 32, 5417-5426 (1993).
    [CrossRef] [PubMed]
  10. P. G. Verly, J. A. Dobrowolski, and R. R. Willey, "Fourier-transform method for the design of wideband antireflection coatings," Appl. Opt. 31, 3836-3846 (1992).
    [CrossRef] [PubMed]
  11. A. V. Tikhonravov and J. A. Dobrowolski, "Quasi-optimal synthesis for antireflection coatings: a new method," Appl. Opt. 32, 4265-4275 (1993).
    [CrossRef] [PubMed]
  12. A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, "Application of the needle optimization technique to the design of optical coatings," Appl. Opt. 35, 5493-5508 (1996).
    [CrossRef] [PubMed]
  13. A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, "Optical coating design approaches based on the needle optimization technique," Appl. Opt. 46, 704-710 (2007).
    [CrossRef] [PubMed]
  14. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1988).
  15. J. A. Dobrowolski, "Comparison of the Fourier transform and flip-flop thin-film synthesis methods," Appl. Opt. 25, 1966-1972 (1986).
    [CrossRef] [PubMed]
  16. A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, "Key role of the coating total optical thickness in solving design problems," Proc. SPIE 5250, 312-321 (2004).
    [CrossRef]
  17. R. Willey, "Predicting achievable design performance of broadband antireflection coatings," Appl. Opt. 32, 5447-5451 (1993).
    [CrossRef] [PubMed]
  18. A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and S. A. Yanshin, "Structural properties of antireflection coatings," in Proceedings of Optical Interference Coatings on CD-ROM, presentation WB5 (Optical Society of America, 2007).
  19. R. Willey, "Refined criteria for estimating limits of broad-band AR coatings," Proc. SPIE 5250, 393-399 (2004).
    [CrossRef]
  20. T. V. Amotchkina, "Analytical properties of spectral characteristics of antireflection optical coatings," Vestnik MGU Ser. 3 Fiz. Astron. (posted 16 July 2007, in press).
  21. A. V. Tikhonravov, P. W. Baumeister, and K. V. Popov, "Phase properties of multilayers," Appl. Opt. 36, 4382-4392 (1997).
    [CrossRef] [PubMed]

2007 (3)

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and S. A. Yanshin, "Structural properties of antireflection coatings," in Proceedings of Optical Interference Coatings on CD-ROM, presentation WB5 (Optical Society of America, 2007).

T. V. Amotchkina, "Analytical properties of spectral characteristics of antireflection optical coatings," Vestnik MGU Ser. 3 Fiz. Astron. (posted 16 July 2007, in press).

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, "Optical coating design approaches based on the needle optimization technique," Appl. Opt. 46, 704-710 (2007).
[CrossRef] [PubMed]

2006 (3)

J. A. Dobrowolski, Y. Guo, T. Tiwald, P. Ma, and D. Poitras, "Toward perfect antireflection coatings. 3. Experimental results obtained with the use of Reststrahlen materials," Appl. Opt. 45, 1555-1562 (2006).
[CrossRef] [PubMed]

N. Kaiser, "Optical coatings road-map," presented at the International Workshop on Optical Coatings, Ottawa, Canada, 11 May 2006.

U. B. Schallenberg, "Antireflection design concepts with equivalent layers," Appl. Opt. 45, 1507-1514 (2006).
[CrossRef] [PubMed]

2004 (3)

R. Willey, "Refined criteria for estimating limits of broad-band AR coatings," Proc. SPIE 5250, 393-399 (2004).
[CrossRef]

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, "Key role of the coating total optical thickness in solving design problems," Proc. SPIE 5250, 312-321 (2004).
[CrossRef]

D. Poitras and J. A. Dobrowolski, "Toward perfect antireflection coatings. 2. Theory," Appl. Opt. 43, 1286-1295 (2004).
[CrossRef] [PubMed]

2003 (1)

U. B.Schulz, U. B. Schallenberg, and N. Kaiser, "Symmetrical periods in antireflective coatings for plastic optics," Appl. Opt. 42, 1346-1351 (2003).
[CrossRef] [PubMed]

2002 (1)

J. A. Dobrowolski, D. Poitras, P. Ma, H. Vakil, and M. Acree, "Toward perfect antireflection coatings: numerical investigation," Appl. Opt. 41, 3075-3083 (2002).
[CrossRef] [PubMed]

1997 (1)

A. V. Tikhonravov, P. W. Baumeister, and K. V. Popov, "Phase properties of multilayers," Appl. Opt. 36, 4382-4392 (1997).
[CrossRef] [PubMed]

1996 (2)

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, "Optimal single-band normal-incidence antireflection coatings," Appl. Opt. 35, 644-658 (1996).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, "Application of the needle optimization technique to the design of optical coatings," Appl. Opt. 35, 5493-5508 (1996).
[CrossRef] [PubMed]

1993 (3)

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

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

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

1992 (1)

1988 (1)

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

1986 (1)

1946 (1)

I. V. Grebenshchikov, L. G. Vlasov, B. S. Neporent, and N. V. Suikovskaya, Antireflection Coating of Optical Surfaces (State Publishers of Technical and Theoretical Literature, 1946) (in Russian).

Acree, M.

J. A. Dobrowolski, D. Poitras, P. Ma, H. Vakil, and M. Acree, "Toward perfect antireflection coatings: numerical investigation," Appl. Opt. 41, 3075-3083 (2002).
[CrossRef] [PubMed]

Amotchkina, T. V.

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and S. A. Yanshin, "Structural properties of antireflection coatings," in Proceedings of Optical Interference Coatings on CD-ROM, presentation WB5 (Optical Society of America, 2007).

T. V. Amotchkina, "Analytical properties of spectral characteristics of antireflection optical coatings," Vestnik MGU Ser. 3 Fiz. Astron. (posted 16 July 2007, in press).

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, "Key role of the coating total optical thickness in solving design problems," Proc. SPIE 5250, 312-321 (2004).
[CrossRef]

Baumeister, P. W.

A. V. Tikhonravov, P. W. Baumeister, and K. V. Popov, "Phase properties of multilayers," Appl. Opt. 36, 4382-4392 (1997).
[CrossRef] [PubMed]

DeBell, G. W.

Dobrowolski, J. A.

J. A. Dobrowolski, Y. Guo, T. Tiwald, P. Ma, and D. Poitras, "Toward perfect antireflection coatings. 3. Experimental results obtained with the use of Reststrahlen materials," Appl. Opt. 45, 1555-1562 (2006).
[CrossRef] [PubMed]

D. Poitras and J. A. Dobrowolski, "Toward perfect antireflection coatings. 2. Theory," Appl. Opt. 43, 1286-1295 (2004).
[CrossRef] [PubMed]

J. A. Dobrowolski, D. Poitras, P. Ma, H. Vakil, and M. Acree, "Toward perfect antireflection coatings: numerical investigation," Appl. Opt. 41, 3075-3083 (2002).
[CrossRef] [PubMed]

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, "Optimal single-band normal-incidence antireflection coatings," Appl. Opt. 35, 644-658 (1996).
[CrossRef] [PubMed]

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

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

J. A. Dobrowolski, "Comparison of the Fourier transform and flip-flop thin-film synthesis methods," Appl. Opt. 25, 1966-1972 (1986).
[CrossRef] [PubMed]

Grebenshchikov, I. V.

I. V. Grebenshchikov, L. G. Vlasov, B. S. Neporent, and N. V. Suikovskaya, Antireflection Coating of Optical Surfaces (State Publishers of Technical and Theoretical Literature, 1946) (in Russian).

Guo, Y.

J. A. Dobrowolski, Y. Guo, T. Tiwald, P. Ma, and D. Poitras, "Toward perfect antireflection coatings. 3. Experimental results obtained with the use of Reststrahlen materials," Appl. Opt. 45, 1555-1562 (2006).
[CrossRef] [PubMed]

Kaiser, N.

N. Kaiser, "Optical coatings road-map," presented at the International Workshop on Optical Coatings, Ottawa, Canada, 11 May 2006.

U. B.Schulz, U. B. Schallenberg, and N. Kaiser, "Symmetrical periods in antireflective coatings for plastic optics," Appl. Opt. 42, 1346-1351 (2003).
[CrossRef] [PubMed]

Kokarev, M. A.

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, "Key role of the coating total optical thickness in solving design problems," Proc. SPIE 5250, 312-321 (2004).
[CrossRef]

Ma, P.

J. A. Dobrowolski, Y. Guo, T. Tiwald, P. Ma, and D. Poitras, "Toward perfect antireflection coatings. 3. Experimental results obtained with the use of Reststrahlen materials," Appl. Opt. 45, 1555-1562 (2006).
[CrossRef] [PubMed]

J. A. Dobrowolski, D. Poitras, P. Ma, H. Vakil, and M. Acree, "Toward perfect antireflection coatings: numerical investigation," Appl. Opt. 41, 3075-3083 (2002).
[CrossRef] [PubMed]

Neporent, B. S.

I. V. Grebenshchikov, L. G. Vlasov, B. S. Neporent, and N. V. Suikovskaya, Antireflection Coating of Optical Surfaces (State Publishers of Technical and Theoretical Literature, 1946) (in Russian).

Poitras, D.

J. A. Dobrowolski, Y. Guo, T. Tiwald, P. Ma, and D. Poitras, "Toward perfect antireflection coatings. 3. Experimental results obtained with the use of Reststrahlen materials," Appl. Opt. 45, 1555-1562 (2006).
[CrossRef] [PubMed]

D. Poitras and J. A. Dobrowolski, "Toward perfect antireflection coatings. 2. Theory," Appl. Opt. 43, 1286-1295 (2004).
[CrossRef] [PubMed]

J. A. Dobrowolski, D. Poitras, P. Ma, H. Vakil, and M. Acree, "Toward perfect antireflection coatings: numerical investigation," Appl. Opt. 41, 3075-3083 (2002).
[CrossRef] [PubMed]

Popov, K. V.

A. V. Tikhonravov, P. W. Baumeister, and K. V. Popov, "Phase properties of multilayers," Appl. Opt. 36, 4382-4392 (1997).
[CrossRef] [PubMed]

Schallenberg, U. B.

U. B. Schallenberg, "Antireflection design concepts with equivalent layers," Appl. Opt. 45, 1507-1514 (2006).
[CrossRef] [PubMed]

U. B.Schulz, U. B. Schallenberg, and N. Kaiser, "Symmetrical periods in antireflective coatings for plastic optics," Appl. Opt. 42, 1346-1351 (2003).
[CrossRef] [PubMed]

Schulz, U. B.

U. B.Schulz, U. B. Schallenberg, and N. Kaiser, "Symmetrical periods in antireflective coatings for plastic optics," Appl. Opt. 42, 1346-1351 (2003).
[CrossRef] [PubMed]

Suikovskaya, N. V.

I. V. Grebenshchikov, L. G. Vlasov, B. S. Neporent, and N. V. Suikovskaya, Antireflection Coating of Optical Surfaces (State Publishers of Technical and Theoretical Literature, 1946) (in Russian).

Sullivan, B. T.

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, "Optimal single-band normal-incidence antireflection coatings," Appl. Opt. 35, 644-658 (1996).
[CrossRef] [PubMed]

Thelen, A.

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

Tikhonravov, A. V.

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and S. A. Yanshin, "Structural properties of antireflection coatings," in Proceedings of Optical Interference Coatings on CD-ROM, presentation WB5 (Optical Society of America, 2007).

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, "Optical coating design approaches based on the needle optimization technique," Appl. Opt. 46, 704-710 (2007).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, "Key role of the coating total optical thickness in solving design problems," Proc. SPIE 5250, 312-321 (2004).
[CrossRef]

A. V. Tikhonravov, P. W. Baumeister, and K. V. Popov, "Phase properties of multilayers," Appl. Opt. 36, 4382-4392 (1997).
[CrossRef] [PubMed]

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, "Optimal single-band normal-incidence antireflection coatings," Appl. Opt. 35, 644-658 (1996).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, "Application of the needle optimization technique to the design of optical coatings," Appl. Opt. 35, 5493-5508 (1996).
[CrossRef] [PubMed]

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

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

Tiwald, T.

J. A. Dobrowolski, Y. Guo, T. Tiwald, P. Ma, and D. Poitras, "Toward perfect antireflection coatings. 3. Experimental results obtained with the use of Reststrahlen materials," Appl. Opt. 45, 1555-1562 (2006).
[CrossRef] [PubMed]

Trubetskov, M. K.

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and S. A. Yanshin, "Structural properties of antireflection coatings," in Proceedings of Optical Interference Coatings on CD-ROM, presentation WB5 (Optical Society of America, 2007).

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, "Optical coating design approaches based on the needle optimization technique," Appl. Opt. 46, 704-710 (2007).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, "Key role of the coating total optical thickness in solving design problems," Proc. SPIE 5250, 312-321 (2004).
[CrossRef]

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, "Optimal single-band normal-incidence antireflection coatings," Appl. Opt. 35, 644-658 (1996).
[CrossRef] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell, "Application of the needle optimization technique to the design of optical coatings," Appl. Opt. 35, 5493-5508 (1996).
[CrossRef] [PubMed]

Vakil, H.

J. A. Dobrowolski, D. Poitras, P. Ma, H. Vakil, and M. Acree, "Toward perfect antireflection coatings: numerical investigation," Appl. Opt. 41, 3075-3083 (2002).
[CrossRef] [PubMed]

Verly, P. G.

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, "Optimal single-band normal-incidence antireflection coatings," Appl. Opt. 35, 644-658 (1996).
[CrossRef] [PubMed]

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

Vlasov, L. G.

I. V. Grebenshchikov, L. G. Vlasov, B. S. Neporent, and N. V. Suikovskaya, Antireflection Coating of Optical Surfaces (State Publishers of Technical and Theoretical Literature, 1946) (in Russian).

Willey, R.

R. Willey, "Refined criteria for estimating limits of broad-band AR coatings," Proc. SPIE 5250, 393-399 (2004).
[CrossRef]

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

Willey, R. R.

Yanshin, S. A.

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and S. A. Yanshin, "Structural properties of antireflection coatings," in Proceedings of Optical Interference Coatings on CD-ROM, presentation WB5 (Optical Society of America, 2007).

Appl. Opt. (8)

J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, "Optimal single-band normal-incidence antireflection coatings," Appl. Opt. 35, 644-658 (1996).
[CrossRef] [PubMed]

J. A. Dobrowolski, D. Poitras, P. Ma, H. Vakil, and M. Acree, "Toward perfect antireflection coatings: numerical investigation," Appl. Opt. 41, 3075-3083 (2002).
[CrossRef] [PubMed]

J. A. Dobrowolski, Y. Guo, T. Tiwald, P. Ma, and D. Poitras, "Toward perfect antireflection coatings. 3. Experimental results obtained with the use of Reststrahlen materials," Appl. Opt. 45, 1555-1562 (2006).
[CrossRef] [PubMed]

U. B.Schulz, U. B. Schallenberg, and N. Kaiser, "Symmetrical periods in antireflective coatings for plastic optics," Appl. Opt. 42, 1346-1351 (2003).
[CrossRef] [PubMed]

U. B. Schallenberg, "Antireflection design concepts with equivalent layers," Appl. Opt. 45, 1507-1514 (2006).
[CrossRef] [PubMed]

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

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

A. V. Tikhonravov, P. W. Baumeister, and K. V. Popov, "Phase properties of multilayers," Appl. Opt. 36, 4382-4392 (1997).
[CrossRef] [PubMed]

Appl. Opt. (6)

Proc. SPIE (1)

R. Willey, "Refined criteria for estimating limits of broad-band AR coatings," Proc. SPIE 5250, 393-399 (2004).
[CrossRef]

Proc. SPIE (1)

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and M. A. Kokarev, "Key role of the coating total optical thickness in solving design problems," Proc. SPIE 5250, 312-321 (2004).
[CrossRef]

Vestnik MGU Ser. 3 Fiz. Astron. (1)

T. V. Amotchkina, "Analytical properties of spectral characteristics of antireflection optical coatings," Vestnik MGU Ser. 3 Fiz. Astron. (posted 16 July 2007, in press).

Other (4)

N. Kaiser, "Optical coatings road-map," presented at the International Workshop on Optical Coatings, Ottawa, Canada, 11 May 2006.

I. V. Grebenshchikov, L. G. Vlasov, B. S. Neporent, and N. V. Suikovskaya, Antireflection Coating of Optical Surfaces (State Publishers of Technical and Theoretical Literature, 1946) (in Russian).

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

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and S. A. Yanshin, "Structural properties of antireflection coatings," in Proceedings of Optical Interference Coatings on CD-ROM, presentation WB5 (Optical Society of America, 2007).

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Figures (8)

Fig. 1
Fig. 1

Refractive-index profiles of the first five designs from Table 1 .

Fig. 2
Fig. 2

Reflectances of the first five designs from Table 1.

Fig. 3
Fig. 3

Average residual reflectances R a v of AR coatings designed for the AR spectral band from 400 to 800 nm with different refractive-index ratios: ρ H L = 1.4 , ρ L a = 1.45 (circles), ρ H L = 1.5 , ρ L a = 1.45 (triangles), ρ H L = 1.7 , ρ L a = 1.45 (squares), ρ H L = 1.7 , ρ L a = 1.38 (diamonds). The solid curves, calculated from Eq. (3), are in excellent agreement with the experimental data.

Fig. 4
Fig. 4

Average reflectances R a v of AR coatings designed for the AR spectral band from 400 to 1200 nm with different refractive-index ratios: ρ H L = 1.4 , ρ L a = 1.45 (circles), ρ H L = 1.5 , ρ L a = 1.45 (triangles), ρ H L = 1.7 , ρ L a = 1.45 (squares), ρ H L = 1.7 , ρ L a = 1.38 (diamonds). The solid curves, calculated from Eq. (3), are in excellent agreement with the experimental data.

Fig. 5
Fig. 5

Average values of residual reflectances R a v of AR coatings designed for the AR spectral band from 400 to 1600 nm with different refractive-index ratios: ρ H L = 1.4 , ρ L a = 1.45 (circles), ρ H L = 1.5 , ρ L a = 1.45 (triangles), ρ H L = 1.7 , ρ L a = 1.45 (squares), ρ H L = 1.7 , ρ L a = 1.38 (diamonds). The solid curves, calculated from Eq. (3), are in excellent agreement with the experimental data.

Fig. 6
Fig. 6

Minimum achievable values of the average residual reflectance R a v for the relative widths of the AR spectral band λ u / λ l = 2 and for different refractive-index ratios ρ H L and ρ L a .

Fig. 7
Fig. 7

Minimum achievable values of the average residual reflectance R a v for the relative widths of the AR spectral band λ u / λ l = 3 and for different refractive-index ratios ρ H L and ρ L a .

Fig. 8
Fig. 8

Minimum achievable values of the average residual reflectance R a v for the relative widths of the AR spectral band λ u / λ l = 4 and for different refractive-index ratios ρ H L and ρ L a .

Tables (3)

Tables Icon

Table 1 Parameters of the AR Designs for the 400–1200 nm Spectral Region

Tables Icon

Table 2 Integrated Results for a Series of Optimal AR Designs for Different Combinations of Refractive Indices and AR Spectral Bandwidths

Tables Icon

Table 3 Parameters R and b Found by the Least-Squares Method for the Twelve Sets of R a v Values Presented in Figs. 35

Equations (4)

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R a v = 1 λ u λ l λ l λ u R ( λ ) d λ ,
T c = λ u 2 [ 1 + 2 π arcsin ( ρ H L 1 ρ H L + 1 ) ] .
R a v = R b 1 / M .
Δ R a v R ln b M 2 .

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