Abstract

A new empirical expression for estimating minimum achievable residual reflectance of antireflection (AR) coatings is presented. The expression gives an accurate approximation of the minimum residual reflectance for normal- and oblique-incidence AR coatings in the visible and infrared spectral ranges.

© 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 use of Reststrahlen materials,” Appl. Opt. 45, 1555-1562 (2006).
    [CrossRef] [PubMed]
  7. U. Schulz, U. 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. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1988), pp. 17-20.
  10. 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]
  11. R. Willey, “Predicting achievable design performance of broadband antireflection coatings,” Appl. Opt. 32, 5447-5451(1993).
    [CrossRef] [PubMed]
  12. R. Willey, “Refined criteria for estimating limits of broad-band AR coatings,” Proc. SPIE 5250, 393-399 (2004).
    [CrossRef]
  13. A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and S. A. Yanshin, “Structural properties of antireflection coatings,” in Proceedings in Optical Interference Coatings on CD-ROM, presentation WB5 (Optical Society of America, 2007).
  14. A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and J. A. Dobrowolski, “Estimation of the average residual reflectance of broadband antireflection coatings,” Appl. Opt. 47, C1-C7 (2008).
    [CrossRef]
  15. T. V. Amotchkina, “Computational experiments of the design of antireflection coatings and an empirical expression for the minimum residual reflectance,” Num. Meth. Programming 9, 90-92 (2008).
  16. A. V. Tikhonravov and J. A. Dobrowolski, “Quasi-optimal synthesis method for antireflection coatings: a new method,” Appl. Opt. 32, 4265-4275 (1993).
    [CrossRef] [PubMed]
  17. A. V. Tikhonravov, “Some theoretical aspects of thin film optics and their applications,” Appl. Opt. 32, 5417-5426 (1993).
    [CrossRef] [PubMed]
  18. 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]
  19. A. V. Tikhonravov and M. K. Trubetskov, optilayer thin film software, http://www.optilayer.com.

2008 (2)

T. V. Amotchkina, “Computational experiments of the design of antireflection coatings and an empirical expression for the minimum residual reflectance,” Num. Meth. Programming 9, 90-92 (2008).

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and J. A. Dobrowolski, “Estimation of the average residual reflectance of broadband antireflection coatings,” Appl. Opt. 47, C1-C7 (2008).
[CrossRef]

2006 (2)

2004 (3)

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

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

2003 (1)

2002 (1)

1996 (1)

1993 (3)

1992 (1)

Acree, M.

Amotchkina, T. V.

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and J. A. Dobrowolski, “Estimation of the average residual reflectance of broadband antireflection coatings,” Appl. Opt. 47, C1-C7 (2008).
[CrossRef]

T. V. Amotchkina, “Computational experiments of the design of antireflection coatings and an empirical expression for the minimum residual reflectance,” Num. Meth. Programming 9, 90-92 (2008).

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

Dobrowolski, J. A.

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.

Kaiser, N.

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

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

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.

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.

Schallenberg, U.

Schallenberg, U. B.

Schulz, U.

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.

Thelen, A.

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1988), pp. 17-20.

Tikhonravov, A. V.

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and J. A. Dobrowolski, “Estimation of the average residual reflectance of broadband antireflection coatings,” Appl. Opt. 47, C1-C7 (2008).
[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]

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, “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 method for antireflection coatings: a new method,” Appl. Opt. 32, 4265-4275 (1993).
[CrossRef] [PubMed]

A. V. Tikhonravov and M. K. Trubetskov, optilayer thin film software, http://www.optilayer.com.

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

Tiwald, T.

Trubetskov, M. K.

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and J. A. Dobrowolski, “Estimation of the average residual reflectance of broadband antireflection coatings,” Appl. Opt. 47, C1-C7 (2008).
[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]

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

A. V. Tikhonravov and M. K. Trubetskov, optilayer thin film software, http://www.optilayer.com.

Vakil, H.

Verly, P. G.

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 in Optical Interference Coatings on CD-ROM, presentation WB5 (Optical Society of America, 2007).

Appl. Opt. (11)

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]

A. V. Tikhonravov and J. A. Dobrowolski, “Quasi-optimal synthesis method 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]

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

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]

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

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

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

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

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, and J. A. Dobrowolski, “Estimation of the average residual reflectance of broadband antireflection coatings,” Appl. Opt. 47, C1-C7 (2008).
[CrossRef]

Num. Meth. Programming (1)

T. V. Amotchkina, “Computational experiments of the design of antireflection coatings and an empirical expression for the minimum residual reflectance,” Num. Meth. Programming 9, 90-92 (2008).

Proc. SPIE (2)

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]

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

Other (5)

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

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), pp. 17-20.

A. V. Tikhonravov and M. K. Trubetskov, optilayer thin film software, http://www.optilayer.com.

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Figures (2)

Fig. 1
Fig. 1

Experimentally found R values in the case of ρ L a = 1.38 for various AR spectral ranges: λ u / λ l = 2 (black circles), λ u / λ l = 2.5 (triangles), λ u / λ l = 3 (squares), λ u / λ l = 3.5 (diamonds), and λ u / λ l = 4 (gray circles). The solid curves are calculated from Eq. (6).

Fig. 2
Fig. 2

Experimentally found R values in the case of ρ L a = 1.45 for various AR spectral ranges: λ u / λ l = 2 (black circles), λ u / λ l = 2.5 (triangles), λ u / λ l = 3 (squares), λ u / λ l = 3.5 (diamonds), and λ u / λ l = 4 (gray circles). The solid curves are calculated from Eq. (6).

Tables (4)

Tables Icon

Table 1 Comparison of R Values Calculated from Eq. (6) and Average Residual Reflectance R 10 of 10-Cluster Optimal AR Designs a

Tables Icon

Table 2 Comparison of R Values Calculated from Eq. (6) and Average Residual Reflectance R a v of 10-Cluster Optimal AR Designs a

Tables Icon

Table 3 Comparison of R Values Calculated from Eq. (6) and Average Residual Reflectance R 10 of 10-Cluster Optimal AR Designs a

Tables Icon

Table 4 Comparison of R Values Calculated from Eq. (6) and Average Residual Reflectance R 10 of 10-Cluster Optimal AR Designs in the Infrared Spectral Ranges a

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

R a v = 1 λ u λ l λ l λ u R ( λ ) d λ ,
ρ H L = n H / n L , ρ L a = n L / n a , ρ s a = n s / n a .
T c = λ u 2 [ 1 + 2 π arcsin ( ρ H L 1 ρ H L + 1 ) ] .
TOT = T c · M ,
R a v = R b 1 / M ,
R = f 1 ( ρ L a , ρ s a ) ) · [ π 4 ( π 120 ) 1 / ( λ u / λ l 1 ) ] ( 1 1 / ρ H L 2 ) ρ L a 1 , f 1 = ρ s a 2 ( 1 ρ L a 2 ) 2 + ( ρ L a 2 ρ s a 2 ) 2 ( ρ L a 2 + ρ s a ) 2 ( 1 + ρ s a ) 2 .
n s = n 2 n a 2 sin 2 θ , n s s = n s 2 n a 2 sin 2 θ , n a s = n a cos θ ,

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