Abstract

We consider the problem of designing antireflection coatings by numerical optimization methods with merit functions in different metrics. The dependence of the optimal solution on the metrics is investigated through numerical experiments on several types of coatings, using the average and maximal values of the reflected power in the regions of interest. The results confirm existing statements and provide a few new findings, e.g., some specific metrics can yield particularly better solutions than others.

© 2013 Optical Society of America

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  1. A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their applications,” Appl. Opt. 32, 5417–5426 (1993).
    [CrossRef]
  2. 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]
  3. P. Baumeister, “Design of multilayer filters by successive approximations,” J. Opt. Soc. Am. 48, 955–957 (1958).
    [CrossRef]
  4. J. A. Dobrowolski and R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876–2893 (1990).
    [CrossRef]
  5. A. V. Tikhonravov, “On the synthesis of optical thin films using optimality conditions,” Vestn. Mosk. Univ. Fiz. Astron. 23, 91–93 (1982).
  6. G. Bao, “Inverse and optimal design problems in diffractive optics,” in Proceedings of the International Conference on Inverse Problems, Y. Hon, M. Yamamoto, J. Cheng, and J. Lee, eds. (World Scientific, 2003), pp. 37–46.
  7. G. Bao and E. Boneetier, “Optimal design of periodic diffractive structures,” Appl. Math. Optim. 43, 103–116 (2001).
    [CrossRef]
  8. G. Bao and D. C. Dobson, “Modeling and optimal design of diffractive optical structures,” Surv. Math. Ind. 8, 37–62 (1998).
  9. G. Bao, D. C. Dobson, and J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1043 (1995).
    [CrossRef]
  10. D. C. Dobson, “Optimal shape design of blazed diffraction gratings,” Appl. Math. Optim. 40, 61–78 (1999).
    [CrossRef]
  11. D. C. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Eur. J. Appl. Math. 4, 321–339 (1993).
    [CrossRef]
  12. G. Bao and K. Huang, “Optimal design of guided mode grating resonance filters,” IEEE Photon. Technol. Lett. 16, 141–143 (2004).
    [CrossRef]
  13. G. Bao and K. Huang, “Computational design for guided-mode grating resonances,” J. Opt. Soc. Am. A 22, 1408–1413 (2005).
    [CrossRef]
  14. G. Bao and K. Ramdani, “Resonant frequencies for diffraction gratings,” Appl. Math. Lett. 15, 755–760 (2002).
    [CrossRef]
  15. G. Bao, K. Huang, and G. Schmidt, “Optimal design of nonlinear diffraction gratings,” J. Comput. Phys. 184, 106–121 (2003).
    [CrossRef]
  16. J. A. Dobrowolski, “Completely automatic synthesis of optical thin film systems,” Appl. Opt. 4, 937–946 (1965).
    [CrossRef]
  17. Sh. A. Furman and A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Edition Frontières, 1992).
  18. H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Taylor & Francis, 2001).
  19. P. G. Verly, “Modified needle method with simultaneous thickness and refractive-index refinement for the synthesis of inhomogeneous and multilayer optical thin films,” Appl. Opt. 40, 5718–5725 (2001).
    [CrossRef]
  20. C. Sanderson, “Armadillo: C++ linear algebra library,” http://arma.sourceforge.net .
  21. A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, M. A. Kokare, N. Kaiser, O. Stenzel, S. Wilbrandt, and D. Gäbler, “New optimization algorithm for the synthesis of rugate optical coatings,” Appl. Opt. 45, 1515–1524(2006).
    [CrossRef]
  22. R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, 1987).
  23. G. Erdmann and F. Santosa, “Minimax design of optically transparent and reflective coatings,” J. Opt. Soc. Am. A 21, 1730–1739 (2004).
    [CrossRef]
  24. A. Premoli and M. L. Rastello, “Minimax refining of optical multilayer systems,” Appl. Opt. 31, 1597–1605 (1992).
    [CrossRef]
  25. A. Premoli and M. L. Rastello, “Minimax refining of wideband antireflection coatings for wide angular incidence,” Appl. Opt. 33, 2018–2024 (1994).
    [CrossRef]
  26. S. G. Johnson, “The NLopt nonlinear-optimization package,” http://ab-initio.mit.edu/nlopt .
  27. D. C. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
    [CrossRef]
  28. D. Kraft, “Algorithm 733: Tomp Fortran modules for optimal control calculations,” ACM Trans. Math. Softw. 20, 262–281 (1994).
    [CrossRef]
  29. P. G. Verly, “Optical coating synthesis by simultaneous refractive-index and thickness refinement of inhomogeneous films,” Appl. Opt. 37, 7327–7333 (1998).
    [CrossRef]
  30. T. V. Amotchkina, “Empirical expression for the minimum residual reflectance of normal- and oblique-incidence antireflection coatings,” Appl. Opt. 47, 3109–3113 (2008).
    [CrossRef]
  31. 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, C124–C130 (2008).
    [CrossRef]
  32. R. Willey, “Predicting achievable design performance of broadband antireflection coatings,” Appl. Opt. 32, 5447–5451 (1993).
    [CrossRef]
  33. R. Willey, “Refined criteria for estimating limits of broad-band AR coatings,” Proc. SPIE 5250, 393–399 (2004).
  34. 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]
  35. A. V. Tikhonravov and M. K. Trubetskov, “Modern design tools and a new paradigm in optical coating design,” Appl. Opt. 51, 7319–7332 (2012).
    [CrossRef]

2012 (1)

2008 (2)

2006 (1)

2005 (1)

2004 (3)

G. Erdmann and F. Santosa, “Minimax design of optically transparent and reflective coatings,” J. Opt. Soc. Am. A 21, 1730–1739 (2004).
[CrossRef]

G. Bao and K. Huang, “Optimal design of guided mode grating resonance filters,” IEEE Photon. Technol. Lett. 16, 141–143 (2004).
[CrossRef]

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

2003 (1)

G. Bao, K. Huang, and G. Schmidt, “Optimal design of nonlinear diffraction gratings,” J. Comput. Phys. 184, 106–121 (2003).
[CrossRef]

2002 (1)

G. Bao and K. Ramdani, “Resonant frequencies for diffraction gratings,” Appl. Math. Lett. 15, 755–760 (2002).
[CrossRef]

2001 (2)

1999 (1)

D. C. Dobson, “Optimal shape design of blazed diffraction gratings,” Appl. Math. Optim. 40, 61–78 (1999).
[CrossRef]

1998 (2)

G. Bao and D. C. Dobson, “Modeling and optimal design of diffractive optical structures,” Surv. Math. Ind. 8, 37–62 (1998).

P. G. Verly, “Optical coating synthesis by simultaneous refractive-index and thickness refinement of inhomogeneous films,” Appl. Opt. 37, 7327–7333 (1998).
[CrossRef]

1996 (2)

1995 (1)

1994 (2)

D. Kraft, “Algorithm 733: Tomp Fortran modules for optimal control calculations,” ACM Trans. Math. Softw. 20, 262–281 (1994).
[CrossRef]

A. Premoli and M. L. Rastello, “Minimax refining of wideband antireflection coatings for wide angular incidence,” Appl. Opt. 33, 2018–2024 (1994).
[CrossRef]

1993 (3)

1992 (1)

1990 (1)

1989 (1)

D. C. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[CrossRef]

1982 (1)

A. V. Tikhonravov, “On the synthesis of optical thin films using optimality conditions,” Vestn. Mosk. Univ. Fiz. Astron. 23, 91–93 (1982).

1965 (1)

1958 (1)

Amotchkina, T. V.

Bao, G.

G. Bao and K. Huang, “Computational design for guided-mode grating resonances,” J. Opt. Soc. Am. A 22, 1408–1413 (2005).
[CrossRef]

G. Bao and K. Huang, “Optimal design of guided mode grating resonance filters,” IEEE Photon. Technol. Lett. 16, 141–143 (2004).
[CrossRef]

G. Bao, K. Huang, and G. Schmidt, “Optimal design of nonlinear diffraction gratings,” J. Comput. Phys. 184, 106–121 (2003).
[CrossRef]

G. Bao and K. Ramdani, “Resonant frequencies for diffraction gratings,” Appl. Math. Lett. 15, 755–760 (2002).
[CrossRef]

G. Bao and E. Boneetier, “Optimal design of periodic diffractive structures,” Appl. Math. Optim. 43, 103–116 (2001).
[CrossRef]

G. Bao and D. C. Dobson, “Modeling and optimal design of diffractive optical structures,” Surv. Math. Ind. 8, 37–62 (1998).

G. Bao, D. C. Dobson, and J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1043 (1995).
[CrossRef]

G. Bao, “Inverse and optimal design problems in diffractive optics,” in Proceedings of the International Conference on Inverse Problems, Y. Hon, M. Yamamoto, J. Cheng, and J. Lee, eds. (World Scientific, 2003), pp. 37–46.

Baumeister, P.

Boneetier, E.

G. Bao and E. Boneetier, “Optimal design of periodic diffractive structures,” Appl. Math. Optim. 43, 103–116 (2001).
[CrossRef]

Cox, J. A.

DeBell, G. W.

Dobrowolski, J. A.

Dobson, D. C.

D. C. Dobson, “Optimal shape design of blazed diffraction gratings,” Appl. Math. Optim. 40, 61–78 (1999).
[CrossRef]

G. Bao and D. C. Dobson, “Modeling and optimal design of diffractive optical structures,” Surv. Math. Ind. 8, 37–62 (1998).

G. Bao, D. C. Dobson, and J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1043 (1995).
[CrossRef]

D. C. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Eur. J. Appl. Math. 4, 321–339 (1993).
[CrossRef]

Erdmann, G.

Fletcher, R.

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, 1987).

Furman, Sh. A.

Sh. A. Furman and A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Edition Frontières, 1992).

Gäbler, D.

Huang, K.

G. Bao and K. Huang, “Computational design for guided-mode grating resonances,” J. Opt. Soc. Am. A 22, 1408–1413 (2005).
[CrossRef]

G. Bao and K. Huang, “Optimal design of guided mode grating resonance filters,” IEEE Photon. Technol. Lett. 16, 141–143 (2004).
[CrossRef]

G. Bao, K. Huang, and G. Schmidt, “Optimal design of nonlinear diffraction gratings,” J. Comput. Phys. 184, 106–121 (2003).
[CrossRef]

Kaiser, N.

Kemp, R. A.

Kokare, M. A.

Kraft, D.

D. Kraft, “Algorithm 733: Tomp Fortran modules for optimal control calculations,” ACM Trans. Math. Softw. 20, 262–281 (1994).
[CrossRef]

Liu, D. C.

D. C. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Taylor & Francis, 2001).

Nocedal, J.

D. C. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[CrossRef]

Premoli, A.

Ramdani, K.

G. Bao and K. Ramdani, “Resonant frequencies for diffraction gratings,” Appl. Math. Lett. 15, 755–760 (2002).
[CrossRef]

Rastello, M. L.

Santosa, F.

Schmidt, G.

G. Bao, K. Huang, and G. Schmidt, “Optimal design of nonlinear diffraction gratings,” J. Comput. Phys. 184, 106–121 (2003).
[CrossRef]

Stenzel, O.

Sullivan, B. T.

Tikhonravov, A. V.

Trubetskov, M. K.

Verly, P. G.

Wilbrandt, S.

Willey, R.

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

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

ACM Trans. Math. Softw. (1)

D. Kraft, “Algorithm 733: Tomp Fortran modules for optimal control calculations,” ACM Trans. Math. Softw. 20, 262–281 (1994).
[CrossRef]

Appl. Math. Lett. (1)

G. Bao and K. Ramdani, “Resonant frequencies for diffraction gratings,” Appl. Math. Lett. 15, 755–760 (2002).
[CrossRef]

Appl. Math. Optim. (2)

G. Bao and E. Boneetier, “Optimal design of periodic diffractive structures,” Appl. Math. Optim. 43, 103–116 (2001).
[CrossRef]

D. C. Dobson, “Optimal shape design of blazed diffraction gratings,” Appl. Math. Optim. 40, 61–78 (1999).
[CrossRef]

Appl. Opt. (14)

A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, M. A. Kokare, N. Kaiser, O. Stenzel, S. Wilbrandt, and D. Gäbler, “New optimization algorithm for the synthesis of rugate optical coatings,” Appl. Opt. 45, 1515–1524(2006).
[CrossRef]

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, C124–C130 (2008).
[CrossRef]

T. V. Amotchkina, “Empirical expression for the minimum residual reflectance of normal- and oblique-incidence antireflection coatings,” Appl. Opt. 47, 3109–3113 (2008).
[CrossRef]

A. V. Tikhonravov and M. K. Trubetskov, “Modern design tools and a new paradigm in optical coating design,” Appl. Opt. 51, 7319–7332 (2012).
[CrossRef]

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

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

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

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

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

A. Premoli and M. L. Rastello, “Minimax refining of wideband antireflection coatings for wide angular incidence,” Appl. Opt. 33, 2018–2024 (1994).
[CrossRef]

P. G. Verly, “Optical coating synthesis by simultaneous refractive-index and thickness refinement of inhomogeneous films,” Appl. Opt. 37, 7327–7333 (1998).
[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]

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]

P. G. Verly, “Modified needle method with simultaneous thickness and refractive-index refinement for the synthesis of inhomogeneous and multilayer optical thin films,” Appl. Opt. 40, 5718–5725 (2001).
[CrossRef]

Eur. J. Appl. Math. (1)

D. C. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Eur. J. Appl. Math. 4, 321–339 (1993).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

G. Bao and K. Huang, “Optimal design of guided mode grating resonance filters,” IEEE Photon. Technol. Lett. 16, 141–143 (2004).
[CrossRef]

J. Comput. Phys. (1)

G. Bao, K. Huang, and G. Schmidt, “Optimal design of nonlinear diffraction gratings,” J. Comput. Phys. 184, 106–121 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Math. Program. (1)

D. C. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[CrossRef]

Proc. SPIE (1)

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

Surv. Math. Ind. (1)

G. Bao and D. C. Dobson, “Modeling and optimal design of diffractive optical structures,” Surv. Math. Ind. 8, 37–62 (1998).

Vestn. Mosk. Univ. Fiz. Astron. (1)

A. V. Tikhonravov, “On the synthesis of optical thin films using optimality conditions,” Vestn. Mosk. Univ. Fiz. Astron. 23, 91–93 (1982).

Other (6)

G. Bao, “Inverse and optimal design problems in diffractive optics,” in Proceedings of the International Conference on Inverse Problems, Y. Hon, M. Yamamoto, J. Cheng, and J. Lee, eds. (World Scientific, 2003), pp. 37–46.

Sh. A. Furman and A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Edition Frontières, 1992).

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Taylor & Francis, 2001).

C. Sanderson, “Armadillo: C++ linear algebra library,” http://arma.sourceforge.net .

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, 1987).

S. G. Johnson, “The NLopt nonlinear-optimization package,” http://ab-initio.mit.edu/nlopt .

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Figures (5)

Fig. 1.
Fig. 1.

Dependence of Pave and Pmax on p for three types of AR coatings.

Fig. 2.
Fig. 2.

Refractive-index profiles and power of reflection in the omnidirectional case for p=98 (top) and p=99 (bottom).

Fig. 3.
Fig. 3.

Refractive-index profiles and reflected power for the broadband case at p=1, 100 and p= (top to bottom).

Fig. 4.
Fig. 4.

Refractive-index profiles and reflected power for the omnidirectional case at p=1, 100 and p= (top to bottom).

Fig. 5.
Fig. 5.

Refractive-index profiles (top) and reflected power (bottom) in the broadband-omnidirectional case at p=1, 100 and , respectively.

Equations (10)

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

{1Mi=1M[R(ki)]2}12,1Mi=1MR(ki),max1iMR(ki),
Mi=[cosδiisinδiηiiηisinδicosδi][BC]=(i=1DMi)[1ηD+1]r=η0BCη0B+CR=r*r,
δi=kniticosθiηi={nicosθi,s-polarizationnicosθi,p-polarizationθi=sin1(sinθInIni).
i=1j1Mi,i=j+1DMi.
fp={{1Mi=1M[R(ki)]p}1p,ifp<,max1iMR(ki),ifp=,
mintmax1iMR(t,ki)
mintssubject toR(t,ki)s0i,
fp={{1Mi=1M[R(ki)]p}1p,ifp<,max1iMR(ki)I(ki),ifp=,
fp={{1Ni=1N[R(θi)cos(θi)]p}1p,ifp<,max1iNR(θi)cos(θi),ifp=,
fp={{1MNi=1Mj=1N[R(ki,θj)cos(θj)]p}1p,ifp<,max1iM,1jNR(ki,θj)cos(θj),ifp=,

Metrics