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Applied Optics

Applied Optics


  • Editor: Joseph N. Mait
  • Vol. 51, Iss. 34 — Dec. 1, 2012
  • pp: 8277–8295

Numerical methods for the design of gradient-index optical coatings

Stephan W. Anzengruber, Esther Klann, Ronny Ramlau, and Diana Tonova  »View Author Affiliations

Applied Optics, Vol. 51, Issue 34, pp. 8277-8295 (2012)

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We formulate the problem of designing gradient-index optical coatings as the task of solving a system of operator equations. We use iterative numerical procedures known from the theory of inverse problems to solve it with respect to the coating refractive index profile and thickness. The mathematical derivations necessary for the application of the procedures are presented, and different numerical methods (Landweber, Newton, and Gauss–Newton methods, Tikhonov minimization with surrogate functionals) are implemented. Procedures for the transformation of the gradient coating designs into quasi-gradient ones (i.e., multilayer stacks of homogeneous layers with different refractive indices) are also developed. The design algorithms work with physically available coating materials that could be produced with the modern coating technologies.

© 2012 Optical Society of America

OCIS Codes
(310.1620) Thin films : Interference coatings
(310.5696) Thin films : Refinement and synthesis methods

ToC Category:
Thin Films

Original Manuscript: August 7, 2012
Revised Manuscript: September 28, 2012
Manuscript Accepted: October 12, 2012
Published: November 30, 2012

Stephan W. Anzengruber, Esther Klann, Ronny Ramlau, and Diana Tonova, "Numerical methods for the design of gradient-index optical coatings," Appl. Opt. 51, 8277-8295 (2012)

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  1. H. Anders and R. Eichinger, “Die optische Wirkung und praktische Bedeutung inhomogener Schichten,” Appl. Opt. 4, 899–905 (1965). [CrossRef]
  2. R. Jacobsson and J. O. Mårtensson, “Evaporated inhomogeneous thin films,” Appl. Opt. 5, 29–34 (1966). [CrossRef]
  3. E. Delano, “Fourier synthesis of multilayer filters,” J. Opt. Soc. Am. 57, 1529–1533 (1967). [CrossRef]
  4. M. Glio, “Design of a nonpolarizing beam splitter inside a glass cube,” Appl. Opt. 31, 5345–5349 (1992). [CrossRef]
  5. J. Ciosek, J. A. Dobrowolski, G. A. Clarke, and G. Laframboise, “Design and manufacture of all-dielectric nonpolarizing beam splitters,” Appl. Opt. 38, 1244–1250 (1999). [CrossRef]
  6. 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]
  7. W. Wang, S. Xiong, and Y. Zhang, “Design and analysis of all-dielectric broadband nonpolarizing parallel-plate beam splitters,” Appl. Opt. 46, 3185–3188 (2007). [CrossRef]
  8. P. G. Verly, “Optical coating synthesis by simultaneous refractive-index and thickness refinement of inhomogeneous films,” Appl. Opt. 37, 7327–7333 (1998). [CrossRef]
  9. 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]
  10. V. Janicki, D. Gäbler, S. Wilbrandt, R. Leitel, O. Stenzel, N. Kaiser, M. Lappschies, B. Görtz, D. Ristau, C. Rickers, and M. Vergöhl, “Deposition and spectral performance of an inhomogeneous broadband wide-angular antireflective coating,” Appl. Opt. 45, 7851–7857 (2006). [CrossRef]
  11. A. Thelen, “Nonpolarizing edge filters,” J. Opt. Soc. Am. 71, 309–314 (1981). [CrossRef]
  12. A. Thelen, “Nonpolarizing edge filters: part 2,” Appl. Opt. 23, 3541–3543 (1984). [CrossRef]
  13. V. Pervak, A. V. Tikhonravov, M. K. Trubetskov, J. Pistner, F. Krausz, and A. Apolonski, “Band filters: two-material technology versus rugate,” Appl. Opt. 46, 1190–1193 (2007). [CrossRef]
  14. S. Wilbrandt, O. Stenzel, and N. Kaiser, “All-oxide broadband antireflection coatings by plasma ion assisted deposition: design, simulation, manufacturing and re-optimization,” Opt. Express 18, 19732–19742 (2010). [CrossRef]
  15. T. V. Amotchkina, M. K. Trubetskov, V. Pervak, and A. V. Tikhonravov, “Design, production, and reverse engineering of two-octave antireflection coatings,” Appl. Opt. 50, 6468–6475 (2011). [CrossRef]
  16. D. Ristau, H. Ehlers, T. Gross, and M. Lappschies, “Optical broadband monitoring of conventional and ion processes,” Appl. Opt. 45, 1495–1501 (2006). [CrossRef]
  17. D. Rats, D. Poitras, J. M. Soro, and L. Martinu, “Mechanical properties of plasma-deposited silicon-based inhomogeneous optical coatings,” Surf. Coat. Technol. 111, 220–228 (1999). [CrossRef]
  18. R. Vernhes, O. Zabeida, J. E. Klemberg-Sapieha, and L. Martinu, “Single-material inhomogeneous optical filters based on microstructural gradients in plasma-deposited silicon nitride,” Appl. Opt. 43, 97–103 (2004). [CrossRef]
  19. O. Arnon and P. Baumeister, “Electric field distribution and the reduction of laser damage in multilayers,” Appl. Opt. 19, 1853–1855 (1980). [CrossRef]
  20. O. Arnon, “Loss mechanisms in dielectric optical interference devices,” Appl. Opt. 16, 2147–2151 (1977). [CrossRef]
  21. M. Lappschies, B. Görtz, and D. Ristau, “Application of optical broadband monitoring to quasi-rugate filters by ion-beam sputtering,” Appl. Opt. 45, 1502–1506 (2006). [CrossRef]
  22. D. Ristau, H. Ehlers, S. Schlichting, and M. Lappschies, “State of the art in deterministic production of optical thin films,” Proc. SPIE 7101, 71010C1–71010C14 (2008). [CrossRef]
  23. M. Jupé, M. Lappschies, L. Jensen, K. Starke, and D. Ristau, “Applications of mixture oxide materials for fs optics,” in Optical Interference Coatings (Optical Society of America, 2007), p. TuA6.
  24. A. Melninkaitis, T. Tolenis, L. Mažulé, J. Mirauskas, V. Sirutkaitis, B. Mangote, X. Fu, M. Zerrad, L. Gallais, M. Commandré, S. Kičas, and R. Drazdys, “Characterization of zirconia- and niobia-silica mixture coatings produced by ion-beam sputtering,” Appl. Opt. 50, C188–C196 (2011). [CrossRef]
  25. C.-J. Tang, C.-C. Jaing, K.-S. Lee, and C.-C. Lee, “Residual stress in Ta2O5-SiO2 composite thin-film rugate filters prepared by radio frequency ion-beam sputtering,” Appl. Opt. 47, C167–C171 (2008). [CrossRef]
  26. C. Polenzky, C. Rickers, and M. Vergöhl, “Properties of cosputtered SiO2-Ta2O5-mixtures,” Thin Solid Films 517, 3126–3129 (2009). [CrossRef]
  27. C.-J. Tang, C.-C. Jaing, K.-H. Lee, and C.-C. Lee, “Effect of thermal annealing on the optical properties and residual stress of graded-index-like films deposited by radio-frequency ion-beam sputtering,” Appl. Opt. 50, C62–C68 (2011). [CrossRef]
  28. B. J. Pond, J. I. DeBar, C. K. Carniglia, and T. Raj, “Stress reduction in ion beam sputtered mixed oxide films,” Appl. Opt. 28, 2800–2805 (1989). [CrossRef]
  29. M. Vergöhl, C. Rickers, F. Neumann, and C. Polenzky, “Temperature-resistant layered system,” U.S. patent 7, 985,489 B2 (26July2011).
  30. J. A. Dobrowolski and D. Lowe, “Optical thin film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039–3050 (1978). [CrossRef]
  31. G. Boivin and D. St. Germain, “Synthesis of gradient-index profiles corresponding to spectral reflectance derived by inverse Fourier transform,” Appl. Opt. 26, 4209–4213 (1987). [CrossRef]
  32. B. G. Bovard, “Rugate filter design: the modified Fourier transform technique,” Appl. Opt. 29, 24–30 (1990). [CrossRef]
  33. X. Cheng, B. Fan, J. A. Dobrowolski, Li Wang, and Z. Wang, “Gradient-index optical filter synthesis with controllable and predictable refractive index profiles,” Opt. Express 16, 2315–2321 (2008). [CrossRef]
  34. H. Fabricius, “Gradient-index filters: designing filters with steep skirts, high reflection, and quintic matching layers,” Appl. Opt. 31, 5191–5196 (1992). [CrossRef]
  35. A. V. Tikhonravov, B. T. Sullivan, and M. V. Borisova, “Discrete-Fourier-transform approach to inhomogeneous layer synthesis,” Appl. Opt. 33, 5142–5150 (1994). [CrossRef]
  36. P. G. Verly and J. A. Dobrowolski, “Iterative correction process for optical thin film synthesis with the Fourier transform method,” Appl. Opt. 29, 3672–3684 (1990). [CrossRef]
  37. P. G. Verly, “Fourier transform technique with frequency filtering for optical thin-film design,” Appl. Opt. 34, 688–694 (1995). [CrossRef]
  38. P. G. Verly, “Fourier transform technique with refinement in the frequency domain for the synthesis of optical thin films,” Appl. Opt. 35, 5148–5154 (1996). [CrossRef]
  39. D. Poitras, S. Larouche, and L. Martinu, “Design and plasma deposition of dispersion-corrected multiband rugate filters,” Appl. Opt. 41, 5249–5255 (2002). [CrossRef]
  40. P. V. Bulkin, P. L. Swart, and B. M. Lacquet, “Fourier-transform design and electron cyclotron resonance plasma-enhanced deposition of lossy graded-index optical coatings,” Appl. Opt. 35, 4413–4419 (1996). [CrossRef]
  41. S. Larouche and L. Martinu, “Dispersion implementation in optical filter design by the Fourier transform method using correction factors,” Appl. Opt. 46, 7436–7441 (2007). [CrossRef]
  42. P. G. Verly, “Hybrid approach for rugate filter design,” Appl. Opt. 47, C172–C178 (2008). [CrossRef]
  43. W. H. Southwell and R. L. Hall, “Rugate filter sidelobe suppression using quintic and rugated quintic matching layers,” Appl. Opt. 28, 2949–2951 (1989). [CrossRef]
  44. B. G. Bovard, “Rugate filter theory: an overview,” Appl. Opt. 32, 5427–5442 (1993). [CrossRef]
  45. W. E. Johnson and R. L. Crane, “Introduction to rugate filter technology,” Proc. SPIE 2046, 88–108 (1993). [CrossRef]
  46. W. H. Southwell, “Extended-bandwidth reflector designs by using wavelets,” Appl. Opt. 36, 314–318 (1997). [CrossRef]
  47. A. G. Imenes and D. R. McKenzie, “Flat-topped broadband rugate filters,” Appl. Opt. 45, 7841–7850 (2006). [CrossRef]
  48. W. H. Southwell, “Gradient-index antireflection coatings,” Opt. Lett. 8, 584–586 (1983). [CrossRef]
  49. J. H. Kim and Y. J. Lee, “Optimization of gradient-index antireflection coatings,” J. Opt. Soc. Korea 4, 86–88 (2000). [CrossRef]
  50. S. Dutta Gupta and G. S. Agarwal, “A new approach for broad-band omnidirectional antireflection coatings,” Opt. Express 15, 9614–9624 (2007). [CrossRef]
  51. D. Poitras and J. A. Dobrowolski, “Toward perfect antireflection coatings. 2. Theory,” Appl. Opt. 43, 1286–1295 (2004). [CrossRef]
  52. M. Chen, H.-C. Chang, A. S. P. Chang, S.-Y. Lin, J.-Q. Xi, and E. F. Schubert, “Design of optical path for wide-angle gradient-index antireflection coatings,” Appl. Opt. 46, 6533–6538 (2007). [CrossRef]
  53. T. Eisenhammer, M. Lazarov, M. Leutbecher, U. Schöffel, and R. Sizmann, “Optimization of interference filters with genetic algorithms applied to silver-based heat mirrors,” Appl. Opt. 32, 6310–6315 (1993). [CrossRef]
  54. R. Bertram, M. F. Ouellette, and P. Y. Tse, “Inhomogeneous optical coatings: an experimental study of a new approach,” Appl. Opt. 28, 2935–2939 (1989). [CrossRef]
  55. S. Martin, J. Rivory, and M. Schoenauer, “Synthesis of optical multilayer systems using genetic algorithms,” Appl. Opt. 34, 2247–2254 (1995). [CrossRef]
  56. S. I. Park and Y. J. Lee, “Design of multilayer antireflection coatings,” J. Korean Phys. Soc. 32, 676–680 (1998).
  57. P. L. Swart, A. P. Kotzé, and B. M. Lacquet, “Effects of the nature of the starting population on the properties of rugate filters designed with the genetic algorithm,” J. Lightwave Technol. 18, 853–859 (2000). [CrossRef]
  58. J.-M. Yang and C.-Y. Kao, “Efficient evolutionary algorithm for the thin-film synthesis of inhomogeneous optical coatings,” Appl. Opt. 40, 3256–3267 (2001). [CrossRef]
  59. J.-M. Yang and C.-Y. Kao, “An evolutionary algorithm for synthesis of multilayer coatings at oblique light incidence,” J. Lightwave Technol. 19, 559–570 (2001). [CrossRef]
  60. S. Larouche and L. Martinu, “Optical filters with prescribed optical thickness and refined refractive indices,” Appl. Opt. 47, 4140–4146 (2008). [CrossRef]
  61. S. Larouche and L. Martinu, “Step method: a new synthesis method for the design of optical filters with intermediate refractive indices,” Appl. Opt. 47, 4321–4330 (2008). [CrossRef]
  62. A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Application of constrained optimization to the design of quasi-rugate optical coatings,” Appl. Opt. 47, 5103–5109 (2008). [CrossRef]
  63. P. G. Verly, A. V. Tikhonravov, and M. K. Trubetskov, “Efficient refinement algorithm for the synthesis of inhomogeneous optical coatings,” Appl. Opt. 36, 1487–1495 (1997). [CrossRef]
  64. A. V. Tikhonravov, M. K. Trubetskov, T. V. Amotchkina, M. A. Kokarev, 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]
  65. A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems (Springer-Verlag, 1996).
  66. A. K. Louis, Inverse und schlecht gestellte Probleme (Teubner, 1989).
  67. B. Kaltenbacher, A. Neubauer, and O. Scherzer, Iterative Regularization Methods for Nonliner Ill-Posed Problems(Walter de Gruyter, 2008).
  68. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Klewer, 2000).
  69. D. A. Tonova, “Inverse profiling by ellipsometry: a Newton-Kantorovitch algorithm,” Opt. Commun. 105, 104–112 (1994). [CrossRef]
  70. D. A. Tonova and A. A. Konova, “Characterization of inhomogeneous dielectric coatings with arbitrary refractive index profiles by multiple angle of incidence ellipsometry,” Thin Solid Films 397, 17–23 (2001). [CrossRef]
  71. F. Natterer, The Mathematics of Computerized Tomography (SIAM, 2001).
  72. R. Ramlau and W. Ring, “A Mumford–Shah level-set approach for the inversion and segmentation of X-ray tomography data,” J. Comput. Phys. 221, 539–557 (2007). [CrossRef]
  73. E. Klann, R. Ramlau, and W. Ring, “A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data,” Inverse Probl. Imaging 5, 137–166 (2011). [CrossRef]
  74. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
  75. D. Mumford and J. Shah, “Optimal approximations by piecewise smooth functions and associated variational problems,” Commun. Pure Appl. Math. 42, 577–685 (1989). [CrossRef]
  76. A. Chambolle, “Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations,” SIAM J. Appl. Math. 55, 827–863 (1995). [CrossRef]
  77. T. F. Chan and L. A. Vese, “Image segmentation using level sets and the piecewise constant Mumford-Shah model” (UCLA CAM Report 00-14, 2000).
  78. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  79. S. Furman and A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Edition Frontieres, 1992).
  80. W. J. Gunning, R. L. Hall, F. J. Woodberry, W. H. Southwell, and N. S. Gluck, “Codeposition of continuous composition rugate filters,” Appl. Opt. 28, 2945–2948 (1989). [CrossRef]
  81. R. Leitel, O. Stenzel, S. Wilbrandt, D. Gäbler, V. Janicki, and N. Kaiser, “Optical and non-optical characterization of Nb2O5-SiO2 compositional graded-index layers and rugate structures,” Thin Solid Films 497, 135–141 (2006). [CrossRef]
  82. J. Weber, H. Bartzsch, and P. Frach, “Sputter deposition of silicon oxynitride gradient and multilayer coatings,” Appl. Opt. 47, C288–C292 (2008). [CrossRef]
  83. H. W. Engl and P. Kügler, “Nonlinear inverse problems: theoretical aspects and some industrial applications,” in Multidisciplinary Methods for Analysis Optimization and Control of Complex Systems, V. Capasso and J. Périaux, eds. (Springer, 2005), pp. 3–47.
  84. B. Eicke, “Iteration methods for convexly constrained ill-posed problems in Hilbert space,” Numer. Funct. Anal. Optim. 13, 413–429 (1992). [CrossRef]
  85. R. Ramlau, “TIGRA, An iterative algorithm for regularizing nonlinear ill-posed problems,” Inverse Probl. 19, 433–465 (2003). [CrossRef]
  86. R. Ramlau and G. Teschke, “A Tikhonov-based projection iteration for non-linear ill-posed problems with sparsity constraints,” Numer. Math. 104, 177–203 (2006). [CrossRef]
  87. R. Ramlau, “Regularization properties of Tikhonov regularization with sparsity constraints,” Electron. Trans. Numer. Anal. 30, 54–74 (2008).
  88. V. Janicki, J. Sancho-Parramon, and H. Zorc, “Refractive index profile modelling of dielectric inhomogeneous coatings using effective medium theories,” Thin Solid Films 516, 3368–3373 (2008). [CrossRef]
  89. M. Scherer, “Magnetron sputter-deposition on atom layer scale,” Vak. Forsch. Prax. 21, 24–30 (2009). [CrossRef]
  90. I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
  91. A. K. Louis, P. Maass, and A. Rieder, Wavelets, Theory and Applications (Wiley, 1997).
  92. S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1998).
  93. G. Strang and T. Nguyen, Wavelets and Filter Banks(Wellesley-Cambridge, 1996).
  94. A. Cohen, Numerical Analysis of Wavelet Methods (North-Holland, 2003).
  95. D. L. Donoho, and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994). [CrossRef]
  96. D. L. Donoho and M. Johnstone, “Minimax estimation via wavelet shrinkage,” Ann. Stat. 26, 879–921 (1998). [CrossRef]
  97. D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Wavelet shrinkage: asymptopia?” J. R. Stat. Soc. Ser. B 57, 301–369 (1995).
  98. U. B. Schallenberg, “Antireflection design concepts with equivalent layers,” Appl. Opt. 45, 1507–1514 (2006). [CrossRef]
  99. A. V. Tikhonravov and J. A. Dobrowolski, “Quasi-optimal synthesis for antireflection coatings: a new method,” Appl. Opt. 32, 4265–4275 (1993). [CrossRef]
  100. 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]
  101. A. V. Tikhonravov, “Some theorethical aspects of thin film optics and their applications,” Appl. Opt. 32, 5417–5426 (1993). [CrossRef]

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