DOI: https://doi.org/10.26089/NumMet.2024s01

Inverse Sturm–Liouville Problem and its application to inverse problems in thin film optics

Authors

  • Alexander V. Tikhonravov
  • Andrey A. Shkalikov

Keywords:

inverse problem
Sturm-Liouville problem
transformation operators
optical coatings
spectral characteristics

Abstract

The application of the results of the inverse Sturm–Liouville problem for the development of the theory and numerical methods for solving inverse problems in thin film optics is shown. A brief overview of the most effective methods for designing optical coatings is given. It is shown that with the help of developed methods the most demanding types of coatings with a large number of optimized parameters can be designed. The unique spectral properties of these coatings are achieved on the basis of the developed theory.

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Published

2024-12-16

Issue

2024: Special issue

Section

Methods and algorithms of computational mathematics and their applications

Author Biographies

Alexander V. Tikhonravov

Lomonosov Moscow State University,
Research Computing Center
• Head of Laboratory

Andrey A. Shkalikov

Lomonosov Moscow State University,
Faculty of Mechanics and Mathematics
• Professor

References

  1. V. Ambarzumian, “Über eine Frage der Eigenwerttheorie,” Z. Physik 53 (9-10), 690-695 (1929).
    doi 10.1007/BF01330827
  2. G. Borg, “Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe: Bestimmung der Differentialgleichung durch die Eigenwerte,” Acta Math. 78, 1-96 (1946).
    doi 10.1007/BF02421600
  3. V. A. Marchenko, “Some Problems in the Theory of Second-Order Differential Operators,” Dokl. Akad. Nauk SSSR 72, 457-460 (1950).
  4. V. A. Marchenko, “Some Questions of the Theory of One-Dimensional Linear Differential Operators of the Second Order,” Tr. Mosk. Mat. Obs. 1, 327-420 (1952).
  5. V. A. Marchenko, Sturm-Liouville Operators and Their Applications (Naukova Dumka, Kiev, 1977) [in Russian].
  6. A. N. Tikhonov, “On the Uniqueness of the Solution of the Problem of Electric Prospecting,” Dokl. Akad. Nauk SSSR 69, 797-800 (1949).
  7. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for Solving the Korteweg-deVries Equation,” Phys. Rev. Lett. 19 (19), 1095-1097 (1967).
    doi 10.1103/PhysRevLett.19.1095
  8. B. M. Levitan, Inverse Sturm-Liouville Problems (Nauka, Moscow, 1984; VSP Verlag, Zeist, 1987).
  9. V. A. Yurko, Inverse Spectral Problems and Their Applications (Saratov Pedagog. Inst., Saratov, 2001) [in Russian].
  10. V. V. Kravchenko, Direct and Inverse Sturm-Liouville Problems: A Method of Solution (Birkh854user, Cham, 2020).
    doi 10.1007/978-3-030-47849-0
  11. A. M. Savchuk and A. A. Shkalikov, “Inverse Problems for Sturm-Liouville Operators with Potentials in Sobolev Spaces: Uniform Stability,” Funkts. Anal. Prilozh. 44 (4), 34-53 (2010) [Funct. Anal. Its Appl. 44 (4), 270-285 (2010)].
    doi 10.1007/s10688-010-0038-6
  12. A. Tikhonravov, Optical Coatings: Design, Characterization, Monitoring (SPIE Press, Bellingham, United States, 2024).
  13. A. V. Tikhonravov, “The Synthesis of Laminar Media with Specified Amplitude and Phase Properties,” Zh. Vychisl. Mat. Mat. Fiz. 25 (11), 1674-1688 (1985) [USSR Comput. Math. Math. Phys. 25 (6), 55-64 (1985)].
    doi 10.1016/0041-5553(85)90009-6
  14. A. V. Tikhonravov and M. K. Trubetskov, “Modern Design Tools and a New Paradigm in Optical Coating Design,” Appl. Opt. 51 (30), 7319-7332 (2012).
    doi 10.1364/AO.51.007319

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Copyright (c) 2024 А. В. Тихонравов, А. А. Шкаликов

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