Comparison transfer matrix methods and scattering matrix method for investigation the optical properties of multilayer structures

Authors

  • Nikolay V. Egorov St. Petersburg State University, 7–9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation https://orcid.org/0000-0003-4721-1377
  • Arthur G. Fedorov M. K. Ammosov North-Eastern Federal University, 58, ul. Belinskogo, Yakutsk, 677027, Russian Federation https://orcid.org/0000-0002-8905-9564
  • Vasiliy V. Trofimov St. Petersburg State University, 7–9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation

DOI:

https://doi.org/10.21638/spbu10.2024.401

Abstract

This article presents an analysis of transfer matrix method (TMM) and scattering matrix method (SMM) for determining reflection and transmission coefficients of thin films. Investigated single layer structures of semiconductor materials (Si, Ge, GaAs), noble metals (Ag, Au, Cu) and multilayer structure of Si. Numeric results were getting in two diapason wavelengths: λ = 0.2067–0.8267 µm and λ = 0.2–20 µm. In this work obtained with TMM and SMM the reflection and transmission coefficient of layer structures. Numerical results of reflection coefficients of all investigation structures were exactly match with literature data. But results we got for the transmission coefficients did not match of literature data for the both of method. This mismatch is investigated, as we assume from some of normalization coefficient, corresponding a refractive index of right side of medium which we didn’t take into account.

Keywords:

transfer matrix methods, scattering matrix method, reflection coefficient, transmission coefficient, layer structure

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References

Литература

Egorov N. V., Antonova L. I., Karpov A. G., Trofimov V. V., Fedorov A. G. Theoretical and experimental evaluation of the electrical parameters of a holographic microscope // Journal of Surface Investigation. X-ray, Synchrotron and Neutron Techniques. 2020. Vol. 14. P. 1061–1065. https://doi.org/10.1134/S1027451020050250

Egorov N. V., Karpov A. G., Antonova L. I., Fedorov A. G., Trofimov V. V., Antonov S. R. Technique for investigating the spatial structure of thin films at a nanolevel // Journal of Surface Investigation. X-ray, Synchrotron and Neutron Techniques. 2011. Vol. 5. N 5. P. 992–995. https://doi.org/10.1134/S1027451011100089

Abeles F. Sur la propagation des ondes electromagnetiques dans les milieux stratifies // Ann. Phys. (Paris). 1948. N 3. P. 504–520. https://doi.org/10.1051/anphys/194812030504

Koji O., Hatsuo I. Matrix formalism for calculation of electric field intensity of light in stratified multilayered films // Applied Optics. 1990. Vol. 29. N 13. P. 1952–1959. https://doi.org/10.1364/ao.29.001952

Charalambos C. K., Dimitrios I. S. General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference // Applied Optics. 2002. Vol. 41. N 19. P. 3978–3987. https://doi.org/10.1364/AO.41.003978

Aspnes D. E., Studna A. A. Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV // Physical Review B. 1983. Vol. 27. P. 985–1009. https://doi.org/10.1103/PhysRevB.27.985

Polyanskiy M. N. Refractiveindex.info database of optical constants // Sci. Data. 2024. Vol. 11. Art. N 94. https://doi.org/10.1038/s41597-023-02898-2

Johnson P. B., Christy R. W. Optical constants of the noble metals // Physical Review B. 1972. Vol. 6. N 12. P. 4370–-4379. https://doi.org/10.1103/PhysRevB.6.4370

Shkondin E., Takayama O., Aryaee P. M. E., Liu P., Larsen P. V., Mar M. D., Jensen F., Lavrinenko A. V. Large-scale high aspect ratio Al-doped ZnO nanopillars arrays as anisotropic metamaterials // Opt. Mater. Express. 2017. Vol. 7. P. 1606–1627. https://doi.org/10.1364/OME.7.001606

Ciesielski A., Skowronski L., Trzinski M., Szoplik T. Controlling the optical parameters of self-assembled silver films with wetting layers and annealing // Appl. Surf. Sci. 2017. Vol. 421B. P. 349–356. https://doi.org/10.1016/j.apsusc.2017.01.039

Querry M. R. Optical constants. 1985. Contractor Report CRDC-CR-85034.

Amotchkina T., Trubetskov M., Hahner D., Pervak V. Characterization of e-beam evaporated Ge, YbF3, ZnS, and LaF3 thin films for laser-oriented coatings // Applied Optics. 2020. Vol. 59. P. A40–A47. https://doi.org/10.1364/AO.59.000A40

Olmon R. L., Slovick B., Johnson T. W., Shelton D., Oh S.-H., Boreman G. D., Raschke M. B. Optical dielectric function of gold // Physical Review. 2012. Vol. B86. Art. N 235147. https://doi.org/10.1103/PhysRevB.86.235147

Papatryfonos K., Angelova T., Brimont A., Reid B., Guldin S., Smith P. R., Tang M., Li K., Seeds A. J., Liu H., Selviah D. R. Refractive indices of MBE-grown AlxGa1-xAs ternary alloys in the transparent wavelength region // AIP Adv. 2021. Vol. 11. Art. N 025327. https://doi.org/10.1063/5.0039631

Dyakov S. A., Tolmachev V. A., Astrova E. V., Tikhodeev S. G., Timoshenko V. Yu., Perova T. S. Numerical methods for calculation of optical properties of layered structures // Proceedings of SPIE 7521. International Conference on Micro- and Nano-Electronics. 2009. Art. N 75210G. https://doi.org/10.1117/12.862566

Yuk Kei Ko D., Inkson J. C. Matrix method for tunneling in heterostructures: Resonant tunneling in multilayer systems // Physical Review B. 1988. Vol. 38. N 14. P. 9945–9951. https://doi.org/10.1103/PhysRevB.38.9945

Lifeng Li. Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings // Journal of Opt. Soc. Amer. A. 1996. Vol. 13. N 5. P. 1024–1035. https://doi.org/10.1364/JOSAA.13.001024

Tikhodeev S. G., Yablonskii A. L., Muljarov E. A., Gippius N. A., Teruya I. Quasiguided modes and optical properties of photonic crystal slabs // Physical Review B. 2002. Vol. 66. Art. N 045102(17). https://doi.org/10.1103/PhysRevB.66.045102

Whittaker D. M. Scattering-matrix treatment of patterned multilayer photonic structures // Physical Review B. 1989. Vol. 60. N 4. P. 2610–2618. https://doi.org/10.1103/PhysRevB.60.2610


References

Egorov N. V., Antonova L. I., Karpov A. G., Trofimov V. V., Fedorov A. G. Theoretical and experimental evaluation of the electrical parameters of a holographic microscope. Journal of Surface Investigation. X-ray, Synchrotron and Neutron Techniques, 2020, vol. 14, pp. 1061–1065. https://doi.org/10.1134/S1027451020050250

Egorov N. V., Karpov A. G., Antonova L. I., Fedorov A. G., Trofimov V. V., Antonov S. R. Technique for investigating the spatial structure of thin films at a nanolevel. Journal of Surface Investigation. X-ray, Synchrotron and Neutron Techniques, 2011, vol. 5, no. 5, pp. 992–995. https://doi.org/10.1134/S1027451011100089

Abeles F. Sur la propagation des ondes electromagnetiques dans les milieux stratifies. Ann. Phys. (Paris), 1948, no. 3, pp. 504–520. https://doi.org/10.1051/anphys/194812030504

Koji O., Hatsuo I. Matrix formalism for calculation of electric field intensity of light in stratified multilayered films. Applied Optics, 1990, vol. 29, no. 13, pp. 1952–1959. https://doi.org/10.1364/ao.29.001952

Charalambos C. K., Dimitrios I. S. General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference. Applied Optics, 2002, vol. 41, no. 19, pp. 3978–3987. https://doi.org/10.1364/AO.41.003978

Aspnes D. E., Studna A. A. Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV. Physical Review B, 1983, vol. 27, pp. 985–1009. https://doi.org/10.1103/PhysRevB.27.985

Polyanskiy M. N. Refractiveindex.info database of optical constants. Sci. Data, 2024, vol. 11, art. no. 94. https://doi.org/10.1038/s41597-023-02898-2

Johnson P. B., Christy R. W. Optical constants of the noble metals. Physical Review B, 1972, vol. 6, no. 12, pp. 4370–4379. https://doi.org/10.1103/PhysRevB.6.4370

Shkondin E., Takayama O., Aryaee P. M. E., Liu P., Larsen P. V., Mar M. D., Jensen F., Lavrinenko A. V. Large-scale high aspect ratio Al-doped ZnO nanopillars arrays as anisotropic metamaterials. Opt. Mater. Express, 2017, vol. 7, pp. 1606–1627. https://doi.org/10.1364/OME.7.001606

Ciesielski A., Skowronski L., Trzinski M., Szoplik T. Controlling the optical parameters of self-assembled silver films with wetting layers and annealing. Appl. Surf. Sci., 2017, vol. 421B, pp. 349–356. https://doi.org/10.1016/j.apsusc.2017.01.039

Querry M. R. Optical constants, 1985. Contractor Report CRDC-CR-85034.

Amotchkina T., Trubetskov M., Hahner D., Pervak V. Characterization of e-beam evaporated Ge, YbF3, ZnS, and LaF3 thin films for laser-oriented coatings. Applied Optics, 2020, vol. 59, pp. A40–A47. https://doi.org/10.1364/AO.59.000A40

Olmon R. L., Slovick B., Johnson T. W., Shelton D., Oh S.-H., Boreman G. D., Raschke M. B. Optical dielectric function of gold. Physical Review, 2012, vol. 86, art. no. 235147. https://doi.org/10.1103/PhysRevB.86.235147

Papatryfonos K., Angelova T., Brimont A., Reid B., Guldin S., Smith P. R., Tang M., Li K., Seeds A. J., Liu H., Selviah D. R. Refractive indices of MBE-grown AlxGa1-xAs ternary alloys in the transparent wavelength region. AIP Adv., 2021, vol. 11, art. no. 025327. https://doi.org/10.1063/5.0039631

Dyakov S. A., Tolmachev V. A., Astrova E. V., Tikhodeev S. G., Timoshenko V. Yu., Perova T. S. Numerical methods for calculation of optical properties of layered structures. Proceedings of SPIE 7521. International Conference on Micro- and Nano-Electronics, 2009, art. no. 75210G. https://doi.org/10.1117/12.862566

Yuk Kei Ko D., Inkson J. C. Matrix method for tunneling in heterostructures: Resonant tunneling in multilayer systems. Physical Review B, 1988, vol. 38, no. 14, pp. 9945–9951. https://doi.org/10.1103/PhysRevB.38.9945

Lifeng Li. Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings. Journal of Opt. Soc. Amer. A, 1996, vol. 13, no. 5, pp. 1024–1035. https://doi.org/10.1364/JOSAA.13.001024

Tikhodeev S. G., Yablonskii A. L., Muljarov E. A., Gippius N. A., Teruya I. Quasiguided modes and optical properties of photonic crystal slabs. Physical Review B, 2002, vol. 66, art. no. 045102(17). https://doi.org/10.1103/PhysRevB.66.045102

Whittaker D. M. Scattering-matrix treatment of patterned multilayer photonic structures. Physical Review B, 1989, vol. 60, no. 4, pp. 2610–2618. https://doi.org/10.1103/PhysRevB.60.2610

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Published

2024-12-30

How to Cite

Egorov, N. V., Fedorov, A. G., & Trofimov, V. V. (2024). Comparison transfer matrix methods and scattering matrix method for investigation the optical properties of multilayer structures. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 20(4), 432–445. https://doi.org/10.21638/spbu10.2024.401

Issue

Vol. 20 No. 4 (2024)

Section

Applied Mathematics

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Articles of "Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.