Relations between Laplace’s spheroidal harmonics associated with different spheroidal coordinates are derived. The transition matrices for the functions of the 1st kind are lower triangular and are related by inversion. The matrices for the functions of the 2nd kind are the transposed ones for the functions of the 1st kind. The series for the functions of the 1st kind are finite, and those for the 2nd kind are infinite. In the latter case the region of convergence is considered. Using the derived relations, the rigid solution to the electrostatic problem for the multi-layered scatterers with nonconfocal spheroidal boundaries of the layers is obtained and the Rayleigh approximation is constructed, as well as an approximate approach to a similar light scattering problem, which provides reliable results far beyond the range of applicability of the Rayleigh approximation, is suggested.
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References
V. G. Farafonov, V. B. Il’in, V. I. Ustimov, and A. R. Tulegenov, “On the ellipsoidal model for small nonspherical particles,” Opt. Spektr., 122, 506–516 (2017).
V. G. Farafonov, V. B. Il’in, M. S. Prokopiev, A. R. Tulegenov, and V. I. Ustimov, “On the spheroidal model of light scattering by nonspherical particles,” Opt. Spektr., 126, 443–449 (2019).
V. G. Farafonov, V. I. Ustimov, V. B. Il’in, and M. V. Sokolovskaya, “Ellipsoidal model for small multilayer particles,” Opt. Spektr., 124, 241–249 (2018).
V. G. Farafonov, A. A. Vinokurov, and S. V. Barkanov, “Electrostatic solution and Rayleigh approximation for small nonspherical particles in a spheroidal basis,” Opt. Spektr., 111, 1026–1038 (2011).
V. G. Farafonov and M. V. Sokolovskaya, “Construction of the Rayleigh approximation for axisymmetric multilayer particles using the eigenfunctions of the Laplace operator,” Zap. Nauchn. Semin. POMI, 409, 187–211 (2012).
C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles [Russian translation], Mir, Moscow (1986).
V. G. Farafonov, “Light scattering by multilayer ellipsoids in Rayleigh approximation,” Opt. Spektr. 88, 441–444 (2000).
V. G. Farafonov and V. B. Il’in, “On the applicability of a spherical basis for spheroidal layered scatterers,” Opt. Spektr., 115, 836–843 (2013).
V. G. Farafonov, V. I. Ustimov, and M. V. Sokolovskaya, “Condition of applicability EBCM for small multilayer particles,” Opt. Spektr., 120, 470–483 (2016).
V. G. Farafonov, V. I. Ustimov, and V. B. Il’in, “Light scattering by small multilayer nonconfocal spheroids using suitable spheroidal bases,” Opt. Spektr., 125, 786–794 (2018).
V. I. Komarov, L. I. Ponomarev, and S. Yu. Slavyanov, Spheroidal and Coulomb Spheroidal Functions [in Russian], Nauka, Moscow (1976).
P. M. Morse and H. Feshbach, Methods of Theoretical Physics [Russian translation], In. Lit., Moscow (1958).
G. Jansen, “Transformation properties of spheroidal multipole moments and potentials,” J. Phys. A: Math. Gen., 33, 1375–1394 (2000).
V. A. Antonov and A. S. Baranov, “Connection between expansions of an external potential in spherical functions and spheroidal harmonics,” Zh. Tekh. Fiz., 72, 80–82 (2002).
H. Bateman and A. Erdelyi, Higher Transcendental Functions [Russian translation], Nauka, Moscow (1973).
V. G. Farafonov, “Rayleigh conjecture and the domain of applicability of the method of extended boundary conditions in electrostatic problems for nonspherical particles,” Opt. Spektr., 117, 949–962 (2014).
V. G. Farafonov, N. V. Voshchinnikov, and E. G. Semenova, “Some relations between wave spheroidal and spherical functions,” Zap. Nauchn. Semin. POMI, 426, 203–217 (2014).
V. G. Farafonov, “Diffraction of a plane electromagnetic wave by a dielectric spheroid,” Differents. Uravn., 19, 1765–1777 (1983).
V. V. Klimov, Nanoplasmonics [in Russian], Fizmatlit, Moscow (2009).
H. C. van de Hulst, Light Scattering by Small Particles [Russian translation], In. Lit., Moscow (1961).
V. G. Farafonov and V. I. Ustimov, “On the properties of the T-matrix in the Rayleigh approximation,” Opt. Spektr., 119, 1020–1032 (2015).
V. G. Farafonov and V. I. Ustimov, “Light scattering by small multilayer particles: a generalized method for separating variables,” Opt. Spektr., 124, 255–263 (2018).
V. G. Farafonov, V. I. Ustimov, and V. B. Il’in, “Rayleigh approximation for multilayer nonconfocal spheroids,” Opt. Spektr., 126, 450–457 (2019).
B. Posselt, V. G. Farafonov, V. B. Il’in, and M. S. Prokopjeva, “Light scattering by multilayered ellipsoidal particles in the quasistatic approximation,” Meas. Sci. Technol., 13, 256–262 (2002).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 483, 2019, pp. 199–242.
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Farafonov, V.G., Ustimov, V.I. & Il’in, V.B. Relations between Spheroidal Harmonics and the Rayleigh Approximation for Multilayered Nonconfocal Spheroids. J Math Sci 252, 702–730 (2021). https://doi.org/10.1007/s10958-021-05192-x
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DOI: https://doi.org/10.1007/s10958-021-05192-x