Spherical symmetry breaking in electric, magnetic and toroidal multipole moment radiations in spherical toroidal resonant cavities and optimum-efficiency antennas

E. Ley Koo, H. Torres-Bustamante, A. Góngora T.


This Letter reports the breaking of the spherical symmetry in the complete electromagnetic multipole expansion when its sources are distributed on spherical toroidal surfaces, identifying the specic geometrical and physical changes from
the familiar case of sources on a spherical surface. In fact, for spherical toroids dened by concentric spherical rings and symmetric conical rings, the boundary conditions at the latter are not compatible in general with integer values for the orbital angular momentum label of the multipole moments: the polar angle eigenfunctions become Legendre functions of order λ and associativity m represented as innite series with a denite parity, and their complementary associated radial functions are spherical Bessel functions of the same order λ. Consequently, the corresponding multipole sources for the electric, magnetic and toroidal moments and their connections are identied within the Debye formalism, and the
appropriate outgoing wave Green functions are constructed in the new basis of eigenfunctions of the Helmholtz equation. Our familiarity with the exact solutions, for the cases of the complete sphere and of cylindrical toroids, allow us to give a preliminary account of the electromagnetic elds for the spherical toroids via the integration of their sources and the Green function for resonant cavities and optimum effciency antennas.


Electromagnetic multipole moment expansion

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V. Savinov, V. A. Fedotov, and N. I. Zheludev, Toroidal dipolar excitation and macroscopic electro- magnetic properties of metamaterials, Phys. Rev. B 89 (2014) 205112. https://doi.org/10.1103/PhysRevB.89.205112.

N. Papasimakis, V. A. Fedotov, V. Savinov, T. A. Raybould, and N. I. Zheludev, Electromagnetic toroidal excitations in matter and free space, Nat. Mater., 15 (2016) 263. https://doi.org/10.1038/nmat4563.

N. Talebi, S. Guo, and P. A. van Aken, Theory and applications of toroidal moments in electrodynam- ics: their emergence, characteristics, and technological relevance, Nanophotonics 7 (2018) 93. https://doi.org/10.1515/nanoph-2017-0017.

I. Fernandez-Corbaton, S. Nanz, and C. Rockstuhl, On the dynamic toroidal multipoles from local- ized electric current distributions, Sci. Rep. 7 (2017) 7527. https://doi.org/10.1038/s41598-017-07474-4.

V. A. Fedotov, Electrodynamics beyond common multipoles, in Proceedings of the 2019 International Conference on Electromagnetics in Advanced Applications, edited by IEEE (IEEE, Granada, 2019), https://doi.org/10.1109/ICEAA.2019.8878911.

N. Talebi, Near-Field-Mediated Photon-Electron Interactions,(Springer, Switzerland, 2019), https://doi.org/10.1007/978-3-030-33816-9, pp. 81-118.

A. Gongora T. and E. Ley-Koo, Complete electromagnetic multipole expansion including toroidal moments, Rev. Mex. Fis. E 52 (2006) 188.

E. Ley-Koo, A. Gongora T., and H. Torres-Bustamante, Addendum to complete electromagnetic multipole expansion including toroidal moments Rev. Mex. Fis E 52 (2006) 138 Rev. Mex. Fis. E 17 (2020) 138. https://doi.org/10.31349/RevMexFisE.17.138.

E. Ley-Koo, C. Esparza-Lopez, and H. Torres-Bustamante, Exact solutions for electromagnetic fields inside and outside a spherical surface with magnetic/electric dipole distributed sources, Rev. Mex. Fis. E 64 (2018) 139. https://doi.org/10.31349/RevMexFisE.17.138.

H. Torres-Bustamante and E. Ley-Koo, Puntos cuánticos y radiación electromagnética multipolar en toroides cilíndricos, LXIII Congreso Nacional de Física, Sociedad Mexicana de Física, 2020.

E. Ley-Koo, Recent progress in confined atoms and molecules: Superintegrability and symmetry breakings, Rev. Mex. Fis. E 64 (2018) 326. https://doi.org/10.31349/RevMexFis.64.326.

DOI: https://doi.org/10.31349/RevMexFis.67.174


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REVISTA MEXICANA DE FÍSICA, year 67, issue 2, March-April 2021. Bimonthly Journal published by Sociedad Mexicana de Física, A. C. Departamento de Física, 2º Piso, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, Alcaldía Coyacán, C.P. 04510 , Ciudad de México. Apartado Postal 70-348. Tel. (+52)55-5622-4946, https://rmf.smf.mx/ojs/rmf, e-mail: rmf@ciencias.unam.mx. Chief Editor: José Alejandro Ayala Mercado. INDAUTOR Certificate of Reserve: 04-2019-080216404400-203, ISSN: 2683-2224 (on line), 0035-001X (print), both granted by Instituto Nacional del Derecho de Autor. Responsible for the last update of this issue, Technical Staff of Sociedad Mexicana de Física, A. C., Fís. Efraín Garrido Román, 2º. Piso, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, Alcaldía Coyacán, C.P. 04510 , Ciudad de México. Date of last modification, March 1st., 2021.

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