DOI: https://doi.org/10.15407/rpra25.02.147

Yu. V. Svishchov


Purpose: The behavior of spectral characteristics (eigenfrequencies, eigenmodes, Q-factors) of a spherical particle with the arbitrary values of permittivity and permeability is considered. The aim of this work is to study some important laws of behavior of the spectral characteristics of a particle with both positive and negative values of the real and imaginary parts of material parameters. The emphasis is made on the electric type eigenmodes.

Design/ methodology/approach: To achieve this goal, the corresponding spectral problem solution is given. The method of solution is based on the electromagnetic field representation as the expansion in vector spherical wave functions.

Findings: The dependences of the first eigenfrequencies of a spherical particle on the relative permittivity ε1 and relative permeability µ1, which real and imaginary parts can take both positive and negative values, are calculated. The eigenmodes are divided into two families: internal and external eigenmodes. The internal eigenmodes in each of the quadrants of plane (µ1, ε1) have an independent classification based on the structure of eigenmodes. Unlike internal eigenmodes, external eigenmodes have a single classification in plane (µ1, ε1). By their structure, external eigenmodes have the form of surface plasmon eigenmodes, which are distributed in the vicinity of a particle surface or outside it. In the first quadrant of plane (µ1, ε1), they repeatedly interact with internal eigenmodes that leads either to a hybridization of eigenmodes or to an exchange of types of eigenmodes. In the third quadrant of plane (µ1, ε1), the external eigenmodes can interact with each other. The anomalous behavior of the spectral characteristics of a spherical particle corresponds to the phenomenon of inter-type coupling of eigenmodes, being previously known and well described in the scientific literature.

Conclusions: The results of the studies made it possible to establish new patterns of behavior of the spectral characteristics of a spherical particle with arbitrary values of its permittivity and permeability.

Key words: spherical particle, dielectric ball, metamaterial, eigenfrequencies, eigenmodes

Manuscript submitted  23.03.2020

Radio phys. radio astron. 2020, 25(2): 147-157


1. MIE, G., 1908. Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen. Annalen der Physik [online]. vol. 330, is. 3, pp. 377–445. DOI: https://doi.org/10.1002/andp.19083300302

2. DEBYE, P., 1909. Der Lichtdruck auf Kugeln von beliebigem Material. Annalen der Physik [online]. vol. 335, is. 11, pp. 57–136. DOI: https://doi.org/10.1002/andp.19093351103

3. GASTINE, M., COURTOIS, L. and DORMAN, J. L., 1967. Electromagnetic resonances of free dielectric spheres. IEEE Trans. Microw. Theory Tech. vol. 15, is. 12, pp. 694–700. DOI: https://doi.org/10.1109/TMTT.1967.1126568

4. WOLFF, I., 2018. Electromagnetic Fields in Spherical Microwave Resonators: H-Modes and E- Modes in Lossless Open Dielectric Spheres. Version 05.2018. [online preprint]. Research Gate, May 2018. [viewed 5 April 2020]. Available from: https://www.researchgate.net/publication/325335243 DOI: https://doi.org/10.23919/PIERS.2018.8597646

5. KLIMOV, V. V., 2002. Spontaneous emission of an excited atom placed near a “left-handed” sphere. Opt. Commun. vol. 211, is. 1-6, pp. 183–196. DOI: https://doi.org/10.1016/S0030-4018(02)01802-3

6. SVISHCHOV, YU. V., 2019. The eigenmode interaction in a spherical dielectric resonator. Radiof. Elektron. vol. 24, is. 4, pp. 11–19. (in Russian). DOI: https://doi.org/10.15407/rej2019.04.011

7. SVISHCHOV, YU. V., 2019. Interaction of eigenmodes in a spherical particle with negative values of its material parameters. Radio Phys. Radio Astron. vol. 24, is. 3, pp. 206–217. (in Russian). DOI: https://doi.org/10.15407/rpra24.03.206

8. MELEZHIK, P. N., POEDINCHUK, A. E., TUCHKIN, YU. A. and SHESTOPALOV, V. P., 1988. On the Analytical Nature of the Phenomenon of Intertype Relationship of Natural Oscillations. Dok. Akad. Nauk SSSR. vol. 300, no. 6, pp. 1356–1359. (in Russian).

9. WEI, J. and XIAO, M., 2007. Electric and magnetic losses and gains in determining the sign of refractive index. Opt. Commun. vol. 270, is. 2. pp. 455–464. DOI: https://doi.org/10.1016/j.optcom.2006.09.039

10. AFANAS’EV, S. A., SANNIKOV, D. G. and SEMENTSOV, D. I., 2013. The refractive index sign chosen for amplifying and lossy metamaterials. J. Commun. Technol. Electron. vol. 58, is. 1, pp. 1–11. DOI: https://doi.org/10.7868/S0033849413010014

11. GRIGORENKO, A. N., 2006. Negative Refractive Index in Artificial Metamaterials. Opt. Lett. vol. 31, is. 16, pp. 2483–2485. DOI: https://doi.org/10.1364/OL.31.002483

12. STRATTON, J., 1941. Electromagnetic Theory. New York, London: McGraw-Hill Book Company, Inc.

13. MORSE, P. M. and FESHBACH, H., 1960. Methods of Theoretical Physics. Vol. 2. Moscow, Russia: Inostrannaya Literatura Publ. (in Russian).


spherical particle; dielectric ball; metamaterial; eigenfrequencies; eigenmodes

Full Text:


Creative Commons License

Licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0) .