Yu. V. Svishchov


PACS number: 41.20.-q

Purpose: The behavior of spectral characteristics (eigenfrequencies, natural oscillations, Q-factors) of a spherical particle whose permittivity and permeability simultaneously take on negative values (a “left-handed” sphere) is considered. It is known that in the vicinity of some values of the material parameters a transformation of natural oscillations in such a particle is observed. The purpose of this work is to study the patterns of the behavior of spectral characteristics of a particle under the conditions of transformation of natural oscillations.

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

Findings: The dependences of the eigenfrequencies of a spherical particle on the permittivity and permeability, which can simultaneously take on negative values, are calculated. A method for classification of natural oscillations is proposed. It is based on the structure of natural oscillations. It is shown that the anomalous behavior of the spectral characteristics of a spherical particle corresponds to the previously known and well-described phenomenon of inter-type coupling of oscillations in the scientific literature. For the control parameter of this phenomenon, both the relative permittivity and the relative permeability of the particle can be used. It is established that the natural oscillations of a spherical particle being distributed in the vicinity of the particle surface or outside it interact. As a result, when changing the material parameters of a particle, either hybridization or exchange of oscillation types is observed.

Conclusions: The results of the research allowed us to establish new laws of the anomalous behavior of the spectral characteristics of a spherical particle with simultaneously negative values of its permittivity and permeability. 

Key words: metamaterial, spherical particle, eigenfrequency, interaction
of eigenmodes  

Manuscript submitted 30.05.2019

Radio phys. radio astron. 2019, 24(3): 206-217 


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

2. 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).

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

4. DEBYE, P., 1909. Der Lichtdruck auf Kugeln von beliebigem Material. Annalen der Physik. vol. 335, Is. 11, pp. 57–136. DOI: 10.1002/andp.19093351103

5. 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: 10.1109/TMTT.1967.1126568

6. 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 June 2019]. Available from:

7. MÜLLER, C., 1969. Foundations of the Mathematical Theory of Electromagnetic Waves. Berlin, Heidelberg: Springer-Verlag. DOI: 10.1007/978-3-662-11773-6

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

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

10. KORN, G. A. and KORN, T. M., 2000. Mathematical Handbook for Scientists and Engineers. 2th ed. New York: Dover Publ.





metamaterial; spherical particle; eigenfrequency; interaction of eigenmodes

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