# Scattering of Plane Electromagnetic Waves by Lattices of Spherical Dielectric Resonators with Degenerate Lower Types of Natural Oscillations

## DOI:

https://doi.org/10.20535/RADAP.2023.91.12-17## Keywords:

dielectric resonator, lattice, coupling coefficient, c-function, scattering amplitude## Abstract

The problem of scattering of plane electromagnetic waves on lattices of spherical dielectric resonators (DRs) with low magnetic oscillations is considered. The results of theoretical calculations of complex coefficients of mutual coupling of spherical dielectric resonators in open space for cases of excitation of degenerate types of oscillations are presented. The expressions found coincide with those obtained earlier for the special case of oscillations of resonators excited along or perpendicular to the line connecting their centers. The main regularities of the change in the coupling coefficients with variations in the coordinates of the resonators in the transverse plane are considered. Analytical expressions for c-functions are found for the field of fundamental magnetic oscillations of a resonator and a plane wave in open space. On the basis of the obtained formulas, with the help of the perturbation theory, the characteristics of the scattering of plane waves on a square lattice of spherical DRs with basic degenerate magnetic oscillation types are calculated and studied. The distribution of the scattering field in the wave zone of the grating is studied for different angles of incidence. The regions of variation of the angles of incidence are determined, in which the scattering amplitude of a lattice constructed on the basis of spherical DRs differs most noticeably from DR lattices of other shapes with nondegenerate types of oscillations. The polarization characteristics of scattered waves in the far zone of the lattice are calculated. It is noted that, in contrast to the lattices of pseudorotating cylindrical DRs with the main magnetic types of oscillations, lattices based on spherical resonators are characterized by a more complex distribution of the polarization of scattered waves. In the wave zone of the lattice, scattered waves of all three types of polarization, linear, circular, elliptical, can be observed. The obtained results significantly expand the possibilities of developers, since allow us to create electrodynamic models of lattices, as well as other devices in the millimeter and infrared ranges, built on the basis of the use of spherical resonators with oscillations of the main types. Such lattices can be used in antennas, passive reflectors, and other devices of modern optical communication systems.

## References

**References**

Almpanis E., Zouros G. P., Tsakmakidis K. L. (2021). Active THz metasurfaces for compact isolation. *Journal of the Optical Society of America B*, Vol. 38, Iss. 9, pp. 191-197. doi:10.1364/JOSAB.430160.

Abujetas D. R., Sánchez-Gil J. A. (2021). Near-Field Excitation of Bound States in the Continuum in All-Dielectric Metasurfaces through a Coupled Electric/Magnetic Dipole Model. *Nanomaterials*, Vol. 11, Iss. 4, 998. doi:10.3390/nano11040998.

Molet P., Gil-Herrera L. K., Garcia-Pomar J. L., Caselli N., Blanco B., Lуpez C., Mihi A. (2020). Large area metasurfaces made with spherical silicon resonators. *Nanophotonics*, Vol. 9, Iss. 4, pp. 943–951. doi:10.1515/nanoph-2020-0035.

Panagiotidis E., Almpanis E., Stefanou N., Papanikolaou N. (2020). Multipolar interactions in Si sphere metagratings. *Journal of Applied Physics*, Vol. 128, Iss. 9, 093103. DOI:10.1063/5.0012827.

Kwon S.-H., Kim Y., Moon K., Hong S., Lee Y. J., Shin E. (2019). Far-Field Analysis on Reflecting Colors of Dielectric Nanosphere Metasurface. *Journal of Nanomaterials*, Vol. 2019, Article ID 6532967. doi:10.1155/2019/6532967.

Brettin A., Abolmaali F., Limberopoulos N. I., Green A., Anisimov I., Urbas A. M., Astratov V. N. (2018). Towards fabrication of mid-IR FPAs with enhanced sensitivity and reduced dark current by using integration with microspherical arrays. *NAECON 2018 - IEEE National Aerospace and Electronics Conference*, pp. 533-535. DOI: 10.1109/NAECON.2018.8556727.

Hoang T. X., Nagelberg S. N., Kolle M., Barbastathis G. (2017). Fano resonances from coupled whispering–gallery modes in photonic molecules. *Optics Express*, Vol. 25, Iss. 12, pp. 13125-13144. doi:10.1364/OE.25.013125.

Shen F., An N., Tao Y., Zhou H., Jiang Z., Guo Z. (2016). Anomalous forward scattering of gain-assisted dielectric shell-coated metallic core spherical particles. *Nanophotonics*, Vol. 6, Iss. 5, pp. 1063–1072. doi:10.1515/nanoph-2016-0141.

Wang H., et al. (2013). Computational Modeling and Experimental Study on Optical Microresonators Using Optimal Spherical Structure for Chemical Sensing. *Advanced Chemical Engineering Research*, Vol. 2, Iss. 3, pp. 45-50.

Mitsui T., et al. (2011). Influence of micro-joints formed between spheres in coupled-resonator optical waveguide. *Optics Express*, Vol. 19, Iss. 22, pp. 22258-22267. doi:10.1364/OE.19.022258.

Xifre-Perez E., Domenech J. D., Fenollosa R., Munoz P., Capmany J., Mesenguer F. (2011). All silicon waveguide spherical microcavity coupler device. *Optics Express*, Vol. 19, Iss. 4, pp. 3185-3192. doi:10.1364/OE.19.003185.

Astratov, V. N. (2010). Fundamentals and Applications of Microsphere Resonator Circuits. In: Chremmos, I., Schwelb, O., Uzunoglu, N. (eds) Photonic Microresonator Research and Applications. *Springer Series in Optical Sciences*, Vol 156. doi:10.1007/978-1-4419-1744-7_17.

Chiasera A., Dumeige Y., Feron P., Ferrari M., Jestin Y., et al. (2010). Spherical whispering-gallery-mode microresonators. *Laser&Photonics Reviews*, Vol. 4, Iss. 3, pp. 457-482. DOI:10.1002/lpor.200910016.

Cai X., Zhu R., Hu G. (2008). Experimental study for metamaterials based on dielectric resonators and wire frame. *Metamaterials*, Vol. 2, Iss. 4, pp. 220-226. doi:10.1016/j.metmat.2008.08.001.

Rusakou K. I., Gladyshchuk A. A., Chugunov S. V., Rakovich Y. P., Donegan J. F., Rogach A. L., and Gaponik N. (2008). Photonic molecule modes in coupled spherical microcavities with CdTe nanocrystals, Proc. *SPIE* 7009, Second International Conference on Advanced Optoelectronics and Lasers, 70090H. doi:10.1117/12.793330.

Astratov V. N., Ashili S. P. (2007). Percolation of light through whispering gallery modes in 3D lattices of coupled microspheres. *Optics Express*, Vol. 15, Iss. 25, pp. 17351-17361. doi:10.1364/OE.15.017351.

Kieu K., Mansuripur M. (2007). Fiber laser using a microsphere resonator as a feedback element. *Optics Letters*, Vol. 32, Iss. 3, pp. 244-246. doi:10.1364/OL.32.000244.

Chen Z., Taflove A., Backman V. (2006). Highly efficient optical coupling and transport phenomena in chains of dielectric microspheres. *Optics Letters*, Vol. 31, Iss. 3, pp. 389-391. doi:10.1364/OL.31.000389.

Boriskina S. V. (2006). Theoretical prediction of a dramatic *Q*-factor enhancement and degeneracy removal of whispering gallery modes in symmetrical photonic molecules. *Optics Letters*, Vol. 31, Iss. 3, pp. 338-340. doi:10.1364/OL.31.000338.

Gorodetsky M. L. and Ilchenko V. S. (1999). Optical microsphere resonators: Optimal coupling to high-*Q* whispering-gallery modes. *Journal of the Optical Society of America B*, Vol. 16, Iss. 1, pp. 147–154. doi:10.1364/JOSAB.16.000147.

Mackowsky D. W., Mishenko M. I. (1996). Calculation of the *T* matrix and the scattering matrix for ensembles of spheres. *Journal of the Optical Society of America A*, Vol. 13, Iss. 11, pp. 2266-2278. doi:10.1364/JOSAA.13.002266.

Trubin A. (2016). Lattices of Dielectric Resonators. Springer Series in Advanced Microelectronics, Vol. 53. *Springer*, 171 p. DOI:10.1007/978-3-319-25148-6.

*Handbook of mathematical functions*. Ed. by M. Abramowitz and I. Stegun. National bureau of standards. (1964). 830 p.

Blum K. (2011). *Density Matrix Theory and Applications*. Springer, 343 p.

Trubin A. A. (2015). Coupling coefficients of the Spherical dielectric microresonators with whispering gallery modes. *Visnyk NTUU KPI Seriia - Radiotekhnika Radioaparatobuduvannia*, No. 62, pp. 49-61. doi: 10.20535/RADAP.2015.62.49-61.

Trubin A. A. (2022). Scattering of plane waves on pseudo-rotatable lattices of cylindrical dielectric resonators. *Modern Challenges in Telecommunications. 16 International Scientific and Technical Conference*, pp. 69-72.

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*Visnyk NTUU KPI Seriia - Radiotekhnika Radioaparatobuduvannia*, (91), pp. 12-17. doi: 10.20535/RADAP.2023.91.12-17.

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