photonics hv Article Exceptional Points through Variation of Distances between Four Coaxial Dielectric Disks Konstantin Pichugin 1 , Almas Sadreev 1,2, * and Evgeny Bulgakov 1   Citation: Pichugin, K.; Sadreev, A.; Bulgakov, E. Exceptional Points through Variation of Distances between Four Coaxial Dielectric Disks. Photonics 2021, 8, 460. https://doi.org/10.3390/ photonics8110460 Received: 4 August 2021 Accepted: 15 October 2021 Published: 21 October 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Federal Research Center KSC SB RAS, Kirensky Institute of Physics, 660036 Krasnoyarsk, Russia; knp@tnp.krasn.ru (K.P.); bulgakov100@yandex.ru (E.B.) 2 Department of Electronic Engineering, College of Information Science and Technology, Jinan University, Guangzhou 510632, China * Correspondence: almas@tnp.krasn.ru Abstract: By variation of a refractive index and aspect ratio of the isolated disk, we achieved exceptional points (EPs) at which the resonant frequencies and resonant modes coalesce. However, in practice, that kind of variation presents a technological problem. We considered the method to avoid the problem by substitution of two disk’s dimers. In each dimer, variation of the distance between disks was equivalent to a variation of the aspect ratio of the dimer. Moreover, the variation of the distance between dimers provides the second parameter that gives rise to a vast number of EPs. We recovered the initial resonant eigenmode by encircling multiple EPs two, three, and four times in the two-dimensional parametric space of distances. Keywords: resonant eigenmodes; multiple exceptional points; encircling of exceptional points 1. Introduction A dielectric particle embedded into open space is specified by eigenfrequencies, which are complex owing to open boundary conditions for the solutions of the homogeneous Maxwell equations. The most drastic difference between closed systems and open systems is that the latter exhibit exceptional points (EPs) where both the complex eigenfrequencies and eigenmodes coalesce [1,2]. Many works on EPs and their applications are associated with parity-time (PT) symmetric optical systems with a balanced gain and loss. In that case, EPs can be easily found by tuning a single parameter, namely, the amplitude of the balanced gain and loss [3–8]. Since it is not always easy or desirable to keep a balanced gain and loss, it is of significant interest to explore EPs and their applications in non-PT-symmetric optical systems. In the photonic system, the appearance of EPs can be exploited to a broad range of interesting applications, including lasing [9], asymmetric mode switching [10], nonreciprocal light transmission [11,12], and ultrasensitive sensing [13]. Currently, there have been studies concerning EPs for resonant states in extended periodic dielectric structures sandwiched between two homogeneous half-spaces [14–17], dual-mode planar optical waveguides [10] and plasmonic waveguides [18], layered struc- tures [3,19], two infinitely long dielectric cylinders [20–24], and even a single rod with a deformed cross-section [22,25–28]. As for compact dielectric resonators, we distinguished the studies of EPs in the high-Q microcavities with their boundary shape continuous vari- ables: a 2d chaotic deformed billiard [29], a compact-coated dielectric sphere [30], and a spheroid [31]. However, in practice, an achievement of the EP in such compact optical cavity by continuous deformation of its shape is technologically challenging. In the present study, we considered two dimers that each consisted of two coaxial disks. This introduced two-fold scale parameters: the distance between the disks in each dimer and the distance between the dimers. The first scale introduced a coupling strength between disks in a dimer, and the second scale introduced a coupling strength between distant dimers. This approach made it easy to conduct experimentally two-fold variation Photonics 2021, 8, 460. https://doi.org/10.3390/photonics8110460 https://www.mdpi.com/journal/photonics