Non-Hermitian degeneracies of internal–external
mode pairs in dielectric microdisks
CHANG-HWAN YI,
1
JULIUS KULLIG,
1,2
MARTINA HENTSCHEL,
2
AND JAN WIERSIG
1,
*
1
Institut für Physik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany
2
Institut für Physik, Technische Universität Ilmenau, D-98693 Ilmenau, Germany
*Corresponding author: jan.wiersig@ovgu.de
Received 22 January 2019; revised 19 February 2019; accepted 19 February 2019; posted 19 February 2019 (Doc. ID 357992);
published 25 March 2019
Open quantum and wave systems can exhibit non-Hermitian degeneracies called exceptional points, where both
the eigenvalues and the corresponding eigenstates coalesce. Previously, such exceptional points have been inves-
tigated in dielectric microcavities in terms of optical modes which are well confined inside the cavity. However,
beside these so-called “internal modes” with a relatively high quality factor, there exists another kind of mode
called “external modes,” which have a large decay rate and almost zero intensity inside the cavity. In the present
paper, we demonstrate the physical significance of the external modes via the occurrence of exceptional points of
internal–external mode pairs for transverse electric polarization. Our numerical studies show that these excep-
tional points can be achieved by either a boundary deformation of the microdisk or by introducing absorption
into a circular cavity.
© 2019 Chinese Laser Press
https://doi.org/10.1364/PRJ.7.000464
1. INTRODUCTION
Openness is a generic property of realistic quantum and wave
systems. Hence, their dynamics is in general described by a
non-Hermitian Hamiltonian with complex eigenvalues. An
interesting feature of this non-Hermitian physics is the excep-
tional point (EP) in parameter space where both the eigenvalues
(complex frequencies) and the corresponding eigenstates
(modes) coalesce [1–3]. So far, EPs have been studied both
theoretically and experimentally in various research fields such
as hydrogen atoms [4], photonic lattices [5], microlasers [6,7],
microwave resonators [8–10], and optical microcavities [11–14].
One of the fascinating applications of EPs is an enhancement of
the sensitivity of sensors [15–19].
Optical microdisk cavities are ideal model systems to study
non-Hermitian physics [20]. In recent years, it was demon-
strated that EPs in microcavities can be caused by several mech-
anisms, e.g., by boundary deformations [12,21–24] or external
perturbations like nanoscatterers or nanofiber tips [11,13,25].
However, in these studies, the involved modes are always so-
called internal modes whose intensity is well confined inside
the cavity, resulting in a small decay rate, i.e., a large Q-factor.
In the language of formal quantum mechanics, they are called
Feshbach resonances [26–28].
Along with the internal modes, there is another class of
optical modes known as “external modes,” which have a large
decay rate and almost no intensity inside the cavity. The
internal and external modes have been classified in previous
studies [28–30] according to their behavior in the limit of
infinite refractive index n; i.e., in the limit of vanishing open-
ness of the system. In this limit, the external modes remain
unbound and are therefore referred to as “shape resonances”
[26–28]. The internal modes, however, can become bound
states with real-valued frequency in this limit.
In spite of the pioneer works proving the interactions
between internal and external modes [30], there have been
no investigations conceiving the possibility of EPs associated
with external modes. Therefore, the aim of this paper is to re-
veal the existence of such EPs involving internal and external
mode pairs. To demonstrate this, we introduce two kinds of
systematic perturbations of a circular cavity: (i) a deformation
of the cavity’
s boundary, and (ii) an absorbing material.
The paper is organized as follows. In Section 2, the optical
modes of the dielectric microdisk cavity are briefly explained. In
Section 3, a transition between the strong and weak coupling
regime in the circular cavity is analyzed. Section 4 demonstrates
EPs through the variation of a deformation parameter. In
Section 5, absorption is used to generate EPs in the circular
cavity. Section 6 summarizes and concludes the paper.
2. MODES IN THE DIELECTRIC MICRODISK
For the convenience of the reader, we repeat some known facts
of optical modes in quasi-two-dimensional microdisks [20].
The damped time-harmonic solutions of Maxwell’s equations
in the dielectric microdisk cavity with effective refractive index
464
Vol. 7, No. 4 / April 2019 / Photonics Research
Research Article
2327-9125/19/040464-09 Journal © 2019 Chinese Laser Press
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