Exceptional points (EPs) in open optical systems are rigorously studied using
the resonant-state expansion (RSE). A spherical resonator, specifically a
homogeneous dielectric sphere in a vacuum, perturbed by two point-like defects
which break the spherical symmetry and bring the optical modes to EPs, is used
as a worked example. The RSE is a non-perturbative approach encoding the
information about an open optical system in matrix form in a rigorous way, and
thus offering a suitable tool for studying its EPs. These are simultaneous
degeneracies of the eigenvalues and corresponding eigenfunctions of the system,
which are rigorously described by the RSE and illustrated for perturbed
whispering-gallery modes (WGMs). An exceptional arc, which is a line of
adjacent EPs, is obtained analytically for perturbed dipolar WGMs. Perturbation
of high-quality WGMs with large angular momentum and their EPs are found by
reducing the RSE equation to a two-state problem by means of an orthogonal
transformation of a large RSE matrix. WGM pairs of opposite chirality away from
EPs are shown to have the same chirality at EPs. This chirality can be observed
in circular dichroism measurements, as it manifested itself in a
squared-Lorentzian part of the optical spectra, which we demonstrate here
analytically and numerically in the Purcell enhancement factor for the
perturbed dipolar WGMs.Comment: 24 pages. 13 figures (3 in Appendix). To be submitted in Physical
Review A. Authors: K S Netherwood (primary author), H Riley (initial concept
work), E A Muljarov (theme leader