Abstract
In this thesis, we study a class of spectral degeneracies that occur in non-Hermitian eigenvalue problems of resonant states on dielectric structures. The eigenvalue problems considered here are those related to electromagnetic wave solutions to Maxwell’s equations where the frequency (or the free-space wavenumber k) is considered as the eigenvalue. For open dielectric structures, these eigenvalue problems are non-Hermitian due to radiation losses. At a non-Hermitian degeneracy, two or more eigenvalues and—unlike the Hermitian case—their corresponding eigenfunctions coalesce. Non-Hermitian spectral degeneracies, also known as exceptional points (EPs), have recently found many interesting and useful applications such as sensing enhancement, exotic laser operation, unidirectional wave propagation, and optical parametric amplification.Here, we are concerned with two-dimensional (2D) structures on which Maxwell’s equations can be reduced to Helmholtz equations. Resonant states on these 2D structures are solutions to Helmholtz equations with outgoing radiation conditions. In general, a resonant solution has a complex eigenvalue and its field profile blows up at infinity. For a periodic (or uniform) slab, resonant modes are usually studied above the light line, i.e. Re(k) > |β|, where β is the Bloch wavenumber (or propagation constant). However, we show that resonant mode bands continue to exist below the light line where they end at some exceptional points. These EPs pose themselves as intrinsic ones, i.e., they always exist below the light line as branching points linking resonant and improper (unguided) modes.
Mostly above the light line, another class of EPs, related to the accidental coalescence of two resonant modes, may also exist. When the eigenvalues and eigenfunctions of two resonant modes coalesce at some particular values of the system parameters, a second-order EP is created. We study these EPs on a periodic slab that has a two-segment periodic permittivity profile. We show that EPs appear as families that depend continuously on the system parameters. We further consider the geometry-based limit of the periodic slab to a uniform slab; as the thickness of one segment tends to zero. This limit provides a way to
link the EP families to some points on the light line, approximately.
More details and accurate results on the EP limits can be obtained if the geometry of the structure is kept fixed while the uniform slab is approached via material variation. We study such EP limits on a slab with a periodic array of circular cylinders and calculating the EPs using the Dirichlet-to-Neumann map method. New third-order EP families are revealed below the light, and the EP limits are accurately found. We also emphasize the existence of a single fourth-order EP that separates the second and third-order EP families.
Finally, we study EPs that may also exist on non-periodic dielectric structures. We mainly consider EPs of resonant states for finite sets of parallel infinitely-long circular dielectric cylinders with subwavelength radii. For systems with two, three and four cylinders, we present examples for second and third-order EPs and highlight their topological features. This study provides insight into understanding EPs on more complicated photonic structures and can be used as a simple platform to explore potential applications of EPs in nanophotonics.
| Date of Award | 15 Jul 2019 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Ya Yan LU (Supervisor) |