Abstract
Optical waveguides, like slab waveguides, ridge waveguides, and optical fibers, are classical optical devices that can guide the propagation of light. They can also be used to make up interferometers, filters, and sensors, etc. Classical wave phenomena such as guidance and leakage of light in waveguides have been widely investigated in the last century. In this thesis, some novel wave phenomena, including complex modes in waveguides, modes in parity–time (PT) symmetric waveguides, and bound states in the continuum (BICs), are studied in detail. This research provides an insight into understanding waveguides from a novel optical view and can be adopted to explore potential photonic applications.Optical properties of waveguides depend highly on their eigenmodes such as guided, resonant and leaky modes. Modes of waveguides are eigensolutions of Maxwell’s equations without sources. An eigenmode in a waveguide is associated with a free space wavenumber k0 and a propagation constant β for the waveguide axis. Sometimes β is called as Bloch wavenumber if the dielectric function ε is periodic along the axis. For a guided mode, light can propagate along the waveguide axis and be confined around the waveguide core. In an open lossless dielectric waveguide, guided modes typically have a real k0 and a real β. For lossy waveguides, guided modes with a real k0 must have a complex β due to absorption loss. In fact, there may exist complex modes that are also confined around the waveguide core but have a complex β and a real k0 even though the waveguide consists of lossless dielectric materials. In this thesis, we consider circular fibers and strip waveguides, study the formation mechanism of complex modes, and calculate the dispersion relations for several complex modes in each waveguide. This study provides a basis for realizing potential applications of complex modes.
A PT-symmetric waveguide has a complex ε with the symmetric real and anti-symmetric imaginary parts keeping the balanced gain and loss. It is known that a PT-symmetric waveguide can support guided modes with a real β and a real k0 if the imaginary part of ε is relatively small. These guided modes in PT-symmetric waveguides are widely studied for a fixed k0 with varying imaginary parts of ε but their dispersion curves have been rarely addressed. In this thesis, we analyze dispersion curves in a PT-symmetric uniform slab waveguide. It is shown that the number of cutoff points and dispersion curves is finite. For special ε, there exist some dispersion curves which are not tangential to the lightline at cutoff points. We also find third order exceptional points (EPs) in this waveguide.
PT-symmetric waveguides can also support other eigenmodes such as leaky modes and resonant modes. A leaky or resonant mode with a real k0 or β in a waveguide satisfies an outgoing condition and decays exponentially with increasing propagation length or time t respectively due to the radiation loss. It has a complex β or k0 and a divergent field in the lateral direction. Research on dispersion curves of eigenmodes is of fundamental importance since they demonstrate intrinsic optical properties of waveguides. In this thesis, various eigenmodes are calculated from a dispersion equation with the help of Riemann sheets. Our research then provides a complete picture for different kinds of eigenmodes. We study the evolution of dispersion curves by varying imaginary parts of ε for a PT-symmetric uniform slab waveguide. Different dispersion curves are characterized by unusual features like EPs and local behaviors near cutoff points.
A BIC is a special leaky or resonant mode which has a real β, a real k0 and bound mode profile in the lateral direction. It can also be regarded as a guided mode but there exist propagating waves carrying power to or from infinity. BICs can be classified into generic BICs and non-generic BICs by a precise definition. In this thesis, we consider BICs in rotationally symmetric periodic (RSP) waveguides. A fiber grating and a one-dimensional (1D) periodic array of spheres are examples of RSP waveguides. A BIC in an RSP waveguide can be characterized by an azimuthal index m additionally. In this thesis, we investigate the robustness of BICs in RSP waveguides. The question is whether a BIC in a RSP waveguide with a reflection symmetry along its axis z, can continue its existence when the waveguide is perturbed by small but arbitrary structural perturbations that preserve the periodicity and the reflection symmetry in z. It is shown that generic BICs with m = 0 and m ≠ 0 are robust and non-robust, respectively, and a non-robust BIC with m ≠ 0 can continue to exist if the perturbation contains one tunable parameter. The theory is established by proving the existence of a BIC in the perturbed structure mathematically, where the perturbation is small but arbitrary, and contains an extra tunable parameter for the case of m ≠ 0.
BICs can also exist in a strip waveguide with lateral leakage channels if slab guided modes can be supported away from the waveguide core. A BIC is surrounded by a family of resonant or leaky modes with quality factors (Q-factors) or leakage losses approaching infinity or zero respectively. In this thesis, for a strip waveguide, we construct a perturbation theory for resonant and leaky modes near BICs. It is shown that for a generic BIC with propagation constant β∗, we have Q ∼ (β − β∗)-2 for nearby resonant modes and Im(βl ) ∼ (k − k∗)2 for nearby leaky modes, where β, βl, and β∗ are the propagation constants of the resonant mode, leaky mode and the BIC respectively, k and k∗ are the real free space wavenumbers of the leaky mode and the BIC, respectively. For a non-generic BIC, we can get Q ∼ (β − β∗)−4 for nearby resonant modes and Im(βl) ∼ (k − k∗)4 for nearby leaky modes at least. The robustness theory built for generic BICs is not applicable to non-generic BICs in such a waveguide. In order to analyze the local behavior at non-generic BICs under symmetric perturbations, for a waveguide which has a lateral mirror symmetry, we construct a bifurcation theory and show that typically two BICs bifurcate from a non-generic BIC. This means that non-generic BICs are actually merging BICs which have been found in many periodic structures.
| Date of Award | 12 Sept 2023 |
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| Original language | English |
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| Supervisor | Ya Yan LU (Supervisor) |