Transition radiation, non-Hermiticity and temporal modulation in photonic and lattice systems

Abstract

This dissertation is a compilation of three independent, but nonetheless connected papers. The first one extends the concept of transition radiation into photonic crystals, that is, the radiation emitted when a charge cross the boundary of two different photonic crystals. To simplify the problem and for practical considerations, the focus is put on the edge waves emitted that travel along the boundary, if there is any supported. For frequencies where only edge waves exist, the field and thus the power spectrum are calculated conveniently using the Bloch mode expansion thanks to the remaining translational symmetry of the system. An example is then given where topological protected edge waves between two topologically distinct photonic crystal domains are excited by a moving charge. For this specific problem, the concept of quasi-phase matching is introduced which provides a very good understanding to the power spectrum. The second paper demonstrates that eigendecomposition is not suited for the analysis of non-Hermitian systems. Instead, the Jordan decomposition should be used. While in the strict sense the Jordan decomposition is also not stable to compute numerically, if the system hosts a few number of spectrally isolated states, then it makes sense to separate this invariant subspace and complete the ``partial Jordan decomposition". A numerical recipe on how to achieve this is presented, based on the stable and well-studied Schur decomposition. An example of a non-Hermitian quadrupole insulator is then given to show how this ``partial Jordan decomposition" is helpful to our understanding of the response of the system to drives with frequencies close to the eigenfrequencies of those spectrally isolated states. The third paper originates from the apparent discrepancy between Floquet crystals and the so-called ``photonic time crystals", both are systems periodically varying in time, but the former having bandgaps in frequency while the latter having bandgaps in wavevector. This discrepancy is resolved by realising that if photonic systems are described by first order (in time) differential equations instead of second order ones, then varying permittivity alone is actually a non-Hermitian perturbation, even though the permittivity is always real. The paper goes on to discuss some criteria of possible bandgaps opening in frequency or wavevector or both directions, in either solid state systems or photonic systems. Multiple examples are given and finally the difference between a gap in frequency and one in wavevector, as well as an interface in time and one in space is discussed.