Abstract : Topological photonics is an emerging area in material sciences that explores and utilizes the topological invariants of photonic materials, which take integer values and are robust against system disorder. The area has attracted significant interest in recent years due to the ability of topological photonic materials to transport photon energy in a very robust manner, yet its mathematical development is in its infancy. This minisymposium aims to bring together mathematicians and scientists interested in this interdisciplinary area to share recent results and exchange ideas.
Michael Weinstein (Dept of Applied Physics & Applied Mathematics, and Dept. of Mathematics, Columbia University, New-York, United States)
Abstract : We analyse the propagation of transverse electric (TE) waves in a two dimensional honeycomb photonic medium.
This medium consists of a homogeneous bulk of fixed permittivity and an array of high permittivity dielectric inclusions centered at the vertices of a honeycomb lattice. Our mathematical results, supported by numerical simulations, give detailed local information about the conical crossings of dispersion surfaces (Dirac points) as well as global information about the high contrast behavior of dispersion surfaces.
[03902] Super band gaps and interface modes in one-dimensional quasicrystals
Format : Talk at Waseda University
Author(s) :
Bryn Davies (Imperial College London)
Lorenzo Morini (Cardiff University)
Richard Craster (Imperial College London)
Abstract : Quasicrystalline photonic crystals show significant potential (large spectral gaps and robustness) but are underutilised in applications due to the lack of efficient modelling techniques. In this work, we show that periodic (supercell) approximations give accurate predictions of the main spectral gaps of Fibonacci quasicrystals. This is based on characterising the growth of the underlying recursion relation and corroborates the existence of previously observed “super band gaps”. We also present a strategy for creating interface modes.
[03872] Mathematical theory for the interface mode in a waveguide bifurcated from a Dirac point
Format : Talk at Waseda University
Author(s) :
Jiayu QIU (Hong Kong University of Science and Technology)
Junshan LIN (Auburn University)
Peng XIE (Hong Kong University of Science and Technology)
Hai ZHANG (Hong Kong University of Science and Technology)
Abstract : In this talk, we present our new results on the existence of a bound state in a waveguide consisting of two semi-infinite periodic structures separated by an interface. The two periodic structures are perturbed from the same periodic medium with a Dirac point, and possess a common band gap enclosing the Dirac point. Using the layer potential technique and asymptotic analysis, we are able to overcome the difficulty imposed by the sharp interface.
[05154] Bloch Waves for Maxwell's Equations in High-Contrast Photonic Crystals
Format : Talk at Waseda University
Author(s) :
Robert Paul Viator (Swarthmore College)
Robert Lipton (Louisiana State University)
Silvia Jimenez Bolanos (Colgate University)
Abiti Adili (University of Massachusetts - Lowell)
Abstract : We investigate the Bloch spectrum of a 3-dimensional high-contrast photonic crystal. The Bloch eigen-
values, for fixed quasi-momentum, are expanded in a power series in the material contrast parameter in the high-
contrast limit, together with a convergence radius, obtained by decomposing an appropriate
vectorial Sobolev space into three mutually orthogonal curl-free subspaces. We also identify the limit spectrum in the periodic case. Time permitting, we will describe some geometries which admit this spectral structure.
[05207] Pseudo-magnetism and Landau Levels in strained 2-dimensional photonic crystals
Format : Online Talk on Zoom
Author(s) :
Michael I Weinstein (Columbia University)
Abstract : The principal use of photonic crystals is to engineer the photonic density of states, which controls light-matter
coupling. We show theoretically that a strained 2D honeycomb photonic crystal can generate artificial electromagnetic fields and highly degenerate Landau
levels, having high density of states. Since the tight-binding approximation is generally
not applicable to photonics, we employ a multiscale analysis
of the full continuum 2D Helmholtz wave equation and derive effective Dirac equations
with pseudo-magnetic and pseudo-electric potentials for the dynamics of wave-packets. The deformation
can be chosen to induce a constant pseudo-magnetic field, for which the effective Hamiltonian
has Landau level spectrum.
Our numerical simulations of the full continuum wave equations show mildly dispersive Landau levels,
which we show can be “flattened" by adjusting the pseudo-electric potential.
This theory is joint work with J. Guglielmon and M.C. Rechtsman.
The predictions have recently been corroborated in optical experiments, in joint work
with M. Barsukova, Z. Zhang, B. Zhen, L. He, F. Grisé, R. McEntaffer, S. Vaidya, J. Guglielmon and M.C. Rechstman
[05298] Quantized Fractional Thouless Pumping of Solitons
Format : Online Talk on Zoom
Author(s) :
Mikael Rechtsman (Penn State Univ)
Abstract : I will present my group’s recent work on the fractional pumping of solitons in photonic Thouless pumps. Specifically, I will show that the displacement (in unit cells) of solitons in Thouless pumps is strictly quantized to the Chern number of the band from which the soliton bifurcates in the low power regime; whereas in the intermediate power regime, nonlinear bifurcations lead to fractional quantization of soliton motion. This fractional quantization can be predicted from multi-band Wannier functions associated with the states of the pump.
[04812] Topological insulators in semiclassical regime
Format : Online Talk on Zoom
Author(s) :
Alexis Drouot (University of Washington)
Abstract : I will study a semiclassical Dirac equation that originates in the field of topological insulators. The semiclassical regime allows to make sense of propagating and counter-propagating companion states. We'll derive speed and profile for wavepackets corresponding to topological edge states, and show that the counter-propagating state disperses strongly -- an unusual phenomena in the analysis of coherent states.
[05311] Topological-cavity surface-emitting laser
Format : Online Talk on Zoom
Author(s) :
Ling Lu (Institute of Physics, CAS)
Abstract : Contrary to the perception that the Nobel-winning topological physics has not found useful applications, we show that the textbook design of daily-life semiconductor lasers are equivalent to standard topological models in 1D. By upgrading to the 2D vortex zero mode, we invent the topological-cavity surface-emitting lasers (TCSEL) whose performance far exceeds that of the commercial counterparts. Finally, we demonstrate the monopole cavity in 3D with the optimal single-mode behavior, completing the kink-vortex-monopole trilogy of topological defect modes.
