Non-reciprocal wave systems

Time-modulated systems

A popular way to break recipricity is to consider material parameters that vary in time. The creation of such "dynamic" materials has led to the development of space-time metamaterials. These materials have exotic wave scattering, resonance and transmission properties. For example, when excited with an incident wave of a single frequency, the scattered field consists of a family of coupled harmonics at frequencies differing by the frequency of temporal modulation. Moreover, the lack of energy conservation can cause scattering coefficients to blow up at certain parameter values. We have developed an integral operator approach to characterise time-modulated scattering problems and, for high-contrast scatterers, have obtained small-volume asymptotic formulas analogous to the classical results for the static (unmodulated) case [7]. This gives a concise mathematical framework for understanding the exotic physics of these time-modulated systems.

Non-Hermitian skin effects

Systems with non-reciprocal coupling or damping terms can exhibit the so-called "skin-effect" whereby all the eigenmodes are localised at one edge of the system. This powerful effect has great potential for developing broadband wave control and guiding devices, removing the typical frequency selectivity of tranditional metamaterial solutions. Our work has developed the fundamental theory to understand non-Hermitian skin effects in differential models [3], including characterising the phenomenon in three-dimensional partial differential systems [5]. We have studied, in particular, the robustness of the skin effect in response to random defects. We have characterised the minimal magnitude of defects large enough to break the skin effect [4] and developed effective models that describe the transition from skin to bulk localisation that occurs at this critical defect size [6]. 

Parity-time symmetry and exceptional points

Introducing energy sources and damping to a wave system breaks time-reversal symmetry and can lead to exotic properties. One class of systems that have received substantial interest is those with parity-time symmetry, as these typically have resonances that are either real or in conjugate pairs. Our work extended these principles to three-dimensional PDE models. We proved the existence of asymptotic exceptional points of arbitrarily high order in high-contrast PDE models [1, 2].

These systems have applications in creating sensors with enhanced sensitivity. Near to an N-th order exceptional point, a local perturbation (due to e.g. an earthquake or the presence of a small particle such as a virus) eigenfrequencies will be perturbed in proportion to the 1/N-th power of the strength of the perturbation (as opposed to in direct proportion to the perturbation). This eigenfrequency shift can be measured and will be greatly enhanced for small perturbations, yielding enhanced sensitivity [1].