Anomalous Goos-Hänchen shift and group delay around scattering singularities in complex crystals

We demonstrate that scattering singularities can be achieved in parity-time (PT) symmetric complex crystals, even when the PT phase remains unbroken. In the two-dimensional space defined by the incident angle θ and the angular frequency ω of optical waves, the singularity is characterized by isolated reflection and transmission peaks. The phase around the singularity forms a vortex with a topological charge of ±1. The two observable spatiotemporal features of the phase gradient, namely the Goos-Ha¨nchen shift Δ and the group delay τ_g, exhibit anomalous behavior, with both potentially being negative near the singularity. The emergence of negative Δ and τ_g is correlated with the topological charge of the vortex. The negative values of Δ and τ_g can be attributed to an analog of backward-amplified pulse propagation in the complex crystals, which acts as an equivalent gain medium. Some other features of the singularities are also discussed, for example, the singularity originates from a boundary effect and can be engineered by altering the geometry of the non-Hermitian elements. This work highlights a potential framework in the study of novel spatiotemporal effects in non-Hermitian systems.