Calculating the spectral response strength of non-Hermitian systems with an exceptional point directly from wave simulations

Exceptional points are spectral degeneracies in open systems, which have attracted considerable attention in recent years. One reason for the attention is that physical systems with an exceptional point exhibit a strong spectral response to perturbations. This response can be quantified by a single quantity known as the spectral response strength. Up to now, the theory for the spectral response strength is restricted to effective non-Hermitian Hamiltonians acting on a finite-dimensional Hilbert space, which often appear naturally within coupled-mode theory. We introduce here a scheme for the computation of the spectral response strength directly from numerical results of wave simulations, which correspond to an infinite Hilbert-space dimension. Our approach is based on the relation of the spectral response strength to the Petermann factors of eigenstates in a system near the exceptional point. To illustrate our theory, we consider three different photonic systems: a microring dimer, waveguide-coupled microrings, and a weakly deformed microdisk. A comparison to results provided by effective Hamiltonians based on coupled-mode theory and perturbation theory demonstrates very good agreement.