Continuous coherent perfect absorption and lasing at an exceptional point of anti-parity-time symmetric photonic structures

We consider a type of hypothetical compound materials in which its refractive index in spatial distribution meet $n(-x)=-n^{*}(x)$, belonging to anti-parity-time (APT) symmetric structures. Additionally, we demand balanced real positive- and negative- permeabilities with $\mu(-x)=-\mu(x)$. By introducing parametrization into APT symmetric transfer matrix, together with reciprocity theorem, we propose a generic parametric space to display its associated scattering results including symmetry phase, exceptional point, and symmetry broken phase. The outcome is irrespective of any system complexity, geometries, materials, and operating frequency. With the parametric space, we find that APT symmetric system not only enables coherent perfect absorption or lasing occurred at an exceptional point, but also realize a simultaneous coherent perfect absorption-lasing. Since APT-symmetric system is constructed by balanced positive and negative index materials, the phase accumulated from optical path length is null, resulting in an assignment of mode order lost. To verify our analysis, several designed heterostructures are demonstrated to support our findings.