A Complex-Frequency Framework for Kerker Unidirectionality in Photonic Resonators

Electromagnetic scattering exhibits various anomalous behaviors, such as the forward or backward scattering effects known as the Kerker conditions. These anomalies appear in non-Hermitian photonic resonators and are mathematically described by operators with complex eigenfrequencies. Here, an effective Hamiltonian approach is proposed to analyze the complex eigenvalues of scattering systems. By imposing zero scattering constraints (i.e., transmission or reflection zeros) on the Hamiltonian of a two-mode photonic system, the complex eigenvalues corresponding to transmissionless and reflectionless modes are calculated and it is revealed that the Kerker anomalies are a result of spontaneous parity-time ($\mathcal{PT}$)-symmetry-breaking in the complex plane. This formalism provides a generic method to study non-Hermitian scattering and can yield novel physical insights into coupled-mode resonating systems. Furthermore, it provides universal design rules that yield photonic devices with unidirectional scattering and full $2\pi$ phase modulation.