PT-symmetric time delay oscillator modelling beyond the weak coupling limit via a scattering matrix formulation

Abstract

Parity-time (PT) symmetry in time-delay oscillators such as lasers and optoelectronic oscillators provides a potential route to enhanced spectral purity, including reduced phase noise and improved sidemode suppression. Existing theoretical descriptions are typically based on coupled-mode formulations derived under slowly varying envelope and near-degeneracy assumptions, which restrict their validity to weak coupling, small gain/loss contrast, and small detuning. In this work, a non-perturbative formulation of PT symmetric time-delay oscillators is developed based on a delay-difference equation and a scattering matrix representation of the coupling network. The approach treats propagation delay explicitly and does not rely on modal truncation, remaining valid for arbitrary coupling strength, gain/loss imbalance, and resonance detuning. The exact eigenvalue structure of the system is obtained in closed form, yielding a complete characterization of the unbroken and broken PT symmetric regimes as well as the associated exceptional points. A dimensionless order parameter is introduced that governs the symmetry transition over the full parameter space. It is further shown that conventional coupled-mode theory is recovered as an asymptotic limit of the exact formulation for small parameters. The results provide a unified and physically transparent framework for analysing PT symmetric delay systems beyond the weak-coupling limit, with direct implications for the design and optimisation of low-noise oscillators and photonic systems.