Propagation dynamics and modulation instability control in inhomogeneous nonlinear Schrödinger equation with two-photon absorption

This study offers detailed analysis of modulation instability (MI) dynamics in inhomogeneous nonlinear Schrödinger (NLS) media, including two-photon absorption (TPA) through analytical modeling and numerical simulations. In contrast to traditional MI studies conducted in uniform environments, our methodology systematically investigates how engineered spatial inhomogeneity through customized nonlinearity coefficients, group velocity dispersion (GVD) parameters, and distance-dependent profiles can be utilized to regulate nonlinear wave stability. Starting with a fixed TPA coefficient, we demonstrate that an increase in TPA not only mitigates instability but also causes significant pulse broadening, steering the system into a dissipative phase. Contour and surface analyses reveal a unique stabilization mechanism in the MI spectrum resulting from the interplay between non-linearity and TPA. When the TPA is established, fluctuations in GVD and nonlinearity coefficients yield a diverse array of structural behaviors, encompassing total sideband suppression, localized stability regions, and periodically recurring stable bands. By expanding this framework to inhomogeneous media, we illustrate that distance-dependent group velocity dispersion and nonlinearity profiles facilitate spatial localization and periodic modulation instability evolution, providing a customizable platform for wave manipulation. Numerical propagation studies indicate that elevated TPA values, in conjunction with graded GVD profiles, produce droplet-like spatiotemporal patterns, reflecting a sophisticated and manageable equilibrium between dissipation and dispersion. The results present novel opportunities for utilizing spatially designed optical medium to attain nonlinear light propagation, with potential applications in high-power pulse shaping, optical signal processing, and dissipative photonics.