Dynamical systems in nonlinear optics: Maxwell-Bloch models

John M. Arnold

A spatially discretised Reduced Maxwell Bloch model is investigated, which models nonlinear optical propagation through a thin slab of resonant atomic material placed in a recirculating cavity. The model may be used to determine the stationary states of the cavity field, which may then be subjected to stability analysis. The analysis identifies the structure of the canonical integrable discrete model that approximates the electromagnetic model.

Smail B. Bougouffa, S. Al‐Awfi

Kolade Matthew Owolabi, Sonal Jain, Eben Maré

Classical Maxwell–Bloch models are often inadequate for describing electromagnetic wave propagation in complex optical media where memory effects, anomalous relaxation processes, and long-range electromagnetic interactions are important. To address these limitations, this work develops a fractional nonlocal Maxwell–Bloch framework that simultaneously incorporates Caputo fractional temporal memory and nonlocal electromagnetic coupling. The principal novelty lies in the unified treatment of temporal memory and spatial nonlocality within the Maxwell–Bloch framework, together with the corresponding analysis of their effects on optical-wave dynamics. The proposed model captures hereditary polarization dynamics and spatially distributed electromagnetic interactions that are not represented in the standard local integer-order formulation. The analytical part establishes the well-posedness structure of the governing system, including existence, boundedness, and stability properties of solutions in appropriate functional spaces. A spectral stability analysis is further employed to characterize fractional instability thresholds and transitions in the resulting wave propagation. Numerical simulations demonstrate the emergence of localized fractional optical structures, memory-dependent pulse propagation, nonlinear wave interactions, and complex dynamical regimes. A systematic parametric study further shows that variations in the fractional order significantly modify the electric-field profile, energy evolution, and pulse amplitude. The results demonstrate that the combined effects of fractional temporal memory and spatial nonlocality can substantially alter the propagation and stability of electromagnetic waves compared with the classical Maxwell–Bloch limit. The framework therefore provides a unified mathematical and computational basis for investigating memory-dependent and spatially nonlocal optical dynamics.

Y. R. Shen

Giovanni Giacomelli, Serhiy Yanchuk, Antonio Politi

The acclaimed Maxwell-Bloch (or Arecchi-Bonifacio) equations are a valid dynamical model, effectively describing wave propagation in nonlinear optical media: from the amplification in input-output devices to multimode instabilities arising in laser systems. However, the inherent spatial variability of the physical observables represents an obstacle to fast simulations and analysis, especially whenever networks of active elements have to be considered. In this paper, we propose an approach which, stripping the spatial dependence of its role as a generator of dynamical richness, allows for a compelling simple portrait. It leads to (a few) ordinary differential equations in input-output configurations, complemented by a time-delayed feedback in closed-loop setups. Such a scheme reproduces accurately the dynamics, paving the way to a plain treatment of the wealth of phenomena described by the Maxwell-Bloch equations.