Unified approach to paraxial propagation in uniform media and media with linear or quadratic refractive index distribution

Abstract The application of Lie algebraic methods to wave propagation offers a powerful tool for unraveling the underlying principles governing optical phenomena. We present explicit equations and matrix representations that allow simple calculations between three different media (free space, linear, and quadratic refractive index distributions) for paraxial light propagation. These equations provide a unified framework for understanding the dynamics of wavefronts in the (1 + 1) D and (1 + 2) D scenarios, shedding light on the interplay between the geometry of the medium and the evolution of optical fields. In particular, our analysis reveals that only potentials up to second degree polynomials, such as x and x 2 , can be effectively described within this framework. Our findings offer practical implications for the design and analysis of optical systems.