Unified geometrical and paraxial approach toHermite-Gaussian and Laguerre-Gaussian beams inquadratic and free-space media via Niederer’s pointtransformation

We present an elementary derivation of Niederer’s coordinate point transformation within the framework of geometrical optics, establishing a direct connection between scalar paraxial optics and the Bohmian hydrodynamic formulation for free space and quadratic optical media. Unlike conventional approaches that rely on the sophisticated mathematical machinery of Lie groups and Lie algebras, our derivation stems naturally from fundamental physical and geometrical principles of light rays. Using this transformation, we determine explicit analytical relations between the hydrodynamic properties (trajectories, linear momentum, and angular momentum) of beams in both environments, demonstrating that their quantum trajectories and geometrical rays share the exact same functional mapping form. This mapping framework is successfully applied to structured fields, including Hermite– Gaussian and Laguerre–Gaussian beams and their superpositions. We prove that the coincidence between the zeros of the quantum potential and the classical geometric caustics holds rigorously in quadratic media. Furthermore, the behavior of quantum trajectories near beam vortices lays the groundwork for generalizing Berry’s one- dimensional Madelung trajectory results to the two-dimensional cases of both free particles and harmonic oscillators. Finally, we point out how these mapping relations pave the way for investigating point transformations in uniform magnetic fields and opening new avenues for the hydrodynamic inverse-design of structured paraxial beams in non-uniform media.