We present a framework for the paraxial wave equation based on propagation-dependent unitary transformations closely related to the Lewis-Ermakov invariant. This approach establishes a formal equivalence between free-space propagation and the dynamics in a quadratic gradient index (GRIN) medium. We show that the Gaussian modulation of physical beams induces an effective harmonic confinement, resulting in a non-removable non-commutativity between the dynamical invariant and the free-space Hamiltonian. This conceptual shift recasts free-space propagation as an intrinsic parametric oscillator problem, providing a rigorous and enabling bridge to map advanced quantum control protocols—such as Shortcuts to Adiabaticity—into spatial light engineering. Our formalism provides a unified description of mode families across various coordinate systems and enables the analytical study of complex phase dynamics and experimentally testable beam control protocols in free space.
Huerta-Sandoval et al. (Sun,) studied this question.