Unitary transformation approach to the paraxial wave equation

TL;DR

This work develops an operator-based framework for paraxial beam propagation by introducing propagation-dependent unitary transformations that map the paraxial equation to two decoupled harmonic oscillators with a time-dependent scaling (). The scaling function () obeys the Ermakov equation, yielding an effective GRIN-like medium where the Gaussian envelope induces harmonic confinement and non-commutativity with free-space dynamics. The methodology produces explicit, exact solutions in rectangular, circular-cylindrical, and elliptic-cylindrical coordinates, including Hermite-Gauss, Laguerre-Gauss, and Ince-Gauss modes, and reveals the instantaneous Lewis-Ermakov invariant governing propagation. A key result is that any finite-energy beam behaves as if subject to a quadratic potential, clarifying the connection between structured light and oscillator invariants, with broader relevance to quantum systems with time-dependent quadratic Hamiltonians. The framework provides a versatile, exact description of beam evolution and supports potential adiabatic-control extensions for mode suppression and beam shaping.

Abstract

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. In this context, the dynamical invariant and the free-space Hamiltonian do not commute at the initial propagation stage due to Gaussian modulation, which imposes an effective quadratic confinement. Exact commutativity would only be possible for an infinitely wide, nonsquare-integrable optical field; therefore, any finite-energy beam propagates as if it were subject to a quadratic GRIN-like potential. The unitary transformation approach reveals how the Gaussian envelope of physical beams leads to effective harmonic confinement and connects the propagation dynamics to oscillator-like invariants. This method enables the derivation of stationary solutions in different coordinate systems by mapping to an effective quadratic-like medium and establishes a direct link to the zero-frequency Ermakov equation and the Lewis-Ermakov invariants.

Figures (1)

• Figure 1: Transverse intensity distributions of paraxial beams at different propagation distances $z$. Subfigures (a$_1$–a$_3$) correspond to Hermite–Gauss beams with indices $(n,m)=(5,5)$ at $z = 0.0$ m, $0.5$ m, and $1.0$ m, respectively. Subfigures (b$_1$–b$_3$) show Laguerre–Gauss beams with $(n,m)=(3,2)$ at the same propagation distances, while (c$_1$–c$_3$) display the corresponding patterns for Ince–Gauss beams with $(n,m)=(7,5)$. Each case highlights the characteristic transverse profiles of the beam families, which are preserved during propagation up to a scaling transformation. The simulations are performed using red light with wavelength $\lambda = 633$ nm (He–Ne laser), an initial waist $W_0 \approx 200~\mu$m ($b_0=2/W_0^2$), and a Rayleigh range $z_R \approx 0.20$ m. Each case highlights the characteristic transverse profiles of the beam families, which are preserved during propagation up to a scaling transformation.