Paraxial wave equation of light bundles: Gaussian beams and caustic avoidance

Abstract

Infinitesimal light bundles on curved spacetimes can be studied via a Hamiltonian formalism, similar to the Newtonian paraxial rays. In this work, we assign a classical wave function to a thin null bundle and study its evolution equation. This is achieved via the usage of the Schr\"odinger operators within a procedure analogous to the one in the semi-classical regime of quantum mechanics. The correspondence between the metaplectic operators and the symplectic phase space transformations of the geodesic deviation variables is at the core of our method. It allows for the introduction of unitary operators. We provide two solutions of the null bundle wave function which differ by their origin: (i) a point source, and (ii) a finite source. It is shown that while the former wave function includes the same information as the standard thin null bundle framework, the latter is a Gaussian beam. The Gaussianity of the intensity profile of our beam depends on the spacetime curvature and not on the random processes. We show that this beam avoids the caustics of an instantaneous wavefront. Our results are applicable for any spacetime and they can be used to model light propagation from coherent sources while averting the mathematical singularities of the standard thin null bundle formalism. This is especially relevant when estimating cosmological distances in a realistic inhomogeneous universe.