Gaussian beams and caustic avoidance in gravitational optics

TL;DR

This work develops a covariant Gaussian-beam formalism for gravitational optics by recasting light propagation as a wavefunction on a reduced, symplectic phase space of null bundles. It reveals that point-source beams recover standard thin-null-bundle dynamics, while finite-size sources yield Gaussian beams whose complex wavefront curvature couples amplitude and phase through a wave potential, enabling wave-like behavior without caustic singularities. The framework preserves power conservation and provides a practical, covariant method to compute cosmological distances in spacetimes with caustics, with analytic demonstration in Barriola–Vilenkin monopole spacetime. This approach offers a versatile, wave-optics–oriented tool for modeling coherent sources in curved backgrounds and suggests Gaussian-beam decompositions as a robust alternative to Fourier-based methods in gravitational lensing and cosmology.

Abstract

In this study, we consider a beam summation method adapted from the semiclassical regime of quantum mechanics to study the classical properties of thin light bundles in gravity. In Newtonian paraxial optics, this method has been shown to encapsulate the wave properties of the light beams. In our case, the wave function assigned to the light bundle can be viewed as a coarse-grained description that captures information about the dynamics of superposed bundles within the geometric optics regime. We investigate two solutions of the null bundle wave function that differ by their origin: (i) a point source and (ii) a finite source. It is shown that while the wave function in the point source case contains the same information as the standard thin null bundle framework, the finite source case corresponds to a Gaussian beam. The novel aspect of this work arises from our geometric construction of covariant Gaussian beams, which can be applied in any spacetime. Additionally, the effects of a finite source on cosmological distances are discussed. With this framework, one can model light propagation from coherent sources while avoiding the mathematical singularities of the standard thin null bundle formalism. We explicitly demonstrate the caustic-avoidance property of Gaussian beams in the analytically tractable example of a Barriola-Vilenkin monopole spacetime.

Figures (5)

• Figure 2: Screen-projected solutions of the geodesic deviation equation of a null bundle.
• Figure 3: A null bundle with central null geodesic $\Theta (v)$. The red curve represents $\Sigma(\lambda)$ given by Synge's spacelike world function, $\sigma (m,n)$. The outermost null geodesic $\zeta (v)$ can be uniquely obtained through $\Theta (v)$ and $\Sigma(\lambda)$.
• Figure 4: The sketch of Gaussian beams and the transverse screen.
• Figure 5: Apparent magnitude, m, versus observation direction, $\Theta$. For an observer located at $r_0=0.77$, we consider sources at $r=1$ distributed on the observer's sky for each $\Theta$ value. Affine distances to those fixed sources are obtained through the solution of the geodesic given in (\ref{['eq:Monopole_sol_geod_r']}). The strength of the monopole deficit solid angle is chosen as $\kappa=1/3.7$. The red curve represents the magnitude calculated for a vertex beam. The blue, green and black curves are those which correspond to Gaussian beams with initial widths $0.3, 0,2$ and $0.1$ respectively. The smaller the initial width is, the closer the Gaussian beam magnitude is to the one of a vertex bundle.
• Figure 6: Luminosity distance, $D_L$, versus observation direction, $\Theta$. The locations of the observer and the source in addition to the value of $\kappa$ are same as in the caption of Figure \ref{['fig:Monopole_mag_vs_angle_all']}. The red curve represents the estimated $D_L$ for a standard vertex bundle. Blue, green and black curves are those which correspond to the $D_L$ of Gaussian beams with initial widths $0.3, 0.2$ and $0.1$ respectively. The smaller the initial width is, the smaller the estimated luminosity distance becomes.