# SU(2) symmetry of spatiotemporal Gaussian modes propagating in the isotropic dispersive media

## TL;DR

This paper reveals that the SU(2) symmetry of spatiotemporal Gaussian modes explains their observed patterns and dynamics in isotropic media, deriving analytical expressions and identifying regimes of dispersion affecting mode evolution.

## Contribution

It introduces the SU(2) symmetry framework for spatiotemporal Gaussian modes and derives analytical expressions for their propagation in isotropic media, linking Gouy phase to dispersion regimes.

## Key findings

- Multi-petal patterns explained by SU(2) symmetry.
- Analytical expressions for modes with arbitrary indices.
- Dispersion regimes cause distinct propagation behaviors.

## Abstract

The far-field intensity distribution of spatiotemporal Laguerre-Gaussian (STLG) modes propagating in free space exhibits a multi-petal pattern analogous to that observed in tilted Hermite-Gaussian modes. Here, we show that this phenomenon can be explained by the SU(2) symmetry of spatiotemporal Gaussian modes, which can support the irreducible representation of SU(2) group and enable the construction of the spatiotemporal model Poincar\'e sphere. We have also derived analytical expressions for the STLG mode with an arbitrary radial and angular indices propagating in the isotropic media. The propagation dynamics can be understood as a unitary transformation generated by a conserved quantity, where the rotation angle is exactly the intermodal Gouy phase of the spatiotemporal modes in the same order subspace. This spatiotemporal Gouy phase depends on the ellipticity of the wave packets and the group velocity dispersion (GVD) of the media. The phase, as a function of propagation distance, is categorized into three distinct regimes: normal dispersion, anomalous dispersion, and zero dispersion. Interestingly, in the regime of anomalous dispersion, the non-monotonic behavior induces to both distortion and revival of the intensity distribution, thereby establishing a phase-locked mechanism that is analogous to the Talbot effect.