# Dynamics of the optical swallowtail catastrophe

## TL;DR

This paper explores how optical caustics, especially swallowtail and butterfly patterns, evolve dynamically, revealing universal behaviors and transformations through analytical and experimental methods.

## Contribution

It provides a universal analytical expression for the propagation of optical swallowtail caustics and demonstrates their dynamic transformation to higher-order catastrophes.

## Key findings

- Swallowtail beams can transform into butterfly caustics.
- Certain swallowtail beams decay into cusp catastrophes.
- Analytical and experimental validation of caustic dynamics.

## Abstract

Perturbing the external control parameters of nonlinear systems leads to dramatic changes of its bifurcations. A branch of singular theory, the catastrophe theory, analyses the generating function that depends on state and control parameters. It predicts the formation of bifurcations as geometrically stable structures and categorizes them hierarchically. We evaluate the catastrophe diffraction integral with respect to two-dimensional cross-sections through the control parameter space and thus transfer these bifurcations to optics, where they manifest as caustics in transverse light fields. For all optical catastrophes that depend on a single state parameter, we analytically derive a universal expression for the propagation of all corresponding caustic beams. We reveal that the dynamics of the resulting caustics can be expressed by higher-order optical catastrophes. We show analytically and experimentally that particular swallowtail beams dynamically transform to higher-order butterfly caustics, whereas other swallowtail beams decay to lower-order cusp catastrophes.