Non-Paraxial Wave Analysis of 3D Airy Beams

TL;DR

This paper develops a non-paraxial wave theory for 3D Airy beams, providing accurate local solutions beyond the paraxial approximation and describing the diffraction catastrophe near the beam's main lobe.

Contribution

It introduces a spectral integral approach and hyperbolic umilic canonical integrals to analyze 3D Airy beams beyond the paraxial regime, including finite-energy modifications.

Findings

Exact spectral integral solutions valid far beyond paraxial zone

Identification of hyperbolic umilic diffraction catastrophe near the main lobe

Extension of theory to ultra wide band (UWB) regime

Abstract

The 3D Airy beam (AiB) is thoroughly explored from a wave-theory point of view. We utilize the exact spectral integral for the AiB to derive local ray-based solutions that do not suffer from the limitations of the conventional parabolic equation (PE) solution, and are valid far beyond the paraxial zone and for longer ranges. The ray topology near the main lobe of the AiB delineates a hyperbolic umilic diffraction catastrophe, consisting of a cusped double-layered caustic, but this caustic is deformed in the far range where the field loses its beam shape. The field in the vicinity of this caustic is described uniformly by a hyperbolic umilic canonical integral which is structured explicitly on the local geometry of the caustic as obtained from the initial field distribution. In order to accommodate the finite-energy AiB we also modify the canonical integral by adding a complex loss parameter. The canonical integral is calculated using a series expansion and the results are used to identify the validity zone of the conventional PE solution. The analysis is performed within the framework of the non-dispersive AiB where the aperture field is scaled with frequency such that the ray skeleton is frequency-independent. This scaling enables an extension of the theory to the ultra wide band (UWB) regime and ensures that the pulsed field propagates along the curved beam trajectory without dispersion, as will be demonstrated in a subsequent publication.