Nonparaxial wave analysis of three-dimensional Airy beams

Area of Science:

• Optics and Photonics
• Wave Physics
• Nonlinear Optics

Background:

• The conventional parabolic equation (PE) solution for optical beams has limitations in accuracy, especially beyond the paraxial zone and for longer propagation ranges.
• Three-dimensional Airy beams (AiBs) exhibit unique propagation characteristics that require advanced theoretical frameworks for comprehensive understanding.

Purpose of the Study:

• To develop and analyze novel local ray-based solutions for three-dimensional Airy beams (AiBs) derived from the exact spectral integral.
• To overcome the limitations of the conventional parabolic equation (PE) solution for AiBs, extending validity to non-paraxial regimes and longer distances.
• To characterize the complex caustic structures and field behavior near these features using canonical integrals.

Main Methods:

• Utilizing the exact spectral integral of the AiB to derive local ray-based solutions.
• Analyzing the ray topology and identifying hyperbolic umbilic catastrophes and cusped double-layered caustics.
• Employing hyperbolic umbilic canonical integrals to uniformly describe the field in the vicinity of caustics.
• Modifying canonical integrals with a complex loss parameter for finite-energy AiBs and calculating using series expansion.
• Analyzing nondispersive AiBs with frequency-scaled aperture fields for frequency-independent ray skeletons.

Main Results:

• Derived local ray-based solutions valid far beyond the paraxial zone and for longer ranges, surpassing conventional PE limitations.
• Identified hyperbolic umbilic catastrophes and cusped double-layered caustics near the main lobe of the AiB.
• Demonstrated that the field in the far zone deforms, losing its beam shape, and is uniformly described by a modified hyperbolic umbilic canonical integral.
• Determined the validity zone of the conventional PE solution through series expansion calculations.
• Established a framework for nondispersive AiBs enabling extension to the ultrawideband regime without dispersion.

Conclusions:

• The developed ray-based solutions provide a more accurate and comprehensive description of AiB propagation compared to traditional PE methods.
• The study elucidates the complex wave structures, including caustics, associated with AiBs, offering insights into their behavior in various regimes.
• The theoretical framework supports the analysis of finite-energy AiBs and opens possibilities for dispersion-free propagation in ultrawideband systems.