Dressing the cusp: how paraxial sharp-edge diffraction theory solves a basic issue in catastrophe optics

Area of Science:

• Classical Optics
• Theoretical Optics
• Mathematical Physics

Background:

• Catastrophe optics offers a novel framework for describing light diffraction.
• Practical application has been hindered by challenges in describing optical singularities (caustics).
• Previous solutions were limited to the simplest caustic, the fold.

Purpose of the Study:

• To solve the fundamental problem of dressing the cusp caustic in catastrophe optics.
• To extend the description of diffraction patterns beyond the fold singularity.
• To demonstrate a practical and effective method for implementing this solution.

Main Methods:

• Utilized an ancient mathematical theorem to address the cusp singularity.
• Employed the paraxial version of the boundary diffraction wave theory.
• Integrated catastrophe optics with boundary diffraction wave theory.

Main Results:

• Achieved a complete solution for the dressing problem of the cusp caustic.
• Demonstrated the effectiveness and ease of implementation of the developed algorithm.
• Provided a significant example showcasing the algorithm's capabilities.

Conclusions:

• The study successfully solves a long-standing theoretical challenge in catastrophe optics.
• This advancement facilitates a more comprehensive understanding of light diffraction phenomena.
• The presented method offers a practical tool for optical research and applications.