On the numerical evaluation of umbilic diffraction catastrophes

Area of Science:

• Optics and Photonics
• Computational Physics
• Mathematical Physics

Background:

• Diffraction catastrophes, including umbilics and cuspoids, classify structurally stable optical diffraction patterns.
• Evaluating these complex phenomena often requires sophisticated mathematical techniques.

Purpose of the Study:

• To introduce a simple computational approach for evaluating umbilic diffraction catastrophes.
• To extend the analysis of umbilic fields to complex control parameters.

Main Methods:

• Expanding double integral representations of hyperbolic and elliptic umbilics into convergent power series.
• Applying the Weniger transformation to these series expansions.
• Deriving exact expressions for the 'on-axis' umbilic field.

Main Results:

• The study presents a novel computational method for analyzing umbilic diffraction catastrophes.
• Exact 'on-axis' umbilic field expressions were derived, extending previous results.
• Numerical experiments demonstrated the approach's effectiveness and ease of use.

Conclusions:

• The proposed computational method provides a simplified and effective tool for studying umbilic diffraction catastrophes.
• This approach facilitates a deeper understanding of complex optical diffraction patterns.