A Hybrid Zernike-Lyapunov Framework for Aberration-Based Statistical Wavefront Reconstruction of Chaotic Optical Surfaces

TL;DR

This paper introduces a unified theoretical framework combining chaotic wavefront dynamics with classical aberration theory, enabling systematic design and analysis of optical surfaces with chaotic geometries for advanced wavefront control.

Contribution

It develops a hybrid Zernike-Lyapunov expansion framework that links chaos parameters to optical performance, bridging wave chaos and aberration theory.

Findings

Establishes mathematical equivalences between Lyapunov exponents and aberration coefficients.

Derives analytical relationships between surface chaos and optical metrics.

Demonstrates the framework's validity through phase space analysis.

Abstract

We present a comprehensive theoretical framework that unifies chaotic wavefront dynamics with classical aberration theory through a Statistical Wavefront Reconstruction Framework (SWRF) formalism. By establishing rigorous connections between ray trajectory deflections and wave-optical phase perturbations through the eikonal equation, we decompose chaotic wavefront perturbations into modified Zernike-Lyapunov hybrid expansions, establishing mathematical equivalences between Lyapunov exponents, fractal dimensions, and traditional aberration coefficients. This chaotic aberration theory enables systematic incorporation of non-integrable wavefront dynamics into deterministic design frameworks, providing a rigorous foundation for controlled chaos in optical systems. We derive analytical relationships connecting surface chaos parameters to optical performance metrics, demonstrate the framework's validity through phase space analysis, and establish convergence criteria for the chaotic expansion. The theory reveals how chaotic surface geometries can be intentionally designed to achieve specific optical functionalities, including beam homogenization, speckle reduction, and novel wavefront shaping capabilities, while maintaining mathematical rigor comparable to classical aberration analysis.