Nonlinear absorption with Hermite-Gaussian beams

This study develops a theoretical model to describe nonlinear absorption in Z-scan experiments for arbitrary nonlinearity order n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} and incident Hermite-Gaussian beam profiles. A generalized expression is derived for the normalized optical transmittance as a function of beam mode indices and system parameters. Using the weak nonlinearity approximation, an analytical solution is obtained. Critically, the model demonstrates how transmittance decreases with increasing spatial complexity of the incident beam mode structure, offering new physical insights. The formulation is validated by verifying it correctly reproduces previous results for a fundamental Gaussian beam. The generalized transmittance expression provides a robust framework to accurately model and analyze Z-scan data captured with diverse laser sources and beam profiles. Through fitting experimental curves, key nonlinear material properties such as absorption coefficients can be precisely determined. Overall, the developed formalism establishes a rigorous yet widely applicable methodology to advance understanding of nonlinear beam propagation and optical characterization techniques.