A Highly Inclusive Theory of Q-Switched Lasers

Abstract

A highly inclusive theory of Q-switched lasers is presented which applies to both actively and passively Q-switched lasers. The theory expands on the author’s previous publications on energy optimization to include: (1) saturable absorbers exhibiting excited state absorption and therefore a finite Figure of Merit (FOM); (2) unequal beam areas in the gain and absorbing media; and (3) both pulsed and CW pumping. Design curves for optimizing the output energy for pulse pumped lasers and CW pumped lasers at a desired pulse rate are provided. As in the author’s previous work, the optimization equation is obtained utilizing Lagrange Multipliers and yields the value of the parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\rho = ln(n_{i}/n_{f})$ </tex-math></inline-formula> which maximizes the single pulse energy, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n_{i}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n_{f}$ </tex-math></inline-formula> are the laser inversion population densities before and after Q-switching. Using this initial result, the associated optimum absorber transmission and mirror reflectivity, pulse energy, peak power, and pulse width can be computed sequentially via simple formulas. The aforementioned parameters can all be expressed as functions of two variables, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z_{eff}$ </tex-math></inline-formula> , which are slight modifications of the variables <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and z previously introduced by the author for passively Q-switched lasers having infinite FOM and equal beam areas The term “plane wave theory” has sometimes been used by this and other authors in prior publications to describe a uniform pump profile in the laser medium, but we will refer to this special case more accurately as the “uniform pump theory”. The theory is extended here in the final section to include non-uniform pumping. The common case of a Q-switched laser end-pumped by a quasi-Gaussian CW laser is considered in detail and the theoretical results are found to agree well with experimental data.