$M^2$ as a Quantitative Measure for Beam Quality

Beam quality is a fundamental aspect for evaluating the performance of laser sources. $M^2$-measurements serve as the gold standard for beam quality assessment since the 1990s. The measured $M^2$-parameter indicates similarity to the pure fundamental Gaussian mode, characterized by the ideal $M^2=1$, by describing a beams' divergence. $M^2$-values close to 1 are considered to correspond to nearly fundamental sources. However, in terms of the higher-order mode contribution of a laser, it acts as a qualitative measure that does not permit a quantitative statement. Here, we introduce a framework to assess the fundamental mode content of a laser beam using $M^2$-measurements and establish a direct link between beam quality and its mode composition. Our results significantly enhance the utility of $M^2$-measurements in evaluating laser sources, coupling efficiencies, focusing performance, and long-distance propagation. This repositions $M^2$ from a qualitative figure to a quantitative tool in modern photonics.