\begin{document}${\boldsymbol{\beta}}$\end{document} factor of fundamental mode of fiber laser beam

Abstract

Owing to the advantages of high conversion efficiency, compactness and reliability, the fiber lasers are widely applied to many scientific areas, such as optical fiber communication, sensing and industrial processing. Beam quality is an important criterion for evaluating the performances of high-energy laser beam systems. Therefore, researchers have been constantly searching for the methods of evaluating the beam quality while pursuing higher output power. Until now, the researchers have proposed many definitions of beam quality. In practice, the evaluation parameters of beam quality include focused spot size, Strehl ratio, far-field divergence angle, diffraction limited <i>β</i> factor, energy circle rate, beam parameter product, and <i>M</i>² factor. Among them, the <i>M</i>² factor is the most suitable for the assessment of beam quality in both the near-field and far-field, which avoids the inaccuracy of the measurement of the beam quality only by the far-field radius or the far-field divergence angle. Thus, the <i>M</i>² factor is recognized as an important standard for evaluating beam quality by the International Organization for Standardization (ISO). However, it proves that the <i>M</i>² factor is not suitable for non-Gaussian distribution spot. On the other hand, in applications of high-energy laser beam transmission and laser industrial manufacturing, people pay more attention to the focusability of laser energy. In this case, the diffraction limited <i>β</i> factor is more suitable for evaluating beam quality. In this paper, we investigate the beam quality of LP₀₁ mode of fiber laser by <i>β</i> factor, and a circular and solid homogenous beam with the energy of 99% of LP₀₁ mode is considered as an ideal beam. The relationship between <i>β</i> factor and the parameters of LP₀₁ mode in a step-index fiber is studied theoretically. It is found that the value of the beam quality <i>β</i> factor is lower than 1 when the normalized frequency <i>V</i> is bigger than 1.8, and the far-field energy focusability of LP₀₁ mode is better than the case of ideal beam. Besides, the value of <i>β</i> factor decreases with the increase of normalized frequency <i>V</i>, core radius <i>a</i> or numerical aperture <i>NA</i>. In addition, the relationship between <i>M</i>² factor and <i>β</i> factor is non-linear.