M2 factor of four-petal Gaussian beam

Based on the second-order moments, this paper derives an analytical expression of the $M^{2}$ factor of four-petal Gaussian beam. The results show that the $M^{2}$ factor is only determined by the beam order $n$. The corresponding numerical calculations are also given. As the beam order increases, the augment of $M^{2}$ factor is disciplinary. As the expression of $M^{2}$ factor is expressed in series form and becomes more complicated, a new concise formula of $M^{2}$ factor is also presented by using curve fitting of numerical calculations. When $3 \le n \le 200$, the maximum error rate of fitting formula will not exceed 2.6{\%} and the average error rate is 0.28{\%}. This research is helpful to the applications of four-petal Gaussian beam.