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.