With the rapid development of laser technology, it has been widely applied in important fields such as medicine, biology, materials and national defense. The amplitude of a laser beam generally has a Gaussian distribution, and such an uneven energy limits its further application. Thus, beam shaping techniques have been proposed to transform Gaussian beams into flat top beams with a uniform energy distribution. Researchers have proposed various beam shaping methods, among which shaping using liquid crystal spatial light modulators has been widely investigated for its controllable transmittance function, good flexibility and real-time performance. Traditional phase distribution algorithms suffer from the problems of being easily trapped in local extrema, being sensitive to the initial value of the phase, and not being able to obtain high utilization of energy and high beam top uniformity at the same time. In this paper, the phase distribution function algorithm where beam is shaped using liquid crystal spatial light modulators is optimized by using the combination of lowliest place elimination (LPE), genetic algorithm (GA) and Gerchberg-Saxton (GS) algorithm. The hybrid method is called LPE-GSGA algorithm, which further improves the output beam top uniformity without sacrificing the utilization of energy, or even improving it. Meanwhile, it reduces the dependence of conventional algorithms on initial values to a certain extent and has important applications in flat top beam shaping with high utilization of energy and high beam top uniformity.

The LPE-GSGA algorithm designed in this paper uses the strong global search capability of the GA algorithm to help the GS algorithm to jump out of local extrema. Also, LPE is introduced to retain individuals with good phase points and accelerate convergence. Sum of squares for error e_ss and fitting coefficient η are used as evaluating indicators to describe the quality of output beams. The algorithm can be divided into two processes: the first is the iterations of all initial phase groups using GS algorithm, and the second is the calculation of the comprehensive evaluation index where some phase individuals with good indexes are selected to enter the next generation phase group directly and the remaining phase individuals experience selection, crossover (Fig. 1), mutation and LPE to enter the next generation phase population until the number of individuals in the phase population is 1. The flow chart of the process is shown in Fig. 2.

We calculate the output beam's information use LPE-GSGA algorithm through simulation, show its iterative process (Figs. 3 and 4) and further compare it with those of the GS, generalized adaptive additive (GAA), weighted Gerchberg-Saxton (GSW) and GSGA algorithms under the same input and evaluation metrics (Table 1). The e_ss and η calculated by LPE-GSGA algorithm are superior to those obtained with other algorithms. Compared with GS algorithm, the LPE-GSGA algorithm shows great advantages with 10.1% reduction in e_ss and 0.85% improvement in fitting coefficient η. From the point of initial value dependence, the variances of e_ss and η of 50 sets of results figured by LPE-GSGA algorithm are much lower than those of the other algorithms, with the variance of e_ss being about 74% lower than that of the GS algorithm, and a nearly one order of magnitude reduction of variance of η. The role of each process is also discussed: process 1 makes use of the fast convergence ability of the GS algorithm to obtain the local extrema quickly, and process 2 uses the screening of the LPE and the global search ability of the GA algorithm to help the GS algorithm obtain better iterative initial phase values, reduce its dependence on the initial values, and thus obtain better phase distributions.

The LPE-GSGA phase distribution algorithm based on the LPE, GS algorithm and genetic algorithm is proposed in this paper. Based on the algorithm, we get the quality of the output beam by simulation which is superior to those of the GS, GAA, GSW and GSGA algorithms, and solve the problem of initial values dependence. Additionally, the improved algorithm diminishes the number of intensity abrupt change points on the top of output beam, the number of sidelobes, and the sidelobe amplitude. In a word, we demonstrate the effectiveness of the LPE-GSGA algorithm in improving the quality of the output flat top beam and getting a flat top beam with high utilization of energy and high beam top uniformity.