Results and Discussions Several important results are obtained as follows.1) Within the pre-trained range, the trained BP neural network is applied to predict the corresponding DOE phase coefficients. The randomly selected 20 sets of DOE system coefficients are taken as the test samples. The prediction accuracies of the DOE phase coefficients are all above 99.9% (Fig. 6). The uniformity of the laser irradiance generated by the 4 sets of DOEs which are designed by machine learning and energy mapping method respectively is almost the same (Fig. 7 and Fig. 8).2) The influences of system parameters on the prediction accuracy are studied while the system parameters are beyond the training range. Also, the influence of each parameter on the prediction accuracy is analyzed. When all the parameters are expanded by 80% and 55% based on the pre-trained range, the accuracy of prediction is above 99.5% and 97.5%, respectively (Fig. 10). It is shown that the size of the target plane has the most obvious influence on the prediction accuracy when the size of the target plane is smaller than the predetermined size (Fig. 11).

Diffractive optical elements (DOEs) are widely applied in light distribution control such as laser beam shaping, structured light illumination, and beam splitter. Various methods can be utilized to design DOEs, such as Gerchberg-Saxton (GS) algorithm, simulated annealing algorithm (SAA), and Genetic algorithm (GA). These traditional methods can design DOE effectively for a group of initial parameters, such as beam waist radius, wavelength, size of target plane, and distance between DOE and target plane. However, when any parameter is changed, the new phase profile needs to be recalculated, which is time-consuming, especially by global optimization algorithms such as SAA and GA. To overcome the disadvantages, this paper employs a machine learning algorithm to design the DOEs with continuous phase distribution. The mapping relationship of system parameters such as waist radius, size of target plane, and distance between DOE and target plane with DOE phase coefficients is constructed by the neural network. With this relationship, the DOE phase coefficients can be predicted automatically when a set of system parameters are given. It overcomes the limitation of the traditional design methods which need to recalculate the phase distribution when the parameters are changed.

Machine learning algorithm is employed to design the DOE with continuous phase distribution, which can be used for laser shaping. Firstly, the gird energy mapping method is applied to calculate the phase distribution data of the DOEs with a set of initial parameters including waist radius, size of target plane, and distance between DOE and target plane. The DOE enables the laser to generate uniform laser irradiance distribution. Secondly, the phase distribution data of DOE elements are fitted into a polynomial. Then 10000 sets of initial parameters are generated. With the 10000 sets of initial parameters, the phase distribution data of 10000 sets of DOEs are calculated by grid energy mapping. The initial parameters of the DOE and DOE phase coefficients are taken as input and output data to train the neural network, respectively. The trained network constructs a mapping relationship between system parameters and phase coefficients. With this relationship, the DOE phase coefficients can be predicted automatically when a set of system parameters are given.

The machine learning algorithm is employed to design DOEs with continuous phase distribution. The parameters of the DOE and DOE phase coefficients are acquired automatically as input and output data respectively to train the neural network. The trained network constructs a mapping relationship between system parameters and phase coefficients. With the relationship, the DOE phase coefficients can be predicted automatically when a set of system parameters are given. The results show that the prediction accuracy of the phase coefficient is above 99.9% within the trained range of the system parameters. When all parameters are expanded by 80% and 55% in both forward and reverse directions based on the pre-trained range, the prediction accuracy remains above 99.5% and 97.5%, respectively. It is also shown that the size of the target plane has the most obvious influence on the prediction accuracy when the size of the target plane is smaller than the predetermined size. In future work, the method may be extended to design the DOE with discontinuous phase distribution.