  We introduce deep learning technique to predict the beam propagation factor
M^2 of the laser beams emitting from few-mode fiber for the first time, to the
best of our knowledge. The deep convolutional neural network (CNN) is trained
with paired data of simulated near-field beam patterns and their calculated M^2
value, aiming at learning a fast and accurate mapping from the former to the
latter. The trained deep CNN can then be utilized to evaluate M^2 of the fiber
beams from single beam patterns. The results of simulated testing samples have
shown that our scheme can achieve an averaged prediction error smaller than 2%
even when up to 10 eigenmodes are involved in the fiber. The error becomes
slightly larger when heavy noises are added into the input beam patterns but
still smaller than 2.5%, which further proves the accuracy and robustness of
our method. Furthermore, the M^2 estimation takes only about 5 ms for a
prepared beam pattern with one forward pass, which can be adopted for real-time
M^2 determination with only one supporting Charge-Coupled Device (CCD). The
experimental results further prove the feasibility of our scheme. Moreover, the
method we proposed can be confidently extended to other kinds of beams provided
that adequate training samples are accessible. Deep learning paves the way to
superfast and accurate M^2 evaluation with very low experimental efforts.
