Objective  Visible and ultraviolet laser output can be realized by solid-state lasers using second-harmonic generation (SHG) technology, which have a wide range of applications in many fields, including atmospheric exploration, biomedicine and industrial processing. The spatial walk-off effect that occurs inside the crystal in the SHG process can cause the spatial phase mismatch between the fundamental laser and the frequency-doubling laser, which lead to obvious loss of SHG efficiency restricting the SHG conversion efficiency. In order to further optimize the frequency doubling efficiency, the SHG theoretical model combined with influencing factors including the spatial walk-off effect needs to be improved. Therefore, combined with the spatial walk-off effect and SHG laser analysis, a hybrid model is proposed to study the parameters in the conversion process of the internal walk-off effect of nonlinear crystals. In this model, the influence of various parameters such as SHG crystal length, laser power density, SHG crystal type, laser wavelength on the SHG efficiency are carefully analyzed, which is of great significance for the optimization of SHG structure and the improvement of conversion efficiency.

  Methods  Based on the ideal SHG conversion efficiency model, this study discussed the energy conversion and beam separation in the process of space walk-off effect, and establishes the SHG conversion efficiency model of two different pump structures, including focused beam and parallel transmission beam (Fig.1-2). The specific effects of various parameter changes of SHG crystals and fundamental laser have been discussed.

  Results and Discussions   According to the established frequency-doubling efficiency model, the influence of the departure effect on the frequency-doubling efficiency has been discussed (Fig.3). The frequency-doubling efficiency was analyzed by using the fundamental frequency optical parameters of fundamental frequency light, including beam quality, beam radius, focal length of focusing lens, beam waist size and divergence angle (Fig.4-5). The frequency-doubling efficiency was analyzed by taking the crystal parameters of frequency-doubling crystal type and crystal cutting angle as variables, and it was found that the theoretical optimal frequency-doubling efficiency of KDP, LBO and KTP crystals was 58%, 80% and 97% (Fig.6). In addition, the influence of frequency-doubling crystal length on frequency-doubling efficiency is studied, and the optimal length selection model of frequency-doubling crystal is obtained (Fig.7).

  Conclusions  A SHG efficiency model of nonlinear crystals with the walk-off effect is established. Aiming at the two different situations of parallel transmission pump beam and focused pump beam, the nonlinear transformation efficiency model of the SHG process is improved through the simulation analysis of the spatial walk-off effect, and the influence of various pump parameters and crystal parameters changes on the SHG efficiency has been discussed. The theoretical optimal SHG efficiency of KDP, LBO and KTP crystals is 58%, 80% and 97% due to the influence of different spatial walk-off distances. Compared with the traditional SHG efficiency model, this model has a more complete study of the spatial walk-off process, and comprehensively refines the analysis of various factors in the SHG process, which can more accurately predict the change of SHG efficiency. It would be helpful to achieve SHG efficiency optimization by adjusting pump parameters and crystal parameters in the future. The model can also be used to realize the selection of the optimal length of SHG crystal and the best SHG crystal type in different conditions, which can be applied to the nonlinear crystal selection process of scientific research and commercial short-wave solid-state lasers to achieve conversion efficiency improvement and cost reduction.