The development of ultrafast femtosecond laser sources in different spectral regions has aroused great interest, which can be applied to fields such as spectroscopy and material processing. However, due to the limitations of solid-state and fiber gain media fluorescence bandwidth, the wavelength coverage range of mode-locked ultrafast lasers is usually restricted to a limited area in the near-infrared and mid-infrared regions. Second harmonic generation (SHG) provides a very effective method for circumventing this limitation and is commonly adopted to generate shorter wavelength light fields. In particular, single-pass SHG provides the simplest device that can be easily implemented in the most compact and cost-effective system. However, the spatial walk-off effect caused by the birefringence of crystals is often an important factor limiting the conversion efficiency. Additionally, when pulsed light is utilized for SHG, the group velocity mismatch will give rise to temporal walk-off effect in the time domain, which will also limit the conversion efficiency improvement. Therefore, it is necessary to study how to achieve optimal conversion efficiency in the presence of spatio-temporal walk-off effects. Generally, conversion efficiency can be enhanced by increasing the crystal length, improving the peak power of the fundamental pulse, and adjusting the focusing parameter. Our study mainly concentrates on optimizing the focusing parameter. The focusing parameter is a crucial factor that is relatively easy to manipulate in experiments. It directly influences the intensity distribution of the beam and the interaction within the crystal, thus playing a significant role in improving the conversion efficiency. We investigate the influence of the spatio-temporal walk-off effect on the generation of 405 nm pulsed light during the type-I single pass SHG, in which an 810 nm femtosecond pulse pumps a BIBO crystal. By conducting theoretical analysis, the optimal focusing conditions for efficient conversion efficiency under different spatio-temporal walk-off effects are provided. We offer important theoretical guidance for optimizing the preparation of second harmonic light fields under different spatio-temporal walk-off effects.

To investigate the influence of spatio-temporal walk-off effects on conversion efficiency, we define the temporal walk-off parameter A and the spatial walk-off parameter B [Eqs. (1) and (2)]. We assume that the fundamental field is Gaussian both spatially and temporally, and ignore the effects of crystal absorption and group velocity dispersion. The larger values of A and B lead to more severe walk-off effects. Subsequently, we utilize the built theoretical model hmA,B,ξ [Eq. (3)] to characterize the conversion efficiency. From this function, it is intuitively evident that the spatio-temporal walk-off parameters A and B, as well as the focusing parameter, influence the conversion efficiency. Therefore, for different spatio-temporal walk-off parameters, we can determine how the focusing parameter affects the conversion efficiency. In the experiment, to study the conversion efficiency under different focusing parameters, we employ four lenses with different focal lengths (250, 100, 50, and 30 mm) to change the waist radius of the fundamental light. In each case, the conversion efficiency under different fundamental light power is measured and analyzed.

As depicted in Fig. 3, the variation of hmA,B,ξ with ξ shows distinct trends for different A and B values. However, there is a peak in general, which indicates that the conversion efficiency will reach its optimal under a certain focusing parameter ξm. Under the fixed value of A, as the value of B increases, the curve moves downward as a whole. The efficiency decreases and then ξm decreases, which implies that a weak focus is required to enhance the efficiency. When B is fixed, as the value of A increases, the curve also moves downward overall. At this time the efficiency decreases, but ξm increases, which means that strong focus is needed to improve the efficiency. Therefore, for the existence of spatio-temporal walk-off effects, the optimal focusing parameter can be obtained by theoretical calculation. In the experiments, we analyze the results under different focusing parameters (Fig. 5). Under the focusing parameter of 1.21, the maximum conversion efficiency of approximately 51% is obtained, and the experimental data is consistent with the theoretical simulation results, which validates the correctness of the theory. Additionally, we analyze the influence of spatial walk-off effect on the spatial profile of second harmonic light (Table 2). In weak focusing conditions, the ellipticity of second harmonic light is 94.9%, indicating that it is less affected by the spatial walk-off. In strong focusing conditions, the ellipticity decreases to 80.7% under the influence of spatial walk-off.

In the process of critical phase matching frequency doubling pumped by pulsed light, there exists a spatial walk-off effect caused by birefringence and a temporal walk-off effect caused by the group velocity mismatch. We conduct a study on how to improve the conversion efficiency by optimizing the focusing parameter in the presence of spatio-temporal walk-off effects both theoretically and experimentally. The results demonstrate that the optimal focusing parameter gradually increases with the growing temporal walk-off. As the spatial walk-off rises, the optimal focusing parameter gradually decreases. In the experiment, 405 nm pulsed light is generated by a single pass SHG of pulsed light with a duration of 140 fs and a central wavelength of 810 nm. By optimizing the focusing parameter, the conversion efficiency of 51% is achieved. Although the conversion efficiency in our study is not the highest among relevant studies, mainly due to the differences in the types and lengths of the employed crystals, our systematic analysis of the optimal focusing parameter under different spatio-temporal walk-off effects provides valuable guidance for similar studies.