# Optical Vortex-Pumped KTiOAsO\({}_{4}\) Narrow-Linewidth Picosecond-Pulsed Parametric Oscillator
## 3 Results and Discussion
### Spatial Overlap Efficiency  
Previous efforts have confirmed that due to the conservation of OAM within an OPO cavity, the transfer of OAM among the pump, signal, and idler fields can be regulated by modulating cavity losses or by modifying the spatial overlap of respective fields [40]. Generally, in ultrafast OPOs, a cavity design capable of synchronizing with a high-repetition-rate pump field is necessary. Z-shaped or ring-shaped cavities and a large number of mirrors with various curvature radii are typically employed. This may result in a resonant field with significant astigmatism and diffraction losses, causing the resonant field to oscillate in Gaussian mode, while the non-resonant field possesses the OAM characteristics of the pump field [41]. The closer the overlap of the pump field as any one mode of space approaches 1, the lower the loss, and the more readily the cavity oscillates in that mode. In this study, we adjusted the spatial overlap efficiency between the pump and parametric field to precisely control the transfer of the pump field's OAM. The formula for the spatial intensity overlap efficiency (\(\eta\)) between the pump and parametric field within the KTA crystal is presented below, as derived from reference [42]:  
\[\eta=\frac{\left|\iint E_{p}(r,\varphi)E_{s,i}^{*}(r,\varphi)rdrd\varphi\right| ^{2}}{\iint\left|E_{p}(r,\varphi)\right|^{2}rdrd\varphi*\iint\left|E_{s,i}(r, \varphi)\right|^{2}rdrd\varphi} \tag{1}\]  
where \(E_{p}\) represents the electric field of the pump field, \(E_{s}\) represents the electric field of the signal field, and \(E_{i}\) represents the electric field of the idler field. The pump, idler, and signal fields are assumed to lase with a vortex mode (\(\ell\) = 1) within the cavity, which can be expressed as  
\[E_{p,s,i}=\frac{1}{\omega_{0_{p,s,i}}}*\frac{\sqrt{2}r}{\omega_{p,s,i}}*exp \left(-\frac{r^{2}}{\omega_{p,s,i}^{2}}\right)\,exp(i\varphi) \tag{2}\]  
where \(\omega_{p,s,i}\) is the pump, signal, or idler beam radius, and \(\omega_{0_{p},s,i}\) is the fundamental mode size of the pump, signal, or idler field. In this theoretical model, the mode sizes of the signal (\(\omega_{s}\) = 92 \(\upmu\)m) and idler fields (\(\omega_{i}\) = 180 \(\upmu\)m), which were assumed to be fixed, at the center of the KTA crystal were determined based on the parameters of the OPO cavity mirrors using the ABCD resonator model. Subsequently, the spatial overlap efficiency between the pump field and the signal or idler fields as a function of the pump beam radius (\(\omega_{p}\)) was calculated and is presented in Figure 3. It can be observed that when the pump beam radius ranged from 80 to 120 \(\upmu\)m; the spatial overlap between the first-order vortex mode (LG\({}_{01}\)) of the signal field and the pump field was higher compared to other modes, thereby reducing cavity loss and enabling the signal beam to oscillate in the vortex mode. As the pump spot size gradually increased (\(\omega_{p}\) > 142 \(\upmu\)m), the spatial overlap between the first-order signal beam and the pump beam deteriorated, leading to increased diffraction loss within the cavity and forcing the signal beam to oscillate in the Gaussian (GS) mode. Due to the conservation of OAMand the improved spatial overlap between the LG\({}_{01}\) of the idler beam and the pump beam, the pump beam transferred the OAM to the idler field output. In the experiment, we utilized a focusing lens with a focal length of 100 mm to focus the pump beam of the vortex mode to a minimum spot size of 89.7 \(\upmu\)m. When \(d\) was set to 45 mm, the signal beam was generated in the vortex mode, while the idler beam was emitted in the Gaussian mode. For \(d\) = 43 mm or \(d\) = 47 mm, the pump beam radius at the crystal increased (\(\omega_{p}\) > 142 \(\upmu\)m), resulting in the idler beam acquiring OAM from the pump beam and the signal beam being emitted in the fundamental mode.