# Optical Vortex-Pumped KTiOAsO\({}_{4}\) Narrow-Linewidth Picosecond-Pulsed Parametric Oscillator
## 3 Results and Discussion  
### Modes of the Output Beam  
Generally, there are two methods for determining the order of the output optical vortex. One is that when an optical vortex beam of order \(\ell\) passes through a tilted lens, it will be transformed into a multi-lobed beam with \(\ell+1\) lobes [36]. The other is that when an optical vortex beam of order \(\ell\) passes through a Mach-Zehnder interferometer, it will be transformed into a pair of forked fringes with a \(\ell+1\) leg structure oriented in opposite directions [37]. Based on this, the beam output mode can be ascertained. As depicted in Figure 2, the spatial intensity profiles and the profiles of the output signal and idler fields are presented for different focus positions of the pump vortex beam when passing through a Mach-Zehnder interferometer or a tilted lens. Notably, \(d\) denotes the distance between  
Figure 1: (**a**) Experimental configuration of the optical vortex-pumped KTA picosecond-pulsed parametric oscillator. \(\lambda/2\), HWP; (**b,c**) show the spatial intensity profile of the pump vortex beam after the SPP conversion and when passing through a tilted lens. the pump beam focusing lens and M\({}_{2}\). By adjusting the position of the focusing lens, \(d\) was set to 43 cm, 45 cm, and 47 cm. Under these conditions, the focal point of the pump vortex beam was positioned at the front, middle, and rear of the KTA crystal, respectively. Figure 2a,b reveal that the intensity profile of the output signal field was Gaussian, while that of the idler field was annular, respectively, when the pump vortex beam was focused on the front face of the crystal (\(d\) = 43 mm, and the pump beam radius here was measured to be 146 \(\upmu\)m). As shown in Figure 2a1, when the signal field was characterized using the Mach-Zehnder interference technique, a series of fringes were generated. The straight-line fringes, a characteristic pattern for the plane phase front, signified that the resonant signal beam was output in a Gaussian mode. When the idler field was analyzed with the tilted lens method, two lobes were produced, as observed in Figure 2(b1), confirming that the order of the idler field was \(\ell_{i}\) = 1, which was the same as that of the pump vortex field. These findings suggest that the OAM of the pump field was transferred to the idler field. When the focus of the pump vortex was located at the center of the KTA crystal (\(d\) = 45 mm, and the pump beam radius here was measured to be 90 \(\upmu\)m), the generated signal and idler fields exhibited distinct spatial profiles. In this situation, the idler field possessed a Gaussian spatial intensity profile (Figure 2c), whereas the signal field had an annular spatial intensity profile (Figure 2d). The Mach-Zehnder and tilted lens characterizations were also applied to these outputs, and the resulting spatial intensity profiles were displayed in Figure 2(c1) and Figure 2(d1), respectively. It can be analyzed that the OAM of the pumping field has been transferred to the signal field. When the pump vortex beam was focused on the end face of the KTA crystal (\(d\) = 47 mm, and the pump beam radius here was measured to be 153 \(\upmu\)m), the spatial intensity profiles of the generated signal and idler fields were similar to those when the pump beam was focused on the front surface of the KTA crystal. The signal field featured a Gaussian spatial intensity profile (Figure 2e), while the idler field exhibited an annular spatial intensity profile (Figure 2f). The spatial intensity profiles of these beams when passing through the Mach-Zehnder and tilted lens setups are, respectively, presented in Figure 2(e1),(f1). In this case, the OAM of the pump field was transferred to the idler field.