# Spatial-Temporal Self-Focusing of Partially Coherent Pulsed Beams in Dispersive Medium
## 3 Spatial-Temporal Self-Focusing of PCP Beams in Dispersive Medium
### Water Case  
Figures 1a-c and 2a-c give the evolution of normalized spatial-temporal intensity of PCP beams with different values of time \(t\) in the \(x\)-\(z\) plane, while the corresponding on-axis profile are given in Figures 1d and 2d, respectively. Figure 1 is the case of rect weight function case, which corresponds to the pulses with circular spatial and sinc temporal coherence distribution. Figure 2 is the case of  
Gaussian weight function, which corresponds to pulses with non-uniform coherence distribution. We choose water as the dispersive medium with refractive index \(n_{\text{g}}=c/v_{\text{g}}=1.3425\) (water, \(20\,^{\circ}\)C). As can be seen, there is an intensity maximum upon propagation, i.e., the spatial-temporal self-focusing takes place upon the water medium propagation. In addition, detection time \(t\) has an important effect on the intensity distribution of pulsed beams. With increasing time \(t\), the value of the peak intensity becomes small and the focal spot shifts slightly far from the source plane. Physically, the spatial-temporal intensities of the pulsed beams depend on the spatial positions and detection time \(t\), respectively. As can be seen from Equation (20), detection time \(t\) has a close relationship with pulse duration \(T(z)\). Furthermore, as shown in Equation (18), \(T(z)\) is dependent of \(z\). That is why the detection time affects intensity peak and position of peak intensity. Figure 1: Evolution of normalized spatial-temporal intensity \(I(x,\,t,\,z)\) of PCP beams with rect weight function with different values of time \(t\): (**a**) \(t=0\), (**b**) \(t=1\) ps and (**c**) \(t=2\) ps in the \(x\)-\(z\) plane. (**d**) On-axis normalized spatial-temporal intensity versus propagation distance \(z\). Figure 2: Evolution of normalized spatial-temporal intensity \(I(x,\,t,\,z)\) of PCP beams with Gaussian weight function with different values of time \(t\): (**a**) \(t=0\), (**b**) \(t=1\) ps and (**c**) \(t=2\) ps in the \(x\)-\(z\) plane. (**d**) On-axis normalized spatial-temporal intensity versus propagation distance \(z\). Figures 3a-c and 4a-c give the evolution of normalized spatial-temporal intensity of PCP beams with different values of \(x\) in the \(t\)-\(z\) plane, while the corresponding on-axis and off-axis profile are given in Figures 3d and 4d, respectively. Figure 3 is the case of rect weight function, while Figure 4 is the case of Gaussian weight function. As can be seen, the spatial-temporal self-focusing occurs not only at on-axis (\(x=0\)) but also the off-axis (\(x\
eq 0\)). And, with increasing \(x\), the focal spot shifts slightly towards the source plane. Physically, the spatial-temporal intensities of the pulsed beams depend on the spatial positions and detection time \(t\), respectively. When detection time \(t\) is fixed, the intensity distribution has a marked impact on position coordinates, which can be seen easily from Equations (17) and (19). Figure 4: Evolution of normalized spatial-temporal intensity \(I(x,t,z)\) of PCP beams with Gaussian weight function with different values of \(x\): (**a**) \(x=0\), (**b**) \(x=0.2\) cm and (**c**) \(x=0.4\) cm in the \(t\)-\(z\) plane. (**d**) Normalized spatial-temporal intensity versus propagation distance \(z\) with \(t=0\). Figure 3: Evolution of normalized spatial-temporal intensity \(I(x,t,z)\) of PCP beams with rect weight function with different values of \(x\): (**a**) \(x=0\), (**b**) \(x=0.2\) cm and (**c**) \(x=0.4\) cm in the \(t\)-\(z\) plane. (**d**) Normalized spatial-temporal intensity versus propagation distance \(z\) with \(t=0\). Figures 5 and 6 give the contour graph of normalized spatial-temporal intensity distribution of PCP beams at the \(z\) plane with different values of propagation distance \(z\). Figure 5 is the case of rect weight function, while Figure 6 is the case of Gaussian weight function. It is shown that at the \(z=0\) plane, the pulse duration and beam spatial width is quite large.