# Spatial-Temporal Self-Focusing of Partially Coherent Pulsed Beams in Dispersive Medium
## 2 Theory  
Consider a statistically PCP source located in the plane \(z=0\), radiating a beam-like field that propagates into the positive half-space \(z>0\). The statistical properties of the 2D source at points \(\rho_{1}=(x_{1}^{\prime},y_{1}^{\prime})\) and \(\rho_{2}=(x_{2}^{\prime},y_{2}^{\prime})\), at different instant time \(t_{10}\) and \(t_{20}\), can be characterized by the mutual coherent function (MCF), which is defined as a two-point two-time correlation function without spatiotemporal coupling:  
\[\Gamma_{0}(\rho_{1},t_{10};\rho_{2},t_{20})=R(\rho_{1},\rho_{2})T(t_{10},t_{20}) \tag{1}\]  
where  
\[R(\rho_{1},\ \rho_{2})=\exp(-\frac{\rho_{1}^{2}+\rho_{2}^{2}}{2w_{0}^{2}})\sin \zeta\left(-\frac{\rho_{2}^{2}-\rho_{1}^{2}}{\sigma^{2}}\right) \tag{2}\]  
\[T(t_{10},t_{20})=\exp(-\frac{t_{10}^{2}+t_{20}^{2}}{2T_{0}^{2}})\sin\zeta\left( -\frac{t_{20}^{2}-t_{10}^{2}}{T_{c}^{2}}\right)\exp[-i\omega_{0}\left(t_{20}-t _{10}\right)] \tag{3}\]  
where \(\text{sinc}x=\sin\pi x/\pi x\) and \(\omega_{0}\) is the carrier frequency of pulsed beams. \(w_{0}\) and \(\sigma\) denotes beam width and spatial coherent parameter, respectively. \(T_{0}\) and \(T_{c}\) represent pulse duration and temporal coherent length, respectively. The beams generated by this kind of pulsed source expressed by Equation (1) are regarded as the spatially and temporally PCP beams with circular spatial and sinc temporal coherence distribution. The spatial part of mutual coherent function Equation (1) is the same as that in reference [39], which has a circular coherence distribution. Equation (1) can be also written as:  
\[\Gamma_{0}(\rho_{1},t_{10};\rho_{2},t_{20})=\int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty}p_{R}(v_{1})p_{T}(v_{2})H^{*}(\rho_{1},t_{10},v_ {1},v_{2})H(\rho_{2},t_{20},v_{1},v_{2})dv_{1}dv_{2} \tag{4}\]  
where  
\[p_{R}(v_{1})=\sigma^{2}\text{rect}(\sigma^{2}v_{1}) \tag{5}\]  
\[p_{T}(v_{2})=T_{c}^{2}\text{rect}(T_{c}^{2}v_{2}) \tag{6}\]  
\[H(\rho,t_{0};v_{1},v_{2})=H_{R}(\rho,v_{1})H_{T}(t_{0},v_{2}) \tag{7}\]  
\[H_{R}(\rho,v_{1})=\exp(-\frac{\rho^{2}}{2w_{0}^{2}})\exp(-2\pi iv_{1}\rho^{2}) \tag{8}\]  
\[H_{T}(t_{0},v_{2})=\exp(-\frac{t_{0}^{2}}{2T_{0}^{2}})\exp(-2\pi iv_{2}t_{0}^{ 2})\exp(-i\omega_{0}t_{0}) \tag{9}\]where \(p_{\rm R}(v_{1})\) and \(p_{\rm T}(v_{2})\) are rect weight functions of \(H(\cdot)\). In the plane \(z>0\), by means of the extended Huygens-Fresnel integral with paraxial approximation, the propagation properties of PCP beams with circular spatial and sinc temporal coherence distribution in dispersive medium in spatiotemporal domain can be characterized by the following integral formula [1,62]:  
\[ \begin{array}{ll}\Gamma(\mathbf{r}_{1},\mathbf{r}_{2},t_{1},t_{2},z)&=\left(\frac{k}{ 2\pi i2}\right)^{2}\frac{\omega_{00}}{2\pi i2}\int\int\int\int\int_{-\infty}^{ \infty}\Gamma_{0}(\mathbf{\rho}_{1},t_{10};\rho_{2},t_{20})\\ &\times\exp\biggl{[}-ik\frac{(\mathbf{r}_{1}-\mathbf{\rho}_{1})^{2}-(\mathbf{r}_{2}-\mathbf{ \rho}_{2})^{2}}{2z}\biggr{]}d^{2}\rho_{1}d^{2}\rho_{2}\\ &\times\exp\biggl{[}-ia\omega_{0}\frac{(t_{1}-t_{10})^{2}-(t_{2}-t_{20})^{2}}{ 2z}\biggr{]}dt_{10}dt_{20},\end{array}  \tag{10}\]  
where \(k=n(\omega)/c\) denotes the wave number, in which \(n(\omega)\) is refractive index of medium and \(c\) is the speed of light in vacuum. In this paper, we assume that the medium is a linear dispersive medium whose refractive index is given by \(n(\omega)=n_{\rm a}\omega+n_{\rm b}\), where \(n_{\rm a}\) and \(n_{\rm b}\) are constants. And \(a=\omega_{0}\beta_{2}z\), with \(\beta_{2}\) representing the group velocity dispersion, which is related to \(n_{\rm a}\) by \(\beta_{2}=n_{\rm a}/c\). And \(n_{\rm b}=2\beta_{2}\omega_{0}c\) - \(c/v_{\rm g}\), where \(v_{\rm g}\) is the group velocity of the pulse [63]. Here, the time coordinate is a retarded time, which is calculated in a frame moving with group velocity of the pulses.