# Intensity Evolution of Cosine-Gaussian-Correlated Schell-Model Pulse Scattered by a Medium
## 2 Theory  
The temporal mutual coherence function for a CGSM pulse has the following form [19]:  
\[\Gamma(t_{1},t_{2})=\left\langle E^{*}(t_{1})E(t_{2})\right\rangle=\Gamma_{0} \exp\left[-\frac{t_{1}^{2}+t_{2}^{2}}{4T_{0}^{2}}-\frac{(t_{2}-t_{1})^{2}}{2T_{ c}^{2}}+i\omega_{0}(t_{1}-t_{2})\right]\cos\left(\frac{n\sqrt{2\pi}(t_{2}-t_{1})}{T_ {c}}\right) \tag{1}\]  
where \(E(t_{1})\) and \(E(t_{2})\) are the complex analytic signals of pulse realizations at time \(t_{1}\) and \(t_{2}\), respectively; the asterisk is the complex conjugate; \(\Gamma_{0}\) is a positive constant; \(T_{0}\) represents the pulse duration; \(T_{c}\) is the temporal-coherence length denoting the temporal correlation of the pulse; \(\omega_{0}\) describes the carrier frequency of the pulse [25]; \(n\) is a constant greater than zero and does not need to be an integer. Obviously, for the case of \(n=0\), a CGSM pulse is simplified to a GSM pulse. By performing Fourier transform on Equation (1), the cross-spectral density function for a CGSM pulse is given by [11]  
\[W(\omega_{1},\omega_{2})=W_{0}\exp\left[-\frac{(\omega_{1}-\omega_{0})^{2}+( \omega_{2}-\omega_{0})^{2}}{2\Omega_{0}^{2}}\right]\exp\left[-\frac{(\omega_{ 1}-\omega_{2})^{2}}{2\Omega_{c}^{2}}\right]\times\exp\left[-\frac{\sqrt{2\pi} nb}{T_{c}}\right]\cosh\left[b(\omega_{1}+\omega_{2}-2\omega_{0})\right], \tag{2}\]  
where  
\[W_{0}=\frac{\sqrt{2}T_{0}}{2n\Omega_{0}}\Gamma_{0} \tag{3}\]  
\[\Omega_{0}=\sqrt{\frac{1}{2T_{0}^{2}}+\frac{2}{T_{c}^{2}}} \tag{4}\]  
\[\Omega_{c}=\frac{\sqrt{2}T_{c}}{2T_{0}}\Omega_{0} \tag{5}\]  
\[b=\frac{n\sqrt{2\pi}}{\Omega_{0}^{2}T_{c}}, \tag{6}\]  
where \(\Omega_{0}\) denotes the spectral width of the CGSM pulse. \(\Omega_{c}\) represents the spectral coherence width of the pulse. \(\cosh(x)\) is a hyperbolic cosine function. Suppose a CGSM pulsed beam is incident on a scatterer along the direction described by unit vector \(\mathbf{s}_{0}\) (Figure 1). The cross-spectral density of a CGSM pulse at two points with position vectors \(\mathbf{r}_{1}\)' and \(\mathbf{r}_{2}\)' can be written as [26]  
\[ \begin{array}{l}W^{(i)}(\mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime}, \omega_{1},\omega_{2})=W_{0}\exp\left[-\frac{(\omega_{1}-\omega_{0})^{2}+( \omega_{2}-\omega_{0})^{2}}{2\Omega_{0}^{2}}\right]\exp\left[-\frac{(\omega_{ 1}-\omega_{2})^{2}}{2\Omega_{c}^{2}}\right]\\ \times\exp\left[-\frac{\sqrt{2\pi}nb}{T_{c}}\right]\cosh\left[b(\omega_{1}+ \omega_{2}-2\omega_{0})\right]\exp\left[i(k_{2}\mathbf{s}_{0}\cdot\mathbf{r}_ {2}^{\prime}-k_{1}\mathbf{s}_{0}\cdot\mathbf{r}_{1}^{\prime})\right]\end{array}  \tag{7}\]  
Figure 1: Notation relating to scattering a cosine-Gaussian-correlated Schell-model (CGSM) pulse by a medium. In the scattering process, the correlation function of the scattering potential is used to describe the scattering properties of a random medium [27]. In [10], it was assumed that the resonance frequency of molecules or atoms in the medium is approximately represented by the carrier frequency of the pulse. Therefore, the correlation function of the scattering potential is expressed as [28; 29]  
\[C_{F}(\mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime},\omega)=\left\langle F^{ \ast}(\mathbf{r}_{1}^{\prime},\omega)F(\mathbf{r}_{2}^{\prime},\omega)\right\rangle _{m}=\left\langle F^{\ast}(\mathbf{r}_{1}^{\prime},\omega_{0})F(\mathbf{r}_{2 }^{\prime},\omega_{0})\right\rangle_{m}=C_{F}(\mathbf{r}_{1}^{\prime},\mathbf{ r}_{2}^{\prime},\omega_{0}), \tag{8}\]  
where \(F(\mathbf{r}^{\prime},\omega)\) is defined as the scattering potential of the medium, and\(<\)\(>\)denotes averaging over the ensemble of a random medium [30]. According to the first-order Born approximation, we obtained the formula for the cross-spectral density function of the scattered pulse [31; 32]:  
\[