# Correlation of Intensity Fluctuations for Scattering of a Partially Coherent Plane-Wave Pulse
## 2 Correlation of Intensity Fluctuations
In the space-time domain, the temporal coherence function of the pulses takes the form [14; 15]:  
\[\Gamma(t_{1},t_{2})=\Gamma_{0}\exp[-\frac{t_{1}^{2}+t_{2}^{2}}{2T^{2}}-\frac{ \left(t_{1}-t_{2}\right)^{2}}{2T_{c}^{2}}-i\omega_{0}(t_{1}-t_{2})], \tag{8}\]  
where \(\omega_{0}\) is the central frequency of the pulse, \(T\) denotes the pulse duration, and \(T_{c}\) describes the temporal coherence length of the pulse. The cross-spectral density function of the pulses in thespace-frequency domain can be readily determined by employing the Fourier transform [14; 15], which yields:  
\[W(\omega_{1},\omega_{2})=W_{0}\exp[-\frac{\left(\omega_{1}-\omega_{0}\right)^{2}+ \left(\omega_{2}-\omega_{0}\right)^{2}}{2\Omega^{2}}-\frac{\left(\omega_{1}- \omega_{2}\right)^{2}}{2\Omega_{c}^{2}}], \tag{9}\]  
where \(W_{0}=\Gamma_{0}T/2\pi\Omega\) and the spectral width \(\Omega\) and the spectral coherence width \(\Omega_{c}\) of the pulses are connected to the temporal parameters by the equations \(\Omega^{2}=1/T^{2}+2/T_{c}^{2}\) and \(\Omega_{c}=T_{c}\Omega/T\). Suppose that the incident pulses propagate in a direction specified by a real unit vector \(\mathbf{s}_{0}\) (as shown in Figure 1), the cross-spectral density function of the incident light at a pair of points, specified by position vectors \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\), is given by the formula:  
\[ \begin{array}{ll}W^{(i)}(\mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime},\omega_{1},\omega_{2})&=W_{0}\exp[-\frac{\left(\omega_{1}-\omega_{0}\right)^{2}+\left( \omega_{2}-\omega_{0}\right)^{2}}{2\Omega^{2}}-\frac{\left(\omega_{1}-\omega _{2}\right)^{2}}{2\Omega_{c}^{2}}]\\ &\times\exp[i(k_{2}\mathbf{s}_{0}\cdot\mathbf{r}_{2}^{\prime}-k_{1}\mathbf{s}_{0} \cdot\mathbf{r}_{1}^{\prime})],\end{array}  \tag{10}\]  
with \(k_{i}=\omega_{i}/c_{i}\) (\(i=1,2\)) and \(c\) being the speed of light in vacuum. We assumed that the medium is a weak scatterer, so that the scattering may be analyzed within the accuracy of the first-order Born approximation. The scattering potential of the medium, at a point specified by a position vector \(\mathbf{r}_{i}^{\prime}\) within the scatterer, is characterized by \(F(\mathbf{r}_{i}^{\prime},\omega_{i})\). The resonance frequencies of the medium, i.e., the frequencies of its atomic or molecular transitions, were assumed not to lie within the spectral brand of the incident light. Thus, over the effective frequency range of the incident light, the scattering potential may be approximated by \(F(\mathbf{r}_{i}^{\prime},\omega_{0})\). For a deterministic medium, the scattering potential \(F(\mathbf{r}_{i}^{\prime},\omega_{0})\) is a well-defined function of position. The cross-spectral density function of the scattered light in the far zone, at two points specified by position vectors \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\), is given by the approximate far-zone formula  
\[ \begin{array}{ll}W^{(s)}(\mathbf{r}_{1},\mathbf{r}_{2},\omega_{1},\omega_{2})&= \int\limits_{D}\int\limits_{D}W^{(i)}(\mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime},\omega_{1},\omega_{2})F^{*}(\mathbf{r}_{1}^{\prime},\omega_{0})F(\mathbf{r}_{2}^{ \prime},\omega_{0})\\ &\times\frac{1}{r^{2}}\exp[ir(k_{2}-k_{1})-i(k_{2}\mathbf{s}_{2}\cdot\mathbf{r}_{2}^{ \prime}-k_{1}\mathbf{s}_{1}\cdot\mathbf{r}_{1}^{\prime})]d^{3}\mathbf{r}_{1}^{\prime} d^{3}\mathbf{r}_{2}^{\prime}.\end{array}  \tag{11}\]  
For a random medium, the scattering potential is, of course, a random function of position. The correlation function of the scattering potential at a pair of points, specified by position vectors \(\mathbf{r}_{1}^{\prime}\) and \(\mathbf{r}_{2}^{\prime}\) in the scattering medium, is defined by the formula:  
\[C_{F}(\mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime},\omega_{0})=\langle F^{*}(\mathbf{r} _{1}^{\prime},\omega_{0})F(\mathbf{r}_{2}^{\prime},\omega_{0})\rangle, \tag{12}\]  
Figure 1: Scattering problem geometry. \(\mathbf{s}_{0}\) and \(\mathbf{s}\) represent the incident and scattering directions, respectively.