# Intensity Evolution of Cosine-Gaussian-Correlated Schell-Model Pulse Scattered by a Medium
## 3 Intensity Properties of CGSM Pulse Scattered by a Medium  
On the basis of Equation (15), the intensity involution of a CGSM pulse scattered by a random medium can be studied as follows. Figure 2 shows changes in the normalized intensity of the scattered pulse with the scattering angle for four different values of parameter \(n\). In the following numerical calculations, unless specified otherwise, parameters were chosen as: \(T_{0}=T_{c}=5\) fs, \(\lambda_{0}\)= 800 nm, \(\sigma_{R}=10\)\(\lambda_{0}\), \(\sigma_{\tau}=\lambda_{0}\), and \(t=\tau/c\). Figure 2 shows that the intensity properties of the scattered CGSM pulse were closely related to parameter \(n\). For a GSM pulse, the intensity distribution of the scattered pulse has Gaussian distribution. With an increase of parameter \(n\), the effective angular width of the scattered pulse decreases. Here, the effective angular width of the scattered pulse is defined as the 1/\(e\) point of the normalized intensity of the scattered pulse [37]. Figure 3a illustrates the variations of the normalized intensity of the scattered pulse with scattering angle \(\theta\), \(n=2\). In order to display the impact of initial pulse duration \(T_{0}\) on the intensity of the scattered pulse, Figure 3b gives the changes of the effective angular width of the scattered pulse with initial pulse duration \(T_{0}\). It is clear from Figure 3a that the effective angular width of the scattered pulse increased when the initial pulse duration \(T_{0}\) increased. Furthermore, Figure 3b shows that, for small values of parameter \(n\), the effective angular width of the scattered pulse varied rapidly with initial pulse duration \(T_{0}\); however, it changed slowly for large values of parameter \(n\). Figure 2: Normalized intensity of scattered pulse as function of scattering angle \(\theta\) for four different values of parameter \(n\). Figure 4 shows (a) the behavior of the normalized intensity of the scattered pulse as a function of scattering angle \(\theta\), and (b) the effective angular width of the scattered pulse against temporal-coherence length \(T_{c}\) of the initial pulse. The effective angular width of the scattered pulse increased with increasing temporal-coherence length \(T_{c}\) of the initial pulse. However, the effective angular width of the scattered pulse changed rapidly with temporal-coherence length \(T_{c}\) for large values of parameter \(n\). Especially with an increase of temporal-coherence length \(T_{c}\) of the initial pulse, the effective angular width of the scattered pulse remained nearly unchanged for the case of \(n=0\), and it converged to a constant 0.241 for the case of \(n=0\), 2, 6, 10. The reason for this phenomenon is as follows. When temporal-coherence length \(T_{c}\) of the initial pulse is infinite, the intensity of the scattered pulse takes the following form:  
\[I^{(s)}(r_{\mathbf{s}},t)=\frac{W_{0}C_{0}T_{0}(2\pi\sigma_{R}\sigma_{r})^{3} }{r^{2}\sqrt{2(T_{0}^{2}+m+a)\Big{[}T_{0}^{2}+(m-a)\Big{]}}}\exp\left[-\frac{( t-\frac{t}{\varepsilon})^{2}}{2(T_{0}^{2}+m+a)}\right]\exp\left[-\frac{2\omega_{0}^{2 }T_{0}^{2}(m-a)}{T_{0}^{2}+(m-a)}\right], \tag{16}\]  
which is independent of temporal-coherence length \(T_{c}\) of the initial pulse and parameter \(n\). Therefore, the effective angular width of the scattered pulse converged to a constant when temporal-coherence length \(T_{c}\) of the initial pulse increased for different \(n\) parameters. Figure 5 illustrates the influence of effective radius \(\sigma_{R}\) of the medium on the normalized intensity of the scattered pulse. As shown in Figure 5, the effective angular width of the scattered pulse increased as effective radius \(\sigma_{R}\) of the medium decreased. For large values of parameter \(n\), effective radius \(\sigma_{R}\) of the medium had little impact on the effective angular width of the scattered pulse.