\frac{4 \int_{-\infty}^{\infty} \left( x - \Delta_{t} \right)^2 \left\vert {H_y}^{(t)}(x,0) \right\vert^2 dx }
 {\int_{-\infty}^{\infty} \left\vert {H_y}^{(t)}(x,0) \right\vert^2 dx}.\end{aligned}$$ These expressions are just the first and second moments of a distribution function and give the average and the variance for a given statistical distribution function. The first moment can also be interpreted as the centroid of the area under the distribution curve. The half-widths of the reflected and transmitted beams, $W_r$ and $W_t$, are obtained using $$W_r = \sqrt{\beta_{r}} \cos\theta_i,~~~W_t = \sqrt{\beta_{t}} \cos\theta_i.$$

In the expressions given above, the reflection and transmission coefficients play a crucial role. In order to calculate $r(k_x)$ and $t(k_x)$ in the wave vector space, we resort to the invariant imbedding method, which we summarize here briefly. For a $p$-polarized wave propagating in a nonmagnetic ($\mu
= 1$) layered structure along the $xz$ plane, the complex amplitude of the magnetic field, $H = H(z)$, satisfies the wave equation $$\label{p-wave_eq}
{d^2 H \over dz^2} - {1 \over \epsilon(z)} {d \epsilon \over dz} {dH
\over dz} + \left[ {k_0}^2 \epsilon(z) - {k_x}^2 \right] H = 0,$$ where $k_0$ ($ = \omega / c$) is the vacuum wave number and $\epsilon(z)$ is the dielectric permittivity. We consider a plane wave of 