half-width $W$ is incident from the region where $z>L$ at an angle $\theta_i$. For a $p$-polarized beam propagating in the $xz$ plane, the $y$ component of the magnetic field associated with the incident beam at the $z=L$ plane can be written as $${H_y}^{(i)}(x,L) = H_0 \exp \left( -{x^2 \over {W_x}^2} + i k_{x0} x
\right),$$ where $W_x$ $( = W / \cos\theta_i)$ is the half-width in the $x$ direction. The center of the incident beam is at $x = 0$. The parameter $k_{x0}$ ($= k_1 \sin\theta_i$) is the $x$ component of the wave vector corresponding to the incident angle $\theta_i$ and $k_1$ is the wave number in the incident region, which corresponds to the prism. The superscript $(i)$ refers to the incident beam.

We consider the incident Gaussian beam as a linear combination of plane waves and write its field as $${H_y}^{(i)}(x,L) = {1 \over \sqrt{2\pi}} \int_{-\infty}^{\infty}
\tilde{H}\left(k_x\right) \exp \left( i k_x x  \right) dk_x,$$ where the Fourier transform $\tilde{H}\left(k_x\right)$ is given by $$\tilde{H}(k_x) = \frac{1}{\sqrt{2}}H_0 W_x\exp\left[-\frac{{W_x}^2}{4}\left(
k_x-k_{x0}\right)^2\right].$$ The variable $k_x$ can be parameterized as $k_x=k_1\sin\theta$. We write the reflection and transmission coefficients corresponding to each Fourier component as $r(k_x)$ and $t(k_x)$ respectively. Then the field profiles for the reflected 