= & i {\epsilon(l) \over \epsilon_1}
t(l) - {i \over 2} \left[ {\epsilon(l) \over \epsilon_1} - 1 \right]
\left[ 1 - {\epsilon_1 \over \epsilon(l)}
\tan^2 \theta \right] \left[ 1 + r(l) \right] t(l).\end{aligned}$$ These equations are integrated from $l=0$ to $l=L$ using the initial conditions given by the Fresnel formulas $$\begin{aligned}
&&r(0) = {\epsilon_2 \sqrt{\epsilon_1} \cos \theta - \epsilon_1
\sqrt{\epsilon_2 - \epsilon_1 \sin^2 \theta} \over \epsilon_2
\sqrt{\epsilon_1} \cos \theta + \epsilon_1 \sqrt{\epsilon_2 -
\epsilon_1 \sin^2 \theta}}, \nonumber\\
&&t(0) = {2 \epsilon_2 \sqrt{\epsilon_1} \cos \theta \over
\epsilon_2 \sqrt{\epsilon_1} \cos \theta + \epsilon_1
\sqrt{\epsilon_2 -
\epsilon_1 \sin^2 \theta}}.\end{aligned}$$ The reflection and transmission coefficients $r(L)$ and $t(L)$ calculated above are employed into the integral in (4), multiplying each Fourier component.

The GH shift for reflected plane waves is computed and compared with those of beams, using Artmann’s formula $$\Delta = -{d\phi \over dk_x} = -{\lambda \over 2\pi
\sqrt{\epsilon_1} \cos\theta} {d\phi \over d\theta}$$ where $\lambda$ is the wavelength and $\phi$ is the phase of the reflection coefficient satisfying $r = |r| e^{i \phi}$.

Numerical results and discussion
================================

![\[fig1\] Schematic of the Otto configuration considered in this paper. ](fig1){width="15cm"}

We assume that a $p$-polarized beam is incident from a prism onto a dielectric-metal bilayer, 