Time Delay of Scattering Processes

It is shown that Smith's formulation of the collision lifetime matrix can be applied to the relativistic steady-state solutions of Maxwell's equations for the total-reflection case. In this way, a delay time $\Delta t$ is obtained which, though probably unobservable as a time process ($\Delta t\sim {10}^{-15\mathrm{}}$ sec), leads to measurable results in a suitable stationary experiment. By multiplying the delay time $\Delta t$ by the group velocity of the wave in the vacuum, we obtain the displacement $\Delta x$ of the wave in the vacuum parallel to the surface. If the incident wave is then collimated by a slit, the center of the reflected beam, on reappearing in the dielectric, will show a parallel displacement relative to the beam reflected at the mirror which is given by $\Delta x_{\parallel }=\left(\frac{\lambda }{\pi }\right)\left[\frac{\left({sin\theta }_i\right)}{{({{sin}^2\theta }_i-\frac{1}{n^2})}^{\frac{1}{2}}}\right]$, where $\lambda$ is the wavelength of the light in the dielectric of refractive index $n$, and ${\theta }_i$ is the incident angle, greater than or equal to the critical one. This is just the result of the experiment performed as early as 1947 by Goos and Hänchen, who found the parallel displacement to be given by ${\left(\Delta x_{\parallel }\right)}_{\exp }=0.52\mathrm{}\lambda \left[\frac{1}{{({{sin}^2\theta }_i-\frac{1}{n^2})}^{\frac{1}{2}}}\right]$.