Signal arrival

Let us try to establish at what time $t_2$ a signal first arrives at position $x$ inside the dielectric medium whose amplitude is comparable with that of the wave incident at time $t=0$ on the surface of the medium ($x=0$). Let us term this event the ``arrival'' of the signal. It is plausible from the discussion in Section 4.11 regarding the stationary phase approximation that signal arrival corresponds to the situation where the point of stationary phase in $\omega$-space corresponds to a pole of the function $F(\omega)$. In other words, when $\omega_s$ approaches the frequency $2\pi/\tau$ of the incident signal. It is certainly the case that the stationary phase approximation yields a particularly large amplitude signal when $\omega_s\rightarrow 2\pi/\tau$. Unfortunately, as has already been discussed, the method of stationary phase becomes inaccurate under these circumstances. However, calculations involving the more robust method of steepest decent¹⁴ confirm that in most cases the signal amplitude first becomes significant when $\omega_s=2\pi/\tau$. Thus, the signal arrival time is

$$t_2 = \frac{x}{v_g(2\pi/\tau)},$$ (794)

where $v_g(2\pi/\tau)$ is the group velocity calculated using the frequency of the incident signal. It is clear from Fig. 12 that

$$t_0 < t_1 < t_2.$$ (795)

Thus, the main signal arrives later than the Sommerfeld and Brillouin precursors.

Figure 14: A sketch of the signal amplitude as a function of time as seen inside some dielectric medium subject to an incident wave which starts at some specific time

The final picture which emerges from our investigations is summarized in Fig. 14. The main signal arrives at the group velocity corresponding to the frequency of the incident wave. However, it is possible to detect the arrival of the signal before this, given sufficiently accurate detection equipment. In fact, the first information regarding the arrival of the incident wave at the vacuum/dielectric interface propagates at the velocity of light in a vacuum.