The group velocity

The point of stationary phase, defined by $\partial\phi/\partial\omega =0$, satisfies the condition

$$\frac{c}{v_g} = \frac{ct}{x},$$ (768)

where

$$v_g = \frac{d \omega}{d k}$$ (769)

is conventionally termed the group velocity. Thus, the signal seen at position $x$ and time $t$ is dominated by the frequency range whose group velocity $v_g$ is equal to $x/t$. In this respect, the signal incident at the surface of the medium ($x=0$) at time $t=0$ can be said to propagate through the medium at the group velocity $v_g(\omega)$.

The simple one-resonance dispersion relation (4.86) yields

$$\frac{c}{v_g} \simeq n(\omega) \left[1 +\frac{\omega^2}{\ome... ...+\frac{\omega^2} {\omega^2-\omega_0^{~2}-\omega_p^{~2}}\right]$$ (770)

in the limit $g\rightarrow 0$, where

$$n(\omega) =\frac{c k}{\omega} = \sqrt{\frac{\omega_0^{~2} + \omega_p^{~2}-\omega^2}{\omega_0^{~2}-\omega^2}}.$$ (771)

The variation of $c/v_g$ and the refractive index $n$ with frequency is sketched in Fig. 12. With $g=0$ the group velocity is less than $c$ for all $\omega$, except for $\omega_0<\omega< \omega_1\equiv\sqrt{\omega_0^{~2} +\omega_p^{~2}}$, where it is purely imaginary. Note that the refractive index is also complex in this frequency range. The phase velocity $v_p=c/n$ is subluminal for $\omega<\omega_0$, imaginary for $\omega_0 \leq \omega\leq \omega_1$, and superluminal for $\omega>\omega_1$.

Figure 12: The typical variation of the functions $c/v_g(\omega )$ and $n(\omega )$. Here, $\omega_1=(\omega_0^{~2} + \omega_p^{~2})^{1/2}$.

The frequency range which contributes to the amplitude at time $t$ is determined graphically by finding the intersection of a horizontal line with ordinate $ct/x$ with the solid curve in Fig. 12. There is no crossing of the two curves for $t<t_0 \equiv x/c$, thus no signal arrives before this time. For times immediately following $t_0$ the point of stationary phase is seen to be at $\omega\rightarrow\infty$. In this large $\omega$ limit the point of stationary phase is given by

$$\omega_s \simeq \omega_p \sqrt{\frac{t_0}{2(t-t_0)}}.$$ (772)

Note that $\omega=-\omega_s$ is also a point of stationary phase. It is easily demonstrated that

$$\phi_s \simeq -2\sqrt{\xi(t-t_0)},$$ (773)

and

$$\phi_s'' \simeq -2\,\frac{(t-t_0)^{3/2}}{\xi^{1/2}},$$ (774)

with

$$F(\omega_s) \simeq \frac{t-t_0}{\tau\,\xi}.$$ (775)

Here, $\xi$ is given by Eq. (4.109). The stationary phase approximation (4.133) gives

$$f_s \simeq \sqrt{\frac{\pi \,\xi^{1/2} } {(t-t_0)^{3/2}}} \,... ... e}^{-2{\rm i} \sqrt{\xi(t-t_0)} +3\pi {\rm i}/4} +{\rm c.c.},$$ (776)

where c.c. denotes the complex conjugate of the preceding term (this contribution comes from the second point of stationary phase located at $\omega=-\omega_s$). The above expression reduces to

$$f_s \simeq \frac{ 2\sqrt{\pi}}{\tau} \,\frac{(t-t_0)^{1/4}}{\xi^{3/4}} \cos\!\left[2\sqrt{\xi(t-t_0)} - 3\pi/4\right].$$ (777)

It is easily demonstrated that the above formula is the same as the expression (4.118) for the Sommerfeld precursor in the large argument limit $t-t_0 \gg 1/\xi$. Thus, the method of stationary phase yields an expression for the Sommerfeld precursor which is accurate at all times except those immediately following the first arrival of the signal.