Group Velocity

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

$$\frac{c}{v_g} = \frac{c\,t}{x},$$ (918)

where

$$v_g = \frac{d \omega}{d k}$$ (919)

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 dielectric dispersion relation (871) yields

$$\frac{c}{v_g} \simeq n(\omega) \left(1 +\frac{\omega^{\,2}}{\omeg... ...{\,2}} +\frac{\omega^{\,2}} {\omega^{\,2}-\omega_0^{\,2}-\omega_p^{\,2}}\right)$$ (920)

in the limit $g\rightarrow 0$ , where

$$n(\omega) =\frac{c \,k}{\omega} = \sqrt{\frac{\omega_0^{~2} + \omega_p^{\,2}-\omega^{\,2}}{\omega_0^{\,2}-\omega^{\,2}}}.$$ (921)

The variation of $c/v_g$ , and the refractive index $n$ , with frequency is sketched in Figure 12. With $g=0$ , the group velocity is less than $c$ for all $\omega$ , except for $\omega_0<\omega< \omega_1\equiv(\omega_0^{\,2} +\omega_p^{\,2})^{1/2}$ , 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: 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 that contributes to the amplitude at time $t$ is determined graphically by finding the intersection of a horizontal line with ordinate $c\,t/x$ with the solid curve in Figure 12. There is no crossing of the two curves for $t<t_0 \equiv x/c$ . Thus, no signal can arrive before this time. For times immediately after $t=t_0$ , the point of stationary phase is seen to be at $\omega\rightarrow\infty$ . In this large-$\omega$ limit, the point of stationary phase satisfies

$$\omega_s \simeq \omega_p \sqrt{\frac{t_0}{2\,(t-t_0)}}.$$ (922)

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

$$\phi_s \simeq -2\sqrt{\xi\,(t-t_0)},$$ (923)

and

$$\phi_s'' \simeq -2\,\frac{(t-t_0)^{\,3/2}}{\xi^{\,1/2}},$$ (924)

with

$$F(\omega_s) \simeq \frac{t-t_0}{\tau\,\xi}.$$ (925)

Here, $\xi$ is given by Equation (894). The stationary phase approximation (918) yields

$$f_s \simeq \sqrt{\frac{\pi \,\xi^{\,1/2} } {(t-t_0)^{\,3/2}}} \,\... ...\xi}\, {\rm e}^{-2\,{\rm i} \sqrt{\xi\,(t-t_0)} +3\pi\, {\rm i}/4} +{\rm c.c.},$$ (926)

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 previous expression reduces to

$$f_s \simeq \frac{ 2\sqrt{\pi}}{\tau} \,\frac{(t-t_0)^{1/4}}{\xi^{\,3/4}} \cos\!\left[2\sqrt{\xi\,(t-t_0)} - 3\pi/4\right].$$ (927)

It is readily shown that the previous formula is the same as expression (903) 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 that is accurate at all times except those immediately following the first arrival of the signal.