Group Velocity

(918) |

where

is conventionally termed the

The simple one-resonance dielectric dispersion relation (871) yields

(920) |

in the limit , where

(921) |

The variation of , and the refractive index , with frequency is sketched in Figure 12. With , the group velocity is less than for all , except for , where it is purely imaginary. Note that the refractive index is also complex in this frequency range. The phase velocity is subluminal for , imaginary for , and superluminal for .

The frequency range that contributes to the amplitude at time is determined graphically by finding the intersection of a horizontal line with ordinate with the solid curve in Figure 12. There is no crossing of the two curves for . Thus, no signal can arrive before this time. For times immediately after , the point of stationary phase is seen to be at . In this large- limit, the point of stationary phase satisfies

(922) |

Note that is also a point of stationary phase. It is easily demonstrated that

(923) |

and

(924) |

with

(925) |

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

(926) |

where c.c. denotes the complex conjugate of the preceding term (this contribution comes from the second point of stationary phase located at ). The previous expression reduces to

(927) |

It is readily shown that the previous formula is the same as expression (903) for the Sommerfeld precursor in the large argument limit . 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.