Cutoffs

Suppose that a cutoff is located at , so that

in the immediate vicinity of this point, where . It is evident, from the WKB solutions, (6.17) and (6.18), that the cutoff point lies at the boundary between a region ( ) in which electromagnetic waves propagate, and a region ( ) in which the waves are evanescent. In a physically realistic solution, we would expect the wave amplitude to decay (as decreases) in the evanescent region . Let us search for such a wave solution.

In the immediate vicinity of the cutoff point, , Equations (6.3) and (6.19) yield

where

(6.21) |

Equation (6.20) is a standard equation, known as

The second solution, , is unphysical, because it blows up as . The physical solution, , has the asymptotic behavior

(6.22) |

in the limit , and

(6.23) |

in the limit .

Suppose that a unit amplitude plane electromagnetic wave, polarized in the -direction, is launched from an antenna, located at large positive , toward the cutoff point at . It is assumed that at the launch point. In the non-evanescent region, , the wave can be represented as a linear combination of propagating WKB solutions:

(6.24) |

The first term on the right-hand side of the previous equation represents the incident wave, whereas the second term represents the reflected wave. The complex constant is the

However, we have another expression for the wave in this region:

(6.26) |

where is an arbitrary constant. The previous equation can be written

A comparison of Equations (6.25) and (6.27) reveals that

(6.28) |

We conclude that at a cutoff point there is total reflection of the incident wave (because ) with a phase-shift.