Wave-Guide Systems with Negative Phase Velocities

Abstract

IN all smooth wave-guides and many loaded wave-guides the phase velocity has the same sign as the energy velocity. Some systems having complex forms of loading have in the past been attributed with zero or even negative group-velocities, based on apparently anomalous dispersion curves. Such phenomena have been clouded by doubts about the conception of group velocity and its relationship to energy velocity. It has recently been shown that in a periodic structure with negligible attenuation, energy and group velocities are identical1. If a system is analysed in terms of the positive solution for phase velocity, a zerogroup-velocity indicates a form of resonance with zero net power flow, and a negative group-velocity indicates a negative net energy velocity. Since in any experiment the net energy velocity is taken to be positive, systems can be devised in which the phase-velocity is negative. Dispersion curves showing guide and air wave-lengths as ordinates and abscissae must have a positive slope, and in these anomalous systems this requires a negative sign to be attributed to the measured guide wave-lengths. When interaction with charged particles is required, it is of supreme importance to determine the correct direction of the phase-velocity.