On the Group Front and Group Velocity in a Dispersive Medium Upon Refraction From a Nondispersive Medium

Abstract

Conventional definitions of the velocities associated with the propagation of the modulated wave are both confusing and insufficient to describe the behavior of the wave packet clearly in a multi-dimensional dispersive medium. There exist infinite solutions to the general equation of wave-front movement, suggesting that there are infinite phase velocities. Therefore, the introduction of “normal phase velocity” becomes necessary to unambiguously define the phase velocity in the direction perpendicular to the wave front. Similarly, there exist infinite solutions to the equation describing the group-front movement in the case of a wave packet, resulting in infinite “group-front velocities.” The “normal group-front velocity” is defined as the smallest speed at which the group-front travels and is in the direction perpendicular to the group front. We show that the group velocity (i.e., the velocity of energy flow) is one of the group-front velocities and, in general, is not the same as the normal group-front velocity. Hence, the direction in which the wave packet travels is not necessarily normal to the group front. Examples are used to demonstrate the behavior of a wave packet that is refracted from vacuum to a positive index medium (PIM) or a negative index medium (NIM).