Off-Axis Wave-Optics Transmission in a Lens-Like Medium with Aberration

Transmission through a lens-like medium consisting of a dielectric rod in which the dielectric constant decreases radially with quadratic law has been studied because, first, it is closely related to light transmission in a periodic sequence of gaseous lenses, 1,2 ' 3 second, it helps to understand the filamentary nature of the oscillations in ruby lasers,4 and third, if the dielectric constant is complex, it describes the gain medium in a gaseous laser.5 Whenever the radial dependence of the dielectric constant is quadratic, any paraxial beam propagates undulating periodically about the axis, and the field reproduces itself after each period. This is true only within 149 150 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1967 some approximations.* Part of the object of this paper is to find the distance over which the approximations hold and how it depends upon the beam shape and displacement from the axis. The mathematical description of Gaussian beams in such an aberration-free lens-like quadratic medium has been achieved via two techniques. The first consists of expanding the input beam in terms of on-axis normal modes' and summing them up everywhere. The second technique consists of solving differential equations that elegantly relate the location of the beam axis, the spot size, and the curvature of the phase front at any place along the beam.7 Nevertheless, we show in this paper that if the lens-like medium has aberrations, no matter how small, the electromagnetic field of a paraxial beam changes shape radically as it travels along the guide.