The Parabolic Cylinder Functions of Miller's Second Kind for Complex Parameter.

This report describes the derivation of some properties of the Parabolic Cylinder Functions ea,z and Ea,z. These functions are a natural choice for expressing the solutions of the coupled-mode equations for the Tapered Coupler which is important in integrated optics and fiber optics coupling. The parabolic cylinder functions Wa, or - z, ea,z and ea,z, originally defined only for real arguments, are considered when a and z are complex. Due to a choice of notation the original defining expressions do not remain valid when the parameter, a, is complex as required for the tapered coupler. Alternative expressions which are generally valid are obtained from the function of the first kind, Ua,z. It is found that these functions of the second kind are not analytic throughout the finite plane of a, as is allowed by the defining differential equation. Rather, thay have multiple branches four and the half integer points along the imaginary axis are all branch points. Three-term recurrence relations of which the couples-mode equations for the tapered coupler are a special case are obtained for the functions Ea,z, but cannot be uniformly valid throughout the complex a-plane. Within either the right or left half of the complex plane, and on the real axis, these functions have useful applications, but their singular properties may discourage their full acceptance as standard parabolic cylinder functions.