The Parabolic Cylinder Functions of Miller's Second Kind for Complex Parameter.
Abstract
This report describes the derivation of some properties of the Parabolic Cylinder Functions e(a,z) and E*(a,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 W(a, + or - z), e(a,z) and e*(a,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, U(a,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 E(a,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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Feb 29, 1976
- Accession Number
- ADA025314
Entities
People
- Robert B. Smith
Organizations
- University of Washington