Half-Way Duality in Electromagnetics Using an Explicit Expression for the Half-Curl Operator

Abstract

Duality between electric and magnetic fields and parameters has long been recognized as a useful shortcut for inferring solutions to certain problems from their known duals. In this article, we suggest the use of “half-way dual” fields for extending the range of problems that can be addressed in this way. To this end, a workable expression for the fractional curl operator of order one-half is derived. While general expressions for fractional derivatives are purely formal and hard to implement, the special case of electromagnetic fields enables substantial simplification that leads to this closed form expression. It is then used to formulate a four-field set of Maxwell-type equations that include two half-way dual fields in addition to the original $\mathbf {E}$ and $\mathbf {H}$ . As an example, the solution to a reactive surface is inferred from its perfect electric conductor (PEC) half-way dual.