Proposal Number: ECS-9612634 Principal Investigator: Nader Engheta Title: Theoretical study of applications of fractional calculus in electromagnetism In this research, a theoretical investigation using the fractional calculus approach to waves and geometries of interest will be developed. The fractional calculus approach enables the study of waves and structures with an intermediate character. It is an area which has been developed with major contributions from the PI. Under the present study, the following topics will be investigated: (1) "fractional" intermediate sources and (2) applications of fractional calculus in electromagnetic scattering. Dual investigations of geometries and boundaries for which an incident plane waves generates scattered waves with fractional characteristics, and, conversely the scattering of incident waves with fractional characteristics from various geometries will be carried out. In each of these areas, a new corresponding development of the theory itself will be investigated (1') extension of the concept of fractional order multipoles for complex sources involving electric and magnetic fractional-order multipoles and/or aperture fields and (2') the development of a "fractional" curl operator, both for analysis of the electromagnetic scattering interactions as stated in (2) and for investigation the basic concepts involved in the duality of the fractional character of the waves and the geometries.