Novel Fractional-Dimensional Approach to Electromagnetics

Fractional-dimensional approach has attracted widespread attention in recent years motivated by its fundamental importance and possible practical applications in electromagnetic (EM)modeling of complex, anisotropic, heterogeneous, disordered or fractal media. In this approach, a complex, rough structure lying in a real space is mapped into a simple structure embedded in a non-integer (fractional)space in an effective manner. Such an approach can provide unparalleled efficiency when handling complex EM structures. In this work, we introduce the basic theory of fractional electrodynamics via generalization of electromagnetic equations to non-integer (fractional)dimensional spaces including the definition of multi-poles and electric, magnetic field of charges in fractional spaces, the derivation of fractional Maxwell equations, the exact solutions of fractional Laplaces, Poissons and Helmholtzs equations in planar-, cylindrical-, and spherical-coordinates and the exact solutions of related electrostatic and electromagnetic wave propagation problems in generalized fractional spaces. We demonstrate the usefulness of this self-consistent theoretical framework in efficient modeling of numerous disordered systems in electronics and photonics. For all these examples, the complexity of the problem is reduced by solving the corresponding fractionalized governing equations under appropriate boundary conditions. The effectiveness of this approach is demonstrated by good agreement of calculated results with full-wave EM simulations or experiments.