SCATTERING PROPERTIES OF THE STRIP WITH FRACTIONAL BOUNDARY CONDITIONS AND COMPARISON WITH THE IMPEDANCE STRIP

Abstract—In this paper two-dimensional problem of plane-wave diffraction by a “fractional strip ” is studied. “Fractional strip ” is introduced as a strip with fractional boundary conditions (FBC) involving fractional derivatives of the field components. FBC describe intermediate boundary between perfect electric conductor (PEC) and perfect magnetic conductor (PMC). It is shown that “fractional strip ” has scattering properties similar to the well-known impedance strip. For one important case of fractional order equal to 0.5 the solution of the wave diffraction problem by a “fractional strip ” can be found analytically. Detailed comparison analysis of the physical characteristics of the scattered fields for both fractional and impedance strips is presented. The relation between the fractional order and the value of impedance is derived. It is shown that in a wide range of input parameters the physical characteristics of the “fractional strip” are similar to the strip with pure imaginary impedance. 1. FORMULATION OF THE PROBLEM Consider a two-dimensional problem of electromagnetic wave diffrac-tion by a strip located at the plane y = 0 and infinite along the axis z. The width of the strip is 2a. In this paper E-polarization case is discussed. An incident plane wave is described by the function Ei(x, y) = zEi(x, y) = ze−ık(xα0+y 1−α20) where α0 = cosθ0, θ0 is 190 Ivakhnychenko, Veliev, and Ahmedov the incidence angle and k = 2πλ is the wave number. Time dependence is assumed to be e−ıωt and deprecated throughout the paper. Boundary conditions are fractional boundary conditions (FBC) defined as application of fractional derivative to the electric field component [1] DνkyEz(x, y)|y→±0 = 0, −a < x < a (1) For convenience the fractional differentiation is applied in respect to a dimensionless variable ky. Operator Dνyf(y) in (1) is defined by the integral of Riemann-Liouville [2] Dνyf(y) ≡ −∞Dνyf(y) = 1 Γ(1 − ν