Absnacr -The purpose of this paper is to investigate some applications of the fractionalized curl operator in the scattering problems. Using the technique of fractionalizing a linear operator we obtain the presentation for the fractional curl operator for the functions of hvo variables expressed via exponents. We use ibis presentation in a plane wave reflection at oblique incidence on an impedance infinite surface. Applying fractional curl operator to a fired solution we obtain the fields that describe, in certain sense, an intermediate of "fractional" solution behveen the original solution and its dual solution. These "fractional" fields represent the solution of reflection problem from an anisotropic surface. The relation behveen the impedance of such an anisotropic surface and the original impedance are presented in this paper.
I. INTRODUCTION
2In recent years, it has been studied the possible applications of tools of fractional calculus in electromagnetics problems. The idea of fractional derivatives and integrals can be extended to fractionalization of some other operators commonly used in electromagnetics. In [I] it was first proposed a fractionalization of the curl operator. New operator curl" was introduced where parameter 0 < a < 1 is real. If a = 0 , one gets the identity operator. If a = 1 , one gets the conventional curl operator. In [I] the presentation of the fractional curl operator was obtained in explicit form for a function of one variable. But in certain reflection problems we should extend it for the functions of two variables. Following the technique proposed in [I], we shall obtain the fractional curl operator when the function is represented via exponents Such functions characterize obliquely incident plane waves. This presentation will be used in the problem of a plane wave incidence on the boundary. We are going to study the fractional fields obtained by applying fractional curl operator to the solution of the problem with fixed impedance. We shall also generalize impedance boundary conditions with the aid of the fractional curl operator.
DEFINITION OF FRACTIONAL CURL OPERATORThe operator '* curl '' is one of the commonly used operators in electrodynamics. Using the spatial Fourier transform from (x,y,z)-space into k-domain (k,,k,,k,), the curl operator of a three-dimensional vector field = F,f + F j + F , Z can be expressed as a cross product of vector k with the vector F, ( p ) in k-domain, that is 6 (curlF)(k,,ky, k;) c i i x FI ( P ) . operator L" is considered as the fractionalized operator (from L ) if [I] Consider a-general linear operator L that acts within the space C" of n-dimensional vectors. The new I) if a = 1 : L" I o = , = L ; 2) if a = 0 : L" lo=o= I -the identity operator; 3) L!'LB = LpLa =La'@. Operator Lo can be defined asoperator that has the same eigenvectors {&} as operator L has, but with the eigenvalues of {(U,)"), where {um] are the eigenvalues of L . An arbihxy vector can be represented as a linear combination of eigenvectors & with some coefficients, i.e., as f7...
