1 Sufficient conditions for zero backscattering by a uniaxial dielectric-magnetic scatterer endowed with magnetoelectric gyrotropy Hamad M. Alkhoori, Akhlesh Lakhtakia, James K. Breakall, and Craig F. Bohren Abstract—As vector wavefunctions are available to represent incident and scattered fields in an isotropic dielectric-magnetic medium endowed with magnetoelectric gyrotropy, a transition matrix can be conceptualized to relate the scattered field co- efficients to the incident field coefficients for scattering by an arbitrary scatterer composed of a linear medium. The elements of the transition matrix must satisfy certain conditions for zero backscattering. For a scatterer composed of a uniaxial dielectric- magnetic medium endowed with magnetoelectric gyrotropy, the extended boundary condition method can be formulated to determine the transition matrix. Numerical results obtained thereby lead to the formulation of a sufficient set of three zero- backscattering conditions: (i) The scatterer is a body of revolution with the incident plane wave propagating along the axis of revolution. (ii) The impedances of both mediums are identical. (iii) The magnetoelectric-gyrotropy vectors of both mediums are aligned along the axis of revolution, whether or not both magnetoelectric-gyrotropy vectors are co-parallel. Index Terms—Electromagnetic scattering, Dielectric anisotropy, Magnetic anisotropic, Magnetoelectric gyrotropy I. I NTRODUCTION Backscattered fields are used in monostatic radar systems [1]. The higher the backscattering efficiency [2], the more detectable the scatterer is, and vice versa. Hence, designing scatterers with low backscattering efficiency is important in stealth applications. Zero backscattering means that the scatterer is invisible to monostatic radar. Conditions for zero backscattering have been found in some instances [3]–[6] when the exterior medium is free space, the scatterer is a body of revolution, and the incident field is a plane wave propagating parallel to the axis of revolution. Weston [3] showed that if the total external electric field and the total external magnetic field on the surface of the scatterer satisfy an isotropic impedance boundary condition, then backscattering is zero if the surface impedance equals the intrinsic impedance of free space. Yee and Chang [4] extended Weston’s requirement of impedance match to an anisotropic H. M. Alkhoori is with the Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802, USA (email: hma157@psu.edu). A. Lakhtakia is with NanoMM—Nanoengineered Metamaterials Group, Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA 16802, USA (email: akhlesh@psu.edu). J. K. Breakall is with the Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802, USA (email: jkb1@psu.edu). C. F. Bohren is with the Department of Meteorology and Atmospheric Science, The Pennsylvania State University, University Park, PA 16802, USA (email: cfb@psu.edu). impedance boundary condition. Uslenghi [5] showed that the requirement of impedance match will be effective in producing zero backscattering if the scatterer is composed of an isotropic chiral material [7]. Finally, Lindell et al. [6] proved that the same requirement holds for any anisotropic dielectric-magnetic scatterer, and it may not be a body of revolution in some instances. The effect of the exterior medium on conditions for zero backscattering was not unveiled in these works, although extension to an isotropic dielectric-magnetic medium as the exterior medium is straightforward. The exterior medium is usually air, which is the same as free space, but consideration of exterior mediums other than free space is beneficial for designing composite materials [8], [9], coatings [10], [11], and radomes [12]. In this paper, we present a new approach for zero backscat- tering that is valid for scatterers of any electrical size. Both the interior and the exterior mediums can also have mag- netoelectric properties [13], [14] of a special type [15] that is Lorentz nonreciprocal [16], [17]. Our approach requires that the vector wavefunctions to express the incident and the scattered fields be known, and that a linear relationship between the scattered field coefficients and the incident field coefficients can be found. This linear relationship is codified in terms of a transition matrix [18]. An expression for the backscattering efficiency can be obtained from the expansion of the scattered field. For the backscattering efficiency to vanish, certain relations have to be satisfied by the elements of the transition matrix. We consider an isotropic dielectric-magnetic medium en- dowed with magnetoelectric gyrotropy as the exterior medium. This medium is described by the frequency-domain constitu- tive relations D(r)= ε 0 ε re E(r) - ( Γ e × I ) • H(r) B(r)= μ 0 μ re H(r)+ ( Γ e × I ) • E(r) ) (1) where ε 0 and μ 0 are the permittivity and permeability of free space, respectively; ε re and μ re are real functions of the angular frequency ω; the Cartesian components of the magnetoelectric-gyrotropy vector Γ e are real; and I is the identity dyadic. Because the magnetoelectric term (Γ e × I ) is antisymmetric, the medium is classified as Lorentz nonrecip- rocal [14], [17]. Effectively homogeneous materials described by Eqs. (1) can potentially be fabricated by properly dispersing electrically small wires, loops, and other inclusions of different materials and shapes in some host material [19]–[22]. Also,