Scattering by a DNG Wedge Vito G. Daniele † Piergiorgio L.E. Uslenghi & † Dipartimento di Elettronica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy, e-mail: vito.daniele@polito.it, Website: http://www.eln.polito.it/staff/daniele/. & Department of Electrical and Computer Engineering, University of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois 60607- 7053, USA. E-mail: uslenghi@uic.edu Abstract – The scattering of a plane electromagnetic wave by a double-negative (DNG) metamaterial wedge of arbitrary aperture angle is considered, in the frequency domain. The boundary-value problem is solved by separation of variables, and the modal expansion coefficients are determined as residues of a suitable meromorphic function. 1 INTRODUCTION The penetrable wedge considered in this work is made of a lossless DNG metamaterial whose negative permittivity and negative permeability have values opposite to those in the surrounding space. It has been shown by Ziolkowski and Heyman [1] that, on the basis of causality, a DNG lossless medium has a negative index of refrection and a positive intrinsic impedance. Consequently, the propagation constant inside the DNG wedge is negative and the opposite of the propagation constant k in the medium outside the wedge, whereas the intrinsic impedance Z is the same in both media. A cross section of the wedge in a plane z = constant perpendicular to the edge is shown in Fig. 1. The primary plane wave is incident perpendicularly to the edge z of the wedge and propagates in a direction forming the angle with the negative x-axis, where may be restricted to the range < . In circular cylindrical coordinates, the faces of the wedge are = and = - , and the DNG material occupies the angular region – < < . Figure 1: Geometry of the problem 2 SOLUTION BY SEPARATION OF VARIABLES We consider explicitly the case of electric field parallel to the edge of the wedge (E-polarization). The solution for H-polarization is easily obtained by duality. For E- polarization, the total field may be written as: ˆ z E z = E , 0 z E E H ρ ϕ = = = , (1) z E j H Zk ρ ρ ϕ ∂ = ∂ , z E j H Zk ϕ ρ ∂ − = ∂ , (2) where 1 = for the angular region 2 α ϕ π α ≤ ≤ − and 2 = for the angular region α ϕ α − ≤ ≤+ ; in both regions 0 ρ ≤ <∞ . In the case of plane wave incidence, the field components z E and H ρ , that are both needed to impose the boundary conditions, may be written in the form: ( ) 2 ( ) sin cos j z l E J k e a b π ν ν ν ν ν ρ νϕ νϕ = + (3) ( ) 2 ( ) cos sin j l j H J k e a b Zk π ν ρ ν ν ν ν ν ρ νϕ νϕ ρ = − (4) where and a b ν ν are constants , 1 ϕ ϕ π = − , 2 ϕ ϕ = , and J ν is the Bessel function of order ν . In the following the primary electric field is the plane wave ( ) 1 cos o jk o Ee ρ ϕϕ − . By taking into account that 2 1 1 ( ) ( ) ( ) j J k J k e J k πν ν ν ν ρ ρ ρ = − = and imposing the boundary conditions, one finds the following homogeneous equation: 1-4244-0767-2/07/$20.00 ©2007 IEEE 68