PIERS ONLINE, VOL. 4, NO. 1, 2008 69 Lossless DNG-DPS Bilayer Structures for Tunneling and Zero Reflection Homayoon Oraizi and Majid Afsahi Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran Abstract— Distributions of the electromagnetic field and power flux inside and outside of a lossless DNG-DPS bilayer structure are investigated by the Transmission Line Transfer Matrix Method (TLTMM) and appropriate conditions are determined that under which complete wave tunneling occurs with no reflection at any angle of incidence and at all frequencies. This structure may have applications in antenna radomes. 1. INTRODUCTION Double negative (DNG) media are made of small metallic rods and split ring resonators in the microwave frequency bands [1–3]. It is shown that the DNG-DPS bilayer structures exhibit unusual and interesting wave propagation [4]. By an iterative method [5] and by a full wave analysis method [6], it just mentioned without details that under appropriate conditions the reflection from a lossless DNG-DPS bilayer structure becomes zero. Also, field and power flux distribution is determined inside and outside of a lossless MNG-ENG bilayer by a full wave method [6]. In this paper, which is complementary to [5, 6], field and power flux distribution is deter- mined inside and outside of a DNG-DPS bilayer under TM plane wave incidence by the TLTMM method [7, 8]. The dependence of reflection coefficient on angle of incidence and wave frequency is investigated. Furthermore, the necessary and sufficient conditions for zero reflection of loosless DNG-DPS bilayer structure are obtained. 2. PROBLEM FORMULATION Consider a lossless DNG-DPS bilayer, with thicknesses d 1 and d 2 . A TM plane wave is incident on it at an arbitrary angle of incidence in the y-z plane with the following electric and magnetic incident field: -→ H i =ˆ xH 0 e -(γoyy+γ0zz) -→ E i = (ˆ y γ 0z jωε 0 - ˆ z γ 0y jωε 0 )H 0 e -(γoyy+γ0zz) (1) as shown in Fig. 1. The transmission line transfer matrix method (TLTMM) is introduced in [7, 8] for the numerical analysis of multilayered structures with arbitrary number of layers at any angle of incidence and any frequency with an arbitrary wave polarization. In TLTMM the original problem shown in Fig. 1 is treated by the equivalent transmission line model shown in Fig. 2, characterized by the characteristic impedance (Z n ) and propagation constant (γ nz ) in the z -direction, which are functions of the angle of incidence, frequency and wave polarization, namely Z n = ‰p μ n /ε n cos θ n for TM p μ n /ε n sec θ n for TE γ nz = jω √ μ n ε n cos θ n (2) which θ n is the angle of incidence in the n’th medium. The Snell’s law at the boundary between two consecutive layers in terms of incidence angles θ i are: γ n sin θ n = γ n+1 sin θ n+1 , n =0, 1, 2 (3) It is necessary to select the correct sign for the characteristic impedances (Z = p μ/ε) and prop- agation constants (γ z = jω √ με cos θ) in metamaterials [9]. The necessary and sufficient conditions for tunneling may be obtained from the equivalent transmission lines as Z 1 = Z 2 , β 1 d 1 = -β 2 d 2 (4)