110 IEEE TRANSACTIONS ONELECTROMAGNETIC COMPATIBILITY, VOL. 54, NO. 1, FEBRUARY2012 Scattering Properties of Carbon Nanotube Arrays Apostolos I. Sotiropoulos, Ioannis-Gerasimos V. Plegas, Stavros Koulouridis, Member, IEEE, and Hristos T. Anastassiu, Senior Member, IEEE Abstract—Interaction of carbon nanotubes (CNTs) with electro- magnetic waves has not been extensively investigated, nor is well understood. In this paper, we examine the scattering behavior of finite CNTs arrays at terahertz frequencies. To this end, we employ a method of moment (MoM) analysis in conjunction with the thin wire approximation capable of analyzing complex geometries in a few minutes on a standard portable PC. MoM analysis and relevant numerical code is evaluated against published results and excellent agreement is observed. To fully understand scattering properties of CNT arrays, we keep the structures simple, yet valuable results are extracted. Arrays comprise parallel elements and are illuminated by a plane wave. Apart from scattering properties and patterns, current profiles are also quantified and plotted. This analysis and the results extracted are expected to prove useful in the future as a reference for electromagnetic shielding applications. Index Terms—Carbon nanotubes (CNTs), electromagnetic in- teraction, THz, arrays, method of moments (MoM). I. INTRODUCTION A LTHOUGH carbon nanotubes (CNTs) are fairly novel ma- terials, originally developed as recently as 1991 [1], they have already been applied to many areas of material science. However, electromagnetic compatibility (EMC) applications are still not sufficiently widespread. Despite the extensive study of their mechanical properties, their interaction with electromag- netic waves is not yet fully understood. Mathematical models have been developed and radiation capabilities have been inves- tigated in [2]–[5], where an integral equation formulation was invoked for the characterization of a single CNT finite dipole. Furthermore, potential EMC applications have been addressed in [6] and [7]. Also, various conclusions with respect to optical properties of CNT arrays have been drawn experimentally in [8]. Given the fact that CNTs can be artificially oriented so that they form arrays of parallel dipoles [8], transmission and scattering properties of such arrays are of particular interest. Indeed, large amount of such dipoles embedded in a surface, made out of a composite material, is expected to cause significant alteration of its original transmission and scattering properties. Hence, such Manuscript received April 30, 2011; revised August 30, 2011 and November 27, 2011; accepted December 3, 2011. Date of publication January 23, 2012; date of current version February 17, 2012. A. I. Sotiropoulos and I.-G. V. Plegas are with the Department of Electrical and Computer Engineering, University of Patras, Rio 26500, Greece, and also with Hellenic Air Force, 15561 Holargos, Greece (e-mail: apssotiro@upatras.gr; mplegas@upatras.gr). S. Koulouridis is with the Department of Electrical and Computer En- gineering, University of Patras, Rio 26500, Greece (e-mail: koulouridis@ ece.upatras.gr). H. T. Anastassiu is with the Department of Informatics and Communications, Technological and Educational Institute of Serres, Serres 62124, Greece (e-mail: hristosa@teiser.gr). Digital Object Identifier 10.1109/TEMC.2011.2179940 CNT arrays are potentially useful as EM shielding devices, if properly designed. In view of such promising future prospects, the scattering behavior of single CNTs was investigated in [9], whereas Hao and Hanson [10] presented an analogous analy- sis for infinite planar arrays. Moreover, in [11], a preliminary treatment of finite planar arrays was carried out. Dipole antennas formed by CNTs have extensively been an- alyzed. Yet, limited attempts have been reported so far to study the variation of the scattered field as a function of the elements’ separating distance and the array geometry. The purpose of this paper is the mathematical simulation of the scattering properties of CNT parallel element arrays, when they are illuminated by a plane wave, and the assessment of their potential shielding capabilities. Preliminary work has been presented in [11]. The dependence of the scattered field on the geometric layout is also discussed. To the best of the authors’ knowledge, this is the first time in the literature that this particular problem is investigated, which explains the general lack of reference results. Yet, for a few elementary cases, limited comparisons with reference data are provided, validating (even partially) the model used herein. The fundamental integral equation this analysis is based on is the Hallen/Pockligton-type equation, suitably modified for the CNT case. In contrast to perfectly conducting materials, usually involved in scattering problems, CNTs have finite conductance, whose numerical values can be extracted via a semiclassical, partly quantum-mechanical approach [2]. We assume that CNT dipoles are thin enough, so that the integral equation kernel is of the reduced type, which corresponds to a thin wire approx- imation. We discretize the equation via pulse basis functions and, subsequently, solve it through a standard point matching scheme (method of moments—MoM). Given a particular in- cident field (e.g., a plane wave), the pertinent linear system is solved, the currents on the tubes are computed, and finally, scat- tered fields can be obtained. A large number of parallel tubes (array elements) are employed in the array devices analyzed herein; however, the overall size is restricted by the computa- tional resources available. II. MATHEMATICAL BACKGROUND Assume an array of Q parallel, thin CNTs, each one of length 2L, lying along the z-axis of a convenient coordinate system (see Fig. 1). Also assume that the CNT radii are equal to a q , q = 1, ... , Q. Let the midpoint of the qth dipole lie at (x q 0 ,y q 0 ,z q 0 ). The overall geometry resembles an array of standard linear dipoles. The actual value of L lies typically in the order of micrometers, whereas a q is of the order of nanome- ters. Also, the CNT conductivity σ is finite and complex, and its values have been estimated in [2] as a function of frequency. 0018-9375/$31.00 © 2012 IEEE