IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 12, DECEMBER 2011 3013 Field Propagation in Circular Hollow Waveguides With Non-Ideal Metallic Conductors From Microwaves to Terahertz Frequencies Carlos A. Leal-Sevillano, Student Member, IEEE, Jorge A. Ruiz-Cruz, José R. Montejo-Garai, and Jesús M. Rebollar Abstract—A general and rigorous formulation is proposed for the analysis of hollow metallic waveguides from the gigahertz to the terahertz band. The analysis is based on a hybrid mode formulation and the Drude model for the dielectric permittivity of metallic conductors. The obtained results for the circular waveguide are compared with the classical microwave approach (surface impedance approximation or Leontovich condition). The validity range of the surface impedance approximation in both the propagation constant and the electromagnetic field pattern is studied. As a consequence, a direct relation between the error in the propagation constant and the electromagnetic field configu- ration is shown. Moreover, this formulation shows the evolution in the field pattern: from modes at microwaves to the so-called Surface Plasmon Polariton at terahertz frequencies. Index Terms—Drude, hybrid mode, Leontovich, modal solution, surface impedance, Surface Plasmon Polariton (SPP), surface wave, Terahertz, waveguides. I. INTRODUCTION T HE frequency band centered in the terahertz spectral re- gion has focused the attention of many researchers in the last ten years for security, chemical identification and medical imaging applications [1]–[3]. Its use in radio astronomy is also well-known [4]. This spectral region between the microwave frequencies and lower infrared offers specific advantages in terms of resolution, penetration and classification. Therefore, the development of devices and techniques for using these specific features is very attractive. Waveguides for microwave through terahertz frequencies present an increasing interest [5]–[9], since they are the core of many circuits and subsystems. A full-wave characterization of waveguiding systems must be done in order to optimize the full performance of the structures made up of waveguide Manuscript received March 23, 2011; revised September 02, 2011; accepted September 18, 2011. Date of publication November 01, 2011; date of current version December 14, 2011. This work was supported in part by the Spanish government program TEC2010-17795, the CONSOLIDER CSD2008-00068, and a Ph.D. grant from Universidad Politécnica de Madrid. C. A. Leal-Sevillano, J. R. Montejo-Garai, and J. M. Rebollar are with the Departamento de Electromagnetismo y Teoría de Circuitos, ETSI Telecomu- nicación, Universidad Politécnica de Madrid, 28040 Madrid, Spain (e-mail: caleal@etc.upm.es; jr@etc.upm.es; jmrm@etc.upm.es). J. A. Ruiz-Cruz is with the Escuela Politécnica Superior, Universidad Autónoma de Madrid, 28049 Madrid, Spain (e-mail: Jorge.RuizCruz@uam.es). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2011.2170179 sections. Furthermore, an estimation of the error when the classical microwave approach used in the design of waveguide components at the terahertz band is imperative. Transmission media are made up of a combination of ma- terials: dielectrics and metallic conductors. These materials are usually homogeneous, linear, isotropic, non-dispersive and present a magnetic permeability equal to the vacuum perme- ability. The classical approximation used in microwave engineering for calculating the propagation constants of a waveguide with real conductors is based on the perturbation of the solution of the ideal waveguide (the same geometry but with perfect con- ductors) [10]. The ideal waveguide is solved first, obtaining the electromagnetic fields of the modes with their propagation con- stants . For taking into account the ohmic losses introduced by the real conductor, an attenuation constant is added to the ideal . An estimation of the value of is obtained applying the surface impedance [11], [12] or the Leontovich condition [13]. It is also possible to make a correction in the imaginary part of the propagation constant [10], although at microwave frequen- cies this correction may be neglected. Losses due to dielectric materials may be included by simply considering a complex di- electric permittivity in the analysis. Dielectric materials used as transmission media behave like good dielectrics up to frequencies at the visible region; see for example the step index fiber optic waveguides normally used for wide-band communications. At the near-infrared band, some di- electric materials show resonance frequencies and must be de- scribed by more elaborated models with time dispersion char- acteristics, like the Helmholtz-Drude model [14]. Metallic conductor materials normally used in communica- tions subsystems, such as gold, cooper or silver, behave as very good conductors at microwave and millimeter-wave bands. The conduction currents are much greater than the displacement currents and the conductors are well described by the constant value of conductivity . This is the underlying reason that allows to analyze waveguides with enough accuracy by means of the aforementioned perturbation method. However, when frequency increases to the terahertz band, metallic conduc- tors change their behaviors significantly. In fact, the constant value of used for characterizing the metallic conductors at microwave frequencies may not be accurate enough and more evolved models are needed for their correct characterizations [15], [16]. In the past, metallic conductor materials have been 0018-9480/$26.00 © 2011 IEEE