1820 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 47, NO. 12, DECEMBER 1999 Coordinate-Independent Dyadic Formulation of the Dispersion Relation for Bianisotropic Media Eng Leong Tan and Soon Yim Tan Abstract—This paper presents a coordinate-independent dyadic formulation of the dispersion relation for general bianisotropic media. The dispersion equation is expanded with the aid of dyadic operators including double-dot, double-cross and dot-cross or cross-dot products. From the dispersion relation, the Booker quartic equation is derived in a form well-suited for studying multilayered structures. Several deductions are made in con- junction with the bianisotropic media satisfying reciprocity and losslessness conditions. In particular, for reciprocal bianisotropic media, the dispersion equation is biquadratic in wave vector while for lossless bianisotropic media, all dispersion coefficients are of real values. As an application example, the dispersion equation for gyrotropic bianisotropic media is considered in detail. Index Terms—Bianisotropic media, dispersion relation. I. INTRODUCTION I N the study of plane wave propagation and interaction with various media, the dispersion relation plays a funda- mentally important role. Since the past few decades, many papers have been devoted to the derivation and utilization of the dispersion relations for several complex media. In [1], the dispersion relation for both electrically and magnetically anisotropic media has been developed along with the Booker quartic equation for the refractive-index vector component normal to a given plane. In [2] and [3], the dispersion relations for lossless positive bianisotropic media and general anisotropic media follow readily from the wave normal and ray surface equations derived using coordinate-independent dyadic formulation. In [4], the dispersion equation and Booker quartic equation obtained in coordinate-free forms have been applied to solve the problem of wave reflection from an anisotropic medium. A more comprehensive treatment on the coordinate-free approach to wave propagation and reflection from anisotropic media can be found in [5]. For arbitrary general bianisotropic media, a detailed derivation of the dis- persion relation has been carried out in Cartesian coordinates in [6]. Although the coefficients of the dispersion equation have been reduced largely by symmetry considerations, their remaining expressions are still fairly lengthy and cumbersome. These lengthy expressions tend to obscure the insights into the properties of the dispersion relation, for instance, when reciprocity and losslessness, which represent two important medium conditions, are under consideration [7]. Manuscript received July 21, 1998; revised September 28, 1999. The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, 639798 Republic of Singapore. Publisher Item Identifier S 0018-926X(99)09984-6. In this paper, based on the coordinate-independent dyadic formulation, we present the dispersion relation for plane wave propagation in general bianisotropic media. The dispersion equation is expanded as a full quartic equation utilizing various dyadic operations such as the double-dot and double- cross products defined in [8]. Moreover, the less commonly used dot-cross or cross-dot operators are also exploited and incorporated into the dispersion equation to extract the anti- symmetric vectors of dyadics. Some of the identities associated with these operators are listed in Appendix A. From the dispersion relation, the Booker quartic equation is derived in coordinate-free form as well. Several deductions are made in conjunction with the bianisotropic media satisfying reciprocity and losslessness conditions. To demonstrate the application of general dispersion equation, the gyrotropic bianisotropic media which comprise many recently proposed materials are considered in particular [9]–[14]. The explicit expressions of the dispersion coefficients are given in detail in Appendix B. II. FORMULATION A. Dispersion Equation A homogeneous linear bianisotropic medium can be char- acterized by the constitutive relations of the form [7] (1) (2) where and are, respectively, the permittivity and perme- ability dyadics, while and are the magneto-electric pseu- dodyadics. Substituting (1)–(2) into the source-free Maxwell equations and considering a plane wave with space-time dependence factor of , we find the dispersion relation from the conditions for nontrivial solutions of electromagnetic fields as (3) where and is the identity dyadic. The factor has been introduced to normalize the dispersion equation and these determinant terms also make the resulting equation applicable in the limit of singular or [6]. Equation (3) relates the wave vector and the angular frequency in the most compact form. In practical computation, it is sometimes more convenient to expand (3) into an algebraic equation in terms of or its components. For this purpose, one 0018–926X/99$10.00  1999 IEEE