2594 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 5, MAY 2012 A Simple and Direct Time Domain Derivation of the Dyadic Green’s Function for a Uniformly Moving Non-Dispersive Dielectric-Magnetic Medium Tatiana Danov and Timor Melamed Abstract—The present contribution is concerned with deriving the relativistic electric and magnetic time-dependent dyadic Green’s functions of an isotropic dielectric-magnetic medium (at the frame-at-rest) that is moving in a uniform relativistic velocity under the framework of the Minkowski constitutive relations. The results presented here correct previous reports in the literature [R. Compton, J. Math. Phys. 7, 2145 (1966)] and [C. Tai, J. Math. Phys. 8, 646 (1967)]. By applying a simple space-time and fields-sources transformation, scalarization of the vectorial problem is obtained. Thus the electromagnetic dyads are evaluated from the Green’s function of the scalar (time-dependent) wave equation in free space. Emphasis is placed on a simple and direct time domain derivation. Index Terms— ˇ Cerenkov radiation, dyadic Green’s function, moving medium, special relativity. I. INTRODUCTION AND STATEMENT OF THE PROBLEM In the late sixties, Compton [1] and Tai [2] have investigated the relativistic electric and magnetic time-dependent (TD) dyadic Green’s functions (GFs) of an isotropic dielectric-magnetic medium that is moving in a fast uniform velocity. In [1] the dyadic GFs were derived by applying a four-fold (space-time) Fourier transform to the vector and scalar potentials. The dyadic GFs were obtained by applying operators to a scalar GF for which closed form analytic solution was given. In [2] a temporal Fourier transform was applied for obtaining the necessary scalar GF for the anisotropic medium (in the laboratory frame). Similar results were obtained in [3] by applying retarded potentials. Analytic solutions for the time-harmonic dyadic GFs in a moving medium have been a subject of continuous research for the past decades [4]–[7], and in recent years [8]–[11]. In the present contribution, we derive the dyadic GFs directly from the TD Maxwell’s equations by using an elegant space-time and fields-sources normalization that transforms the vectorial problem into the standard procedure of obtaining the scalar isotropic (free-space) GF. Furthermore, the resulting GFs in the over phase-speed regime ( ˇ Cerenkov radiation) are different from the ones that were obtained in [1]–[3] [see discussion following (30)]. Our aim is to obtain the TD dyadic GFs for medium that is moving in a uniform translation velocity where we assume with no loss of generality, that the velocity is in the direction of the -axis. Unit vectors in the conventional cartesian coordinate system are denoted by hat over bold fonts. The medium is assumed to be a linear isotropic non-dispersive dielectric-magnetic in the comoving frame, with and denoting its relative permittivity and permeability with respect to vacuum, respectively. Manuscript received September 19, 2011; revised November 05, 2011; ac- cepted November 08, 2011. Date of publication March 02, 2012; date of current version May 01, 2012. The authors are with the Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel (e-mail: timo- rmel@ee.bgu.ac.il). Color versions of one or more of the figures in this communication are avail- able online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2012.2189745 Maxwell’s TD curl equations that are corresponding to the Minkowski constitutive relations are given by [1] (1) where is the speed of light in vacuum, (2) and is the diagonal matrix (3) Here and henceforth underlined boldfaced letters denote dyadics/ matrices. II. DYADIC GREEN’S FUNCTIONS A. General Formulation The TD electric and magnetic Green’s dyads, and , perform a mapping from the current density sources to the electric and magnetic vector fields by (4) In order to transform Maxwell’s equations in (1) to an isotropic form we define (5) By inserting in (5) into Maxwell’s equations in (1) we obtain (6) Note that the rotor operator in (6) is applied only on the first ( -) argu- ment of or . We distinguish two medium speed regimes in which the medium is moving in a speed that is either under or over the medium’s (at rest) phase-speed. These speed regimes are identified by either or for which in (3) is positive or negative, respectively. In order to obtain a uniform formulation for both the under and the over phase-speed regimes, we apply here an analytic continuation of the fields for the complex plane and define the normalized -coordinate (cf. [12], [13]) (7) The branch cut of in (7) is chosen such that (8) in order to ensure the validity of the following derivation for the two speed regimes . The analytic continuation of the real field 0018-926X/$31.00 © 2012 IEEE