IEEE MICROWAVE AND GUIDED WAVE LETTERS, VOL. 7, NO. 11, NOVEMBER 1997 371 Systematic Derivation of Anisotropic PML Absorbing Media in Cylindrical and Spherical Coordinates F. L. Teixeira, Student Member, IEEE, and W. C. Chew, Fellow, IEEE Abstract— A simple and systematic derivation of anisotropic perfectly matched layers (PML’s) in cylindrical and spherical coordinates is presented. The derivation is based on the analytic continuation of Maxwell’s Equations to complex space. Through field transformations, results for Cartesian anisotropic PML media are recovered and, more importantly, a generalization of the anisotropic PML to cylindrical and spherical systems is obtained, providing further clarification on the PML concept. As expected, these new PML media are cylindrically and spherically layered, respectively. Index Terms—Absorbing boundary conditions, anisotropic me- dia, perfectly matched layer. I. INTRODUCTION T HE perfectly matched layer (PML) absorbing boundary condition, first derived for Cartesian coordinates and planar interfaces [1], was recently extended to cylindrical [2]–[5] and spherical coordinates [2], [3]. In [2] and [3] this was achieved through an analytic continuation of the frequency-domain Maxwell’s equations to complex space. As a result, the resultant fields inside the PML are not Maxwellian and the question naturally arises if it is possible to derive a Maxwellian anisotropic PML medium on cylindrical and spherical coordinates, as done for the Cartesian case [6], [7]. An anisotropic-medium formulation has the advantage of providing a physical basis for possible engineered artificial materials [8] and an easier interfacing with methods other than the finite-difference time-domain (FDTD), e.g., the finite- element method (FEM) [9], [10]. Here, a systematic analytical approach to derive the consti- tutive tensors for anisotropic PML formulations on Cartesian, cylindrical, and spherical coordinates from the complex space Maxwell’s Equations is developed. The relation between the anisotropic PML fields and the complex space PML fields on each of these systems is elucidated by presenting the pertinent mapping equations. From the constitutive tensors obtained, it explains why a previously proposed set of tensors for cylindrical and spherical Manuscript received June 18, 1997. This work was supported by the Air Force Office of Scientific Research under MURI Grant F49620-96-1-0025, the Office of Naval Research under Grant N00014-95-1-0872, the National Science Foundation under Grant NSF ECS93-02145, and by a CAPES Graduate Scholarship. The authors are with the Center for Computational Electromagnetics, Elec- tromagnetics Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2991 USA. Publisher Item Identifier S 1051-8207(97)08185-3. anisotropic PML media [10], [11] provides only an approxi- mately matched layer. It should be noted that, for the cylindrical case, an al- ternative derivation of anisotropic PML was carried out on [4] through a graphical construction. The constitutive tensors obtained there coincide with those of our analysis. II. FROM COMPLEX SPACE TO ANISOTROPIC PML In [2] and [3] it was shown how the analytic continuation of the frequency-domain Maxwell’s equations to complex space achieves the reflectionless absorption of the electromagnetic waves. This motivated the development of PML-FDTD al- gorithms in cylindrical and spherical grids [3]. In analogy to the Cartesian PML case [6], [7], the objective here is to derive a mapping of the non-Maxwellian fields of [3] to a set of Maxwellian fields on cylindrical and spherical anisotropic PML media and to determine the constitutive parameters of such media. To introduce this general approach on a simpler setting and for completeness, we first discuss it briefly for Cartesian coordinates. The -component of the Faraday equation on complex space [2] reads as ( convention) (1) where ( ) are the stretching variables [2], [12]. The fields in (1) do not satisfy Maxwell’s equations when (i.e., inside the PML), and to make this fact more explicit the superscript is added onto the field variables. However, if we multiply (1) by and using the fact that and commute when , we arrive at (2) If we then repeat the same procedure for the other components of the curl equations and introduce a new set of fields and , then this new set of fields obeys Maxwell’s equations on an anisotropic medium of constitutive parameters and , with (3) as obtained in [7]. This is the most general form for the con- stitutive tensors on the Cartesian anisotropic PML formulation 1051–8207/97$10.00  1997 IEEE