IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 2, MARCH 2002 733 Application of Anisotropic PML in Mode-Matching Analysis of Open-Ended Waveguides Zhongxiang Shen, Member, IEEE, Choi Look Law, Member, IEEE, and Robert H. MacPhie, Life Fellow, IEEE Abstract—The anisotropic perfectly matched layer (PML) is em- ployed in the mode-matching analysis of an open-ended rectan- gular waveguide radiating into a layered half space. In the analysis, the layered half space is approximated by a large rectangular wave- guide coated with anisotropic PMLs along its walls. Closed form expressions of the eigenvalues and the electromagnetic fields for both TE and TM modes in the anisotropic PML-filled rectangular waveguide are first derived and then used in the mode-matching analysis of the resultant waveguide-junction problem. Numerical results for the aperture admittance and reflection coefficient are presented and compared with available data in the literature. Good agreement is achieved. Index Terms—Anisotropic PML, mode-matching method, open- ended waveguide, waveguide junction. I. INTRODUCTION O PEN-ENDED waveguides have found wide applications in aeronautics, large phased array systems, thermography, diathermy, and the measurement of material properties [1]. Theoretical and experimental studies of a single open-ended waveguide have occupied the attention of numerous researchers for several decades. Reflection and radiating properties of open- ended rectangular waveguides were investigated by a number of researchers using different methods [2]–[6]. Among the many proposed methods [2]–[6], most of them are hybrid techniques involving complicated integrals. Mutual coupling between open-ended rectangular waveguides were also studied by Bird [7], [8]. It is noted that most of the published works are limited to the situation that a homogeneous half space was assumed. This paper applies the anisotropic perfectly matched layer (PML) [9], [10] to the analysis of an open-ended rectangular waveguide radiating into a layered half space. First, the layered half space is replaced by a large rectangular waveguide filled with anisotropic PMLs along its four walls. Then, the open- ended waveguide radiating problem reduces to that of a wave- guide junction, which is solved by the mode-matching method to obtain the reflection coefficient or the aperture admittance of the open-ended waveguide. It should be mentioned that the Manuscript received July 5, 2001; revised October 25, 2001. This work was supported in part by the National Science and Technology Board of Singapore under Project NSTB/43/11/4-3. Z. Shen is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: ezxshen@ntu.edu.sg). C. L. Law is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: ecllaw@ ntu.edu.sg). R. H. MacPhie is with the Department of Electrical and Computer Engi- neering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: r.macphie@ece.uwaterloo.ca). Publisher Item Identifier S 0018-9464(02)02778-4. Fig. 1. Cross-section of a rectangular waveguide filled with anisotropic PMLs along its wall. eigenvalues and modal functions in the PML-coated rectangular waveguide can be readily found in closed-form with the real length and width replaced by complex length and width. The layered dielectric medium can be easily dealt with in the present analysis by its equivalent transmission line. Numerical results for the aperture admittance agree well with available data in the literature. II. MODES IN A PML-COATED RECTANGULAR WAVEGUIDE The analysis method described in the paper begins with the replacement of the layered half space with a large rectangular waveguide with all the four side walls coated with anisotropic PMLs. Fig. 1 shows the cross section of a rectangular waveguide filled with anisotropic PMLs along its walls. The interior size of the rectangular waveguide is of with and being the thicknesses of the coating PMLs in the - and -directions, respectively. The constitutive relations for the coating material have the forms: and . Here, the expression of is as follows [9] for region II (1a) for region III (1b) 0018-9464/02$17.00 © 2002 IEEE