312 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 4, 2005 Including PML-Based Absorbing Boundary Conditions in the MLFMA Davy Pissoort, Dries Vande Ginste, and Frank Olyslager, Fellow, IEEE Abstract—In this letter, the multilevel fast multipole algorithm is extended to the use of complex coordinates. These complex coor- dinates appear, e.g., when extracting the -parameters of an elec- tromagnetic crystal device terminated by perfectly matched layer based absorbing boundary conditions with the multiple scattering technique. The coordinates of the centers of the boxes on the dif- ferent levels in the multilevel multipole algorithm are chosen so that they follow the path according to which the coordinates of the cylinders’ centers become complex. Therefore, a new real coordi- nate is introduced along this path. The boxes are first constructed based on this real coordinate and after that the corresponding com- plex coordinates are calculated. The described scheme is applied to the analysis of a multiplexer-demultiplexer device. Index Terms—Electromagnetic crystals, fast multipole methods, perfectly matched layers (PMLs), photonic crystals. I. INTRODUCTION R ECENTLY, a new perfectly matched layer (PML)-based absorbing boundary condition was introduced for the termination of periodic waveguides in integral equation like simulation techniques. In [1], this PML-based absorbing boundary condition was applied to the characterization of two-dimensional (2-D) electromagnetic crystal (EC) devices with the multiple scattering technique (MST) [2], [3]. Two-di- mensional (2-D) ECs consist of parallel homogeneous dielectric cylinders residing on a periodic lattice in a homogeneous back- ground medium. The underlying periodic structure of an EC gives rise to the appearance of frequency ranges (electromag- netic bandgaps) for which the electromagnetic fields cannot propagate inside the crystal. Disruption of the crystal period- icity by introducing crystal defects induces a field localization that allows the design of various interesting devices. Most often, the goal of the simulations is the knowledge of the -parameters of the EC device. Therefore, it is advantageous to be able to model EC waveguide appendages that are infinitely long. Exploiting the complex coordinate interpretation of a PML [4], [5], the EC waveguides that constitute the ports of the EC device are terminated by adding a couple for which the cylinders have complex coordinates. If the waveguide enters the complex plane in a smooth way, reflections caused by the periodicity dis- turbance are low and at the same time a significant absorption is achieved. The main disadvantage of an integral equation technique is that it requires the solution of a dense linear system of equa- Manuscript received April 21, 2005; revised June 23, 2005. The authors are with the Department of Information Technology (INTEC), Ghent University, B-9000 Gent, Belgium (e-mail: Davy.Pissoort@intec.ugent. be). Digital Object Identifier 10.1109/LAWP.2005.854000 Fig. 1. Transformation coordinate system. tions whose dimension is proportional to the number of cylin- ders in the EC device. Recently, there has been a considerable interest in the development of fast schemes to iteratively solve such systems [6]. A popular fast scheme is the multilevel fast multipole algorithm (MLFMA), which is the multilevel version of the fast multiple method (FMM). With this scheme, the cost per iteration scales nearly linearly with the number of cylinders. In this letter, the FMM is extended to the use of the PML-based absorbing boundary conditions. II. MULTIPLE SCATTERING TECHNIQUE This section details the MST for characterizing 2-D EC de- vices comprising identical, -invariant, homogeneous, di- electric/magnetic circular cylinders with radius and consti- tutive parameters embedded in a homogeneous back- ground medium with constitutive parameters (Fig. 1). Let , with a global position vector in the -plane, denote a known incident -polarized electric field generated by impressed sources in the absence of any cylin- ders. The difference between the total electric field , defined as the field observed in the presence of the cylinders, and the incident field is called the scattered field . To describe these electric fields, a set of equivalent directed elec- tric currents are introduced on the surface of every cylinder . Let denote the electric field generated by in an unbounded medium with pa- rameters . For dielectric/magnetic cylinders, appropriate 1536-1225/$20.00 © 2005 IEEE