This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1 Recursively Convolutional CFS-PML in 3-D Laguerre-FDTD Scheme for Arbitrary Media Guo-Qiang He , Johan H. Stiens, Wei Shao , Member, IEEE, and Bing-Zhong Wang, Senior Member, IEEE Abstract— Weighted Laguerre finite-difference time-domain (Laguerre-FDTD) method is an efficient and unconditionally stable time-domain numerical method. The complex frequency- shifted perfectly matched layer (CFS-PML) is the main absorb- ing boundary condition in the Laguerre-FDTD method. The implementations of CFS-PML in the Laguerre-FDTD method, which have been reported, include the use of auxiliary variables and auxiliary differential equations. These implementations make CFS-PML sensitive to its parameters and timescale parameter s of weighted Laguerre polynomials and result in potential instability and poor absorption issues in numerical simulations. In this paper, we proposed an efficient and stable implementation of CFS-PML based on recursive convolution in the Laguerre- FDTD method. The accuracy of the proposed implementation is theoretically validated. Its numerical dispersion is theoretically derived for choosing the key parameters of CFS-PML in the Laguerre-FDTD scheme. The numerical dispersion demonstrates that the timescale factor s needs to match the simulated signal frequency and the CFS-PML parameter α is chosen as 0.5sε 0 to obtain the minimum dispersive error. Numerical examples validate the theoretical predictions and verify that the proposed implementation of CFS-PML is robust and retains the advantages of CFS-PML in classical FDTD method, such as effectively absorbing evanescent as well as guided waves. It requires non- splits of electromagnetic field components, no auxiliary variables, and no modifications when applying it to inhomogeneous, lossy, and dispersive media. The CFS-PML formulas can directly be converted into computer codes of the Laguerre-FDTD method. Index Terms— Absorbing boundary conditions (ABCs), com- plex frequency-shifted perfectly matched layer (CFS-PML), finite-difference time domain (FDTD), Laguerre, recursive convolution. I. I NTRODUCTION T HE weighted Laguerre finite-difference time-domain (Laguerre-FDTD) method, like the classical FDTD method, solves both electric fields and magnetic fields using Manuscript received September 18, 2017; revised December 11, 2017; accepted January 6, 2018. This work was supported by the Vrije Universiteit Brussel through the SRP-Project M3D2 and the ETRO-IOF Project. The work of G.-Q. He was supported by the Vrije Universiteit Brussel. (Corresponding author: Guo-Qiang He.) G.-Q. He is with the Faculty of Engineering, Department of Electronics and Informatics ETRO-IR, Vrije Universiteit Brussel, 1050 Brussels, Belgium (e-mail: ghe@etrovub.be). J. H. Stiens is with the Faculty of Engineering, Department of Electronics and Informatics ETRO-IR, Vrije Universiteit Brussel, 1050 Brussels, Belgium, and also with the SSET Department, IMEC, B-3001 Leuven, Belgium (e-mail: jstiens@etrovub.be). W. Shao and B.-Z. Wang are with the Institute of Applied Physics, University of Electronic Science and Technology of China, Chengdu 610051, China (e-mail: weishao@uestc.edu.cn; bzwang@uestc.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2018.2801849 the coupled scalar equations of Faraday’s law and Ampere’s law and utilizing Yee space lattice and central-difference method to numerically discretize partial space derivatives [1]. Compared to the FDTD method, the difference is that the Laguerre-FDTD method uses the weighted Laguerre polyno- mials as temporal basis functions and the Galerkin method as the temporal testing procedure to deal with the time variable and partial time derivatives, and then forms a marching-on- in-order scheme [2]. The marching-on-in-order scheme of the Laguerre-FDTD method has been proven to be uncon- ditionally stable. The implicitly unconditional stability makes the Laguerre-FDTD method efficient in dealing with multi- scale structures and models involving extremely small time steps [3]–[6]. For open electromagnetic problems, absorbing boundary conditions (ABCs) are necessary to truncate the computa- tional domains. Many efforts have been done on investigating how the ABCs can be implemented in the Laguerre-FDTD scheme. Chung et al. [2] first introduced the uncondition- ally stable scheme for the FDTD method using weighted Laguerre polynomials and presented the Mur first-order ABC for the Laguerre-FDTD method. Later, a second-order ABC is introduced into the Laguerre-FDTD method [7]. The perfectly matched layer (PML) for the Laguerre-FDTD method is combined with the TF/SF boundary to solve a scattering problem [8]. Compared to the UPML in [9], the PML has better absorption [8]. The complex frequency- shifted (CFS) PML was recently introduced into the Laguerre- FDTD method [10], [11]. In these implementations of the CFS-PML, splitting electric and magnetic field components are used in the derivation of update formulas. For the dis- persive materials, the highest order of temporal derivative in auxiliary differential equation (ADE) would be more than two. Xi et al. [12] introduced another 2-D CFS-PML scheme using ADE for the Laguerre-FDTD method. However, the updating formulas of electromagnetic fields are complicated. The high- order CFS-PML in the Laguerre-FDTD method is introduced in [13]. All the implementations of CFS-PML in the Laguerre- FDTD scheme use ADE. The CFS-PML using ADE is referred to here as ADE-PML. These implementations of CFS-PML potentially have unstable risk and poor absorption issue, which will be illustrated with numerical examples later. In this paper, a convolutional CFS-PML (CPML) is pro- posed for the Laguerre-FDTD method. The implementation of the convolutional CFS-PML is based on a complex stretched coordinate PML formula and a recursive convolution [14]. The implementation of the CPML in the Laguerre-FDTD 0018-9480 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.