Recursively Convolutional CFS-PML in 3-D Laguerre-FDTD Scheme for Arbitrary Media

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 absorbing 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 $\alpha $ is chosen as $0.5s\varepsilon _{0}$ 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 nonsplits 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.