Well-Designed Termination Wall of Perfectly Matched Layers for ATS-FDTD Method

. This paper presents a well-designed termination wall for the perfectly matched layers (PML). This termination wall is derived from Mur’s absorbing boundary condition (ABC) with special difference schemes. Numerical experiments illustrate that PML and the termination wall works well with ATS-FDTD(Shi et al. 2015). With the help of termination wall, perfectly matched layers can be decreased to two layers only; meanwhile, the reflection error still reaches -60[dB] when complex waveguide is simulated by ATS-FDTD.


Introduction
It is well known that the perfectly matched layers (PML) [1][2][3][4] are wildly used for simulating Maxwell's equations in unbounded domain.The overall performance of PML depends on its all parameters, and the optimal parameters have been studied by many scientists [5][6][7][8][9].
To avoid causing much rebound at PEC, the traditional termination wall of PML, the PML are improved by some termination walls [8][9][10][11].In [8], a lossy version of the oneway Engquist Majda ABC was used for the termination of magnetic-field component of PML and the electric-field component of PML was terminated by an according transverse impedance with specific direction   , which lead to the inconvenience of this method.In [9][10][11], unified lossy version of Mur's second-order ABC and dispersive boundary condition were developed to be the PML termination for wave equation which could be derive from Maxwell equations.The thickness of PML is still a restriction; none of the above improved PML can reach optimal thickness, two layers, with good performance.
In this paper, a special single-layer Mur's ABC is designed as the PML termination for solving 2D Maxwell equations by ATS-FDTD [12] method.The staggered grids are used in this paper and the electromagnetic field components are placed on the boundary of each cell,    ,+1/2 ,    +1/2, and    , , which is different from the traditional Yee's grid [13].The magnetic-field component of PML is terminated by first-order Mur's ABC where the difference schemes will be designed very differently.This termination wall, PML-ABC, is very convenient and effective to be implemented with the ATS-FDTD method to solve Maxwell equation.

The PML-ABC and ATS-FDTD
The equations of PML [8]   . ( The boundaries of PML are given by Mur's ABC: which  = 1/ √  is the speed of light in vacuum.This paper uses the staggered grid technique: where ℎ and Δ are the size of grid.By ATS-FDTD method, the approximations can be expressed as where are the value at the former time level and  is positive integer and should be determined by The unknown coefficients    ,+1/2 ,    +1/2, ,    , ,    , , and    , are determined by recursion formulas ( 11)-( 21).
In vacuum, the unknown coefficients are obtained by where The coefficients of    +1/2, in PML are given by From the equation in (1), the schemes for coefficients of    , and    , in PML are written as The schemes for coefficients of   and   at the Mur's ABC are given by where two-order difference schemes are used for one-order Mur's ABC.

Numerical Results
To show the good performance of the PML-ABC with implementation of the ATS-FDTD, three examples are given in this section.The conductivity  in PML is defined as [1]  =   (   ) where (≤ ) is the distance of PML layer from the inside PML interface.The reflection error at observation point in the time domain is defined as where    denotes the numerical results and  .
denotes the reference results, free of reflection errors and computed with a large enough domain. and size of grid are calculated by (10).
Example I.In the first example, comparison between PML-ABC and the traditional PML, PML-PEC, is presented.The vacuum domain contains 101 × 101 cells and the timeharmonic source is located at the center of the vacuum.Observation point is located one cell diagonally away from a corner of the PML interface.
Figures 1-3 show the reflection errors with varying variables where  .. denotes the numerical results computed with the PML-ABC boundary and  .. denotes the numerical results computed with the PML-PEC boundary.Figure 1 illustrates the reflection error of PML-ABC with different thick .From Figure 1 we can find that the reflection error decline along with increasing . Figure 2 shows comparison between PML-ABC and PML-PEC with different   , where =12 for PML-PEC.From Figure 2 we can find that the reflection errors of PML-ABC are less than that of PML-PEC.The performance of PML-ABC is slightly affected by   , International Journal of Antennas and Propagation  while the performance of PML-PEC is distinctly affected by   .Figure 3 shows the reflection errors of PML-ABC are slightly affected by power .
Figures 1-3 illustrate the good performance of PML-ABC, reflection error around -60[dB], with only two PML layers.
Figures 4-7 show that the ATS-FDTD method works well with the PML-ABC.
Figure 4 shows the numerical magnetic-field component at time steps =32 where the outgoing wave does not cause an obvious echo from the PML-ABC.Figure 5 presents the error reflection around -60[dB] at point 1( 2 ,  2 ) and 2( 15 ,  15 ).Example III.In the third example, variables and coefficients are the same as those in Example II.Here, three timeharmonic sources are placed around ( 51 ,  51 ) with isogonal location and 10 cells away from center.With initial phases 0, 2/3, 4/3, these three sources circle around ( 51 ,  51 ) with angular speed of (/18)/.Figure 6 presents the numerical magnetic-field component at time steps  = 20, where the complex outgoing wave goes through the PML-ABC and does not cause an obvious echo.Figure 7 shows reflection errors around -60[dB] at points A1 and A2.
Example IV.In the fourth example, variables and coefficients are the same as those in Example II while the vacuum domain contains 150 × 150 cell.Here, time-harmonic source is placed in the center.A circle with conductivity  = 10√/ and 40 cells away from center is placed at ( 75 ,  35 ).And a circle with electric permittivity 10.25 and 40 cells away from center is placed at ( 35 ,  75 ).Figure 8 presents the numerical magnetic-field component at time steps  = 100, where the complex outgoing wave goes through the PML-ABC and does not cause an obvious echo.Figure 9 shows reflection errors around -60[dB] at points A3 ( 2 ,  75 ) and A4 ( 75 ,  2 ).

Conclusions
This paper presents a well-designed termination wall for PML on staggered grid where all electromagnetic field components are placed on the boundary of each cell.Numerical experimentations show that the PML-ABC works well with ATS-FDTD method to solve Maxwell equation in free-domain.
The PML-ABC can be decreased to two layers with reflection error -60[dB] when simulate complex waveguide.

Figure 5 :Figure 6 :
Figure 5: Reflection error as a function of time step for Example II.

Figure 7 :Figure 8 :
Figure 7: Reflection error as a function of time step for Example III.

Figure 9 :
Figure 9: Reflection error as a function of time step for Example IV.