Boundary conditions

Open boundary condition for electromagnetic waves

For the TE case, the original Berenger’s Perfectly Matched Layer (PML) writes

\[\begin{split}\begin{aligned} \varepsilon _{0}\frac{\partial E_{x}}{\partial t}+\sigma _{y}E_{x} = & \frac{\partial H_{z}}{\partial y}\label{PML_def_1} \\ \varepsilon _{0}\frac{\partial E_{y}}{\partial t}+\sigma _{x}E_{y} = & -\frac{\partial H_{z}}{\partial x}\label{PML_def_2} \\ \mu _{0}\frac{\partial H_{zx}}{\partial t}+\sigma ^{*}_{x}H_{zx} = & -\frac{\partial E_{y}}{\partial x}\label{PML_def_3} \\ \mu _{0}\frac{\partial H_{zy}}{\partial t}+\sigma ^{*}_{y}H_{zy} = & \frac{\partial E_{x}}{\partial y}\label{PML_def_4} \\ H_{z} = & H_{zx}+H_{zy}\label{PML_def_5}\end{aligned}\end{split}\]

This can be generalized to

\[\begin{split}\begin{aligned} \varepsilon _{0}\frac{\partial E_{x}}{\partial t}+\sigma _{y}E_{x} = & \frac{c_{y}}{c}\frac{\partial H_{z}}{\partial y}+\overline{\sigma }_{y}H_{z}\label{APML_def_1} \\ \varepsilon _{0}\frac{\partial E_{y}}{\partial t}+\sigma _{x}E_{y} = & -\frac{c_{x}}{c}\frac{\partial H_{z}}{\partial x}+\overline{\sigma }_{x}H_{z}\label{APML_def_2} \\ \mu _{0}\frac{\partial H_{zx}}{\partial t}+\sigma ^{*}_{x}H_{zx} = & -\frac{c^{*}_{x}}{c}\frac{\partial E_{y}}{\partial x}+\overline{\sigma }_{x}^{*}E_{y}\label{APML_def_3} \\ \mu _{0}\frac{\partial H_{zy}}{\partial t}+\sigma ^{*}_{y}H_{zy} = & \frac{c^{*}_{y}}{c}\frac{\partial E_{x}}{\partial y}+\overline{\sigma }_{y}^{*}E_{x}\label{APML_def_4} \\ H_{z} = & H_{zx}+H_{zy}\label{APML_def_5}\end{aligned}\end{split}\]

For \(c_{x}=c_{y}=c^{*}_{x}=c^{*}_{y}=c\) and \(\overline{\sigma }_{x}=\overline{\sigma }_{y}=\overline{\sigma }_{x}^{*}=\overline{\sigma }_{y}^{*}=0\), this system reduces to the Berenger PML medium, while adding the additional constraint \(\sigma _{x}=\sigma _{y}=\sigma _{x}^{*}=\sigma _{y}^{*}=0\) leads to the system of Maxwell equations in vacuum.

Propagation of a Plane Wave in an APML Medium

We consider a plane wave of magnitude (\(E_{0},H_{zx0},H_{zy0}\)) and pulsation \(\omega\) propagating in the APML medium with an angle \(\varphi\) relative to the x axis

\[\begin{split}\begin{aligned} E_{x} = & -E_{0}\sin \varphi e^{i\omega \left( t-\alpha x-\beta y\right) }\label{Plane_wave_APML_def_1} \\ E_{y} = & E_{0}\cos \varphi e^{i\omega \left( t-\alpha x-\beta y\right) }\label{Plane_wave_APML_def_2} \\ H_{zx} = & H_{zx0}e^{i\omega \left( t-\alpha x-\beta y\right) }\label{Plane_wave_AMPL_def_3} \\ H_{zy} = & H_{zy0}e^{i\omega \left( t-\alpha x-\beta y\right) }\label{Plane_wave_APML_def_4}\end{aligned}\end{split}\]

where \(\alpha\) and\(\beta\) are two complex constants to be determined.

Introducing ([Plane_wave_APML_def_1]), ([Plane_wave_APML_def_2]), ([Plane_wave_AMPL_def_3]) and ([Plane_wave_APML_def_4]) into ([APML_def_1]), ([APML_def_2]), ([APML_def_3]) and ([APML_def_4]) gives

\[\begin{split}\begin{aligned} \varepsilon _{0}E_{0}\sin \varphi -i\frac{\sigma _{y}}{\omega }E_{0}\sin \varphi = & \beta \frac{c_{y}}{c}\left( H_{zx0}+H_{zy0}\right) +i\frac{\overline{\sigma }_{y}}{\omega }\left( H_{zx0}+H_{zy0}\right) \label{Plane_wave_APML_1_1} \\ \varepsilon _{0}E_{0}\cos \varphi -i\frac{\sigma _{x}}{\omega }E_{0}\cos \varphi = & \alpha \frac{c_{x}}{c}\left( H_{zx0}+H_{zy0}\right) -i\frac{\overline{\sigma }_{x}}{\omega }\left( H_{zx0}+H_{zy0}\right) \label{Plane_wave_APML_1_2} \\ \mu _{0}H_{zx0}-i\frac{\sigma ^{*}_{x}}{\omega }H_{zx0} = & \alpha \frac{c^{*}_{x}}{c}E_{0}\cos \varphi -i\frac{\overline{\sigma }^{*}_{x}}{\omega }E_{0}\cos \varphi \label{Plane_wave_APML_1_3} \\ \mu _{0}H_{zy0}-i\frac{\sigma ^{*}_{y}}{\omega }H_{zy0} = & \beta \frac{c^{*}_{y}}{c}E_{0}\sin \varphi +i\frac{\overline{\sigma }^{*}_{y}}{\omega }E_{0}\sin \varphi \label{Plane_wave_APML_1_4}\end{aligned}\end{split}\]

Defining \(Z=E_{0}/\left( H_{zx0}+H_{zy0}\right)\) and using ([Plane_wave_APML_1_1]) and ([Plane_wave_APML_1_2]), we get

\[\begin{split}\begin{aligned} \beta = & \left[ Z\left( \varepsilon _{0}-i\frac{\sigma _{y}}{\omega }\right) \sin \varphi -i\frac{\overline{\sigma }_{y}}{\omega }\right] \frac{c}{c_{y}}\label{Plane_wave_APML_beta_of_g} \\ \alpha = & \left[ Z\left( \varepsilon _{0}-i\frac{\sigma _{x}}{\omega }\right) \cos \varphi +i\frac{\overline{\sigma }_{x}}{\omega }\right] \frac{c}{c_{x}}\label{Plane_wave_APML_alpha_of_g}\end{aligned}\end{split}\]

Adding \(H_{zx0}\) and \(H_{zy0}\) from ([Plane_wave_APML_1_3]) and ([Plane_wave_APML_1_4]) and substituting the expressions for \(\alpha\) and \(\beta\) from ([Plane_wave_APML_beta_of_g]) and ([Plane_wave_APML_alpha_of_g]) yields

\[\begin{split}\begin{aligned} \frac{1}{Z} = & \frac{Z\left( \varepsilon _{0}-i\frac{\sigma _{x}}{\omega }\right) \cos \varphi \frac{c^{*}_{x}}{c_{x}}+i\frac{\overline{\sigma }_{x}}{\omega }\frac{c^{*}_{x}}{c_{x}}-i\frac{\overline{\sigma }^{*}_{x}}{\omega }}{\mu _{0}-i\frac{\sigma ^{*}_{x}}{\omega }}\cos \varphi \nonumber \\ + & \frac{Z\left( \varepsilon _{0}-i\frac{\sigma _{y}}{\omega }\right) \sin \varphi \frac{c^{*}_{y}}{c_{y}}-i\frac{\overline{\sigma }_{y}}{\omega }\frac{c^{*}_{y}}{c_{y}}+i\frac{\overline{\sigma }^{*}_{y}}{\omega }}{\mu _{0}-i\frac{\sigma ^{*}_{y}}{\omega }}\sin \varphi\end{aligned}\end{split}\]

If \(c_{x}=c^{*}_{x}\), \(c_{y}=c^{*}_{y}\), \(\overline{\sigma }_{x}=\overline{\sigma }^{*}_{x}\), \(\overline{\sigma }_{y}=\overline{\sigma }^{*}_{y}\), \(\frac{\sigma _{x}}{\varepsilon _{0}}=\frac{\sigma ^{*}_{x}}{\mu _{0}}\) and \(\frac{\sigma _{y}}{\varepsilon _{0}}=\frac{\sigma ^{*}_{y}}{\mu _{0}}\) then

\[\begin{aligned} Z = & \pm \sqrt{\frac{\mu _{0}}{\varepsilon _{0}}}\label{APML_impedance}\end{aligned}\]

which is the impedance of vacuum. Hence, like the PML, given some restrictions on the parameters, the APML does not generate any reflection at any angle and any frequency. As for the PML, this property is not retained after discretization, as shown subsequently in this paper.

Calling \(\psi\) any component of the field and \(\psi _{0}\) its magnitude, we get from ([Plane_wave_APML_def_1]), ([Plane_wave_APML_beta_of_g]), ([Plane_wave_APML_alpha_of_g]) and ([APML_impedance]) that

\[\label{Plane_wave_absorption} \psi =\psi _{0}e^{i\omega \left( t\mp x\cos \varphi /c_{x}\mp y\sin \varphi /c_{y}\right) }e^{-\left( \pm \frac{\sigma _{x}\cos \varphi }{\varepsilon _{0}c_{x}}+\overline{\sigma }_{x}\frac{c}{c_{x}}\right) x}e^{-\left( \pm \frac{\sigma _{y}\sin \varphi }{\varepsilon _{0}c_{y}}+\overline{\sigma }_{y}\frac{c}{c_{y}}\right) y}\]

We assume that we have an APML layer of thickness \(\delta\) (measured along \(x\)) and that \(\sigma _{y}=\overline{\sigma }_{y}=0\) and \(c_{y}=c.\) Using ([Plane_wave_absorption]), we determine that the coefficient of reflection given by this layer is

\[\begin{split}\begin{aligned} R_{APML}\left( \theta \right) = & e^{-\left( \sigma _{x}\cos \varphi /\varepsilon _{0}c_{x}+\overline{\sigma }_{x}c/c_{x}\right) \delta }e^{-\left( \sigma _{x}\cos \varphi /\varepsilon _{0}c_{x}-\overline{\sigma }_{x}c/c_{x}\right) \delta }\nonumber \\ = & e^{-2\left( \sigma _{x}\cos \varphi /\varepsilon _{0}c_{x}\right) \delta }\end{aligned}\end{split}\]

which happens to be the same as the PML theoretical coefficient of reflection if we assume \(c_{x}=c\). Hence, it follows that for the purpose of wave absorption, the term \(\overline{\sigma }_{x}\) seems to be of no interest. However, although this conclusion is true at the infinitesimal limit, it does not hold for the discretized counterpart.

Discretization

\[\begin{split}\begin{aligned} \frac{E_x|^{n+1}_{j+1/2,k,l}-E_x|^{n}_{j+1/2,k,l}}{\Delta t} + \sigma_y \frac{E_x|^{n+1}_{j+1/2,k,l}+E_x|^{n}_{j+1/2,k,l}}{2} = & \frac{H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j+1/2,k-1/2,l}}{\Delta y} \\ % \frac{E_y|^{n+1}_{j,k+1/2,l}-E_y|^{n}_{j,k+1/2,l}}{\Delta t} + \sigma_x \frac{E_y|^{n+1}_{j,k+1/2,l}+E_y|^{n}_{j,k+1/2,l}}{2} = & - \frac{H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j-1/2,k+1/2,l}}{\Delta x} \\ % \frac{H_{zx}|^{n+3/2}_{j+1/2,k+1/2,l}-H_{zx}|^{n}_{j+1/2,k+1/2,l}}{\Delta t} + \sigma^*_x \frac{H_{zx}|^{n+3/2}_{j+1/2,k+1/2,l}+H_{zx}|^{n}_{j+1/2,k+1/2,l}}{2} = & - \frac{E_y|^{n+1}_{j+1,k+1/2,l}-E_y|^{n+1}_{j,k+1/2,l}}{\Delta x} \\ % \frac{H_{zy}|^{n+3/2}_{j+1/2,k+1/2,l}-H_{zy}|^{n}_{j+1/2,k+1/2,l}}{\Delta t} + \sigma^*_y \frac{H_{zy}|^{n+3/2}_{j+1/2,k+1/2,l}+H_{zy}|^{n}_{j+1/2,k+1/2,l}}{2} = & \frac{E_x|^{n+1}_{j+1/2,k+1,l}-E_x|^{n+1}_{j+1/2,k,l}}{\Delta y} \\ % H_z = & H_{zx}+H_{zy}\end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} E_x|^{n+1}_{j+1/2,k,l} = & \left(\frac{1-\sigma_y \Delta t/2}{1+\sigma_y \Delta t/2}\right) E_x|^{n}_{j+1/2,k,l} + \frac{\Delta t/\Delta y}{1+\sigma_y \Delta t/2} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j+1/2,k-1/2,l}\right) \\ % E_y|^{n+1}_{j,k+1/2,l} = & \left(\frac{1-\sigma_x \Delta t/2}{1+\sigma_x \Delta t/2}\right) E_y|^{n}_{j,k+1/2,l} - \frac{\Delta t/\Delta x}{1+\sigma_x \Delta t/2} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j-1/2,k+1/2,l}\right) \\ % H_{zx}|^{n+3/2}_{j+1/2,k+1/2,l} = & \left(\frac{1-\sigma^*_x \Delta t/2}{1+\sigma^*_x \Delta t/2}\right) H_{zx}|^{n}_{j+1/2,k+1/2,l} - \frac{\Delta t/\Delta x}{1+\sigma^*_x \Delta t/2} \left(E_y|^{n+1}_{j+1,k+1/2,l}-E_y|^{n+1}_{j,k+1/2,l}\right) \\ % H_{zy}|^{n+3/2}_{j+1/2,k+1/2,l} = & \left(\frac{1-\sigma^*_y \Delta t/2}{1+\sigma^*_y \Delta t/2}\right) H_{zy}|^{n}_{j+1/2,k+1/2,l} + \frac{\Delta t/\Delta y}{1+\sigma^*_y \Delta t/2} \left(E_x|^{n+1}_{j+1/2,k+1,l}-E_x|^{n+1}_{j+1/2,k,l}\right) \\ % H_z = & H_{zx}+H_{zy}\end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} E_x|^{n+1}_{j+1/2,k,l} = & e^{-\sigma_y\Delta t} E_x|^{n}_{j+1/2,k,l} + \frac{1-e^{-\sigma_y\Delta t}}{\sigma_y \Delta y} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j+1/2,k-1/2,l}\right) \\ % E_y|^{n+1}_{j,k+1/2,l} = & e^{-\sigma_x\Delta t} E_y|^{n}_{j,k+1/2,l} - \frac{1-e^{-\sigma_x\Delta t}}{\sigma_x \Delta x} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j-1/2,k+1/2,l}\right) \\ % H_{zx}|^{n+3/2}_{j+1/2,k+1/2,l} = & e^{-\sigma^*_x\Delta t} H_{zx}|^{n}_{j+1/2,k+1/2,l} - \frac{1-e^{-\sigma^*_x\Delta t}}{\sigma^*_x \Delta x} \left(E_y|^{n+1}_{j+1,k+1/2,l}-E_y|^{n+1}_{j,k+1/2,l}\right) \\ % H_{zy}|^{n+3/2}_{j+1/2,k+1/2,l} = & e^{-\sigma^*_y\Delta t} H_{zy}|^{n}_{j+1/2,k+1/2,l} + \frac{1-e^{-\sigma^*_y\Delta t}}{\sigma^*_y \Delta y} \left(E_x|^{n+1}_{j+1/2,k+1,l}-E_x|^{n+1}_{j+1/2,k,l}\right) \\ % H_z = & H_{zx}+H_{zy}\end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} E_x|^{n+1}_{j+1/2,k,l} = & e^{-\sigma_y\Delta t} E_x|^{n}_{j+1/2,k,l} + \frac{1-e^{-\sigma_y\Delta t}}{\sigma_y \Delta y}\frac{c_y}{c} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j+1/2,k-1/2,l}\right) \\ % E_y|^{n+1}_{j,k+1/2,l} = & e^{-\sigma_x\Delta t} E_y|^{n}_{j,k+1/2,l} - \frac{1-e^{-\sigma_x\Delta t}}{\sigma_x \Delta x}\frac{c_x}{c} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j-1/2,k+1/2,l}\right) \\ % H_{zx}|^{n+3/2}_{j+1/2,k+1/2,l} = & e^{-\sigma^*_x\Delta t} H_{zx}|^{n}_{j+1/2,k+1/2,l} - \frac{1-e^{-\sigma^*_x\Delta t}}{\sigma^*_x \Delta x}\frac{c^*_x}{c} \left(E_y|^{n+1}_{j+1,k+1/2,l}-E_y|^{n+1}_{j,k+1/2,l}\right) \\ % H_{zy}|^{n+3/2}_{j+1/2,k+1/2,l} = & e^{-\sigma^*_y\Delta t} H_{zy}|^{n}_{j+1/2,k+1/2,l} + \frac{1-e^{-\sigma^*_y\Delta t}}{\sigma^*_y \Delta y}\frac{c^*_y}{c} \left(E_x|^{n+1}_{j+1/2,k+1,l}-E_x|^{n+1}_{j+1/2,k,l}\right) \\ % H_z = & H_{zx}+H_{zy}\end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} c_x = & c e^{-\sigma_x\Delta t} \frac{\sigma_x \Delta x}{1-e^{-\sigma_x\Delta t}} \\ c_y = & c e^{-\sigma_y\Delta t} \frac{\sigma_y \Delta y}{1-e^{-\sigma_y\Delta t}} \\ c^*_x = & c e^{-\sigma^*_x\Delta t} \frac{\sigma^*_x \Delta x}{1-e^{-\sigma^*_x\Delta t}} \\ c^*_y = & c e^{-\sigma^*_y\Delta t} \frac{\sigma^*_y \Delta y}{1-e^{-\sigma^*_y\Delta t}}\end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} E_x|^{n+1}_{j+1/2,k,l} = & e^{-\sigma_y\Delta t} \left[ E_x|^{n}_{j+1/2,k,l} + \frac{\Delta t}{\Delta y} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j+1/2,k-1/2,l}\right) \right] \\ % E_y|^{n+1}_{j,k+1/2,l} = & e^{-\sigma_x\Delta t} \left[ E_y|^{n}_{j,k+1/2,l} - \frac{\Delta t}{\Delta x} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j-1/2,k+1/2,l}\right) \right] \\ % H_{zx}|^{n+3/2}_{j+1/2,k+1/2,l} = & e^{-\sigma^*_x\Delta t} \left[ H_{zx}|^{n}_{j+1/2,k+1/2,l} - \frac{\Delta t}{\Delta x} \left(E_y|^{n+1}_{j+1,k+1/2,l}-E_y|^{n+1}_{j,k+1/2,l}\right) \right] \\ % H_{zy}|^{n+3/2}_{j+1/2,k+1/2,l} = & e^{-\sigma^*_y\Delta t} \left[ H_{zy}|^{n}_{j+1/2,k+1/2,l} + \frac{\Delta t}{\Delta y} \left(E_x|^{n+1}_{j+1/2,k+1,l}-E_x|^{n+1}_{j+1/2,k,l}\right) \right] \\ % H_z = & H_{zx}+H_{zy}\end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} E_x|^{n+1}_{j+1/2,k,l} = & E_x|^{n}_{j+1/2,k,l} + \frac{\Delta t}{\Delta y} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j+1/2,k-1/2,l}\right) \\ % E_y|^{n+1}_{j,k+1/2,l} = & E_y|^{n}_{j,k+1/2,l} - \frac{\Delta t}{\Delta x} \left(H_z|^{n+1/2}_{j+1/2,k+1/2,l}-H_z|^{n+1/2}_{j-1/2,k+1/2,l}\right) \\ % H_{zx}|^{n+3/2}_{j+1/2,k+1/2,l} = & H_{zx}|^{n}_{j+1/2,k+1/2,l} - \frac{\Delta t}{\Delta x} \left(E_y|^{n+1}_{j+1,k+1/2,l}-E_y|^{n+1}_{j,k+1/2,l}\right) \\ % H_{zy}|^{n+3/2}_{j+1/2,k+1/2,l} = & H_{zy}|^{n}_{j+1/2,k+1/2,l} + \frac{\Delta t}{\Delta y} \left(E_x|^{n+1}_{j+1/2,k+1,l}-E_x|^{n+1}_{j+1/2,k,l}\right) \\ % H_z = & H_{zx}+H_{zy}\end{aligned}\end{split}\]