diff options
Diffstat (limited to 'Docs/source/latex_theory/PML')
| -rw-r--r-- | Docs/source/latex_theory/PML/PML.tex | 24 |
1 files changed, 14 insertions, 10 deletions
diff --git a/Docs/source/latex_theory/PML/PML.tex b/Docs/source/latex_theory/PML/PML.tex index 6956b5182..7c5f09619 100644 --- a/Docs/source/latex_theory/PML/PML.tex +++ b/Docs/source/latex_theory/PML/PML.tex @@ -1,5 +1,9 @@ \input{newcommands} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Boundary conditions} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \subsection{Open boundary condition for electromagnetic waves} For the TE case, the original Berenger's Perfectly Matched Layer (PML) writes @@ -10,7 +14,7 @@ For the TE case, the original Berenger's Perfectly Matched Layer (PML) writes \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} +H_{z} = & H_{zx}+H_{zy}\label{PML_def_5} \end{eqnarray} This can be generalized to @@ -21,7 +25,7 @@ This can be generalized to \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} +H_{z} = & H_{zx}+H_{zy}\label{APML_def_5} \end{eqnarray} For $c_{x}=c_{y}=c^{*}_{x}=c^{*}_{y}=c$ and $\overline{\sigma }_{x}=\overline{\sigma }_{y}=\overline{\sigma }_{x}^{*}=\overline{\sigma }_{y}^{*}=0$, @@ -39,7 +43,7 @@ angle $\varphi$ relative to the x axis 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} +H_{zy} = & H_{zy0}e^{i\omega \left( t-\alpha x-\beta y\right) }\label{Plane_wave_APML_def_4} \end{eqnarray} @@ -55,7 +59,7 @@ and (\ref{APML_def_4}) gives \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} +\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{eqnarray} @@ -64,7 +68,7 @@ and (\ref{Plane_wave_APML_1_2}), we get \begin{eqnarray} \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} +\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{eqnarray} @@ -75,14 +79,14 @@ and (\ref{Plane_wave_APML_alpha_of_g}) yields \begin{eqnarray} \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 + + & \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{eqnarray} 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{eqnarray} -Z = & \pm \sqrt{\frac{\mu _{0}}{\varepsilon _{0}}}\label{APML_impedance} +Z = & \pm \sqrt{\frac{\mu _{0}}{\varepsilon _{0}}}\label{APML_impedance} \end{eqnarray} @@ -93,7 +97,7 @@ retained after discretization, as shown subsequently in this paper. Calling $\psi$ any component of the field and $\psi _{0}$ its magnitude, we get from (\ref{Plane_wave_APML_def_1}), (\ref{Plane_wave_APML_beta_of_g}), -(\ref{Plane_wave_APML_alpha_of_g}) and (\ref{APML_impedance}) that +(\ref{Plane_wave_APML_alpha_of_g}) and (\ref{APML_impedance}) that \begin{equation} \label{Plane_wave_absorption} @@ -104,7 +108,7 @@ its magnitude, we get from (\ref{Plane_wave_APML_def_1}), (\ref{Plane_wave_APML_ 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 (\ref{Plane_wave_absorption}), we determine -that the coefficient of reflection given by this layer is +that the coefficient of reflection given by this layer is \begin{eqnarray} 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 \\ @@ -187,7 +191,7 @@ H_z = & H_{zx}+H_{zy} 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}} +c^*_y = & c e^{-\sigma^*_y\Delta t} \frac{\sigma^*_y \Delta y}{1-e^{-\sigma^*_y\Delta t}} \end{align} \end{subequations} |