FDTD-CPML

FDTD with CPML (Convolutional Perfectly Matched Layer)

Pre: Coefficients and The Set of PML

There are two types of form of coordinate stretched factor. The first is: $$s_w=\kappa_w + \frac{\sigma_w}{j\omega\epsilon_0},$$ and the other is: $$s_w=\kappa_w + \frac{\sigma_w}{a_w + j\omega\epsilon_0}.$$ Here, we adopt the second one.

$$\alpha = \frac{\sigma_w}{\epsilon_0 \kappa_w}$$

$$c_{w}=\frac{1}{\kappa_w}\left\{ exp\left[ -\frac{\sigma_w\Delta t}{\epsilon_0 \kappa_w} \right]-1 \right\}$$

$$\sigma_{max}=\frac{m+1}{\sqrt{\epsilon_r}150\pi\delta}$$

$$\sigma_z(z) = \frac{\sigma_{max}|z-z_0|^m}{d^m}$$

$k_{max}=5 ~--~ 11$, There is 8. $d$ is the width of the PML layer. $m$ is a integer, we set $m=4$.

$$\kappa_z(z) = 1 + (\kappa_{max} - 1)\frac{|z-z_0|^m}{d^m}$$

$$CA(m) = \frac{ 1-\frac{\sigma(m)\Delta t}{2\epsilon(m)} }{ 1+\frac{\sigma(m)\Delta t}{2\epsilon(m)} }$$

$$CB(m)=\frac{ \frac{\Delta t}{\epsilon(m)} }{ 1+\frac{\sigma(m)\Delta t}{2\epsilon(m)} }$$

$$CP(m) = \frac{ 1-\frac{\sigma_m(m)\Delta t}{2\mu(m)} }{ 1+\frac{\sigma_m(m)\Delta t}{2\mu(m)} }$$

$$CQ(m) = \frac{ \frac{\Delta t}{\mu(m)} }{ 1+\frac{\sigma_m(m)\Delta t}{2\mu(m)} }$$

if $\sigma=\sigma_m=0$, so,$CA(m)=1, CB(m)=\frac{\Delta t}{\epsilon}, CP(m)=1, CQ(m)=\frac{\Delta t}{\mu}$

$$\kappa = \sqrt{\omega^2 \mu\epsilon}$$

3D equations, continual form

$$\frac{\partial D}{\partial t} = \hat{x}\left( \frac{1}{\kappa_y}\frac{\partial H_z}{\partial y}-\frac{1}{\kappa_z}\frac{\partial H_y}{\partial z}+\zeta_y(t)*\frac{\partial H_z}{\partial y}-\zeta_z(t)*\frac{\partial H_y}{\partial z} \right)\\ + \hat{y}\left( \frac{1}{\kappa_z}\frac{\partial H_x}{\partial z}-\frac{1}{\kappa_x}\frac{\partial H_z}{\partial x}+\zeta_z(t)*\frac{\partial H_x}{\partial z}-\zeta_x(t)*\frac{\partial H_z}{\partial x} \right)\\ + \hat{z}\left( \frac{1}{\kappa_x}\frac{\partial H_y}{\partial x}-\frac{1}{\kappa_y}\frac{\partial H_x}{\partial y}+\zeta_x(t)*\frac{\partial H_y}{\partial x}-\zeta_y(t)*\frac{\partial H_x}{\partial y} \right)$$

$$-\frac{\partial B}{\partial t} = \hat{x}\left( \frac{1}{\kappa_y}\frac{\partial E_z}{\partial y}-\frac{1}{\kappa_z}\frac{\partial E_y}{\partial z}+\zeta_y(t)*\frac{\partial E_z}{\partial y}-\zeta_z(t)*\frac{\partial E_y}{\partial z} \right)\\ + \hat{y}\left( \frac{1}{\kappa_z}\frac{\partial E_x}{\partial z}-\frac{1}{\kappa_x}\frac{\partial E_z}{\partial x}+\zeta_z(t)*\frac{\partial E_x}{\partial z}-\zeta_x(t)*\frac{\partial E_z}{\partial x} \right)\\ + \hat{z}\left( \frac{1}{\kappa_x}\frac{\partial E_y}{\partial x}-\frac{1}{\kappa_y}\frac{\partial E_x}{\partial y}+\zeta_x(t)*\frac{\partial E_y}{\partial x}-\zeta_y(t)*\frac{\partial E_x}{\partial y} \right)$$

$$\psi_{H_{wv}}(n) = c_w \frac{\partial E_v(n)}{\partial w} + \exp(-\alpha\Delta t)\psi_{H_{wv}}(n-1)$$

$$\psi_{E_{wv}}(n) = c_w \frac{\partial H_v(n)}{\partial w} + \exp(-\alpha\Delta t)\psi_{E_{wv}}(n-1)$$

2D equations, continual form, TM($E_z, H_x, H_y$)

$$\frac{\partial}{\partial t}(\epsilon E_z)= \frac{1}{\kappa_x}\frac{\partial H_y}{\partial x}-\frac{1}{\kappa_y}\frac{\partial H_x}{\partial y}+\zeta_x(t)*\frac{\partial H_y}{\partial x}-\zeta_y(t)*\frac{\partial H_x}{\partial y} = \frac{1}{\kappa_x}\frac{\partial H_y}{\partial x}-\frac{1}{\kappa_y}\frac{\partial H_x}{\partial y}+\psi_{E_{xy}}-\psi_{E_{yx}}$$

$$-\frac{\partial}{\partial t}(\mu H_x)= \frac{1}{\kappa_y}\frac{\partial E_z}{\partial y}-\frac{1}{\kappa_z}\frac{\partial E_y}{\partial z}+\zeta_y(t)*\frac{\partial E_z}{\partial y}-\zeta_z(t)*\frac{\partial E_y}{\partial z} = \frac{1}{\kappa_y}\frac{\partial E_z}{\partial y}-\frac{1}{\kappa_z}\frac{\partial E_y}{\partial z}+\psi_{H_{yz}}-\psi_{H_{zy}}$$

$$-\frac{\partial}{\partial t}(\mu H_y)= \frac{1}{\kappa_z}\frac{\partial E_x}{\partial z}-\frac{1}{\kappa_x}\frac{\partial E_z}{\partial x}+\zeta_z(t)*\frac{\partial E_x}{\partial z}-\zeta_x(t)*\frac{\partial E_z}{\partial x} = \frac{1}{\kappa_z}\frac{\partial E_x}{\partial z}-\frac{1}{\kappa_x}\frac{\partial E_z}{\partial x}+\psi_{H_{zx}}-\psi_{H_{xz}}$$

2D equations, discrete form, TM($E_z, H_x, H_y$)

1. Normal FDTD

$$E_z^{n+1}(i,j) = E_z^n(i,j) +\frac{\Delta t}{\epsilon}\left[ \frac{H_y^{n+1/2}(i+\frac{1}{2},j)-H_y^{n+1/2}(i-\frac{1}{2},j)}{\kappa_x \Delta x}-\frac{H_x^{n+1/2}(i,j+\frac{1}{2})-H_x^{n+1/2}(i,j-\frac{1}{2})}{\kappa_y \Delta y} \right] +\frac{\Delta t}{\epsilon}\left[ \psi_{E_{xy}}^{n+\frac{1}{2}}\left(i,j\right)-\psi_{E_{yx}}^{n+\frac{1}{2}}\left(i,j\right) \right]$$

$$H_x^{n+1/2}\left( i, j+\frac{1}{2} \right) = H_x^{n-1/2}\left( i, j+\frac{1}{2} \right) -\frac{\Delta t}{\mu}\frac{E_z^n(i,j+1)-E_z^n(i,j)}{\kappa_y \Delta y} + \frac{\Delta t}{\mu}\left[ \psi^n_{H_{zy}}\left( i, j+\frac{1}{2} \right)-\psi^n_{H_{yz}}\left( i, j+\frac{1}{2} \right) \right]$$

$$H_y^{n+1/2}\left( i+\frac{1}{2},j \right) = H_y^{n-1/2}\left( i+\frac{1}{2},j \right) + \frac{\Delta t}{\mu} \frac{E_z^n(i+1,j)-E_z^n(i,j)}{\kappa_x \Delta x} + \frac{\Delta t}{\mu} \left[ \psi^n_{H_{xz}}\left( i+\frac{1}{2},j \right)-\psi^n_{H_{zx}}\left( i+\frac{1}{2},j \right) \right]$$

2. Convolutions

$$\psi_{E_{xy}}^{n+1/2}(i,j) = c_x \frac{H_y^{n+1/2}(i+1/2,j)-H_y^{n+1/2}(i-1/2,j)}{\Delta x} + \exp(-\alpha\Delta t)\psi_{E_{xy}}^{n-1/2}(i,j)$$

$$\psi_{E_{yx}}^{n+1/2}(i,j) = c_y \frac{H_x^{n+1/2}(i,j+1/2)-H_x^{n+1/2}(i,j-1/2)}{\Delta y} + \exp(-\alpha\Delta t)\psi_{E_{yx}}^{n-1/2}(i,j)$$

$$\psi_{H_{wv}}(n) = c_w \frac{\partial E_v(n)}{\partial w} + \exp(-\alpha\Delta t)\psi_{H_{wv}}(n-1)$$

$$\psi_{H_{zy}}^{n}\left( i,j+\frac{1}{2} \right) = 0$$

$$\psi_{H_{yz}}^n \left( i,j+\frac{1}{2}\right) = c_y\frac{E_z^n(i,j+1)-E_z^n(i,j)}{\Delta y} + \exp(-\alpha\Delta t)\psi_{H_{yz}}^{n-1} \left( i,j+\frac{1}{2}\right)$$

$$\psi_{H_{xz}}^n \left( i+\frac{1}{2},j\right) = c_x\frac{E_z^n(i+1,j)-E_z^n(i,j)}{\Delta x} + \exp(-\alpha\Delta t)\psi_{H_{xz}}^{n-1} \left( i,j+\frac{1}{2}\right)$$

$$\psi_{H_{zx}}^{n}\left( i+\frac{1}{2},j \right) = 0$$

2D Computation steps

1. $E\rightarrow H_{yz} ~\& ~ H_{xz}$
2. $E, H_{yz}, H_{xz} \rightarrow H$
3. $H\rightarrow E_{yx}, E_{xy}$
4. $H, E_{xy}, E{yx} \rightarrow E$

1D, TEM($E_x, H_y$)

$$E_{x}^{n+1}\left( k \right) = E_{x}^{n}\left( k \right)+\frac{\Delta t}{\epsilon}\left[ -\frac{H_y^{n+\frac{1}{2}}\left(k+\frac{1}{2}\right)-H_y^{n+\frac{1}{2}}\left(k-\frac{1}{2}\right)}{\kappa_z(z)\Delta z} \right] +\frac{\Delta t}{\epsilon}\left[ \psi_{E_{yz}}^{n+\frac{1}{2}}\left(k\right)-\psi_{E_{zy}}^{n+\frac{1}{2}}\left(k\right) \right]$$

$$H_y^{n+\frac{1}{2}}\left(k+\frac{1}{2} \right) = H_y^{n-\frac{1}{2}}\left( k+\frac{1}{2} \right) -\frac{\Delta t}{\mu}\left[ \frac{E_x^n\left(k+1\right) - E_x^n\left(k\right)}{\kappa_z \Delta z}\right] -\frac{\Delta t}{\mu}\left[ \psi_{H_{zx}}^n\left(k+\frac{1}{2}\right)-\psi^n_{H_{xz}}\left(k+\frac{1}{2}\right) \right]$$

$$\psi_{E_{yz}}^{n+\frac{1}{2}}\left( k \right)=0$$

$$\psi_{H_{xz}}^{n}\left( k+\frac{1}{2} \right)=0$$

$$\psi_{E_{zy}}^{n+\frac{1}{2}}\left( k \right)=\exp(-\alpha \Delta t)\psi_{E_{zy}}^{n-\frac{1}{2}}\left( k \right) +c_z(m)\frac{ H_y^{n+1/2}(k+1/2)-H_y^{n+1/2}(k-1/2) }{ \Delta z }$$

$$\psi_{H_{zx}}^n\left(k+\frac{1}{2}\right) = \exp(-\alpha\Delta t)\psi_{H_{zx}}^{n-1}\left( k+\frac{1}{2} \right) +c_z(m)\frac{ E_{x}^{n}(k+1)-E_{x}^{n}(k) }{ \Delta z } $$