s<sub>ei</sub> and s<sub>mi</sub> are the anisotropic components of the synthesized electric and magnetic conductivities in the CPML region. κ<sub>ei</sub> , κ<sub>mi</sub>, α<sub>ei</sub> and α<sub>mi</sub> are all assumed to be positive real and κ<sub>ei</sub>, κ<sub>mi</sub> ≥ 1. Similar equations hold for the Y and Z components of the electric and magnetic fields in the CPML layers. The requirement for zero reflection at PML-PML interfaces imposes the following condition:
:<math> s_{ei} = s_{mi} \quad \Rightarrow \quad \kappa_{ei} = \kappa_{mi} ,\quad \frac{\sigma_{ei}}{\varepsilon_0} = \frac{\sigma _{mi}}{\mu_0}, \quad \frac{\alpha_{ei}}{\varepsilon_0} = \frac{\alpha_{mi}}{\mu _0} </math><!--[[Image:FDTD99.png]]-->
The tilde notation above denotes the Fourier transform of the field components in the frequency domain. Transforming the above equations back to the time domain, one encounters convolution on the right hand side due to the frequency dependence of the stretched coordinate metrics:
:<math> \left(\varepsilon_x \frac{\partial E_x}{\partial t} + \sigma_{ex}E_x \right) = \breve{s}_{ey}(t) \frac{\partial H_z}{\partial y} - \breve{s}_{ez}(t)\frac{\partial H_y}{\partial z} </math>Â :<math> \left(\mu_x \frac{\partial H_x}{\partial t} + \sigma_{mx}H_x \right) = -\breve{s}_{my}(t)\frac{\partial E_z}{\partial y} + \breve{s}_{mz}(t)\frac{\partial E_y}{\partial z} </math><!--[[Image:FDTD98.png]]-->
where Å¡<sub>ei</sub>(t) and Å¡<sub>mi</sub>(t) denote functions of time that are indeed the inverse Laplace transform of s<sub>ei</sub><sup>-1</sup> and s<sub>mi</sub><sup>-1</sup>,respectively, given by: