The CPML is formulated in the stretched coordinate space. The CPML layers are assumed to terminate the FDTD computational domain. The X components of Maxwell's frequency-domain curl equations can then be written in the following form:
[[Image:FDTD95.png]]<math> (j\omega\epsilon_x + \sigma_{ex})\tilde{E}_x = \frac{1}{s_{ey}} \frac{\partial \tilde{H}_z}{\partial y} - \frac{1}{s_{ez}} \frac{\partial \tilde{H}_y}{\partial z} </math>
where s:<submath>ei(j\omega\mu_x + \sigma_{mx}) \tilde{H}_x = -\frac{1}{s_{my}} \frac{\partial \tilde{E}_z}{\partial y} + \frac{1}{s_{mz}} \frac{\partial \tilde{E}_y}{\partial z} </submath> and s<sub!-- [[Image:FDTD95.png]] -->mi</sub> are the stretched coordinate metrics defined by
where s<sub>ei</sub> and s<sub>mi</sub> are the stretched coordinate metrics defined by:Â :<math> s_{ei} = \kappa_{ei} + \frac{\sigma_{ei}}{\alpha_{ei} + j\omega\varepsilon_0}, \quad i=x,y,z </math>Â :<math> s_{mi} = \kappa_{mi} + \frac{\sigma_{mi}}{\alpha_{mi} + j\omega\mu_0}, \quad i=x,y,z </math><!-- [[Image:FDTD96(2).png]]-->
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: