The wave front is given by the contour of . If one
defines by ,
is nearly a unit vector and approximated by
In CAIN, when the relevant particle is at (t,x,y,s), or at in laser coordinate, the laser field is considered to be locally a plane wave with the power density , wave number k, and the propagation direction .
There is some problem on the polarization because eq.(60) does not exactly satisfy the Maxwell equation. For simplicity, the basis for polarization is defined in the following manner: is the unit vector along and . (This is irrelevant if only the longitudinal polarization is needed.)
The Lorentz transformation is a little complicated because eq.(60) is far from a covariant form. The particle coordinates and the external fields are transformed immediately when LORENTZ command is invoked and the transformation parameters are forgotten. In the case of lasers, the transformation is not done immediately but instead the transformation parameters are stored. When the laser is called at every time step for each particle, the particle coordinates are Lorentz transformed back to the frame where the laser was defined, and the calculated parameters (A, , ) are transformed to the current Lorentz frame. Therefore, the Lorentz transformation is a little time-consuming.