Laser Geometry

    Define a coordinate system attached to a laser. Let be the unit vector along the direction of propagation, and introduce a unit vector perpendicular to and another unit vector . The three vectors () form an orthonormal frame. Define the components of these vectors in the original frame () as

Then, is a orthogonal matrix. Let be the spatial coordinate in this frame. Define the origin of as the laser focus and let be its coordinate in the original frame and the time when the laser pulse center passes the origin. Introduce a time coordinate whose origin is . Now, the relation between (t,x,y,s) and is







A plane wave is written in the form, in the coordinate,

where is the wave number, is the unit vector along the propagation direction of the wave component, and is a complex vector perpendicular to . A laser beam is considered to be a superposition of plane waves with slightly different and k. If the distribution of around and that of k are Gaussian and if one ignores the dependence of , the laser field can be approximated by
 
where

where (i=1,2) is the Rayleigh length   and is the r.m.s. pulse length.

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.