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Laser Geometry

    Define a coordinate system attached to a laser. Let tex2html_wrap_inline7626 be the unit vector along the direction of propagation, and introduce a unit vector tex2html_wrap_inline7628 perpendicular to tex2html_wrap_inline7626 and another unit vector tex2html_wrap_inline7632. The three vectors (tex2html_wrap_inline9912) form an orthonormal frame. Define the components of these vectors in the original frame (tex2html_wrap_inline9914) as
Then, tex2html_wrap_inline9916 is a tex2html_wrap_inline9918 orthogonal matrix. Let tex2html_wrap_inline9920 be the spatial coordinate in this frame. Define the origin of tex2html_wrap_inline9922 as the laser focus and let tex2html_wrap_inline9924 be its coordinate in the original frame and tex2html_wrap_inline7708 the time when the laser pulse center passes the origin. Introduce a time coordinate tex2html_wrap_inline9624 whose origin is tex2html_wrap_inline7708. Now, the relation between (t,x,y,s) and tex2html_wrap_inline9934 is


A plane wave is written in the form, in the tex2html_wrap_inline9934 coordinate,
where tex2html_wrap_inline9938 is the wave number, tex2html_wrap_inline9940 is the unit vector along the propagation direction of the wave component, and tex2html_wrap_inline9942 is a complex vector perpendicular to tex2html_wrap_inline9940. A laser beam is considered to be a superposition of plane waves with slightly different tex2html_wrap_inline9940 and k. If the distribution of tex2html_wrap_inline9940 around tex2html_wrap_inline7626 and that of k are Gaussian and if one ignores the tex2html_wrap_inline9940 dependence of tex2html_wrap_inline9942, the laser field can be approximated by
where tex2html_wrap_inline9960 (i=1,2) is the Rayleigh length   and tex2html_wrap_inline7764 is the r.m.s. pulse length.

The wave front is given by the contour of tex2html_wrap_inline9966. If one defines tex2html_wrap_inline9968 by tex2html_wrap_inline9970, tex2html_wrap_inline9968 is nearly a unit vector and approximated by


In CAIN, when the relevant particle is at (t,x,y,s), or at tex2html_wrap_inline9934 in laser coordinate, the laser field is considered to be locally a plane wave with the power density tex2html_wrap_inline9978, wave number k, and the propagation direction tex2html_wrap_inline9982.

There is some problem on the polarization because eq.(60) does not exactly satisfy the Maxwell equation. For simplicity, the basis tex2html_wrap_inline9984 tex2html_wrap_inline9986 for polarization is defined in the following manner: tex2html_wrap_inline9988 is the unit vector along tex2html_wrap_inline9990 and tex2html_wrap_inline9992. (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, tex2html_wrap_inline9996, tex2html_wrap_inline9998) are transformed to the current Lorentz frame. Therefore, the Lorentz transformation is a little time-consuming.

next up previous contents index
Next: Linear Compton Scattering Up: Laser Previous: Laser

Toshiaki Tauchi
Thu Dec 3 17:27:26 JST 1998