In the following, is an approximation in the frame where the initial electron and laser collide head-on and the electron is ultra-relativistic.
Number of photons per unit time is
The terms involving and simultaneously
are ignored, i.e., the correlation of polarization between final particles
is ignored.
The ultra-relativistic approximation has been applied in
the terms related to electron helicity ( and/or ). (Note that the electron
helicity is a Lorentz invariant quantity only in the ultra-relativistic limit.)
The sum over the final electron and photon helicities gives
The functions are defined by
, , , are identical to , , , divided by in Tsai's paper[5], although the expressions in his paper look much more complicated.
Once x and n are given, the final momentuma are given, in any frame, by
Here, is the azimuthal angle in a head-on frame (therefore its
distribution is uniform in [0,]) and
and are given by
where is the completely anti-symmetric tensor
().
These vectors satisfy
The vector in eq.(139) is ill-defined when and
are colinear in the original frame. In such a case the spatial
part of is an arbitrary unit vector perpendicular to .