In the following, is an approximation in the frame where the initial electron and laser collide head-on and the electron is ultra-relativistic.

*p*,,*p*',*k*- 4-momenta of initial electron, laser photon, final electron and emitted photon, respectively.
- ,,,
- Energies of initial electron, laser photon, final electron and emitted photon, respectively.
- Laser energy parameter: .
*n*- Number of absorbed laser photon. The kinetic relation
holds exactly. Here,
*q*is defined as

(*p*is replaced by*p*' for*q*') and is called quasimomentum. *x*- , (0<
*x*<1) *v*-
*v*=*x*/(1-*x*),*x*=*v*/(1+*v*). () . - Maximum
*v*for given*n*: . - Maximum
*x*for given*n*: . - Laser helicity (-1 or +1)
- ,
- Initial and final electron helicities ()
- Final photon helicity
- ,
- `Detector helicity' of the final particles. See section 65 of [3].
- is the effective energy of initial electron in the laser field.
- Final photon angle.

- The argument of the Bessel functions in the following
expressions:

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 .

Thu Dec 3 17:27:26 JST 1998