3D Effects

The basic FEL process remains the same when three dimensional effects are included in the discussion, although the 1D FEL parameter might not be valid anymore. However an effective 3D FEL parameter can be found, which satisfies the gain length equation and the electron beam requirements.Eqs.~\ref{eqlg}, and \ref{eqbeamcond} -- \ref{eqsatpower}.

Two major effects contribute to the extended, three dimensional model of the FEL process, which are:

1. the spread in the transverse velocity of the electron beam and
2. the diffraction of the radiation field.

A measure for the transverse spread in the electron beam is the normalized emittance _n [22], which is the area in transverse phase space, covered by the electron beam distribution. It is an invariant in linear beam optics. If the beam is more focused the spread in the transverse momenta is increased. Focusing along the undulator is necessary to prevent the growth of the transverse electron beam size, reducing the electron density and decreasing the gain. Electrons, which drift away from the axis are deflected back to the undulator axis. As a result electrons performs an additional oscillation even slower than the "slow'' oscillation due to the periodic magnetic field of the undulator. Because part of the electron energy goes into this so-call betatron oscillation, the electron is slower in longitudinal direction than an electron without a betatron oscillation. The spread in the betatron oscillation, which scales with the normalized emittance, has a similar impact as an energy spread. Thus the requirement for the transverse emittance is

where the beta-function ([22]) is a measure for the beam size of the electron beam. Stronger focusing would increase the electron density ( becomes smaller) and consequently the parameter. But in the same time the impact of the emittance effect is enhanced, reducing the amount of electrons, which can stay in phase with the radiation field. It requires the optimization of the focusing strength to obtain the shortest possible gain length. A rough estimate for the optimum -function is ~ L_g. Using the constraint shows that the beam emittance =_n/ has to be smaller than the photon beam ''emittance'' /4. If this condition is fulfilled the electron beam does not diverge faster than the radiation field and all electrons stay within the radiation field.

The diffraction of the radiation field transports the phase information of the radiation field and the bunching of the electron beam in the transverse direction. This is essential to achieve transverse coherence of the radiation field, in particular if the FEL starts from the spontaneous radiation (SASE FEL). SASE FELs do not guarantee the transverse coherence as a seeding laser signal of an FEL amplifier would do. The betatron oscillation, which yield a change in the transverse position of the electron, contributes to the growth of transverse coherence as well.

As a degrading effect, radiation field escapes from the electron beam in the transverse direction. The field intensity at the location of the electron beam is reduced and the FEL amplification is inhibited. The compensation for field losses due to diffraction is the FEL amplification itself. After the lethargy regime the FEL achieves equilibrium between diffraction and amplification. The transverse radiation profile becomes constant and the amplitude grows exponentially. This ''quasi-focusing'' of the radiation beam is called ''gain guiding'' [23] and the constant profile of the radiation field is an eigenmode of the FEL amplification.

Similar to the eigenmodes of the free-space propagation of a radiation field (e.g. Gauss-Hermite modes), there are an infinite number of FEL-eigenmodes [24]. Each couples differently to the electron beam and, thus, have different growth rate or gain lengths. That mode, which has the largest growth rate, dominates after several gain lengths and the radiation field becomes transversely coherent.

At saturation gain guiding vanishes and the electron beams radiates into multiple modes. Typically the fundamental FEL-eigenmode is similar to that of free-space propagation and the resulting reduction in transverse coherence at saturation is negligible.

The characteristic measures for diffraction and FEL amplification are the Rayleigh length z_R and gain length L_g. To calculate z_R we approximate the radiation size at its waist by the transverse electron beam size as the effective size of the radiation source, resulting in z_R=k/2$. For z_R << Lg the FEL amplification is diffraction limited with a gain length significantly larger than in the 1D model. In the opposite limit (z_R >> Lg), the 1D model becomes valid.