Hp. Freund, NONLINEAR-THEORY OF SHORT-WAVELENGTH FREE-ELECTRON LASERS, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 52(5), 1995, pp. 5401-5415
The nonlinear evolution of free-electron laser (FEL) amplifiers is stu
died for infrared and shorter wavelengths. The configuration of intere
st consists in the propagation of an energetic electron beam through a
drift tube in the presence of a periodic wiggler magnetic field with
planar symmetry. A three-dimensional formulation is derived in which t
he electromagnetic field is represented as an expansion of Gaussian op
tical modes. Since the wiggler model is characterized by planar symmet
ry, the Gauss-Hermite modes are used for this purpose. A set of nonlin
ear differential equations is derived for the evolution of the amplitu
de and phase of each mode, and they are solved simultaneously in conju
nction with the three-dimensional Lorentz force equations for an ensem
ble of electrons in the presence of the magneto-static wiggler, self-e
lectric and self-magnetic fields due to the charge and current distrib
utions of the beam, and the electromagnetic fields. It is important to
note that no wiggler average is used in the integration of the electr
on trajectories. This permits the self-consistent modeling of effects
associated with (1) the injection of the beam into the wiggler, (2) em
ittance growth due to inhomogeneities in the wiggler and radiation fie
lds as well as due to the self-fields, (3) the effect of wiggler imper
fections, and (4) betatron oscillations. The optical guiding of the ra
diation field is implicitly included in the formulation. This approach
has important practical advantages in analyzing FELs, since it is nec
essary only to characterize the beam upon injection into the wiggler,
and the subsequent evolution is treated self-consistently. Numerical s
imulations are performed for two examples corresponding to an infrared
FEL at wavelengths near 3.5 mu m, and an x-ray FEL operating in the n
eighborhood of 1.4 Angstrom wavelengths corresponding to the proposed
linear coherent light source (LCLS) at the Stanford Linear Accelerator
Center. Results for both cases indicate that the more severe limiting
factor on the performance of the FEL is the beam emittance. For the i
nfrared example, the transition to the thermal regime occurs for an ax
ial energy spread of Delta gamma(z)/gamma(0) approximate to 0.19%, and
optimal performance is obtained for Delta gamma(z)/gamma(0)<0.1% and
gamma is the relativistic factor. This restriction is more severe for
the LCLS parameters, for which the thermal transition is found for Del
ta gamma(z)/gamma(0) approximate to 0.05% and optimal performance requ
ires Delta gamma(z)/gamma(0) less than or equal to 0.01%. Wiggler impe
rfections are found to be a much less important constraint on FEL desi
gn. Simulations indicate that there is no coherent ''walkoff'' of the
beam from the symmetry axis due to wiggler imperfections, and that the
radiation field is sufficiently guided by the interaction that no sev
ere degradation is found in the extraction efficiency or growth rate f
or moderate levels of wiggler fluctuations.