Download velocity components and the magnetic fields are in the transverse

velocity components and the magnetic fields
are in the transverse plane. Therefore, the
v´B term
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is directed along the axis of the
undulator (z-direction). It is this part of the
Newton force equation that is responsible for
the ponderomotive force. As the undulator
field varies as eikuz and the radiation field as
ei(kz-wt), we find that the v´B term varies as
ei((ku+k)z-wt). Hence it has a phase velocity that is
less than the speed of light in vacuum. Therefore
an electron beam can move synchronous with
this ponderomotive force. The corresponding
radiation field is said to be resonant with the
electron beam.
Let us now consider a low gain FEL, such that
the radiation wave is only a little bit amplified
for a single pass through the undulator. This
means that the ponderomotive force does not
change much when the beam and radiation
wave propagate through the undulator. Initially,
when the electron beam is injected into the
undulator, the electrons are homogeneously
distributed within the beam. Suppose a
resonant radiation wave is co-propagating with
the electrons. This means that electrons will
be at rest with respect to the ponderomotive
force, however they have arbitrary phases. This
is schematically shown in fig. 3a, which shows
one period of the potential that is associated
with the ponderomotive force. Thus, depending
on the relative phase, some electrons will be
accelerated by the ponderomotive force, while
others will be decelerated. The interaction
with the radiation and undulator field will
induce a coherent energy modulation on
the beam, which will evolve into a density
modulation upon further propagation (faster
moving electrons will overtake slower moving
electrons). As the ponderomotive force is
periodic with periodicity 2p/(k+ku)»l� (since
usually k>>ku), the density modulation will
take place on the scale of the wavelength of
the radiation field (see fig. 3b). Hence the
beam will become bunched with the bunches
approximately a wavelength apart. For a wave
at resonance, the bunches will be formed at the
nodes of the ponderomotive force (bottom of
the potential well). Because of the symmetry,
on average as many electrons are accelerated as
decelerated.
Thus no net energy is transferred between
the electrons and the radiation field and the
field is not amplified. So how can we amplify
a radiation wave with the electron beam? The
answer is by increasing the electron velocity
slightly above the resonant value (see fig. 3c),
i.e., by detuning the laser slightly. This means
that the electrons are initially not at rest with
respect to the ponderomotive potential when
they enter the undulator, they slip slowly
over the potential. Still the electrons feel
Figure 3. One period of the ponderomotive potential. At the undulator’s
entrance (z=0), the electrons are distributed uniformly over all phases relative to
ponderomotive potential and at rest when the electron beam energy g equals
the resonant energy g (a) or slightly slipping over the potential when detuned
(c). The interaction with the ponderomotive potential results in the formation
of bunches which are formed at the bottom of the while at resonance (b) or at a
decelerating phase if detuned (d).
the periodic ponderomotive force, and the
electrons are still accelerated and decelerated
depending on the relative phase. However, due
to the higher initial velocity of the electrons,
the bunch is not formed at the bottom of the
well but at a decelerating phase (fig. 3d). Now
all the electrons ‘fall’ towards the bottom of the
well, and energy is coherently transferred to
the radiation field, because the emission of the
individual electrons are in phase and interfere
constructively.
Note, that the initial energy g of the electrons
should not be to far away from the resonant
energy gr as only near resonance the electrons
will become trapped (i.e., are contained within
a single well of the ponderomotive potential)
and bunching will be near optimal. For too
high initial energies, the electrons will not be
trapped and this results in a poor bunching
and a weak interaction. Also note that at the
time the electrons reach the bottom of the
well, they should leave the undulator as at this
time maximum energy is transferred to the
Figure 4. Schematic view of the
bunching process, amplification of the
radiation wave and saturation.