Download undulator field and electromagnetic field results in a beat wave

Figure 2. Illustration of FEL resonance
condition. Er is the electric field
component of the radiation field. The
radiation field slips one wavelength
as the electron traverses one
undulator period. The case for a
linear undulator is shown for which
the magnetic field is perpendicular
to the paper. The dashed arrow (Er)
indicates a reference phase.
undulator field and electromagnetic field
results in a beat wave, which is essentially an
interference pattern. It is the interaction of
the electrons with this beat wave that results
in bunching and thus gives rise to stimulated
emission in free-electron lasers.
The beat wave has the same frequency w as
the radiation but its wavenumber is the sum
of the wavenumbers of the undulator field,
ku=2p/lu, and the radiation field, k=2p/
l. Therefore it travels slower than the light
wave, hence it is called a ponderomotive wave.
This ponderomotive wave is the effective
field experienced by the electrons when they
travel though the undulator. Since it has a
phase velocity which is less than the speed of
light in vacuum, the electrons can be made
synchronous with this wave. Electrons which
move at the same speed as the phase velocity
of the ponderomotive wave are said to be in
resonance. For this condition, the following
relation holds
(1)
As wr = ckr, this relation can be rewritten as
(2)
where wu=vzku is the undulator frequency
and bz=vz/c is the normalised velocity of the
electrons in the propagation direction. Note,
that eq. 2 can also be derived from the Doppler
shift. First consider how the static undulator
field in the laboratory frame is viewed in the
rest frame of the electrons. This is a pseudoelectromagnetic wave with a frequency gzwu
propagating with a phase velocity vz towards the
electrons at rest. As a result of the interaction
with this field, the electrons in the rest frame
will undergo a transverse oscillation with the
same frequency and produce dipole radiation,
which is now a real electromagnetic wave. This
dipole radiation is propagating opposite to the
impinging pseudo-field from the undulator.
Therefore, an observer in the laboratory frame
sees this dipole radiation as a Doppler shifted
wave, the frequency of which is given by eq. 2.
Eq. 2 gives the fundamental tuning relation
of undulator based FELs and relates the
emitted radiation wavelength to the period
of the undulator and the axial velocity of
the electrons, which is determined by the
total electron energy and the strength of the
undulator field. It shows that, in principle, any
desired wavelength can be generated by an
FEL. However as the wavelength decreases,
the electron energy needs to be increased.
A different way of looking at the resonance
condition is the following (see fig. 2). If the
resonance condition is fulfilled, exactly one
wavelength of light will pass over an electron
as the electron traverses one undulator period.
In this case, the transverse velocity retains
it orientation with respect to the electric
field component of the radiation field over
many undulator periods. Therefore the work
done by the electron on the light wave will
have the same sign over a large number of
undulator periods, allowing for significant
energy exchange. Effectively this means that
the ponderomotive force is stationary with
respect to the electrons (i.e., moving at the
same velocity as the electrons).
In order to understand the bunching of
the electron beam, and hence the stimulated
emission, we consider the evolution of the
motion of an electron and its energy that is
given by the Newton-Lorentz equations
(3)
where e and m are the charge and mass
respectively of an electron, Er , Br are the electric
and magnetic components of the radiation field
respectively and Bu is the undulator magnetic
field. The equation for the momentum shows
that the electron trajectory is mostly due to the
undulator field (as |Bu| >> |Br|). The induced
transverse velocity allows the electron to couple
with the radiation field and exchange energy.
From eq. 3 it is clear that the v´B term in the
Newton force equation couples the radiation
induced velocity component to the undulator
magnetic field and vice versa. Also note that
both the radiation and undulator induced