11 Application of Accelerators and Storage Rings
685
11.1.2 Free-Electron Lasers
11.1.2.1 One Dimensional FEL Theory
The interaction of electrons with electromagnetic waves in an undulator of an FEL
is sketched in Fig. 11.8. Important aspects of the FEL process can be described in
a model, where electromagnetic fields do not depend on the transverse coordinates
x,y and the trajectories of particles with different transverse initial conditions are
just transversely shifted. This does not exclude transverse motion of particles.
For the description of the interaction of electrons with waves we need only three
types of state quantities. Two of them are particle coordinates in longitudinal phase
space: the ponderomotive phase ψ, see below and the relative energy offset η. A
third quantity ˆ
E x stands for the complex amplitude of the electric field of the plane
electromagnetic wave, and z, the length along the undulator axis, is the independent
coordinate. To simplify the FEL equations, we are not interested in oscillations with
the undulator period λ u , but in the variation of our quantities from period to period
or in average versus one period. Therefore the FEL equations for a bunch with N
particles (of index ν) per period are in principle of the following type
d
dz ψ ν = f ψ
ψ 1 · · · ψ N , η 1 · · · η N , ˆ
E x , z
d
dz η ν = f η
ψ 1 · · · ψ N , η 1 · · · η N , ˆ
E x , z
d
dz
ˆ
E x = f E
ψ 1 · · · ψ N , η 1 · · · η N , ˆ
E x , z
.
(11.61)
If the undulator parameters are z independent, and with some approximations
they can be written as
d
dz ψ ν ∼ η ν
d
dz η ν ∼ Re
ˆ
E x exp (iψ ν )
d
dz
ˆ
E x ∼ b = N −1
exp (−iψ ν ) .
(11.62)
The first equation relates the relative position in the bunch (ponderomotive phase)
to the energy offset. This is just a linearized version of the equation of motion,
that describes that a particle with more energy (higher η) is deflected less by the
undulator, performs oscillations with smaller amplitude, has a shorter trajectory
through one period and increases its phase (longitudinal position) relative to a
particle without deviation from reference energy.
The second equation represents the change of particle energy caused by the
electromagnetic wave. This depends on the phase of the particle ψ ν relative to the
phase of the wave arg
ˆ
E x
and on the absolute amplitude
ˆ
E x
. This clarifies the
definition of the ponderomotive phase: ψ ν is ν 2π/λ l with ν the length shift
relative to a reference particle of reference energy (η ref = 0), and wavelength λ l
into forward direction according to Eq. (11.42).
685
11.1.2 Free-Electron Lasers
11.1.2.1 One Dimensional FEL Theory
The interaction of electrons with electromagnetic waves in an undulator of an FEL
is sketched in Fig. 11.8. Important aspects of the FEL process can be described in
a model, where electromagnetic fields do not depend on the transverse coordinates
x,y and the trajectories of particles with different transverse initial conditions are
just transversely shifted. This does not exclude transverse motion of particles.
For the description of the interaction of electrons with waves we need only three
types of state quantities. Two of them are particle coordinates in longitudinal phase
space: the ponderomotive phase ψ, see below and the relative energy offset η. A
third quantity ˆ
E x stands for the complex amplitude of the electric field of the plane
electromagnetic wave, and z, the length along the undulator axis, is the independent
coordinate. To simplify the FEL equations, we are not interested in oscillations with
the undulator period λ u , but in the variation of our quantities from period to period
or in average versus one period. Therefore the FEL equations for a bunch with N
particles (of index ν) per period are in principle of the following type
d
dz ψ ν = f ψ
ψ 1 · · · ψ N , η 1 · · · η N , ˆ
E x , z
d
dz η ν = f η
ψ 1 · · · ψ N , η 1 · · · η N , ˆ
E x , z
d
dz
ˆ
E x = f E
ψ 1 · · · ψ N , η 1 · · · η N , ˆ
E x , z
.
(11.61)
If the undulator parameters are z independent, and with some approximations
they can be written as
d
dz ψ ν ∼ η ν
d
dz η ν ∼ Re
ˆ
E x exp (iψ ν )
d
dz
ˆ
E x ∼ b = N −1
exp (−iψ ν ) .
(11.62)
The first equation relates the relative position in the bunch (ponderomotive phase)
to the energy offset. This is just a linearized version of the equation of motion,
that describes that a particle with more energy (higher η) is deflected less by the
undulator, performs oscillations with smaller amplitude, has a shorter trajectory
through one period and increases its phase (longitudinal position) relative to a
particle without deviation from reference energy.
The second equation represents the change of particle energy caused by the
electromagnetic wave. This depends on the phase of the particle ψ ν relative to the
phase of the wave arg
ˆ
E x
and on the absolute amplitude
ˆ
E x
. This clarifies the
definition of the ponderomotive phase: ψ ν is ν 2π/λ l with ν the length shift
relative to a reference particle of reference energy (η ref = 0), and wavelength λ l
into forward direction according to Eq. (11.42).