686
M. Dohlus et al.
The third equation assumes that only one spectral line (usually the fundamental
wavelength) is excited and is characterized by the amplitude ˆ
E x . This amplitude is
driven by the microbunching of the particle distribution with the same wavelength.
Therefore the Fourier coefficient of that wavelength, the bunching factor b, determines the increase of ˆ
E x . This approach of a resonant and slowly varying amplitude
(SVA) is used in most FEL programs (as in ALICE or GENESIS [11, 12]).
For a systematic presentation of FEL theory see [7, 13–16] and the literature
quoted therein.
11.1.2.1.1 Low-Gain FEL Theory
The low-gain approximation assumes that the electromagnetic wave amplitude does
not change during the passage of one electron bunch through the undulator. This
assumption may sound like a contradiction to the purpose of the FEL of amplifying
the intensity of a radiation field. The approximation makes it possible, however, to
estimate in a simple way the amount of amplification as a cumulated effect during
a single passage of the undulators. Therefore only the change of the ponderomotive
phase and of the particle energy are considered. This causes microbunching and a
net gain or loss of the particle energy. The microbunching can reach saturation if
the electromagnetic wave is strong enough or the undulator sufficiently long. The
net change of particle energy is an indirect method to calculate the change of field
energy by utilizing energy conservation. The gain function G = 0 is the ratio
of the intensity change to I 0 , the intensity of the electromagnetic wave assumed
for low-gain theory. Strong bunching is not in contradiction to low-gain operation if
G is small, but it requires presence of a very large initial intensity. This is normally
accumulated over many round trips of the radiation within an optical cavity, with
low-gain amplification at each round trip. The technical realization of such a FEL
“oscillator” is thus based on the existence of an optical cavity consisting of low-loss
mirrors.
The light wave co-propagating with the electron beam is taken as a plane wave
E x (z, t) = E 0 cos (k l z − ω l t + ψ 0 ) with wavelength λ l and wave number k l =
2π
λ l
.
The motion of a particle in a planar undulator is described by Eqs. (11.37) and
(11.40):
x (z(t)) ≈
K
γ k u
sin (k u z(t))
z(t) ≈ v z t −
K 2
8γ 2 k u
sin (2ω u t) .
The transverse velocity is
v x (t) ≈
Kv z
γ
cos (k u z(t)) ,
(11.63)
M. Dohlus et al.
The third equation assumes that only one spectral line (usually the fundamental
wavelength) is excited and is characterized by the amplitude ˆ
E x . This amplitude is
driven by the microbunching of the particle distribution with the same wavelength.
Therefore the Fourier coefficient of that wavelength, the bunching factor b, determines the increase of ˆ
E x . This approach of a resonant and slowly varying amplitude
(SVA) is used in most FEL programs (as in ALICE or GENESIS [11, 12]).
For a systematic presentation of FEL theory see [7, 13–16] and the literature
quoted therein.
11.1.2.1.1 Low-Gain FEL Theory
The low-gain approximation assumes that the electromagnetic wave amplitude does
not change during the passage of one electron bunch through the undulator. This
assumption may sound like a contradiction to the purpose of the FEL of amplifying
the intensity of a radiation field. The approximation makes it possible, however, to
estimate in a simple way the amount of amplification as a cumulated effect during
a single passage of the undulators. Therefore only the change of the ponderomotive
phase and of the particle energy are considered. This causes microbunching and a
net gain or loss of the particle energy. The microbunching can reach saturation if
the electromagnetic wave is strong enough or the undulator sufficiently long. The
net change of particle energy is an indirect method to calculate the change of field
energy by utilizing energy conservation. The gain function G = 0 is the ratio
of the intensity change to I 0 , the intensity of the electromagnetic wave assumed
for low-gain theory. Strong bunching is not in contradiction to low-gain operation if
G is small, but it requires presence of a very large initial intensity. This is normally
accumulated over many round trips of the radiation within an optical cavity, with
low-gain amplification at each round trip. The technical realization of such a FEL
“oscillator” is thus based on the existence of an optical cavity consisting of low-loss
mirrors.
The light wave co-propagating with the electron beam is taken as a plane wave
E x (z, t) = E 0 cos (k l z − ω l t + ψ 0 ) with wavelength λ l and wave number k l =
2π
λ l
.
The motion of a particle in a planar undulator is described by Eqs. (11.37) and
(11.40):
x (z(t)) ≈
K
γ k u
sin (k u z(t))
z(t) ≈ v z t −
K 2
8γ 2 k u
sin (2ω u t) .
The transverse velocity is
v x (t) ≈
Kv z
γ
cos (k u z(t)) ,
(11.63)