Page 109 - High Power Laser Handbook

78 G a s , C h e m i c a l , a n d F r e e - E l e c t r o n L a s e r s High-Power Fr ee-Electr on Lasers 79

In fact, if the process is permitted to continue, the electrons can travel
up the other side of the well and take energy back from the wave. A
net transfer of energy from the electrons to the wave can only happen
if the electrons are moving slightly faster than the ponderomotive
wave. In this case, the electrons “surf” the wave but with roles
reversed, as if the surfer were pushing the ocean wave rather than the
other way around. The speed of this wave is wavelength dependent.
The resonant wavelength (the one in which the electrons are traveling
at the same velocity) is given by

l
2
l = s w 1 ( + K ) (4.1)
g 2 2
where l is the radiated wavelength; l is the wiggler wavelength; and
s
w
K is the strength parameter of the wiggler field, given by K = 93.4B rms (T)
l (m), with B being the rms wiggler field (K is of order 1 and compen-

w
sates for the fact that the electron trajectories in the wiggler are not
exactly parallel to the axis). For example, if B rms = 0.2T and l = 0.05 m,
w
the resonant wavelength would be around 1.2 µm for an electron energy
of 100 MeV. The resonant wavelength turns out to be the one in which
the electrons slip backward exactly one optical wavelength for each
wiggler period. When this occurs, a net transfer of energy between the
electrons and the optical wave can occur, because the electrons’ direction
of transverse motion ends up always in the same direction as the trans-
verse field of the optical wave (qE·dl is always positive; see Fig. 4.2).
In practical terms, K is a measure of the strength on the wiggler
interaction and needs to be of order 1 to give reasonable gain.
It can be seen from a cursory examination of Eq. (4.1) that once
constructed with a fixed wavelength, the wiggler has a limited range
of control over the output wavelength either through the field strength
in K by means of a power supply (if the wiggler is electromagnetic) or
by changing the gap of a permanent magnet wiggler. The output
wavelength can also be controlled through the input electron beam
energy—hence, the statement that FELs can provide lasing at any
wavelength. There are, however, practical and physics performance
limitations to the operating range, which will become clearer in the
discussions that follow.
4.2.3 Gain and Bandwidth
The small signal gain of an FEL is given by
2
g = 31.8 (I/I )(N /g)Bη η η (4.2)
A
I f µ
2
2
2
where I = 17 kA, B = 4ξ[J (ξ) – J (ξ)] , and ξ = K /[2(1 + K )]. The last
0
A
1
three terms are degradations due to finite emittance, energy spread,
and optical electron beam overlap. Here I is the peak current, N is the
number of wiggler periods, and J is a Bessel function.