Page 493 - Tunable Lasers Handbook

9 Tunable Free-Electron Lasers 453
2.3 Harmonic Operation

Operation of a FEL at an odd harmonic of the fundamental wavelength was
first proposed by Madey and Taber [32]. The full theory of harmonic lasing was
given by Colson in 1981 [33]. The gain at the harmonic can actually be higher
than that of the fundamental. If one is using this approach to lase at a short
wavelength without raising the energy of the accelerator. the wiggler parameter
K must be greater than unity for the harmonic gain to be higher than the gain at
the fundamental. The gain at the harmonic is much more sensitive to degradation
by the energy spread and emittance of the electron beam. as well as the wiggler
field quality, so in practice the harmonic gain is rarely higher than the gain at the
fundamental for most existing systems.
Experimental verification of third harmonic lasing was demonstrated in
1987 at Stanford [33], in 1988 at LANL [35], and in 1992 at Orsay [28]. Lasing
at harmonics higher than the third has not yet been demonstrated. Warren has
proposed that operation at very high harmonics may be a good way to operate a
compact FEL [36]. The analysis below is a summary of his approach. An
approximate gain formula for a FEL with a linearly polarized wiggler takes the
form
.
g =O-OO~QQN~~I~~I~~~~~ (3)

where I is the peak current, Np = N,K/yis the number of betatron periods in the
wiggler, Q is a factor that depends on the wiggler parameter and the harmonic
number h:

where the variable 5 is given by

5= K' (5)
2( 1 +K' ) '

qy is the gain degradation due to the energy spread,

q, is the gain degradation due to the rms emittance E (the emittance is a measure
of the transverse phase space area occupied by the electron beam distribution)