Using Beam Echo Effect for Generation of Short-Wavelength Radiation Page: 4 of 11
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laser wavelength, and one can locally consider a longitudinally uniform beam, neglecting
variation of the beam current over the distance of several laser wavelength. We assume
an initial Gaussian beam energy distribution with the variance UE and use the variable
p = (E-Eo)/oE for the dimensionless energy deviation of a particle. The initial distribution
function of the beam, normalized by unity, is f (p) = N(27)-1/2cp2/2, where N is the number
of particles per unit length of the beam.
After passage through the undulator, the beam energy is modulated with the ampli-
tude AE, so that the final dimensionless energy deviation p' is related to the initial one
p by the equation p' = p + A sin(Kz), where A = AE/E, K = Lo/c, and z is the longi-
tudinal coordinate in the beam. The distribution function after the interaction with the
laser becomes f(z, p) = (27r)-1/2 exp [-(p - A sin ()2/2] where we now use the dimensionless
variables ( = Kz. Sending then the beam through a dispersive system with the dispersive
strength R56, converts the longitudinal coordinate z into z', z' = z+R5p UE/Eo, and makes
the distribution function
f ((, p) = N exp (p - A sin(( - Bp)) , (1)
/2,r 2
where B = R5CKUE/Eo. Integration of f over p gives the distribution of the beam density
N as a function of the coordinate (,
N(() = No dpf ((, p) . (2)
Noting that this density is a periodic and even function of ( one can expand it into Fourier
series
N(() = 3
N 1 + b cos(k() , (3)
k= 1
where the coefficient bk is the amplitude of the harmonic k. Calculations with the function
(1) give an analytical expression for bk [4]
bk = 26-jB2k2 Jk(ABk) , (4)
where Jk is the Bessel function of order k.
It follows from this formula that if A < 1, then bk decays exponentially when k increases.
Indeed, for a given k, in order for the exponential factor in Eq. (4) not to suppress the
result, the value of B should be such that B < 1/k. Taking into account that the Bessel4
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Stupakov, G. Using Beam Echo Effect for Generation of Short-Wavelength Radiation, article, October 31, 2008; [Menlo Park, California]. (https://digital.library.unt.edu/ark:/67531/metadc894883/m1/4/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.