Measuring the Electron Beam Energy in a Magnetic Bunch ... - DESY

Bunch compressors are frequently designed to accommodate a range of energies
and compression schemes which are defined to first-order by parameters denoted R 56 and
R 16 after their location in a six-dimensional transfer matrix used to calculate the beam
transport [12]. In the introduction, analytic formulas for these parameters were derived
for a symmetric, single chicane. Physically, they relate the change in position to the
change in energy deviation, δ, according to
∂
R = z
56
∂δ
∂
R = x
16
(2.3.2)
∂δ
They are related to one another by
R
s
R16
( s')
= ∫ '
(2.3.3)
r(
s')
56
ds
0
where the integral is along the reference trajectory, s, and r(s’) is the bending radius of
the magnets. From Eq. 2.3.3, we can see that for smaller bending radii, the R 56 increases
and for drift spaces where r is infinite, the R 56 vanishes.
It is useful to be able to make a quick estimation of the energy chirp of an electron
bunch after an accelerator section and calculate the resulting bunch length change or x
position spread change. To do this, let us first write down the energy of an electron
subject to an accelerating module with an acceleration voltage of U and a phase of
φ=k rf Δs+φ 0 with k rf =2π/λ rf , in terms of the wavelength of the accelerating RF, and φ 0
equal to the phase for which the longitudinal position is equal to that of the reference
trajectory, Δs=0
δ = E − E eU cosϕ
(2.3.5)
f i
=
where E i is the initial energy of the particle and E f is the energy after the accelerating
module. We can describe the energy chirp produced by the accelerating RF by doing a
Taylor expansion of δ about a small longitudinal position change, Δs,
1
2
δ () s = δ ( Δs)
+ ( s − Δs)
δ '( Δs)
+ ( s − Δs)
δ ''( Δs)
… (2.3.6)
2
2
= R δ initial
+ R Δs
+ R Δ .
66 65 655
s
The first term describes the initial energy spread over the position change, the second
term describes the linear chirp acquired over the position spread and the third term
describes the quadratic chirp acquired. The indices of the R coefficients describe the
coordinates of the values in the beam transport matrix. The first index coordinate equal to
six corresponds to energy deviations and the second index coordinate equal to five
12