30
P. R. Willmott
bunch length of 100 µm) and increases n e by a factor of approximately 20–50. The
increased energy spread of the chirped electron beam after bunch compression must
be removed through ‘dechirping’, which can be achieved in one of several ways [18,
19].
1.5.2 The SASE Process
As the electrons begin to propagate down the undulator, they initially emit X-rays
independently and stochastically and are bathed in this radiation. They will interact
with this EM field. We begin our discussion of SASE by determining the magnitude
of these forces in conventional undulators at synchrotrons.
Classical electromagnetism tells us that the power transmitted per unit area by an
EM plane wave is given by
P
A
=
E 0 H 0
2
,
(1.37)
where A is the cross-section of the beam, E 0 is the amplitude of the electric field
component and H 0 is the amplitude of the magnetic intensity, which, in vacuum, is
related to the magnetic field strength amplitude B 0 by
H 0 =
B 0
μ 0
,
(1.38)
where μ 0 = 4π × 10
−7 m kg C
−2 is the permeability of free space (given here in SI
units). Also from classical electromagnetic theory, it emerges that
B 0 =
E 0
c
.
(1.39)
From (1.37) to (1.39), we thus obtain an areal power density
P
A
=
E
2
0
2μ 0 c
.
(1.40)
A typical synchrotron undulator may have a source size of A = 100 × 10 µm
2 , produce light bunches of 50-ps duration, each containing 5 × 10
6 1-Å photons (within
the full width of a harmonic). This equates to an areal power density of the order
of 1.6 × 10
11 W m
−2 . From (1.39) and (1.40), one calculates an electric field amplitude E 0 ≈ 10
7 V m
−1 and B 0 ≈ 40 mT. This latter value is nearly two orders of
magnitude smaller than that imposed by the undulator’s magnet array. We can thus
conclude that, in a conventional third- or fourth-generation synchrotron facility, the
P. R. Willmott
bunch length of 100 µm) and increases n e by a factor of approximately 20–50. The
increased energy spread of the chirped electron beam after bunch compression must
be removed through ‘dechirping’, which can be achieved in one of several ways [18,
19].
1.5.2 The SASE Process
As the electrons begin to propagate down the undulator, they initially emit X-rays
independently and stochastically and are bathed in this radiation. They will interact
with this EM field. We begin our discussion of SASE by determining the magnitude
of these forces in conventional undulators at synchrotrons.
Classical electromagnetism tells us that the power transmitted per unit area by an
EM plane wave is given by
P
A
=
E 0 H 0
2
,
(1.37)
where A is the cross-section of the beam, E 0 is the amplitude of the electric field
component and H 0 is the amplitude of the magnetic intensity, which, in vacuum, is
related to the magnetic field strength amplitude B 0 by
H 0 =
B 0
μ 0
,
(1.38)
where μ 0 = 4π × 10
−7 m kg C
−2 is the permeability of free space (given here in SI
units). Also from classical electromagnetic theory, it emerges that
B 0 =
E 0
c
.
(1.39)
From (1.37) to (1.39), we thus obtain an areal power density
P
A
=
E
2
0
2μ 0 c
.
(1.40)
A typical synchrotron undulator may have a source size of A = 100 × 10 µm
2 , produce light bunches of 50-ps duration, each containing 5 × 10
6 1-Å photons (within
the full width of a harmonic). This equates to an areal power density of the order
of 1.6 × 10
11 W m
−2 . From (1.39) and (1.40), one calculates an electric field amplitude E 0 ≈ 10
7 V m
−1 and B 0 ≈ 40 mT. This latter value is nearly two orders of
magnitude smaller than that imposed by the undulator’s magnet array. We can thus
conclude that, in a conventional third- or fourth-generation synchrotron facility, the