Precision Measurement of the Undulator K Parameter using Spontaneous Radiation Page: 2 of 4
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more convenient comparison of the slopes at AO/w 0.
The LCLS undulator consists of 33 almost identical seg-
ments, each 3.4 m long. Each segment is provided with
"roll-away" capability; and can be independently displaced
up to 8 cm horizontally, effectively turning it 'off'. The
segments are also constructed with a 4.5 mrad cant angle
of the poles, which allows K-value adjustment by small
horizontal displacements (about 1.5 mm per 0.1%), us-
ing the same "roll-away" mechanism. If the slope of the
high-energy edge of the spectrum is measured with suffi-
cient precision as a function of the horizontal displacement,
(equivalent to scanning AK/K), two undulator K values
can be set equal within the required precision (0.015%),
and this relative correction might be applied repetitively
over the full 130-m undulator to adjacent, or nearly adja-
cent, segment pairs.
METHOD
The method proposed here requires retracting all but two
adjacent, or nearly adjacent, undulator segments from the
beamline so that all x-rays detected come only from the
segments under test. The electron trajectory must then
be brought to essentially beam-based alignment quality, so
that the kinks in the trajectory between segments are less
than iprad. Beam-based alignment is done by mechani-
cally moving the quadrupoles to obtain a dispersion-free
trajectory. The quadrupoles are mechanically tied to the
undulator segments and both move together, so this step
also insures that the undulator segments are brought verti-
cally to within about 100 pm of the ideal position before
starting.
On each machine pulse, a small portion of the x-ray
spectrum is sampled in the region of the high-energy edge
of the first harmonic, using a silicon crystal spectrome-
ter, set for diffraction at a fixed Bragg angle from the
(111) crystal planes. (The LCLS electron beam-angle jitter
should be < iprad, which is small compared to the Darwin
width of the crystal reflection.) As a result of the natural
electron energy jitter (~ 0.1% rms), the photon spectrum is
randomly sampled. The electron energy jitter is measured
on each pulse (see below) and the inferred photon energy
shift is then associated with the detector data; the underly-
ing spectrum is then reconstructed by plotting the detector
data against the inferred photon spectrum shift.1
About 100 pulses will be needed to reconstruct a spec-
trum. After a spectrum is collected in this manner for a
given arrangement of two adjacent undulator segments, the
K value of the second undulator is changed by 0.05% by
translating it Ax 0.75 mm, and then a new spectrum is
obtained. This process is repeated for 9 separate K values,
ranging over about 0.2%.
The electron energy jitter is precisely measured by two
1Electron energy loss from radiation is 0.005% per segment and
will be taken into account in setting the appropriate K values. Wakefield
losses are expected to be even less. Both types of energy losses are ignored
in the following discussion.beam position monitors (BPMs) located upstream of the
undulator, at points of high horizontal momentum disper-
sion. The BPMs are separated in betatron phase advance
by 2wr and have opposite sign dispersion, such that the dif-
ference in their position readback values is proportional to
the relative electron energy variation and completely insen-
sitive to incoming betatron oscillations. With dispersion of
125 mm at each BPM, and a 5-pm rms single-pulse po-
sition resolution, the relative electron energy resolution is
(5 pm)/(125 mm)/ 2 3 x 10-5, and the corresponding
photon energy resolution is twice this, or 6 x 10-5.
Since the spectrum shifts towards AO/w > 0 for
AK/K < 0 (see Fig. 3), the data tends to be poorly
centered on the spectrum edge for AK/K $ 0. To im-
prove resolution, we adjust the mean electron energy by
-K (AK/K) for each new setting so that the energy
always varies around the center of the edge. These small
adjustments are possible using the BPM-based feedback
loop, which maintains the desired average electron energy,
but cannot remove the random pulse-to-pulse jitter.
The slope of each high-energy spectrum edge is found
by fitting the data for each K value. The Ax at which
the slope is steepest corresponds to equal K values in the
two segments. At a 10 Hz machine rate, this process will
require 90 seconds, plus the time required to translate the
undulators nine times, for a total of about 4 minutes per un-
dulator pair. A description of a simulation of this process,
including realistic errors, follows.
SIMULATION
A simulation is performed using a computer-generated,
two-undulator, spectrum integrated over all angles, at
nine values of AK/K: (-0.2% to +0.2% in steps of
0.05%). To simulate measured data, the perfect, computer-
generated spectrum is sampled at random values of twice
the electron energy error (Ao/w 2AE/E). The elec-
tron energy varies randomly in a Gaussian distribution with
0.1% rms. In practice, either the average electron beam en-
ergy or the Bragg angle can be adjusted to best center the
data on the high-energy edge of the spectrum.
A cubic spline is used to interpolate the computer-
generated spectrum for each randomly selected energy. An
error of 6 x 10-5 rms is added to the photon energy to ac-
count for the BPM-based electron energy measurement res-
olution. An error is also added to the number of photons de-
tected at that energy, assuming the bunch charge randomly
varies from pulse to pulse, but a toroid charge monitor, ca-
pable of resolving the relative charge variation to within
0.5% rms, is used to normalize the data. In addition, the
beam angle is assumed to vary by 0.5prad rms (one-half
the nominal rms beam divergence), adding another source
of undetermined energy error based on small variations of
the Bragg angle. Detector noise is also added assuming a
noise level of 100 photons with respect to the the peak sig-
nal of 105 photons. And finally, a photon statistics error is
included, which is proportional to the inverse square-root
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Welch, J. J.; Arthur, J.; Emma, P.; Hastings, J. B.; Huang, Z.; Nuhn, H. D. et al. Precision Measurement of the Undulator K Parameter using Spontaneous Radiation, article, April 17, 2007; [Menlo Park, California]. (https://digital.library.unt.edu/ark:/67531/metadc886124/m1/2/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.