Calculations of angular distribution of undulator radiation to evaluate
angular separation of sliced electrons

Philip Heimann July 18, 2002

The goal of this note is to use calculated angular distributions of
undulator radiation in order to evaluate the method of angular
separation of sliced electrons (S. Khan and H.A. Durr EPAC 2002). In
this method dispersion prime is introduced into the accelerator, and the
first beamline mirror is displaced by 1 mrad to collect the radiation
from the sliced electrons.

Calculations were made with the program SRW from the ESRF. Cases of the
1^st^, 2^nd^ and 3^rd^ harmonic at K values between 1 and 3.09 were
considered. The U46 undulator was chosen since it is planned to be the
first insertion device used in the femtosecond undulator beamline. All
calculations were made in the far field approximation.

Figure 1 shows angular distributions of the 3^rd^ harmonic at K values
of 1, 1.8 and 3.09. The horizontal axis is the position at 10 m from the
source so that 10 mm is equivalent to 1 mrad. The vertical acceptance is
chosen to collect 4 of the central cone. The K = 1.8 plot is very
similar to Khan's figure 5 confirming our results. A strong K dependence
is seen. At low K the background is quite low while at high K the
background is large. This effect can be understood from the wiggler
approximation, where K/ determines the maximum radiation angle.

Figure 2 is a compilation of different photon energies for the 1^st^,
2^nd^ and 3^rd^ harmonics: each point represents the background at 1
mrad divided by the peak harmonic intensity. A background of 1 x 10^-4^
or less is considered acceptable because the intensity of the sliced
x-ray pulse compared with the ps pulse is \~ 3 x 10^-4^. With a low
enough background, the 1^st^ harmonic can cover the photon energy range
from 310 to 600 eV, the 2^nd^ harmonic from 530 to 860 eV and the 3^rd^
harmonic from 580 to 1480 eV. On the positive side, a continuous energy
range is provided from 310 to 1480 eV. On the negative side, it is not
possible to access the the C K edge and lower photon energies.

An additional issue is the use of off-axis radiation. Off-axis radiation
provides a wider bandwidth for dispersive x-ray absorption experiments.
For example, the bandwidth of the U46 3^rd^ harmonic central cone is 1 %
insufficient for XANES or EXAFS spectra. Figure 3 shows the fundamental
at K = 1.4 and different photon energies: 377 eV the on-axis harmonic
energy, 368 eV the peak intensity (1 x 0.3 mrad acceptance), 335 eV and
319 eV the low photon energy half maximum. Two evaluations of the
background are given: at 1 mrad and integrated from 0.8 to 1.2 mrad
(compared with the peak integrated from --0.2 to 0.2 mrad). In general,
the integrated background is about 10 times higher than the value at 1
mrad because the background steeply increases to lower angle. The
background for the red-shifted photon energies 319 and 335 eV is
unacceptable. The method of angular separation is successful only with
the central cone radiation.

Finally, angular separation restricts the peak magnet field of a future
insertion device to about 1 T. This future insertion device should
provide both hard as well as soft x-rays. Figure 4 displays the
performance of a undulator with u of 2.1 cm and peak field of 1 T. For
comparison is shown a wiggler with u of 3 cm and peak field of 1.5 T,
where the wiggler radiation was accepted from 0.5 x 0.3 mrad aperture.
Both of the insertion devices would be superconducting magnets. The
cross over in performance is at 3 keV with the undulator radiation
giving acceptable intensities up to 5 keV. A future insertion device
compatible with angular separation gives limited hard x-ray performance.

Figure 1

![](media/image1.wmf){width="8.552083333333334in"
height="5.590277777777778in"}

![](media/image2.wmf){width="8.34375in"
height="5.673611111111111in"}Figure 2

Figure 3

![](media/image3.wmf){width="8.40625in" height="6.269444444444445in"}

![](media/image4.wmf){width="7.966666666666667in"
height="5.340277777777778in"}Figure 4