Analytical Calculation Method for the Bandpass Filter

The influence of the chromatic lenses on the undulator radiation (UR) can be quantified by the calculation of the radiation emitted by a thick (finite emittance) electron beam with a varying divergence and beam size. The beam size of the undulator radiation from a thick electron beam can be determined by a convolution of the single-electron emission with the electron beam distribution (see section 3.5.2). As described in section 3.4.1), both quantities can be approximated by a Gaussian distribution (which for the UR is only possible for the resonance frequencies). In this case, the UR beam size of a thick electron beam resulting from the convolution can be written as the quadratic sum of the beam size of the single-electron emission and the beam size of the electron beam. In order to determine the undulator radiation at an observation point after a free drift, the radiation has to be propagated. As shown in section 3.5.3, carrying out the convolution in the center of the undulator and propagating the resulting radiation is equivalent to first separately propagating the single-electron UR and the electron beam to the observation point and then performing the convolution.

Explicitly, this means that the undulator beam sizeat the position of the detectorcan be calculated by a convolution of the single-electron UR beam size with the electron beam size, bothat the position of the detectorwhich in the Gaussian approximation can be written as

Σx,y(s) =

q σ2

x,y(s) +σr2(s), (6.19)

where σx,y(s) is the horizontal/vertical electron beam size at the position s measured

from the undulator center. The single-electron UR beam size σr(s) is mainly given by

the natural UR divergence for distances much larger than the undulator length (sL). For a free drift, the beam size can be written as (see section 3.5.3)

Σx,y(s) = r σ2 x,y(s) + λL 2π2 + λ 2Ls 2_{, (6.20)}

whereL is the undulator length and λ the wavelength of the undulator emission. For a givenλandL(which are determined by undulator parameters and the electron energy), the undulator beam size only varies with the electron beam size at the detector. It can be seen that the on-axis flux which depends on the undulator beam size (see equation 6.17) thus can be modified by changing the electron beam size. Since the electron beam size can be adjusted by the magnetic lenses, they implicitly determine the on-axis flux intensity: The small size of an electron beam focused at the position of the detector leads to a small UR beam size and thus a high on-axis photon flux. The size of the electron beam for a particular energy is given by the specific setup of the quadrupole lenses: in our case the lenses are set-up to collimate a particular energy. Since the undulator radiation is observed at a relatively long distance ('3 m) downstream of the last lens, electron energies, slightly below the energy that is collimated, are focused at the position of the detector. Owing to the chromaticity of the quadrupole lenses, only electrons in a small bandwidth around this energy have a small electron beam size and therefore a small UR beam size at the detector. Both the energy-dependent spatial electron beam

6.2. Magnetic Quadrupole Lenses as Energy-Bandpass Filter for the Undulator Radiation

Electron energy [MeV]

Bea

m

siz

e

[µm]

Electron energy [MeV]

No

rmali

ze

d

on-axis

flux [arb. units]

0 1 2 3 4 5 170 190 210 230 250 0100 150 200 250 300 0.2 0.4 0.6 0.8 1.0 electron beam

undulator beam analyticalsimulation

a

b

Figure 6.17. _|Electron and undulator beam areas as well as resulting energy band-pass filter through the effect of the magnetic lenses. a) The electron beam area (Ael = π σx·σy) at the position of the detector is calculated for various

electron energies considering the effect of the magnetic lenses (blue). The red curve shows the undulator beam area analytically calculated by the convolution of the elec- tron beam size with the size of the single electron emission (AUR =πΣx·Σy). The

“wiggle” of the blue curve at_≈200₋215MeV is due to an astigmatic focus of the electron beam. b) shows the system response curve of our setup, which corresponds to the calculated energy-dependent on-axis undulator flux at the position of the de- tector. The narrow bandwidth filter is due to the energy-dependent electron-beam divergence introduced by the the magnetic lenses as explained in the main text. The red curve is the result of the simulation of the undulator code SRW that includes the focusing effect of the gold-mirror (for details of the calculation refer to sec. 6.2.5). The green curve is a result of analytical calculations of the on-axis flux after a free drift using equation 6.17. Both consider the energy-dependent electron beam sizes given by the effect of the magnetic lenses and the wavelength-dependent UR beam size given by equation 6.20. In the green curve, the focusing effect of the the mirror is not included. The red curve has a FWHM-bandwidth of 9% around 211 MeV and the green curve a bandwidth of 15% FWHM around 209 MeV. Both the curves in

aand inb are calculated for a lens setup that collimates an electron energy of 220 MeV. The natural focusing of the undulator is not considered in these curves.

area (Ael = π σx·σy) and the spatial undulator beam area (AUR = πΣx·Σy) at the

position of the detector for a lens setup that collimates an electron energy of 220 MeV can be seen in figure 6.17a.

Electron bunches with identical beam currents but different electron energies produce the same angle-integrated undulator spectral flux Φn(as it is independent of the electron

energy (see equation (6.15)). However, the undulator radiation flux from electrons of energies within the small bandwidth that are focused to a small beam size at the detector is not as smeared out as that from electrons outside this energy band. This results in a higher on-axis flux of the radiation emitted by the focused electron energies.

Figure 6.17b shows the result of computations of this energy-bandpass filter for the setup used in this experiment: An electron beam with an energy that is focused to the smallest spot at the detector (for this lens setup _' 210 MeV) yields the highest on-axis flux, whereas deviations of energies of a few tens of MeV causes the on-axis flux to drop sharply. In order to determine the undulator spectrum and the (spectral) fraction of the electron beam that primarily contributes to the measured undulator spectrum, the detected electron spectrum has to be filtered by this curve (see figure 6.22) which is therefore called the system response curve.