Radiation emitted by transverse-gradient undulators

Conventional undulators are used in synchrotron light sources to produce radiation with a narrow relative spectral width as compared to bending magnets or wigglers. The spectral width of the radiation produced by conventional undulators is determined by the number of undulator periods and by the energy spread and emittance of the electron beam. In more compact electron sources like for instance laser plasma accelerators the energy spread becomes the dominating factor. Due to this effect these electron sources cannot in general be used for high-gain free electron lasers (FELs). In order to overcome this limitation, modified undulator schemes, so-called transverse gradient undulators (TGUs), were proposed and a first superconducting TGU was built at Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany. In this paper simulations of the expected synchrotron radiation spectral distribution are presented. An experimental test with that device is under preparation at the laser wakefield accelerator at the JETI laser at the University of Jena, Germany.

Figure 1

Transverse gradient undulator geometries, coordinate systems and basic geometry parameters: left, transversely tapered TGU; right, cylindric TGU.

Figure 2

Magnetic flux density amplitude as a function of transverse position $x$ for a transversely tapered and a cylindric TGU. Common parameters for both cases are $B_{\mathrm{pol}}=1.3\mathrm{T}$ and ${\lambda }_{\mathrm{u}}=10.5\mathrm{mm}$. For the transversely tapered undulator $x_c=13\mathrm{mm}$ and an opening angle $\xi =200\mathrm{mrad}$ are assumed, for the cylindric undulator $r_{\mathrm{cyl}}=30\mathrm{mm}$, $h_{\mathrm{gap}}=1.1\mathrm{mm}$ and $x_c=6.47\mathrm{mm}$, corresponding to the example case discussed below.

Figure 3

Model of the cylindric superconducting TGU, shortened to ten periods for better visibility. The superconducting coils are depicted in red, the copper coil former in green.

Figure 4

Geometry of the correction coils generating an $x$-dependent magnetic field correcting for the ponderomotive drift in the example-SCTGU (left). The correction field generated by this coil configuration in the deflection plane is shown in the right part of the figure (solid line). The single data points shown in the graph indicate the required correction field calculated from the displacement of the electrons after one period, if uncorrected.

Figure 5

Basic proof of the TGU concept: Electron energy dispersion and resulting radiation spectra for an electron energy band of $\mathrm{\Delta }E=120\mathrm{MeV}\pm 10\%$ for the cases (a) planar undulator with $K=1.07$, (b) ideal linear TGU with $K_0=1.07$ and $\alpha =139.7{\mathrm{m}}^{-1}$ and linear dispersion [Eq. (4)], (c) the same with nonlinear dispersion according to Eq. (12), (d) idealized cylindric SCTGU with optimized dispersion. For all cases a beam current of 10 pA, equally distributed over the 21 monoenergetic zero-emittance beamlets, is assumed.

Figure 6

The effect of the ponderomotive particle drift in TGUs: (a) linear TGU without trajectory correction and (b) with correction, (c) cylindric SCTGU without trajectory correction and (d) with correction.

Figure 7

Comparison of the required dispersion and the resulting radiation spectra (assuming zero-emittance beamlets) between the case of 100 periods out of an infinitely long SCTGU and the case of a finite 100-period SCTGU with matching periods. See the text for details.

Figure 8

Comparison of the optimized spectral dispersion of the electron beam and resulting radiation spectra for an observation point in infinite and 2 m distance, respectively.

Figure 9

Optimized electron dispersion and normalized peak intensity of the resulting radiation for each electron energy as a function of distance to the observation point for two different transverse positions of the observation point. For details on the intensity normalization refer to the text.

Figure 10

Left: Spectra of the photon flux through a pinhole with $1.7\mathrm{mrad}\times 1.7\mathrm{mrad}$ angular acceptance for (a) a planar undulator, (b) the example SCTGU observed at infinite distance, (c) the SCTGU at 2 m distance; right: incoherently added spectra for the three cases assuming a Gaussian energy distribution of the electrons with ${\sigma }_E=6\mathrm{MeV}$.

Figure 11

Left: Phase space distribution in $x$ and $y$ of a 120 MeV electron bunch at the entrance, center and exit of the idealized cylindric TGU. The electron bunch parameters were chosen as ${\beta }_x^{\text{waist}}=0.5\mathrm{m}$, ${\beta }_y=0.7\mathrm{m}=\text{const}$ and the geometric emittance ${\epsilon }_x={\epsilon }_y={10}^{-8}\mathrm{m}\mathrm{rad}$. The color scale refers to the relative deviation of the peak energy of the radiation emitted by the respective particle with respect to that of the reference particle. Right: Incoherently added radiation spectrum emitted by the particle bunch as a whole for different values of ${\epsilon }_{x,y}$, observed at infinite distance.

Figure 12

Incoherently added radiation spectra emitted by particle bunches with $E=120\mathrm{MeV}$, ${\epsilon }_x={10}^{-8}\mathrm{m}\mathrm{rad}$ for two different choices of ${\beta }_x^{\text{waist}}$, observed at $10^5\mathrm{m}$ distance.

Figure 13

Dynamic acceptance levels for 120 MeV electrons entering the TGU in terms of the relative deviation of the mean wavelength of the emitted radiation from that emitted by the reference particle (left) and in terms of the peak intensity of the emitted radiation (right).

Figure 14

Dynamic acceptance for maximum, central and minimum electron energy for an acceptance level of 0.5% relative deviation of the radiated wavelength (after convolution resulting in 1.1% bandwidth of the incoherently added spectrum) and 50% of the radiated peak intensity as compared to the wavelength and peak intensity radiated by the reference particle.

Figure 15

Radiation spectra expected for the real SCTGU-P40 with finite beam emittance and observation at 2 m distance with (a) a photon optics for a parallel beam (zero angular acceptance) and (b) a point detector. The corresponding zero-emittance and planar undulator spectra in the forward direction (parallel beam) are added for comparison. For the undulator and beam parameters refer to Table 2.