Transition radiation wakefields for a beam passing through a metallic foil

In the x-ray, free-electron laser project, the Linac Coherent Light Source (LCLS), a proposal has been made to generate a shorter light pulse by placing a spoiler foil in the middle of a compressor chicane: The foil has a small slot, which selects out the small fraction of particles passing through it (“target particles”) to lase. In this report, using the method of field matching, we obtain longitudinal and transverse impedances and wakefields for several models of the proposed LCLS spoiler foil. We consider the model of a pencil beam and of a cylindrically symmetric, bi-Gaussian beam that is wider than it is long. Third, we generate a Green function that allows us to consider asymmetric beams also. For target particles of the tilted, tri-Gaussian beam that is found at the LCLS spoiler location we obtain approximate analytical formulas and numerical results for wakefield kicks in the three directions. We find that the kicks, after correction using a simple dipole and quadrupole, are all within tolerances.

Figure 1

A sketch in the $x\mathrm{\text{−}}z$ plane of the beam and the foil with its slot. The beam pipe is round but the slot is flat, with half-width $\Delta x$. The beam is tilted in $x\mathrm{\text{−}}z$ by angle $\alpha$ ($\ll 1$), with the longitudinal slice rms length, ${\sigma }_{zs}$. Note that $a\gg {\sigma }_x\gg {\sigma }_z$, ${\sigma }_y\gg {\sigma }_{zs}$.

Figure 2

A sketch of the geometry of our cylindrically symmetric model showing the two regions, I and II, and the relative positions of the drive and test particles.

Figure 3

The function $f(x)$ defined by Eq. (27).

Figure 4

For Gaussian bunches, the function ${\sigma }_z\mathcal{W}(s)$ for three values of $a/{\sigma }_z$.

Figure 5

A sketch of the model for the transverse wake calculation. The beam pipe and beam are both cylindrically symmetric.

Figure 6

For Gaussian bunches, the function $-{\sigma }_z{\mathcal{W}}_r(r,s)$ for several values of $r/{\sigma }_z$.

Figure 7

For LCLS spoiler parameters: ${\overline{\mathcal{W}}}_y$ vs $y$, for $x=-\Delta x$, 0, and $\Delta x$.