Analytical theory of coherent synchrotron radiation wakefield of short bunches shielded by conducting parallel plates

We develop a general model of coherent synchrotron radiation (CSR) impedance with shielding provided by two parallel conducting plates. This model allows us to easily reproduce all previously known analytical CSR wakes and to expand the analysis to situations not explored before. It reduces calculations of the impedance to taking integrals along the trajectory of the beam. New analytical results are derived for the radiation impedance with shielding for the following orbits: a kink, a bending magnet, a wiggler of finite length, and an infinitely long wiggler. All our formulas are benchmarked against numerical simulations with the CSRZ computer code.

Figure 1

Parallel plates located at $y=\pm \frac{1}{2}a$ and a plane orbit in the mid-plane shown in black. The coordinate system is chosen so that the orbit lies in the $xz$ plane and the $z$-axis is directed along the tangent vector to the trajectory at $x=z=0$.

Figure 2

A part of a circular orbit with coordinate system.

Figure 3

Kink orbit shown in blue corresponds to a short magnet that deflects the orbit by a small angle ${\theta }_0\ll 1$.

Figure 4

Impedance of a kink as a function of parameter $w=ka{\theta }_0$.

Figure 5

Comparison of the analytical theory (dots) and numerical simulations (solid lines) for the case of a kink orbit. The blue line shows the real part and the red line shows the imaginary part of $Z$ computed by the code. The dots show $\mathrm{Re}Z$ computed using Eq. (54) (the imaginary part is equal to zero and is not shown).

Figure 6

The orbit for a bending magnet of length $L$ consists of a straight line $z<0$, a circular arc occupying the region $0<z<L$ and a straight line in the region $z>L$ tilted at angle ${\theta }_0$. Panels (a), (b), (c), and (d) show four different situations for relative locations of the leading point $z$ (shown by the red dot) and that of the trailing point $z^{\prime }$ (shown by the blue dot). The beam moves from left to right.

Figure 7

Comparison of analytical calculations (shown by dots) with computer simulations (shown by solid lines): $\mathrm{Re}Z$ (blue) and $\mathrm{Im}Z$ (red).

Figure 8

Wiggler of length $L_w=N_w{\lambda }_w$ with the orbit shown by blue line.

Figure 9

Wiggler impedance: comparison of analytical calculations (shown by dots) with computer simulations (shown by solid lines). $\mathrm{Re}Z$ is shown in blue and $\mathrm{Im}Z$ in red.

Figure 10

Comparison of analytical calculations (left panel) with computer simulations (right panel) for the NSLS-II wiggler impedance: $\mathrm{Re}Z$ (blue) and $\mathrm{Im}Z$ (red).

Figure 11

Wakefields for a Gaussian bunch with ${\sigma }_z=0.5\mathrm{mm}$ (the bunch profile is shown by red dotted line): calculated with CSRZ (blue line) and with the analytical model (red line) using impedances shown in Fig. 10. The dashed magenta shows the numerically calculated wake for the aspect ratio $b/a=16$. The bunch head is to the right; positive wake corresponds to the energy loss.