Calculation of coherent synchrotron radiation impedance using the mode expansion method

We study an impedance due to coherent synchrotron radiation (CSR) generated by a short bunch of charged particles passing through a dipole magnet of finite length in a vacuum chamber of a given cross section. In our method we decompose the electromagnetic field of the beam over the eigenmodes of the toroidal chamber and derive a system of equations for the expansion coefficients in the series. The general method is further specialized for a toroidal vacuum chamber of a rectangular cross section where the eigenmodes can be computed analytically. We also develop a computer code that calculates the CSR impedance for a toroid of rectangular cross section. Numerical results obtained with the code are presented in the paper.

Figure 1

A section of a vacuum chamber with toroidal bend between points A and B (a quarter of the pipe is cut out to expose the internal parts of the drawing). The thick black solid line is the central line of the pipe. Shown is a local coordinate system at the position with the longitudinal coordinate equal to $s$.

Figure 2

Cross section of a rectangular toroidal chamber with $a$ denoting the width in the horizontal ($x$) and $b$ denoting the height in the vertical ($y$) directions. The origin of the coordinate system is located at the center of the rectangle.

Figure 3

Steady state wakefields in the toroid for the aspect ratio $A=3$ and various values of $b$: (a) $b=4\mathrm{cm}$, (b) $b=2\mathrm{cm}$, (c) $b=1\mathrm{cm}$, and (d) $b=0.5\mathrm{cm}$.

Figure 4

Steady state wakefields in the toroid for the aspect ratio $A=1$ and various values of $b$: (a) $b=4\mathrm{cm}$, (b) $b=2\mathrm{cm}$, (c) $b=1\mathrm{cm}$, and (d) $b=0.5\mathrm{cm}$.

Figure 5

Steady state wakefields in the toroid for the aspect ratio $A=0.5$ and various values of $b$: (a) $b=4\mathrm{cm}$, (b) $b=2\mathrm{cm}$, (c) $b=1\mathrm{cm}$, and (d) $b=0.5\mathrm{cm}$.

Figure 6

Wakefields for the case $A=3$ and $b=2\mathrm{cm}$ at various distances from the entrance to the toroid: (a) distances 4, 8, 12, and 16 cm from the entrance (a larger distance corresponds to a larger amplitude of the wake), (b) distances 84, 88, 92, and 96 cm from the entrance.

Figure 7

Wakefields for the case $A=1$ and $b=2\mathrm{cm}$ at various distances from the entrance to the toroid: (a) distances 4, 8, 12, and 16 cm from the entrance (a larger distance corresponds to a larger amplitude of the wake), (b) distances 84, 88, 92, and 96 cm from the entrance.

Figure 8

Comparison of the wakefields at the entrance to the toroidal segment with $A=3$ and $b=4\mathrm{cm}$ with the free-space case. The four pairs of lines 1, 2, 3, and 4 correspond to distances 5.7, 11.3, 17.0, and 22.6 cm from the entrance point, respectively. The blue lines show the wakefield in free space and the black lines show the wakefield in the toroidal segment.

Figure 9

Wakefields for the case $A=3$ and $b=2\mathrm{cm}$ after the beam exits from the toroidal section at various distances $\Delta s$ from the exit: blue line—$\Delta s=0$; red broken line—$\Delta s=2\mathrm{m}$; and thin black line—$\Delta s=4\mathrm{m}$.

Figure 10

Integrated wake for the case $A=3$ and $b=2\mathrm{cm}$ and three lengths of the toroidal segment: 5 (blue line), 10 (broken red line), and 20 (thin black line) cm.