Shielding and synchrotron radiation in toroidal waveguide

We develop a new approach to the calculation of the synchrotron radiation in a toroidal vacuum chamber. Using a small parameter $ϵ=\sqrt{a/R}$, where $a$ is the characteristic size of the cross section of the toroid and $R$ is the bending radius, we simplify Maxwell’s equations assuming that the characteristic frequency of the modes $\omega \sim c/aϵ$ and neglect terms of higher order in $ϵ$. For a rectangular cross section of the waveguide, we find an analytical solution of the equations and analyze their asymptotics at very high frequency. We then obtain an equation which gives radiation into each synchronous mode. We demonstrate the flexibility of the new method by calculating the frequencies and the loss factors for the lowest modes in square and round waveguides.

Figure 1

Smooth toroidal vacuum camera for circular (left) and rectangular (right) cross sections with $R$ the major radius of the toroid. It is assumed that a particle’s orbit goes through the center of the toroidal chamber and has the same radius $R$. A segment is removed for illustration purposes.

Figure 2

Solution of the dispersion equation $\Lambda =0$ for various mode numbers $m$ indicated on the plot. $E_r$ mode: magenta (also indicated by symbol $r$); $E_z$ mode: blue (indicated by symbol $z$). The dashed lines show the approximate solutions discussed in the subsection of 4.

Figure 3

Field patterns for a square cross section of the waveguide: (a) $E_r$ mode, $m=0$, $p=1$; (b) $E_r$ mode, $m=0$, $p=2$; (c) $E_z$ mode, $m=1$, $p=1$; (d) $E_r$ mode, $m=0$, $p=3$; (e) $E_z$ mode, $m=1$, $p=2$; (f) $E_r$ mode, $m=0$, $p=4$. For the $E_z$ modes, the graphs show distribution of the $E_z$ component of the field; and for the $E_r$ modes, the $E_r$ distribution is shown. The color coding palette is shown in the upper right corner of the picture, where the bottom and the top of the palette correspond to lower and higher values, respectively. Note that the mode profiles are not normalized, and the patterns of different modes cannot be compared with each other.

Figure 4

Field profiles for a round cross section of the waveguide for the three lowest antisymmetric modes: (a) $k{R}^{-1/2}{a}^{3/2}=2.12$; (b) $k{R}^{-1/2}{a}^{3/2}=3.95$; (c) $k{R}^{-1/2}{a}^{3/2}=4.82$. The left plot in each pair of graphs shows distribution of the $E_r$ in the mode, and the right one shows the distribution of $E_z$. The color coding palette is shown in the upper right corner of the picture.