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=√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 ω~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.