Coherent synchrotron radiation by electrons moving on circular orbits

We study coherent synchrotron radiation by electrons in the Frenet-Serret coordinate system with a constant curvature $1/\rho$. Based on the Hamiltonian in the Courant-Synder theory of particle accelerators, we find in general that the transverse force is essentially the Lorentz force but with a substitution of the transverse magnetic field $B_{x,y}\to (1+x/\rho )B_{x,y}$, where $x$ and $y$ are the transverse positions. The curvature term provides us a key to derive the point-charge wakefield explicitly in terms of the incomplete elliptic integrals of the first and second kind, resulting in a steady-state theory of the coherent synchrotron radiation in two-dimensional free space.

Figure 1

The curvilinear coordinate system.

Figure 2

A source particle (at A) moves along a circular orbit and radiates electromagnetic fields that reach a testing particle (at D).

Figure 3

The scaled longitudinal field $E_s{\rho }^2/e{\gamma }^4$ (left) and transverse force $F_x{\rho }^2/e^2{\gamma }^3$ (right) as a function of $\mu =3{\gamma }^3\xi$ and $\nu ={\gamma }^2\chi$ with $\gamma =500$.

Figure 4

The scaled longitudinal potential ${\psi }_s{\rho }^2/e\gamma$ (left) and transverse potential ${\psi }_x{\rho }^2/e^2$ as a function of $\mu =3{\gamma }^3\xi$ and $\nu ={\gamma }^2\chi$ with $\gamma =500$.

Figure 5

The longitudinal (left) and transverse (right) wakes at various transverse positions $\chi =-{\sigma }_{\chi }$ (black), 0 (blue), and ${\sigma }_{\chi }$ (red) for a Gaussian bunch with ${\sigma }_x={\sigma }_z=10\mu \mathrm{m}$ on a curve with radius $\rho =1\mathrm{m}$ and $\gamma =500$.