Courant-Snyder formalism of longitudinal dynamics

Originating from the stochastic nature of the photon emission process, the diffusion of the electron longitudinal coordinate exists even if the global phase slippage of a storage ring is zero, as we cannot zero all the local phase slippages simultaneously. This quantum diffusion is viewed as the most fundamental limit of the lowest bunch length realizable in an electron storage ring from the single-particle dynamics perspective. Here, we present an analysis of this effect using the Courant-Snyder parametrization in the longitudinal dimension. Analytical formulas for the longitudinal emittance, energy spread, and bunch length are derived. The same formalism is used to discuss the application of multiple radio frequency systems for longitudinal strong focusing. The presented work is expected to be useful in the development of novel light source mechanisms like steady-state microbunching, where a short bunch length or small longitudinal emittance is needed.

Figure 1

The horizontal dispersion $D_x$, longitudinal beta function ${\beta }_z$, and $F(0,s)$ of the MLS lattice used for analysis and simulation. In this plot, the zero-length rf is placed at $s_{\mathrm{rf}}=0\mathrm{m}$ and $V_{\mathrm{rf}}=80\mathrm{MV}$ is applied. The dipoles are shown at the bottom as blue rectangles. Each dipole has a length of 1.2 m and bends the electron trajectory for an angle of $\pi /4$.

Figure 2

Bunch length saturation and energy spread and longitudinal emittance divergence when the bunch length is pushed to the limit given by quantum diffusion, in the simulation by increasing the rf voltage. The lattice optics and related parameters are shown in Fig. 1 and Table 1. In each simulation, 1000 electrons are tracked for 50000 turns, which corresponds to about five longitudinal damping times, and the averaged result of the last 10000 turns are used to get the ${\sigma }_z$, ${\sigma }_{\delta }$, and ${\epsilon }_z$. The observation point is at the rf.

Figure 3

The emittance ratio $\frac{{\epsilon }_z}{{\epsilon }_{z\mathrm{S}}}$ as a function of the rf place $s_{\mathrm{rf}}$ (top) and the $r_{56}$ artificially inserted in front of the rf whose location is fixed at $s_{\mathrm{rf}}=0\mathrm{m}$ (bottom). The lattice optics and related parameters are shown in Fig. 1 and Table 1. The rf voltage applied is $V_{\mathrm{rf}}=80\mathrm{MV}$.

Figure 4

Schematic layout of a storage ring using two rf systems for longitudinal strong focusing and an example beam distribution evolution in the longitudinal phase space. Note that the tilted angles of the beam distribution and bunch length ratios at different places do not strictly correspond to the parameters in Table 2 but only to present the qualitative characteristics.