Symplectic modeling of beam loading in electromagnetic cavities

Simulating beam loading in radio frequency accelerating structures is critical for understanding higher-order mode effects on beam dynamics, such as beam break-up instability in energy recovery linacs. Full wave simulations of beam loading in radio frequency structures are computationally expensive, while reduced models can ignore essential physics and can be difficult to generalize. We present a self-consistent algorithm derived from the least-action principle which can model an arbitrary number of cavity eigenmodes and with a generic beam distribution. It has been implemented in our new Open Library for Invesitigating Vacuum Electronics (OLIVE).

Figure 1

Energy gain and eigenmode energy decomposition for the case of one macroparticle on axis.

Figure 2

Total particle-field energy and individual mode energies for the four-particle configuration.

Figure 3

Total particle-field energy and individual mode energies for the four-particle configuration in a longer cavity.

Figure 4

Total system energy evaluated for 1000 particles and 6 modes in a cavity of length $\sim 1.8\mathrm{km}$. 1 million steps of length $\frac{\lambda }{30}$ produced relative energy ($\frac{\mathrm{\Delta }E}{E}$) losses on the order of ${10}^{-8}$.

Figure 5

The change in Hamiltonian evaluated after a single step reveals 2nd order scaling with step size $h$.

Figure 6

Timings for a single step weighted per particle and per mode are shown for upwards of 256 modes and 12800 particles. For large systems, we find $\mathrm{\Delta }t\sim 0.5\mu \mathrm{s}$ per-particle-per-mode.