Microbunching of Electron Beam in a Magnetic Compressor Yunhai Cai Beam Physics Meeting, SLAC August 5, 2011

Single-Particle Dynamics in x-z planes Inside a magnetic chicane, the linear transport matrix M is given by Due to symplectic condition, there are six independent parameters in the matrix. In particular, R51 and R52 are linear combinations of h and h’. R56 from t to s can be written Note that it vanishes when t=s.

A Linear Integral Equation A bunching factor b[k(s);s] is given by a solution of integral equation: where the kernel is and S. Heifets, G. Stupakov, and S. Krinsky, PRSTAB, 5, 064401 (2002) Z. Huang and K-J Kim, PRSTAB, 5, 074401 (2002)

A General Transformation Without impedance, the initial bunching damps down according to where This suggests to us to make a general transformation:

A Simplified Integral Equation A bunching factor b[k(s);s] is given by a solution of integral equation: where the kernel is Tracking through a dipole magnet: 0 1 2 n is the bunching before it inters the dipole. Note that there is no iteration at any point. A linear map (DA) is tracked using the Ruth integrator between each step along with the bunching factor for calculating the kernel.

Berlin Benchmark: A Simple Chicane Z. Huang, M. Borland, P. Emma, K-J, Kim, SLAC-PUB-9538. Thank Zhirong for providing the results of the elegant simulation and many helpful discussions.

Coasting Beam to Guassian Beam Parameters of a Guassian beam is given by where h is the energy chirp of the beam. The density distribution of a Guassian beam can be written Here we have z>0 for the head of bunch and p=-d for canonical coordinates. When h is large, one can show that it is very close to the coasting distribution.

Integral Equation for a Bunched Gaussian Beam A bunching factor b[k(s);s] can be described by an integral equation: where the kernel is given by and There is a coupling among various wavelengths.

Without Impedance The initial bunching still damps down according to where Provided that we have the initial bunching:

Bunching Factor by a Guassian Beam In general, we have Given the bunching factor b(k), the perturbation e(z) depends on the bunch length sz. So there is an ambiguity in calculation of a gain since the bunch length may not be well defined in some cases, for example, the bunch length with or without wake. In LEGO calculation, we always use the ratio of the bunching factors to defined the gain.

Gain of White Noise Gain of white noise from a Guassian beam are the same as those for the coasting beam provided one identifies the peak current of the Guassian beam to the current of the coasting beam (Boussard criterion).