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

2. 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.

3. 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, (2002)
Z. Huang and K-J Kim, PRSTAB, 5, (2002)

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

5. 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: 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.

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

7. 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.

8. 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.

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

10. 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.

11. 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).