D. Rubin, S.Isaacman, A.Long
March 20, 2005.
Wiggler nonlinearities
Introduction
The superconducting damping wigglers have strong nonlinearities.
The vertical cubic nonlinearity that is characteristic of
all wigglers, introduces a significant
amplitude dependent tune shift. Dependence of vertical field on horizontal
displacement results from the finite pole width. The wiggler map in our
standard machine
model includes all of the field nonlinearity. We explore the effect of wiggler nonlinearity
on luminosityy by replacing the precised wiggler map with a linearized verison.
We find no significant different in specific luminosity.
Wiggler Model
In our machine model
the wiggler mapping is based on an analytic representation of the
field tabulated by a finite element calculation. The analytic
form yields a taylor map. A third order map adequately reproduces
the computed field nonlinearity. The map is symplectified to preserve
phase space. The wiggler mapping is illustrated
in the plot
of phase space transfer function. The 3 graphs in the left hand column
show x' out vs x in, and the 3 on the right are y' out vs y in.
The graphs in the center row are in the horizontal and vertical
midplane respectively. The upper and lower plots on the left are
displaced 20mm above and below the midplane, and the upper and
lower plots on the right are displaced 30mm to right and left of the
vertical midplane. The blace line is computed by integration throught the
field table, the red line by runge kutta integration through the
analytic representation and the red line by symplectic integration
based on the taylor map. Note that the three techniques give essentially
the same result.
A linearized map is created by simply neglecting all higher
order terms in the taylor series. The transfer
function
for the linearized
map shows that the result isvertical focusing and horizontal defocusing.
Here the red line corresponds to the runke kutta integration through the
analytic representation and the green and black lines to
the simple first order taylor mapping,
and to the symplectic integration through the
taylor mapping.
Because the sextupole component of the wigglers is excised, the
lattice sextupole distribution is reoptimized. The lattice with
linearized wigglers that we use for the simulation is
hibetainj_20040628_v04_o1_s2.lat
Simulation
The simulated luminosity vs current for the lattice with linearized
wigglers is plotted
along with that of the standard 12 wiggler lattice.
There is no significant difference.
