Transport formalism

Taylor map:

6

1

6

1

6

11112 )()()()(

j j kkjijkjiji swswTswRsw Third order

,,',,',,,,,, 654321 tyyxxwwwwwww

Linear matrix elements Second order matrix elements

Truncated maps

Violation of the symplectic condition !

Lie algebraic treatment

Symplectic MapsLie Transformations

:: f

if

eM

Mww

Dragt-Finn factorization :[A. Dragt et al., Ann. Rev. Nucl. Part. Sci. 38 (1988) 455]

...:::: 43 ff eeM Linear matrix

::

::

4

3

f

f

e

e produces Tijk and higher order terms (sextupole effects)

produces third order and higher order terms (octupoles effects)

Numerical methods for nonlinear optimization : PARTICLE TRACKING,Dynamic aperture scans, particle spectra…

generators

Tracking codes:Simulations to show the feasibility

Optics lattice

MADPlacetSAD

…

Entrance:IP:

Multiparticle tracking

Beam-beaminteraction

Guinea-Pig

performance

[T. Asaka and J. Resta Lopez, CLIC-Note-637]

transport

Lie

Importance of the benchmarking of codes

Nanometer-Size Beams in CLIC

Beam profile at the IP:

Some problems: Residual horizontal dispersion at the IP

Nominal: σx=40.12 nm; σy=0.55 nmSimulations: σx≈47.3 nm; σy≈0.65 nm

Nanometer-Size Beams in CLICPhase space at the IP:

Particles with lower energy than the nominal one (1500 GeV) contribute strongly to the tails of the transversal phase space

Chromatic effects in phase spaceChromatic aberrations study by means of tracking from matched initial ellipses at 1σ for the transversal plane X

Red line: center ellipse

movement in phase space 32666

226626

31666

216616

)(

)(

UTREEx

UTREEx

c

c

up to third order !

Chromatic effects in phase spaceChromatic aberrations study by means of tracking from matched initial ellipses at 1σ (figure on the left) and 3σ (figure on the right) for the transversal plane Y

The particles at high position amplitude of several sigmas contribute to the population of the long tails. For the case of the ellipses at 3σ in the vertical phase space, it is possible to observe a strong deformation of the shape caused by the sextupoles located in the FFS.

Limits of the LuminosityL/L0PlacetWithout SR With SR

•Tolerable bandwidth up to 1 % energy spread•The synchrotron radiation is a very important limitation factor for the luminosity

Collimation issues in CLIC

Beam-beam effects

Simulations with Guinea-Pig: it includes beam-beam effects

Disruption parameters:Dy= 3.5 (CLIC)Dy=19.4 (ILC)

Luminosity versus vertical offset

Analytic calculation considering a rigid gaussian beam:

2

24

0

y

yL

eL

Input LINAC BDS Beam-Beam Output

Input LINAC BDS Beam-Beam Output

Placet

MatmerlinPlacet

Guinea-Pig

Guinea-Pig

Simulink

Simulink

FB

FB

ILC integrated simulations Updated simulations:

G. White version (2005):

Ground motion and FB system

Nominal: L=2x1034 cm-2s-1

85 % of the nominal luminosity