Collimator wakefields

1
Collimator wake-fields

• Wake fields in collimators
• General information
• Types of wake potentials
• Geometric
• Steep
• Tapered
• Dielectric layers
• Resistive
• Uniform
• Transient
• Surface roughness
• Merlin simulations
• Geometric
• Resistive

2
General information about Wake fields applicable
to collimators and waveguides

• General theory
• When charged particle is moving through linac or
  any other structure in accelerator it generates
  fields in the structure.
• Reasons of the fields can be different limited
  resistivity of the pipes, dielectric layers on
  the walls, macro obstacles or micro obstacles due
  to surface roughness.
• The generated fields will affect any trailing
  particle following the leading one. In general
  the fields depend on both particles positions so
  the momentum change is (Stupakov)
• And we can define wakes
• We can integrate over the structure fields
  generated by the bunch to get total effect of
  fields on particles from the same or the next
  bunch. Total effects are usually expressed in
  loss and kick factors. Loss factor is the
  integrated energy (momentum) loss by the bunch
  over passing the structure. Kick factor is
  usually average transverse angle gained by the
  bunch passing the structure.
• Fields of the leading particle are calculated
  assuming there are no transverse displacements of
  the particle during passing the structure due to
  the Wake fields.

3
General information about Wake fields applicable
to collimators and waveguides

• Superposition
• Superposition principle allows us to calculate
  one particle fields and then integrate over all
  the other particles in the bunch so
• We can split bunch of relativistic particles into
  transverse slices.
• Si slice position in the bunch.
• Wake fields of the slice can be received by
  integration over transverse distribution of the
  particles in the slice.
• Wake fields of the bunch can be received by
  convolution over longitudinal distribution of the
  particles in the bunch.
• Slices cannot mix so we can express all the field
  components in a frequency domain in a following
  way
• In ultrarelativictic limit we have vc

4
General information about Wake fields applicable
to collimators and waveguides

• Uniform structures
• For uniform structures we don't need to integrate
  over longitudinal direction of the structure as
  the fields don't depend on this so we need only
  get the field itself.
• Wake fields are estimated over the unit length of
  the structure.
• Different units of measurements for Wakes used
  here (1/L). Consequently to get usual loss or
  kick factors we need to multiply by L.

5
Geometric wakefields

• Linked with macro obstacles in the structures
  such collimators as a whole.
• In steep collimators diffraction theory
  applicable so we have
• Circular case (Chao)
• With higher modes and correction confirmed by
  MAFIA and Merlin simulations
• Rectangular case (Stupakov)
• No higher modes yet and for m1
• Tapered cases
• Circular (Yakoya)
• Rectangular (Stupakov)

6
Dielectric wakefields

• Investigated as new collective acceleration
  methods (Park, Hirshfield (2000))
• Usual approach is similar to resistive wakes -
  field matching technique on boundaries
• The difference is that we dont need to count
  energy loss in the walls but have waves with
  different speed in different media.
• Cn-normalization constant that can be received by
  orthonormality relation and from the source
  currents.

7
Resistive wakefields

• Linked with finite conductivity of the metal
  walls of the pipe.
• Analysis usually limited by uniform structures
• Circular case well investigated
• Chao
• Bane, Sands with corrections for small s and
  for a.c. conductivity
• Flat and elliptic cases (Piwinski Yokoya)
• Rectangular case and elliptic case (Gluckstern)
  only monopole and dipole
• Form factors in comparison with round case
• Transition effects were studied recently for
  short bunches and for the same circular case by
  Ivanyan, Tsakanov Glukstern Stupakov
• It was demonstrated how the potential evolves to
  uniform one received in Bane, Sands paper when
  there is a transition from infinite to finite
  conductivity pipe at some z.

8
Resistive wakefields

• Rectangular case again
• Known approach for rectangular case
  perturbation theory (Gluckstern Palumbo).
• Following Chaos analysis but in Cartesian
  coordinates we can get similar expression for all
  the fields. Ignoring space charge in
  ultrarelativistic limit we will get constant
  longitudinal electric field in the pipe.
  Integrating this over frequency we will get the
  same expression for 0 order longitudinal wake as
  in circular case.
• 1st order solutions based on Poisson equation and
  Leontovich boundary condition.
• Form factors received for rectangular case are
  quite similar to elliptical
• In fact we can utilize standard eigenfunctions
  approach developed for dielectric layered
  waveguides for rectangular case. Different set of
  eigenfunction - so we need to use different
  orthonormality relations to get normalization
  constant. This is under investigation now.

9
Merlin simulation

• Merlin - optical code with phenomenological wake
  fields to get the effect on beam transport
  through the linac and BDS
• Merlin had a process for handling wake fields. We
  inserted Wake functions.
• Bunch slices for wakes integration prepared and
  integration already implemented
• Momentum change on slice i

10
Merlin simulation

• Geometric wake for steep collimator, kick factor
• SLAC experiment Merlin simulation
  with up to 50 modes

11
Merlin simulation

• Resistive
• SLAC experiment Merlin simulations
  for 5 modes (1-Cu, 2- Ti)

12
Merlin simulation

• Dielectric wakes, in progress