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Beam-Wave Coupling

- Wakefields occur where ever there is coupling between the beam and RF at any frequency or mode.
- Like acceleration, beam-wave coupling can only occur when the RF is resonant with the beam.
- This means either the wave must have a phase velocity equal to the speed of the beam or the beam must only experience the RF a fraction of the time.

In multi-cell cavities or in waveguide a dispersion diagram can be used to find resonances.

Resonances occur where ever the light line (Phase Advance=kzL =wL/vbeam) crosses the modes dispersion line (ie phase velocity=beam velocity).

Generation of RF Current

The negative potential difference causes the electrons to slow down and the energy is absorbed into the cavity

C

A

The lower energy electrons then pass through the cavity and force the electrons within the metal to flow back to the opposite side

A bunch of electrons approaches a resonant cavity and forces the electrons to flow away from the bunch.

B

Pancake effect

- Due to lorentz contraction the electric fields of the bunch are almost entirely perpendicular to the bunch.
- This means wakefields cannot affect charges in front of itself only behind.

Bunch Spectrum

- A charged bunch can induce wakefields over a wide spectrum given by, fmax=1/T. A Gaussian bunch length has a Gaussian spectrum.
- On the short timescale (within the bunch) all the frequencies induced can act on following electrons within the bunch.
- On a longer timescale (between bunches) the high frequencies decay and only trapped low frequency (high Q) modes participate in the interaction.

Coupling Impedance

Narrowband Impedance

Broadband Impedance

Cut-off

Impedance

frequency

The fourier transform of a wakefield is the coupling impedance. It has three regions (shown above).

The broadband impedance region doesn’t look like it has much of an effect but it covers a huge frequency spectrum so it’s integral can dwarf the narrowband region.

Fundamental Theorem of Beam Loading

- A charge, q, traverses a cavity and induces a voltage, Vc, in a mode of the cavity.
- What proportion of Vc is seen by the charge?
- If the voltage in the mode was initially zero and has voltage Vq when the charge leaves the cavity, the average voltage is Vq/2. This is the voltage seen by the charge.
- This is known as the fundamental theorem of beam loading.
- If a cavity has an initial voltage, Vc, in a mode and a bunch passes and induced an additional voltage Vq, the bunch will loose energy

Mode Excitation

If we have a mode in a cavity with initial voltage Vc and a bunch traverses the cavity inducing an additional, superimposed, voltage Vq, The modal energy is

And by conservation of energy

As Vq is proportional to q we can group the terms in powers of q and q2. The q2 term gives

As the power of q term must balance a must be 2p times an integer, hence

Short range wakes

Short range wakes are the wakefields acting inside the bunch, where the charge in the head induce wakes that act on the tail of the bunch.

For short length bunches, this is dominated by the narrowband impedance.

The wake is not a step for a real bunch (shown above) as the charge density varies as a Gaussian.

Single Bunch Wake

A mode excited by a single bunch will decay exponentially with time due to ohmic heating and external coupling.

Wake (V)

The single bunch will excite several modes each with different beam coupling and damping rates.

Bunch Separation km

Dipole modes

Dipole mode have a transverse magnetic and/or transverse electric fields on axis. They have zero longitudinal field on axis. The longitudinal electric field increases approximately linearly with radius near the axis.

Electric Magnetic

Wakefields are only induced by the longitudinal electric field so dipole wakes are only induced by off-axis bunches.

Once induced the dipole wakes can apply a kick via the transverse fields so on-axis bunches can still experience the effect of the wakes from preceding bunches.

Panofsky-Wenzel Theorem

If we rearrange Farday’s Law ( )and integrating along z we can show

Inserting this into the Lorentz (transverse( force equation gives us

for a closed cavity where the 1st term on the RHS is zero at the limits of the integration due to the boundary conditions this can be shown to give

This means the transverse voltage is given by the rate of change of the longitudinal voltage

Multibunch Wakefields

- For multibunch wakes, each bunch induces the same frequencies at different amplitudes and phases.
- These interfere to increase or decrease the fields in the cavity.
- As the fields are damped the wakes will tend to a steady state solution.

Long Range Wakefields

- The long range wakes are found by summing over all modes and all bunches
- This is known as the sum wake.
- The kick from a dipole mode can then be found using the equation below

The time taken to converge to this value is dependant on the modal Q factors.

The long range wakefield is a sum of damped periodic oscillations and hence converges to a finite value.

Long Range Transverse WakesHorizontal kick for 4 offset.

Vertical kick for 4 offset.

9-cell

9-cell

Resonances modal Q factors.

- As you are summing the contribution to the wake from all previous bunches, resonances can appear. For monopole modes we sum
- Hence resonances appear when
- It is more complex for dipole modes as the sum is
- This leads to two resonances at +/-some Δfreq from the monopole resonant condition.

Effect of frequency errors modal Q factors.

Cavity

Deflected beam

The effect of frequency errors can be estimated by introducing small variations in the bunch spacing.

X’

Undeflected beam

X’

Wakefield Kick

Tolerance

As can be seen if the beam hits a resonance with a HOM the wakefields increase significantly.

% variation in bunch separation

Damping modal Q factors.

- As the wakes from each bunch add together it is necessary to damp the wakes so that wakes from only a few bunches add together.
- The smaller the bunch spacing the stronger the damping is required (NC linacs can require Q factors below 50).
- This is normally achieved by adding external HOM couplers to the cavity.
- These are normally quite complex as they must work over a wide frequency range while not coupling to the operating mode.
- However the do not need to handle as much power as an input coupler.

I modal Q factors.

Cs

R

Coaxial HOM couplersHOM couplers can be represented by equivalent circuits. If the coupler couples to the electric field the current source is the electric field (induced by the beam in the cavity) integrated across the inner conductor surface area.

If the coaxial coupler is bent at the tip to produce a loop it can coupler to the magnetic fields of the cavity. Here the voltage source is the induced emf from the time varying magnetic field and the inductor is the loops inductance.

L

V

R

L modal Q factors.

L

I

Cs

R

I

Cs

R

Cf

Loop HOM couplersInductive stubs to probe couplers can be added for impedance matching to the load at a single frequency or capacitive gaps can be added to loop couplers.

Also capacitive gaps can be added to the stub or loop inductance to make resonant filters.

The drawback of stubs and capacitive gaps is that you get increase fields in the coupler (hence field emission and heating) and the complex fields can give rise to an electron discharge know as multipactor (see lecture 6).

As a result these methods are not employed on high current machines.

F-probe couplers modal Q factors.

F-probe couplers are a type of co-axial coupler, commonly used to damp HOM’s in superconducting cavities.

Their complex shapes are designed to give the coupler additional capacitances and inductances.

These additional capacatances and inductances form resonances which can increase or decrease the coupling at specific frequencies.

Capacative gaps

Inductive stubs

Output antenna

Log[S21]

The LRC circuit can be used to reduce coupling to the operating mode (which we do not wish to damp) or to increase coupling at dangerous HOM’s.

frequency

waveguide 2 modal Q factors.

waveguide 1

w2/2

w1/2

Waveguide CouplersWaveguide HOM couplers allow higher power flow than co-axial couplers and tend to be used in high current systems. They also have a natural cut-off frequency.

They also tend to be larger than co-axial couplers so are not used for lower current systems.

To avoid taking the waveguides through the cryomodule, ferrite dampers are often placed in the waveguides to absorb all incident power.

load modal Q factors.

choke

cavity

Choke DampingFor high gradient accelerators, choke mode damping has been proposed. This design uses a ferite damper inside the cavity which is shielded from the operating mode using a ‘choke’. A Choke is a type of resonant filter that excludes certain frequencies from passing.

The advantage of this is simpler (axially-symmetric) manufacturing

Beampipe HOM Dampers modal Q factors.

For really strong HOM damping we can place ferrite dampers directly in the beampipes. This needs a complicated engineering design to deal with the heating effects.

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