Cavity%20BPM

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Cavity BPM

• A. Liapine, UCL

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A Black Box View
Beam
z
out
BPM signal is a mixture of decaying harmonic
signals with different amplitudes and decay
times. Some of the amplitudes depend mostly on
the bunch charge, some have a strong offset
dependence
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Excursion Into Waveguides (1)
The electromagnetic field is known to propagate
through a waveguide as a wave (or a mixture of a
few waves) with a fixed configuration. This
configuration depends on the frequency of
oscillations, waveguide type and excitation type.

• electric field is shown with red lines, magnetic
  with blue ones
• wave is Transverse Electric - the electric field
  has no longitudinal component (in some literature
  it is marked as H-wave)
• the direction of the propagation is given by E x
  H
• nodes and antinodes of transversal components of
  E and H coincide in case of vacuum filling

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Excursion Into Waveguides (2)

• indexes show the numbers of antinodes of the
  field for both axes - x, y for a rectangular, f,
  r for a circular waveguide
• the number of antinodes for the f direction is a
  doubled index (the field must be continuous among
  f)

• magnetic coupling uses a loop acting to the
  magnetic field. The coupling strength depends on
  the magnetic flux through the loop i.e.
  inductivity of the loop
• electric coupling uses an antenna , the coupling
  depends on its capacitance
• electromagnetic coupling is a sum of two
  electric and magnetic, they may sometimes even
  cancel each other

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Circular Waveguide
We solve the wave equation
in the cylindrical coordinate system
Look for the solutions in a form
Transversal components follow from the
Maxwells equations
integrating by parts. Solutions are
The boundary condition gives the critical k
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Cylindrical Cavity Resonator
A cylindrical cavity is a piece of a circular
waveguide cut transversally with conductive
planes at z0 and zL. At these planes the sum of
the transversal components of the electric field
has to be 0
This boundary condition says us that
In that way we get the equations describing all
the standing waves possible in the cavity
called eigenmodes and coinciding frequencies
called eigenfrequencies
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Dipole Mode
A bunch propagating through the cavity interacts
with its eigenmodes exciting electromagnetic
oscillations in the cavity. The excitation of the
modes, which have a node at r0, is very
sensitive to the beam offset, what is used for
the beam position detection. The first dipole
mode TM110 is used because it is the strongest
one among the others. The phase of the excited
field depends on the direction of the offset.
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Useful Definitions
It is convenient to represent a cavity as an RLC
circuit, usually loaded to external load by means
of an ideal transformator. The impedance R is
called the shunt impedance.
The internal quality factor is introduced to
indicate the decay of the oscillations due to the
losses in the cavity walls.
The voltage in the cavity is calculated among a
certain path, beam trajectory in our case.
The external quality factor indicates the decay
due to the power coupled out of the cavity.
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Single Bunch Excitation (1)
The excitation is proportional to the voltage
seen by the bunch
We use the definition of the normalized shunt
impedance
and get the excited voltage and stored energy as
The energy given by the bunch to the mode n is
The voltage excited in the cavity is two times
higher
Using the definition of the external Q we get the
output power
With we get the output voltage
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Single Bunch Excitation (2)
The dipole mode electric field in the cavity
Extension to the beam pipe region
Fit both fields in order to get constants at ra
Using an integral
we get
And the field in the beampipe is
We need the voltage, so integrating and using
again we get
The voltage is linear vs. offset!
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Excitation Summary

• The bunch excites the eigenmodes of the cavity
  passing through it
• The dipole mode excitation has a significant
  dependence on the beam offset, the phase depends
  on the offset direction
• The excited signal decays exponentially,
  depending on how much power is lost in the walls
  and coupled out
• The excitation is almost linear in the beam pipe
  range

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Multibunch Excitation
Exponential decay of the energy stored in the
cavity is given by
Were the loaded Q value is used. It takes into
account walls losses and output power
If the mode frequency is a harmonic of the bunch
repetition rate, an infinite bunch train produces
a voltage
The error can be calculated as
The sum of this series is
A fixed error gives a high limit for the loaded Q
value
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Beam Incline Impact
The ratio of the voltages does not depend on the
bunch charge
Incline component of the dipole mode is excited
if the beam trajectory is inclined with respect
to the z axis of the cavity. We compare the
excitation calculating the voltages for the both
cases.
Equivalent offset for a 5.5 GHz cavity (x 0.5
mrad)
Approximating the Bessel function we get
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Monopole Modes Impact
Monopole modes have the highest excitation among
all other modes. The difference to the dipole
mode excitation maybe 100 dB and more. The first
two monopole modes surround the dipole mode
resonance.
Due to the finite Q values these modes have
components at the dipole mode frequency. These
components can not be filtered out and need a
mode selective solution. A mode selective
coupling realizing the difference in the field
structure of dipole and monopole modes is used in
all the latest designs.
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Polarization and Cross-Talk
?
The excited dipole mode field can be repre-sented
as a combination of two polarizations.
?
Need to align the polarizations to x, y and
separate them in frequency.
?
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Thermal Noise
The spectral noise power density integrated over
the bandwidth of the narrowest filter in the
electronics gives us the level of the noise
component
Following the path of the signal in the
electronics and taking into account the losses
and the internal noise of the electronics we can
estimate the resolution limit
The final estimation has to take into account
also the discretization noise.
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Impacts Summary

• The energy stored in the cavity decays
  exponentially. If the decay is not fast enough,
  the previous bunch signal contributes to the next
  bunch signal.
• An inclined beam excites the dipole mode even if
  it passes through the centre. The phase
  difference between position and incline
  components is 900.
• Monopole modes are strongly excited and therefore
  generate large backgrounds.
• Asymmetries cause a coupling between x and y
  signals.

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Analog Signal Processing

• The readings are waveforms in GHz range, so we
  need a downconversion electronics. Basically, two
  methods are available
• homodyne receiver
• heterodyne receiver.
• An accurate direct conversion is not possible
  because of the high frequency.

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Homodyne Receiver
The signal is downconverted to the direct
current in one stage. Just a few components are
needed, the losses are low.
HR is very sensitive to the isolations between LO
and RF ports of the mixer. I/Q mixer is usually
used.
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Heterodyne Receiver
Downconversion is realized in several stages.
That gives a better possibility for the filtering
and amplification of the signal. The mirror
frequency issue does not seem to be really
dangerous in our case.
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Im afraid thats all I can say

• Check also
• http//www.hep.ucl.ac.uk/liapine/