Cavity BPM

A. Liapine, UCL

Beam

A Black Box View

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 nanoBPM Meeting, KEK, March 2005 2

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 nanoBPM Meeting, KEK, March 2005 3

Excursion Into Waveguides (2)

► ► indexes show the numbers of antinodes of the field for both axes - x, y for a rectangular, φ, r for a circular waveguide the number of antinodes for the φ direction is a doubled index (the field must be continuous among φ) ► ► ► 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 nanoBPM Meeting, KEK, March 2005 4

Circular Waveguide

We solve the wave equation in the cylindrical coordinate system Look for the solutions in a form integrating by parts. Solutions are: Transversal components follow from the Maxwell’s equations The boundary condition gives the critical k: nanoBPM Meeting, KEK, March 2005 5

Cylindrical Cavity Resonator

A cylindrical cavity is a piece of a circular waveguide cut transversally with conductive planes at z=0 and z=L.

At these planes the sum of the transversal components of the electric field has to be 0: This boundary condition says us that …called eigenmodes and coinciding frequencies called eigenfrequencies In that way we get the equations describing all the standing waves possible in the cavity… nanoBPM Meeting, KEK, March 2005 6

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 r=0, is very sensitive to the beam offset, what is used for the beam position detection.

The first dipole mode TM direction of the offset.

110 is used because it is the strongest one among the others.

The phase of the excited field depends on the nanoBPM Meeting, KEK, March 2005 7

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.

L R C n The voltage in the cavity is calculated among a certain path, beam trajectory in our case.

The internal quality factor is introduced to indicate the decay of the oscillations due to the losses in the cavity walls.

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 The energy given by the bunch to the mode n is and get the excited voltage and stored energy as 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 nanoBPM Meeting, KEK, March 2005 9

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 r=a 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 nanoBPM Meeting, KEK, March 2005 11

Multibunch Excitation

Exponential decay of the energy stored in the cavity is given by V Were the loaded Q value is used. It takes into account walls losses and output power V t 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 t A fixed error gives a high limit for the loaded Q value nanoBPM Meeting, KEK, March 2005 12

Beam Incline Impact

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.

The ratio of the voltages does not depend on the bunch charge Equivalent offset for a 5.5 GHz cavity (x’ = 0.5 mrad) Approximating the Bessel function we get nanoBPM Meeting, KEK, March 2005 13

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.

 nanoBPM Meeting, KEK, March 2005 15

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 90 0 .

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. nanoBPM Meeting, KEK, March 2005 20

I’m afraid that’s all I can say…

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Check also http://www.hep.ucl.ac.uk/~liapine/

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