monalisa: the precision of absolute distance interferometry measurements

MONALISA:The precision of absolute distance interferometry

measurements

for AcceleratorScience

John Adams Institute

J A I

Matthew Warden, Paul Coe, David Urner, Armin ReicholdPhoton 08, Edinburgh

Why are we interested in optical metrology?

• Particle accelerators contain systems of magnetic lenses and prisms to focus and steer the beam

• beam trajectory affects accelerator performance• When magnets move the trajectory is altered• optical metrology to monitor magnet positions• Absolute distance interferometry (ADI) used

ComparisonPreliminaries Concept Results Conclusions

The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

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Coherent ADI with a reference interferometer

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ComparisonPreliminaries Concept Results Conclusions 2/14



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Typical signals

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Typical signals

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Introducing the Cramér-Rao bound:

A tool to help understand measurement uncertainty

Methods to measure uncertainty

How precisely can this distance ratio be measured?

• Empirical: variance of repeated measurements

• Can see how this varies with certain parameters, e.g. signal to noise ratio

• Analytical: Cramér-Rao bound

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What is the Cramér-Rao Bound?

• Statistical tool• Used in signal analysis• e.g. to find uncertainty of frequency estimation

• ADI measurements involve frequency estimation!



How does it work?

Parameters

FrequencyPhaseAmplitude

• Calculation revolves around variations in the likelihood of getting the data you got, given certain parameter values

• Narrow range of likely parameters Low uncertainty• Wide range of likely parameters High uncertainty

• Lower bound on uncertainty of unbiased estimators



Results:

Cramér-Rao bound calculations

Cramér-Rao Bound – Linear Tuningwith perfect reference interferometer

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Cramér-Rao Bound – Linear TuningrefD

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Cramér-Rao Bound – Non-Linear Tuning

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(Cramér-Rao Bound – No phase quadrature)

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Given (fairly loose) restrictions on signal spectra:

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(Cramér-Rao Bound – No phase quadrature)

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Given (fairly loose) restrictions on signal spectra:

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Fourier Transform Technique

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How these result should and should not be used

• Calculates minimum uncertainty for simplified situation• In real life, other sources of error could be dominant• So may not achieve this lower uncertainty limit

• This result useful for:– Occasions when the considered random errors are dominant– Benchmark for testing analysis algorithms

• Potential to extend model to other random error sources



Comparison with simulated and experimental uncertainties

Simulation

• Wish to check an analysis method to see if it acheives the CRB

• Analysis method is just a linear fit to interferometer phases, calculated from phase quadrature readouts



Comparison with simulationUncertainty vs:

Signal to noise ratio

Optical path difference

Number of samples

Frequency scan range

Frequency scan linearity

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Comparison with experiment



• Can experimental uncertainty reach the predicted lower bound?

• Not here, not yet!• …But the uncertainty

scales as predicted!

Conclusions

• Uncertainty often measured empirically

• Alternative: statistical method• Helps understand sources of uncertainty• Provide benchmark for analysis algorithms

• Calculated Cramér-Rao bound for certain situations

• Tested analysis method against them

• Need to include more sources of uncertainty

Group Website: www-pnp.physics.ox.ac.uk/~monalisa



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References

Statistical Inference, Prentice Hall, 1995, ISBN 0-13-847260-2

Paul H. Garthwaite, Ian T. Jolliffe, Byron Jones

Single-Tone Parameter Estimation from Discrete-Time Observations, David C. Rife,

IEEE Transactions on information theory, Vol 20, No 5, Sept 1974

“Names are not always what they seem. The common Welsh name BZJXXLLWCP is pronounced Jackson.”

- Mark Twain

ADI Absolute Distance Interferometry

FSI Frequency Scanning Interferometry

WSI Wavelength Shifting Interferometry

FMCW Frequency Modulated Continuous Wave

OFDR Optical Frequency Domain Reflectometry

VSW Variable Synthetic Wavelength

Names…

Methods with all these names rely on the same basic principles.


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SimulationPreliminaries Introducing the CRB Results Conclusions

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A typical signal

What is this tool? How does it work?

The Cramér-Rao Bound

• Statistical tool• Used in signal analysis e.g.

to find uncertainty in frequency estimation

• ADI measurements involve frequency estimation!

Analogy: least squares fitting

Without phase quadrature

Hilbert Transformor

Fourier Transform Technique

Comparison with simulationVaried:

Number of samples

Signal to noise ratio

Frequency scan range

Frequency scan linearity

Optical path difference

monalisa: the precision of absolute distance interferometry measurements

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