monalisa: the precision of absolute distance interferometry measurements
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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
1/14
Coherent ADI with a reference interferometer
measmeas fc
D
2
measD
inte
nsity
time
time
lase
r fr
eque
ncy
f
measf
ComparisonPreliminaries Concept Results Conclusions 2/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
Coherent ADI with a reference interferometer
measmeas fc
D
2
refD
measDtime
inte
nsity
Typical signals
inte
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time
refref fc
D
2time
lase
r fr
eque
ncy
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measf
reff
ComparisonPreliminaries Concept Results Conclusions 2/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
Coherent ADI with a reference interferometer
measmeas fc
D
2
refD
measD
ref
meas
ref
meas
f
f
D
DR
time
inte
nsity
Typical signals
inte
nsity
time
refref fc
D
2time
lase
r fr
eque
ncy
f
measf
reff
ComparisonPreliminaries Concept Results Conclusions 2/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
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
ref
meas
ref
meas
f
f
D
DR
ComparisonPreliminaries Concept Results Conclusions 3/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
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!
ComparisonPreliminaries Concept Results Conclusions 4/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
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
ComparisonPreliminaries Concept Results Conclusions 5/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
Results:
Cramér-Rao bound calculations
Cramér-Rao Bound – Linear Tuningwith perfect reference interferometer
measD
inte
nsity
time
c
NDSNRR measmeas
R 611
4
1
ComparisonPreliminaries Concept Results Conclusions 6/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
Cramér-Rao Bound – Linear TuningrefD
measDtime
inte
nsity
inte
nsity
time
c
NDSNRDSNRR refrefmeasmeas
R 6111
4
121
22
ComparisonPreliminaries Concept Results Conclusions 7/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
)(2
111
4
121
22
std
c
NDSNRDSNRR refrefmeasmeas
R
Cramér-Rao Bound – Non-Linear Tuning
refD
measDtime
inte
nsity
inte
nsity
time
ComparisonPreliminaries Concept Results Conclusions 8/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
(Cramér-Rao Bound – No phase quadrature)
refD
measDtime
inte
nsity
inte
nsity
time
Given (fairly loose) restrictions on signal spectra:
)(2
122
4
121
22
std
c
NDSNRDSNRR refrefmeasmeas
R
ComparisonPreliminaries Concept Results Conclusions 9/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
(Cramér-Rao Bound – No phase quadrature)
time
inte
nsity
inte
nsity
time
Given (fairly loose) restrictions on signal spectra:
time
inte
nsity
inte
nsity
time
Hilbert Transformor
Fourier Transform Technique
)(2
122
4
121
22
std
c
NDSNRDSNRR refrefmeasmeas
R
ComparisonPreliminaries Concept Results Conclusions 9/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
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
ComparisonPreliminaries Concept Results Conclusions 10/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
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
ComparisonPreliminaries Concept Results Conclusions 11/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
Comparison with simulationUncertainty vs:
Signal to noise ratio
Optical path difference
Number of samples
Frequency scan range
Frequency scan linearity
)(2
111
4
121
22
std
c
NDSNRDSNRR refrefmeasmeas
R
ComparisonPreliminaries Concept Results Conclusions 12/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
Comparison with experiment
ComparisonPreliminaries Concept Results Conclusions 13/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
• 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
ComparisonPreliminaries Concept Results Conclusions 14/14
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
c
NDSNRR measmeas
R 611
4
1
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.
Coherent ADI with a reference interferometer
OPD
c
2
ddc
OPD
2
SimulationPreliminaries Introducing the CRB Results Conclusions
refOPD21
measOPD21
ref
meas
ref
meas
d
d
OPD
OPDR
time
inte
nsity
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
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