Transcript The precision of absolute distance interferometry measurements

John Adams Institute
JAI
for Accelerator
Science
MONALISA:
The precision of absolute
distance interferometry
measurements
Matthew Warden, Paul Coe, David Urner, Armin Reichold
Photon 08, Edinburgh
Preliminaries
Concept
Results
Comparison
Conclusions
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
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
1/14
Concept
Preliminaries
Results
Comparison
Conclusions
2/14
Coherent ADI with a reference interferometer
intensity
Dmeas
f meas
laser frequency
time
  f
Dmeas
c

f meas
2 
time
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
Concept
Preliminaries
Results
Comparison
Conclusions
2/14
Coherent ADI with a reference interferometer
Dref
intensity
Typical signals
f ref
time
intensity
Dmeas
f meas
laser frequency
time
  f
time
c
Dmeas 
f meas
2 
c
Dref 
f ref
2 
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
Concept
Preliminaries
Results
Comparison
Conclusions
2/14
Coherent ADI with a reference interferometer
Dref
intensity
Typical signals
f ref
time
intensity
Dmeas
f meas
laser frequency
time
  f
time
c
Dmeas 
f meas
2 
c
Dref 
f ref
2 
Dmeas f meas
R

Dref
f ref
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
Introducing the
Cramér-Rao bound:
A tool to help understand
measurement uncertainty
Preliminaries
Concept
Results
Comparison
Conclusions
3/14
Methods to measure uncertainty
How precisely can this
distance ratio be measured?
Dmeas f meas
R

Dref
f ref
• Empirical: variance of repeated
measurements
• Can see how this varies with
certain parameters, e.g. signal to
noise ratio
• Analytical: Cramér-Rao bound
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
Preliminaries
Concept
Results
Comparison
Conclusions
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!
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
4/14
Preliminaries
Concept
Results
Comparison
Conclusions
How does it work?
Parameters
Frequency
Phase
Amplitude
• 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
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
5/14
Results:
Cramér-Rao bound calculations
Preliminaries
Concept
Results
Comparison
Conclusions
Cramér-Rao Bound – Linear Tuning
with perfect reference interferometer
intensity
Dmeas
time
R
1

R 4

 1 6c
1


 SNR D  N 
meas meas 

The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
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Preliminaries
Concept
Results
Comparison
Conclusions
Cramér-Rao Bound – Linear Tuning
intensity
Dref
time
intensity
Dmeas
time

 
R
1 
1
1






R 4  SNRmeas Dmeas   SNRref Dref


2




2




1
2
1
6c
N 
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
7/14
Preliminaries
Concept
Results
Comparison
Conclusions
Cramér-Rao Bound – Non-Linear Tuning
intensity
Dref
time
intensity
Dmeas
time
2

 
R
1 
1
1




R 4  SNRmeas Dmeas   SNRref Dref






2




1
2
1
c
N 2std ( )
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
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Preliminaries
Concept
Results
Comparison
Conclusions
9/14
(Cramér-Rao Bound – No phase quadrature)
intensity
Dref
time
intensity
Dmeas
time
Given (fairly loose) restrictions on signal spectra:
2

 
R
1 
2
2




R 4  SNRmeas Dmeas   SNRref Dref






2




1
2
1
c
N 2std ( )
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
Preliminaries
Concept
Results
Comparison
Conclusions
9/14
intensity
(Cramér-Rao Bound – No phase quadrature)
Hilbert Transform
or
Fourier Transform Technique
intensity
time
time
Given (fairly loose) restrictions on signal spectra:
2

 
R
1 
2
2




R 4  SNRmeas Dmeas   SNRref Dref






2




1
2
1
c
N 2std ( )
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
Preliminaries
Concept
Results
Comparison
Conclusions
10/14
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
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
Comparison with simulated and
experimental uncertainties
Preliminaries
Concept
Results
Comparison
Conclusions
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
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
11/14
Preliminaries
Concept
Results
Comparison
Conclusions
Comparison with simulation
Uncertainty vs:
Signal to noise ratio
Optical path difference
Number of samples
Frequency scan range
Frequency scan linearity

 
R
1 
1
1






R 4  SNRmeas Dmeas   SNRref Dref


2




2




1
2
1
c
N 2std ( )
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
12/14
Preliminaries
Concept
Results
Comparison
Conclusions
Comparison with experiment
• Can experimental
uncertainty reach the
predicted lower
bound?
• Not here, not yet!
• …But the uncertainty
scales as predicted!
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
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Preliminaries
Concept
Results
Comparison
Conclusions
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Conclusions
• Uncertainty often measured empirically
• Alternative: statistical method

1 

• Helps understand sources of uncertainty R 4 
• Provide benchmark for analysis algorithms
R
 1
6c


SNRmeas Dmeas  N 
1
• 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
The precision of absolute distance interferometry measurements - Matt Warden – Photon 08
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…
“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
Methods with all these names rely on the same basic principles.
Preliminaries
Introducing the CRB
Results
Simulation
Conclusions
Coherent ADI with a reference interferometer
1
1
intensity
A typical signal
time
2
2
OPDref
OPDmeas
2

OPD
c
c d
OPD 
2 d
OPDmeas d meas
R

OPDref
d ref
What is this tool?
How does it work?
The Cramér-Rao Bound
Analogy: least squares fitting
• Statistical tool
• Used in signal analysis e.g.
to find uncertainty in
frequency estimation
• ADI measurements involve
frequency estimation!
Without phase quadrature
Hilbert Transform
or
Fourier Transform Technique
Comparison with simulation
Varied:
Number of samples
Signal to noise ratio
Frequency scan range
Frequency scan linearity
Optical path difference