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MONALISA: The precision of absolute distance interferometry measurements

MONALISA: The precision of absolute distance interferometry measurements. Matthew Warden , Paul Coe, David Urner, Armin Reichold Photon 08, Edinburgh. Concept. Results. Comparison. Conclusions. 1/14. Preliminaries. Why are we interested in optical metrology?.

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MONALISA: The precision of absolute distance interferometry measurements

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  1. MONALISA:The precision of absolute distance interferometry measurements Matthew Warden, Paul Coe, David Urner, Armin Reichold Photon 08, Edinburgh

  2. Concept Results Comparison Conclusions 1/14 Preliminaries 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

  3. Concept Results Comparison Conclusions 2/14 Preliminaries Coherent ADI with a reference interferometer intensity time laser frequency time The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

  4. Concept Results Comparison Conclusions 2/14 Preliminaries Coherent ADI with a reference interferometer Typical signals intensity time intensity time laser frequency time The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

  5. Concept Results Comparison Conclusions 2/14 Preliminaries Coherent ADI with a reference interferometer Typical signals intensity time intensity time laser frequency time The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

  6. Introducing the Cramér-Rao bound:A tool to help understand measurement uncertainty

  7. Concept Results Comparison Conclusions 3/14 Preliminaries 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 The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

  8. Concept Results Comparison Conclusions 4/14 Preliminaries 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

  9. Concept Results Comparison Conclusions 5/14 Preliminaries 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

  10. Results:Cramér-Rao bound calculations

  11. Concept Results Comparison Conclusions 6/14 Preliminaries Cramér-Rao Bound – Linear Tuningwith perfect reference interferometer intensity time The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

  12. Concept Results Comparison Conclusions 7/14 Preliminaries Cramér-Rao Bound – Linear Tuning intensity time intensity time The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

  13. Concept Results Comparison Conclusions 8/14 Preliminaries intensity time intensity time Cramér-Rao Bound – Non-Linear Tuning The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

  14. Concept Results Comparison Conclusions 9/14 Preliminaries (Cramér-Rao Bound – No phase quadrature) intensity time intensity time Given (fairly loose) restrictions on signal spectra: The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

  15. Concept Results Comparison Conclusions 9/14 Preliminaries intensity intensity Hilbert Transform or Fourier Transform Technique time time intensity intensity time time (Cramér-Rao Bound – No phase quadrature) Given (fairly loose) restrictions on signal spectra: The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

  16. Concept Results Comparison Conclusions 10/14 Preliminaries 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

  17. Comparison with simulated and experimental uncertainties

  18. Concept Results Comparison Conclusions 11/14 Preliminaries 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

  19. Concept Results Comparison Conclusions 12/14 Preliminaries Comparison with simulation Uncertainty vs: Signal to noise ratio Optical path difference Number of samples Frequency scan range Frequency scan linearity The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

  20. Concept Results Comparison Conclusions 13/14 Preliminaries 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

  21. Concept Results Comparison Conclusions 14/14 Preliminaries 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 The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

  22. 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

  23. Names… “Names are not always what they seem. The common Welsh name BZJXXLLWCP is pronounced Jackson.” - Mark Twain Methods with all these names rely on the same basic principles.

  24. Preliminaries Introducing the CRB Results Simulation Conclusions Coherent ADI with a reference interferometer A typical signal intensity time

  25. 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

  26. Without phase quadrature Hilbert Transform or Fourier Transform Technique

  27. Comparison with simulation Varied: Number of samples Signal to noise ratio Frequency scan range Frequency scan linearity Optical path difference

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