MONALISA: The precision of absolute distance interferometry measurements Matthew Warden, Paul Coe, David Urner, Armin Reichold Photon 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 Comparison Preliminaries ConceptResultsConclusions The precision of absolute distance interferometry measurements - Matt Warden – Photon 08 1/14

Coherent ADI with a reference interferometer intensity time laser frequency Comparison Preliminaries ConceptResultsConclusions2/14 The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

Coherent ADI with a reference interferometer time intensity Typical signals intensity time laser frequency Comparison Preliminaries ConceptResultsConclusions2/14 The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

Coherent ADI with a reference interferometer time intensity Typical signals intensity time laser frequency Comparison Preliminaries ConceptResultsConclusions2/14 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 Comparison Preliminaries ConceptResultsConclusions3/14 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! Comparison Preliminaries ConceptResultsConclusions4/14 The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

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 Comparison Preliminaries ConceptResultsConclusions5/14 The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

Results: Cramér-Rao bound calculations

Cramér-Rao Bound – Linear Tuning with perfect reference interferometer intensity time Comparison Preliminaries ConceptResultsConclusions6/14 The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

Cramér-Rao Bound – Linear Tuning time intensity time Comparison Preliminaries ConceptResultsConclusions7/14 The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

Cramér-Rao Bound – Non-Linear Tuning time intensity time Comparison Preliminaries ConceptResultsConclusions8/14 The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

(Cramér-Rao Bound – No phase quadrature) time intensity time Given (fairly loose) restrictions on signal spectra: Comparison Preliminaries ConceptResultsConclusions9/14 The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

(Cramér-Rao Bound – No phase quadrature) time intensity time Given (fairly loose) restrictions on signal spectra: time intensity time Hilbert Transform or Fourier Transform Technique Comparison Preliminaries ConceptResultsConclusions9/14 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 Comparison Preliminaries ConceptResultsConclusions10/14 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 Comparison Preliminaries ConceptResultsConclusions11/14 The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

Comparison with simulation Uncertainty vs: Signal to noise ratio Optical path difference Number of samples Frequency scan range Frequency scan linearity Comparison Preliminaries ConceptResultsConclusions12/14 The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

Comparison with experiment Comparison Preliminaries ConceptResultsConclusions13/14 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 Comparison Preliminaries ConceptResultsConclusions14/14 The precision of absolute distance interferometry measurements - Matt Warden – Photon 08

References Statistical Inference, Prentice Hall, 1995, ISBN 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 ADIAbsolute Distance Interferometry FSIFrequency Scanning Interferometry WSIWavelength Shifting Interferometry FMCWFrequency Modulated Continuous Wave OFDROptical Frequency Domain Reflectometry VSWVariable Synthetic Wavelength Names… Methods with all these names rely on the same basic principles.

Coherent ADI with a reference interferometer Simulation PreliminariesIntroducing the CRBResults Conclusions time intensity 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 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