Ellipse fitting for interferometry Part 1: static methods

Collett, M. J.; Tee, Garry J.

doi:10.1364/josaa.31.002573

Cited by 9 publications

(18 citation statements)

References 16 publications

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“…Note also that the difference in mean values between the two methods is statistically significant (3σ for individual circuits, and hence over 20σ for the overall results). Figures 5 and 6 summarize results from a series of data sets with varying differential phase shifts, each fitted circuit by circuit by the dynamic method and four different static ones: Bookstein's [12], type-specific [4,5], conic pencil [13,14], and self-normalizing [1]. The preceding figures correspond to a polarizer angle of 42.0°; note that the small (about 7 mV) overshoot of the static self-normalizing method apparent in Figs.…”

Section: Resultsmentioning

confidence: 99%

“…For all but the first, an obvious choice is to carry forward the result of fitting the previous segment. For the first segment (and as an alternative choice for others), we may use a static method: in general the self-normalizing method [1] will be the most accurate, but for poor-quality data, the greater stability of the type-specific method [4,5] may sometimes be useful.…”

Section: Windowing and Preliminary Fittingmentioning

confidence: 99%

“…From the preliminary estimate of the ellipse parameters, we may obtain an initial set of phase valuesθ i using the corresponding elliptical coordinates, as summarized in Section 2.D of [1]. Specifically, we have…”

Section: Phase Unwrapping and Smoothingmentioning

confidence: 99%

“…In the first work in this series [1], we surveyed a number of different methods of fitting an ellipse to measured data points, specifically considering their suitability for interferometric applications. Readers are referred to that work and references therein for general background to the problem.…”

Section: Introductionmentioning

confidence: 99%

“…The "static" methods treated in [1] use only the locations of the data points, independent of the sequence in which they were measured. Recognizing that the data are in fact a well-behaved time series of measurements allows us significantly to improve both the accuracy and the statistical reliability of the fitted ellipses.…”

Section: Introductionmentioning

confidence: 99%

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Ellipse fitting for interferometry Part 3: dynamic method

Collett

Watkins

2015

J. Opt. Soc. Am. A

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We describe a dynamically based method for fitting an ellipse to noisy data, which has for interferometric applications a number of advantages over conventional static methods (originally developed for image processing). Our method relies on the observation that each data point belongs to an ordered time series and thus has a well-defined phase parameter. We demonstrate that for real experimental data it can achieve much greater accuracy than static methods. The precision of the fit is limited only by the statistical reliability of the data, even in extreme cases such as ellipses with a minor axis smaller than the measurement noise.

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Section: Resultsmentioning

confidence: 99%

Section: Windowing and Preliminary Fittingmentioning

confidence: 99%

Section: Phase Unwrapping and Smoothingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ellipse fitting for interferometry Part 3: dynamic method

Collett

Watkins

2015

J. Opt. Soc. Am. A

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Contributed Review: A review of compact interferometers

Watchi

Cooper

Ding

et al. 2018

Review of Scientific Instruments

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Compact interferometers, called phasemeters, make it possible to operate over a large range while ensuring a high resolution. Such performance is required for the stabilization of large instruments dedicated to experimental physics such as gravitational wave detectors. This paper aims at presenting the working principle of the different types of phasemeters developed in the literature. These devices can be classified into two categories: homodyne and heterodyne interferometers. Improvement of resolution and accuracy has been studied for both devices. Resolution is related to the noise sources that are added to the signal. Accuracy corresponds to distortion of the phase measured with respect to the real phase, called non-linearity. The solutions proposed to improve the device resolution and accuracy are discussed based on a comparison of the reached resolutions and of the residual non-linearities.

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The statistical uncertainty of the Heydemann correction: a practical limit of optical quadrature homodyne interferometry

Köning

Wimmer

Witkovský

2015

Meas. Sci. Technol.

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Although the Heydemann correction is widely used to demodulate the phase of quadrature homodyne interferometer and encoder signals, the related measurement uncertainty is considered only in a few publications. The statistical uncertainty of the phase, which is the contribution of the high-frequency noise in the input signals, due to the use of a simplified fitting approach in the Heydemann correction, cannot be determined exactly. Therefore, neglecting the correlation between the input signals and the fit parameters, an approximation is provided and verified using Monte Carlo simulations and an iterative fitting method that allows exact determination of the statistical uncertainty of the phase. For the high-quality signals encountered in dimensional metrology applications, only minor differences of a few percent in the uncertainty estimates are observed. Furthermore, for cases in which a number of points used in the fit is sufficiently large, a new, simple, analytic expression for the statistical uncertainty of the phase is derived. It represents a practical limit of optical quadrature displacement interferometry, which already has been reached experimentally. To achieve an expanded measurement uncertainty of 10 pm a signal-to-noise greater than 104 has to be realized experimentally.

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Ellipse fitting for interferometry Part 1: static methods

Cited by 9 publications

References 16 publications

Ellipse fitting for interferometry Part 3: dynamic method

Ellipse fitting for interferometry Part 3: dynamic method

Contributed Review: A review of compact interferometers

The statistical uncertainty of the Heydemann correction: a practical limit of optical quadrature homodyne interferometry

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