Ellipse fitting by nonlinear constraints to demodulate quadrature homodyne interferometer signals and to determine the statistical uncertainty of the interferometric phase

摘要

Optical interferometers are widely used in dimensional metrology applications. Among them are many quadrature homodyne interferometers. These exhibit two sinusoidal interference signals shifted, in the ideal case, by 90° to allow a direction sensitive detection of the motion responsible for the actual phase change. But practically encountered signals exhibit additional offsets, unequal amplitudes and a phase shift that differs from 90°. In order to demodulate the interference signals the so called Heydemann correction is used in almost all cases, i.e. an ellipse is fitted to both signals simultaneously to obtain the offsets, amplitude and the phase lag. Such methods are typically based on a simplified least squares fit that leads to a system of linear equations, which can be solved directly in one step. Although many papers related to this subject have been published already only a few of them consider the uncertainties related to this demodulation scheme. In this paper we propose a new method for fitting the ellipse, based on minimization of the geometric distance between the measured and fitted signal values, which provides locally best linear unbiased estimators (BLUEs) of the unknown model parameters, and simultaneously also estimates of the related statistical uncertainties, including the uncertainties of estimated phases and/or displacements.