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

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.