Enhanced ellipse fitting in a two-detector homodyne quadrature laser interferometer

Abstract

The choice of fitting methods for elliptically scattered data obtained with displacement-measuring homodyne quadrature laser interferometers significantly influences the accuracy of the interferometer. This is especially important when the data contain a lot of noise or provide only a segment of the ellipse. The ellipse parameters extracted by the fitting are used either to correct the data or the basic arctangent phase-unwrapping function in order to enhance the accuracy of the measured displacement by reducing the common nonlinearities. We propose the use of linear, ellipse-specific, least-squares fitting that is further bias-corrected using a linear algorithm. This stable fitting method provides a good balance between the accuracy of the fit and the computational efficiency, and never returns corrupt, non-ellipse parameters. It is therefore applicable for an online, uniform fringe subdivision when there is a demand for sub-nanometric resolution. An experimental confirmation of the improvement over traditional fitting methods was carried out with a single-pass, two-detector homodyne quadrature laser interferometer. We were able to operate the interferometer with nanometric accuracy, provided the data draw out at least a quarter-arc of an ellipse.

Ellipse fitting for interferometry Part 1: static methods

“…Extracting quantities of interest, such as the interferometric phase shift, then requires an accurate determination of the parameters of the ellipse (e.g., [1][2][3][4][5]). Mathematical treatments of methods to fit an ellipse to noisy data (e.g., [6][7][8][9]) have mainly been in the context of image processing or computer vision.…”

“…For image processing, this minimization is often (though not always) itself the desired end; for interferometry, it is only ever a proxy for the accurate determination of certain ellipse parameters. 5. In images, most ellipses may be expected to be of lowto-moderate eccentricity; in high-precision interferometry, there is particular interest in extremely eccentric ellipses, corresponding to very small phase shifts.…”

Detection of Interference Phase by Digital Computation of Quadrature Signals in Homodyne Laser Interferometry

“…Currently, the problems with scale linearity (i) are well explored [12,13] and there are several methods to deal with them [14–16]. …”

Dual-probe homodyne quadrature laser interferometer

“…It can be accurately determined because the wavelength λ of the laser light in air is well defined for displacements smaller than 100 μm with a resolution of 0.1 nm, without compensating for the environmental changes of the index of refraction. The displacement difference ut is obtained from the signals x and y by unwrapping with the arctangent function after the five unknowns: A x , A y , x 0 , y 0 , and p 0 are determined and used to correct the signals [9]. Ideally, the interferometer has to be aligned so that the signals x and y are in quadrature.…”

“…Moreover, the DPHQLI does not need to be precalibrated when the measured displacement exceeds one half of the wavelength of the employed laser. We reduced this requirement to one eighth of the wavelength (80 nm in the case of 632.8 nm He-Ne light) using an enhanced ellipse fitting [9].…”