## Date of Award

12-14-2015

## Document Type

Open Access Thesis

## Department

Mechanical Engineering

## First Advisor

Joshua A. Tarbutton

## Abstract

Heterodyne displacement measuring interferometry provides important metrology for applications requiring high resolution and accuracy. Heterodyne Michelson interferometers use a two-frequency laser source and separate the two optical frequencies into one fixed length and one variable length path via polarization. Ideally these two beams are linearly polarized and orthogonal so that only one frequency is directed toward each path. An interference signal is obtained by recombining the light from the two paths; this results in a measurement signal at the heterodyne (split) frequency of the laser source. This measurement signal is compared to the optical reference signal. Motion in the measurement arm causes a Doppler shift of the heterodyne frequency which is measured as a continuous phase shift that is proportional to displacement. In practice, due to component imperfections, undesirable frequency mixing occurs which yields periodic errors. Ultimately, this error can limit the accuracy to approximately the nanometer level. Periodic error is typically quantified using a Fourier transform-based analysis of constant velocity motions. However, non-constant velocity profiles lead to non-stationary signals that require alternate analysis techniques for real-time compensation. The objective of this study is to design a new discrete time continuous wavelet transform (DTCWT)-based algorithm, which can be implemented in real time to quantify and compensate periodic error for constant and non-constant velocity motion in heterodyne interferometer. It identifies the periodic error by measuring the phase and amplitude information at different orders (the periodic error is modeled as a summation of pure sine signals), reconstructs the periodic error by combining the magnitudes for all orders, and compensates the periodic error by subtracting the reconstructed error from the displacement signal measured by the interferometer. The algorithm is validated by comparing the compensated results with a traditional frequency domain approach for constant velocity motion. For a linear displacement signal where first and second order periodic errors (amplitudes 4 nm and 2.5 nm, respectively) are superimposed during a constant velocity (50 mm/min) displacement, the wavelet-based algorithm demonstrates successful reduction of the first order periodic error amplitude to 0.24 nm (a 94% decrease) and a reduction of the second order periodic error to 0.3 nm (an 88% decrease).

## Rights

© 2015, Chao Lu

## Recommended Citation

Lu, C.(2015). *Wavelet Analysis of Periodic Error in Heterodyne Interferometry.* (Master's thesis). Retrieved from https://scholarcommons.sc.edu/etd/3268