IMPROVEMENT OF A PERIODIC ERROR COMPENSATION ALGORITHM BASED ON THE CONTINUOUS WAVELET TRANSFORM Chao Lu 1 , Jonathan D. Ellis 2,3 , Tony L. Schmitz 4 , and Joshua A. Tarbutton 1 1 Department of Mechanical Engineering University of South Carolina, Columbia, SC, USA 2 Department of Mechanical Engineering, 3 The Institute of Optics University of Rochester, Rochester, NY, USA 4 Department of Mechanical Engineering and Engineering Science University of North Carolina at Charlotte, Charlotte, NC, USA INTRODUCTION Heterodyne displacement measuring interferometry provides important metrology for applications requiring high resolution and accuracy, such as in the semiconductor manufacturing industry and for linear stage calibration. 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 misalignment of optical components, component imperfections, and elliptical polarization, undesirable frequency mixing occurs which yields periodic errors [1-3]. Typically, 1 st , 2 nd and even higher order periodic errors occur, which correspond to the number of periods per fringe displaced, as shown in Figure 1. A displacement fringe corresponds to the wavelength divided by the interferometer fold factor which is determined by the interferometer setup. Ultimately, this error can limit the accuracy to approximately the nanometer level. Many studies have investigated the measurement and compensation of periodic error, including frequency domain [4] and time domain approaches [5, 6]. For frequency domain approach, the periodic error can be measured by calculating the Fourier transform of the time domain data collected during constant velocity target displacement. However, this method is not well suited to non-constant velocity profile and sub-fringe positioning ranges. An alternate digital algorithm which can be applied in real- time for constant or non-constant velocity motions is also available for measuring and compensating 1 st and 2 nd order periodic error. But this method leaves 3 rd and higher order periodic errors as residual errors. In this research, a real-time continuous wavelet transform (CWT) based algorithm, which was developed in previous work [7], is improved and used to compensate 1 st , 2 nd and higher order periodic errors (modeled as pure sine signals). Moreover this algorithm can compensate errors in both constant and non-constant velocity motions. FIGURE 1. Example of 1 st , 2 nd , and 3 rd order periodic error as a function of fringes. Typically, 1 st order error has a larger magnitude than higher order errors. CONTINUOUS WAVELET TRANSFORM The wavelet transform can be used to analyze time series data that contains non-stationary (variable period) power at multiple frequencies [8]. Wavelet functions refer to either orthogonal or non-orthogonal wavelets. The choice of the appropriate wavelet transform (continuous or discrete) and wavelet function is based on