Using higher steps phase-shifting algorithms and linear least-squares fitting in white-light scanning interferometry Ming-Hsing Shen a , Chi-Hung Hwang b , Wei-Chung Wang a,n a Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan 30013, ROC b Instrument Technology Researcher Center, National Applied Research Laboratories, Hsinchu, Taiwan, ROC article info Article history: Received 12 October 2013 Received in revised form 1 August 2014 Accepted 11 September 2014 Keywords: Phase-shifting algorithm Linear least-squares fitting Local linear condition Micro-profile measurement White-light scanning Interferometry abstract White-light scanning interferometry (WLSI) has been a well-established tool for measuring the profile of objects. Since the white-light source is continuous in spectrum, the phase ambiguity problem can be avoided. Specimens with discontinuous profile can therefore be possibly measured by WLSI. In this paper, using higher steps, i.e. nine and eleven steps, phase-shifting algorithms (PSAs) based on local linear conditions of envelope function were proposed. The actual zero optical path difference position was retrieved by extracting the phase value and phase compensation. Maximum intensity peak positions at five points were used to calculate the center wavelength of the light source by both phase unwrapping and linear least-squares fitting. Both simulated and experimental results showed that the proposed nine- and eleven-step PSAs have good linearity and robustness to accurately measure the profile and center wavelength of the light source. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction In the development of micro electro mechanical system (MEMS) technology and optical components, surface profile is the important information for optimizing manufacturing process [1]. To measure the three-dimensional profile accurately, various methods of both contact and non-contact were proposed and investigated. The contact meth- ods perform measuring point-wisely with probes physically contact the object surface and stack the acquired data for future analysis [2]. The contact measurement process not only time-consuming but also may cause electrical and/or mechanical damage on the testing surface. On the contrary, the advantages of non-contact methods such as optical interferometry methods are nondestructive, high-resolution and whole-field. For typical optical interferometry, monochromatic light sources are frequently adopted; however, the fringe pattern interpolation is always a challenge even with the help of phase stepping method and the phase ambiguity is the main restriction for height measurement. Therefore, monochromatic interferometry is usually applied to measure samples with smoother profile. Different from monochromatic interferometry, the white light scanning interferometry (WLSI) uses broadband with short coher- ent optical length is often applied to measure micro-specimens with discontinuous profile. The interference signal of WLSI can be obtained only when the optical path difference (OPD) between object beam and reference beam is close to zero, and maximum signal could be obtained only when the OPD of reference and object beams is vanished. Therefore, WLSI can avoid the problem of phase ambiguity. The surface profile can be determined by the locations of zero optical path difference (ZOPD). In real applica- tions, ZOPDs locate at peak positions of WLSI signal envelopes. As listed in Table 1, during the development of WLSI, optical systems such as Michelson and Mirau interferometers [3–25] were proposed to determine the surface height of rough and mirror-like surfaces. Linnik interferometer has been rarely employed. Five and seven steps phase-shifting algorithms (PSAs) were occasionally used. Regarding the signal processes, especially the ZOPD position evaluation of the WLSI, Kino and Chim [7] suggested a ZOPD extraction algorithm based on Fourier transform. Their method can be used for smooth (mirror-like) surface measurement and often applied in microscope-based WLSI system of Linnik and Mirau optical setups. Hilbert transform was proposed to determine the peak positions of an interferometry signal envelope in 1992 [8]. Also Dresel et al. [9] introduced a three-dimensional sensor based Michelson interferometer to scan the rough object surface and called this method coherence radar. Determining ZOPD posi- tions by Fourier transform and Hilbert transform are based on the filter operator. Chen et al. [10] used gravity of WLSI signals to determine ZOPD positions of symmetric signals. Sandoz [11] derived seven-step PSA based on the local linearization together with the phase compensation to estimate actual ZOPD positions. Larkin [12] utilized five-step PSA to efficiently perform a non- linear algorithm for precise ZOPD position determination. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/optlaseng Optics and Lasers in Engineering http://dx.doi.org/10.1016/j.optlaseng.2014.09.004 0143-8166/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author. Tel.: þ886 3 5721585. E-mail address: wcwang@pme.nthu.edu.tw (W.-C. Wang). Optics and Lasers in Engineering 66 (2015) 165–173