Development of phase-shifting device for implementation of three-step interferometric method with arbitrary phase shifts of reference wave

Abstract

Two basic approaches to the implementation of procedures of a wavefront stepwise and smooth phase shifts in a two-beam optical interferometer are considered. Advantages and lacks of both procedures are analyzed. It is noted that the smooth phase shift procedure removes the oscillations of a phase shifting element (PSE) that occur during the stepwise procedure. A phase-shifting device that is used to realize the three-step phase shifting interferometry (PSI) method with two arbitrary phase shifts of the reference wave within the angular interval (0, π) is developed. This device does not need the calibration procedure, because the three step PSI method allows defining any phase shift angle by calculating the correlation coefficient between two recorded interferograms. The correlation coefficient can be considered as a normalized scalar product of two interferograms that can be represented by centered multidimensional vectors or as a cosine of a phase shift angle between these vectors. Therefore, the developed device is much simpler and cheaper than its calibrated prototypes. It contains a PSE consisting of a piezoelectric transducer (PZT) and a mirror attached to the PZT, and an electronic unit for smooth linear motion of the mirror in the reference beam of the interferometer. Basic characteristics of the developed electronic unit are considered. The operating principle of the phase-shifting device electronic unit is to provide the smooth rising voltage supply initiated a smooth change of the PZT size. Therefore, the mirror attached to the PZT also moves smoothly. The PSE is located in the reference beam of the two-beam interferometer. If the reference beam in the interferometer enters the mirror in the direction perpendicular to the plane of the mirror, the beam wavefront moves to a distance twice that of the mirror. To verify the reliability of the developed phase-shifting device, we have elaborated the experimental setup dedicated for formation, registration and processing of interferograms of test surfaces and based on the Twyman-Green interferometer architecture. In this setup, the three-step PSI method with arbitrary phase shifts of a reference wave is fulfilled by recording of three interferograms during smooth phase shift of the mirror attached to the PZT. Comparative analysis of stepwise and smooth phase shift procedures for interferograms recording in the experimental setup has shown that their modulation transfer functions (MTFs) are similar for small exposures of interferograms. In particular, it is shown that for interferogram’s small exposure times, whose ratio to the voltage increase time in the angular interval (0, π) does not exceed 0.06, the MTF of the interferogram recorded with the smooth phase shift of the reference beam does not differ from the interferogram MTF obtained by stepwise phase shift. Key words: phase shifting interferometry, phase shift, phase shifting element, interferogram, modulation transfer function