A self-scanned 1024 element photodiode array and minicomputer are used to measure the phase (wavefront) in the interference pattern of an interferometer to lambda/100. The photodiode array samples intensities over a 32 x 32 matrix in the interference pattern as the length of the reference arm is varied piezoelectrically. Using these data the minicomputer synchronously detects the phase at each of the 1024 points by a Fourier series method and displays the wavefront in contour and perspective plot on a storage oscilloscope in less than 1 min (Bruning et al. Paper WE16, OSA Annual Meeting, Oct. 1972). The array of intensities is sampled and averaged many times in a random fashion so that the effects of air turbulence, vibrations, and thermal drifts are minimized. Very significant is the fact that wavefront errors in the interferometer are easily determined and may be automatically subtracted from current or subsequent wavefrots. Various programs supporting the measurement system include software for determining the aperture boundary, sum and difference of wavefronts, removal or insertion of tilt and focus errors, and routines for spatial manipulation of wavefronts. FFT programs transform wavefront data into point spread function and modulus and phase of the optical transfer function of lenses. Display programs plot these functions in contour and perspective. The system has been designed to optimize the collection of data to give higher than usual accuracy in measuring the individual elements and final performance of assembled diffraction limited optical systems, and furthermore, the short loop time of a few minutes makes the system an attractive alternative to constraints imposed by test glasses in the optical shop.

Digital wavefront measuring interferometer for testing optical surfaces and lenses

We describe a generalized phase-shifting interferometry for which the reference phases are directly evaluated at each time that the interference fringe data are read. The reference phases are obtained from the additional straight fringes on the interfering plane by the fast-Fourier-transform method. According to error estimation, the repeatabilities in the measurements of optical surfaces are λ/500 rms, when the generalized algorithm with eight data acquisitions is used.

Generalized phase-shifting interferometry

Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: An overview

Many optical measurement techniques provide fringe patterns as their results. The decodification processes that employ one or several fringe patterns to automatically retrieve the phase are generally designated as phase-evaluation methods. In this work, an overview of these methods will be schematically presented. Their particular performances will be compared, stressing their main advantages and drawbacks. An important group of these methods employs the well-known phase-shifting algorithms as a tool for calculating the phase. In the general form of these algorithms, the principal value of the optical phase is computed by an inverse trigonometric function whose argument is a combination of phase-shifted intensity values, provided by the modulation of one or several fringe patterns. These algorithms will be also studied in the general context of the phase-evaluation methods.

https://iopscience.iop.org/article/10.1088/0957-0233/10/3/005/pdf

Phase-evaluation methods in whole-field optical measurement techniques

A new phase-shifting interferometry analysis technique has been developed to overcome the errors introduced by nonlinear, irregular, or unknown phase-step increments. In the presence of a spatial carrier frequency, by observation of the phase of the first-order maximum in the Fourier domain, the global phase-step positions can be measured, phase-shifting elements can be calibrated, and the accuracy of phase-shifting analysis can be improved. Furthermore, reliance on the calibration accuracy of transducers used in phase-shifting interferometry can be reduced; and phase-retrieval errors (e.g., fringe print-through) introduced by uncalibrated fluctuations in the phase-shifting phase increments can be alleviated. The method operates deterministically and does not rely on iterative global error minimization. Relative to other techniques, the number of recorded interferograms required for analysis can be reduced.

Fourier-transform method of phase-shift determination

In the fringe-scanning methods such as the three- and four-bucket methods, the phase of a reference wave is generally changed at equal intervals from 0 to 2π. In practice, however, it is difficult or almost impossible for some interferometers, such as the dynamic zone plate interferometer, to realize an equal phase shift and to make a full scanning of 2π, which are necessary for discrete Fourier transform analysis. We show that even in such a case, phases in an interferogram can be obtained by using a Fourier analysis in a continuous space, and we analyze the accuracy of the phase determination in case of additive noise.

Satoshi Kinoshita

Papers

Accuracy of phase determination with unequal reference phase shift