Publisher Summary This chapter describes the phase-measurement interferometry techniques. For all techniques, a temporal phase modulation is introduced to perform the measurement. By measuring the interferogram intensity as the phase is shifted, the phase of the wavefront can be determined with the aid of electronics or a computer. Phase modulation in an interferometer can be induced by moving a mirror, tilting a glass plate, moving a grating, rotating a half-wave plate or analyzer, using an acousto-optic or electro-optic modulator, or using a Zeeman laser. Phase-measurement techniques using analytical means to determine phase all have some common denominators. There are different equations for calculating the phase of a wavefront from interference fringe intensity measurements. The precision of a phase-measuring interferometer system can be determined by taking two measurements, subtracting them, and looking at the root-meansquare of the difference wavefront. The chapter discusses the simulation results. The elimination of the errors that reduce the measurement accuracy depends on the type of measurement being performed. Phase-measurement interferometry (PMI) can be applied to any two-beam interferometer, including holographic interferometers. Applications can be divided into: surface figure, surface roughness, and metrology.

V Phase-Measurement Interferometry Techniques

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

Phase shifting algorithms for fringe projection profilometry: A review

Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review

A compact and efficient algorithm for digital envelope detection in white light interferograms is derived from a well-known phase-shifting algorithm. The performance of the new algorithm is compared with that of other schemes currently used. Principal criteria considered are computational efficiency and accuracy in the presence of miscalibration. The new algorithm is shown to be near optimal in terms of computational efficiency and can be represented as a second-order nonlinear filter. In combination with a carefully designed peak detection method the algorithm exhibits exceptionally good performance on simulated interferograms.

https://nontrivialzeros.net/KGL_Papers/23_Neat_Algorithm_JOSAA96.pdf

Efficient nonlinear algorithm for envelope detection in white light interferometry

Conventional phase-shifting algorithms based on a least-squares estimate use N samples over an incomplete period of the sampled waveform We introduce a class of phase-shifting algorithms having N + 1 samples symmetrically disposed over one full period of the sampled waveform Fourier analysis techniques are used to derive these algorithms and modify them to improve their performance in the presence of phase-shift errors The algorithms can be used in phase measurement systems having periodic, but not necessarily sinusoidal, waveforms

Design and assessment of symmetrical phase-shifting algorithms

In phase measurement systems that use phase shifting techniques, phase errors that are due to nonsinusoidal waveforms can be minimized by applying synchronous phase shifting algorithms with more than four samples. However, when the phase shift calibration is inaccurate, these algorithms cannot eliminate the effects of nonsinusoidal characteristics. It is shown that, when a number of samples beyond one period of a waveform such as a fringe pattern are taken, phase errors that are due to the harmonic components of the waveform can be eliminated, even when there exists a constant error in the phase shift interval. A general procedure for constructing phase shifting algorithms that eliminate these errors is derived. It is shown that 2j + 3 samples are necessary for the elimination of the effects of higher harmonic components up to the jth order. As examples, three algorithms are derived, in which the effects of harmonic components of low orders can be eliminated in the presence of a constant error in the phase shift interval.

https://nontrivialzeros.net/KGL_Papers/16_Hibino_Phase_Shifting_Non_sinusoidal_JOSAA_1995.pdf

Phase shifting for nonsinusoidal waveforms with phase-shift errors

In phase-shifting interferometry spatial nonuniformity of the phase shift gives a significant error in the evaluated phase when the phase shift is nonlinear. However, current error-compensating algorithms can counteract the spatial nonuniformity only in linear miscalibrations of the phase shift. We describe an error-expansion method to construct phase-shifting algorithms that can compensate for nonlinear and spatially nonuniform phase shifts. The condition for eliminating the effect of nonlinear and spatially nonuniform phase shifts is given as a set of linear equations of the sampling amplitudes. As examples, three new algorithms (six-sample, eight-sample, and nine-sample algorithms) are given to show the method of compensation for a quadratic and spatially nonuniform phase shift.

http://nontrivialzeros.net/KGL_Papers/26_Hibino_Phase_Nonlinear_JOSAA_1997.pdf

Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts

Stroboscopic holographic interferometry: Application of digital techniques

Sol–gel silica coatings prepared for high-power laser applications were optimized by a two-step structure modification through treatment of the sol with poly(ethylene glycol) (PEG), followed by treatment of the coating with ammonia. The effect of each treatment on the laser-induced damage threshold (LIDT) at 1064 nm wavelength was studied, mainly in terms of the change in the coating’s structure. Both of these treatments increased the LIDT, and the greatest LIDT was achieved by combining the two treatments. The LIDT increase can be attributed to the decrease in structural defects such as nodular defect and the variety of pore-distribution, resulting from PEG and ammonia treatments, respectively. The procedure of producing coated glass with high LIDT for 1064 nm laser irradiation was suggested, involving substrate etching, sol modification, and coating treatment.

Bob F. Oreb

Papers

Design and assessment of symmetrical phase-shifting algorithms

Phase shifting for nonsinusoidal waveforms with phase-shift errors

Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts

Stroboscopic holographic interferometry: Application of digital techniques

Increased Laser-Damage Resistance of Sol–Gel Silica Coating by Structure Modification