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

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

Self-Calibration: Reversal, Redundancy, Error Separation, and ‘Absolute Testing’

Publisher Summary This chapter describes the advanced evaluation techniques in interferometry The chapter discusses the methods that rely on the adjustment of a fringe pattern The chapter examines the historically older interpolation methods where the phase information is extracted from the positions of the extrema The chapter also discusses more general methods that deliver the phase in a more continuous form A comparison of the different methods for phase evaluations necessitates a consideration of the error sources and limitations These error sources for the case of phase sampling interferometry (PSI) have been discussed The error causes are classified into statistical errors (measuring errors that can be diminished in their influence by averaging) and systematic errors (errors that cannot be quenched) The interferometers used in the assessment of test data are rather complex, thus, there is some necessity for calibration The interferometer aberrations, even with ideal test samples, are caused by deviations of optical elements and to also by the state of adjustment The chapter identifies the types of error sources—namely, environmental errors, errors that are dependent on the method, and errors typically encountered in interferometry

IV Advanced Evaluation Techniques in Interferometry

We propose a method to obtain the shape of a large plane surface by connecting phase distributions measured by a small-aperture interferometer. These separately measured phase distributions cannot be connected directly because the object will tilt or have vertical displacement during the measurements. To correct these errors, the measurements are made so that the adjacent interferograms have common
areas, and these interferograms are connected to minimize the difference of the phase distributions in the common areas. A matrix equation is derived to obtain coefficients to correct tilt and vertical displacement, and the accuracy of connection increases in proportion to an exponent of 1.5 of the width of the common area.

Measurement of large plane surface shapes by connecting small-aperture interferograms

Several conventional methods for absolute sphericity testing of optical surfaces are known. One method has been used previously for real time interferometry, but a detailed investigation has not been carried out so far. In this work the known conventional methods are reviewed and assessed for suitability in real time interferometry. One method has been applied to phase stepping interferometry and examined. Adjustment peculiarities, experimental results of deviation measurements on normal surfaces, and application of normals are reported. Measuring errors, especially coherent noise, are analyzed and a special subsequent digital spatial filtering technique for diminution of noise is described.

Absolute sphericity measurement

In diffraction-limited optical systems the number of optical components is rather high. Local changes in the refractive index of the order of 10−6 must be detected to guarantee an adequate performance of the system as a whole. Therefore, the Twyman-Perry method for the evaluation of refractive-index deviations from uniformity—further developed by Roberts and Langenbeck—has been programmed for a digital Twy-man-Green interferometer. If the glass block to be tested is slightly wedge shaped (wedge angle ~10−3 rad), the measurement can be carried out without coating the glass block. With the help of four interferograms the variations of the refractive index apart from a linear slope, as well as the thickness variations of the glass block, can be measured and displayed.

Homogeneity testing by phase sampling interferometry.

If phase-measurement interferometry (PMI) is applied for a high numerical aperture (NA), specific errors can occur. The resulting phase no longer varies linearly with the height difference in the test surface. Thus, for height measurements, a constant correction factor or NA scaling factor, which depends only on the NA and the illumination of its stop, does not exist here. This is shown for several pupil illuminations (intensity ∼ cos2 α, intensity ∼ cos α, intensity constant; α is the incident angle). For example (NA = 0.9, pupil illumination intensity is constant, wedge angle ≈ 0, PMI phase increments = π/2 for normal incidence), the correction factor can vary from 1.20 (height difference ≈ 0) to 1.37 (height difference ≈ λ/4), i.e., by 17%.

Errors in phase-measurement interferometry with high numerical apertures

A further-developed three-plate rotation method [Appl. Opt. 31, 3767 (1992)] for the absolute testing of flats is used as a measuring method for establishing flatness standards. A special phase-stepping Fizeau interferometer was built that permits flats of diameters up to 200 mm to be tested. First results show a mean error between 0.002 and 0.003 λ in the determination of the absolute flatness deviations (λ = 632.8 nm). These deviations are obtained at 500-600 points on each plate. A number of experimental conditions connected with the thermal and mechanical stability of the plates must be fulfilled.

Establishing a flatness standard

Prior to any interferometric surface shape testing, it is necessary to calibrate the interferometer in order to avoid systematic measurement errors. One common calibration method employs a standard specimen which has been previously calibrated by a so-called absolute testing method. Another method is the direct application of absolute testing methods to the interferometer. The authors have in the past developed and used absolute interferometric testing methods to calibrate flat, spherical and cylindrical surfaces, as well as complete interferometers. Now they have developed new methods, suitable also for toric and conical surfaces and the interferometers. The basic principles of the calibration methods are described. The correctness of the methods and the influence which measurement errors have on the results are demonstrated by computer simulation.

Karl-Edmund Elssner

Papers

Absolute sphericity measurement

Homogeneity testing by phase sampling interferometry.

Errors in phase-measurement interferometry with high numerical apertures

Establishing a flatness standard

Absolute interferometric calibration of toric and conical surfaces