A fast-Fourier-transform method of topography and interferometry is proposed. By computer processing of a noncontour type of fringe pattern, automatic discrimination is achieved between elevation and depression of the object or wave-front form, which has not been possible by the fringe-contour-generation techniques. The method has advantages over moire topography and conventional fringe-contour interferometry in both accuracy and sensitivity. Unlike fringe-scanning techniques, the method is easy to apply because it uses no moving components.

Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry

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

Imaging with visible light today uses numerous contrast mechanisms, including bright- and dark-field contrast, phase-contrast schemes and confocal and fluorescence-based methods. X-ray imaging, on the other hand, has only recently seen the development of an analogous variety of contrast modalities. Although X-ray phase-contrast imaging could successfully be implemented at a relatively early stage with several techniques, dark-field imaging, or more generally scattering-based imaging, with hard X-rays and good signal-to-noise ratio, in practice still remains a challenging task even at highly brilliant synchrotron sources. In this letter, we report a new approach on the basis of a grating interferometer that can efficiently yield dark-field scatter images of high quality, even with conventional X-ray tube sources. Because the image contrast is formed through the mechanism of small-angle scattering, it provides complementary and otherwise inaccessible structural information about the specimen at the micrometre and submicrometre length scale. Our approach is fully compatible with conventional transmission radiography and a recently developed hard-X-ray phase-contrast imaging scheme. Applications to X-ray medical imaging, industrial non-destructive testing and security screening are discussed.

Hard-X-ray dark-field imaging using a grating interferometer

Publisher Summary This chapter describes the self-imaging phenomenon and its applications. The self-imaging phenomenon requires a highly spatially coherent illumination. It disappears when the lateral dimensions of the light source are increased. When the source is made spatially periodic and is placed at the proper distance in front of the periodic structure, a fringe pattern is formed in the space behind the structure. The chapter discusses the theoretical and applicational aspects of the self-imaging phenomenon—that is, the property of the Fresnel diffraction field of some objects illuminated by a spatially coherent light beam. The applications of self-imaging are summarized in four main groups—namely, (1) image processing and synthesis, (2) technology of optical elements, (3) optical testing, and (4) optical metrology. The chapter describes the double diffraction systems using spatially incoherent illumination. The first periodic structure plays the role of a periodic source composed of a multiple of mutually incoherent slits. Depending on whether the periods of two periodic structures are equal, the Lau or the generalized Lau effect is discussed. Various applications of incoherent double-grating systems are described in the fields of optical testing, image processing, and optical metrology. After examining some cases of coherent and incoherent illumination, the general issue of spatial periodicities of optical fields and its relevance to the replication of partially coherent fields in space is discussed.

I The Self-Imaging Phenomenon and its Applications

The X-ray phase tomography of biological samples is reported, which is based on X-ray Talbot interferometry. Its imaging principle is described in detail, and imaging results obtained for a cancerous rabbit liver and a mouse tail with synchrotron radiation are presented. Because an amplitude grating is needed to construct an X-ray Talbot interferometer, a high-aspect-ratio grating pattern was fabricated by X-ray lithography and gold electroplating. X-ray Talbot interferometry has an advantage that it functions with polychromatic cone-beam X-rays. Finally, the compatibility with a compact X-ray source is discussed.

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Phase Tomography by X-ray Talbot Interferometry for Biological Imaging

The theoretical interpretation of the shearing interferometer based on the moire method using the fourier image of the grating is described. To obtain a pattern with good contrast, the observing plane must coincide with the normal fourier image plane of the grating or with the reversed fourier image plane. The information obtained by this method is the first partial derivative and under certain conditions the second partial derivative of the distortion from the reference wavefront, which is planar or spherical. Applications to measurement of the phase gradient and lens aberration are shown.

Shearing interferometer using the grating as the beam splitter.

The double grating shearing interferometry method for determination of the degree of light collimation is described. High accuracy is obtained by performing the observation of fringes in the area of the size twice as big as the one usually assumed in shearing interferometry experiments. The conditions under which such a detection mode is feasible are derived. They represent at the same time a very strong argument proving the highly diffractive (wave optics) character of the classical Ronchi test.

Collimation test by double grating shearing interferometer

Collimation method using fourier imaging and moire techniques

Speckle interferometry is used to measure the deformation of an object with a rough surface. In particular, the precise measurement can be easily performed with electronic-speckle-pattern interferometry (ESPI) using fringe scanning technology. Then, the measurement accuracy is influenced by the ratio between the speckle size and the pixel size of a CCD. Sometimes, this causes a problem concerning optical dislocations. An in-plane displacement is measured by the optical measurement arrangement using the two collimated beams. The measurement is performed by ESPI technology with the spatial fringe analysis method under an optimal measurement condition. Then, the optimum condition is discussed as the measurement parameters concerning the speckle's size and the passband of the bandpass filter. In the experimental measurement of in-plane displacement, it is shown that the optimal measurement condition can evade the problem of optical dislocations. At the same time, it is confirmed that a precise measurement can be performed.

In-plane displacement measurement using electronic-speckle-pattern-interferometry-based on spatial fringe analysis method

Fringe analysis is used in engineering fields as a precise measurement technology. In this paper, a method for improving the accuracy of spatial fringe analysis is discussed. A new method, based on fringe scanning and interpolation techniques, is put forward. Two-dimensional fringe analysis based on this new method is also proposed. Measured results obtained using the new two-dimensional method and those obtained using the FFT method are compared. Results showed that while the accuracy of the two methods is equivalent, the calculation time for the new method is five times as speedy as that for the FFT method.

Shunsuke Yokozeki

Papers

Shearing interferometer using the grating as the beam splitter.

Collimation test by double grating shearing interferometer

Collimation method using fourier imaging and moire techniques

In-plane displacement measurement using electronic-speckle-pattern-interferometry-based on spatial fringe analysis method

High precision two-dimensional spatial fringe analysis method