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

This article describes the principles and major applications of digital recording and numerical reconstruction of holograms (digital holography). Digital holography became feasible since charged coupled devices (CCDs) with suitable numbers and sizes of pixels and computers with sufficient speed became available. The Fresnel or Fourier holograms are recorded directly by the CCD and stored digitally. No film material involving wet-chemical or other processing is necessary. The reconstruction of the wavefield, which is done optically by illumination of a hologram, is performed by numerical methods. The numerical reconstruction process is based on the Fresnel–Kirchhoff integral, which describes the diffraction of the reconstructing wave at the micro-structure of the hologram. In the numerical reconstruction process not only the intensity, but also the phase distribution of the stored wavefield can be computed from the digital hologram. This offers new possibilities for a variety of applications. Digital holography is applied to measure shape and surface deformation of opaque bodies and refractive index fields within transparent media. Further applications are imaging and microscopy, where it is advantageous to refocus the area under investigation by numerical methods.

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Digital recording and numerical reconstruction of holograms

We have developed a digital holographic microscope (DHM), in a transmission mode, especially dedicated to the quantitative visualization of phase objects such as living cells. The method is based on an original numerical algorithm presented in detail elsewhere [
	 Cuche
	 
	, Appl. Opt.38, 6994 (1999)]. DHM images of living cells in culture are shown for what is to our knowledge the first time. They represent the distribution of the optical path length over the cell, which has been measured with subwavelength accuracy. These DHM images are compared with those obtained by use of the widely used phase contrast and Nomarski differential interference contrast techniques.

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Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy.

Off-axis holograms recorded with a CCD camera are numerically reconstructed with a calculation of scalar diffraction in the Fresnel approximation. We show that the zero order of diffraction and the twin image can be digitally eliminated by means of filtering their associated spatial frequencies in the computed Fourier transform of the hologram. We show that this operation enhances the contrast of the reconstructed images and reduces the noise produced by parasitic reflections reaching the hologram plane with an incidence angle other than that of the object wave.

/pdf/spatial-filtering-for-zero-order-and-twin-image-elimination-579r0qo7p6.pdf

Spatial filtering for zero-order and twin-image elimination in digital off-axis holography

Advantages of the lensless Fourier holography setup for the reconstruction of digitally recorded holograms in holographic interferometry are presented. This very simple setup helps to achieve a maximum lateral resolution of the object under investigation. Also, the numerical-reconstruction algorithm is very simple and fast to compute. A mathematical model based on Fourier optics is used to describe discretization effects and to determine the lateral resolution. The recording and the reconstruction processes are regarded as an optical imaging system, and the point-spread function is calculated. Results are verified by an experimental setup for a combined shape and deformation measurement.

Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology

Digital Holography

A procedure is proposed to calibrate a generic structured light system, consisting of one camera and one projector. The proposed pro- cedure is based on defining a unique coordinate system for both devices in the structured light system, and thus, a rigidity constraint is introduced into the transformation process. This constraint is used to derivate a simple function for the simultaneous estimation of the parameters, result- ing in parameters that are more reliable. The performance of the pro- posed procedure is shown on examples of the calibration of two different structured light systems. © 2004 Society of Photo-Optical Instrumentation Engi- neers. (DOI: 10.1117/1.1635373)

Accurate procedure for the calibration of a structured light system

Accurate 3D shape measurement is of big importance for industrial inspection. Because of the robustness, accuracy and ease of use optical measurement techniques are gaining importance in industry. For fast 3D measurements on big surfaces fringe projection is commonly used: A projector projects fringes onto the object under investigation and the scattered light is recorded by a camera from a triangulation angle. Thus, it is possible reaching a depth resolution of about one by 10.000 of the measurement field size (e.g. 100 μm for a 1 m sized field). For non- or low scattering objects it is common to put scattering material like particle spray onto the object under investigation. Objects where this is not allowed are often regarded as problematic objects for full field non-coherent optical measurement techniques. The solution is to switch from fringe projection to fringe reflection. The fringe reflection technique needs a simple setup to evaluate a fringe pattern that is reflected from the surface under investigation. Like for fringe projection the evaluated absolute phase identifies the location of the originating fringe. This allows identifying the reflection angles on the object for every camera pixel. The results are high resolution local gradients on the object which can be integrated to get the 3D shape. The achievable depth resolution compared to fringe projection is much better and reaches to a depth resolution down to 1 nm for smooth surfaces. We have proven the ability, robustness and accuracy of the technique for various technical objects and also fluids. A parallel paper of this conference 'Evaluation Methods for Gradient Measurement Techniques' picks up further processing of the evaluated data and explains in more detail the performed calculations. This paper mainly concentrates on the fringe reflection principle, reachable resolution and possible applications.

Werner Jüptner

Papers

Digital recording and numerical reconstruction of holograms

Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology

Digital Holography

Accurate procedure for the calibration of a structured light system

High-resolution 3D shape measurement on specular surfaces by fringe reflection