High Precision Contouring with Moiré and Related Methods: A Review

Abstract: Optical methods of contouring that utilise image recordings by cameras are based on a fundamental discipline, projective geometry. The 3‐D world is projected in 2‐D utilising a camera modelled in the technical literature by the pinhole camera. To get back 3‐D information, the fundamental property measured is parallax. Parallax is a vector resulting from the difference of the projective coordinates of a point in space when projected onto a plane from two different points. The oldest method used for measuring parallax is photogrammetry. It is assumed to be the most precise technique, with the capability of obtaining 10^?5 of the largest dimension of the measured object. This study summarises the state‐of‐the‐art methods based on projecting a spatial carrier. Starting with the concept of moiré as a form of photogrammetry, the different optical techniques for parallax determination are discussed. Although the moiré method has reached 1 μm accuracy in laboratory work, a question remains: can moiré become a standardised contouring technique yielding 10^?6 m accuracy? This study is devoted to the analysis of high accuracy contour measurements, through both theoretical derivations and experimental verifications.     

Influence of several grating sensor parameters on the modulation depth of moiré signals under incoherent illumination

Theoretical expressions for the modulation depth of moiré signals under incoherent illumination are derived. Consequently, the modulation depth of a moiré signal is related to the following factors: the geometric shape and the size of the light source; the diffraction effect of the grating, which relates to the number of lines in the grating; the line and the space ratio; the grating pair gap; the geometric shape and the size of the receiving window; etc. In addition, the influence of the grating pair on the period and the inclination of moiré fringes under noncollimated illumination are discussed, and the changes in the moiré signal modulation depth under noncollimated illumination with that under collimated illumination are made. Finally, some experimental results are given to verify the theoretical expressions. This research is useful for the actual design of grating sensors.     

Colour imperfections in structured light projection for freeform measurement – a rapid test

Three-dimensional (3D) optical measuring systems using structured light are often limited in practice by imperfections in the light patterns projected onto the object under test. An understanding of the fine structure of the patterns enables limits to be placed on the accuracy and precision of 3D measurements that can be obtained with such systems. A simple technique has been studied that reveals imperfections in patterns of parallel lines projected in white light. Colour fringing has been revealed using a reference grid in the form of a diffusely reflective grating or a lenticular array, to obtain magnification by means of moiré effects. The moiré magnifier gives a representation of the average error over an area. It is a very simple and robust device which may be convenient to use in an industrial production environment to provide a rapid check for the presence of colour fringing in light patterns with regular structures.     

Automated small tilt-angle measurement using Lau interferometry

A technique for a tilt-angle measurement of reflecting objects based on the Lau interferometry coupled with the moire readout has been proposed. A white-light incoherent source illuminates a set of two gratings, resulting in the generation of the Fresnel image due to the Lau effect. The Fresnel image is projected onto a reflecting object. The image reflected from the object is superimposed onto an identical grating, which results in the formation of a moiré fringe pattern. The inclination angle of moiré fringes is a function of tilt angle of the object. Theory and experimental arrangement of the proposed technique is presented and results of the investigation are reported.     

Effects of a Small Inclination Misalignment in Talbot Interferometry by use of Gratings with Arbitrary Line Orientation. I. Theoretical Analysis

We discuss the effects of a general small inclination misalignment, which is formed by rotation of the beam-splitter grating around an axis that is laid on the grating plane and that has an arbitrary angle with respect to the line direction of the grating, between the two grating planes on the moiré fringes in Talbot interferometry. It is shown that the small inclination angle has a significant influence on measurement results based on Talbot interferometry because both the period and the slope of the moiré fringes are sensitive to the angle, especially when the rotation axis is nearly parallel to the lines of the grating. Simple and practical detection methods for the small inclination angle are proposed, and the effects of the inclination angle on the contrast in the moiré fringes are also briefly discussed.     

Effects of Unparallel Grating Planes in Talbot Interferometry. II

The properties of the moiré fringes in Talbot interferometry are analyzed for a small inclined angle beta between the two grating planes, which is produced by rotation of the beam splitter grating about the axis perpendicular to the lines of the grating. Theoretical analyses indicate that the tilt angle of the resultant moiré fringes is less sensitive to beta than when the small inclined angle is formed by rotation of the beam splitter grating about the axis parallel to the lines direction of the grating as described earlier [Appl. Opt. 38, 4111 (1999)] and that contrast of the moiré fringes decreases with an increase in beta or in the spatial frequency of the grating and may result in impaired measurement accuracy. The validity of the theoretical analyses is illustrated by experiments.     

Effects of unparallel grating planes in Talbot interferometry

The properties of moiré fringes in Talbot interferometry are analyzed when the angle between the two grating planes is small. The results indicate that the tilt angle of the moiré fringes, observed just behind the test grating, is sensitive to the small angle. Based on this sensitivity, several features of parallelism of the two grating planes are presented, and the influence of the small angle when checking the beam collimation of a lens is also discussed. The validity of the theoretical analysis is illustrated by experiment.     

The Development and the Present Status of Moiré Topography

The process of the development and the present status of optical moiré topography at the University of Shizuoka are described and a historical review is presented. The main techniques described are the basic method and the grating hologram type of moiré topography.     

Analysis of the tilt angle of the moiré fringes produced by two inclined gratings in talbot interferometry

Abstract

Features of the moiré fringes produced by the inclined grating planes in Talbot interferometry are studied under the illumination of a plane wave. Inclinations of the two grating planes are introduced by rotating the beam-splitter and the detector gratings around the axes parallel to the line directions of each grating, when the line directions of the two grating are different. Theoretical analyses indicate that the tilt angle of the moiré fringes is sensitive to the inclination angle, and the analyses are supported by experimental results. Some simple and practical methods for judging and removing the inclinations are also discussed.     

Influence of unparallel grating planes in talbot interferometry with cross gratings of different spatial frequencies in two directions

The effects of an arbitrary small inclination between two cross gratings on the moiré fringes in Talbot interferometry are discussed when the frequencies of the grating differ in two perpendicular directions. We show that the small angles, alpha and beta, by which the beam-splitter cross grating is rotated around the two axes parallel to the two perpendicular line directions of the cross grating, have a greater influence on the moiré fringes with cross gratings than that with one-dimensional gratings. A simple and practical detection method for the angles between the two unparallel grating planes in Talbot interferometry is also proposed. The theoretical analyses are proved by experimental results.     

Using random features of dot-matrix holograms for anticounterfeiting

The images on a dot-matrix hologram contain many two-dimensional (2D) dots with different grating orientations and different grating pitches. Because the zeroth-order light nondiffracted by different grating structures has the same progress direction, the nondiffracted light can be diffracted to a 2D spot spectrum by the 2D dot structure. The 2D spot spectrum depends on grating depths and dot sizes. Although ordinary noises are troublesome for 2D spots, noises caused by special dot arrangement defects or special grating moiré fringes are useful in checking holograms. Since the features of grating depths, dot sizes, dot arrangement defects, and grating moiré fringes can be randomly changed on a case by case basis, 2D spot spectra in different cases are different. The aforementioned random features are used to identify dot-matrix holograms.     

Collimation testing by use of the Lau effect coupled with moiré readout

We present an easy, simple, and inexpensive technique for checking the quality of the collimation of optical beams using the Lau effect combined with moiré readout. The experimental arrangement consists of a modified Lau-based interferometer in which a white-light incoherent source illuminates a set of two gratings. A collimating lens is placed between the two gratings such that the self-images of the second grating are formed. The third grating is positioned at one of the self-imaging planes forming moiré fringes. The type of the moiré fringe demonstrates the quality of collimation of the optical beam. The necessary theoretical background is presented and the results of our experimental investigation are reported. The technique can also be used for accurate determination of the focal length of a collimating lens using low-cost components.     

Digital moiré fringe-scanning method for centering a circular fringe image

A digital moiré fringe-scanning method for centering a circular fringe image is proposed. The image of a nondiffracting beam, whose cross section is a circular fringe, is first downloaded onto a computer. The image is then superposed with a digital circular grating, whose center is close to the center of the image, to generate circular moiré fringes. Changing the phase of a digital grating can cause moiré fringe scanning. The global center of the image can be calculated by use of sets of the scanned picture. Because all the image data are used for the calculation, the effect of random noise on centering is greatly reduced and the center position resolution can reach the order of a subelement of a CCD. The measurement of spatial straightness is discussed.     

Profiling of objects with height steps by wavelet analysis of shadow moiré fringes

A temporal wavelet analysis algorithm is proposed for shadow-moiré-based three-dimensional surface profiling on objects having discontinuous height steps. A grating is positioned close to an object, and its shadow is observed through the grating. The moiré fringe patterns vary when the grating is in-plane rotating. A series of fringe patterns are captured by a CCD camera at different rotating angles. Phase values are evaluated point by point with the continuous wavelet transform. From the phase values of each point on the object, the distance between the object and the grating can be retrieved. The surface profile is obtained without temporal or spatial phase unwrapping. This technique is applicable to objects with discontinuous height steps, which are impossible to measure with conventional shadow moiré topography. Two specimens are tested to demonstrate the validity of the method: One is an object with a height step of 1.6 mm, and another is a small coin with unevenness of less than 0.2 mm. The experimental results are compared with test results by use of the mechanical stylus method.     

Fourier‐series‐based virtual fields method combining with moiré interferometry for characterising elastic modulus distribution of laser repaired GH4169

Laser cladding is an effective technology for repairing damaged components with high efficiency and low cost. Characterisation for the mechanical properties of the repaired structure is of great importance to evaluate its reliability. In this study, the mechanical properties of laser repaired GH4169 are investigated by Fourier‐series‐based virtual fields method and moiré interferometry. The elastic modulus distribution of the whole structure is identified to understand the mechanical performance of the repaired zone and base material. A uniaxial tensile test is performed and the full‐field deformation is measured by moiré interferometry. With the measured strain field, the distribution of modulus in repaired structures can be calculated. In addition, simulation experiments of moiré interferometry are conducted to optimise the parameters of Fourier‐series‐based virtual fields method to improve the accuracy of the proposed method. On the basis of these results, experiments on the repaired GH4169 material are performed and the variation of modulus in the whole structure is identified. The results verify that the proposed method is effective for characterising modulus distribution of the laser repaired structures, and will have a good prospect for further application.     

3-D Contouring of Diffuse Objects by Talbot-projected Fringes

Abstract

Talbot fringe projection, a moiré technique, is applied to three-dimensional contouring of diffuse targets for absolute shape measurement. The basic system relies on depth coding the test target surface by projecting the Talbot image of a linear grating. A second grating, similar to that used for the Talbot image, is employed to obtain the moiré fringes. These fringes represent surface contours of equal depth. Using a phase measurement technique and digital image processing algorithms, the surface shape information is obtained from the contour maps. Experimental results, merits and limitations of the system are discussed.     

Study of the surface structure of butterfly wings using the scanning electron microscopic moiré method

Scanning electron microscopic (SEM) moiré method was used to study the surface structure of three kinds of butterfly wings: Papilio maackii Menetries, Euploea midamus (Linnaeus), and Stichophthalma howqua (Westwood). Gratings composed of curves with different orientations were found on scales. The planar characteristics of gratings and some other planar features of the surface structure of these wings were revealed, respectively, in terms of virtual strain. Experimental results demonstrate that SEM moiré method is a simple, nonlocal, economical, effective technique for determining which grating exists on one whole scale, measuring the dimension and the whole planar structural character of the grating on each scale, as well as characterizing the relationship between gratings on different scales of each butterfly wing. Thus, the SEM moiré method is a useful tool to assist with characterizing the structure of butterfly wings and explaining their excellent properties.     

Phase-shifting moireí method with an atomic force microscope

Using a phase-shifting technique with an atomic force microscope (AFM), we propose a phase-shifting AFM scanning moiré method. The phase shifting is realized in four steps from 0 to 2pi by a piezoscanner in the AFM. The measurement method and experimental techniques are described in detail. For demonstration this method is applied to determine the phase distribution in the AFM moiré of a 1200-line/mm holographic grating used to measure the thermal deformation in a Quad FlatPack electronic package.     

Multiplicative moiré two-beam phase-stepping and fourier-transform methods for the evaluation of multiple-beam fizeau patterns: a comparison

A phase-evaluation method of multiple-beam Fizeau patterns that combines two-beam phase-stepping algorithms with the moiré effect was previously reported [Appl. Opt. 34, 3639-3643(1995)]. The method is based on a multiplicative moiréimage-formation process obtained by the direct superposition of high-frequency multiple-beam Fizeau carrier fringes upon a transmission grating (working as a phase modulator). We present a comparison between this multiplicative moiré two-beam phase-stepping method and the well-known Fourier-transform method for the topographic measurement of an undoped silicon wafer. The discrepancy between the two methods yields a rms phase-difference value of the order of(~2pi/90).     

A study of the displacement field around embedded fibre optic sensors

Deformation fields around optical fibres embedded in carbon fibre/epoxy composite specimens have been measured using moiré interferometry. The inclusion of the optical fibre resulted in large strain gradients. Calculated displacements from finite element analysis were compared to the experimental results. The numerical analysis showed that the displacement field on the specimen surface is smoothed out through the moiré grating thickness, an effect which is most pronounced at the material interfaces. With this influence taken into consideration a reasonable good quantitative agreement between the experiments and the finite element analysis was obtained. The finite element analysis also showed that the grating stiffness did not affect the measured displacements as long as the grating had a lower stiffness than the specimen.