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.     

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.     

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 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.     

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.     

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.     

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.     

Error analysis of the phase-shifting technique when applied to shadow moiré

An exact solution for the intensity distribution of shadow moiré fringes produced by a broad spectrum light is presented. A mathematical study quantifies errors in fractional fringe orders determined by the phase-shifting technique, and its validity is corroborated experimentally. The errors vary cyclically as the distance between the reference grating and the specimen increases. The amplitude of the maximum error is approximately 0.017 fringe, which defines the theoretical limit of resolution enhancement offered by the phase-shifting technique.     

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.     

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.     

Interpretation of Fringes Produced by Time‐Averaged Projection Moiré

Abstract: New exact formulas describing the observed shift of projected moiré grating lines on a surface of an object are derived for the paraxial model. These formulas enable to construct more accurate explicit relationship among the amplitude of oscillation, the pitch of the projected grating and the order of the fringe. Analytical derivations and numerical illustrations are used throughout the text to explain the process of formation of double‐exposure and time‐averaged projection moiré fringes.     

Minimizing stochastic moiré in frequency-modulated halftones by means of green-noise masks

In color halftoning, moiré is the low-frequency spatial artifact generated by the interference of superimposed primary color dot screens that adds an unwanted artificial texture to the printed image. When these overlapping dot screens are irregular, as in the case of stochastic dot screens, this interference pattern follows a random spatial distribution resulting in 'stochastic" moiré. This stochastic moiré is at its most visible when the overlapping dither patterns have the same relative spacing between dots. We study the occurrence of stochastic moiré in green-noise halftones where dither patterns are composed of clusters of varying sizes and where the visibility of stochastic moiré6 can be reduced by varying the coarseness of dither patterns between the component cyan, magenta, yellow, and black colors.     

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).     

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.     

基于莫尔现象的印刷品防伪技术

目的利用一个动态的光栅(解码片)与一个静态的光栅(印刷品)产生莫尔条纹来放大隐藏信息,实现信息解码。方法对莫尔现象的形成和放大原理进行分析,结合印刷加网技术特点,确定影响单、双向隐藏信息的加密参数,用印刷机输出加密样张。结果得到了印刷加网中的不同网点形状、网点面积率、加网线数和网点角度等参数对信息隐藏效果和解码效果的影响。结论研究成果中隐藏信息加密参数匹配可以获得良好的解码效果,对防伪图像、印刷条件的适用性较强,为防伪印刷品的制作提供技术支持。     

Improved phase-shifting method for automatic processing of moiré deflectograms

An improved moiré deflectometry phase-shifting technique is presented. A squared grating is used to multiplex the information of the deflections in two orthogonal directions in one image. This procedure avoids the need to rotate the gratings to obtain complete deflection information. However, the use of these gratings makes impossible the application of standard phase-shifting algorithms. Specifically, the problems associated with the nonsinusoidal profile of the moiré fringes and the low-modulation areas produced by the square gratings are solved. A modified moiré deflectometry phase-shifting method is designed to deal with these problems. In addition, a method to obtain the zero order of the prismatic effect is developed. The technique configures a complete and automatic method of mapping ray deflections. From them the refractive power maps can be derived. Experimental results obtained with a progressive-addition lens are shown.     

Generation of carrier fringes in holography and shearography

Double-exposure holography and double-exposure shearography are often used together with the carrier fringe technique, which requires additional shifting of the light source in a prescribed manner between exposures. In the holographic carrier fringe technique, difficulty in prescribing a suitable movement of the light source may be alleviated through visualization of the moiré fringes that are reconstructed by slight displacement of two overlaid families of ellipsoids in a holodiagram. Because shearography is the first differential of holography, it is often impractical to perform two successive optical differentiations on the ellipsoids to visualize the shearographic carrier fringes. A simple method of discerning holographic and shearographic carrier fringes is described. The method is based on the hyperboloids in a holodiagram that represent Young's (interference) fringes produced by the interference of two point sources. The hyperboloids are analogous to holographic carrier fringes, whereas the moiré patterns reconstructed from two overlaid hyperboloids are analogous to shearographic carrier fringes. Use of this method for explaining the formation of deformation fringes in plate bending, as well as the effect of light-source movement on the deformation fringes, is also illustrated.     

Effects of a Small Inclination Misalignment in Talbot Interferometry by Use of Gratings with Arbitrary Line Orientation. II. Experimental Verification

The theoretical results of the effects of a small inclination misalignment, which is formed by rotation of the beam-splitter grating around the axis on the grating plane when the axis has an arbitrary angle with respect to the line direction of the grating, between the two grating planes on the moiré fringes in the Talbot interferometry are verified by experiment. The experimental results coincide well with theoretical ones. Consequently, the effect of the small arbitrary inclination on practical measurements based on the Talbot interferometry is further explained by an example that examines the beam collimation of a lens, and the advantages and limitations of the effect are also discussed.     

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.     

Unification of formulation of moiré fringe spacing in parametric equation and Fourier analysis methods

Inspections of moiré fringe characteristics, such as period and orientation, conventionally are done by two approaches; namely, parametric equation and Fourier analysis methods. In some cases these methods yield different results. This inconsistency is removed by revising the derivation of the indicial equation for moiré fringes by the parametric equation method.