Multiple-wave lateral shearing interferometry for wave-front sensing

Multiple-wave achromatic interferometric techniques are used to measure, with high accuracy and high transverse resolution, wave fronts of polychromatic light sources. The wave fronts to be measured are replicated by a diffraction grating into several copies interfering together, leading to an interference pattern. A CCD detector located in the vicinity of the grating records this interference pattern. Some of these wave-front sensors are able to resolve wave-front spatial frequencies 3 to 4 times higher than a conventional Shack-Hartmann technique using an equivalent CCD detector. Its dynamic is also much higher, 2 to 3 orders of magnitude.     

Modal wavefront reconstruction based on Zernike polynomials for lateral shearing interferometry: comparisons of existing algorithms

Four modal methods of reconstructing a wavefront from its difference fronts based on Zernike polynomials in lateral shearing interferometry are currently available, namely the Rimmer-Wyant method, elliptical orthogonal transformation, numerical orthogonal transformation, and difference Zernike polynomial fitting. The present study compared these four methods by theoretical analysis and numerical experiments. The results show that the difference Zernike polynomial fitting method is superior to the three other methods due to its high accuracy, easy implementation, easy extension to any high order, and applicability to the reconstruction of a wavefront on an aperture of arbitrary shape. Thus, this method is recommended for use in lateral shearing interferometry for wavefront reconstruction.     

Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears

A method is proposed for exact discrete reconstruction of a two-dimensional wave front from four suitably designed lateral shearing experiments. The method reconstructs any wave front at evaluation points of a circular aperture exactly up to an arbitrary constant for noiseless data, and it shows excellent stability properties in the case of noisy data. Application of large shears is allowed, and high resolution of the reconstructed wave front can be achieved. Results of numerical experiments are presented that demonstrate the capability of the method.     

The one-step phase-shifting technique for wave-front interferometry

Abstract

Hitherto, phase-shifting schemes in wave-front interferometry required a minimum of two precise phase steps in order to derive the phase. This paper outlines and demonstrates a novel technique entailing one precise phase step. Numerical analyses of phase-drift errors indicate that the proposed technique compares favourably with performances of the three- and four-bucket algorithms currently in use. Furthermore, it minimizes the period of computation by necessitating the use of three processing frames.     

Implementation of tridirectional large lateral shearing displacement interferometry in temperature measurement of a diffused ethylene flame

A tridirectional large lateral shearing displacement interferometric system has been proposed and used to reconstruct the temperature field of a quasi-axisymmetric diffused ethylene flame in two-dimensional (2D) and three-dimensional (3D) hypotheses. In comparison with the thermocouple results, the 2D reconstructed results affords a quantitative analysis with an average discrepancy between 20 and 40 K in the full field, except in the closer part inside the peak temperature location where a high soot volume fraction exists. The 3D reconstructed results affords qualitative analysis and exhibits some asymmetrical characters, but an obvious error occurs at 1 cm height where it is not suitable to use the universal correction coefficient.     

Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry

We propose a simple multifrequency spatial-carrier and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry (LSI). In this system a wedge-shaped plate lateral shearing interferometer is used and, owing to the presence of tilt, a finite number of fringes parallel to the direction of the shear appears; hence a significant spatial-carrier frequency is generated at the focus position. We further enhance the spatial-carrier frequency either by changing the wavelength of the laser light or by slight defocusing. A synthetic interferogram with low spatial-carrier frequency is obtained by use of laser light of two wavelengths simultaneously in the lateral shear interferometer. We obtain the phase-shifted fringe patterns from the same setup by simply moving the wedge plate in an in-plane parallel direction, using a linear translator. The fringe projection system was tested for measurement of the three-dimensional shape of a discontinuous object. The present system has many advantages; e.g., it is a common-path interferometry and hence is insensitive to external vibrations, is compact in size, and is relatively inexpensive.     

High-precision analysis of a lateral shearing interferogram by use of the integration method and polynomials

Interferograms obtained with ordinary interferometers, such as the Fizeau interferometer or the Twyman-Green interferometer, show the contour maps of a wave front under test. On the other hand, lateral shearing interferograms show the difference between a wave front under test and a sheared wave front, that is, the inclination of the wave front. Therefore the shape of the wave front under test is reconstructed by means of analyzing the difference. To reconstruct the wave front, many methods have been proposed. An integration method is usually used to reconstruct the wave front under test rapidly. However, this method has two disadvantages: The analysis accuracy of the method is low, and part of the wave front cannot be measured. To overcome these two problems, a new, to our knowledge, integration method, improved by use of polynomials, is proposed. The validity of the proposed method is evaluated by computer simulations. In the simulations the analysis accuracy achieved by the proposed method is compared with the analysis accuracy of the ordinary integration method and that of the method proposed by Rimmer and Wyant. The results of the simulations show that the analysis accuracy of the newly proposed method is better than that of the integration method and that of the Rimmer-Wyant method.     

Fundamental performance comparison of a Hartmann and a shearing interferometer wave-front sensor

The performance of ground-based optical imaging systems is severely degraded from the diffraction limit by the random effects of the atmosphere. Adaptive-optics techniques have been used to compensate for atmospheric-turbulence effects. A critical component in the adaptive-optics system is the wave-front sensor. At present, two types of sensors are common: the Hartmann-Shack wave-front sensor and the shearing interferometer wave-front sensor. In this paper we make a direct performance comparison of these two sensors. The performance calculations are restricted to common configurations of these two sensors and the fundamental limits imposed by shot noise and atmospheric effects. These two effects encompass the effects of extended reference beacons and sensor subaperture spacings larger than the Fried parameter r(0). Our results indicate comparable performance for good seeing conditions and small beacons. However, for poor seeing conditions and extended beacons, the Hartmann sensor has lower error levels than the shearing interferometer.     

Reflective grating interferometer: a folded reversal and shearing wave-front interferometer

The reflecting grating interferometer (RGI) is a folded and reversal wave-front interferometer sensitive only to asymmetrical aberrations such as third-order coma. The RGI can isolate and evaluate coma both in nearly collimated and in noncollimated beams. We propose a RGI with a different optical configuration that includes a lateral shearing in addition to folding and reversal operations. With lateral shear, the RGI also becomes sensitive to other terms of third-order aberrations such as defocusing, astigmatism, and spherical aberration. Optical path difference equations for interpreting interferograms and numerical simulations are presented to show how the interferometer works in the shearing configuration. Its potential applications are described and discussed.     

Gram-Schmidt orthonormalization of Zernike polynomials for general aperture shapes

We present analytical derivations of aberration functions for annular sector apertures. We show that the Zernike functions for circular apertures can be generalized for any aperture shape. Interferogram reduction when Zernike functions were used as a basis set was performed on annular sectors. We have created a computer program to generate orthogonal aberration functions. Completely general aperture shapes and user-selected basis sets may be treated with a digital Gram-Schmidt orthonormalization approach.     

Risley prisms to control wave-front tilt and displacement in a vectorial shearing interferometer

A pair of thin prisms is used to deviate a light beam without changing the image orientation in a vectorial shearing interferometer. The relative angle between prisms determines the displacement of the wave front and its tilt. The direction of the beam displacement is controlled by means of changing the relative angle between prisms. This system is employed to control the displacement of a sheared wave front as a vector quantity and to introduce a controlled amount of tilt in what we believe is a novel interferometric shearing system. The predicted performance of this wave-front director is confirmed experimentally.     

Measurement of longitudinal displacement using lateral shearing cyclic path optical configuration setup and phase shifting interferometry

We present a technique for the measurement of longitudinal displacement using a lateral shearing cyclic path optical configuration (CPOC) setup and phase shifting interferometry. In the technique, a plane mirror mounted on a linear translation stage, placed slightly away from the focal plane of a lens, introduces a longitudinal focal shift to the incident focusing beam. The resulting spherical wavefront emerging from the lens is sheared into two orthogonally polarized beams using the CPOC setup. By applying polarization phase shifting interferometry (PPSI), the longitudinal focal shift of the beam focus is calculated by determining the slope of the optical path difference variation between the sheared beams. Similarly, the additional focal shift introduced due to longitudinal translation of the mirror, by an unknown amount, is determined using PPSI. Half of the difference between the two longitudinal focal shifts measured gives the longitudinal displacement of the mirror. The technique can be used for an extended range of distance measurement. The novelty of the technique is the introduction of CPOC for the distance measurement. The advantages of the technique compared to other related methods are discussed.     

Thickness measurement of transparent glass plates using a lateral shearing cyclic path optical configuration setup and polarization phase shifting interferometry

We present a measurement technique to determine the thickness of a transparent glass plate (GP) by using a lateral shearing cyclic path optical configuration (CPOC) setup and polarization phase shifting interferometry (PPSI). In the technique, the GP introduces a longitudinal shift in the focus of the beam and, as a result, a spherical wavefront emerges from the lens, which is otherwise set for producing a collimated beam. Using CPOC, two laterally sheared orthogonally polarized beams are generated from the incident spherical wavefront. By applying PPSI, the slope of the optical path difference variation between the laterally sheared interfering beams is evaluated, and the radius of the spherical wavefront and the longitudinal shift of the beam focus are calculated. The thickness of the GP is determined from the standard relation between the longitudinal shift of the focus introduced by the GP and the thickness of the GP. Results obtained for a GP of 9.810mm thickness are presented.     

Analysis of coherent symmetrical illumination for electronic speckle pattern shearing interferometry

In an effort to find a non-contact technique capable of providing measurements of in-plane strain, a speckle shearing interferometer was designed using symmetrical coherent illumination. This paper presents an analysis of the sensitivity to displacement and strain of this interferometer, together with an analysis of the phase-stepping of the resultant fringe patterns. New notation is introduced alongside this analysis to define the interference components in speckle shearing interferometers using multiple illumination beams. Experimental results show fringe patterns and phase stepping in support of the theoretical analysis.     

Multiple-beam lateral shear interferometry for optical testing

Lateral shear interferometry is used to obtain the lateral aberrations of a lens. The zeroth-order fringe in an interferogram obtained from a wedge-plate lateral shear interferometer, however, directly displays the lateral aberration curve of a test lens. Nevertheless, the intensity distribution, is cosinusoidal. Multiple-beam interferometry results in sharpened fringes; hence the multiple-beam wedge-plate shear interferometer displays the lateral aberration curve of a lens sharply, provided the shear is small. For large shear, some new artifacts appear in the interferogram, which are also explained.     

Complex division as a common basis for calculating phase differences in electronic speckle pattern interferometry in one step

We point out that all formulas for calculating the phase map ofobject deformations in one step can be described by the same simpleformalism of a complex division.     

轮胎剪切散斑干涉包裹相位图缺陷识别方法

目前轮胎剪切散斑干涉缺陷识别技术主要依靠人工辅助识别,难以满足大批量工业生产检测的需要。该文提出基于剪切散斑图角点特征的轮胎缺陷识别方法,实现轮胎缺陷自动识别。首先通过对激光剪切散斑干涉包裹相位图缺陷特征、去噪方法等方面进行分析,并根据激光散斑干涉包裹相位图角点特征,提出基于角点特征的包裹相位图缺陷识别方法。实验结果表明:在两组包裹相位图缺陷识别实验中,该文方法对条纹质量较好、较差的包裹相位图,缺陷识别效果准确、定位精度高,该方法简单、高效,有助于提高轮胎缺陷检测速度与效率,可实现自动化轮胎缺陷检测。