Determination of the zero-order fringe position in digital speckle pattern interferometry

A method for determining the position of the zero order fringe in a metrological experiment with digital speckle pattern interferometry is proposed. It is based on an averaging procedure with shifted images obtained before and after a load is applied. This technique is a complement to the phase-shifting methods. Experimental examples are shown. © 1997 Optical Society of America

Interferometry can be used to produce a fringe pat tern that represents the field surface displacement of an object in response to some change in mechanical loading. 1 Digital speckle pattern interferometry (DSPI) is one of the most modem techniques for depicting such fringe patterns.It combines real-time processing with the flexibility of software handling.2DSPI was developed by combining the well-known techniques of holographic and speckle interferometry by using an image hologram setup and following the methods of double-exposure holography.It utilizes a CCD camera interfaced to a computer to process the data.For comparison measurements, a reference frame stored in memory is continuously subtracted from incoming data, and then the intensity difference is displayed.The fringes represent the correlation between the speckle patterns.However, for quantitative pur poses, interferograms must be analyzed so that the results can be presented in the required numerical form.3Progress has been made in fringe pattern analysis with a number of electronic aids.These devices allowed a substantial improvement in the accuracy of fringe location within an interferogram.
The wide availability of digital image processing equipment prompted a number of studies to investigate different alternatives of automatic interfero gram analyzing.Modem techniques have reached a point at which they can provide useful results, especially with phase-stepping methods.4By introduc ing discrete shifts in the position of the fringes, we can calculate the phase map at pixel location ( i , j ) in a relatively simple way.The phase is changed when a mirror with a computer-controlled piezoelectric translator is moved.
The successful interpretation of an interferogram depends to a great extent on the ability to assign the right order numbers to the flanges.This fringe la beling is used to determine the displacement of a point relative to a certain origin when the sensitivity vector of the experiment is known.Order ambiguity is then a major source of errors, both for displacement measurements and for generating and interpreting contour fringes.Analogical5 and digital6 holo graphic contouring have used multiple illumination sources to modify the fringe structure as a means for reducing order assignment ambiguity.
Another technique for contouring with multiple digital images is to synthesize a particular fringe profile, namely, that of an approximate delta func tion.7That fringe corresponds to the zero-order in terference fringe, showing the locus of places with a zero optical path difference between the two illuminating beams.Using such a single-fringe projec tion, we can eliminate order ambiguity.
We present in this study a method for displaying the fringe of zero displacement by using the same principle.Several illuminating beams are provided for that purpose by the movement of a lens controlled by a stepping motor.A series of images are captured and stored with different object illuminations before and after a load is applied to the object.DSPI fringes are obtained by subtracting the corresponding pairs of images, and the results are then incoherently added, resulting in a single dark fringe showing the where N(P) is the number of fringes in the image between a fixed point (a point that suffered no dis placement between exposures) and P, X is the light wavelength, eB is a unit vector from P in the observation direction, is a unit vector from P in the illumination direction, and S is the sensitivity vector.We assume the displacements field to be continuous, and we also suppose that in the image a connected path exists between a fixed point and P.
A displacement d(P) then produces a phase change 8.If it is great enough (corresponding to a phase difference greater than 2tt), the intensity A7 will he somewhere between the extreme values 7max and Jmin.This 7min is zero whenever 8 2mr; n -0,1, 2 , . . . .To construct the phase map, we often re quire the position of at least one point that suffered no displacement.It is used as an origin for counting fringes to every point of interest.If the sensitivity vector is known, the number of fringes between the origin mentioned and a generic point P determines only one component of the displacement.Three in dependent equations are required as a minimum for solve for three components of d: accuracy of the measurement.9The sign of every component remains nevertheless ambiguous.
It is evident that the existence of a nondeformed point is not a priori warranted.Whole-field dis placements have no such a point.Even if such a point exists, it may be located outside the observation field of the experiment.The phase change 8 is then measured with 2mr ambiguity.
When d 0, 8 0 for all sensitivity vectors.This is the only case that gives a zero phase shift for such general geometries.That is, the d 0 locus will be a dark fringe in DSPI, which is obtained through the subtraction of the before and the after states.The converse is true in analogic holography.
If several (in principle, infinite) sensitivity vectors are used, the only dark fringe in the DSPI image common to all observation points, or illuminations or combinations of both, should correspond to the locus d 0, if it exists.When, as in practice, only a finite number of sensitivity vectors are available, several isolated zero intensity fringes will appear.An a priori knowledge of the expected geometry of the dis placements could then be used to decide which represents the zero order.
With the technique we propose that a set of images be stored with a progressive increment in the angle directing the object-beam illumination.This incre ment is obtained by careful movements of a lens con trolled by a stepping motor.The illumination directions must be chosen so that every possible dis placement produces a phase change, that is, three sensitivity vectors should constitute a basis.Then the load is applied to the object and a similar set of images is recorded, repeating the same illuminating directions of the object beams as before.For each pixel (i, j ) the computer assigns a z value to the re sulting image calculated as

448
where zk is the gray level in the recorded image cor responding to the &th illumination direction before loading, zk' is the corresponding value after loading, (4) where 70 and 7r are the intensities of the object and the reference beams, respectively, <p is the random phase of the speckle pattern produced by surface roughness, and 8 is the phase change caused by the modified optical path produced, for example, by a displacement between exposures of the correspond ing object point.In principle, if the optical system is stable, 70, 7r, and cp are constants for a given point.Therefore A7 is a function only of 8.For an object point, P(x, y, z), suffering a displacement, d d(x, y, z ), 8 is given by9 locus where no displacement occurred.Accurate re positioning of the lens at its original starting point before object loading is required for the successful application of the method.If this condition is not fulfilled, the dark fringe blurs out its contrast but does not change its position.Experimental verifica tions of the proposal are presented.

Background of the Technique
The basic DSPI equation describing the intensity dif ference in an image point between a reference frame and the current one is given by8 Overdetermined systems combined with the leastsquares calculations are usually used to increase the For a given sensitivity vector there are nonzero displacem ents th a t produce a dark fringe in the im age, nam ely, those corresponding to directions per pendicular to that vector.The sensitivity vector set is determ ined from the observation direction, and the different illum ination directions can then be chosen to show a dark fringe corresponding to a subspace of the displacem ent field, the loci of places th at did not move in a certain direction.If, for example, all the sensitivity vectors are contained in a single plane, points w ith displacem ents perpendicular to that plane w ill appear dark in the resulting im age.As a consequence, regions obeying a defined property can be highlighted if required for a particular experi m ent.

Experiments
Several experim ents were conducted to dem onstrate the validity of our proposal.Figure 1 show s the ex perim ental setup.A linear polarized H e N e laser beam is divided by the beam splitter, BS.One arm serves as a reference beam, and the other is used to illum inate the object through a collim ating lens, which is mounted on a stepping motor.This last introduces the required changes in the illum inating beam direction.In all our experim ents five image pairs w ere stored w ith a step of 30 |xm betw een im ages.
In the case o f Fig. 2 a clamped planar m etallic surface, deformed under a load applied from the rear on its center, and an unperturbed m etal strip are compared.Figure 2 2) to these im ages is dis played in Fig. 2(c).The m etal strip rem ains totally black, indicating th at no deformation occurred to it.The corresponding phase mapping is shown in Fig.

2(d).
In Fig. 3 a plane object is rotated around an axis.In Fig. 3(a) the D SPI interference fringes are shown, w hereas Fig. 3(b) shows the actual rotation axis after our procedure is applied.In this example, the axis is the zero order fringe.Figure 3(c) is the result of the phase mapping of Fig. 3(a) and the position of the rotational axis obtained from Fig. 3

(b). A lack o f repeatability o f positioning betw een the acquisition of the reference state and deformed state frames m ay occur. If such an error introduces an extra optical path o f less than A./2, no fringe location
error appears but a blurring in its contrast.The error depends on the apparatus and the involved setup geometry, but it is not a serious constraint.In our case, s, the m axim um error w as of 0.05 |xm for the whole experiment.This value can be achieved w ith commercially available equipment.On the other hand, the geometric param eters, like the focal length or the observation angle, can be adjusted to bring the appropriate value for a particular experiment.

Conclusions
We have demonstrated the possibility of a zero order fringe assignm ent w ith the aid of DSPI techniques in addition to a m ultiple im age storing procedure.The proposed method is to some extent complementary w ith the phase-stepping techniques.Once the zeroorder fringe is localized, w e can proceed w ith the standard phase m apping m ethods to find the abso lute deformation field distribution.The method re ported here is suitable only for static loading.
(a) shows the object, and the rectangle indicates the observation area.Figure 2(b) show s the display o f the normal DSPI result.The result of applying Eq. ( The financial support of the Consejo Nacional de Investigaciones Científicas y Técnicas (Argentina) and Conselho Nacional de Desenvolvim iento Cienti fico e Tecnológico (Brazil) agreem ent, the Third World Academy of Science grant 93 389, the Alex ander von Humboldt Foundation (Germany), Funda ción Antorchas (Argentina), and M utis Program (Spain) is gratefully acknowledged.