fracture detection by grating moiré and in-plane espi techniques

Optics and Lasers in Engineering 39 (2003) 525–536

Fracture detection by grating moir!e and in-planeESPI techniques

Amalia Mart!ınez*, Ram !on Rodr!ıguez-Vera, J.A. Rayas,H.J. Puga

Centro de Investigaciones en Optica, A.C., Apartado Postal 1-948, C. P. 37000, Le !on, Gto., Mexico

Abstract

Optical interferometry techniques have been used for high-precision displacement

measuring. Commonly, in-plane sensitive arrangements use two symmetrical collimated

wavefronts for object surface illumination. However, this is a limitation when large object

surface, has to be analyzed. In this case spherical illumination is needed. As a consequence of

using non-collimated symmetrical dual-beams the sensitivity vector varies with the local

position on the surface target. Then, this kind of illumination is also capable of detecting a

lightly and systematic out-of-plane component of deformation. In this paper a theoretical

analysis of the sensitivity vector components behavior is made. Each component of the

sensitivity vector to minimize the required displacement component uncertainty is calculated.

This study is important in the stage of planning any interferometric measurement experiment,

particularly, for moir!e grating interferometric technique, which has been used only in

collimated illumination. By using a spherical dual-beam optical setup, the present work shows

results of fracture measuring by using moir!e and speckle interferometric methods. As a result,

advantages and disadvantages of both techniques are discussed and an accuracy study is

reported. r 2002 Published by Elsevier Science Ltd.

Keywords: In-plane; Sensitivity vector; ESPI; Moir!e grating

1. Introduction

Optical methods are desirable because displacement information is obtained overa full-field and they are non-contacting ones. The object under test needs to be

*Corresponding author. Fax: +52-4717-5000.

E-mail address: [email protected] (A. Mart!ınez).

0143-8166/03/$ - see front matter r 2002 Published by Elsevier Science Ltd.

PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 0 4 4 - 1

properly illuminated, and little, if any, surface preparation is required. A fringepattern image is the raw result of such methods. The shape and density of the fringesrelate directly to the stress-relief deformations and may be interpreted to yield strainand stress magnitudes. Essentially, the fringe patterns are maps of the wrapped phasechange of the monochromatic wave field of the light scattered from the deformingsurface of the object under study. Some of the most popular methods of recordingthe wave fields include holographic interferometry [1], moir!e interferometry [2],speckle photography and speckle pattern correlation interferometry [3]. Each ofthem is different, but complementary.In moir!e interferometry, a diffraction grating, typically 1200 lines/mm, is mounted

or scribed on the surface to be studied. The object is illuminated under two opposing,collimated beams both incident on the object at the same angle. Under thisgeometry, the angle of intersection between the two diffracted beams is zero and auniform intensity is observed. However, once the object is deformed, the angle ofintersection is no longer zero and the two diffracted wave fields interfere to form afringe pattern. One constraint on this method is that the diffraction grating must beplaced on the object in the area to be studied; this may require special machining ofthe object or the placement of a prepared diffraction grating using epoxies that couldinfluence the experiment.Electronic Speckle Pattern Interferometry (ESPI) is based in slight changes in

the scattered speckle pattern produced by the deformation of the objectsurface between two video frames. This method exploits the fact that speckleintensity varies randomly on the rough surface target. This is due to the highlaser coherence light used to illuminate rough surface target [3]. Consequently,the wrapped phase is determined by local correlation of the observed specklepattern. The advantages of this method are: there is considerable latitude forplacement of source beams and the observing camera, little or no surface preparationis required and the method is inexpensive to implement. The principal kinds ofmeasurements which are made using ESPI are in-plane and out-plane displacements.The first of these requires two plane wavefronts to be incident at equal and oppositeangles to the object surface. The second case requires an in-line smooth referencebeam. ESPI is less sensitive to environmental disturbances than holographicinterferometry, and does not require the applications of a surface grating (as moir!einterferometry).Under spherical illumination, in-plane data are valid only in or near the center of

the surface target. Although ESPI systems have been designed to detect in-planedisplacements using spherical illumination [4,5], their out-of-plane displacementsensitivity has not been considered. Magnitude of the residual stress in a thinaluminum plate subjected, also, to a uniform one-axial tensile load was determinedby Diaz et al. [4]. The system used measures the in-plane component of thedisplacement vector. The sensitivity of this system was 0.633 mm and target specimenwas a 110� 31.7� 1.0mm3 rectangular cross-section plate. Hertwig et al. [5]extended the range and enhanced the measurement accuracy. They work ondetecting various types of fatigue damage in carbon-fiber-reinforced plastics. Theobtained system sensitivity was 1.03 mm and surface size 6� 4.5� 4.5mm3. Some

A. Mart!ınez et al. / Optics and Lasers in Engineering 39 (2003) 525–536526

reported works used as specimen surface very small surfaces, and this is theirlimitation.On the other hand, error estimation introduced by the use of spherical

illumination in ESPI for both in-plane and out-of-plane arrangements hasbeen reported. For out-of-plane ESPI system, De Veuster et al. [6], computedthe displacement error for different interferometric configurations, showing thatan in-plane component appears. Their study is based on a geometrical point ofview. They analyzed the sensitivity change as a function of illumination andobservation angles, as well as, the shape and position of the object. Theseauthors did not study of the in-plane ESPI system. Albrecht et al. [7] useda vectorial approach to show that the total fringe pattern change Dj dependson the wavelength l; the unit vectors of illumination #n1; #n2 (in-plane) or theillumination vector #n1; observation vector #n3 (out-of-plane) and the displacement~dd : For in-plane sensitive system, their experimental setup uses two mirrors forfolding a point source in order to get the two spherical illumination wavefronts. Useof these mirrors gives us an additional mathematical consideration. By introducing alinear function, point-to-point on the mirrors’ surface, phase error correction iscalculated.Puga et al. [8] presented a general model to predict and correct the displacement

and phase map interpretation error taking into account the target shape,illumination, geometry, and in-plane displacement to out-of-plane configurationsused in ESPI.Although a large literature exists work on the theme of in-plane ESPI, few authors

[6–8] have analyzed non-collimated closely illumination, and no author, as far as weknow, analyzed moir!e grating interferometry.The purpose of this paper is to make a theoretical analysis of the behavior

of sensitivity vector components. First, it is calculated each componentof the sensitivity vector to minimized the required displacement componentuncertainty. In this analysis a vectorial approach is used, which does notrequires additional functions simplifying the mathematical treatment. In order tovisualize easily sensitivity regions on the surface target, the componentvalues are normalized with respect to quadratic magnitude sensitivity vector. It iscalculated the sensitivity percentage to each of its component on the specimen target.This study is important in the stage of planning any interferometric measurementexperiment.Second, in order to make a comparison between moir!e and in-plane electronic

speckle pattern interferometer, an experimental application is described. We presenta quantitative description of deformation fields on a metallic fractured plate. Theoptical system used by both techniques is a dual-beam illumination interferometer,which uses two divergent symmetrical point sources. Although our experimentalresults are obtained for a small object of 6� 6mm2, the measurement maximumerror percentage due to the use of spherical illumination is 0.18%. This is quite small.However, analytical error for a sample of one order of magnitude (60� 60mm2) insize greater gives maximum 13.26%, which it has not be took in account by someworks previously reported [4].

A. Mart!ınez et al. / Optics and Lasers in Engineering 39 (2003) 525–536 527

2. Sensitivity vector analysis

In a dual-beam interferometer, a coherent laser beam is splitted in two armsexpanded to illuminate a surface target (Fig. 1). Relation between the measuredphase difference Dfðx; yÞ and the displacement vector ~dd ¼ ~dd ðu; v;wÞ at a point P ¼Pðx; yÞ is given by [1]

DfðPÞ ¼ ~dd ðPÞ �~eeðPÞ; ð1Þ

where ~eeðPÞ is the sensitivity vector given by

~eeðPÞ ¼2pl½ #n1ðPÞ � #n2ðPÞ� ð2Þ

and #n1 and #n2 are unit vectors that describe the vectorial characteristics ofilluminating beams merging from S1 and S2, respectively. Notice that #n1 and #n2change in direction for each point on the inspected area of the surface target. Then,in a double illumination system, the unit vectors go from each illumination source toeach target point P: The sensitivity vector, in this case, is computed by the differencebetween both unitary vectors. Then, in this case of dual illumination it is observedthat the resulting sensitivity vector does not depend of observation vector.Eq. (1) shows that the measured phase difference Df is equivalent to the

component ds of the displacement vector ~dd in the direction of the sensitivity vector~ee:

n1ˆ

ed

n2ˆ

S2

S1

y

x

z

P

Fig. 1. Geometry of the dual-beam speckle interferometer and moir!e interferometer with spherical

illumination.


However, this direction depends on the location Pðx; yÞ where the phase Df ismeasured.Since, the sensitivity vector magnitude is given by

je,ðPÞj ¼ ðe2x þ e2y þ e2zÞ1=2 ð3Þ

and its square value is

je,ðPÞj2 ¼ e2x þ e2y þ e2z : ð4Þ

If we take j %eðPÞj2 as reference values for each corresponding point, Pðx; yÞ; anddivides each term e2x; e2y; and e2z of Eq. (4) and multiply by 100:

je,ðPÞj2

je,ðPÞj2� 100 ¼

e2x

je,ðPÞj2� 100þ

e2y

je,ðPÞj2� 100þ

e2z

je,ðPÞj2� 100; ð5Þ

then, we can define the function sensitivity to each component as

Sx ¼e2x

je,ðPÞj2

� �� 100; ð6aÞ

Sy ¼e2y

je,ðPÞj2

!� 100; ð6bÞ

Sz ¼e2z

je,ðPÞj2

� �� 100: ð6cÞ

Figs. 2a–c, show a three dimensional-map representing function given by Eqs. (6a)–(6c), respectively. These figures were drawn using source distances, with respect toorigin, of 70 cm and the incidence angle for each one was 91, over a field of view of6� 6mm2. For this particular geometry we have a minimum sensitivity,corresponding to the specimen border point (3, �3), of 99.82% in plane for x-

component, a maximum sensitivity of 33.56� 10�5% in-plane for y-component and0.18% to out-of-plane. This kind of geometry delivers too few sensitivity for out-of-plane. Of course, the y-component sensitivity is also too low due to the horizontalsource position. In such case, this sensitivity percentage should be maximized byvertical repositioning the point sources.In addition, if our object target is 60� 60mm2 in size and using the same

experimental parameters, the minimum sensitivity in-plane corresponding to thetarget border point (30, �30) is 86.74% for x-component, a maximum sensitivity in-plane is 2.10% for y-component, and 11.76% to out-of-plane. All these results areobtained by using the analytical analysis (Eqs. (6a)–(6c)).With the theoretical treatment, we can observe that if the essential component (in

our case x-component) loses slightly its sensitivity and falls to a value of about86.74%, it represents an error of 13.26% for the surface size of 60� 60mm2.


3. Experimental procedure

A 14.50� 3.90� 1 cm3 fractured metallic bar was clamped along its shorter twoedges and mechanically point loaded at the center. A region of 6� 6mm2 around acrack of this specimen object was measured and used to present moir!e and speckleinterferometric results. The object was point loaded from the back by a screw, alongthe optical axis of the observation system. The screw chosen load delivers atranslation of 33 mm. As mentioned above, this load produces in-plane and out-of-plane deformations, which can be detected by the kind object illumination. To makea comparative study of the two methods, we investigated a crack on the specimentarget.

-3-2

-10 1 2

x (mm)-3-2-1012

y (mm)

99.599.699.799.899.9100

Sx (%)

99.599.699.799.899.9100

Sx (%)

-3-2

-10 1 2

x (mm)-3

-2-1012

y (mm)

05E-0050.00010.000150.00020.000250.00030.00035

Sy (%)

05E-0050.0001

0.000150.0002

0.000250.0003

0.00035

Sy (%)

-3-2

-1 01 2

x (mm)-3

-2-1012

y (mm)

00.050.10.150.2

Sz (%)

00.050.1

0.15

0.2

Sz (%)

Fig. 2. Sensitivity function for each sensitivity vector component. Distances of each source with respect

origin are 70 cm, incidence angle of 91. Graphic corresponding to (a) Sx; sensitivity associated to term e2x;(b) Sy; sensitivity associated to term e2y and (c) Sz; sensitivity associated to term e2z :


3.1. Technique of moir !e interferometry

As a first step a grating is recorded on the object surface under test. The surface iscovered by photo resist (S1822 Shipley). The photo resist was deposited by means ofa spinner at 1000 r.p.m. given a 4 mm thickness film. After depositing the photo resistfilm, the plate is baked to a temperature of de 701C during 10min in order to extractresidual humidity.Placing the sensibilized plate in the interference zone of two point sources similar

to those of Fig. 3a, the specimen grating is recorded. The region of 6� 6mm2 wascentered about the observation axis (Fig. 3b). The laser used in this step, is a He–Cd,with a power of 80mW, and wavelength l ¼ 0:440 mm. Measured power, where thesample was placed, was 4.93mW/cm2. The exposition time was 30 s, correspondingto 148mJ/cm2.

CCD

x

zy

digitizerand framegrabber

computer PZTcontroller

PZT mirror

spatial filter

spatial filter

mirror

beamsplitter

laser

load

z

x

y

(a)

(b)

Fig. 3. (a) Traditional experimental arrangement scheme of the speckle and moir!e interferometry for in-

plane deformation measuring. (b) It shows that the region of 6� 6mm2 is centered about the observation

axis.


The incidence angle used, y ¼ 31; gives a specimen grating frequency of 240 lines/mm (T ¼ 4 mm). Using Microposit Shipley MF-319 during one and a half-minuterinsing at the same time at environment temperature carries out developmentprocess. After drying, the sample is cured with UV light exposed during a couple ofminutes (1956mJ/cm2). Last step stabilizes chemically the sample and improvesdiffraction efficiency [9].A similar system is used to obtain the virtual grating (Fig. 3a) but with a 4mJ

stabilized He–Ne laser, under an incidence angle of y ¼ 91 which corresponds to afrequency of 480 lines/mm (T ¼ 2 mm) satisfying the condition f ¼ 2fs [2], where fs isthe specimen grating frequency and f is the virtual grating frequency. We getinterferometric moir!e fringes when the plate is loaded at the center. Fringe patternswere captured by means of a CCD camera, which deliver an image area of8.8�6.6mm2 (2

3in format) and cell size of 11.5 mm (H) by 27 mm (V). Fig. 4 shows a

displacement map obtained by using conventional 4 step phase [10] detecting and thediscrete cosine transform for phase unwrapping [11].

3.2. ESPI technique

A similar configuration was used for the ESPI technique, Fig. 3. The metallic platewas sprayed with a white powder to avoid specular reflection. For deformationmeasuring, an initially (reference) speckle pattern image from the surface target istaken. Then the object is deformed, and another image is taken. By using subtractionmode [3], correlation fringes corresponding to the object deformation are obtained.Again, using 4 steps phase shifting technique, the wrap and unwrap phase map iscalculated, in a similar manner that the moir!e technique. But, since the four acquiredfringe images with this setup are too noise, a kind of filter must be performed. In thiscase, the used filter was the ksmooth function [12], which uses a Gaussian kernelto compute local weighted average of the input vector vy: For each element in

32

10

-1-2

-3

32

10

-1-2

-3

0

10

20

30

40

50

u (�m)

x (mm)y (mm)

Fig. 4. Deformation three-dimensional plot around the region of the crack by moir!e interferometry.


the n-element vector vy; the ksmooth function returns a new vy0 given by

vy0i ¼

Pnj¼1 Kððvxi � vxjÞ=bÞ � vyjPn

j¼1 Kððvxi � vxjÞ=bÞð7Þ

where

KðtÞ ¼1ffiffiffiffiffiffi

2pp

ð0:37Þexp �

t2

2 � 0:372

� �ð8Þ

and b is a bandwidth, which can supplies to the ksmooth function. The bandwidth isusually set to a few times the spacing between data point on the x-axis depending onhow big a window it wants to use when smoothing. The behavior of this and otherkind of filters are analyzed for preserving borders [13].As result of the use of the above steps for experimental fringe patterns, Fig. 5

shows a three-dimensional plot of the object deformation measured near the regionof the crack.As we can see from Fig. 2a, we can associate for the point (3, �3) an error

percentage of 0.18% corresponding at same points in Figs. 4 and 5. Similarly we canextent on each graphical point to obtain their corresponding percentage error in thedisplacement measurement. This should be done automatically for obtain a errorfunction to correct on all the displacement field, as in similar way for interpreting ofthe phase map [8].

4. Comparison of the methods

In moir!e interferometry, the displacement sensitivity is defined by the number offringes generated per unit object displacement, which is associated to the frequency f

of the reference grating. Then the sensitivity is increased with the frequency. Theinverse of sensitivity defines the contour interval, 1=f [2]. On the other hand, the

32

10

-1-2

-3

32

10

-1-2

-3

0

10

20

30

40

50

x (mm)y (mm)

u (�m)

Fig. 5. Deformation three-dimensional plot around the region of the crack by ESPI.


displacement resolution reflects the reliability of a displacement measurement. It is,the tolerance within which a measurement is assumed to be reliable. In the usualengineering analysis where the displacement varies smoothly between neighboringfringes, visual interpolation to 1

5or 1

10of a fringe order is reliable [2]. Although both

techniques use a CCD camera for fringe recording, phase-shifting and interpolationincluded for fringe analysis, which limits these definitions, we will follow them fornext discussion.Table 1 shows the defined parameters above for speckle and moir!e interferometry

techniques experimentally achieved. As we can see that the sensitivity and contourinterval for both moir!e and speckle techniques are the same. But resolution is lesserfor ESPI, as have been reported [3] and the measurement range is greater for moir!einterferometry.Fig. 6 exhibits some profiles from the graphics shown in Figs. 4 and 5, which a

mixed displacement components are displayed. Moir!e interferometry and ESPIresults are in excellent agreement. This comparison highlights the sensitivity of ESPI,although the measurement range is significantly less than for moir!e interferometry.However, it is possible to calculate cumulative values with ESPI [5]. It is observedthat too small deformations (Fig. 6), the real value is confuse in ESPI by poor fringequality and speckle noise, inherent to the technique. This is due to ESPI correlationfringes are obtained from the variation of the phase change, Df; due to the localdisplacement, with of the standard deviation of the subtracted or added signals, ifthe contrast of the individual signals is reduced, the standard deviation of thesubtracted or added signals is also reduced, and the visibility of the correlationfringes is decreased.

5. Conclusion

We have analyzed the use of object spherical illumination commonly used in in-plane speckle interferometry and extended to grating moir!e. A set of graphicalresults was given for studying the sensitivity vector variation due to sphericalillumination. As a result, a lightly and systematic out-of-plane component ofdeformation appears, which introduces an in-plane x-component error. It isimportant in the stage of planning a interferometric measurement experiment to

Table 1

Parameters values for speckle and moir!e techniques

Moir!e interferometry ESPI

Sensitivity 0.48 lines/mm 0.48 lines/mmContour interval 2.08mm 2.08mmResolution 0.26mm 0.43mmMeasurement range x: �5 to 5 cm X: �14 to 14 mm

y: �5 to 5 cm Y: �14 to 14 mmz: �5 to 10 cm z: �1113 to 1113mm


minimized the required displacement component uncertainty. By using a sphericaldual-beam optical setup, the present work shows results of fracture measuring byusing both moir!e and speckle interferometric methods. As a result, advantages anddisadvantages of both techniques are discussed and an accuracy study is reported.

Acknowledgements

This work was partially supported by Consejo Nacional de Ciencia y Tecnolog!ıa,CONACYT. Amalia Martinez and H.J. Puga thank grants from CONACYT andConsejo de Ciencia y Tecnolog!ıa del Estado de Guanajuato, CONCYTEG.

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-3 -2 -1 0 1 2 3

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(b)

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fracture detection by grating moiré and in-plane espi techniques

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