Uncertainty analysis of displacements measured by in-plane electronic speckle-pattern interferometry with spherical wave fronts
Post on 02-Oct-2016
Uncertainty analysis of displacements measuredby in-plane electronic speckle-patterninterferometry with spherical wave fronts
Amalia Martnez, Ral Cordero, Juan Antonio Rayas, Hctor Jos Puga, andRamn Rodrguez-Vera
Displacement measurements by optical interferometry depend on the induced phase difference and on theinterferometers sensitivity vector; the latter depends in turn on the illuminating sources and on thegeometry of the optical arrangement. We have performed an uncertainty analysis of the in-plane dis-placements measured by electronic speckle-pattern interferometry with spherical incident wave fronts.We induced the displacements by applying a uniaxial tensile load on a nominally flat elastic sample. Weapproached the displacement uncertainty by propagating the uncertainties that we considered reason-able to assign to the measured phase difference and to the characteristic parameters of the interferom-eters sensitivity vector. Special attention was paid to evaluating contributions to the displacementuncertainty. Moreover, we observed that the uncertainty decreases if the angles of incidence and thesourcetarget distances are increased. 2005 Optical Society of America
OCIS codes: 120.3940, 120.6160, 150.3040.
Applications of speckle and moire interferometry tomeasure the relative displacements induced in sam-ples undergoing mechanical deformation have beenreported.1 Displacement measurements depend onthe induced phase difference and on the interferom-eters sensitivity vector.
With respect to the measurement of the phase dif-ference by the phase-stepping technique, an algo-rithm that can diminish the miscalibration andnonlinearity of the phase steppers has been pro-posed.2 Also, an experimental investigation to evalu-
ate the influence of the perturbing environment onthe displacement measured by phase-shifting moireinterferometry has been reported.3
For determination of the sensitivity vector, the in-plane displacements measured by electronic speckle-pattern interferometry (ESPI) have been comparedwith collimated incident beams and with divergentillumination.4 Moreover, the effects of collimation er-rors on the sensitivity of an ESPI interferometer thatuses collimated illumination have been studied5;these errors change the incident beams from colli-mated to slightly divergent, in turn modifying theinterferometers sensitivity. Furthermore, with anESPI out-of-plane setup, a general model has beendesigned to correct systematic displacement errorscaused by faults in the phase interpretation linked tothe target shape, the illumination geometry, and thein-plane displacement undergone by the sample.6Some authors have developed a theory that describesthe measurement errors that are due to the use ofnoncollimated illumination in an out-of-plane ESPSIsystem.7
If a dual-beam ESPI optical setup with sphericalillumination is used to measure the in-plane displace-ment induced on the illuminated sample, eventualerrors in the determination of the interferometer sen-sitivity are related to inaccurate determination of thelocation of the illumination sources. Although some
A. Martnez (email@example.com), J. A. Rayas, and R. Rodrguez-Vera are with the Centro de Investigaciones en ptica, A. C.,Apartado Postal 1-948, C. P. 37000, Len, Guanajuato, Mexico. R.Cordero is with the Faculty of Mechanical and Production SciencesEngineering, Escuela Superior Politcnica del Litoral, Kilometer30.5 Va Perimetral, Guayaquil, Ecuador, and the Department ofMechanical and Metallurgical Engineering, Pontificia UniversidadCatlica de Chile, Vicua Mackenna 4860, Santiago, Chile. H. J.Puga is with Departamento Ciencias Bsicas, Instituto Tec-nolgico de Len, Avenue Tecnolgico s/n Fracc. Julin de Obregn,Len, Guanajuato, Mexico.
Received 15 June 2004; revised manuscript received 1 November2004; accepted 4 November 2004.
0003-6935/05/071141-09$15.00/0 2005 Optical Society of America
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efforts have been made to quantify and minimize theinfluence of these errors, an evaluation of the uncer-tainty associated with the displacement measured byESPI has still not been reported.
In this paper we describe an uncertainty analysisof the in-plane displacements measured by ESPI withspherical wave fronts. We induced the displacementsby applying a uniaxial tensile load to a nominally flatelastic sample. Although an interferometer withspherical illumination has sensitivity along the threespatial coordinates, the geometry of the optical setupthat we used allowed us to find sensitivity mostlyalong the pulling direction; the other two componentsof the sensitivity vector were relatively small. Wemeasured the displacement induced only along thepulling direction.
We approached the displacement uncertainty bypropagating the uncertainties that we consideredreasonable to assign to the measured phase differ-ence3 and to the parameters that characterize theinterferometers sensitivity (source locations and thetarget alignment). Special attention was paid to eval-uating contributions to the displacement uncer-tainty. The uncertainty evaluation procedure wasbased on the application of the well-known law ofpropagation of uncertainties.810 Results are shown,and detailed discussion of displacement uncertaintyis given.
2. Sensitivity Vector
In a dual-beam interferometer, a coherent laser beamis divided into two arms and expanded to illuminatea surface target (Fig. 1). If the illuminated specimenundergoes mechanical deformation, the relation be-tween the induced phase difference and the rela-tive displacement vector, d dx, dy, dzT, at eachpoint of the illuminated surface is given by11
d e, (1)
where e ex, ey, ezT is the sensitivity vector4:
[n1 n2], (2)
and n1 and n2 are unit vectors that describe the illu-minating beams emerging from sources s1 and s2 inFig. 1. is the wavelength of the beams. Notice thatn1 and n2 change direction at each point on the illu-minated area. It should be observed from Eq. (2) thatthe systems sensitivity does not depend on the ob-servation direction.
The sensitivity vector components are computedby4
(x x1)2 (y y1)2 (z z1)212
(x x2)2 (y y2)2 (z z2)212, (3a)
(x x1)2 (y y1)2 (z z1)212
(x x2)2 (y y2)2 (z z2)212, (3b)
(x x1)2 (y y1)2 (z z1)212
(x x2)2 (y y2)2 (z z2)212, (3c)
where x, y, z, x1, y1, z1, and x2, y2, z2 represent,respectively, the position of point P of the illuminatedsurface and the locations of sources s1 and s2 (Fig. 1).
If the illuminated surface is flat, z in Eqs. (3a)(3c)can be modeled as equal to
where and are the angles of the illuminated sur-face with respect to axis x and to axis y, respectively.
In the theory of elasticity, normal strains x, y, and zare defined as the change in length per unit lengthinduced in a sample by mechanical deformation12:
xdxx , (5a)
ydyy , (5b)
zdzz . (5c)
Consider that we are using an in-plane ESPI setupand that we are measuring the displacement fieldalong coordinates x and y. If the placement of theorigin of the reference system utilized to determinethe spatial coordinates of the field (x, y) agrees withthe origin of the reference system used to measure
Fig. 1. Diagram to define sensitivity vector e of a dual-beaminterferometer.
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the displacements, for values of x and y differentfrom zero, x and y can be evaluated approximatelyby
As we have no sensitivity along z, we can obtainapproximately z from
zdWW , (6c)
where dW is the aggregate displacement inducedalong z and W is the sample thickness. If stress isapplied to a sample along x, although the main de-formation is induced along that direction the normalstrain x is accompanied by simultaneous lateralstrains of opposite sign, which are also proportionalto the applied stress. The lateral strains in the direc-tions of the two mutually perpendicular axes y and zare determined by12
where is Poissons ratio.Combining Eqs. (1), (6), and (7) and solving for dx,
xexey yezW. (8)
Because usually the influence of the transversalsensitivity vector components (ey and ez) on Eq. (8) issmall, the displacements are evaluated by the sim-pler and more common expression
4. Experimental Details and Results
Figure 2 illustrates the ESPI optical setup that weused. The illumination was divided into two beams (anin-plane system) that were directed to the target fromsymmetric positions. According to the reference sys-tem shown in Fig. 2, the locations of the sources weres1 17.39, 0, and 165.5 cm and s2 17.39, 0,and 165.5 cm, such that the angles of incidencemeasured with respect to axis z were both 6; thesourcetarget distance was 166.4 cm. The spatialcoordinates of the points on the specimen surfacewere labeled (x, y, z). The origin of the reference sys-tem that we utilized to determine spatial coordinates(x, y, z) was located at the center of the illuminatedarea (Fig. 2). The illuminated sample was a flat elas-tic surface of 10 cm 9.3 cm and thickness W 0.1 mm located on place xy; this means that, for
each point on the specimens surface, z 0. Poissonsratio of the sample that we used was 0.3. A HeCdlaser of 100 mW and 0.44 m was utilized as theilluminating source. To measure phase difference ,we used the conventional phase-stepping techniqueof four frames.13 The piezoelectric transducer devicethat we used to perform the phase-stepping proce-dure was acting on one of the mirrors of the opticalsetup (Fig. 2). Fringe patterns were captured by aCCD camera of 640 480 pixels and 256 gray levels.
Figure 3 shows the sensitivity vector components ofthe interferometer of Fig. 2 at each point of the illu-minated area; these components were evaluated byEqs. (3). It can be observed that the largest-sensitivity component of this interferometer wasalong x; the other two components of the sensitivityvector were relatively small.
Figure 4(a) shows the wrapped phase map obtainedby application of a tensile load along x of the sample.Figure 4(b) depicts induced displacement dx along thepulling direction calculated from Eq. (8).
5. Uncertainty Propagation
The definition of uncertainty was taken from theGuide to the Expression of Uncertainty in Measure-ment8:
A parameter associated with the result of a mea-surement, that characterises the dispersion of thevalues that could reasonably be attributed to themeasurand.
Fig. 2. (a) Photograph of the optical system used. (b) Schematic ofa dual-beam ESPI optical setup with sensitivity mostly along x, asshown in (a). PZTs, piezoelectric transducers.
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To evaluate their corresponding uncertainties, wemust define the measurements (output quantities)through appropriate measurement models. Whenthese models are linear or weakly nonlinear, it ispossible to express the standard uncertainties of theoutput quantities in terms of the standard uncertain-ties of the input quantities by using the so-calledgeneralized law of propagation of uncertainties(GLPU).9
Consider a vector of input quantities p p1. . .pnT related to an unknown vector of outputquantities q q1. . .qmT through a set of measure-ment models Mp, q 0, where 0 is anm-dimensional zero vector. The n n symmetric in-put uncertainty matrix is
u2(p) u2(p1) u(p1, pn)
u(p1, pn) u2(pn)
where the diagonal terms are the squares of the stan-dard uncertainties of the input quantities and theoff-diagonal terms are their mutual uncertainties.The latter are zero if the quantities are uncorrelated.
The GLPU allows us to obtain mm output un-certainty matrix u2q. This law is expressed as9
where S is the m n global sensitivity matrix:
and Sq and Sp are, respectively, the mm out-put and m n input sensitivity matrices:
As an example, we applied the GLPU to the simplemodel z fx, y 0. Following the formulation es-tablished above, in this case the vector of the inputquantities is p x, yT and the output quantity isq z. Therefore n 2 and m 1. Additionally, theinput uncertainty matrix formed by the standardsand the mutual uncertainties of x and y is known:
u2(p) u2(x) u(x, y)u(x, y) u2(y) . (15)Applying Eqs. (4) and (5), we obtained that Sq 1 and that Sp fx fyT. Therefore, byapplying Eq. (2), we achieved the LPU8 for two inputquantities:
u2(z) (fx)2u2(x) (fy)2u2(y) 2(fx)(fy)u(x, y). (16)
The generalization of this law to more than two inputquantities is straightforward. The fx2u2x andfy2u2y are the contributions of input quantitiesx and y to the square of the uncertainty of outputquantity z.
It should be observed that in the case of a singleoutput quantity the GLPU reduces to the LPU, andthe matrix formulation becomes unnecessary.
There are two approaches to evaluating the stan-dard uncertainties of the input quantities (diagonal
Fig. 3. Estimates of the sensitivity vector components evaluatedby Eqs. (3): (a) ex, (b) ey, (c) ez.
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elements of the input uncertainty matrix). The type Aevaluation8 applies only to quantities that are mea-sured directly several times under repeatable condi-tions. The uncertainty of input quantities that aremeasured only once, those are evaluated from modelsthat involve further quantities, or those that are im-ported from other sources should be evaluated by thetype B method of evaluation.8 In many cases this typeinvolves obtaining an uncertainty as the standarddeviation of the probability-density function (pdf)that is assumed to apply. For example, if a quantityX varies uniformly within a given range of width X,using a uniform pdf is recommended.810 Then itsstandard uncertainty is taken as
u(X)X12. (17)6. Standard Uncertainty of the Sensitivity Vector
According to the formulation established in Section 5,to evaluate the standard and mutual uncertaintiesof the components of the sensitivity vector e ex, ey, ezT, first, combining Eqs. (3) and (12), we builtup a set of three measurement models that we rep-resented compactly as Mp, e 0. Through this setof models, the eight-dimensional vector of inputquantities p x1, x2, y1, y2, z1, z2, , T was relatedto the vector of output quantities e ex, ey, ezT. Itshould be observed that in Eqs. (3) the spatial coor-dinates of the illuminated surface (x, y) were consid-ered given and, therefore, without associateduncertainty; moreover, as the maximum reasonableerror associated with the estimated value of is just0.1 nm, we decided to neglect the contribution of to the uncertainty of the sensitivity vector compo-nents. Thus x, y, and were not considered elementsof input vector p.
Because we assumed that the input quantitieswere uncorrelated, 8 8 input uncertainty matrix
u2p was diagonal. The diagonal terms of u2p werethe squares of the standard uncertainties of the inputquantities x1, x2, y1, y2, z1, z2, , and . and werethe angles of the illuminated surface with respect toaxes x and y, respectively; these angles are the pa-rameters that characterize the alignment of the sam-ple.Because we assumed that the illuminated samplewas located on the plane xy, the estimates of and were zero; however, their associated uncertaintieswere not. We calculated the standard uncertainties of and of by using a uniform pdf. We estimated thewidth of this pdf by assuming a maximum error of3 associated with the estimates of and . There-fore, according to Eq. (17), we took
12 180. (18)As was pointed out above, the estimates of the coordi-nates of the placement of the sources were s1 17.39, 0, and 165.5 cm and s2 17.39, 0, and165.5 cm. These coordinates are the parametersthat characterize the location of the sources. We eval-uated the standard uncertainty associated with theestimates of the coordinates of the placement ofthe sources, s1 x1, y1, z1 and s2 x2, y2, z2, as thestandard deviation of the uniform pdf that we haveassumed to apply to x1, x2, y1, y2, z1, and z2. We esti-mated the width of this pdf by assuming a maximumerror of 1 mm associated with the estimates ofx1, x2, y1, y2, z1, z2. For example, for x1, according toEq. (17), we took
The same equation was applied to x2, y1, y2, z1, and z2.
Fig. 4. (a) Wrapped phase, (b) relative displacement induced along x.
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By applying the GLPU [Eq. (11)], we obtained the3 3 output uncertainty matrix
u2(e) u2(ex) u(ex, ey) u(ex, ez)
u(ex, ey) u2(ey) u(ey, ez)u(ex, ez) u(ey, ez) u2(ez)
. (20)The diagonal terms of this matrix are the squares
of the standard uncertainties of the components ofsensitivity vector e, and the off-diagonal terms aretheir mutual uncertainties.
Figure 5 shows the whole-field values ofuex, uey, and uez, that we obtained. It is useful tocompare Figs. 5 and 3; it can be observed that, al-though ey and ez are relatively small, their corre-sponding uncertainties are not negligible. Figure 6shows the mutual uncertainties uex, ey, uex, ez, anduey, ez; because the values are relatively small, weassume that these mutual uncertainties might beignored in further calculations.
In Fig. 7 is shown the relative standard uncer-tainty of the sensitivity vector component along x,computed as uexex. The plot was built up by use ofthe values depicted in Figs. 5(a) and 3(a). It can beobserved that the relative uncertainty of ex increasesslightly at zones close to the borders. This means thatthe relative standard uncertainty of ex should in-crease with the size of the field.
In Fig. 8 we have plotted the various contributionsto u2ex along x 5 cm. For clarity, these contribu-tions have been labeled CX exX2u2X for ge-neric input quantity X. It can be observed thatCx1 and Cx2 are the largest contributions to u2ex.This means that eventual errors in the determinationof x1 and x2 have more influence on the uncertainty
Fig. 5. Standard uncertainties associated with the estimates ofthe sensitivity vector components: (a) uex, (b) uey, (c) uez.
Fig. 6. Mutual uncertainties of the sensitivity vector components:(a) uex, ey, (b) uex, ez, (c) uey, ez.
1146 APPLIED OPTICS Vol. 44, No. 7 1 March 2005
than faults in the alignment of the sample. Thereforewe conclude that, if a dual-beam interferometer isused, special attention must be paid to accurate de-termination of the location of the illuminatingsources.
To analyze the effects of the incidence angles and ofthe sourcetarget distance on the uex values, weapplied the GLPU to the modelsMp, e 0, keepingconstant the uncertainties of the input quantities but(i) changing the estimates of the coordinates of theilluminating sources such that the angles of incidencemeasured with respect to the z axis were both equalto 50 and (ii) reducing the sourcetarget distance to100 cm. The actual incidence angles were both equalto 6, and the sourcetarget distance was 166.4 cm.The results are depicted in Fig. 9. Comparing Fig.9(b) with Fig. 7, we can observe that the relativestandard uncertainty of ex decreases if the angles ofincidence are incrementally larger. Moreover, whenFig. 9(c) is compared with Fig. 7, it appears that therelative uncertainty of ex increases if the distancebetween the sources and the sample is decreased.Comparing Fig. 9(a) with Fig. 7, we concluded thatthe relative uncertainty increases if the size of theilluminated field is increased, especially at zonesclose to the borders.
7. Displacement Standard Uncertainty
Because the values of the transverse sensitivity vec-tor components ey and ez are relatively small (Fig. 3),the displacements can be evaluated by Eq. (9). How-ever, because the uncertainties of ey and ez were notnegligible (Fig. 5), we concluded that Eq. (9) does notinclude all the relevant uncertainty sources. There-fore we took Eq. (8) as the measurement model withwhich to evaluate the displacement uncertainty.
Equation (8) is a single measurement model; dx is
Fig. 7. Relative standard uncertainty of the sensitivity vectorcomponent along x. The plot was built up by use of the valuesdepicted in Figs. 5(a) and 3(a). The angles of incidence measuredwith respect to axis z were both equal to 6, and the sourcetarget distance was r 166.4 cm.
Fig. 8. Contributions to the square of uex along x 5 cm.Cx1 exx12u2x1, Cy1 exy12u2y1, Cz1 exz12u2z1, Cx2 exx22u2x2, Cy2 exy22u2y2, Cz2 exz22u2z2, C ex2u2, C ex2u2.
Fig. 9. Relative standard uncertainty of the sensitivity vectorcomponent along x. (a). Angle of incidence with respect to samplenormal 6, source distance with respect to sample center r 166.4 cm , and 20 20 cm illuminated area. (b) Angle of inci-dence 50, sourcetarget distance r 166.4 cm, and 10 10cm illuminated area. (c) 6, r 100 cm, and 10 10 cmilluminated area.
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related to the input quantities , ex, ey, ez, and v.Because sample thickness W was relatively small aswell as because of its associated uncertainty, we de-cided to neglect the contribution of W to the uncer-tainty of dx. Thus W was not considered an inputquantity. Moreover, because in this case we have asingle output quantity, the matrix formulation estab-lished in Section 5 is unnecessary; thus, to evaluateudx we applied the LPU [Eq. (16)] to Eq. (8).
The estimates of the sensitivity vector componentswere taken, for each point of the illuminated surface,from Fig. 3. Their corresponding standard and mu-tual uncertainties were taken from Fig. 5 and 6.
The estimates of were obtained by the appliedconventional phase-stepping technique of fourframes.13 The standard uncertainty associated withthe values depends on systematic effects linkedwith the phase-shifting procedure and on the com-bined influence of the optical noise and the perturb-ing environment. Assuming that the first uncertaintywas compensated for adequately, we considered justthe eventual variations in the phase differencecaused by the optical noise and the environment.These variations are assumed to be induced mostlyby the optical noise whose influence on the valuesis presumed high when speckle-based techniques areused.3 We estimated that the reasonable standarduncertainty associated with the measured valuesis0.28 rad.
Furthermore, as was pointed out above, we used0.3 as the estimate of Poissons ratio; we evaluated itsassociated standard uncertainty, assuming a uniformpdf of width equal to 0.05; then, according to Eq. (17),we took
u() 0.0512. (21)
We evaluated the standard uncertainty associatedwith relative displacement dx by applying the LPU toEq. (8). Figure 10 shows the results. This figure canbe compared with Fig. 4; the latter shows the esti-mates of relative displacement dx evaluated with Eq.(8). Because the similarities between Fig. 10 and 4(b)are apparent, we conclude that the displacement un-certainty increases with deformation.
Figure 11 depicts the several contributions to thesquare of udx along line x 5 cm again the contri-butions have been labeled CX dxX2u2X forgeneric input quantity X. As the contribution of thephase difference to the square of udx is muchgreater than those that correspond to other contrib-utors, C is not shown in Fig. 11. From Fig. 11 itcan be observed that the greater contributor is ex.Moreover, although, as shown in Fig. 5, the uncer-tainties in ey and ez were not negligible, their contri-butions to the displacement uncertainty were minor,as was the contribution of Poissons ratio. Hence, be-cause in metrological applications the uncertaintiesassociated with the input quantities used are gener-ally smaller than those assumed in this study, weconcluded that reliable evaluations of the uncertainty
of measurements obtained by ESPI with sphericalillumination can be performed by application of theLPU to Eq. (9).
8. Summary and Conclusions
We carried out an investigation to evaluate the un-certainty of displacements measured by dual-beamelectronic speckle-pattern interferometry with non-collimated illuminating beams. The displacementswere induced by application of a uniaxial tensile loadon a nominally flat elastic sample. Although an in-terferometer with spherical illumination has sensi-tivity along the three spatial coordinates, thegeometry of the optical setup used allowed us to havesensitivity mostly along the pulling direction; theother two components of the sensitivity vector wererelatively small. We measured the displacement in-duced only along the pulling direction.
Displacement measurements depend on the in-duced phase difference and on the interferometerssensitivity vector; the latter depends in turn on thelocation of the illuminating sources and on the align-ment of the target. We approached the displacementuncertainty by propagating the uncertainties that weconsidered reasonable to assign to the measuredphase difference and to the characteristic parametersof the source locations and the target alignment. Spe-cial attention was paid to evaluating the contribu-tions to the displacement uncertainty.
Fig. 10. Standard uncertainties associated with the estimates ofthe displacement component along x.
Fig. 11. Contributions to the square of udx along x 5 cm.Cex dxex2u2ex, Cey dxey2u2ey, Cez dxez2u2ez, Cv dxv2u2v.
1148 APPLIED OPTICS Vol. 44, No. 7 1 March 2005
Taking into account the result obtained3 with re-spect to uncertainty measurements of the phase dif-ference, we observed that the greatest contributor tothe uncertainty in the displacement was the uncer-tainty that we attributed to the phase-differencemeasurements. Moreover, we found that the contri-butions to the displacement uncertainty of the sensi-tivity vector components transverse to the pullingdirection were minor; hence we conclude that theinfluence of these components can generally be ig-nored. This means that it is possible to carry outuncertainty evaluations reliably by propagating onlythe uncertainties of the phase-difference measure-ments and of the sensitivity vector component alongthe pulling direction.
With respect to the sensitivity vector componentalong the pulling direction, we found that its uncer-tainty was greater on the illuminated area close tothe borders; this means that uncertainty increaseswith the size of the illuminated field. Moreover, weobserved that eventual errors in the determination ofthe source positions have more influence on the un-certainty evaluation than faults in the alignment ofthe sample.
Finally, by analyzing the effect of the geometry ofthe optical arrangement on the uncertainty evalua-tion, we found that the uncertainty of the displace-ment measured by ESPI decreases if the angles ofincidence and the sourcetarget distance are in-creased. We observed that the displacement standarduncertainty depends also on the measured displace-ment value. We found that the displacement uncer-tainty increases with deformation.
The authors acknowledge partial economic supportfrom the Consejo de Ciencia y Tecnologa del Estadode Guanajuato (grant CONCYTEG-04-04-K117-011)and the Consejo del Sistema Nacional de EducatinTecnolgica (grant COSNET- 497.03-P). R. R. Cord-ero acknowledges support from Programa Mece de
Educacin Superior de Chile PUC/9903 project andVlaamse Interuniversitaire Raad (VLIR-ESPOL,Componente 6).
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