# Uncertainty analysis of displacements measured by in-plane electronic speckle-pattern interferometry with spherical wave fronts

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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.

1. Introduction

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 (amalia@cio.mx), 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

1 March 2005 Vol. 44, No. 7 APPLIED OPTICS 1141

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:

e2

[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

ex2

(x x1)

(x x1)2 (y y1)2 (z z1)212

(x x2)

(x x2)2 (y y2)2 (z z2)212, (3a)

ey2

(y y1)

(x x1)2 (y y1)2 (z z1)212

(y y2)

(x x2)2 (y y2)2 (z z2)212, (3b)

ez2

(z z1)

(x x1)2 (y y1)2 (z z1)212

(z z2)

(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

zxy, (4)

where and are the angles of the illuminated sur-face with respect to axis x and to axis y, respectively.

3. Displacements

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.

1142 APPLIED OPTICS Vol. 44, No. 7 1 March 2005

the displacements, for values of x and y differentfrom zero, x and y can be evaluated approximatelyby

xdxxp

, (6a)

ydyyp

. (6b)

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

yzx, (7)

where is Poissons ratio.Combining Eqs. (1), (6), and (7) and solving for dx,

we obtain5

dxx

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

dxex. (9)

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.

1 March 2005 Vol. 44, No. 7 APPLIED OPTICS 1143

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)

, (10)

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

u2(q)Su2(p)ST, (11)

where S is the m n global sensitivity matrix:

S[S(q)]1S(p), (12)

and Sq and Sp are, respectively, the mm out-put and m n input sensitivity matrices:

S(q)M1q1

M1qm

Mmq1

Mmqm, (13)

S(p)M1p1

M1pn

Mmp1

Mmpn. (14)

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.

1144 APPLIED OPTICS Vol. 44, No. 7 1 March 2005

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...

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