error in the measurement due to the divergence of the object illumination wavefront for in-plane...

Optics Communications 223 (2003) 239–246

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Error in the measurement due to the divergence of theobject illumination wavefront for in-plane interferometers

Amalia Mart�ııneza,*, R. Rodr�ııguez-Veraa, J.A. Rayasa, H.J. Pugab,*

a Centro de Investigaciones en �OOptica, A.C., Apartado Postal 1-948, C.P. 37000, Le�oon, Gto., M�eexicob Instituto Technol�oogico de Le�oon, Depto. Ciencias B�aasicas, Le�oon, Mexico

Received 28 April 2003; received in revised form 19 June 2003; accepted 19 June 2003

Abstract

Commonly, in-plane sensitive interferometric optical arrangements use two symmetrical collimated wavefronts for

object surface illumination. However, this is a limitation when large objects have to be analyzed. In this case spherical

illumination is needed. Non-collimated symmetrical dual-beam techniques have been performed. This kind of illumi-

nation produces a sensitivity vector varying with the position. Then errors in the measurements are introduced when

collimated illumination is supposed especially in extended target objects. In the present work, an in-plane configuration

for electronic speckle pattern interferometer (ESPI) is used. During the design stage of an interferometer should be

useful to know the components of sensitivity vector in order to minimize the un-required displacement components.

This is main task of this paper. We present theoretical analysis and experimental results for object divergent and

collimated illumination. The errors are obtained by comparing the in-plane displacement calculated supposing constant

sensitivity vector and spatial variation of sensitivity vector for object divergent illumination. This analysis is made for a

flat and cylindrical object surface target.

� 2003 Elsevier B.V. All rights reserved.

PACS: 06.20; 06.30.B

Keywords: Speckle interferometry; Measurement error; Sensitivity vector; In-plane

1. Introduction

In the literature it can be found some ap-

proaches that use noncollimated beams in tech-

niques as grating moir�ee, electronic speckle patterninterferometry (ESPI), holographic interferometry

* Corresponding authors. Present address: Centro de Inves-

tigaciones en �OOptica, A.C. Tel.: +52-477-773-10-17; fax: +52-

477-717-50-00.

E-mail address: [email protected] (A. Mart�ıınez).

0030-4018/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/S0030-4018(03)01673-0

and electronic speckle pattern shearing interfer-

ometry (ESPSI) [1–6]. Ferraro et al. [1] developed

a third-order aberration analysis for a grating

moir�ee interferometer in a noncollimated configu-ration. Their interferometer is made from only twocomponents, a mirror which acts as a reversing

and folding element and a plane reflective diffrac-

tion grating. In this configuration the reflective

grating interferometer diffracts the two half-wave

fronts at )1 and +1 orders along of the normal tothe grating.

ed.

mail to: [email protected]

240 A. Mart�ıınez et al. / Optics Communications 223 (2003) 239–246

On the other hand, error estimation introduced

by the use of spherical illumination in ESPI for

both in-plane and out-of-plane arrangements has

been reported. For out-of-plane ESPI system, De

Veuster et al. [2], computed the displacement error

for different interferometric configurations, show-ing that an in-plane component appears. Their

study is based on a geometrical point of view. They

analyzed the sensitivity change as a function of

illumination and observation angles, as well as, the

shape and position of the object. These authors did

not make a study to in-plane ESPI system. Albr-

echt [3] used a vectorial approach to show that the

total fringe pattern phase change Du depends onthe wavelength k, the unit vectors of illuminationn̂n1, n̂n2 (in-plane) or the illumination vector n̂n1, ob-servation vector n̂n3 (out-of-plane) and the dis-

placement ~dd. For in-plane sensitive system, theirexperimental set-up uses two mirrors for folding

a point source in order to get the two spherical

illumination wavefronts. The fact to use these

mirrors gives us an additional mathematical con-sideration: introducing a linear function, point-to-

point on the mirrors� surface and a phase errorcorrection is calculated.

Puga et al. [4] presented a general model to

predict and correct the displacement measurement

and phase map interpretation error taking in ac-

count the target shape, illumination geometry, and

in-plane displacement to out-of-plane configura-tions used in ESPI.

Ojeda et al. [5] described the formation of a

hologram by using a point source. This work is

basically the Young experiment since during any

Fig. 1. ESPI setup with i

movement of the hologram, there are two spheri-

cal beams corresponding to the point virtual and

real sources, respectively. They describe qualita-

tively some applications without analyzing the

system sensitivity.

Wan Abdullah et al. [6] presents data describingthe measurement inaccuracy due the divergence of

the object illumination wavefront for an out-of-

plane interferometer in the case ESPSI. The errors

are measured by comparing divergent object illu-

mination with collimated illumination, with re-

spect to illumination angle, lateral shear and

shearing direction.

In this paper, the used optical system is a dual-beam illumination interferometer, which uses two

divergent symmetrical point sources, see Fig. 1.

Due to this kind of illumination, the experimental

system sensitivity has spatial variation of the sen-

sitivity vector, which is taken into account to cal-

culate the in-plane displacement to a flat elastic

surface and a cylindrical aluminum shell. The rel-

ative error of the in-plane displacement is com-puted considering the spatial variation and

supposing constant sensitivity vector when the

object is illuminated by spherical wave fronts.

ESPI results for both surfaces, under mechanical

load are shown. The analysis shows that the as-

sociated error for in-plane displacement can be

computed from the relative error associated to

corresponding sensitivity component. We tell, inadvance, the associated relative error to measure-

ment. The calculation is important to large object

where the relative error is considerable. It is shown

that for a plane sample of size 20� 20 cm2, the

n-plane sensitivity.

A. Mart�ıınez et al. / Optics Communications 223 (2003) 239–246 241

maximum error is of approximately 6% in the

borders.

2. Expression for the sensitivity vector

In a dual-beam interferometer, a coherent laser

beam is splitted in two arms expanded to illumi-

nate a surface target (Fig. 2). The relation between

the measured phase difference D/ðx; yÞ and the

displacement vector ~dd ¼~ddðu; v;wÞ at a point

P ¼ P ðx; yÞ is given by [7]

D/ðPÞ ¼~ddðP Þ �~eeðPÞ; ð1Þwhere~eeðP Þ is the sensitivity vector given by

~eeðP Þ ¼ 2pk½n̂n1ðP Þ � n̂n2ðP Þ�; ð2Þ

and n̂n1 and n̂n2 are unit vectors that describe thevectorial characteristics of illuminating beams

emerging from S1 and S2, respectively. Notice thatn̂n1 and n̂n2 change in direction for each point on theinspected area of the surface target. It is observed

from Eq. (2) that the system sensitivity does not

depend of the observation direction.The sensitivity vector components are com-

puted by:

ex ¼2pk

�xp � xs1� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixp � xs1� �2 þ yp � ys1

� �2 þ zp � zs1� �2q

� xp � xs2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixp � xs2� �2 þ yp � ys2

� �2 þ zp � zs2� �2q ;

ð3aÞ

Fig. 2. Diagram to define the sensitivity vector with divergence

illumination.

ey ¼2pk�

yp � ys1� �


� �2 þ zp � zs1� �2q

� yp � ys2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixp � xs2� �2 þ yp � ys2

� �2 þ zp � zs2� �2q ;

ð3bÞ

ez ¼2pk

�zp � zs1� �


� �2 þ zp � zs1� �2q

� zp � zs2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixp � xs2� �2 þ yp � ys2

� �2 þ zp � zs2� �2q ;

ð3cÞ

where ðxp; yp; zpÞ, ðxs1; ys1; zs1Þ and ðxs2; ys2; zs2Þ rep-resent an object point and point sources location

S1 and S2, respectively.

3. Prediction of error in the measurement

The relative error (Er) associated to measure-ment due to the wavefront curvature when is

supposed plane curvature can be written as

Er ¼ud � uc

uc

�� 100; ð4Þ

where ud is the displacement measured by usingvariable sensitivity vector, uc is the displacementmeasured by supposing constant sensitivity vec-tor.

We call in advance, the associated error to the

measurement starting from the sensitivity vector

analysis. Considering a rigid displacement in x-

direction and expressing the Eq. (1) to determine u:

u ¼ D/ex

: ð5Þ

Introducing the Eq. (5) in the Eq. (4), the rel-

ative error can be expressed as

Er ¼exc � exd

exd

�� 100; ð6Þ

where exd is the x-component of the variable sen-sitivity vector, exc is the x-component of the con-stant sensitivity vector.


Then the associated relative error of the in-

plane displacement can be computed from the

relative error associated to sensitivity component

which is expressed by Eq. (6).

3.1. Diverging illuminating

By using the parameters of the source distances

r ¼ 56 cm with respect to origin and a symmetrical

Fig. 3. Relative error for an inspected plane object size of

Fig. 4. Relative error for an inspected cylindrical object size

incidence angle of h ¼ 22�, 3D-map representingthe function of the relative error given by Eq. (6)

over a field of view of 5� 5 cm2 is shown in Fig. 3.

This is illustrated by considering that the object is

plane. Fig. 4 shows the case for cylindrical surface

using the previous parameters. The cylindricalsurface is described by the equation z ¼ 3:05�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4:352 � x2

pand it is placed as it shows in the

Fig. 5. Fig. 6 shows the relative error for a plane

5� 5 cm2 with parameters: h ¼ 22� and r ¼ 56 cm.

of 5� 5 cm2 with parameters: h ¼ 22� and r ¼ 56 cm.

Fig. 7. Diagram to define the sensitivity vector with collimated

illumination.Fig. 5. Diagram to define the inspected area of the cylindrical

object which is defined by z ¼ 3:05�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4:352 � x2

p.


object size of 20� 20 cm2 and the same parame-ters. By analyzing the results shown in Figs. 3 and

4, we can appreciate that the relative error in-

creases to cylindrical shell and decreases to plane

form. Fig. 6 shows the relative error is significant

to large objects.

3.2. Collimated illuminating

Fig. 7 shows the case of collimated beams. The

sensitivity vector components are obtained con-

Fig. 6. Relative error for an inspected plane object size of

sidering the effect of the each point of collimating

lens and its corresponding point in the other lens.

When the object is illuminated symmetrically with

respect to its normal, the sensitivity vector has a

unique component ex. The effect of the out-of-plane component ez is fully nulled for in the sym-metric illumination arrangement for objects with

any shape. The before statements can be got fromEqs. (3a)–(3c). The orthogonal component of in-

plane displacement can be measured by rotating

either the object or the illumination beams by 90�about the normal to the object.

20� 20 cm2 with parameters: h ¼ 22� and r ¼ 56 cm.


The Eq. (3a) can be written as

ex ¼4pksin h; ð7Þ

by introducing in Eq. (1)

D/ ¼ 4pk

� u � sin h; ð8Þ

which is the relation obtained by Leendertz [8].

If we suppose an error in the symmetry of the

two beams with respect to the surface normal, the

interpretation of the contour pattern as fringes of

Fig. 8. Plot of in-plane displacement calculated when (a) a variable

supposed. The experiment was accomplished with divergent illuminat

equal in-plane displacement induces a relative er-

ror in the measurement.

4. Experimental procedure

For the propose of the presentation of data, a

flat elastic surface subjected to only in-plane-xdeformation is tested. The object consisted of

elastic surface of 5� 5 cm2, clamped rigidly at one

end. The free end of the surface is clamped and get

sensitivity vector is used; (b) a constant sensitivity vector is

ion to plane object.


out uniformly. The surface is illuminated by two

point sources placed symmetrically and making an

angle h with respect to the surface normal. Theused incidence angle is h ¼ 22�. The distance ofsources from origin is 56 cm. The principle of

measurement based on speckle interferometryconsists of observing the evolution of a speckle

pattern obtained by the coherent superposition of

two independent fields, as a function of the phase

difference between the two fields. The CCD cap-

Fig. 9. Plot of in-plane displacement calculated when (a) a variable

supposed. The experiment was accomplished with divergent illuminat

ture the intensity before and after deformation of

the object.

Fringe patterns were captured by means of a

CCD camera of 640� 480 pixels and 255 gray

levels. It used phase stepping technique for 4-steps

[9] to get the phase. From Eq. (3a), the expressionfor calculating the in-plane displacement with

variable sensitivity vector exd is introduced.Fig. 8 shows results for displacement in-plane

where the unwrapped phase is used to compute u

sensitivity vector is used; (b) a constant sensitivity vector is

ion to cylindrical shell.


displacement. Taking the same unwrapped phase,

it is computed the in-plane displacement consid-

ering constant sensitivity vector.

Then the relative error (Er) due to the wavefrontcurvature when is supposed plane curvature can be

computed from Eq. (4) and is shown in Fig. 3.Fig. 9 shows experimental results for cylindrical

surface.

5. Conclusion

An ESPI optical system that uses symmetrical

divergent illumination is a simple and suitablemethod to measure in-plane displacement with

good sensitivity. Noncollimated and collimated

wavefronts have been used to examine relative

error due to the use divergence illumination and by

supposing of constant sensitivity vector. We have

shown the relative error increases to large surfaces

and surface shape. This can be calculated in ad-

vance and it should be examined in the stage ofplanning an interferometric measurement experi-

ment. The analysis has demonstrated the use of

sensitivity vector variability let us permit to correct

the measurement and it has clear potential for the

use of noncollimated illumination wavefronts to

large objects. This fact is supported by our ex-

perimental and theoretical results. The sensitivity

vector, in the case of double illumination, is cal-culated by the difference between both illumina-

tion unitary vectors and the system sensitivity not

depend observation direction. Experimental re-

sults have been presented to displacements mea-

surement in x-direction.

Acknowledgements

The authors wish to thank Martha Gutierrez

and Guillermo Garnica for their technical support.

This research has been supported by the Consejo

de Ciencia y Tecnolog�ııa del Estado de Guanajuato(CONCYTEG) through Grant 03-04-K118-039-

anexo 04 and the Consejo Nacional de Ciencia y

Tecnolog�ııa (CONACYT) through Grant 33106-E.

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