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.
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� �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixp � xs1� �2 þ yp � ys1
� �2 þ zp � zs1� �2q
� yp � ys2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixp � xs2� �2 þ yp � ys2
� �2 þ zp � zs2� �2q ;
ð3bÞ
ez ¼2pk
�zp � zs1� �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixp � xs1� �2 þ yp � ys1
� �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.
242 A. Mart�ıınez et al. / Optics Communications 223 (2003) 239–246
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.
A. Mart�ıınez et al. / Optics Communications 223 (2003) 239–246 243
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.
244 A. Mart�ıınez et al. / Optics Communications 223 (2003) 239–246
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.
A. Mart�ıınez et al. / Optics Communications 223 (2003) 239–246 245
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.
246 A. Mart�ıınez et al. / Optics Communications 223 (2003) 239–246
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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