OPTICAL CONFIGURATIONS FOR ESPI
Amalia Martíneza, J. A. Rayasa, R. Corderob
a
Centro de Investigaciones en Óptica, A. C.
Apartado Postal 1-948, C. P. 37000, León, Gto., MÉXICO
e-mail: [email protected]
b
Leibniz Universität Hannover, Herrenhäuser Str. 2, D-30419 Hannover, Germany
ABSTRACT
An ESPI optical arrangement consisting in a combination of three illumination and one reference beams is compared to a
system with three optical arms corresponding to an out-of-plane and two in-plane interferometers. Experimental data for each
component of the displacement vector obtained with both systems are presented and discussed.
I. Introduction
It is well known that when an optically rough surface is illuminated by coherent light, speckles appear in front of the surface.
Displacements and deformations of an input diffuser produce displacements and structural changes in the speckle patterns,
which can be on the basis of the double-exposure imaged speckles, before and after deformation [1]. Then for objects with
optically rough surfaces, electronic speckle pattern interferometry (ESPI) [2] is one of the most appropriate methods for high
sensitivity measurements of out-of-plane [3] and in-plane displacement components [4], strain, and vibration. In addition,
various image-processing techniques could be applied, because the signal is obtained in digital form. The deformation field in
form of correlation fringes is obtained by comparing two images, one representing the reference object state, and the other
representing a deformed object state, either by subtracting or adding them. There are numerous techniques for obtaining the
phase, some are: fast Fourier transform method with phase carrier proposed by Takeda [5], phase shifting [2] and spatial
synchronous detection [2].
Recently, we presented an optical setup that can be switched to get single and double dual illumination [6]. Too it was reported
experimental results for a speckle interferometer with three divergent illumination beams with sources in such a position that
gives the maximum sensitivities for each sensitivity vector components [7]. In this paper, we compare both optical systems
mentioned above. Experimental results of displacements maps obtained for the same sample under identical loading
conditions were reported and discussed.
II. Displacement Measurement
When the propagation vectors of two laser beams forming a speckle interferometer are denoted as ê1 and
r
vector, e , of the speckle interferometer is represented as
r 2π
e=
(eˆ1 − eˆ2 ) .
λ
ê2 , the sensitivity
(1)
The sensitivity vector depends of the geometry of the arrangement and on the wavelength of the laser source. In the doubleillumination method, both ê1 and ê2 correspond to unit vectors that describe the vectorial characteristics of illuminating beams
emerging from sources S1 and S2, respectively, figure 1. In this case the system sensitivity does not depend of the observation
direction. In the reference beam method, one vector corresponds to vectorial characteristics of illuminating and the other
corresponds to the observation vector. In this case, the system sensitivity depends of the observation direction. An
interferogram obtained of the subtraction of two speckle images of a test object recorded before and after deformation of a test
object shows fringes that contour the object deformation. The phase change, ∆φ, in each point P(x,y) of the object surface,
r
r
arising from the object deformation is related to the three-dimensional displacement vector, d , through sensitivity vector e as
r
r
∆φ ( P ) = d ( P ) ⋅ e ( P ) .
(2)
The use of multiple interferometers whose sensitivity vectors are different permits measurements of multiple components
displacements. A sensitivity matrix, consisting of three sensitivity vectors that relate the three measured phase maps to the
three components of displacement, can be derived. Because the sensitivity matrix is not singular, the inverse sensitivity matrix
exists and therefore displacement maps can be calculated from the phase maps. Let us consider the three sensitivity
vectors e1 ( P) , e 2 ( P) and e 3 ( P ) , in which the upper index denote correspondence to each one of the interferometers. In this
case e1 ( P) is associated to dual illumination with x-sensitivity; e 2 ( P ) to dual illumination with y-sensitivity and e 3 ( P)
associated to z-sensitivity. Then, at each point P we have to solve the system of linear equations to obtain
d (u , v, w) . The
solution is
1
 u ( P )   e x


2
 v( P)  =  e x
 w( P )   e 3

  x
e1y
e 2y
e 3y
−1
e1z   ∆φ 1 ( P ) 
 

e z2  ⋅  ∆φ 2 ( P ) 
 

e 3z   ∆φ 3 ( P ) 
d (u , v, w) = E −1 ( P ) ⋅ ∆φ ( P )
(3)
(4)
where E (P ) is the sensitivity matrix. The sensitivity vector components to each case are given by reported in ref. [6]. We have
calculated the sensitivity matrix with all the sensitivity vector components. A second system is used to obtain three sensitivity
vectors. The optical system uses illuminating beams to irradiate the object from three directions [7,8]. Here it provides the
phase change undergone by the speckle object wave in response to the object surface deformation, in an in-line reference
wave arrangement. This type of arrangement consists in superimposing to the speckle wave diffused by the object surface a
smooth reference wave. The latter propagates in the direction of the optical axis of the lens imaging the surface. The phase
change is linearly related to the three components of the displacement field and the coefficients depend on the geometry of the
set up, i. e. on the directions of the illumination and observation vectors. In order to solve the three displacements components,
three independent equations are necessary, namely three geometries.
In this work we compared the relative merits of two different ESPI systems. The first system, whose scheme is sketched in
Fig.1, is composed of three circuits sensitive each to one direction of displacement (we call it in the rest of the paper System A,
for brevity). The second ESPI set-up, which is provided with three illumination beams and one of reference, is shown in Fig. 2
(this it will named System B).
III. Experimental part
Figure 1 illustrates the optical systems used for speckle correlation (System A). An He-CD laser is utilized with a power of 100
mW and λ = .440 µm . The illumination is divided into two beams (in-plane system) that are directed to the object surface from
equal and opposite angles, one of them being reflected from a piezo-electric mirror. This interferometer has the largest
sensitivity component in x-direction. Subsequently this illumination is redirected perpendicular to the first illumination plane to
get the largest sensitivity component in y-direction. Again, the illumination is redirected but using only one object beam of
illumination to obtain an interferometer with the largest sensitivity component in z-direction (out-of-plane system). In order to
evaluate the sensitivity vector components for the in-plane setup, x-sensitivity, the coordinates were: S1=(465 mm, 0 mm, 460
mm) and S2=(-465 mm, 0 mm, 460 mm). In the case of the y-sensitivity, the coordinates were: S3=(0 mm, 470 mm, 460 mm)
and S4=(0 mm, -470 mm, 460 mm). To evaluate the sensitivity vector components for the out-of-plane setup, the following
coordinates were considered: camera position P0=(0 mm ,0 mm,850 mm); and illumination source position S5=(-50 mm, 0
mm,660 mm).
The first two columns in Fig. 3, shows the corresponding sensitivity vector components to the dual illumination associated to
the sources S1-S2, and S3-S4 respectively. The third column associates the components to out-of-plane system.
Fringe patterns were captured by means of a CCD camera of 640 x 480 pixels, and 256 gray levels. It used a phase stepping
technique for 15-steps to get the optical phase [9]. The camera observing the object along its normal receives two optical fields,
one before and the second after the deformation. Three recordings, one for each interferometer are made. The recordings
made before the deformation are one-by-one subtracted from the corresponding set of recordings taken after the deformation.
As a result we end up with three interferograms that represent the displacement vector components. Figure 4 shows the
wrapped phase associated to deformation on x, y, and z-directions respectively.
laser
M
switcher
S2
a)
y
CCD
BS
M
z
S1
x
Object
Plane
M
S3
laser
switcher
y
b)
CCD
z
M
S4
x
Object
Plane
laser
switcher
M
c)
y
S5
CCD
BS
S6
z
M
x
Object
Plane
Figure 1 a) Dual-beam optical setup with x-sensitivity, b) dual illumination interferometer with y-sensitivity and c) interferometer
with one beam to z-sensitivity.
S3
S2
y
S5
z
x
Object
Plane
Figure 2 Scheme of an ESPI system with three illumination beams.
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38.47
20
-21212
-38.47
-2161
38.69
-20271
1095
-38.69
-20319
-1095
622.6
619.2
28415
-622.6
-619.2
28302
ex
ey
eZ
a)
b)
c)
Figure 3 Sensitivity vector components ex, ey and eZ in rad/µm to the interferometer in figure 1: a) sources S1-S2, b) S3- S4 and
c) S5.
a)
b)
c)
Figure 4 Wrapped phase obtained with dual illumination a) sources S1-S2, b) S3- S4 ; Wrapped phase obtained with one
illumination beam c) S5.
Figure 2 shows an ESPI arrangement with three illumination beams which was used to get results reported here (System B).
Light from the laser is divided in two beams. One beam serves as the reference beam. The other beam is used sequencely to
illuminate the object from three different directions, forming with the reference beam direction three different sensitivity vectors.
In order to evaluate the sensitivity vector components for each illumination beam, the following coordinates were considered:
observation position P0=(0 mm,0 mm,850 mm); sources position: S2=(-465 mm, 0 mm, 460 mm) which gives x-sensitivity,
S3=(0 mm, 470 mm, 460 mm) which gives v-sensitivity and S5=(-50 mm, 0 mm, 660 mm) which gives z-sensitivity. Figure 5
shows the corresponding sensitivity vector components associated to each sources S2, S3, and S5 respectively.
The optical paths of each of three beams and that of the reference beam are matched such that the combination at the CCD
faceplate of reference and object beams is coherent. The images are recorded individually for each illumination directionreference beam combination. Two digital images are needed for every illumination direction. The data in each image is
processed in order to recover the object phase information that may later be displayed as phase maps. Figure 6, shows the
wrapped phase associated to the sensitivities to S1 (mix of u and w sensitivities), S3 (mix of v and w sensitivities) and S5
(almost pure w sensitivity) respectively. The total object deformation is obtained from the vector resultant of the data obtained
from the phase maps for each sensitivity vector with the combination of object shape, in our case is considered a plane
surface.
The target object consisted of elastic surface with width l = 70 mm and thickness h = .33 mm which can be considered a
thin plate. One of the edges was clamped rigidly. The other side of the plate is submitted to the action of tensile force in the x
direction and uniformly distributed along the longitudinal side of the plate. The plate is now deflected such the points of the
plate lying initially on a normal to the middle plane of the plate remain on the normal to the middle surface of the plate after
bending, figure 7.
The origin of a rectangular set of coordinates is fixed on the clamped side rigidly and it is used to calculate the components of
the displacement vector from the corresponding phase maps.
r
The displacement vector d is evaluated by decomposing it and the sensitivity vectors into their orthogonal components x, y
and z. From the equation 4 the displacements components at each point of the object are obtained. Figure 8 shows
displacement fields u, v and w calculated from the experiments obtained from both systems.
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2161
1121
10928
1095
-1121
9352
-1095
24521
24465
28415
23881
23829
28302
ex
ey
eZ
a)
b)
c)
Figure 5 Sensitivity vector components ex, ey and eZ in rad/µm relative to the three-beams interferometer in figure 2: a)
sources S2, b) S3 and c) S5.
a)
b)
c)
Figure 6 Wrapped phase obtained with each one of the three interferometers: a) S2, b) S3 and c) S5.
y
z
x
r
Fx
r
Fz
Work
area
r
F
a)
b)
Figure 7 Representation of the object target and the loads applied.
v
(µm)
0.49
u
(µm)
3.6
0.25
2.7
0.02
1.8
-0.21
0.9
-0.44
28
0
28
14
14
0
-14
-28
-14
0
14
28
0
y
(mm)
-14
-28
-14
-28
0
14
28
y
(mm)
-28
x (mm)
x (mm)
w
(µm)
0
a)
b)
-0.31
-0.63
-0.94
-1.25
28
14
0
-14
-28
-14
0
14
28
y
(mm)
-28
x (mm)
c)
Figure 8 Displacements maps associated to a) u(x,y), b) v(x,y) and c) w(x,y) . The dots surface corresponds to results obtained
with the optical system showed in figure 1 and the solid surface for the results got with the interferometer showed in figure 2.
IV. Conclusions
Experimental part carried out using two optical configuration to obtain the practical determination of the (u, v, w) displacement
components attached to every points of a plate subjected to a load. The maximum difference between the displacement maps
of u, v and w was 0.30, 0.23 and 0.27 µm, respectively. It is observed that the obtained differences are quite small in absolute
values. Then the system B can be used without introduce large errors in displacement components measurement.
Acknowledgements
Authors wish to thank for the partial economical support from Consejo Nacional de Ciencia y Tecnologia (CONACYT) and
Consejo de Ciencia y Tecnología del Estado de Guanajuato (CONCYTEG).
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