analysis of coherent symmetrical illumination for electronic speckle pattern shearing interferometry
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This article was downloaded by: [North Dakota State University]On: 01 November 2014, At: 09:00Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Journal of Modern OpticsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tmop20
Analysis of coherent symmetricalillumination for electronic specklepattern shearing interferometryJuan F. Román a , Vicente Moreno b , Jon N. Petzing c & John R.Tyrer ca Bayer Diagnostics , Sudbury, CO10 2XQ, Suffolk, UKb Faculty of Physics , University of Santiago de Compostela ,15782, Santiago de Compostela, Spainc Wolfson School of Mechanical and Manufacturing Engineering ,Loughborough University , Loughborough, Leics, LE11 3TU, UKPublished online: 15 Aug 2006.
To cite this article: Juan F. Román , Vicente Moreno , Jon N. Petzing & John R. Tyrer (2005)Analysis of coherent symmetrical illumination for electronic speckle pattern shearinginterferometry, Journal of Modern Optics, 52:6, 797-812, DOI: 10.1080/0950034042000275405
To link to this article: http://dx.doi.org/10.1080/0950034042000275405
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Analysis of coherent symmetrical illumination for electronic
speckle pattern shearing interferometry
JUAN F. ROMAN*y, VICENTE MORENOz,JON N. PETZING� and JOHN R. TYRER�
yBayer Diagnostics, Sudbury, CO10 2XQ, Suffolk, UKzFaculty of Physics, University of Santiago de Compostela, 15782,
Santiago de Compostela, Spain�Wolfson School of Mechanical and Manufacturing Engineering,Loughborough University, Loughborough, Leics., LE11 3TU, UK
(Received 1 April 2004; in final form 21 July 2004)
In an effort to find a non-contact technique capable of providing measurements ofin-plane strain, a speckle shearing interferometer was designed using symmetricalcoherent illumination. This paper presents an analysis of the sensitivity todisplacement and strain of this interferometer, together with an analysis of thephase-stepping of the resultant fringe patterns. New notation is introducedalongside this analysis to define the interference components in speckle shearinginterferometers using multiple illumination beams. Experimental results showfringe patterns and phase stepping in support of the theoretical analysis.
1. Introduction
Speckle based optical metrology techniques allow whole-field, non-contact andreal-time measurement of displacement and strain components [1]. Electronicspeckle pattern interferometry (ESPI) is used for the measurement of displacementin the three orthogonal axes, allowing the independent extraction of two in-planedisplacement (IP) and one out-of-plane (OOP) displacement components. Electronicspeckle pattern shearing interferometry (ESPSI) provides a method for measuringthe first spatial derivatives of displacement, these being related to mechanicalstrain [2–4].
Displacement and strain can be divided into their orthogonal components,the out-of-plane component and the two X and Y in-plane components. If care istaken with the optical configurations [4–8], then the majority of speckle shearinginterferometers can discretely measure the OOP spatial derivative components.However, this is not the case for discrete measurement of the in-plane spatialderivative components, where the issues are more complex.
Extraction of the in-plane terms is possible using aperture based designs andFourier plane analysis [9] but these systems have yet to be demonstrated as real-time instruments. Extraction of the in-plane terms has also been demonstrated using
*Email: [email protected]
Journal of Modern OpticsVol. 52, No. 6, 15 April 2005, 797–812
Journal of Modern OpticsISSN 0950–0340 print/ISSN 1362–3044 online # 2005 Taylor & Francis Group Ltd
http://www.tandf.co.uk/journalsDOI: 10.1080/0950034042000275405
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sequential measurement, changing illumination angles between each data set, thusmaking it possible to identify the in-plane terms [4, 10–12]. While this approachdoes work, if the object under study exhibits time-varying deformation components,then the approach may not produce the correct result. A solution to this issue hasbeen proposed [13], with a shearing interferometer being designed around threewavelength illumination, but the complexity of the system should perhaps not beunderestimated.
Results have been produced which directly measure in-plane strain components,but under special circumstances, such as plane stress or plane strain conditions [14].Further data [15] has suggested that the use of dual or simultaneous illuminationwavefronts may allow direct analysis of in-plane components for arbitrary objects,but this work has not been developed further.
The prediction of the result of speckle interferometers using more than onesimultaneous illuminating beam requires careful understanding of the manipulationof the optical properties of speckle, as well as analysis of the geometry of the objectand image. The standard notation used to describe the speckle shearing interferom-eter output [1, 2], reduces the expression for the interference of the opticalwavefronts at the observation plane, to a cosine expression that contains additionalphase delays. However, this does not take into account the relative spatial correla-tions of the speckle patterns scattered from different incident beams or scatteredfrom different areas of the object, separated by lateral shear. Furthermore, for a dualbeam system, the standard notation used for speckle shearing interferometry doesnot provide indicators for which wavefronts or which illuminating beams interact,after the lateral shift of the images, or after the absolute value subtraction of theimage patterns.
This work considers in-depth the consequences of using dual beam illuminationfor deformation analysis within a speckle shearing interferometer, extending pre-vious discussions of this work [16]. The approach taken has been to modify theexpression describing the speckle interference pattern so that it is separated intoseveral intensity terms, each one labelled according to which illuminating wavefrontcontributes to it. These labels take into account the polarization state of eachwavefront, in order to indicate which ones cause interference and which onesadd together their intensities. A generalized notation for the treatment of multi-wavefront speckle interferometers is presented in this work, and is introduced alongwith the analysis of the novel interferometer. Initial experimental results arepresented which support the development of the theoretical analysis.
2. Theoretical analysis of dual illumination
The speckle shearing interferometer used for this study was based on the Michelsondesign [1], using two mutually coherent and symmetrically incident beams toilluminate the object, as shown in figure 1. The optical axis of the CCD TV camerabisects the angle made by the two laser illumination beams. For the purposes of thedevelopment of the analysis of the simultaneous dual illumination interferometer, itis necessary to individually label wavefront components and consider their amplitudeand phase contributions. It should also be noted that, as would be expected witha Michelson based optical system, the theoretical development has many initial
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similarities with existing speckle pattern interferometry theory, although modifiedand expanded to take into account the nature of simultaneous illumination.
Both illumination wavefronts in figure 1 are marked as having the same stateof polarization (in this case vertical polarization, perpendicular to the plane ofincidence). Each wavefront has a different label according to the direction ofillumination (left L or right R) and the presence or not of lateral shearing(A or B) applied using the Michelson optics. The L or R label corresponds tothe illumination wavefront and the second letter indicates the mirror from whichthe wavefront was reflected. If we denote the amplitudes (including complexphase) by LA, LB, RA, RB, of the contributory wavefronts arriving at a point(x,y) on the image plane, the intensity will be the result of the product describedby equation (1) :
I x, yÞ ¼ LAþ LBþRAþRBð Þ � LAþ LBþRAþRBð Þ�
ð ð1Þ
with
LA ¼ LAk k � ei�LA ð2Þ
LA�¼ LAk k � e�i�LA ð3Þ
and so on.The product of the amplitudes with their own conjugates, results in the
light intensity at point (x,y), and the result indicates any possible constructive
Figure 1. Speckle shearing interferometry with two mutually coherent symmetricallyincident beams.
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or destructive interference, depending on the roughness of the surface responsible
for the initial random phase �.In our notation the optical phase terms � with the sub-index Að�LA and �RAÞ
indicate that these phase terms corresponds to light scattered from point (x,y)OBJECT
on the object’s surface, arriving at the point (x, y)CCD of the image plane, after
being reflected by mirror A. Optical phase terms with the sub-index Bð�LB and �RBÞ
indicate that these phase terms correspond to light scattered from point
(xþ �x, y)OBJECT on the object’s surface, arriving to the same point (x, y)CCD, on
the image plane by means of the tilt on mirror B.
The result of the product in equation (1) is
I x, yð Þ ¼ LA � LA�þ LA � LB� þ LA �RA�
þ LA �RB�
þ LB � LA�þ LB � LB� þ LB �RA�
þ LB �RB�
þRA � LA�þRA � LB� þRA �RA�
þRA �RB�
þRB � LA�þRB � LB� þRB �RA�
þRB �RB�: ð4Þ
In this expression, the terms of the addition LALA�, LBLB�, RARA� and RBRB�
represent the intensity of the beams LA, LB, RA, RB as shown by
LA � LA�¼ LAk k � ei�LA � LAk k � e�i�LA ¼ LAk k
2, ð5Þ
and similarly for the rest of the terms.
Mirror B has a small tilt to provide the necessary lateral shear between
the images reflected by mirrors A and B. Thus, light incident at the same point
of the CCD camera does not come from the same point of the object. Hence,
LA and RA represent the light waves from illumination wavefront L and R
respectively, reflected by mirror A and incident at point (x, y) of the CCD
plane, coming from the correspondent point (x, y) of the object. Analogously,
LB and RB are the light waves reflected by mirror B (laterally tilted) and
incident at point (x, y) of the CCD plane, but with the observation that this
light comes from point (xþ �x, y) on the object, due to the tilting of mirror B.
This notation allows one to manage the 16 terms resulting from the interference
of four coherent beams that takes place at the image plane.
The rest of the terms can be calculated using the same notation for the
amplitude and phase as demonstrated in equation (6), and similarly for the rest
of the terms.
RA �RB� þRB �RA�¼ RAk k � RBk k � ei�RA � e�i�RB þ RAk k � RBk k � e�i�RA � ei�RB
¼ RAk k � RBk k � ei �RA��RBð Þ þ ei �RB��RAð Þ� �
¼ RAk k � RBk k � 2 � cos �RA � �RBð Þ: ð6Þ
The light intensity registered at a point (x, y) on the CCD camera (image plane)
will be the result of the addition of all the terms in equation (4). After manipulation
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of the mathematical terms, the result can be summarized as
Iðx, yÞCCD ¼ LAk k2þ LBk k2þ RAk k
2þ RBk k2
þ RAk k � RBk k � 2 � cos �RA � �RBð Þ
þ LAk k � LBk k � 2 � cos �LA � �LBð Þ
þ RAk k � LAk k � 2 � cos �RA � �LAð Þ
þ RAk k � LBk k � 2 � cos �RA � �LBð Þ
þ RBk k � LAk k � 2 � cos �RB � �LAð Þ
þ RBk k � LBk k � 2 � cos �RB � �LBð Þ: ð7Þ
This equation represents the intensity of light at the arbitrary point (x, y) on imageplane before any alteration or deformation of the object.
If the object undergoes static deformation, the intensity pattern registered atthe image plane will change accordingly. These movements of the object introducechanges in the optical paths of the four wavefronts (LA, LB, RA and RB) thatcombine to make the image, and the final image plane speckle pattern will vary.
At this stage, it is assumed that the object deformation is smaller than the averagespeckle grain size, thus preserving issues of speckle correlation. Deformations greaterthan the average size of the speckle grains would introduce a loss of correlation,making it very difficult to obtain correlation fringe patterns. In the notationpresented here, this condition imposed on the amount of deformation means thatthe amplitude of the wavefronts at point (x, y) will be the same before and afterdeformation of the object, i.e.
LAk kafter ¼ LAk kbefore: ð8Þ
Furthermore, due to the different angles of illumination onto the opticallyrough surface of the object, and the tilt introduced by mirror B, the intensity andphase of light arriving from mirrors A and B will be different. This implies that:
LAk k 6¼ LBk k 6¼ RAk k 6¼ RBk k ð9Þ
�RA 6¼ �RB ð10Þ
�LA � �RAð ÞBefore 6¼ �LA � �RAð ÞAfter: ð11Þ
While the overall amplitude terms do not change with movement of the object(equation (8)), in-plane and out-of-plane deformations introduce phase changes thatmodify the final speckle pattern. This is recognized when calculating the interferenceinvolving the 16 different terms associated with the non-deformed and deformedstate of the object. To a certain extent the complexity of this analysis has parallelswith work previously completed concerning wedge and aperture based shearinginterferometers [8].
If an arbitrary point (x0,y0,z0) on the object’s surface performs a displace-ment with co-ordinates (u,v,w), the displacement associated to the laterally shiftedobject point (x0þ �x, y0,z0) will be (uþ �u,vþ �v,wþ �w), as shown in figure 2.Each component of the displacement will introduce a change in the optical path,as indicated in table 1.
Speckle shearing interferometry using coherent symmetrical illumination 801
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Hence the intensity registered at the same point (x, y)CCD on the image planeafter deformation of the object can be expressed as
I x,yð ÞAfterCCD
¼ LAk k2þ LBk k2þ RAk k
2þ RBk k2
þ RAk k � RBk k � 2 � cos
�RAþ2�
��u � sin��w � 1þ cos�ð Þð Þ
� �
� �RBþ2�
�� uþ �uð Þ � sin�� wþ �wð Þ � 1þ cos�ð Þð Þ
� �0BB@
1CCA
þ LAk k � LBk k � 2 � cos
�LAþ2�
�u � sin��w � 1þ cos�ð Þð Þ
� �
� �LBþ2�
�uþ �uð Þ � sin�� wþ �wð Þ � 1þ cos�ð Þð Þ
� �0BB@
1CCA
þ RAk k � LAk k � 2 � cos
�RAþ2�
��u � sin��w � 1þ cos�ð Þð Þ
� �
� �LAþ2�
�u � sin��w � 1þ cos�ð Þð Þ
� �0BB@
1CCA
þ RAk k � LBk k � 2 � cos
�RAþ2�
��u � sin��w � 1þ cos�ð Þð Þ
� �
� �LBþ2�
�uþ �uð Þ � sin�� wþ �wð Þ � 1þ cos�ð Þð Þ
� �0BB@
1CCA
þ RBk k � LAk k � 2 � cos
�RBþ2�
�� uþ �uð Þ � sin�� wþ �wð Þ � 1þ cos�ð Þð Þ
� �
� �LAþ2�
�u � sin��w � 1þ cos�ð Þð Þ
� �0BB@
1CCA
þ RBk k � LBk k � 2 � cos
�RBþ2�
�� uþ �uð Þ � sin�� wþ �wð Þ � 1þ cos�ð Þð Þ
� �
� �LBþ2�
�uþ �uð Þ � sin�� wþ �wð Þ � 1þ cos�ð Þð Þ
� �0BB@
1CCAð12Þ
Figure 2. Different position and displacement vectors in ESPSI with two mutually coherentsymmetrically incident beams.
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which may be simplified and rewritten as follows:
I x,yð ÞAfterCCD
¼ LAk k2þ LBk k2þ RAk k
2þ RBk k2
þ RAk k � RBk k �2 � cos �RA��RBþ2�
��u � sin�þ �w � 1þ cos�ð Þð Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
�1
0BB@
1CCA
þ LAk k � LBk k �2 � cos �LA��LBþ2�
���u � sin�þ �w � 1þ cos�ð Þð Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
�2
0BB@
1CCA
þ RAk k � LAk k �2 � cos �RA��LA�4�
��u � sin�|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}�3
0BB@
1CCA
þ RAk k � LBk k �2 � cos �RA��LB�4�
��u � sin��
2�
�� �u � sin�þ
2�
�� �w 1þ cos�ð Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
�4
0BB@
1CCA
þ RBk k � LAk k �2 � cos �RB��LA�4�
��u � sin��
2�
�� �u � sin��
2�
�� �w � 1þ cos�ð Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
�5
0BB@
1CCA
þ RBk k � LBk k �2 � cos �RB��LB�4�
��u � sin��
4�
�� �u � sin�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
�6
0BB@
1CCA ð13Þ
In electronic speckle pattern interferometry, the intensities of the images beforeand after deformation of the object are typically recorded by means of a solidstate camera and then subtracted in absolute terms, pixel by pixel. The resultantimage would have dark speckle fringes at the places where Ibefore(x, y)¼ Iafter(x, y).
Table 1. Optical path changes in double beam interferometer.
Mirror Component ofdisplacement
Optical path change inbeam L
Optical path change inbeam R
A u (X axis) þ u � sin � � u � sin �
A v (Y axis) nil nil
A w (Z axis) �w � (1þ cos �) � w � (1þ cos �)
B uþ �u (X axis) þ (uþ �u) � sin � � (uþ �u) � sin �
B vþ �v (Y axis) nil nil
B wþ �w (Z axis) � (wþ �w) � (1þ cos �) � (wþ �w) � (1þ cos �)
Speckle shearing interferometry using coherent symmetrical illumination 803
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For this to happen with the simultaneous illumination, the six optical phaseinterference terms which appear in equation (13), must not change the result ofthe cosine functions and would have to be equal to an even integer number of �.These interference terms may be rewritten as �1 to �6, this transformation beingdescribed by equations (14) to (19), respectively:
�1 ¼2�
��
@u
@x� sin � þ
@w
@x� 1þ cos �ð Þ
� �� �x ð14Þ
�2 ¼2�
�� �
@u
@x� sin � þ
@w
@x� 1þ cos �ð Þ
� �� �x ð15Þ
�3 ¼ �4�
�� u � sin � ð16Þ
�4 ¼ �4�
�� u � sin � �
2�
�
@u
@xsin � �
@w
@x1þ cos �ð Þ
� ��x ð17Þ
�5 ¼ �4�
�� u � sin � �
2�
�
@u
@xsin � þ
@w
@x1þ cos �ð Þ
� ��x ð18Þ
�6 ¼ �4�
�� u � sin � �
4�
��@u
@x� sin � � �x ð19Þ
where �x is lateral shear or shift introduced by mirror B, measured at the objectplane.
To obtain a dark pixel (hence a dark correlation fringe) within the subtractioncorrelation image, the above interference terms must all be simultaneously equalto an even integer number of �. This can only be achieved if certain conditionsare satisfied simultaneously:
(i) �3 ¼ 2m�, where m is the fringe order and can be any integer number. Thiscondition leads to
2 � u � sin � ¼ m1 � �, ð20Þ
where m1 can be any integer number. This is the equation of the zones of the objectwith equal u (x-axis) in-plane displacement, as seen in in-plane ESPI.
(ii) �6 ¼ 2m�, m any integer number. If condition (i) is already satisfied,after some calculation condition (ii) leads to
2 �@u
@x
� �� �x � sin � ¼ m2 � �, ð21Þ
where m2 can be any integer number. This is the equation that describes the partsof the object with the same in-plane displacement derivative (strain) @u=@xð Þ.
(iii) �1, �2, �4, and �5 ¼ 2m�. These conditions are all equivalent if(i) and (ii) have already been satisfied. After some calculation they
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can be summarised as
2 �@w
@x
� �� �x � 1þ cos �ð Þ ¼ m3 � �, ð22Þ
where m3 can be any integer number. This is the equation of the zones of theobject with the same out-of-plane displacement derivative (slope) @w=@xð Þ.
After digital image subtraction, dark correlation fringes appear at the zoneswhere the three previous conditions are satisfied simultaneously. The resultantpattern corresponds then to the ‘moire’ overlapping of three speckle correlationfringe patterns, these being:
. ESPI in-plane displacement fringe pattern;
. in-plane displacement derivative (strain) fringe pattern; and
. out-of-plane displacement derivative (slope) fringe pattern.
This moire pattern will appear as a complex mosaic of spots, resulting fromthe crossing of fringe patterns, as illustrated on figure 3. The fact that ashearing interferometer results in a moire-related result is not new in itself, withauthors having previously presented photographic based systems using Fourieranalysis [17, 18], to first examine out-of-plane terms but also in-plane terms.However, the issues here are to explore optical configurations which resultdirectly in real-time in-plane derivative systems, which lead to extraction of opticalphase terms and the potential for quantitative evaluation.
3. Analysis of optical phase extraction
Optical phase information extraction is an important issue, because it provides theroute forward for eventual generation of quantified data from the instrumentation.By introducing a controlled optical phase shift in one of the illuminating beams ofthe interferometer, it is possible to perform the phase stepping [19] of the resultantspeckle pattern and fringe pattern. If a phase shift ð�piezoÞ is introduced in the optical
Figure 3. Moire overlapping of two speckle patterns.
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path of, for instance, illumination wavefront L (for this purpose both wavefronts are
equivalent), the effect of this phase shift on the interferometer can be calculated
analytically by adding the optical phase shift term to all the phases of the wavefronts
originated by this beam. With respect to equations (12) and (13), �LA and �LB will
change to �LA þ �piezo and �LB þ �piezo respectively. The resultant speckle pattern
can then be expressed as
I x,yð ÞAfterþPiezoCCD
¼ LAk k2þ LBk k2þ RAk k
2þ RBk k2
þ RAk k� RBk k�2�cos �RA��RBþ2�
��u�sin�þ�w� 1þcos�ð Þð Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
�1
0BB@
1CCA
þ LAk k� LBk k�2�cos �LAþ�PIEZO��LB��PIEZOþ2�
���u�sin�þ�w� 1þcos�ð Þð Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
�2
0BB@
1CCA
þ RAk k� LAk k�2�cos �RA��LA��PIEZO�4�
��u�sin�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
�3
0BB@
1CCA
þ RAk k� LBk k�2�cos �RA��LB��PIEZO�4�
��u�sin��
2�
���u�sin�þ
2�
���w 1þcos�ð Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
�4
0BB@
1CCA
þ RBk k� LAk k�2�cos �RB��LA��PIEZO�4�
��u�sin��
2�
���u�sin��
2�
���w� 1þcos�ð Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
�5
0BB@
1CCA
þ RBk k� LBk k�2�cos �RB��LB��PIEZO�4�
��u�sin��
4�
���u�sin�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
�6
0BB@
1CCA ð23Þ
Following a similar process of analysis as previously discussed, to obtain a dark
pixel (and consequently a dark fringe) after the image correlation subtraction
process (Ibefore(x, y)� Iafterþ piezo(x, y)), all the six interference terms ii must satisfy
several conditions simultaneously:
(i) �Piezo3 ¼ 2m�, where �Piezo
3 ¼ �ð4�=�Þ � u � sin � � �PIEZO, and m can be any
integer number.
This is equivalent to the condition given by
2 � u � sin � þ�piezo
2�� � ¼ m0
1 � � ð24Þ
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where m01 can be any integer number, i.e. the fringe order. As in the case without the
phase shifting, this is the equation of the zones of the object with equal u (x-axis)in-plane displacement, similar to in-plane ESPI with the phase shift introducedby �piezo.
(ii) �Piezo6 ¼ 2m�, where �Piezo
6 ¼ �ð4�=�Þ � u � sin � � �PIEZO � ð4�=�Þ � ð@u=@xÞ�sin � � �x, and m is any integer number.
If the condition (i) has already been satisfied, the first two terms in �Piezo6 are equal
to �Piezo3 and condition (ii) is fulfilled without the effect of the phase stepping. In this
case the result is that the in-plane strain component of the moire pattern does notphase step.
(iii) �Piezo1 , �Piezo
2 , �Piezo4 , and�Piezo
5 ¼ 2m�:
As in the case before introducing phase stepping, these conditions are all equivalentif (i) and (ii) have already been satisfied, and after some calculation they all result inthe condition
2 �@w
@x
� �� �x � 1þ cos �ð Þ ¼ m3 � �, ð25Þ
indicating that the OOP component of the displacement derivative does not phasestep either.
Using similar calculations, it can be demonstrated that the introduction of phasestepping on one of the mirrors on the shearing head causes the OOP displacementderivative to phase step, while the in-plane displacement and in-plane displacementderivative do not phase step.
In summary, the introduction of optical phase stepping in one of the illuminat-ing wavefronts causes the in-plane displacement pattern to phase step, while theother two components of the moire pattern (OOP displacement derivative and IPdisplacement derivative) do not phase step. This property can be used to extractoptical phase information and hence quantitative information, through phasestepping and subsequent phase unwrapping processes.
4. Experimental results
The purpose of the experimentation was first to demonstrate that the simultaneousdual illumination associated with the speckle shearing interferometer produces themoire patterns predicted by the theory, and second that optical phase steppingtechniques can be applied to the interferometer.
The test object chosen for the experimental verification was a split cylinder testor Brazilian disc, under compressive loading. This consists of a flat disc compressedalong its equator and has traditionally been used with techniques such as photo-elasticity or moire interferometry [20] to examine loading and deformation char-acteristics, although more recent work has been demonstrated using holographicinterferometry [21]. A 75mm diameter (6mm thick) Brazilian disc was manufacturedfrom AralditeTM sheet, and was loaded using a compressive test rig linked toa hydraulic DH-Budenburg dead-weight tester. This provided forces up to 5000N,with a precision� 1N.
Speckle shearing interferometry using coherent symmetrical illumination 807
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The Michelson based speckle shearing interferometer used a 50mW Nd–YAG
laser (wavelength � ¼ 532 nm), with a coherence length in excess of 10m. A Pulnix
TM-9701 CCD camera was used at the image plane linked through to a MuTech
Corporation image processing board. Optical phase-stepping was introduced
into the interferometer by means of a piezoelectric actuator supplied by
Piezo-Systems Jena.
Two sets of validating experiments were initially completed, producing different
types of results. The first approach was to apply large loads to the disc causing
predominant motion out-of-plane with limited in-plane motion. The second
approach relied on limited loading of the disc, which resulted in dominant in-plane
motion with limited out-of-plane motion.
Figures 4(a) to 4(d) provide examples of the initial experimental analysis of
the large loading experiments. The simultaneous dual illumination interferometer
produces a speckle based subtraction correlation pattern, which is the result of
the intercrossing of the in-plane displacement rotation signature and the
OOP displacement derivative component. From the three expected subpatterns,
the pure in-plane displacement derivative component is missing, due to the fact
that the Brazilian disc was identified as rotating within the compressive testing
rig. Also, the only component of the moire pattern that is affected by the phase
stepping is the in-plane displacement component, as predicted in the theoretical
analysis.
The second experimental approach produced sets of results showing the disc
without rotation and evidence of in-plane displacement derivative data. Figures 5(a)
to 5(d) provide a sequence of images, which include photoelastic analysis of the disc,
(a)
Figure 4. Same sample subjected to same force: (a) ESPI in plane result; (b) OOP strain;(c) fringe pattern with the new interferometer; (d) results of phase stepping with newinterferometer. Note how patterns (a) and (b) overlap to produce the result with thenew interferometer.
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(d)
Figure 4. Continued.
(c)
(b)
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as well as interferometric analysis of the disc. The photoelasticity results wereobtained by means of a reflection polarizcope (030-series by Measurements GroupInc., North Carolina, USA) consisting of two polarizer/quarter-wave assembliesmechanically coupled for synchronous rotation. This optical set-up produces anisochromatic pattern that highlights the distribution of pure in-plane strain, andwhen overlapped with the ESPI in plane, match the moire pattern obtained withthe new interferometer, figure 5(d).
(b)
Figure 5. Overlapping of the results of photoelasticity (in plane strain), ((a) and (b)) andESPI in plane (c), is analogous to the fringe pattern obtained independently with the two-beam interferometer (d).
(a)
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The results provide qualitative evidence for the theoretical predictions of thespeckle shearing interferometer with two mutually coherent symmetrically incidentbeams.
5. Conclusions
In this work the authors present a refinement of conventional Michelson-shearinginterferometers, where the use of two simultaneously coherent beams contributes to
(d)
Figure 5. Continued.
(c)
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highlight in-plane strain. In order to predict the final fringe pattern it was necessaryto introduce a novel notation. This notation makes it possible to predict theinterference of multiple illumination beams in Michelson-shearing interferometers.The calculations using this notation allow prediction of the fringe patterns for anovel interferometer in both stationary state and in the presence of phase-steppingintroduced in one of the illuminating beams.
Experiments performed on a Brazilian disc sample show the performance of thenovel interferometer and confirm the theoretical predictions for the fringe patterns.The results highlight the pure in-plane strain in the sample studied, in one singlemeasurement with the interferometer. This is an advantage since now a singleexperiment produces a measurement, instead of having to perform alternativemeasurements with in-plane ESPI and conventional ESPSI, as the combination ofresults from two interferometers always produces an increase in the error of themeasurement.
References
[1] P.K. Rastogi, (Editor), Digital Speckle Pattern Interferometry and Related Techniques(John Wiley & Sons, Chichester, 2001).
[2] J.A. Leendertz and J.N. Butters, J. Phys. E 6 1107 (1973).[3] Y.Y. Hung and C.E. Taylor, Exp. Mech. 281 (1974).[4] E. Steinchen and L. Yang, Digital Shearography: Theory and Application of Digital
Speckle Pattern Shearing Interferometry (SPIE Press, New York, 2003).[5] W.S. Wan Abdullah, J.N. Petzing and J.R. Tyrer, J. Mod. Opt. 48 757 (2001).[6] J.S. Ibrahim, J.N. Petzing and J.R. Tyrer, J. Meas. Sci. Technol. (in print).[7] K.F. Wang, A.K. Tieu and E.B. Li, Opt. Laser Technol. 31 549 (1999).[8] T. Santhanakrishnan, Krishna N. Mohan, P. K. Palaniswamy, et al., J. Instrum. Soc.
India 27(1) 16 (1997).[9] R.S. Sirohi, Speckle Metrology (Marcel Dekker Inc., New York, 1993).[10] P.K. Rastogi, J. Mod. Opt. 43 1577 (1996).[11] Y.Y. Hung and J.Q. Wang, Opt. Lasers Eng. 24 403 (1996).[12] H.A. Aebischer and S. Waldner, Opt. Lasers Eng. 26 407 (1997).[13] R. Kastle, E. Hack and U. Sennhauser, Appl. Opt. 38(1) 96 (1999).[14] J.R. Tyer and J.N. Petzing, Opt. Lasers Eng. 26 395 (1997).[15] J.N. Petzing and J.R. Tyrer, Proceedings of Interferometry ’94, SPIE 2342, pp. 27–36
(Warsaw, Poland, 1994).[16] J.F. Roman, J.N. Petzing and Tyrer, J.R., Proceedings of FASIG’98 (Brighton, UK,
April 1998).[17] K. Patorski, Appl. Opt. 27 3567 (1998).[18] P.K. Rastogi, Opt. Lasers Eng. 29 97 (1998).[19] D.W. Robinson and G.T. Reid, Interferogram Analysis: Digital Fringe Pattern
Measurement Techniques (IOP Publishing Ltd., Bristol, 1993).[20] J.W. Dally and W.F. Riley, Experimental Stress Analysis (McGraw-Hill International
Editions, New York, 1991).[21] A. Castro-Montero, A. Jia and S. P. Shah, Am. Concrete Inst. Mater. J. 92 268 (1995).
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