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Developments and applications of conjugate-wave holographic interferometry
Optics and Lasers in Engineering 14 (1991) 39-53
Developments and Applications of Conjugate-wave Holographic Interferometry L. Pirodda Department
of Mechanical
Engineering, University 09100 Cagliari, Italy
of Cagliari, Piazza d’Armi,
L. J. Griffiths 16 Dan-y-Coed, (Received
Aberystwyth,
25 November
Dyfed SY23 2HD, UK
1989; accepted
11 December
1989)
ABSTRACT Conjugate-wave holographic interferometry is an optical technique which was recently proposed by one of the authors for the measurement, in real time, of in-plane deformation. Basically, the technique consists of projecting, from two symmetrical directions, two real images of an object upon the object itself. If the object deforms, interference fringes are observed on its surface. These fringes give a representation of one in-plane component of the point displacement vector and are independent of the out of plane component. In this paper some aspects of the optical principles underlying the technique are analyzed in deeper detail, and more advanced solutions for their practical implementation are proposed. In addition, a few examples of applications on-plane and not-plane specimens are presented.
1 INTRODUCTION Holographic interferometry is extensively employed in experimental mechanics, particularly for the measurement of deformation and strain. In the general case its application requires the separate determination of the three components of a point displacement. This task was very troublesome until the advent of computerized procedures, which are 39 Optics and Lasers in Engineering 0143-8166/90/$03.50 Ltd, England. Printed in Northern Ireland
@ 1990 Elsevier Science Publishers
40
L. Pirodda,
L. J. Grifiths
now popular. Such procedures are, of course, hybrid ones and require an optical system, an interface and a computing system. Ideally the optical system, which we are interested in, should be as simple as possible and its output data should be as close as possible to the final result required. For this reason much attention has been and is being paid to any development in the optical part of the system which can simplify, reduce, or in some cases even eliminate the computational part. Holographic systems, which are sensitive only to one component of the point displacement, clearly tend to satisfy the above requirements. Moire holography’T2 can be quoted among such systems. Moire interferometry3,4,5 and two-beam speckle interferometry,6y7 although not strictly holographic techniques, may also be included. A comparative analysis of these techniques may be found in the above references and additional references listed therein. The method proposed by Pirodda8 and further developed in this paper can be linked to moire holography, which we take as the starting point of our discussion. In moire holography the object is symmetrically illuminated by two plane waves and the two waves scattered in the same frontal direction are holographically recorded. The interference of the four waves scattered by the undeformed and deformed states of the object produce intensity patterns consisting of two sets of fringes, from which the loci of constant value for one in-plane component of the point displacement can be visualized in the form of moire curves of the two sets. The necessity of improving the visibility of these curves calls for some complementary steps, such as the introduction of additional carrier fringes by suitably moving the plate or the reference or the object, followed by optical filtering in order to eliminate the carrier fringes. The new technique somewhat reverses the process described above. The object is illuminated frontally by a plane (or quasi plane) wave, and two waves scattered at symmetrical angles are recorded holographically. Eventually, the conjugates of these waves are produced and impinge on the object, which is observed frontally. If the object is unchanged, the light it scatters reconstructs the conjugate of the original plane wave. If the object is deformed, the reconstructed wave is also distorted and its intensity distribution maps one of the two in-plane components of the displacement. In the following text, after a short review of the principles of the technique, three arrangements representing its practical implementation for the measurement of one, two or three components of the in-plane deformation are presented and discussed. A few examples of applications are also presented in the form of interferograms, and some ancillary aspects of the performance of the technique are discussed, mainly in the Appendix.
Conjugate-wave
2 REVIEW
holographic
interferometry
OF THE PRINCIPLES
41
OF THE METHOD
For the commodity of the reader, we first report from Ref. 8 the basic theory of the method. Figure 1 represents, with its irregularities, a portion of the surface of the object, which is referred to a frame xyz. The xy plane is supposed parallel to the surface (macroscopically considered) if this is plane, or more generally, tangential to it at a given point. If a plane wave impinges on the undeformed object in the normal direction, the amplitudes of the wavefronts o1 and 02, scattered according to the symmetrical directions forming the same angle with the normal and both parallel to the xz plane, vary according to: o1 = exp (i F [X sin 8 + ~“(1 + cos e)]} (1) o,=exp{iF[-xsinB+wU(l+cos0)]} where w’(x, y) is a relative measure of the surface irregularity. We have supposed, for simplicity, that the intensity factor is not affected by the scattering process and is everywhere one. recorded on two The wavefronts (eqn (1)) are holographically holographic plates placed at a sufficient distance from the object and of sufficiently small aperture to justify the assumption that they collect the light from a single direction. After processing, the plates are accurately
Fig. 1.
Basic theoretical
representation.
L. Pirodda,
42
L. J. Griffiths
repositioned and illuminated by the conjugate of the reference beam. This can be done in the simplest way by a plane mirror placed behind the plate if the reference is a collimated beam. The complex conjugates of the amplitudes (eqn (1)) represent the wavefronts diffracted by the plate as they impinge back on the object. If the surface of the object is unchanged, those components of the light which are scattered from it in the z direction again form a plane wave. If the points of the object have undergone displacements s with components u and w according to x and z respectively, the two wavefronts, propagating in the positive direction of z are: h’
= exp {i F [u sin 8 + ~(1 + cos e)]}
(2) i, = exp {i$[-u Their interference
sin 13+ ~(1 + cos S)]]
gives an intensity distribution:
Thus, the intensity maxima and minima are loci of equal value for u, with sensitivity factor:
The intensity fringes obeying eqn (3) have the highest theoretical visibility and are not disturbed by the noise of any other carrying fringe. To observe them, one cannot simply look at the object surface from any direction, since they are not directly formed on that surface by the interference of the two conjugate waves but by the interference of the two correspondingly scattered waves. The fringes may be seen or recorded by placing the eye or the lens of a camera immediately behind a point where the two wavefronts scattered according to z are collected by a lens. Of course, the point and the lens may coincide with the pinhole and lens of the beam expander used in the recording step.
3 PRACTICAL
IMPLEMENTATIONS
OF THE TECHNIQUE
The procedure summarily described above may appear rather complicated. However, it can be easily implemented by means of a practical setup which is only slightly more complex than the usual ones used in
Conjugate-wave
holographic
~~ F: v
M2
Fig. 2(a).
interferometry
Practical setup for recording.
Fig. 2(b).
43
GMIM MINNTMHIOR)
Practical setup for observing.
real-time holography. The setup (Fig. 2(a), (b)) features only one holographic plate and a suitably placed mirror to provide a virtual substitute of the second plate. In the recording step the object is directly illuminated by a point source, F (the pinhole of a beam expander). The conditions of plane illuminating and recorded waves are thus relaxed and the errors involved will be analyzed later. In order to make the subsequent observation easier, a screen with a small circular aperture F (about 5 mm) is placed in front of the beam expander. The reference is a collimated beam and a gimbal mounted mirror provides for its conjugate. When recording, a screen is placed between plate and mirror, in order to eliminate the noise of the reflected light. After exposure the holographic plate is processed for high diffraction efficiency and accurately repositioned. A photo-thermoplastic device with in situ development appears particularly suited for this application and it was in fact employed in the experiments described in Ref. 8. At this stage the illuminating beam is discarded and the (deformed) object is observed by placing the eye or the lens of a camera immediately behind the aperture F (Fig. 2(b)). The conjugate waves impinging back upon the object are in fact preferably scattered according to the conjugate of the illuminating wave and hence are collected in the aperture. Another important function of the aperture is to minimize the noise in the system by shielding the observing lens from light scattered in directions other than the cone of rays converging on F.
44
L. Pirodda,
L. J. Grifiths
The setup in Fig. 2 allows the measurement of the displacement components u according to x. In general, the study of a bi-dimensional deformation field requires both components according to x and y to be measured in a single loading cycle. This problem can be solved by means of the more general setup represented in Fig. 3. Again we have a single actual holographic plate, which is now normal to z and with centre on z, and two pairs of mirrors, with normals in the xz plane and in the yz plane respectively, and symmetrically placed with respect to z. Another mirror, reflecting on opposite surfaces, is placed between the plate and the object. This double mirror has three functions: reflecting the reference on to the holographic plate, reflecting the illuminating beam on to the object, and screening the plate from the light coming directly from the object. In the recording step the hologram of the four symmetrical waves scattered at the same average angle is taken. When observing, by alternatively screening a couple of mirrors, either the u or the v fringes may be separately visualized for the same loading condition. A further generalization in the results may be obtained by means of an arrangement (obtained by a slight modification of the preceding one and hence not illustrated) where the two pairs of symmetrical mirrors are substituted by three mirrors so placed that the planes defined by their normal and the z axis form angles of 120”. In this case all three conjugate waves are allowed to impinge on the object at the same time, on observation. If the object is deformed, the interference of the three scattered waves converging on F produces three sets of fringes from which three components of the in-plane displacement vector can be calculated. The corresponding equations are formally analogous to the equations of three beams moire interferometry (see for instance Ref. 9). Examples of the results obtainable by the technique are shown in the photographs in Figs 4,5, and 6. Figure 4 presents, for comparison with other techniques, a classical case: the ring compressed diametrically. The fringes map the displacement component according to the direction of loading. Figure 5 shows the application on a carbon fibre composite plate having a cut and loaded by in-plane forces acting to open the cut. The average surface is plane, but very rough locally. The fringes map the displacement component normal to the cut. Figure 6 shows an application to a more complex shape, the model of a cylindrical vessel under internal pressure. The fringes map the axial displacement component. In all cases (Figs 4-6) a commercial silver halide holographic plate (Agfa 8E75 HD) was used. After development the plate was bleached and repositioned by means of a simple home made plate holder. The angle 0 was 45”, corresponding to a fringe
Conjugate-wave holographic interferometry
t Fig. 3(a).
HOLOGR.
Setup for simultaneously
PLATE
recording the u and v fringes.
PLATE
t M4
Fig. 3(b).
Setup for observing
the u and v fringes.
46
L. Pirodda,
Fig. 4.
u Displacement
under
diametral
Fig. 6.
L. .I. Grifiths
Fig. 5.
fringes on a ring compression.
u Displacement
u Displacement carbon
fringes
on a cylindrical
fringes fibre plate.
on a
vessel.
sensitivity factor of O-7A. The surfaces of the specimens were opaque white painted. A commercial reflex camera with 50 mm focal length loaded with Hilford HP4 film was used to take the photographs of the fringes. The laser employed was a 5 mW He-Ne. 4 DISCUSSION
AND
REMARKS
It is important to point out that the mirrors providing for a virtual holographic plate, like Ml in Fig. 2, or MA, MB in Fig. 3, can be quite
Conjugate-wave
holographic
interferometry
47
ordinary and inexpensive ones, since any wavefront distortions they introduce in the recording step are compensated in the observation step by the phase conjugation process, in the same way as this process compensates for the irregularities of the surface of the object. This is an obvious advantage, particularly when the object to be studied is large and the use of mirrors of optical quality would be quite expensive. The use of a single holographic plate, instead of two or more, presents an additional advantage, besides the obvious one of simplification. If commercial plates with a base of ordinary glass are used, the lack in uniformity of their thickness introduces phase distortions in the conjugate waves. Such distortions are different and hence produce errors in the recorded intensity pattern in the case of separate plates, while in the case of a single plate the phase distortions are almost the same, the only difference being due to the fact that the conjugate waves are not at the same angle with the plate. Even this effect tends to disappear when the average directions of the conjugate waves are symmetrical with respect to the normal to the plate, like in the case of Fig. 3. A further improvement in this respect may be achieved if in the observation step the diameter of the cross section of the reconstructing beam is reduced (for example by changing the collimating lens) to the minimum value consistent with a good resolution of the real images. In our experiences this diameter was about 1 cm, while, on recording, a reference of about 4 cm had been used. The latter expedient presents two additional advantages. A very small gimbal mounted mirror M4 is needed and the brightness of the conjugate waves is definitely increased, an appreciable gain when a small power laser is used. The mirror M4, providing for the conjugate of the reference, in principle could not be of optimum quality, since, in this case too, any introduced phase changes appear in both conjugate waves and do not affect the intensity of the observed wave. However, as we have just pointed out, the required size for this mirror is fairly small, and the use of a high quality unit is not a problem. Moreover, this makes the operation of collimating the reference easier and the adjustments sometimes needed in the direction of the same, easier. If a small power laser. is used, the power ratio of a variable beam splitter must be reversed between recording and observing in order to use all the power available for the latter operation. This operation may cause a small misalignment, which can be easily corrected by adjusting the plane of M4. Such adjustments may prove also useful in order to keep the fringes at their maximum visibility during the process of deformation of the object. For these reasons the gimbal mount of M4 should be precise and stable.
48
L. Pirodda,
L. J. Grifiths
The operation of collimating the reference beam does not present particular difficulties. The lens, as the other optical components, need not be of top quality. The alignment of the expander, lens, plate and mirror can be easily obtained stepwise and by the simple expedient of observing the reflected spots. The final step, exact collimation, is completed by displacing axially the expander until the light reflected on the mirror M3 and on the splitter form spots of the same size as the on-coming light. We now touch upon two interesting questions regarding the mechanism of fringe formation in the observation stage (Fig. 2(b)), which probably deserve a more exhaustive analysis. One of them is the fact, which we have assumed without any other proof than the experimental one, that the conjugate waves diffracted by the plate, upon being scattered by the object, reconstruct the conjugate of the illuminating waves. The other is the question of fringe localization. Clearly, in our case this does not constitute a problem, as it often comes about in classical holographic interferometry. In our case we have the so called ‘projected’ fringes, produced by the interference of two or more
Fig. 7.
Direct record of the projected fringes in the case of Fig. 5.
Conjugate-wave
holographic
interferometry
49
directional beams (the conjugates of the illuminating waves). The fringes can be observed or recorded by placing a screen or a sensitive medium anywhere in the region of superposition of the beams. Figure 7 shows, as an example, an interferogram concerning the same case as Fig. 5, which was obtained by placing the camera, without the lens, at some distance from the aperture F. Recording by a lens focused at the surface of the object is clearly to be preferred, since the details of the object are not lost and there is a definite correspondence between the fringes and the points of the object. In the case of Fig. 7, on the contrary, only the general form of the object is recognizable through the fringes that cover it, and this form is more or less distorted, depending on the distance of the recorder owing to the phase changes caused by deformation. As we have already pointed out, the fringe sensitivity factor (eqn (4)) has been determined on the assumption that the illuminating and recorded waves were collimated, while in the experimental setups this condition is relaxed. With reference to the scheme of Fig. 8, we may say that eqn (4) is exact only for the points of the object which lie,
Fig. 8.
Schematical
representation
of the basic setup.
50
L. Pirodda, L. J. Grifiths
like 0, on the y axis, while it is approximate for the other points, the approximation decreasing with increasing distance from that axis. In addition there is a correspondingly increasing sensitivity to the out-ofplane component of displacement w. A quantitative analysis of the above described situation is presented in the Appendix. We remark, finally, that the finite aperture a of the holographic plate (Fig. 8), in combination with its finite distance from the object, should have some influence on the sensitivity factor. In fact, we choose the central point C of the plate to ‘represent’ the whole of it, which is somewhat arbitrary, except for the points of the object who are in a plane of symmetry of the plate (the points of the y axis in Fig. 8). A quantitative analysis of this problem, however interesting, would have only an academic character since, as we have pointed out, it is highly advisable in the reconstruction stage to utilize a circular area of the plate with a small diameter, much smaller than the distance plate-object and than the lateral size of the object itself. As a consequence, the variations in the sensitivity factor would be negligible, whichever point of this area is used as a reference for evaluating it.
ACKNOWLEDGEMENT The research presented in this paper has been financially supported the Ministry of the University and Scientific Research, Italy.
by
REFERENCES 1. Sciammarella,
C. A. & Gilbert, J. A. A holographic-moire technique obtain separate patterns for components of displacement. Exp. Mech.,
to 16
(1976) 215.
2. Sciammarella, C. A. Holographic moire, an optical tool for the determination of displacements, strains, contours and slopes of surfaces. Opt. Erg., 21 (1982) 447. Boone, P. M. Surface deformation measurements using deformation following holograms. Nouv. Rev. d’Opt. Appl., 1 (1970). Post, D. Developments in moire interferometry. Opt. Eng., 21 (1982) 458. Pirodda, L. Strain analysis by grating interferometry. Opt. Lasers Eng. 5 (1984) 7.
Archbold, E., Burch J. M. & Ennos, A. Recording of in-plane surface displacement by double exposure speckle photography. Optica Acta, 17 (1970) 883. 7. Ennos A. Speckle interferometry.
North Holland,
1978, p. 235.
In Progress
in Optics XVI,
ed. E. Wolf.
Conjugate-wave
holographic
interferometry
51
for the measure8. Pirodda, L. Conjugate wave holographic interferometry ment of in-plane deformations. Appl. Opt., 26 (1989) 1842-4. 9. Walker, C. A. MoirC interferometry for strain analysis. Opt. Lasers Eng., 8 (1988) 213-62.
APPENDIX
We suppose that the recording step has been carried out in the setup represented in Fig. 2(a), and we reproduce in the scheme in Fig. 8, the geometric elements of this setup which are relevant for the analysis which follows. The points Cl and C2 in the latter figure are the central points of the actual and virtual plate, respectively. The scheme has been simplified by putting C2 at the same distance from 0 as Cl, with no practical consequence as far as the results are concerned. Let, in the observing phase, a point P of the object at distance x from 0 undergo a displacement with in-plane and out-of-plane components u and w respectively. Assuming, as usual, that u and w are of infinitesimal order compared to the distances m, c2p and PF, the corresponding phase changes introduced in the rays ClPF and C2PF may be expressed as follows: ~,=~~(usin~,-usin0~+wcos~,+wcos~,) (Al) ~,=i~(-usin8,-usin8,+wcos&+wcos0,) Setting 8, = f3,,,+ A8; e, + e, em=------.
2
’
and discarding the common terms, calculating the intensity, we have: $: = ~F[u
e2=e,-A8 e, - e, A8=----
(AZ)
2
which
are not meaningful
for
sin (em + A0) + w c0s (em + Ae)] 643)
2Jc & = ‘T [-u sin (em - A6) + w cos (e, - Ae)] Substituting the eqn (A2), performing some trigonometrical and algebraic calculation and again discarding the common terms, we obtain
L. Pirodda,
52
for the amplitudes
L. J. Grifiths
of the two rays:
&=exp
[
&=exp
[
~~(usinB,cos*O-wsin8,sin*6))
I
1
iF(-usinB,cos*B+wsin8,sin*8)
(A4)
and for the resultant intensity: z=2[1+
cos g
L
(2u sin 8, cos A8 + 2w sin 8, sin A@
h
According to eqn (A5), the intensity intervals Au and w satisfy the condition:
maxima
Au cos A8 + Aw sin A8 =
1 (A3
or minima
occur
3L 2 sin 8,
at
646)
and the absolute and relative sensitivity factors for u and w are: Au =
A
Aw=
2 sin 0, cos A8’
A
2sin B,sinAe’
Au -=tanAtI Aw
By means of eqn (A7) it is possible to estimate the error involved when the sensitivity for u is evaluated by eqn (4) and the sensitivity for w is ignored. From Fig. 8 we have: 8, = e - 6,;
tan 6, =
xl1 cos 9 2: 6.” 1 - x/l sin e
8, = e + 6,
tan S, =
xil cos e 21 6, 1 + x/l sin 8
8,=B-$sinBcosB Ae=;cos All the above approximations that Ix/Z1<< 1.
8
are supposed
(AS) valid on the assumption
Setting for instance: Ix/Z\ = 0.1;
8 = 45”
Conjugate-wave
holographic
intetferometry
53
we obtain from the eqns (A7) and (A8): Au=-
A
2 sin 8
1.007;
Aw=-
A
2 sin 8
14.22
The error is only 0.7% as far as the absolute sensitivity for u is concerned, and the latter sensitivity is by more than one order of magnitude larger than the sensitivity for w.
Developments and Applications of Conjugate-wave Holographic Interferometry L. Pirodda Department
of Mechanical
Engineering, University 09100 Cagliari, Italy
of Cagliari, Piazza d’Armi,
L. J. Griffiths 16 Dan-y-Coed, (Received
Aberystwyth,
25 November
Dyfed SY23 2HD, UK
1989; accepted
11 December
1989)
ABSTRACT Conjugate-wave holographic interferometry is an optical technique which was recently proposed by one of the authors for the measurement, in real time, of in-plane deformation. Basically, the technique consists of projecting, from two symmetrical directions, two real images of an object upon the object itself. If the object deforms, interference fringes are observed on its surface. These fringes give a representation of one in-plane component of the point displacement vector and are independent of the out of plane component. In this paper some aspects of the optical principles underlying the technique are analyzed in deeper detail, and more advanced solutions for their practical implementation are proposed. In addition, a few examples of applications on-plane and not-plane specimens are presented.
1 INTRODUCTION Holographic interferometry is extensively employed in experimental mechanics, particularly for the measurement of deformation and strain. In the general case its application requires the separate determination of the three components of a point displacement. This task was very troublesome until the advent of computerized procedures, which are 39 Optics and Lasers in Engineering 0143-8166/90/$03.50 Ltd, England. Printed in Northern Ireland
@ 1990 Elsevier Science Publishers
40
L. Pirodda,
L. J. Grifiths
now popular. Such procedures are, of course, hybrid ones and require an optical system, an interface and a computing system. Ideally the optical system, which we are interested in, should be as simple as possible and its output data should be as close as possible to the final result required. For this reason much attention has been and is being paid to any development in the optical part of the system which can simplify, reduce, or in some cases even eliminate the computational part. Holographic systems, which are sensitive only to one component of the point displacement, clearly tend to satisfy the above requirements. Moire holography’T2 can be quoted among such systems. Moire interferometry3,4,5 and two-beam speckle interferometry,6y7 although not strictly holographic techniques, may also be included. A comparative analysis of these techniques may be found in the above references and additional references listed therein. The method proposed by Pirodda8 and further developed in this paper can be linked to moire holography, which we take as the starting point of our discussion. In moire holography the object is symmetrically illuminated by two plane waves and the two waves scattered in the same frontal direction are holographically recorded. The interference of the four waves scattered by the undeformed and deformed states of the object produce intensity patterns consisting of two sets of fringes, from which the loci of constant value for one in-plane component of the point displacement can be visualized in the form of moire curves of the two sets. The necessity of improving the visibility of these curves calls for some complementary steps, such as the introduction of additional carrier fringes by suitably moving the plate or the reference or the object, followed by optical filtering in order to eliminate the carrier fringes. The new technique somewhat reverses the process described above. The object is illuminated frontally by a plane (or quasi plane) wave, and two waves scattered at symmetrical angles are recorded holographically. Eventually, the conjugates of these waves are produced and impinge on the object, which is observed frontally. If the object is unchanged, the light it scatters reconstructs the conjugate of the original plane wave. If the object is deformed, the reconstructed wave is also distorted and its intensity distribution maps one of the two in-plane components of the displacement. In the following text, after a short review of the principles of the technique, three arrangements representing its practical implementation for the measurement of one, two or three components of the in-plane deformation are presented and discussed. A few examples of applications are also presented in the form of interferograms, and some ancillary aspects of the performance of the technique are discussed, mainly in the Appendix.
Conjugate-wave
2 REVIEW
holographic
interferometry
OF THE PRINCIPLES
41
OF THE METHOD
For the commodity of the reader, we first report from Ref. 8 the basic theory of the method. Figure 1 represents, with its irregularities, a portion of the surface of the object, which is referred to a frame xyz. The xy plane is supposed parallel to the surface (macroscopically considered) if this is plane, or more generally, tangential to it at a given point. If a plane wave impinges on the undeformed object in the normal direction, the amplitudes of the wavefronts o1 and 02, scattered according to the symmetrical directions forming the same angle with the normal and both parallel to the xz plane, vary according to: o1 = exp (i F [X sin 8 + ~“(1 + cos e)]} (1) o,=exp{iF[-xsinB+wU(l+cos0)]} where w’(x, y) is a relative measure of the surface irregularity. We have supposed, for simplicity, that the intensity factor is not affected by the scattering process and is everywhere one. recorded on two The wavefronts (eqn (1)) are holographically holographic plates placed at a sufficient distance from the object and of sufficiently small aperture to justify the assumption that they collect the light from a single direction. After processing, the plates are accurately
Fig. 1.
Basic theoretical
representation.
L. Pirodda,
42
L. J. Griffiths
repositioned and illuminated by the conjugate of the reference beam. This can be done in the simplest way by a plane mirror placed behind the plate if the reference is a collimated beam. The complex conjugates of the amplitudes (eqn (1)) represent the wavefronts diffracted by the plate as they impinge back on the object. If the surface of the object is unchanged, those components of the light which are scattered from it in the z direction again form a plane wave. If the points of the object have undergone displacements s with components u and w according to x and z respectively, the two wavefronts, propagating in the positive direction of z are: h’
= exp {i F [u sin 8 + ~(1 + cos e)]}
(2) i, = exp {i$[-u Their interference
sin 13+ ~(1 + cos S)]]
gives an intensity distribution:
Thus, the intensity maxima and minima are loci of equal value for u, with sensitivity factor:
The intensity fringes obeying eqn (3) have the highest theoretical visibility and are not disturbed by the noise of any other carrying fringe. To observe them, one cannot simply look at the object surface from any direction, since they are not directly formed on that surface by the interference of the two conjugate waves but by the interference of the two correspondingly scattered waves. The fringes may be seen or recorded by placing the eye or the lens of a camera immediately behind a point where the two wavefronts scattered according to z are collected by a lens. Of course, the point and the lens may coincide with the pinhole and lens of the beam expander used in the recording step.
3 PRACTICAL
IMPLEMENTATIONS
OF THE TECHNIQUE
The procedure summarily described above may appear rather complicated. However, it can be easily implemented by means of a practical setup which is only slightly more complex than the usual ones used in
Conjugate-wave
holographic
~~ F: v
M2
Fig. 2(a).
interferometry
Practical setup for recording.
Fig. 2(b).
43
GMIM MINNTMHIOR)
Practical setup for observing.
real-time holography. The setup (Fig. 2(a), (b)) features only one holographic plate and a suitably placed mirror to provide a virtual substitute of the second plate. In the recording step the object is directly illuminated by a point source, F (the pinhole of a beam expander). The conditions of plane illuminating and recorded waves are thus relaxed and the errors involved will be analyzed later. In order to make the subsequent observation easier, a screen with a small circular aperture F (about 5 mm) is placed in front of the beam expander. The reference is a collimated beam and a gimbal mounted mirror provides for its conjugate. When recording, a screen is placed between plate and mirror, in order to eliminate the noise of the reflected light. After exposure the holographic plate is processed for high diffraction efficiency and accurately repositioned. A photo-thermoplastic device with in situ development appears particularly suited for this application and it was in fact employed in the experiments described in Ref. 8. At this stage the illuminating beam is discarded and the (deformed) object is observed by placing the eye or the lens of a camera immediately behind the aperture F (Fig. 2(b)). The conjugate waves impinging back upon the object are in fact preferably scattered according to the conjugate of the illuminating wave and hence are collected in the aperture. Another important function of the aperture is to minimize the noise in the system by shielding the observing lens from light scattered in directions other than the cone of rays converging on F.
44
L. Pirodda,
L. J. Grifiths
The setup in Fig. 2 allows the measurement of the displacement components u according to x. In general, the study of a bi-dimensional deformation field requires both components according to x and y to be measured in a single loading cycle. This problem can be solved by means of the more general setup represented in Fig. 3. Again we have a single actual holographic plate, which is now normal to z and with centre on z, and two pairs of mirrors, with normals in the xz plane and in the yz plane respectively, and symmetrically placed with respect to z. Another mirror, reflecting on opposite surfaces, is placed between the plate and the object. This double mirror has three functions: reflecting the reference on to the holographic plate, reflecting the illuminating beam on to the object, and screening the plate from the light coming directly from the object. In the recording step the hologram of the four symmetrical waves scattered at the same average angle is taken. When observing, by alternatively screening a couple of mirrors, either the u or the v fringes may be separately visualized for the same loading condition. A further generalization in the results may be obtained by means of an arrangement (obtained by a slight modification of the preceding one and hence not illustrated) where the two pairs of symmetrical mirrors are substituted by three mirrors so placed that the planes defined by their normal and the z axis form angles of 120”. In this case all three conjugate waves are allowed to impinge on the object at the same time, on observation. If the object is deformed, the interference of the three scattered waves converging on F produces three sets of fringes from which three components of the in-plane displacement vector can be calculated. The corresponding equations are formally analogous to the equations of three beams moire interferometry (see for instance Ref. 9). Examples of the results obtainable by the technique are shown in the photographs in Figs 4,5, and 6. Figure 4 presents, for comparison with other techniques, a classical case: the ring compressed diametrically. The fringes map the displacement component according to the direction of loading. Figure 5 shows the application on a carbon fibre composite plate having a cut and loaded by in-plane forces acting to open the cut. The average surface is plane, but very rough locally. The fringes map the displacement component normal to the cut. Figure 6 shows an application to a more complex shape, the model of a cylindrical vessel under internal pressure. The fringes map the axial displacement component. In all cases (Figs 4-6) a commercial silver halide holographic plate (Agfa 8E75 HD) was used. After development the plate was bleached and repositioned by means of a simple home made plate holder. The angle 0 was 45”, corresponding to a fringe
Conjugate-wave holographic interferometry
t Fig. 3(a).
HOLOGR.
Setup for simultaneously
PLATE
recording the u and v fringes.
PLATE
t M4
Fig. 3(b).
Setup for observing
the u and v fringes.
46
L. Pirodda,
Fig. 4.
u Displacement
under
diametral
Fig. 6.
L. .I. Grifiths
Fig. 5.
fringes on a ring compression.
u Displacement
u Displacement carbon
fringes
on a cylindrical
fringes fibre plate.
on a
vessel.
sensitivity factor of O-7A. The surfaces of the specimens were opaque white painted. A commercial reflex camera with 50 mm focal length loaded with Hilford HP4 film was used to take the photographs of the fringes. The laser employed was a 5 mW He-Ne. 4 DISCUSSION
AND
REMARKS
It is important to point out that the mirrors providing for a virtual holographic plate, like Ml in Fig. 2, or MA, MB in Fig. 3, can be quite
Conjugate-wave
holographic
interferometry
47
ordinary and inexpensive ones, since any wavefront distortions they introduce in the recording step are compensated in the observation step by the phase conjugation process, in the same way as this process compensates for the irregularities of the surface of the object. This is an obvious advantage, particularly when the object to be studied is large and the use of mirrors of optical quality would be quite expensive. The use of a single holographic plate, instead of two or more, presents an additional advantage, besides the obvious one of simplification. If commercial plates with a base of ordinary glass are used, the lack in uniformity of their thickness introduces phase distortions in the conjugate waves. Such distortions are different and hence produce errors in the recorded intensity pattern in the case of separate plates, while in the case of a single plate the phase distortions are almost the same, the only difference being due to the fact that the conjugate waves are not at the same angle with the plate. Even this effect tends to disappear when the average directions of the conjugate waves are symmetrical with respect to the normal to the plate, like in the case of Fig. 3. A further improvement in this respect may be achieved if in the observation step the diameter of the cross section of the reconstructing beam is reduced (for example by changing the collimating lens) to the minimum value consistent with a good resolution of the real images. In our experiences this diameter was about 1 cm, while, on recording, a reference of about 4 cm had been used. The latter expedient presents two additional advantages. A very small gimbal mounted mirror M4 is needed and the brightness of the conjugate waves is definitely increased, an appreciable gain when a small power laser is used. The mirror M4, providing for the conjugate of the reference, in principle could not be of optimum quality, since, in this case too, any introduced phase changes appear in both conjugate waves and do not affect the intensity of the observed wave. However, as we have just pointed out, the required size for this mirror is fairly small, and the use of a high quality unit is not a problem. Moreover, this makes the operation of collimating the reference easier and the adjustments sometimes needed in the direction of the same, easier. If a small power laser. is used, the power ratio of a variable beam splitter must be reversed between recording and observing in order to use all the power available for the latter operation. This operation may cause a small misalignment, which can be easily corrected by adjusting the plane of M4. Such adjustments may prove also useful in order to keep the fringes at their maximum visibility during the process of deformation of the object. For these reasons the gimbal mount of M4 should be precise and stable.
48
L. Pirodda,
L. J. Grifiths
The operation of collimating the reference beam does not present particular difficulties. The lens, as the other optical components, need not be of top quality. The alignment of the expander, lens, plate and mirror can be easily obtained stepwise and by the simple expedient of observing the reflected spots. The final step, exact collimation, is completed by displacing axially the expander until the light reflected on the mirror M3 and on the splitter form spots of the same size as the on-coming light. We now touch upon two interesting questions regarding the mechanism of fringe formation in the observation stage (Fig. 2(b)), which probably deserve a more exhaustive analysis. One of them is the fact, which we have assumed without any other proof than the experimental one, that the conjugate waves diffracted by the plate, upon being scattered by the object, reconstruct the conjugate of the illuminating waves. The other is the question of fringe localization. Clearly, in our case this does not constitute a problem, as it often comes about in classical holographic interferometry. In our case we have the so called ‘projected’ fringes, produced by the interference of two or more
Fig. 7.
Direct record of the projected fringes in the case of Fig. 5.
Conjugate-wave
holographic
interferometry
49
directional beams (the conjugates of the illuminating waves). The fringes can be observed or recorded by placing a screen or a sensitive medium anywhere in the region of superposition of the beams. Figure 7 shows, as an example, an interferogram concerning the same case as Fig. 5, which was obtained by placing the camera, without the lens, at some distance from the aperture F. Recording by a lens focused at the surface of the object is clearly to be preferred, since the details of the object are not lost and there is a definite correspondence between the fringes and the points of the object. In the case of Fig. 7, on the contrary, only the general form of the object is recognizable through the fringes that cover it, and this form is more or less distorted, depending on the distance of the recorder owing to the phase changes caused by deformation. As we have already pointed out, the fringe sensitivity factor (eqn (4)) has been determined on the assumption that the illuminating and recorded waves were collimated, while in the experimental setups this condition is relaxed. With reference to the scheme of Fig. 8, we may say that eqn (4) is exact only for the points of the object which lie,
Fig. 8.
Schematical
representation
of the basic setup.
50
L. Pirodda, L. J. Grifiths
like 0, on the y axis, while it is approximate for the other points, the approximation decreasing with increasing distance from that axis. In addition there is a correspondingly increasing sensitivity to the out-ofplane component of displacement w. A quantitative analysis of the above described situation is presented in the Appendix. We remark, finally, that the finite aperture a of the holographic plate (Fig. 8), in combination with its finite distance from the object, should have some influence on the sensitivity factor. In fact, we choose the central point C of the plate to ‘represent’ the whole of it, which is somewhat arbitrary, except for the points of the object who are in a plane of symmetry of the plate (the points of the y axis in Fig. 8). A quantitative analysis of this problem, however interesting, would have only an academic character since, as we have pointed out, it is highly advisable in the reconstruction stage to utilize a circular area of the plate with a small diameter, much smaller than the distance plate-object and than the lateral size of the object itself. As a consequence, the variations in the sensitivity factor would be negligible, whichever point of this area is used as a reference for evaluating it.
ACKNOWLEDGEMENT The research presented in this paper has been financially supported the Ministry of the University and Scientific Research, Italy.
by
REFERENCES 1. Sciammarella,
C. A. & Gilbert, J. A. A holographic-moire technique obtain separate patterns for components of displacement. Exp. Mech.,
to 16
(1976) 215.
2. Sciammarella, C. A. Holographic moire, an optical tool for the determination of displacements, strains, contours and slopes of surfaces. Opt. Erg., 21 (1982) 447. Boone, P. M. Surface deformation measurements using deformation following holograms. Nouv. Rev. d’Opt. Appl., 1 (1970). Post, D. Developments in moire interferometry. Opt. Eng., 21 (1982) 458. Pirodda, L. Strain analysis by grating interferometry. Opt. Lasers Eng. 5 (1984) 7.
Archbold, E., Burch J. M. & Ennos, A. Recording of in-plane surface displacement by double exposure speckle photography. Optica Acta, 17 (1970) 883. 7. Ennos A. Speckle interferometry.
North Holland,
1978, p. 235.
In Progress
in Optics XVI,
ed. E. Wolf.
Conjugate-wave
holographic
interferometry
51
for the measure8. Pirodda, L. Conjugate wave holographic interferometry ment of in-plane deformations. Appl. Opt., 26 (1989) 1842-4. 9. Walker, C. A. MoirC interferometry for strain analysis. Opt. Lasers Eng., 8 (1988) 213-62.
APPENDIX
We suppose that the recording step has been carried out in the setup represented in Fig. 2(a), and we reproduce in the scheme in Fig. 8, the geometric elements of this setup which are relevant for the analysis which follows. The points Cl and C2 in the latter figure are the central points of the actual and virtual plate, respectively. The scheme has been simplified by putting C2 at the same distance from 0 as Cl, with no practical consequence as far as the results are concerned. Let, in the observing phase, a point P of the object at distance x from 0 undergo a displacement with in-plane and out-of-plane components u and w respectively. Assuming, as usual, that u and w are of infinitesimal order compared to the distances m, c2p and PF, the corresponding phase changes introduced in the rays ClPF and C2PF may be expressed as follows: ~,=~~(usin~,-usin0~+wcos~,+wcos~,) (Al) ~,=i~(-usin8,-usin8,+wcos&+wcos0,) Setting 8, = f3,,,+ A8; e, + e, em=------.
2
’
and discarding the common terms, calculating the intensity, we have: $: = ~F[u
e2=e,-A8 e, - e, A8=----
(AZ)
2
which
are not meaningful
for
sin (em + A0) + w c0s (em + Ae)] 643)
2Jc & = ‘T [-u sin (em - A6) + w cos (e, - Ae)] Substituting the eqn (A2), performing some trigonometrical and algebraic calculation and again discarding the common terms, we obtain
L. Pirodda,
52
for the amplitudes
L. J. Grifiths
of the two rays:
&=exp
[
&=exp
[
~~(usinB,cos*O-wsin8,sin*6))
I
1
iF(-usinB,cos*B+wsin8,sin*8)
(A4)
and for the resultant intensity: z=2[1+
cos g
L
(2u sin 8, cos A8 + 2w sin 8, sin A@
h
According to eqn (A5), the intensity intervals Au and w satisfy the condition:
maxima
Au cos A8 + Aw sin A8 =
1 (A3
or minima
occur
3L 2 sin 8,
at
646)
and the absolute and relative sensitivity factors for u and w are: Au =
A
Aw=
2 sin 0, cos A8’
A
2sin B,sinAe’
Au -=tanAtI Aw
By means of eqn (A7) it is possible to estimate the error involved when the sensitivity for u is evaluated by eqn (4) and the sensitivity for w is ignored. From Fig. 8 we have: 8, = e - 6,;
tan 6, =
xl1 cos 9 2: 6.” 1 - x/l sin e
8, = e + 6,
tan S, =
xil cos e 21 6, 1 + x/l sin 8
8,=B-$sinBcosB Ae=;cos All the above approximations that Ix/Z1<< 1.
8
are supposed
(AS) valid on the assumption
Setting for instance: Ix/Z\ = 0.1;
8 = 45”
Conjugate-wave
holographic
intetferometry
53
we obtain from the eqns (A7) and (A8): Au=-
A
2 sin 8
1.007;
Aw=-
A
2 sin 8
14.22
The error is only 0.7% as far as the absolute sensitivity for u is concerned, and the latter sensitivity is by more than one order of magnitude larger than the sensitivity for w.