rock probe deformation measured by holographic interferometry

5
Rock probe deformation measured by holographic interferometry W. FINK, P. A. BiiGER & L. SCHEPENS National Physical Research Laboratory, C.S.I.R., Pretoria, South Africa Mechanical pressure-induced deformations of cylindrical rock probes were measured by holographic time-lapse interferometry. As it was generally impossible to find any zeroth-order fringes, a moving-fringe evaluation method as first given by Bench-Bruevich was used. Small displacements parallel to the hologram plane could thus be measured with very high accuracy. The method is applicable to any type of local displacement of surface elements, limited only by the solid angle sub- tended at the hologram by the object. WHEN INVESTIGATING the behaviour of rock ma- terials under mechanical pressure deformations of samples need to be measured by a non-contact method. We found holographic interferometry best suited for precision measurements of object point shifts in arbitrary directions. Both real-time and time-lapse interferometry were tried, but the latter method was preferred for our work because of its relative simplicity and its less stringent require- ments with respect to long-time stability of the experimental set-up. Because it was usually impossible to find any point on the probe or on the pressure-applying apparatus which did not move, the observed fringe pattern itself did not give any conclusive indications about object deformations. Therefore a special evaluation and calculation method was used, permitting a rela- tively simple and versatile evaluation of a time- lapse interference hologram. THEORY Holography enables reproduction of the wavefront of light scattered by three-dimensional objects. It therefore allows direct comparison of the object and the reconstructed wavefronts. Any displacements of the object between the time when the first hologram was taken and the time of comparison cause inter- ference of the two wavefrontsl,2. In the time-lapse case this comparison is made between two wave- fronts recorded at different times on the same holo- graphic plate and subsequently reconstructed simul- taneously. This method avoids critical alignment of the hologram with respect to the object as required for live or real time observations, indeed such an Our referee points out that several aspects of this problem are discussed in recent paper-6 in Optics Technology.-Ed. Fig. 1 Coherent light from a source L is incident on an object point P andscattered from there onto the hologram H. Exposure 1 is made. Then P shifts to Q, and exposure 2 is made. alignment proved to be impossible because of insta- bilities in the pressure-applying device. To draw conclusions from the position and shape of the interference pattern (which in general is not located on the object) the wavefront changes caused by changing the object shape or position must be analysed. Considering the most general conditions (Fig. 1) and using the concept of homologous rays, the phase difference between the beams scattered from point P and its position Q after shifting is3 1 where k is the propagation vector of the light of wavelength X, k = 2n ke, ke the unit vector in propa- gation direction, an% d the displacement vector. This is the basic relation necessary for analysis of static displacements in any holographic interferogram. In general we have no a priori knowledge of the dis- placement vector d except that its absolute value d 146 Optics and Laser Technology August 1970

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Rock probe deformation measured by holographic interferometry

W. FINK, P. A. BiiGER & L. SCHEPENS

National Physical Research Laboratory, C.S.I.R., Pretoria, South Africa

Mechanical pressure-induced deformations of cylindrical rock probes were measured by holographic time-lapse interferometry. As it was generally impossible to find any zeroth-order fringes, a moving-fringe evaluation method as first given by Bench-Bruevich was used. Small displacements parallel to the hologram plane could thus be measured with very high accuracy. The method is applicable to any type of local displacement of surface elements, limited only by the solid angle sub- tended at the hologram by the object.

WHEN INVESTIGATING the behaviour of rock ma- terials under mechanical pressure deformations of samples need to be measured by a non-contact method. We found holographic interferometry best suited for precision measurements of object point shifts in arbitrary directions. Both real-time and time-lapse interferometry were tried, but the latter method was preferred for our work because of its relative simplicity and its less stringent require- ments with respect to long-time stability of the experimental set-up.

Because it was usually impossible to find any point on the probe or on the pressure-applying apparatus which did not move, the observed fringe pattern itself did not give any conclusive indications about object deformations. Therefore a special evaluation and calculation method was used, permitting a rela- tively simple and versatile evaluation of a time- lapse interference hologram.

THEORY

Holography enables reproduction of the wavefront of light scattered by three-dimensional objects. It therefore allows direct comparison of the object and the reconstructed wavefronts. Any displacements of the object between the time when the first hologram was taken and the time of comparison cause inter- ference of the two wavefrontsl,2. In the time-lapse case this comparison is made between two wave- fronts recorded at different times on the same holo- graphic plate and subsequently reconstructed simul- taneously. This method avoids critical alignment of the hologram with respect to the object as required for live or real time observations, indeed such an

Our referee points out that several aspects of this problem are discussed in recent paper-6 in Optics Technology.-Ed.

Fig. 1 Coherent light from a source L is incident on an object point P andscattered from there onto the hologram H. Exposure 1 is made. Then P shifts to Q, and exposure 2 is made.

alignment proved to be impossible because of insta- bilities in the pressure-applying device.

To draw conclusions from the position and shape of the interference pattern (which in general is not located on the object) the wavefront changes caused by changing the object shape or position must be analysed. Considering the most general conditions (Fig. 1) and using the concept of homologous rays, the phase difference between the beams scattered from point P and its position Q after shifting is3

1

where k is the propagation vector of the light of

wavelength X, k = 2n ke, ke the unit vector in propa-

gation direction, an% d the displacement vector. This is the basic relation necessary for analysis of static displacements in any holographic interferogram.

In general we have no a priori knowledge of the dis- placement vector d except that its absolute value d

146 Optics and Laser Technology August 1970

must be much smaller than the distance from the object to the hologram in order not to invalidate equation 1. Principally there are two different methods of evaluating the observed fringe patterns. If zeroth-order fringes can be identified (i.e. fringes which remain fixed relative to the object surface independent from the observation direction), then a multiple hologram approach4 can be used. If there are no zeroth-order fringes, as is the case here, a single hologram method similar to that of Aleksan- drov C Bench-Bruevich must be used. In the fol- lowing, a notation similar to that of Sollids is used,

According to Fig. 2, light is scattered from point P to the four points Ai, i =0, 1,2,3. The vectors of unit length in these directions are

&ie = &/lPAiI = fi/lPil9

and the propagation vectors ki are ki = 2nkie/X. Using eqn 1, the phase differences l’n the directions ki are bi = (lti - ke)d. The image of P is thus built up of rays with different directions (within the aper- ture determined by the hologram size). To avoid averaging the interference, the values of 6 should vary by an amount much smaller than r. A small aperture might need to be introduced so that a sharp image of the object surface and the interference pattern can be obtained simultaneously.

Consider the difference vectors &i between the four vectors ki. Any three of the six possible vectors provide a reference co-ordinate system independent of the illumination direction _k, (as long as they are

Fig. 2 The virtual image of P is observed from four directions A P, A,P, AaP and A

7‘ P.

3 Moving

the detector e.g., the human eye from A, to A, will cause NAoAl fringes to pass P. Re-

peating this procedure by going from A, to Aa and A, will give three difference vectors.& - k~ - & = Ai, i = 1,2,3, which form a basrs for the translation vector d, provided they are not coplanar.

not co-planar). Choosing the three difference vectors formed with k. gives

fl = bi - 6, = (ki - ko)d = Ai d9 i = 1,2,3.

The phase difference fl can be obtained by counting the number NA ,, of fringes which pass the point P when the obse&e$ moves from A, to Ai; SO fi = 2R NA,Ai’

The sign of NA A, remains unknown; it is therefore 0 1

necessary to label arbitrarily two adjacent fringes on either side of P as positive and negative and to stick to that convention for all following measure- ments. If Cri are the angles of d with respect to the vectors Ai, and 2Xi the angles between k. and go, it

-5 follows from geometrical considerations that

2n fi = 2 - SinXi d COS “1 (d = 2). 2

A

The d COSCri are thus the components li of d in the co-ordinate system Al:

NAoAi A d cosai = 11 = --.

SinXi 2

d can thus be found by a measurement of NA A, and a

knowledge of the geometry. To evaluate sinii fe note that Pioo = PiPo COSXi. Thus

-_ _-

COS2Xi =

%a0 + bibo + CiCo

Pip0

As it is more convenient, in most cases, to have d given in an orthogonal system, transformation for- mulae are given below. In the system e, r’:Aa’!?Aa

kA i unity vector of Ai) the vector d is given as d =

[Ii, z,z,]. In the orthogonal system el, e2, ea, $iek =

bik, d is given as d = [di, da, da]. Both systems

shall be centered at P, hence the co-ordinates of the points in question in the system gi are P(0, 0, 0), A, (a,, b,, Co), Ai(ai, bi, Ci); i = 1,2,3. Propagation vectors are

k -0 =[a b, d.]?L- 0’ 9 0

h PO ’

ki = [ai, bi, Cl] e 1; A Pi

Pi=I&I =J ai 2 + bi’ + Ci2

Unit vectors tii are si = Ai/ IAl I =[Ui, Vi, Wi].

E&pressed in co-ordinates, the difference vectors & = ki - kc become

a0 bi bo ci co 277 -,---?---- 1 * - Po Pi PO Pi PO X ’

3

Optics and Laser Technology August 1970 147

their absolute value 141 being

,&+k A$ + (y$) ci c0 + --- ( 11 y2 2n

Pi PO

Fi 2n ‘h= *h

The direction cosines of eAi are thus

“i ao bi b. ci co --- --- ---

Pi oo Pi Po Ui=F;Vi=_;wi=

Pi PO

l-i ri ’

The components of i in the rectangular system $1 can now be calculated as

3 3 3

d, = b kuk; da = L lkvk; d, = x lkwk, k=l k=l k=i

giving d = IdI = Jdf + d$ + d$. To avoid determination of the relative positions of all points PI in the image (1 = 1 . . . . . M, M number of investigated points), and to take into account some magnification m which may be desirable in recon- structing the image, one can proceed as follows: the positions of the points PI ({I, 71, [I) on the object are noted relative to one object point 0 chosen as centre, O(0, 0,O). By reconstructing the hologram, the object points PI are transformed into virtual image points PIv. Considering only a lateral magnification m, the image point co-ordinates will be PIv (51, rn?lI, mQ).

What one has to do in evaluating an interference holo- gram is now to measure or calculate the magnifica- tion m, determine the distance a from a scanning plane (plane in which the co-ordinates of the points Ali are defined) to the point Ov in the image which gives the a-co-ordinate for all the points All (aI1 = a for all 2 and all i), and to determine the co-ordi- nates bli and cIi of Al1 together with the fringe numbers NAoAl 1 . (The index 1 indicates to which

object point PI the Al i and NAoAl i belong to, i going from 1 to 3 for each 1).

mirror 4%

Laser

splitter

Fig. 3 Experimental set-up.

The vector &l = P&, is then

ell= [all-- 51,bll-- w~7Cli-mC~l.

This is analogous with pi = [ai, bi, Ci] of equation 3. Applying the arguments following equation 3, the resulting components of translation of the point PI are found. These are now given in the image as

;&vord$v, d13V. T o get the probe translation one has

%” dll = dliV;d12 = -’ d13V

m ,dls =m

giving dl = id$ = Jdl12 + dla2 + dla2.

EXPERIMENTAL WORK

Hologram construction A schematic drawing of the set-up is given in Fig. 3. The coherent beam origin- ating from the laser is divided by the beam splitter. The object beam is directed by a mirror onto the rock probe, the monitor plate mounted on the top of the pressure applying apparatus, and onto the metal base cylinder of the jack. From all these points, scattered light reaches the hologram plate. There interference takes place with the reference beam which is incident directly onto the hologram. The path lengths of the beams from the beam splitter to the hologram should be equal to within the coherence length of the laser. Using a He-Ne 20W single mode laser the path length is 230*5cm. The probe is 2.2cm in diameter and 4.4cm long.

Evaluation procedure The hologram is reconstructed by illuminating the processed plate with a laser beam similar to the reference beam used in construc- tion. The virtual image is thus observed at the object’s place. The radius of divergence of the beam determines the magnification. As schematically indicated in Fig. 3, observation is done through a transparent screen S, with a rectangular metric grid printed onto it. A similar screen S, is placed paral- lel to S, in the virtual image plane, and is illumi- nated from the back. It is thus possible to determine the lateral magnification m, and to choose certain points in the image of the probe, using a co-ordinate system with the e,-axis perpendicular to S,.

Each point PI” is then observed from four arbitrarily chosen directions which are given by the four points AI,,, Air, AI2 and Al3 as indicated in Fig. 2. To see the image and the fringes sharply at the same time, it might be necessary to do the observation through a small movable aperture. In observing PI” in the direction AloPlv, two adjacent fringes on either side are arbitrarily assigned +l and -1 respectively. Then the fringes are counted which pass Plv while moving from direction AIoPIV to AllPlv. This gives

N%o%l with the proper sign. The same procedure

is repeated by going from AloPIV to AlaPlv and from AloPIV to A13Plv. In doing the same for the other points PI’, the convention about the sign of the fringe movement has of course to be maintained. Process- ing of the observed data was done by using a simple Fortran statement.

148 Optics and Laser Technology A1~w1 1970

F&.

4a

4a & b. A typical example of tran&WonfJ observable with the present device. The hori- zontal arrows indicate a tranhtioninaplane perpendicular to the paper; left-pointhg arrows refer to shifts along the positive Q- axis,i.e.aut of the paper plane.

4a: Initial compression stage. Probe,bctttom metal cylinder and monitor plate were mov- ing upwards and towards the left, while the

CONCLUSION

In principle the described method is applicable to any type of local displacement of surface elements and is independent on the localization of the inter- ference pattern with respect to the object. In prac- tice a limitation is given by the solid angle subtended at the hologram by the object. In any practical case the vectors A, AZ A3 are not very far from being coplanar. Only the components of c in the plane of the hologram can be measured with accuracy.

In the case of cylindrical probes discussed here this limitation is not severe as normally there is a high degree of symmetry around the cylinder axis. Any deformation in the radial direction should therefore be detectable from points lying at the edges of the observed image while poink aiong reveal tangential displacement.

the middle line

4b

I- L I _

upper metal cylider moved almost straight upwards.

41: Asthepressure iBincreased,theprobeiS

bent. Non-equal lengths of tie displacement vectors indicate mechanical stresses in the probe. A further increase of pressure breaks the probe,the displacements being quite irregular.

There is an apparent contradiction: one normally expects the biggest phase shifts to be connected with translations normal to the beam directions. In the case of near normal incidence and observation, a translation normal to the hologram would in fact pro- duce the biggest effect, as long as we are using only one single observation direction.

If it is assumed that in observing the fringe pattern it is possible to determine a shift of one quarter of a fringe with reasonable accuracy, it follows from equation 2 that the smallest shift d which can be mea- sured is

d . --h._i_. mm - 8 sinx

This is true for a shift parallel to the vector 4 con- cerned. Considering a practical case having a

Optics and Laser Technology August 1970 149

5a

5b

Fig. 5a, b, c. Examples of different fringe patterns, a and b corresponding to Fig. 4a, b, respec- tively. 5c shows the probe just before crack- ing. (The fringe pattern corresponding to breakage is not shown because it is localized at a distance from the probe image which did not allow fringes and probe to be photo- graphed sharply at the same time).

10 x 10cma sensitive plate area 8. 5cm distant from the object, dmin will be -b/4. Even with experimental error, it should be feasible to measure translations in the hologram plane appreciably less than lpm. The maximal limit of translation will be set by the closest fringe spacing that can be measured, and therefore will depend on the observing technique.

Acknowledgements. It is a pleasure to acknowledge the stimulating interest of Dr G. J. Ritter and also the valuable assistance of L. Dicks in doing the calculations.

RE FJ3RENCES

1 Horman, M. H. Appl.Opt. vol. 4, 1965. 333. p.

2 Hildebrand, B. P. & Haines, K. A. Appl. Opt. vol. 5, 1966. 172. p.

3 Sollid, J. E. Appl.Opt. vol. 8. 1969. 1587. p.

4 Ennos,A. E. J. Sci. Instrum. vol. 21.1968. p. 731.

5 Aleksandrov, E. B. & Bench-Bruevich, A. M. Sov. Phys.-Techn. Phys. vol. 12.1967. p. 258.

6 Gates, J. W. C. Holographic measurement of surface distortion in three dimensions. Opt. Technol. vol 1.1969. 247-50. p.

150 Optics and Laser Technology August 1970