A speckle interferometer including a beam splitter, a mirror in the object beam arm, a phase-conjugate mirror in the reference beam arm, a converging lens and a photographic film. Laser light scattered retro-reflectively from a rough surface under investigation and passed through an imaging lens illuminates the interferometer. Fringes occur upon sandwiching a pair of exposures of the interference pattern made before and after deformation of the rough surface. The relative magnitude of the displacements from the original position at different points of the surface can be determined from the position of the fringes.

BACKGROUND OF THE INVENTION 
This invention relates to interferometry, and more particularly to an 
interferometry apparatus and method utilizing a phase-conjugate mirror. 
Optical interferometers are known which make use of the interference 
phenomenon known as the "speckle effect"--the speckled pattern of laser 
light when reflected from a rough surface. One prior-art speckle 
interferometer takes the form of a Michelson-type instrument in which the 
mirror of one arm is replaced by a scattering surface. Provided that the 
coherence of the laser light is sufficiently high, speckles are formed by 
interference between light from the reference-beam arm and the light 
scattered retro-reflectively from the rough surface. Translating the 
scattering surface in the direction of the incident light causes each 
speckle to vary in brightness cyclically, from light to dark, 
independently from its neighbor, in a similar way that points on the 
equivalent Michelson mirror would appear to fluctuate in brightness as 
interference fringes sweep across it when the mirror is moved. The visual 
speckle interferometer can thus be used for detecting movement, but not 
for measuring the displacement that has taken place. Further, since a 
plane wave with uniform amplitude serves as the reference for the speckle 
wave, the plane wave cannot be matched to the random variation of 
amplitude over the speckle pattern. Thus, the visibility (or contrast) of 
the interference pattern cannot be made the same throughout the field. 
SUMMARY OF THE INVENTION 
It is therefore an object of the present invention to study the deformation 
of the surface of an object under stress. 
It is a further object of the present invention to measure the relative 
magnitude of the displacements from the original position at different 
points on the surface of an object under stress. 
It is yet another object of the present invention to provide an improved 
interferometry apparatus utilizing the speckle effect wherein the contrast 
of the interference pattern is the same everywhere in the field. 
The objects of the present invention are achieved by an interferometer 
design using optical phase conjugation to yield fringes of good contrast 
and standard interferometric sensitivity. The speckle interferometer 
includes a beam splitter, a mirror in the object beam arm, a 
phase-conjugate mirror in the reference beam arm, a converging lens and a 
photographic film. Laser light scattered retro-reflectively from a rough 
surface under investigation and passed through a collimating lens 
illuminates the interferometer. Fringes occur upon sandwiching a pair of 
exposures of the interference pattern made before and after deformation of 
the rough surface. No additional spatial filtering step is necessary. The 
relative magnitude of the displacements from the original position at 
different points of the surface can be determined from the position of the 
fringes. 
The foregoing as well as other objects, features and advantages of the 
present invention will become more apparent from the following detailed 
description when taken in conjunction with the appended drawings.

DESCRIPTION OF THE PREFERRED EMBODIMENT 
Referring to FIG. 1, the apparatus for studying the deformation of the 
surface of an object under stress, such as a loaded cantilever, includes a 
monochromatic laser source 11 which illuminates the surface 13 through a 
beam splitter 15. The surface 13 is assumed to be optically rough and 
therefore produces a speckle pattern when it is illuminated by the laser 
light. The light scattered by the points x, y of surface 13 is reflected 
by the beam splitter 15 into an imaging lens 17 to provide the input beam 
to the interferometer device. A beam splitter 19 divides the input beam 
into a reference beam 21 and an object beam 23. The reference beam 21 
forms an image of surface 13 in the plane of a phase-conjugate mirror 25 
and is reflected back to the beam splitter 19 by the phase-conjugate 
mirror 25 which reverses the phase of the beam. Suitable phase-conjugate 
mirrors are well-known in the art, and are discussed, for example, in the 
article "Phase Conjugate Optics" by J. AuYeung et. al., Optics News 
(Spring 1979) pp. 13-17. The object beam 23 forms an image of surface 13 
in a plane normal to the beam and intersecting a plane mirror 27 tilted at 
an angle .beta. with respect to the plane and is reflected back to the 
beam splitter 19 by the mirror 27. The two beams 21 and 23 recombine at 
the beam splitter 19 and are photographed on film 29 through a converging 
lens 31 which forms an image of mirrors 25 and 27 at unit magnification 
onto film 29. At the photographic film 29, the speckle patterns produced 
by the two beams are allowed to interfere to produce a resultant image, 
the points x, y of which correspond to the points of surface 13. The 
recorded intensity in the film plane is given by: 
EQU I(x,y)=.vertline.u.sub.o +u.sub.r .vertline..sup.2 (1) 
where u.sub.o and u.sub.r are the speckle patterns produced by the object 
beam 23 and the reference beam 21 respectively and are given by: 
EQU u.sub.o =.vertline.A(x,y).vertline.exp{i[.theta.(x,y)+.alpha.x]}(2) 
EQU u.sub.r =.vertline.A(x,y).vertline.exp{i[-.theta.(x,y)]} (3) 
In these expressions, A(x,y) and .theta.(x,y) are functions of position in 
the film plane. A(x,y) represents the square root of the intensity of the 
speckle patterns. The phase of u.sub.o is .theta.(x,y)+.alpha.x; the phase 
of u.sub.r is -.theta.(x,y). It is assumed that the optical path lengths 
in the interferometer device are matched except for the path difference 
introduced by the tilt of the mirror 27 which gives rise to a phase 
difference in the expression for u.sub.o of (2.pi./.lambda.) 
(2.beta.)x.tbd..alpha.x for small .beta.. The minus sign before 
.theta.(x,y) in the expression for u.sub.r is due to the operation of the 
phase-conjugate mirror 25. 
Substituting equations (2) and (3) into equation (1), the following 
expression is obtained for the recorded intensity: 
EQU 1=2.vertline.A(x,y).vertline..sup.2 {1+cos [2.theta.(x,y)+.alpha.x]}(4) 
Note that in equation (4), the intensity of the speckle pattern 
.vertline.A(x,y).vertline..sup.2 is multiplied by an expression which has 
a modulation index of unity but depends on the phase of the light. Thus, 
in applications where one wants to measure the phase variation across the 
beam of the electric field vector of a light wave of varying amplitude 
such as a speckle pattern, the phase conjugate of the wave may be used 
instead of a plane reference wave in the production of interference 
patterns for the measurement of the phase variation. Since the amplitudes 
of the light wave and its conjugate are the same, the visibility of the 
interference pattern is automatically one. This would not be the case if a 
plane wave with uniform amplitude were used as the reference for the 
speckle pattern. Because of the random variation of amplitude over the 
speckle pattern, the plane wave could not be matched to it. Thus, 
interference with the phase conjugate wave provides a relatively simple 
way to measure the phase of a speckle pattern. 
Referring to the flow chart shown in FIG. 2, the method of studying the 
deformation of the surface 13 of the object under stress includes a first 
step 33 of recording the resultant speckle pattern before the relevant 
deformation of the surface 13. The recorded intensity at the film 29 is 
then given by: 
EQU I.sub.1 =2.vertline.A.vertline..sup.2 [1+cos (2.theta..sub.i +.alpha..sub.i 
x)] (5) 
where the functional dependence of A and .theta. on x and y has been 
suppressed. The expression (2.theta..sub.i +.alpha..sub.i x) is the phase 
factor with the surface 13 in its initial state (i) and the plane mirror 
27 tilted at an angle .beta..sub.i. 
Next, the second step 35 is performed hereby the plane mirror 27 is tilted 
at an angle .beta..sub.d different from .beta..sub.i. 
In the third step 37, the resultant speckle pattern is recorded after the 
relevant deformation of the surface 13. The recorded intensity at the film 
29 is then given by: 
EQU I.sub.d =2.vertline.A.vertline..sup.2 [1+cos (2.theta.+.alpha..sub.d x)](6) 
where the expression (2.theta.+.alpha..sub.d x) is the phase factor with 
the surface 13 in its deformed state (d) and the plane mirror 27 tilted at 
the angle .beta..sub.d. The change in .theta. arises from the altered 
optical path length due to the deformation of the surface 13. The change 
in .alpha.x arises from the altered optical path length due to rotation of 
the plane mirror 27. 
The fourth step 39 comprises superposing a set of transparencies of the two 
recordings in front of a source of light to observe the transmitted light 
intensity. The transmitted light intensity I.sub.T is proportional to the 
product of the separate intensities, i.e., I.sub.T =kI.sub.i 
.multidot.I.sub.d, where k is the proportionality constant. Substituting 
equations (5) and (6), I.sub.T is given by: 
##EQU1## 
where 
EQU B=.theta..sub.i +.theta..sub.d +1/2(.alpha..sub.i +.alpha..sub.d)x (9) 
EQU C=.theta..sub.i -.theta..sub.d +1/2(.alpha..sub.i -.alpha..sub.d)x (10) 
Since the speckles are very small, a local spatial average can be performed 
on each term in equation (8). In an ensemble average sense, cos.sup.2 B 
averages to 1/2 assuming a uniform distribution of the phases 
.theta..sub.i and .theta..sub.d between .pi. and -.pi., while the 
cross-term 2 cos B cos C averages to zero. Then equation (8) becomes 
##EQU2## 
The transmitted light intensity is a speckle pattern having a series of 
bands, alternately light and dark, appearing on it. The visibility (or 
contrast) of the bands is the same throughout the field, namely, the 
visibility 
##EQU3## 
independent of position. Further, it will be appreciated that the factor 
of 2 in front of (.theta..sub.i -.theta..sub.d) in the argument of the 
cosine doubles the sensitivity of the band pattern to a change in phase. 
Each of the bright bands corresponds to a locus of points in the film plane 
for which the cosine term in equation (12) is a maximum. Referring to FIG. 
3, the n.sup.th order bright band (viz., the one for which the argument of 
the cosine term equals n 2.pi.) is schematically illustrated as the curve 
OPQ in the film plane. Point P on the band satisfies the relation 
EQU 2.delta..theta.(x.sub.n (y.sub.n),y.sub.n)+.delta..alpha..multidot.x.sub.n 
(y.sub.n)=n2.pi. (13) 
where .delta..theta.=.theta.-.theta..sub.d, .delta..alpha.=.alpha..sub.i 
-.alpha..sub.d and the subscript n refers to the n.sup.th order band. The 
dependence of .delta..theta. on x and y and the dependence of x on y along 
the band contour has been made explicit. .delta..theta. is a function 
whose amplitude varies in accordance with the normal displacements .DELTA. 
at different parts of the surface 13 from the original position. 
.delta..theta. is zero at points in the film plane corresponding to points 
of the surface 13 where the displacement .DELTA. is zero; a value for 
.delta..theta. of 2.pi. radians corresponds to a normal displacement 
.DELTA. at the surface 13 of one-half the wavelength of the laser light. 
Equation (13) can be rewritten in terms of .DELTA. as follows: 
EQU 4.DELTA.(x.sub.n (y.sub.n),y.sub.n)+2.delta..beta..multidot.x.sub.n 
(y.sub.n)=n.lambda. (14) 
Here, .delta..alpha. has been replaced by 
.delta..alpha.=2.pi./.lambda.(2.delta..beta.) where 
.delta..beta..tbd..beta..sub.i -.beta..sub.d ; and .delta..theta. has been 
replaced by 2.pi./.lambda.(2.DELTA.), where the factor of 2 arises because 
the displacement is seen in reflection. 
Point 0 on the band satisfies the relation 
EQU 4.DELTA.(x.sub.n (0),0)+2.delta..beta..multidot.x.sub.n (0)=n.lambda.(15) 
Subtracting equation (15) from equation (14), the normal displacement of a 
point on the surface 13 corresponding to point P on the band relative to 
the normal displacement of a point on the surface 13 corresponding to 
point O on the band, is given by: 
EQU .DELTA.(x.sub.n (y.sub.n),y.sub.n)-.DELTA.(x.sub.n 
(0),0)=-(1/2).delta..beta.[x.sub.n (y.sub.n)-x.sub.n (0)] (16) 
In the fifth step 41, the difference in tilt angles, .delta..beta. is 
determined. From a region of the film plane where the bands are straight, 
corresponding to no displacement of the surface 13, i.e., .DELTA.(x,y)=0, 
one can find the value of .delta..beta. from the distance between two 
bands. Referring to FIG. 4, the m.sup.th and (m+1)th order bands for such 
a region of the film plane are illustrated as lines O'P'Q' and O"P"Q". 
From equation (14), point P on the m.sup.th order band satisfies the 
relation: 
EQU 2.delta..beta..multidot.x.sub.m =m.lambda. (17) 
where x.sub.m is the x coordinate of the m.sup.th order band. Point P" on 
the (m+1)th order band satisfies the relation 
EQU 2.delta..beta..multidot.x.sub.m+1 =(m+1).lambda. (18) 
where x.sub.m+1 is the x coordinate of the (m+1)th order band. Subtracting 
equation (18) from equation (17), the difference in tilt angles is given 
by: 
##EQU4## 
If the bands are not straight anywhere in the field, .delta..beta. can in 
principle be found from a pair of preliminary recordings taken before the 
surface 13 is deformed. 
The sixth step 43 is performed to determine the normal displacement of a 
point on the surface 13 corresponding to point P on the band, relative to 
the normal displacement of a point on the surface 13 corresponding to 
point O on the band from equation (16) by multiplying .delta..beta./2 by 
the distance between the two points O and P along the x direction. 
Obviously, many modifications and variations of the present invention are 
possible in light of the above teachings. It is therefore to be understood 
that within the scope of the appended claims the invention may be 
practiced otherwise than as specifically described.