JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Fringe interpretation for hologram interferometry of rigid-body motionsand homogeneous deformations

Karl A. StetsonInstrumentation Laboratory, United Aircraft Research Laboratories, East Hartford, Connecticut 06108

(Received 14 July 1973)

Previous theoretical work on hologram interferometry is reduced, in this paper, to a number ofguide lines for fringe interpretation. One of the most helpful is that, for any combination of rigid-body motion and homogeneous deformation, fringes appear on the object as though it intersected aset of equally spaced lamina perpendicular to a specific fringe vector. This elucidates the role thatobject shape plays in determining fringe patterns. Fringe localization for slit-apertured opticalsystems and fringe parallax are compared and the conditions for their equivalence are presented.The line of fringe localization that results from observation with a large two-dimensional aperturein an optical system is shown to have a simple geometrical relationship to the axis of the helicalmotion that may characterize any rigid-body motion. The orientation of fringes in their region oflocalization can be used to differentiate between rigid-body motions and some homogeneousdeformations.

Index Headings: Holography; Interferometry.

In recent years, the problem of extracting displacementfrom the fringes of hologram interferometry has beensolved in a number of different ways. Those who wouldmake use of hologram interferometry have often feltconfused, however, by the variety of approachesavailable and the apparent lack of unity between thesemethods. This has been the inevitable result of thedegree to which holography enlarged the field ofinterferometry and enriched it with new phenomena,and the enthusiasm with which researchers haveapproached it. When all is said and done, however, aunique deformation of an object, for a given illumina-tion, does produce a unique three-dimensional inter-ferogram within a hologram reconstruction, which canbe interpreted to determine the vectorial deformation ofthe object's surface. This paper will discuss the fringesand fringe localization associated with rigid-bodymotions of an object. The objective is twofold. First,this offers a convenient context within which to discussfringe parallax and fringe localization. These twophenomena provide the primary method upon whichmost systems of three-dimensional fringe interpretationsare based. Second, although most practical applicationsof hologram interferometry involve deformation of theobject under test, various parts of the object may moveapproximately as rigid bodies while other parts undergostrain. Thus, rigid-body motions and their fringephenomena can help in understanding more-generaldeformations on a region-by-region basis. Finally, theconcepts developed can be extended to homogeneousdeformation of an object.

The first topics to be considered in this paper are theseparability of object motions and the observer-projec-tion theorem. These concepts are reviewed to establishthe basis for a discussion of the way fringes appear toan observer who looks at the reconstruction of a holo-gram recording wherein the object undergoes a purerotation. Next, helical motions of the object are

1

considered with respect to the fringes they create in thereconstruction of a hologram. This is followed by acomparison of fringe parallax and fringe localizationwith optical systems having large one-dimensionalapertures. Then, the topic of fringe localization foroptical systems having large two-dimensional aperturesis considered. The concepts previously developed arenext applied to homogeneous deformations of objectsand discussions are given of isotropic expansion, linearexpansion, and expansion coupled with orthogonalcontraction. Finally, a set of general formulas arederived for the line of fringe localization for observationwith large two-dimensional apertures.

SEPARABLE MOTIONS OF OBJECTS

This paper will be concerned only with objectmotions that are separable." 2 This means that thevarious points on the object move approximatelyalong various straight lines and according to the sametime function during the exposure of the hologram.For such motions, the movement of the object can beexpressed as a vector displacement L times a functionof time f(t); and further, the fringe loci and fringelocalization (as well as fringe parallax, which isanalogous to fringe localization) are uniquely deter-mined by the geometry of the object motion alone.The time function, f(t), will determine the particulartype of fringe function that multiplies the image of theobject. The fringe function, M(Q), can be expressed asthe Fourier integral of density distribution of theobject's motion and therefore may be called a character-istic function. The argument of the fringe function, Q,is a scalar function, constant values of which generateloci of fringes in space. Thus, it may be called thefringe-locus function. Perhaps the two most commonfringe functions are M=Jo(Q), for sinusoidal motionsof an object, and M=cos(Q), for two discrete positionsof an object. So that this discussion may be consistent

Copyright © 1974 by the Optical Society of America.

VOLUME 64, NUMBER I JANUARY 1974

KARL A. STETSON

Y

x

FIG. 1. Observer-projection theorem. If fringes are observedlocalized in plane Pi, they may be projected to the object plane,PO, radially from the center of transmittance of the aperture ofthe observing optical system.

with previous discussions of these two cases, let L bethe vibration amplitude for the sinusoidal motion andhalf the total displacement between the discretepositions. In the latter case, assume that the objectmoves from - L to + L with the central position beingconsidered as rest. For all double-exposure hologramsof two discrete object positions, the object motion maybe defined as separable because it is immaterial whatpath the object takes between the two positions. Thesame argument applies to concomitant hologram inter-ferometry where, again, only two object positions arebeing considered. Nonseparable motions apply tomultiple-exposure hologram interferometry and totime-average hologram interferometry of continuousmotions, which have been discussed elsewhere.1 ' 3 Forsimplicity, cosinusoidal fringe functions will be con-sidered for the bulk of this paper.

Within the class of object motions that are separable,this paper will be restricted to infinitesimal affinetransformations. Because fringe visibility in holograminterferometry drops nearly to zero except when objectmotions are quite small, rotations, translations andhomogeneous deformations may be added vectoriallywithout regard for sequence. This obviates the need forEuler-angle terminology and allows description of thedeformations as though the object were attached to ascrew of arbitrary axis and pitch, and given a homo-geneous deformation.

OBSERVER-PROJECTION THEOREM

The question is often asked: If I see the fringes infocus off the surface of an object, how do I know whathappens to the object itself? Such fringes can be relatedto the object by the observer-projection theorem.4

To explain this theorem, we must review a fundamentalproperty of these fringes, i.e., that they exist in the eye

of the beholder. It has been shown quite generally2' 5,'that some aperture is required in the optical systemthat reconstructs the hologram, or that images thereconstruction, in order that the fringes may beobserved. An exception that tests the rule was suggestedby Walles.7 Let a small object be located very close to alarge hologram plate, and let it be moved so as to createfringes that would localize well away from the object.There is no doubt that they could be observed in oneof the fields (true or conjugate) reconstructed by thehologram without the aid of any auxiliary opticalsystem or aperture. In this case, however, the objectwould act as the aperture itself, for if it were made largeenough, the fringes could not be seen, no matter how themotion were contrived. Thus it may be held generallytrue that some aperture must operate on the recon-structed field to make it possible to see fringes inhologram interferometry of diffusely reflecting objects.

It is also well established, both by experiment andtheory, that the smaller the aperture, the greater thedepth in space over which the fringes will be visible.With a very small aperture, the fringes may be nearlyunlocalized and may be visible at nearly all planes inspace, including the surface of the object. The observer-projection theorem relates the fringes seen localized inspace with a large two-dimensional aperture to thoseseen on the object surface with a small aperture.Figure 1 illustrates an object, located in the x,y planeof a rectangular coordinate system, that is beingreconstructed by a hologram and observed with a lens.With the full aperture of the lens, fringes are observedwith best visibility at the plane P1 that lies off thesurface of the object. That the fringes are localized inthe plane Pi means that they will appear the same inthat plane, to first-order approximation, if the lensaperture is restricted to any portion thereof. It ispossible to define a center of transmittance of the lensaperture analogous to the center of mass it would haveif it were a plate of the same geometrical boundariesand with mass density proportional to transmittanceper unit area. This point forms a natural center for acoordinate system located in the plane of the lensaperture. If we place a small aperture at the center oftransmittance of the lens, we could, in principle, makeit sufficiently small so that the region to which thefringes were localized became sufficiently deep as toinclude the object surface. The fringes in plane P1would remain unchanged, however, and they would lieon a radial projection from the center of transmittanceto the surface of the object. This is the observer-projec-tion theorem: Fringes observed as being localized offthe surface of the object can be projected to thesurface of the object radially from the center of trans-mittance of the aperture of the observing opticalsystem. For the small numerical apertures that mostcameras would subtend in the object space, the fringesact like real obstructions upon which an optical systemcan focus, with the conventional variation of depth of

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January 1974 FRINGE INTERPRETATION

focus associated with numerical aperture. Unlike realobstructions in the field, however, the fringes losecontrast in their localization plane if the numericalaperture becomes too large. However, the experimentalindications are that the projection theorem may bevalid for focal ratios as low as 4 in the object space forcertain representative cases.3' 8

It has been helpful to relate fringes in space to thesurface of the object for a second reason. On theobject surface, the fringe-locus function can be definedquite simply as

Q= L- (K2-K)= L- K, (1)

where K1 and K2 are the propagation vectors, respec-tively, of the field illuminating the object and of thefield propagating for an object point to the center oftransmittance of the observing system. I K1 | = I K2 1=2-r/A, where A is the wavelength of light. The fringe-locus function is a function of three-dimensional

space, R, by the observer projection theorem, and ofthe propagation vector, K2, that connects any pointon the object to the center of transmittance of theoptical system. Generally, in discussing this functionalrelationship, either the center of transmittance of theoptical system will be fixed and R will vary, or R willbe fixed and the center of transmittance will vary byletting K2 vary. From a metrological point of view,however, it is significant that the fringe loci denoteconstant amounts of object motion in the direction ofthe sensitivity vector, K= K2 - K3.

OF RIGID-BODY MOTIONS

8

K

FIG. 3. Fringe-source projection theorem. If an object rotates,the fringes appear as though projected onto it from the directionof the sensitivity vector and parallel to the rotation axis.

FRINGES DUE TO OBJECT ROTATIONS

Let us begin the discussion of the fringes of rigid-

body motions by considering rotations rather thantranslations, because it is rare that an object willmove purely in translation without some small rota-tional component. Figure 2 illustrates by the vectoro the rotation of an object and the corresponding vec-torial displacement of the object points.

L=OXR.

L

K2

FIG. 2. Rotation of an object. If an object rotates about anaxis 0, and Ro is a vector from the origin to any point on theobject surface, then the displacement of any point on the object,L, is the cross product of 0 and Ro. If K is the vector differencebetween the observation propagation vector, K2, and the illumina-tion vector, K, then the fringes can be thought of as generated bythe scalar product of Ro and a fringe vector KXO.

(2)

(3)

(4)

Figure 2 illustrates, geometrically, the vector KX O,which is perpendicular to both the sensitivity vectorand the axis of rotation. Fringe loci, on the three-dimensional surface of the object, can now be definedas those points on the surface where the spatial-positionvector, R, has a component of constant magnitude inthe direction of KXO. If K is nearly constant over thesurface of the object, then the fringes are seen on theobject as though it were intersected by a set of parallelfringe planes, whose normals are perpendicular to bothK and 0, as shown in Fig. 3.

This may be regarded as a fringe-source projectiontheorem, and it is important not to confuse it with theobserver-projection theorem. In fact, the fringes, asfunctions of three-dimensional space, always projectradially from the center of transmittance of the ob-serving optical system. They are seen on the object,

0

Substituting this into Eq. (1) gives

Q=OX R* K,

which can be rewritten as

Q=KXO R.

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KARL A. STETSON

however, as though they were projected onto theobject surface from the source of the sensitivity vector,K, as lamina parallel to 0. When there is curvatureof the illuminating wavefront and curvature of theperspective with which the object is viewed, the vectorK appears to come from an astigmatic source definedby the centers of curvature of the ellipsoid that wouldimage the source of illumination onto the center oftransmittance, by reflection. Also, the magnitude of Kwould vary across the object surface and so, in general,the source-projection theorem will hold only for smallregions of the object. There are a number of configura-tions, however, for which it will hold for substantially allof the object and facilitate measurement of objectrotation. One of these is retroreflective observation, inwhich the illumination is a diverging spherical wavethat originates near the center of transmittance of theoptical system. The magnitude of K will be constantin all directions of observation and its direction willbe that of the observation vector itself. Only the valueof the vector product, KXO, will vary across the fieldof view. If this variation is negligible, the fringes willappear parallel and will be aligned with the rotationaxis, as discovered by Wilson9 and Waddell1 0 for circularshafts. It is clear from this discussion that their tech-niques can be easily extended to shafts of arbitrarycross section, and thus have great practical utility.

If the fringes are projected to a plane containing theaxis of rotation, then the fringe spacing in the objectspace will yield the angular rotation by

20= (X/2) ((f sine), (5)

where (If is the fringe spacing, 20 is the peak-to-peak,or total, angular rotation, and t is the angle betweenthe retroreflective observation direction and therotation axis. Therefore, if t is known, the magnitudeof rotation can be determined from the fringe spacing.

FRINGES DUE TO HELICAL MOTION

If the object shown in Fig. 2 were to move as thoughattached to a screw, its motion would be described byadding to Eq. (2) a translation vector along the axisof rotation, LT. The resulting fringe-locus function is

Q=KXO.R+K-LT. (6)

If the sensitivity vector is constant over the field ofview, the fringes will have the same appearance as withpure rotation, except for a bulk shift by the number ofcycles generated by the component of axial translationin the direction of the sensitivity vector. If the perspec-tive is curved, as it most often is, then K will exhibit anapproximately linear variation along the rotation axisover fields of view. The resulting linear variation of thefringe-locus function along the axis of rotation will givethe fringes a slant relative to that axis and will indicatea false orientation of the rotation. If an observer lookingthough a hologram at such an object moves toward the

hologram, thus increasing the curvature of perspective,the fringes will increase their slant. This peculiar effectis the best indicator of helical motion, and, onceunderstood, is of great value in fringe interpretation.

The fringes due to helical motion have anotherpeculiar effect related to perspective. Lateral motion ofan observer's eye in a direction parallel to the fringeswill cause the fringes to appear to move across thesurface of the object, at right angles to the motion ofthe observer's eye. The fringes, therefore, do not seemto be anywhere in space. This phenomenon will bedealt with in more detail in a later section.

The only valid method for extracting the apparentorientation of the axis of the helix is to view the fringesfrom a distance such that K is constant over the fieldof view. Under this condition, the arguments of theprevious section apply with respect to the fringe-sourceprojection theorem. There is not enough informationavailable, however, to determine the exact location,orientation, and pitch of the helix from the fringes alone.

FRINGE LOCALIZATION FOR ONE-DIMENSIONAL APERTURES

AND FRINGE PARALLAX

If the optical system through which the fringes ofhologram interferometry are viewed has a long, thinaperture (essentially one dimensional), then thereexists a two-dimensional surface to which the fringes arelocalized.3' 4 The aperture places only one angularconstraint on the fringe-locus function, which is itself afunction of three-dimensional space. Correspondingly,as described in the following section, a large two-dimensional aperture in the observing system restrictsthe fringe localization to the region of a line in space.A general formula has been derived that defines thesurface of fringe localization for a slit aperture,4

L. ap= (X/27w)Dap(Shad{fkap) V40), (7)

where kap is a unit vector pointing along the apertureof the slit, Dap is the distance from the object surfaceto the localization plane along the observation vectorK2, Shad{kap) is the projection of kap onto the objectsurface from the direction of K2, and VR is the gradientoperator in three-dimensional space. The last factor inparentheses in Eq. (7) is proportional to the derivativeof the scalar function Qo in the direction of the shadowprojection of the slit aperture onto the object surface.The function Q0 equals K. L evaluated on the surfaceof the object, which is why the derivative must betaken in that plane. The vector K2 is held constant whenthis spatial derivative is taken. If Q0 is projected offthe object surface to the localization surface with K2held constant over the region of space considered, Qowill appear foreshortened in a plane perpendicular toK2. Let us call this foreshortened function Qi. Qo willbe foreshortened by exactly the same amount that kapwas lengthened by its shadow projection onto the

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January 1974 FRINGE INTERPRETATION

object surface. Thus, we may rewrite Eq. (7) as

L- kap= (X/27r) (fap- VRQ I)Dap, (8)

where the derivative of the foreshortened function,&l, is now taken in the direction of the slit aperture.Equation (8) states that the component of objectmotion parallel to the slit aperture is proportional tothe distance from the object to the localization surface,times the derivative, in the direction of the slit, of thefringe-locus function as it would appear to a distantobserver. For equidistant cosinusoidal fringes, thederivative in Eq. (8) equals 7r divided by the apparentfringe spacing, d(ap, in the direction of the slit. This leadsthe formula

2Lap =Dap/dap, (9)

where 2Lap is the total displacement of the object inthe direction of the slit aperture. The fringe spacingmust have a sign convention such that d, is positive if,when going from one fringe to another in the directionchosen for kapu the fringe-locus function increases.

We are now able to compare fringe localization with aslit aperture to fringe parallax along the direction of theaperture. Figure 4 illustrates the observation of threefringes that appear to be equally spaced when tele-centrically viewed from the direction of K2. In fact,they are not equidistant and do not lie on a straightline. To the telecentric viewer, the fringe-localizationsurface appears curved and lies oblique to both theviewing direction and the object surface. Only one crosssection of the fringe-localization surface is shown inFig. 4, one that is parallel to the slit aperture. For theobject point, 0, corresponding to the central fringe, wemay define a localization distance, Dap, and a fringespacing, dap, as shown in the figure. Equation (9)applies to this localization distance and to this fringespacing, and relates them to the component of objectmotion perpendicular to K2 and parallel to the slit.

Superimposed in Fig. 4 are two perspectives of theobject point 0 such that the outer two fringes are seencovering the object point. Let us bisect the angle anddraw lines perpendicular to the two lines of sight. Wemay now define the distance Do and do as shown inFig. 4. From the formula derived by Sollid," we maywrite

2Lp=XDo/d0, (10)

where 2L, is the component of the total object displace-ment in the plane of motion of the observer andperpendicular to the bisector of the angle between thetwo viewing directions. Equation (10) has the sameform as Eq. (9); however, L, is not generally equal toLap because they are components of the object displace-ment L measured in two different directions. Althoughfringe parallax and fringe localization both allow mea-surements of transverse object motions, they do notalways measure the same component. If the threefringes are equidistant from the object point 0, thenEqs. (9) and (10) become identical. This will cause the

OF RIGID-BODY MOTIONS

FIG. 4. Fringe localization and fringe parallax. Three fringes(solid circles) are seen localized in space with a telecentric opticalsystem with a high numerical aperture in the plane of the diagram.If viewed with the eye, they may be nearly unlocalized, but willexhibit parallax with the object point 0. The ratio of Dap to dapis analogous to the ratio of Do to do, but not necessarily equal to it.Both are proportional to transverse motions of the object at thepoint 0, but not to the same component.

localization plane to appear to be curved, with thepoint 0 at the center of curvature when viewed tele-centrically. It is clear from this, however, that theequality of Eq. (9) to Eq. (10) can hold for only onepoint on the object. So long as the localization plane isapproximately perpendicular to the direction oftelecentric viewing, however, the two relationships areapproximately equivalent, and essentially the samemeasurement is obtained by either method.

It is obvious from the foregoing discussion that if theslit aperture is at right angles to the apparent transversedisplacement of the object, the fringes will be localizedon the object surface. Similarly, if there is a direction ofparallax such that the fringes remain fixed on the objectsurface, that direction must be at right angles to thetransverse displacement of the object.

FRINGE LOCALIZATION FOR TWO-DIMENSIONAL APERTURES

With the aid of the previous derivation for one-dimensional apertures in the observing optical system,it is possible to derive the line of localization thatresults from a large two-dimensional aperture. ConsiderFig. 5. Let us project the object-displacement vectoronto a plane normal to the viewing axis at the objectsurface, and call the resulting vector Lob. Let us alsoproject the fringes onto a similar plane normal to theviewing axis at the region of localization for a slitaperture. Let Of be the angle between the aperturevector and the normal to the fringes, which are assumedto appear equally spaced. Let OL be the angle betweenthe aperture vector and the vector Lob. Then we maywrite,

Lap= I Lob I COSOL

and(11)

dap = df/cosof, (12)

5

KARL A. STETSON

A

k ap

LLob kAp

FIG. 5. Localization distance as a function of slit angle. Lob isthe displacement vector of the object projected onto a planenormal to the line of sight, and df is the fringe spacing projectedonto a similar plane. The unit vector pointing along the slitaperture kL,. and the angle it makes with Lob and df are OL and of,respectively. When these two angles are equal, the distance fromthe object to the fringe-localization region, Dlap, is independentof the orientation of the slit.

from which, by substitution into Eq. (9) and solvingfor D0,,, we obtain

Dap = (2/X) ILob (If(cosOL)/(cosOf). (13)

If Oa = OL, the localization distance becomes independentof the angle of the slit aperture, and the fringes will belocalized with a large, two-dimensional aperture. Whatthe equality of these two angles means, however, isthat the apparent displacement of the object is atright angles to the apparent orientation of the fringes.

We have already established that the fringes causedby the rotation of an object, when observed retroreflec-tively, lie parallel to the axis of rotation. The apparentdisplacement of any point on the object will be tangentto some ellipse centered on the axis of rotation. Thetangent to the ellipse will be normal to the rotationaxis, and therefore to the fringes, for only those pointson the object that lie in the plane defined by therotation axis and the center of transmittance of theoptical system. Thus, for retroreflective observation,the line of localization will appear to be directly infront of, or directly behind, the rotation axis. Thus,observation of the localization line can be used to definethe plane within which the axis of rotation lies. Whatremains now is to determine the orientation of thelocalization line within this plane.

Figure 6 illustrates the common plane of the axis ofrotation and the line of fringe localization for purerotation. The observation is no longer retroreflective,but K1 and K2 are assumed to lie in the same plane asthe axis of rotation and the localization line. FromEq. (13),

D= (2/X) | Lob I(If- (14)

From Eq. (2),

Lob =-rI O| sing, (15)

where ,6 is the angle between the object surface and theaxis intersects the object. From Eq. (5),

df =-X/(41 Ol sink) (16)

may be obtained with use made of the sign conventionfor df. By the law of sines,

l/sinik=r/sint, (17)

where / is the distance along the sensitivity vector fromthe object surface to the rotation axis. If Eqs. (14)-(17)are combined, the result is

l=2D. (18)

Under retroreflective observation, the localization linewill lie halfway between the object surface and therotation axis, along the line of sight. This is sufficientto define completely the orientation of the rotationaxis; therefore the magnitude of rotation can bedetermined from the fringe spacing via Eq. (5). Notethat the axis of rotation and the localization lineintersect the object surface at a common point. It isalso implied in this discussion of Fig. 6 that the sensi-tivity vector is constant over the surface of the object.

If the object moves as if attached to a screw, theneach point on the object will move tangent to somehelix centered on the axis of the screw. For distantretroreflective observation, the fringes will appearparallel to the screw axis. So long as the observationdirection is not perpendicular to the screw axis, therewill be some plane to one side or the other of the screwaxis where the tangent to the helix will be seen normalto its axis. This is illustrated in Fig. 7 where the solidline indicates the screw axis and the dashed line lies inthe plane that contains the localization line. Where the

e

K

FIG. 6. The localization line and the rotation axis. The rotationaxis is 0, and the distance from where it intersects the objectsurface to an object point observed is r. The dashed line indicatesthe line of localization when 0, K,, and K2 are coplanar, underwhich conditions the line of localization lies in the same plane.It is shown in the text that 2D=1.

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January 1974 FRINGE INTERPRETATION

localization line intersects the object surface, thetangent to the helix points directly toward the observer.By definition, the tangent to the helix is perpendicularto its radius, and its radius at this point, ri, may becalculated by geometry to be

ri = (LT/0 I I ) tang. (19)

The addition of the translation along the rotationaxis has the effect of translating the plane that containsthe localization line to one side of the rotation axis bythe amount ri. The plane still lies parallel to the rota-tion axis, for distant observation, and contains thecenter of transmittance of the optical system. Uponcareful examination, it can be seen that the set of Eqs.(14)-(17) can be applied to this case as well, with theI LI in Eq. (15) replaced by the component transverseto the line of sight. The result, then, is that the lineof localization simply translates to the side and main-tains its orientation as an axial translation is added toa rotation, when viewed retroreflectively from a con-siderable distance. This could indicate a false axis of

0

FIG. 7. Fringe-localization line for helical motion. If helicalmotion of the object is observed retroreflectively from a distancewith a large two-dimensional aperture, the plane containing thelocalization line is shifted to one side of that containing theobservation vector and the rotation axis. It moves to where thetangent to the helix appears perpendicular to its axis.

rotation if the true helical nature of the motion werenot known. If the zero-order fringe of the fringe functioncan be identified, it will be seen to lie on the oppositeside of the rotation axis from the localization line. Itwill denote all points on the object that move normal tothe sensitivity vector. Because the fringes run parallelto the rotation axis, the distance from this fringe to therotation axis can be calculated by considering only thepoint on the object that appears to move parallel to thefringes. If this point moves perpendicular to the sensi-tivity vector, then the radial distance to the axis of thescrew, r2 , is

r2= (Lr/ ol 1 ) cotQ. (20)

The quantity, rl+r 2 , can be measured; the angle 4can be determined by the inclination of the localizationline, and the fringe spacing can be measured. From this,the magnitude of the rotation can be calculated byEq. (5), and the magnitude of the axial translation canbe calculated by

2LT= 20 (rl+r2)/ (tan4+cotQ)=0(rl+r,) sin24. (21)

From this, the location of the screw axis can be com-puted from Eq. (19). If the zero-order fringe cannot beidentified, then more information is required to deter-mine the parameters of helical motion. It can be seenthat a second hologram reconstruction, with distinctretroreflective observation from the first, would allowthis reduction quite easily. In general, the localizationlines of any two hologram reconstructions wouldsuffice, provided that they represented sufficientlydifferent perspectives.

HOMOGENEOUS DEFORMATIONS

Homogeneous deformations of an object are those forwhich a segment of the object experiences the samedeformation, regardless of where it is chosen in theobject. If Ro is a vector from the origin to any point onthe object surface, then the homogeneous vectorialdeformation of the object surface is the product of thisvector with a strain tensor,12 which may be representedby a 3X3 matrix, [e]. Thus,

E L][e] Ro], (22)

where the vectors L and Ro have been represented by1X3 matrices. The fringe-locus function may now bewritten as

u= K- L= [K][e][Ro, (23)

where K has been represented by a 3X I matrix. Thetriple matrix product of Eq. (23) is invariant to the

7OF RIGID-1301)Y MOTIONS

KARL A. STETSON

order of operations, and thus it is convenient to definea fringe vector, Kf, that plays a role analogous to thecross product KXO in Eq. (4),

[K1 ]= [K][e]. (24)

Thus,

Q2= Kf1 R0. (25)

If the sensitivity vector is constant over the surfaceof the object, the fringes are seen as though the surfacewere intersected by equidistant fringe planes whosenormals are parallel to Kf. Unlike the fringe vector forrotation, Kf need not be perpendicular to K, althoughthis can happen for certain perspectives of certaindeformations. When Kf is perpendicular to the observa-tion vector, K2 , the fringes will appear to be equidistantacross the surface of the object, regardless of theobject shape, which is like the case of retroreflectivelyobserved rotation. If the object surface is flat, thefringes will appear as equidistant, straight lines,regardless of the deformation or the perspective.Because this is also the case for rigid-body motions,it is not possible to distinguish between rigid-bodymotions and homogeneous deformations by use of asingle holographic interferogram. In a considerablenumber of cases, however, fringe localization willdistinguish the two cases.

Before discussing some particular examples ofhomogeneous deformation, let us expand the fringevector in terms of the components of the sensitivityvector and the strain tensor,

Kf = i (kxex+keyx+kezx)

+3 (kxe,±+kue11 ,+kze25)+k (kxexz+kveyz+kzezz). (26)

Isotropic expansion. For isotropic expansion, theoff-diagonal components of the strain tensor will bezero and the diagonal components will be equal. Thusthe fringe vector will be parallel to the sensitivityvector, and the fringes on the object will be perpendic-ular to the projection of the sensitivity vector onto theobject surface. If the object is observed retroreflectively,and its surface experiences no out-of-plane motion,then consideration of the discussion of Eq. (13) yieldsthe following conclusion. The line of fringe localizationwith a large two-dimensional aperture will lie in theplane, normal to the object surface, that contains thesensitivity vector. Here the fringes are seen lyingperpendicular to the line of localization, whereas forrigid-body motions they were seen parallel to it, forsimilar illumination and observation. Under theseconditions, the coefficient of expansion, e, is related tothe apparent fringe spacing (normal to the line of sight)by

e = X/ (4df cosa), (27)

where a is the angle between the retroreflective line ofsight and the object surface.

Linear expansion. If an object undergoes linearexpansion, the coordinate system may be chosen sothat one of the axes, e.g., the x axis, lies in the directionof expansion. With this choice of coordinates, onlyone component of the stress tensor survives, e-. Thefringe vector, from Eq. (26), points in the direction ofexpansion, regardless of the perspective. A considerablevariety of localization lines can occur, depending uponthe perspective, and no simplification such as that forisotropic expansion, viewed retroflectively, is possible.If the fringes are observed by eye through a largehologram, they will maintain their alignment relativeto the coordinates of the object as the observer moves.Only the fringe spacing on the object will change. Thisphenomena should be sufficient to identify linearexpansion. Once identified, the coefficient of expansioncan be extracted easily from the fringe spacing.

Uniaxial stress of an isotropic material. For uniaxialstress of an isotropic material, it is also convenient toalign the coordinates so that one axis, e.g., the x axis,lies along the direction of stress. Under these conditions,the off-axis elements of the strain tensor are zero, andthe fringe vector becomes

Kf = eik -jvk,, -kvkJj, (28)

where v is Poisson's ratio and e is a coefficientexpansion. The fringe vector will be perpendicularthe observation vector when Kf- K2=0. This gives

k-vk 1k 2 -V - vk-kz 2 = 0.

ofto

(29)

It can be shown that Kf and K are coplanar with the xaxis, and therefore with the direction of stress. Theactual orientation of the fringes and of the localizationline for a two-dimensional aperture will depend uponthe orientation of the object surface, except whenEq. (29) is met. This will occur, for retroreflectiveobservation, when the angles of observation arerestricted to the cone defined by

k12+kt'= (2r/X)2/(1+v), (30)

whose axis is coincident with the x axis. Because theKf and K are coplanar with the x axis, the fringes willappear normal to the x axis when Eq. (29) is fulfilledfor retroreflective observation. By symmetry, only thosepoints on the object surface that lie between the x axis(which runs through the point of zero motion of theobject and thus may lie within the object) and thecenter of transmittance of the optical system willappear to move normal to the fringes. Thus, the lineof fringe localization for a large two-dimensionalaperture will lie in the plane defined by the x axis andthe center of transmittance of the optical system.Again, the fringes will appear to lie perpendicular to thelocalization line. It must be kept in mind, however,

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January 1974 FRINGE INTERPRETATION OF RIGID-BODY MOTIONS

that Eq. (29) need not, in general, be fulfilled, but israther a special condition.

GENERAL FORMULAS FORLOCALIZATION LINES

The fringe vector, K1, which was defined in theprevious section, can be useful in obtaining generalformulas for the line of fringe localization for a largetwo-dimensional aperture. Recall that the condition forthe intersection of the line of sight and the localizationline was that the fringes should appear perpendicularto the apparent motion of the object. If 12 is a unit vectorthat is normal to the surface of the object (which isassumed flat over the region of consideration), thenthe fringes, as lines on the surface of the object, pointalong the vector, KfXA. Consequently, 'the vector(KfX6) X K2 appears normal to the fringes as they areseen on the surface of the object from the perspectiveof K2. Similarly, the vector K2XL is normal to theprojection of L onto a surface perpendicular to k2.Thus, the intersection condition can be expressed as ascalar product that equals zero,

(KfXn)X K2v K2XL=0,

which may be rewritten as

(31)

The line of localization intersects the line of sight fromthe center of transmittance of the observing opticalsystem to those points on the object surface thatsatisfy Eq. (35).

Equation (35) together with Eq. (14) define the lineof localization; however, it is possible to reformulateEq. (14) to conform to the vector notation in thissection. First, let us define an apparent fringe-spacingvector df equal in magnitude to the apparent fringespacing in the region of localization, and pointingnormal to both the fringes and the observation vector.It may be expressed in terms of the fringe vector, thesurface normal, and the observation vector as

df=-ri K2 , (KXA )XK2 /l (KiXn)XK2fl. (38)

With this vector, df, Eq. (14) may be rewritten as

2D=-df L.

Substituting for df in Eq. (39), we get

I K2 P 2(KfXn)XK 2. LD= X-

I (K,, X n) X K2 l 2

(39)

(40)

[(KfX12) X K2 ]X K2.L =0. (32)

Equation (32) may be rewritten as

[A XK 2(Kf K2 )-Kf X K2 (h- K2)]. L= 0. (33)

Equation (33) provides some interesting insights intothe properties of the localization line. For example,when the fringe vector is perpendicular to the observa-tion vector, the first term in the brackets is zero andthe intersection condition becomes

KfxK L==O, (34)

which does not depend upon the orientation of theobject surface.

If the illumination and observation vectors areessentially constant over the surface of the object, it isconvenient to denote the vector in the brackets ofEq. (33) by F, which is space invariant. If the deforma-tion of the object is comprised of a helical motion plus ahomogeneous deformation, the definitions of L may besubstituted into Eq. (33) from Eqs. (2) and (22)to give

|[F][e]+[FXO] -Ro+F'LT0, (35)

whereF=Ax K2 (Kf1 K2)-KfXK2Q(2h K2)

and

EKf= [K][e]+EKXO].

Substitutions may be made into Eq. (4) for Kf fromEq. (37) and for L from Eq. (33).

A complete discussion of the phenomena representedby Eqs. (33) and (40) is beyond the scope of this paper.It must suffice to state that all of the localizationrelationships described in this paper can be obtainedby specialization of these two equations. They arepresented here mainly for completeness and to indicatethe breadth of this formulation.

SUMMARY

The form of the interference fringes seen in holograminterferometry has been derived for rigid-body (helical)motions and for homogeneous deformations of objects.For such cases, the fringes can be described as thescalar product of a fringe vector with a vector denotingthe coordinates of the object surface. The fringes appearon the object as though it were intersected by equallyspaced fringe planes oriented normal to the fringevector. Accordingly, object surfaces that are flat havestraight, equally spaced fringes, when the illuminationand observation directions are essentially constantover the flat surfaces. When the fringe vector is perpen-dicular to the observation vector, fringes on the objectappear straight and equally spaced, regardless of objectshape. Rigid-body motions, observed retroreflectively,are the primary example of this; however, the uniaxialstress of an isotropic medium, viewed under certaincritical perspectives, is another example. The effect ofcurvature of perspective was discussed for helicalmotions, which were the most-general form of rigid-body

()

KARL A. STETSON

motions considered; curvature of perspective causesthe fringes to slant, relative to their orientation withno curvature of perspective. For pure rotation andretroreflective observation, the fringes are seen parallelto the plane containing the rotation axis. Thus, curvedperspective is capable of indicating a false orientationfor the rotation axis.

Fringe localization was considered for observingoptical systems with large one-dimensional apertures,and was compared to fringe parallax. The two phenom-ena both result from motions of the object transverseto the direction of observation; however, they are notstrictly equivalent in general.

Fringe localization was considered for large two-dimensional apertures in the observing optical system,and a condition was derived for the intersection of thelocalization line and the line of sight. This occurs if theapparent motion of the object is normal to the apparentorientation of the fringes. For pure rotation, therotation axis, the localization line, and the line of sightare coplanar, if the object is viewed retroreflectively.The localization line is halfway between the objectsurface and the rotation axis, along the line of sight.For helical motion, the localization line is shifted to theside and maintains its orientation, again under retro-reflective observation, and can indicate a false axis ofrotation. Its displacement from the zero-order fringe,however, can be used to define the motion completely.

Fringe loci and localization were considered forhomogeneous deformations; for a number of cases, a

distinction can be made between rigid-body motionsand homogeneous deformations in terms of how thefringes appear along the line of localization. Forrigid-body motions, with retroreflective observation,they appear parallel to the line of localization, whereasfor expansions of an object they tend to appear perpen-dicular to it.

Finally, the general formulas are presented for theintersection condition between the line of sight and thelocalization line, and for the distance from the localiza-tion line to the object surface along the line of sight.

ACKNOWLEDGMENT

The author would like to express gratitude to R.Erf of these Laboratories for his helpful criticismthis paper.

K.of

REFERENCES

'K. A. Stetson, J. Opt. Soc. Am. 60, 1378 (1970).2K. A. Stetson, Optik 29, 386 (1969).3N. E. Molin and K. A. Stetson, Optik 33, 399 (1971).4K. A. Stetson, Optik 31, 576 (1970).'S. Walles, Ark. Fys. 40, 299 (1969).6S. Walles, Opt. Acta 17, 899 (1970).7S. Walles (private communication).'N. E. Molin and K. A. Stetson, Optik 31, 157 (1970);

Optik 31, 281 (1970).9A. D. Wilson, Appl. Opt. 9, 2093 (1970).'OP. Waddell, Eng. Mater. Des. 16, 401 (1972).''J. E. Sollid, Proc. SPIE 25, 171 (1971).12I. S. Sokolnikoff, Mathematical Theory of Elasticity, 2nd ed.

(McGraw-Hill, New York, 1956), Chs. 1-3.

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