Novel concept for obtaining continuously variable topographic contour mapping using holographic interferometry

Novel concept for obtaining continuously variable topographiccontour mapping using holographic interferometry

Pramod K. Rastogi and Leopold Pflug

A holographic technique is developed which allows one to obtain in real time the contour map of an arbitrarilyshaped object. The method provides a relatively coarse contouring of the object. While still operating at lowsensitivities, the method suffers from a breakdown in its functioning due to a saturation of the phase carryingcapacity of the interferometer. As a consequence, a phase management strategy is introduced, whoseincorporation to the technique enables a substantial improvement of its performance. These developmentsare systematically analyzed and presented. The method is whole field and noncontacting. The live andflexible character of the method enables one to rapidly generate sensitivities tuned to the relief variations ofobjects of broadly varying depths. Experimental results are presented to illustrate the feasibility of theproposals.

1. IntroductionWhen a 3-D object is rotated around an axis perpen-

dicular to the line of sight, the resulting holographicfringe pattern is observed to be sensitive to the out-of-plane component of the object surface.1 This observa-tion apparently seems to have gone unnoticed if onewere to judge by the absence of follow-up work toexploit this idea to achieve 3-D shape measurements.Although these fringes code the topographical varia-tions on the object surface, they do not provide truedepth contours. The lack of enthusiasm could thus beunderstood on the ground that one is generally moreinterested to obtain depth contours, with intersectionplanes lying in a direction normal to the direction ofobservation.

The first part of this work is devoted to the develop-ment of a holographic contour measuring techniquebased on dual beam illumination, on the one hand, andcharacterization of the shape information, on the oth-er, by slightly rotating the object surface. The super-imposition of the two coded patterns one due to eachbeam, results in the formation of a moire pattern.This pattern represents a topographic map of the test-ed object. The method, however, suffers from a break-

The authors are with Swiss Federal Institute of Technology, Lau-sanne, Laboratory of Stress Analysis, CH-1015 Lausanne, Switzer-land.

Received 27 June 1989.0003-6935/90/294392-11$02.00/0.© 1990 Optical Society of America.

down in its functioning due to a saturation of the phasecarrying capacity of the interferometer. The sensitiv-ity attained by the method at this stage is low. Assuch, the method, besides its academic contribution,has only limited practical interest to offer.

As a consequence, the next and more important partof the present study is devoted to developing a waycapable of breathing a new life in the method. To thisintent, a new phase management solution is proposed,application of which enables the method to overcomeits low sensitivity handicap and attain its full potenti-ality. Briefly stated, the solution consists of amplify-ing the contribution of the phase term relative to theobject shape at the expense of the carrier phase term.The process is carried out in a way to affect simulta-neously the phase terms in the two arms of the interfer-ometer. Subsequent optical comparison of the twomodified coded patterns yields a beat pattern withmuch improved sensitivity. The repetition of thisprocedure extends considerably the method's poten-tial and its range.

The live and flexible character of the method en-ables one to generate sensitivities tuned to the reliefvariations of objects of broadly varying depths. It issimple to use and the fringe modulated object imagesare displayed in real time on a TV screen.

In Sec. II, the principle of the method is described.Section III introduces and analyzes the phase manage-ment approach. Section IV schematizes the mecha-nism of fringe formation. Section V discusses themodulation aspect factor. Section VI outlines an opti-cal configuration to obtain fringe contrast improve-ment. The work concludes by providing some experi-mental results.

4392 APPLIED OPTICS / Vol. 29, No. 29 / 10 October 1990

II. Basic ConfigurationThe configuration of the interferometric setup for

contour measurement is shown in Fig. 1. The model tobe contoured is illuminated by two beams defined bywavevectors K1 and K2. The beams are supposed tolie in the x-z plane. Wavefield z scattered towardsthe hologram thus results from the superimposition ofwavefronts 21 and 22, produced by each of the twobeams illuminating the object; 2' is the wavefield scat-tered toward the hologram after the model's positionhas been modified. The observation is carried alongwavevector K.

The first phase of the method consists of exposingthe holographic plate H to the wavefield M. The holo-graphic plate is processed and returned to its originalposition. The object is then rotated by a small amountaround an axis perpendicular to the plane containingthe two illumination beams. The axis of rotation,here, is Oy.

If A1 and A', A2 and A'2, represent the correspondingamplitudes at the point P of the image before and afterobject displacement, the intensity observed at a pointP in the image plane is given by

(I(P)) = (dexp(ibr)(Al + A2) + A1 + AJ22), (1)

where

A = Al exp(il)l (2)

A2 = A2 exp(iq,2 )j

'p1 and p02 being the corresponding phase differencesintroduced by the rotation of the object in the two armsof the interferometer.

Since X1 and 12 are coherent and uncorrelated, onlythe amplitudes corresponding to scattered field pairs(1 1,XI) and (X2,Y) are correlated.

H

lf -* - --- -~

\,, X

x

tizFig. 1. Schematic of the optical configuration employed to obtain

contour maps.

The propagation vectors K1, K2 and K0 are respectivelygiven by

K1 = kZ1; K2 = k1Z2; K, = k&,

where k1 , k2 and kO are unit vectors along their re-spective illumination and observation directions; k isthe magnitude of the propagation vectors defined as

IK1I = IK21 = 1K01 = k = A I (8)

X being the wavelength of the illuminating beams.We assume the rotation of the object around the Oy-

axis to be in a counterclockwise direction. The differ-ence in position vectors in this case is then obtained as

(A1A;) = (A1A ) = (A1A;) = (A1A2 )

= (A2 A;) = (A2A;) = (A2A;) = (AA 1 ) = 0

(1A112) = (IA112) = (IA212) = (IA212) = (I).

The intensity in the reconstructed image thus becon

(I(P)) = 4(I)I-Cos (I12 P + (P2]cos Cos I2 2J

Equation (5) describes the formation of a slowly vaing fringe modulation related to the difference in phi(P1 - (P2-

To examine the form of these phase terms, letassume that the corresponding points P and P', bef(and after object displacement, are defined by positivectors r1 and r2 referred to the origin of a coordinisystem as shown in Fig. 1.

r,= Ix + ly + hlz

r2 = IX2 + jY2 + kZ2J

From Fig. 1, phase differences introduced in the tarms of the interferometer could be written as

sol(P) = (K1 - K) -(r2 - rl)

,p2(P) = (K2 -K) * (r2 -r l)J

r2- = zI sinAO, - kxl sinAO,, (9)

and the terms specifying the difference and sum ofphases become

ies 010(P) = (P(P) - P2(P) = (K1- K2) (r2 - r1)

(5) p10(P) = so(P) + P2(P) = 2 (K 2 2 K.) * (r2 - r)

(1Oa)

(lob)

ry- The intensity distribution of the fringe pattern couldLse then be written as

us (I(P)) = 4(I,)[1 - cos(kzl sinAO, sinOe)

)re X cos(kxl sinAO0 cosOe)]. (11)

on The significance of the moire fringes is described byate Eq. (1Oa). If the two wavevectors K1 and K2 are sup-

posed to make an equal angle 0e with the z-axis, the

contour sensitivity per moire fringe issuing from an(6) increment of the difference in phase 01o(P) equal to 27r

is given by:

(12)2 sinOe sinAO0

(7) In other words, Az provides the increment of heightbetween two adjacent contour planes. These planes

10 October 1990 / Vol. 29, No. 29 / APPLIED OPTICS 4393

(I)

xa)

(I)

2

1 _Fig. 2. Influence of the intensities of the two un-derlying patterns on the formation of the moire

pattern (a) I = 2 (b) I = 0.6I2.b)

Table 1. Maximum Values of Sensitivity Az(mm) Attainable by theMethod

hx 1 line/mm 1.5 lines/mm0

300 1.87 1.24400 1.37 0.9250° 1.07 0.71

intersect the object in a direction perpendicular to theline of sight.

The visualization of the modulated fringe pattern isintimately linked with the lot of carrier fringes. If thecarrier fringes are not of sufficient quality, their visual-ization is poor. In the event the carrier fringes aresubjected to contrast degradation due to object dis-placement or maladjustments, the modulated fringepattern as well succombs to a degradation in its quali-ty. A far from optimal visualization of the modulationfringes is obtained even under the assumption thatboth the interacting fringe patterns have unit visibili-ties. One such example is shown in Fig. 2(b), wherethe intensities of the two patterns instead of beingsame as in Fig. 2(a) are in the ratio of 3:5. Finally, sucha modulated pattern is not at all visualized if the carri-er fringe frequency is not resolved by the observationsystem.

The implementation of this method, however, wouldhave little practical utility if one were constraint to thesensitivity limits echoed in Eq. (12).

Ill. Phase Management SolutionA closer look at Eq. (10) would show that a break-

down in the method's potentialities arises from a satu-ration of the phase carrying capacity of the interferom-eter. This saturation appears even while the twointeracting fringe systems display strong visibilitycharacteristics. Since the fringe frequency in the x-direction, fx, is normally two to four and a half timesmuch higher, within the range 53 > ,e > 25°, than inthe z-direction, the phase demodulation computed bythe method naturally suffers from a weak sensitivity.

The limitation thus arises from the presence of carri-er frequency. The sensitivity of the method tends to

its upper limit as the carrier fringes on a granularbackdrop become hard to resolve. Table I shows sen-sitivities theoretically attainable by the method forthree different angles of illumination and two sets offringe frequencies f,. These sets are chosen to containfrequencies of 1 line/mm and 1.5 lines/mm so as toremain on the higher side of the frequency band withwhich one is normally used to work in holographicinterferometry. Finally, 50° is chosen as the upperlimit for the illumination angles as shadowing effects,depending upon the object, could become quite impor-tant beyond this angle.

The phase management approach consists of in-creasing systematically, in the overall phase, the pro-portion of term relative to the topographical informa-tion at the expense of that contained in Eq. (Ob).

Equation (lOb) suggests to its total independence tothe out-of-plane component of the object surface.The process of sensitivity enhancement consists ofseveral steps. Each of these steps are themselves di-vided into two substeps. The sensitivity of the meth-od is enhanced at the end of each step. These inter-ventions are rapid and performed in real time. Let usnow consider the first step.

A. Implementation of the First Step

The first substep in phase management consists ofreducing the value of phase contained in 'I10. Thephase depletion ATI should be carried out in a way so asnot to alter in any discernable manner the localizationof the original fringe pattern. The phase depletiondoes not affect the beat pattern, which is testified bythe new fringe intensity equation:

I = 4(I,)[1 _ Cos 2) cos!\, 2 _AT (13)

The process of phase loss is equivalent to the reductionof spatial carrier frequency in the observed pattern.Several variants are possible to achieve this goal. Theone chosen by us consists of rotating the collimatedreference beam around the Oy-axis and translating theholographic plate in its plane. In this case the spatialfrequency of the so removed carrier fringes is given by

Sri = X1cosor sinAori, (14)

4394 APPLIED OPTICS / Vol. 29, No. 29 / 1.0 October 1990

where 0r is the angle which the collimated referencebeam makes with the z-axis and A6ri is the tilt provid-ed to the reference beam. If the reference beam isrotated by a small amount in a counterclockwise direc-tion around the Oy-axis, a phase subtraction is ob-served between the carrier phase and that introducedby rotating the reference beam.

The phase term relative to the remaining carrierfringes can be written as

1Fo(P) = 2(1 - xK)(' 2 - K.). (r2 - r1), (15)

where X1 is a constant and whose value is determinedby:

sinAOrl cosO, (16)

sin-AO(1 + cosode)

In order that the reduction of the spatial frequencycorresponding to Eq. (14) does not disturb the localiza-tion plane of the fringes from their position in thevicinity of the object surface, it is necessary that thefollowing condition be satisfied

p cosO, sinAOrl

cos2/

(17)

where TXi is the translation provided to the holograph-ic plate, # the angle at which the object point is ob-served, and p the distance between the plate and ob-ject. In these conditions the reduction of carrierfringes does not lead to a delocalization of the remain-ing carrier fringe pattern on the object surface. Andphase subtraction by this method results in no addi-tional visibility loss in the fringe system.

Reduction of the carrier phase makes the interfer-ometer receptive to the introduction of more phase inits two arms. The second substep, thus, consists ofinjecting additional phase shifts, corresponding to ob-ject rotation, to the phase differences existing in thesystem. This results in an obvious increase of the ratiobetween the term containing the phase correspondingto shape and the term corresponding to I.

Let us assume that point P in its new position isdenoted by position vector r3. The object is supposedto be rotated by an additional angle A0 1. Thus, atthe end of the second substep of the first step, thephase terms can be expressed as

(18a)0,,(P) = (K2 - K1) (r, -r),

P11(P) = 2(K + 2 - K.) (r3 - r) - xl(r 2 -r)

and the distance between two adjacent contour planesnow reduces to:

(19)2 sinOe sinAO01

where

AO01 = AG0 + AO01.

We now go to the next step.

B. Implementation of the Second and Following Steps

The second step of the phase management solutionis similar to the first, except that it extends the range ofsensitivity measurement by another order of magni-tude. The total frequency by now amputed from thecarrier fringe frequency is given by

(21)Sr2 = A-1 cosOr sinAOr2,

provided that the condition,p cosO, sinAO, 2

cos 23

is satisfied, where

AO, 2 = AOri + AOr21

T.2 = T41 + AT, 2 J

(22)

(23)

AOr2 and ATX2 being the increments of rotation andtranslation provided respectively to the referencebeam and holographic plate. If r4 is taken to definethe position of the point P after the object is rotated byanother AO'02, the corresponding phase terms for q andI become

(24a)012(P) = (K2 - K1) (r4 -rl),

T12(p) = 2(K +K2 Ko)

[(1 - 2 )1(r3 - r) - xl(r2 - rl )) + (r4 - r3 )], (24b)

wheresinAO, 2 CoSO,

X2 = (1 - X1) sinAO,, + sinA6'011(1 + cosOe) '(25)

and the set of contour planes are consequently separat-ed by

iZ = X2 sinOe sinAG02

where

AGo2 = AOl + AGo2.

(26)

(27)

This procedure can be repeated several times over. Atthe end of nth repetition, the final phase terms corre-sponding to o and T become

01n(P) = (K2 - K1) (rn+2 -r),

Fln(P) = 2(Ki + K2 - KO) (rn+2 -rn+),

where rn+2 defines the final position of point P.(18b) writing Eq. (28b), we have assumed Xn = 1. '

sensitivity of contouring is finally obtained as

2 sinOe sinAG0On

where

Aon = AO + Ao'01 + AO'02 + * * . + AGo0 n,

(28a)

(28b)

InThe

(29)

(30)

On the whole a total withdrawal of carrier frequency(20) amounting to

(31)Srn = X-1 cosG, sinAGn


0 = 2

-0.2

Fig. 3. Plot of the variation offringe visibility as a function ofW/id for different values of L/d.

is carried out under the assumption that the equality,p cosr sinAlr3

cos / (32

is satisfied, whereAG,,, = A, + G2 + + A;, 1Tx = Tx + AT. 2 + + ATJ

The number of times the procedure can be repeated isfinite. It is limited either by the operator's desire tostop at a given sensitivity adapted to the type of objectbeing addressed or on the grounds of an unacceptablefall in fringe visibility. The later imposes a fundamen-tal limitation on the technique's measurable capabili-ty.

C. LimitationsThe visibility of the carrier fringes falls to zero as

soon as either of the following two conditions is satis-fied:

AG0 > sin- , (34a)

AG0 Ž: sin-1 (34b)r(1 + cose)

where h and denote, respectively, the object depthand mean speckle size in the reconstructed image.

Of these two conditions, Eq. (34a) is related to therelative displacement of the homologous rays at theobservation plane and Eq. (34b) takes into account theshift I between the interfering wavefronts at the pupil

plane. At any given instant the fall in visibility is theresult of the combined assault of both these factors.

The visibility is plotted as a function of the ratio 1/-d for several different values of the ratio L/£, where ddenotes the side length of a square aperture. The plotis shown in Fig. 3.

The 3-D plot in Fig. 4 shows the variation of fringevisiblity on a contour plane, parallel to [O,xy] and 2mm away, with respect to the illumination angle androtation of the object. The object is illuminated withan argon-ion laser emitting a wavelength of 5145 A,and the mean speckle size in the reconstructed image issupposed to be 20 jAm.

IV. Mechanism of Fringe FormationThe mechanism of fringe formation is shown sche-

matically in Fig. 5 using the concept of homologousrays. One of the corner stones of holographic interfer-ometry, this concept was developed initially in thepapers by Vienot et a., 2 Froehly et al.,3 Walles,4.5 andMonneret. 6

Since the superposed wavefronts due to the twobeams are coherent but uncorrelated, two pairs of ho-mologous rays are present in the formation of themodulated pattern: one (lp) owing its origin to theleft and the other (r,,rp) to the right illumination beam.The rotation of the object in the first phase of themethod results in different angular orientations of thehomologous rays p and rp. In the case of a planeobject, the two rays 1p and rp would have necessarilyundergone similar angular displacements. Thefringes are localized in that region of the space, where


-

0.5

0

Visibility

0e Fig. 4. Plot of the variation offringe visibility on a contourplane, parallel to [O,xy] and 2 mmaway, as a function of illuminationangle Ge and rotation AG,, under-

gone by the object.

the distances between the pairs of homologous rays isminimum. Two systems of fringes are, thus, observedwhich have their planes of localizations in the vicinityof the object surface. The modulated fringe patterncorresponding to Eq. (5) owes its presence to the differ-ence in angular displacements of the two rays.

The rays l and r, are angularly displaced by provid-ing an adequate tilt to the reference beam [Fig. 5(c)].The holographic plate is then translated, as shown inFig. 5(d), to cause the pairs (llp) and (r,,rp) to inter-sect in the immediate vicinity of the object surface. Asis evident from the figure, the interventions proposedin Figs. 5(c) and (d) result in an overall decrease inangular orientations between l, and 1p, on the onehand, and r, and rp on the other, while keeping thatbetween 1p and rp untouched.

The interferometer is now ready to receive morephase terms. The process, as described in Fig. 5, isrepeated in a closed loop sequence b - c - d.

V. Modulation Aspect FactorFormation of fringes corresponding to topographical

variations results from an optical recombination ofphase terms arising in the two arms of the interferome-ter. According to the derivation of Eq. (12), the for-mation of these fringes is due to shape dependentphase shifts being in opposite directions across theobject surface. To obtain a better appreciation of themanner in which the method functions, we introducehere a quantity .t,,, modulation aspect factor, which isdefined as the ratio between the spatial frequencies inthe z- and x-directions. This quantity provides a mea-

sure of the relative strengths of the contributions fromthe shape dependent phase and that corresponding toI in the overall phase contained in each arm of theinterferometer.

At the end of the first phase of the method, i.e. beforeputting in application the phase management solution,the ratio l,,, is given by its value X,i:

sinfe

- 1 + cosoe (35)

Three observations follow from Eq. (35). First, MA,, isindependent of the amount of phase that can be intro-duced in the two arms of the interferometer. Second,it has a unique value for each 0 e Third, it is always lessthan unity except for the unrealistic case of 0

e = r/2.The basic principle of the phase management ap-

proach described in this paper is to make the phaseterms corresponding to shape dominate at the expenseof the Kiln terms. This is done by systematically sup-planting the carrier phase in favor of the movement ofmore quantity of shape dependent phase terms in theoverall phase contained in each arm of the interferom-eter.

The shape term is regularly reinforced at each stepby injecting more phases in the two arms. At the endof the qth step of the phase management approach, themodulation aspect factor becomes

sinAG,,qX. = A.,smi q (36)

where Xq is assumed to be equal to 1.It is obvious that to visualize moire, the presence of


beam(I)

modela)

rp

model

c)

hologram

reference

Two pairs of-__ homologous rays

(c, p) and(rc .rp) coincident

beeamr)

b)

I rc and Iccoincident

I/

model

d)

rays rp and rcintersect

I rays Ip and Icintersect

Fig. 5. Mechanism of fringe formation.

carrier fringes is mandatory. Besides being depen- speckle size are taken to be 2 mm and 20 ,m, respec-dent upon fringe visibility and speckle size, the quality tively. The series of values of At,, have been calculatedof the moire visualization depends strongly upon the for fA equals to 1 line/mm on the reconstructed inter-frequency of the interacting patterns. These aspects ferograms.have been considered in Refs. 7 and 8. From Eq. (36) it is evident that that the frequency of

Table II shows the values of modulation aspect fac- the carrier fringes could be modified by activating thetor obtained in the pre- and post-phases of the applica- reference arm of the interferometer. Here it sufficestion of the phase management approach. The phase to note that for comparisons of the At,, terms to havemanagement approach is assumed to be applied until any objective sense, these should be carried out in thethe fringe visibility falls to a value of 0.3. For the context of one single frequency of carrier fringes. Un-purposes of calculating visibility, the object depth and der this condition, one has


Irc and Iclcoincident

//

z

Lx

y

model

TX

Irc and Iccoincident

7

/ 12

----�ZiE1 1Field VidiconLens Camera Fig. 6. Schematic diagram of the multiplicative

holographic moire setup. The reference beamshown here in dots is present only in (b).

Table II. Values of Aa Obtained In the Pre- and Post-phase of the Phase Management Approach for fi = 1

line/mm and ¢ = 20 Am. Brackets Enclose Corresponding Contour Sensitivities Alz(m)

Method Method incorporating phase(Section II) management approach

V 0.9 0.8 0.7 0.6 0.5 0.4 0.3He

300 0.268 1.08 1.96 2.70 3.43 4.13 4.91 5.70(1866) (463) (255) (185) (146) (121) (102) (88)

500 0.466 1.75 3.16 4.36 5.56 6.60 7.82 9.08(1072) (285) (158) (115) (90) (76) (64) (55)

a AZ2 (37)*4a; AZ1 '

where At, and At',, represent values of modulation as-pect factors at any two instants during the manage-ment implementation sequence, and Azi(i = 1,2) arethe corresponding sensitivities attained by the methodat those instants.

For any other frequency of carrier fringe, a differentseries of values for the Aa terms will result for samepair of Se and fringe visibility V. Although informa-tion impressed on the carrier for same such pairs,(0eY)tf(0e)f2... is invariant, the number of carrierfringes participating in the formation of a moire fringewill be different in all the cases. Or, in other words,the quality of visualization of the impressed informa-tion will be different for each pair (0e,V)fi.

Table II shows that the shape modulation reinforce-ment capacity of the method is very high. In thechosen example and for 0e = 30°, the modulation as-pect factor can be multiplied by a factor of around 20compared with its initial value. The rate of incrementof sensitivity is higher for Ge = 300 as compared with Ge= 500.

VI. Contrast EnhancementThis section deals with a brief description of a tech-

nique enabling one to obtain contour fringes in theform of multiplicative moire. These type of fringes bydefinition are of better contrast. However, it is advis-able to apply this technique only once an adequatesensitivity for the object being measured has beenobtained.

Let us suppose that successive applications of thephase management approach have lead us to the obser-

vation of contour fringes with sensitivity given by Eq.(29). Refer to the experimental lay-out shown in Fig.6(a). At this stage, one of the illuminated beams, sayI,, is switched off. The reconstructed image will thenconsist of fringes due to I2 and the initial object wave-field arising from I,. The reconstructed pattern isrecorded by imaging it with a lens on a high-contrastplate, and after developing, it is replaced in its originalposition. The process is facilitated by using an im-mersion tank. An accurate repositioning is achievedby means of observing the pattern due to I1 through therecorded mask. The mask is followed by a field lensand the final image is formed on a TV monitor; I2 isnow switched off and I, is switched on. The intensityobserved behind the mask is given by the product ofthe incident intensity by the transmittance of themask:

(I) = C(Iexp(ibr)(Al + A2) + A12)

X (lexp(ir)(Al + A2) + A212), (38)

where C is a constant. The observed moire suffersfrom a visibility setback due to the presence of noiseterms in Eq. (37). The noise terms arise from thelocking of the reconstructed wavefield due to one beamonto the fringe pattern due to the other beam. Themoire fringe visibility is given by

Vm = 0.44 Vm (39)

where Vm is the moire fringe visibility in the absence ofnoise terms.

The problem of visibility loss can be overcome byintroducing a second reference beam in the system.This is shown in Fig. 6(b). As we will see, this modifi-cation eliminates the noise terms while making the


a) b)

[l [ffilil l! lilf ill ltl~~~~~~~~~~~~~~~~~~~~~~~~~~~l p~~~~~llI! I It~~~~~~~~~~~tII

C)

Fig. 7. Interferograms obtained with left (a) and right (b) illumina-tion beams (At. = 0.354) and Moire (c) resulting from an opticalsubtraction of the patterns shown in (a) and (b). The contour

* sensitivity is given by 1480 pm.

technique slightly more complicated. Two hologramsare recorded on the holographic plate, one each due tothe illumination I and reference J? beam pairs (IdYi)and (12,Y?2). The interference pattern resulting fromthe arm (I,.JR1) is recorded on the holographic plate toserve as a mask. The interference pattern arising fromthe arm (U2,Y?2) is now observed through the mask.The intensity observed can be expressed as

(I) = C(Jexp(iir)A + A112)(1exp(iir)A 2 + A 2), (40)

which on simplification yields(1) = 4C12 1. "In, 12

(I) = 4cI0[cos 2 I 2-] (41)

where C is a constant and Io = 4(h1). Equation 41 isdevoid of any noise terms. The moire fringe visibilityis now given by the usual formula

V = V V2 (42)2

where V and V2 are the visibilities of the two interact-ing patterns, respectively.

a)

C)

4W Ili i I flyiY

1Ul

il i i I s1 11ff 1

*i 115 i5ll!{it IIS~,iS H5i,} I

d)Fig. 8. (a) Interferogram obtained with the right illumination beam(t. = 0.56). (b) Corresponding contouring pattern with an im-proved sensitivity of 974 ,um. (c) Similar to (a) but with L = 1.88.(d) Corresponding contouring pattern with a further enhancement

in sensitivity (Az = 292 ,m).

VIl. Experimental VerificationExperiments were performed to test the feasibility

of the proposals introduced in this paper. The colli-mated illumination beams strike the object at an angleof 39° with respect to the z-axis. The models arecylindrical surfaces with rectangular planforms. The3-D image of the model is recorded and reconstructed.

Figures 7(a) and (b) show fringes corresponding toeach illumination beam. Figure 7(c) shows simulta-neous reconstruction of these patterns. The opticalsubtraction that ensues results in the formation of amoire. The moire pattern represents contour fringeswith a distance of 1480 um between two adjacentplanes. Figure 7 also puts in evidence the inherentdrawback associated with the method. Although thecarrier fringes are starting to get quite dense, the sensi-tivity generated by the method is still low. The valueof the modulation aspect factor for patterns in Figs.7(a) and (b) is 0.354.

The results obtained by applying the approach de-


* #,,,,,,i, 1, t i *l,, ,t

. , ,~ltltt t Stt ti ~ ' tF

~tit ttt ' Ft tt t tlt it t

§* i $ tlt I t tts | t t X IJ t t i I II i i I i1

1 I 7 I f i # ) . 0 :' i I ': :

Ii f , S: i `,

*s 1+5 tii

a) b)Fig. 9. Continuous generation of higher sensitivities: (a) Az = 173

um; (b) Az = 141 um. The results displayed in Figs. 7,8, and 9 were

obtained during one and the same experiment.

.i;m*I 1 i til;tX tIi l>T *l t It 1.

ett tilt i ii f ie t I m (z = 8 A

icribedt i taple tiltare i .. * In ti .

'#tItt{titt II,1 *lll iftk t it t t tilli *t f ti ttIlf t ltttt l'i ti*t i*f S + , t t

1 s tit ii tilt tiltlt illtli X tt

Fig. 10. Contouring pattern obtained on a model of maximumdepth equal to 1 mm (Aiz = 86MAm).

scribed in this paper are shown in Fig. 8 onwards.Figures 8(a) and (b) are obtained after the implemen-tation of the first step. Figure 8(a) shows the fringepattern due to the right illumination beam. Figure8(b) shows the moire pattern with the distance be-tween two adjacent contouring planes now reduced to974 ,um.

The management approach is repeated many timesover to obtain a gradual increase in the sensitivity ofthe method. Figures 8(c) and (d) correspond to thepatterns obtained after implementation of the say, qthstep. Figures 8(c) and (d) are respective analogs ofFigs. 8(a) and (b). However, the distance between twoadjacent contouring planes in Fig. 8(d) is 292 Am.

The values of At,, in patterns shown in Figs. 8(a) and(d) are respectively given by 0.56 and 1.88. Thesevalues have been calculated with respect to the samefrequency f, as that present in the pattern shown inFig. 7(a).

Figures 9(a) and (b) show two contouring patterns

'I; X~~~~~~~~~~~~~~~~I

a) , b)Fig. 11. The contour planes are pivoted around the (a) Ox- and (b)

Oy-axes to provide for an oblique slicing of the addressed model.

obtained during the implementation of the phase ap-proach. The sensitivities attained are respectivelygiven by 173 and 141 Am.

Figure 10 shows a contouring pattern obtained onanother model of similar dimensions as the first butwhere the object is more flat. The distance betweentwo adjacent contouring planes is 86 Am. The visibili-ty of the interacting fringes is still high. Much highersensitivities could thus be generated for objects havingsmaller relief variations.

Finally, Fig. 11 demonstrates the possibility offeredby the method to manipulate in real time the orienta-tion of the contouring plane. The contour planes canbe pivoted around Ox [Fig. 11(a)] or Oy [Fig. 11(b)]axes to provide for an oblique slicing of the object.The model has a maximum depth of 6 mm. Theresults presented in this section are in good agreementwith the true object shapes.

Vill. ConclusionThe ensemble work taken up in this paper could be

resolved into two components. The first componenthas consisted of establishing a technique capable ofproviding contour measurements. The second com-ponent has consisted of applying a new phase manage-ment strategy to the technique and which as a resultenables a substantial improvement of its performance.These developments were systematically presented.The variation of sensitivity is obtained in real time.The technique can, thus, be tailored to match theneeds of the object under observation. But, perhapsfar more important than anything else, the develop-ment of the phase management notion symbolizes theextreme versatility inherent in the holographic setup.

This research was supported by the Swiss NationalFoundation under grant 2000-005598.

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I 1 i i tII t iIB~:*:r }t7Xtflt't

ltIlliff i' t It t I t ill I i

t~tli ta litli ifl It Uffl

it tit iii lt it ff1 tXt f tf E i': hg

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4. S. Walles, "Visibility and Localization of Fringes in HolographicInterferometry of Diffusely Reflecting Surfaces," Ark. Fys. 40,299-403 (1970).

5. S. Walles, "On the Concept of Homologous Rays in HolographicInterferometry of Diffusely Reflecting Surfaces," Opt. Acta 17,899-913 (1970).

6. J. Monneret, "Etude theorique et exprimentale des phenombnesobservables en interf6rom6trie holographique, interpretation desinterferogrammes et applications a la m6trologie des microdepla-cements," D.Sc. thesis, University of Franche Comt6, France(1973).

7. P. K. Rastogi, "Comparative Holographic Moire Interferometryin Real Time," Appl. Opt. 23, 924-927 (1984).

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Novel concept for obtaining continuously variable topographic contour mapping using holographic interferometry