Ultrahigh resolution sandwich holography

Ultrahigh resolution sandwich holography

Adil Dzubur and Dalibor Vuki6evi6

A modified method of fabrication of sandwich holographic interferograms coupled with a digital evaluation

technique has yielded ultrahigh resolution of surface deformation. The computer processing technique per-mits both the subfringe resolution and automatic fringe order determination (i.e., discrimination betweendepression and elevation of the deformed object surface). The technique presented here requires neither

the interactive processing nor the fringe interpolation required by other techniques. Such processes are in-

accurate, especially when there is a small number of interference fringes. With this type of analysis, sand-

wich interferometry is capable of fast, simple, ultrahigh resolution of surface deformation.

I. Introduction

Holographic interferogram fringes are caused by thespatial distribution of phase difference between twointerfering light-wave fields, each representing theholographic reconstruction of two states of an investi-gated object. The phase difference 5 is related at eachpoint to the displacement of the deformed surfacethrough the well-known expression

5 = s * L, (1)

where s and L are the sensitivity and displacementvectors, respectively.1 2

To solve the problem of the determination of thedisplacement vector at each point of the object surface,it is necessary to solve three equations of the same formas Eq. (1) for three noncoplanar directions of the sen-sitivity vector.

If the fringe pattern is viewed from one observationpoint, conditions for solving just one of the threeequations are established, the solution being a projec-tion of the local displacement vector in the direction ofthe local sensitivity vector. Actually, the observedfringe pattern is a projection of the 3-D fringe distri-bution across the object onto the plane perpendicularto the central observation direction. The mathematicaldescription of the light intensity distribution in theobserved fringe pattern is of the form

i(x,y) = io(x,y) + h(x,y) cos[o(x,y)], (2)

with h(x,y) = io(x,y)v(xy), where io(xy) and v(xy) are

Both authors are in Zagreb, Yugoslavia; A. Diubur is with Brodarskilnstitut, and D. Vukicevic is with Institut za Fiziku Sveuilista.

Received 20 December 1983.0003-6935/84/101474-07$02.00/0.© 1984 Optical Society of America.

local irradiance intensity of the holographic recon-struction and the visibility of the fringes, respectively,while (x,y) is the phase distribution of the fringepattern as observed in the plane of projection (x,y).

The determination of the quantity "k(x,y) is of pri-mary interest. Once this quantity becomes known, thecomponent of the displacement vector in the givensensitivity vector direction can be obtained for eachvisible point of the object.

Although there are many methods for quantitativeanalysis of holographic interferograms, those which areor could be computerized are of special interest becausea large number of data points and a large number ofcalculations ought to be handled.

DAndliker et al. 3 introduced a point-by-point methodof heterodyne holographic interferometry, which provedto be the most accurate among the existing ones, havingdeep subfringe sensitivity in the phase difference de-termination.

Hariharan et al. 4 recently described a digital systemfor real-time holographic interferogram analysis.Subfringe sensitivity and good spatial resolution areachieved by direct computer processing of the fringepattern intensity distribution measured by a 100 X 100pixel matrix array sensor camera. The phase distri-bution across the object is determined by processingintensity data for three selected reference beam phaseshifts.

Commonly used procedures for quantitative inter-pretation of holographic interferograms are based onthe fringe peak detection technique. 5 -7 By means ofthis technique the high information content of a holo-graphic interferogram is reduced to a set of isophasecontours h(X,y) = kr; k = 0,1,2, .. , where k is theassigned fringe order number. Phase determinationat points of interest is accomplished by use of some ofthe 2-D interpolation algorithms.

1474 APPLIED OPTICS / Vol. 23, No. 10 / 15 May 1984

These procedures have the following inherent dis-advantages: (a) the sensitivity is 27r, i.e., fixed merelyto the fringe occurrence, thus giving rise to interpolationerrors in the case of interferograms with a low numberof fringes; (b) for a given fringe pattern it is not possibleto distinguish between depression and elevation of thedeformed surface; Abramson's "rose of error" 8 is in fullblossom, and this sign of phase ambiguity can be elim-inated only by the operator's previous estimation; (c)accurate determination of a fringe peak is frequentlyobscured by the variation of i(x,y) and h(x,y) termsof Eq. (2). In addition to these three, some other pos-sible sources of error should be taken into account.While speckle noise could be suppressed by imagesmoothing or by man-machine interactive processing,9misinterpretation of rigid body motion fringes as de-formation fringes is inevitable.

Sandwich holography 0 uniquely allows for the pos-sibility of a posteriori manipulation of the fringe pat-tern in order to eliminate unwanted fringes caused byrigid body motion of the object investigated. Moreover,if rigid body motion fringes do not exist in the firstplace, they can be made to appear by an appropriatetilting of the sandwich interferogram, and this will beshown to be an unexpectedly useful feature. Thesesandwich tilt fringes are equidistant and parallel withthe axis of the sandwich tilt rotation. If this axis ismade parallel to the x axis, the fringe pattern of thedeformed object assumes the form

i(x,y) = i(xy) + h(x,y) cos[27rfoy + (x,y)], (3)

where fo is the spatial carrier frequency imposed by thesandwich tilt. The larger the tilt the higher fo.

11. Method

The phase distribution (x,y) can be extracted fromthe light intensity distribution i(x,y) of a tilted sand-wich interferogram fringe pattern, since sandwich ho-lography enables the spatial heterodyning of the objectdeformation fringe pattern with the spatial carrierfrequency imposed by the sandwich tilt. A light-wavefield topography method, developed by Takeda et al. 11for the analysis of the interference patterns in a Mi-chelson interferometer with tilt, could be successfullyapplied here for the deformation analysis of opaque andlarge bodies.

To transform the light intensity distribution i (x,y)into a form suitable for computer processing, it shouldbe digitized both spatially and in amplitude whileobeying the sampling theorem. Through the digitiza-tion procedure a continuous function i(x,y) is trans-formed into its digital representation i(jk), j =0,1,2, . . ., M-1; k = 0,1,2,. .. , N - 1, where M X Nis the number of pixels in a digitized image, and i(j,k)is now the quantized value of the light intensity at thepixel (j,k). Indices j and k are related to the spatialcoordinates x and y via x = j Ax and y = kAy, where Axand Ay are the sampling intervals in the x and y direc-tions, respectively.

Digital representation of the properly rewritten formof Eq. (3) is given as

i(j,k) = io(j,k) + g(j,k) exp(i2irqok/N)+ g*(j,k) exp(-i2rqok/N), (4)

where

g(j,k) = '/2h(j,k) exp[iou(jh)], (5)

with io(j,k), h (j,k), and 9 (j,k) now being digital repre-sentations of corresponding terms in Eq. (3); an asteriskdenotes a complex conjugate, N is the number of samplepoints in the former y direction, and qo stands for theimposed carrier frequency, namely, fo = qoAf, where Af=1/NAy.By Fourier transforming Eq. (4), with k being the

variable of integration, and keeping constant j, theFourier transform of the intensity distribution alongany j column is obtained:

I(j;q) = Io(j;q) + G(j;q - qo) + G*(j;q + qo), (6)

where I(j;q), Io(j;q), and G(j;q) are the discrete Fouriertransforms of i(jk), io(j,k), and g (j,k), respectively, andq is a discrete frequency related to the variable k.

In comparison with the possible variations of theimposed spatial carrier frequency, i(j,k), io(j,k), and0{j,k) are expected to be slowly varying functions of the

spatial coordinates. The three terms of Eq. (6) aretherefore separated in the discrete frequency domain.

Spatial distribution of the phase 0(j,k) could be de-rived either from the term G(j;q - qo) or the termG*(j;q + q) of Eq. (6).

If the intensity spectrum, Eq. (6), is multiplied by thefollowing rectangular window function:

Wj;q) = l for qo-Aq < q < q + Aq,O otherwise,

(7)

whose width 2Aq is narrow enough, all the contributionsto the spectrum except from the second term of Eq. (6),e.g., G(j;q - qo), will be canceled. By introducing adummy variable, q' = q - qo, the spectrum of g(j,k) isextracted. This substitution is equivalent to a spec-trum shift toward the origin in the frequency domain.

By the inverse Fourier transform of the spectrumobtained as above, the complex function g(j,k) definedby Eq. (5) is obtained, with its real Re[g(j,k)] andimaginary Im[g(j,k)] parts, enabling the phase calcu-lation at each point of the object surface along the jcolumn:

Im[g(jk)]OUM~k = arctan Ik = 0,1,2,. .,N - 1.Re[g(j,k)]

(8)

At this stage of calculation a discontinuous phasedistribution is obtained since the arctan subroutinegives the principal values in the range from -7r to r. Asimple program based on the algorithm given in Ref. 11converts this discontinuous phase distribution curveinto a continuous one. However, this is only a relativephase distribution curve since the initial phase offsetvalue stays undetermined and is thus chosen to be zero.The absolute phase difference distribution can be ob-tained if the absolute value of the phase difference isknown at any particular point of column j. Comparingthis phase value with the calculated one the requiredphase offset is determined. The phase distribution

15 May 1984 / Vol. 23, No. 10 / APPLIED OPTICS 1475

Fig. 1. Principal components and layout of the system for digitalevaluation of the spatially heterodyned sandwich hologram fringe

patterns.

Fig. 2. Double-exposure-like interferogram of a slightly bent can-tilever beam.

correctly corresponding to the surface deformation isfinally derived by adding this phase offset to the cal-culated phase values at each point of the column underconsideration.

For the 2-D phase distribution analysis of all M X Nsample points the following procedure should be ap-plied. The sandwich hologram is tilted, the axis of tiltbeing parallel to the x direction. From the interferencefringe pattern which is now a superposition of the objectsurface deformation pattern and the pattern imposedby the sandwich tilt, a set of M relative phase distri-bution curves for each column c (jk) is calculated.This is accomplished by successive application of thedescribed procedure to all the columns of the digitized

image. To derive the relative 2-D phase distributionacross the entire object, it is further necessary to de-termine just one more 1-D phase distribution curvealong any row of the digitized image. To achieve this,the sandwich interferogram is repositioned back tocancel tilt fringes and is tilted again, but with the tiltrotation axis perpendicular to the former one, i.e., par-allel to the y direction. From intensity variation acrossany row, i(j,ko), ko = const, a phase distribution 0,r(jko)is computed again. By comparing the phase valueskc(j,ko) with rU(,ko) at each point (j, ko), a set of Mphase offset values 00(J) for each j 0,1,2, .. , M - 1is obtained. Thus, a complete 2-D phase distributionis given by the summation 0q(j,k) = k,(jk) + 0o(j),] =0,1,2,. .. ,M - 1; k = 0,1,2, ..., N-1.

Ill. Experimental Results

To test and evaluate the method described for auto-matic analysis of sandwich interferograms, a numberof sandwich interferograms of a simple object weremade. The object was an aluminum cantilever beamwhich could be precisely bent out of the plane of a rigidand heavy supporting frame. The bending force wasapplied to the middle of the beam at a distance of one-third of its length from the free end.

Two plates of the sandwich interferogram were ex-posed, developed, glued together in the usual manner,and fixed in a motor-driven two-axes rotation stage.

The instrumentation setup used in all the measure-ments is illustrated in Fig. 1. Interference fringes ob-served by a TV camera were continuously displayed ona TV monitor. The camera signal was fed simulta-neously into the input of a video frame store (ColoradoVideo, Inc. model 274B). A particular frame was storedin the digital memory, with a spatial resolution of 256X 512 pixels and with an intensity resolution of 8 bits.The current content of the frame store memory wasdisplayed on another TV monitor. A software routineexecuted by a programmable calculator (Hewlett-Packard 9825A) and interfaced to the frame store per-formed all the calculations which provided a plot ofdeformation data at a comfortable speed.

The interferogram of a slightly bent cantilever beamis shown in Fig. 2, representing a fringe pattern as ob-served at zero sandwich tilt. This is achieved by repo-sitioning the sandwich interferogram exactly in thesame position it had during exposure. In this way adouble-exposure-like interference pattern was obtained.Such a pattern could not be obtained by manipulatingthe sandwich interferogram in the rotational stage, sincethe phase difference imposed by the tilt and the tiltangle itself are directly proportional. Practically, itmeans that for a certain tilt angle no new fringes aregenerated, but the original deformation fringes changetheir position across the object.

The interferogram in Fig. 2 is a type of interferogramon which automatic analysis methods based on thefringe peak detection could be applied. A low overallnumber of fringes and speckle noise do not permit highaccuracy of deformation measurements.


s(k) (/j k): j=I18256

224

192

760

128

96

64

32

0 k

0 0 N 0 N co

Fig. 4. Fringe pattern intensity distribution along column j weightedby the hanning window s(k) = [1 - cos(27rk/N)]/2.

I(j.q): 18256

2241

192

160

Fig. 3. Interference pattern of a slightly bent cantilever beam spa-tially heterodyned with the carrier frequency introduced by the

sandwich hologram tilt.

128

96

64

Proper manipulation of a sandwich hologram placedin a two-axes rotation stage is carried out until thenumber of fringes produced is large enough, giving riseto the spatial carrier frequency. A fringe pattern ob-tained in this way which is suitable for analysis is shownin Fig. 3. The TV monitor display contained only asegment of a digitized image, the beam and a part of thesurrounding frame. This pattern was produced byrotating the sandwich interferogram around an axisparallel to the primary deformation fringes.

It is worthwhile to note at this point that, in the casesof poor quality of the digitized image, caused for ex-ample, by low contrast or high speckle noise, some cos-metic improvements could be made by use of estab-lished digital image processing algorithms, such as in-tensity stretch, image smoothing, etc.

The white vertical line across the cantilever beam inFig. 3 indicates a set of 512 points of the digitized imagefor which the phase determination was done. Figure4 is a plot of the intensity distribution along these 512pixels, multiplied by a hanning window function. 1 2

Abscissa k corresponds to the vertical coordinate withthe origin at the bottom of the image. On multiplyingthe intensity distribution with a hanning window,smoothing of the spectral data is achieved, and the in-fluence of sharp edges is eliminated.

Variations in the overall intensity in the depth ofmodulation and in the speckle noise are easily identifiedin Fig. 4. The corresponding amplitude spectrum ob-tained through discrete Fourier transformation of the

32 K. AAL.4_AA LAA.I

a -0 N 70 o M 770 N 0 N IED-N N X

q

Fig. 5. Spatial frequency amplitude spectrum of the fringe patternintensity distribution from Fig. 4. Discrete variable q = 0,1,2,...,N- 1 at the abscissa corresponds to the discrete spatial frequency f =qAfforq SN/2andf= (q-N)AfforN/2 <q <N-1. Thermod-

ulus I(j;q) is normalized to the value 255.

(kj,) j118

1440

1000

r 720

360

°~~~- 3 Va X

Fig. 6. Relative phase distribution caused by deformation as ob-tained from the fringe pattern sample indicated by the white vertical

line in Fig. 3.


H ST -

:1

intensity distribution along 512 sample points of Fig.4 is presented in Fig. 5. Note the separation of the threeterms of Eq. (6). The value of the carrier frequency fo,by which this spectrum is to be shifted toward the origin ;g'of the spatial frequency axis, is estimated either fromthe frequency spectrum plot or from the referencesurface fringe pattern. In the case considered here, thereference surface was the surface of the rigid framearound the beam. Before the shift, the spectrum wasmultiplied by a rectangular window function of the typegiven in Eq. (7), so that the dc term as well as the con- 1 Hitribution of negative frequencies were excluded. Thisspectrum was inversely transformed, and the continu-ous relative phase distribution was calculated. A plotof this phase distribution is given in Fig. 6, with theinitial phase value equal to zero, which was justified herebecause the frame was kept undeformed. Since the G il . .,

Fig. 9. Double-exposure-like interferogram of deformed clamped-supported beam

experimental arrangement has been carefully designedto make only negligible variations of the sensitivityvector, maintaining its direction almost perpendicularto the object surface, the curve representing the phasedistribution also represents the out-of-plane deforma-tion of the beam.

On the basis of a phase distribution like the one inFig. 6, calculated for all vertical lines across the beam,the fringe pattern was generated in the computermemory and displayed on a TV monitor. A high degreeof similarity between the fringe pattern of Fig. 1, dou-ble-exposure-like and the computer simulation shown

ILL in Fig. 7, proves that the correct phase distribution was

Fig. 7. Computer-generated fringe pattern corresponding to zero obtained. A 3-D perspective plot of the beam defor-sandwichtiltangle(comparewithFig.2). mation obtained through the spatial heterodyning

phase determination procedure is shown in Fig. 8.To analyze the sensitivity of this procedure to the sign

of the change of the phase difference, the test object wasrearranged in such a way that out-of-plane movementof the beam upper end, formerly free, was blocked, thusforming a clamped-supported beam.

This beam was again carefully deformed and theobtained double-exposure-like fringe pattern is pre-sented in Fig. 9. If such an interferogram is to bequantitatively interpreted, the fringe order has to beassigned to each fringe, and this could be done only witha prior knowledge of the type of deformation itself.

To apply the described method, the sandwich inter-ferogram was adequately tilted, so that a high enoughspatial carrier frequency was superimposed over the

Fig. 8. Perspective plot of deformed cantilever beam. original deformation fringe pattern, resulting in the


fringe distribution presented in Fig. 10. Followingexactly the same procedure as before, the phase wasdetermined for all the points along the white verticalline. The result is given in Fig. 11, where the solid linecorresponds to computed phase distribution by themethod presented here, while the dashed line representsthe deformation of the clamped-supported beam cal-culated theoretically.

By successive applications of the same algorithm toall the vertical lines across the beam, the fringe patternwas computer generated and is presented in Fig. 12.

Fig. 10. Spatially heterodyned interference pattern of deformedclamped-supported beam.

0(j.A, j= Itj

1440

t000

r 720

360

0o

Fig. 11. Sctained fromline in Fig. '

Fig. 12. Computer-generated fringe pattern which would be ob-tained for zero sandwich tilt angle (compare with Fig. 9).

The results presented explicitly prove that with thismethod it is possible to distinguish between the eleva-tion and depression of a deformed surface. The onlyambiguity left is whether the obtained deformation isthe real one or is the real one's mirror image. Thisambiguity is removed taking the direction of the tilt intoaccount (toward the object or away from it), in order toobtain a spatially heterodyned interference pattern.

IV. Conclusion

A new method for automatic computer determinationof object deformation from its sandwich hologram isdeveloped. By spatial heterodyning the object defor-mation fringes with the spatial carrier frequency su-perimposed by the sandwich tilt, a fringe distributionis obtained, suitable for an application of the Fouriertransformation method of fringe pattern analysis. Itis shown that subfringe sensitivity and automatic fringeorder determination, i.e., discrimination between ele-

/ \ vation and depression, are achieved by this procedure./t \ Thus Abramson's "rose of error" has faded away.

/t' / 'The use of the rigid body motion fringes as a spatialcarrier frequency, employed for heterodyning the de-formation fringe pattern, has also been tested with

k 3 ,success. 13 Therefore, the rigid body motion, an historicadversary of quantitative interferometry, can be turned

lid line, deformation caused phase distribution as ob- into a useful ally.the fringe pattern sample indicated by the white vertical The pronounced flexibility of the method presented7; dashed line, theoretically calculated deformation of a in this paper makes it attractive, especially when

clamped-supported beam. quantitative measurements have to be done on an object


in a difficult environment. It does not require morethan the usual sandwich holography arrangement.Quantitative interpretation can be performed a post-eriori, either by direct reading of the sandwich holo-gram, or by reading interferograms taken from thesandwich hologram with a photographic camera.

The authors thank I. Jovanovi6 for computer in-terfacing and the FFT routine implementation.

References1. W. Schumann and M. Dubas, Holographic Interferometry

(Springer, Berlin, 1979).2. R. K. Erf, Ed., Holographic Nondestructive Testing (Academic,

New York, 1974).3. R. Dindliker, B. Ineichen, and F. M. Mottier, Opt. Commun. 9,

412 (1973).

4. P. Hariharan, B. F. Oreb, and N. Brown, Proc. Soc. Photo-Opt.Instrum. Eng. 370, 189 (1982).

5. Y. Seguchi, Y. Tomita, and M. Watanabe, Exp. Mech. 19, 362(1979).

6. A. Choudry, Appl. Opt. 20, 1240 (1981).7. W. R. J. Funnell, Appl. Opt. 20, 3245 (1981).8. N. Abramson, The Making and Evaluation of Holograms (Aca-

demic, London, 1981).9. S. Nakadate and H. Sato, Proc. Soc. Photo-Opt. Instrum. Eng.

370, 180 (1982).10. N. Abramson, Appl. Opt. 13, 2019 (1974).11. M. Takeda, H. Ina, and S. Kobayashi, J. Opt. Soc. Am. 72, 156

(1982).12. L. R. Rabiner and B. Gold, Theory and Application of Digital

Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975),p. 92.

13. Rigid body motion-assisted double-exposure subfringe inter-ferometry has been developed and will be published soon.

PROMOTE PHYSICS!

ORDER T-SHIRTS

ANDTACKLE SHIRTS

PRICES PER SHIRT(All orders must be prepaid.Quantity discounts apply to totalnumber ordered: any size or style).

Children(Torque design

Tee Shirt Adult only)1-3 $8.00 S6.004-9 7.50 5.5010 or more 7.00 5.00

Tackle Shirt1-3 13.00 11.004-9 12.50 10.5010 or more 12.00 10.00

Please mark appropriate boxeswith quantity desired. Makechecks payable to: Society ofPhysics Students, in U.S. dollarsdrawn on a bank in the U.S.

StyleSmall Medium Large X-Large34-36 38-40 42-44 46

Heisenberg Tee I

Torque Tee

Charm Tee

Heisenberg Tackle 7

Torque Tackle

Charm Tackle

Children's Sizes 6-8 10-12 14-16

Torque Tee lI

Torque Tackle I I

Postage and Handling $2.Total Enclosed S

00


Send orders to:Tee-Shirt Department.Society of Physics Students.335 East 45 Street,New York, NY 10017

Documents

Ultrahigh resolution sandwich holography