Journal of the

OPTICALOf

SOCIETYAMERICA

VOLUME 55, NUMBER 12 DECEMBER 1965

Interferometric Vibration Analysis by Wavefront ReconstructionROBERT L. PowELL AND KARL A. STETSON

Institute of Science and Technology, The University of Michigan, Ann Arbor, Michigan 48107

(Received 29 March 1965)

A study of the Leith-Upatnieks hologram for the time average of the coherent wavefronts scattered froma vibrating object is reported. The image reconstructed by the hologram is found to contain a system ofinterference fringes which map contours of constant vibration amplitude, thus providing a method ofanalysis of the vibration of objects with arbitrary surfaces. Experimental results are presented andinterpreted for a simple periodic vibration; statistical motion of objects is discussed.

I. INTRODUCTION

AIHIGHLY successful two-beam extension of theAGabor hologram method of lensless three-dimen-sional photography has been reported by Leith andUpatnieks." 2' 3 The hologram is a photographic record-ing of the pattern of interference between two beams,one of which is characterized by reflection from, ortransmission through, a subject. The other is an analyticbeam, collimated, diverging spherical, or convergingspherical. These two are usually called the subject andreference beams, respectively. Such a recording willdiffract light to form a replica of the subject beam when-ever the recording is illuminated by a replica of thereference beam.

The process depends for its versatility upon the highcoherence of modern gas lasers and upon the high-resolution of spectroscopic plates such as EastmanKodak 649F. Another equally critical parameter is themechanical stability of the optical elements used inmaking the hologram. Since the high-resolution emul-sions are of low speed, exposure times of ten minutesor more are not uncommon; this places an extremerestriction on the stability of the fringe pattern beingrecorded. Even imperceptible motions may completelyblur out the recording which would occur under stableconditions. If the motions are not too severe, however,they will alter the image formed by diffraction from the

'E. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 10 (1962).2 E. Leith and J. Upatnieks, J. Opt. Soc. Am. 53, 12 (1963).E. Leith and J, Upatnieks, J. Opt. Soc. Am. 54, 11 (1964).

hologram in a manner that is characteristic of the mo-tion itself. This effect is the subject of this paper, andit leads to a method for determining the vibration pat-tern of a vibrating object. The object can be of arbitraryshape, and can be either specularly or diffusely reflecting.

II. EXPERIMENTS

Figure 1 is a schematic diagram of a typical systemfor making holograms. The hologram is recorded on aglass plate coated with 649 F emulsion and the platerecords the fringes formed by interference between thereference beam and radiation scattered from the object.After development, fixing, and drying, the hologram isilluminated by a beam of light which resembles the origi-nal reference beam. The first-order diffraction from thehologram appears to be diverging from a reconstructionof the original object behind the plate. If the hologram

MT

H

FIG. 1. Experimental arrangement for hologram recording.B, Beam from laser; BS, beam splitter; LP, lens and pinholecombination; M, mirror; MT, mount; S, solenoid; FC, film can;H, hologram plate.

1593Copyright © 1965, by the Optical Society of America.

R. L. POWELL AND K. A. STETSON

is illuminated by the conjugate of the original referencebeam, the first-order diffraction appears to converge toa real image of the object in front of the plate. By conju-gate, we mean a wave of the same form as referencebeam but with opposite directions of propagation.

In our experiment, the object is a 35 mm film can,the bottom of which is magnetically coupled to a sole-noid mounted inside the can. The can and solenoid aremounted on a rigid vertical surface so that the bottomof the can is easily visible through the hologram plate.A power-amplifier output is wired to the terminals ofthe solenoid and an audio-signal generator is used as theinput source for the amplifier. With this arrangement,the frequency and amplitude of excitation to the bottomof the can are separately controlled. At various fre-quencies the resonances of the can bottom can bedetected by the intensity of sound propagating fromthe can.

The purpose of this experiment was to determine theeffect of sinusoidal motion on the hologram-makingprocess. Figures 2(a), 2(b), and 2(c) are pictures ofreconstructions from holograms made of the film canat its lowest frequency of resonance. The only changein the system between these three holograms was theamplitude of excitation. The rings are not mechanicalnodes, but rather loci of equal amplitudes of vibration.A second example is shown in Figs. 3 (a), 3 (b), and 3 (c),all of which were taken at the second resonant freqeuncy,but with three different amplitudes. Here the line acrossthe middle of the can is a node of vibration of the can,and the contours to either side are loci of constantamplitude of vibration. Figures 4(a) through 42(g) show

(a)

Cc) I

(b)

(a)

(c)

A(b)

FIG. 3. Reconstructions ofthree holograms of the film-canbottom with three progres-sively increasing amplitudes atthe second resonance of the canbottom.

various other resonances of the can bottom as frequencyof excitation was increased.

III. THEORY

An analysis of information that is recorded on thehologram illustrates the formation of fringes in imagesreconstructed from the hologram made of vibratingobjects. Consider the planes P1 and P2 in Fig. 5, wherePi is just in front of the object and P2 is the plane ofthe hologram plate. Denote the complex scalar fieldscattered from the object as S(x,y,z) for all points to theright of plane PA, including the hologram plate. Con-sider as the source of S the complex amplitude distribu-tion in Pi,

S(Xy,0)3So(xo(,yo).

For the purpose of interpreting the role of a time-varying S in the pattern recorded in the hologram, it issufficient now to indicate formally the change of S dueto a disturbance of So at the object. A position changeof the object relative to its illuminating field disturbsSo and S into a new set of fields. Local distortions of theobject surface likewise disturb So and S. Denote such adisturbance as

So(xo'yoxt) = Sot

S(x,y,z,t) = St.

FiG. 2. Reconstructions ofof three holograms of a 35 mmfilm-can bottom with a pro-gressive increase in amplitudeof excitation at the lowestresonance frequency of the canbottom.

(1)

Next add to S, a mutually coherent reference field,

A (x,y,z)ei (x Y, z).

For our case the reference field is essentially a sphericalwave, diverging from a reference-beam lens-pinholecombination in Fig. 1. The total field E at P2 is

E(x,y,z,t)=Aei¢+S, (2)

where the factor ei t is suppressed. The intensity of this

1594 Vol. 55

l

December1965 VIBRATION ANALYSIS BY

the object field alone. If there is a sufficient differencebetween the directions of propagation of the object fieldand the reference field in the region of space occupied

lb by the hologram, the diffractions of the re-illuminatingbeam produced by the intermodulation terms [thesecond term of (4)] are separated from image formingdiffractions [the third and fourth terms of (4)].

In order to interpret the influence of the time depend-ence of St on the recorded information, it is instructiveto compare the hologram recording (4) with the hypo-thetical recording for a scene of an ensemble of objectreflecting surfaces, distributed along positions that the

(d) actual vibrating object occupies successively during therecording interval. The field at P2 owing to a set of Nsuch objects, and the reference field would be

I NEav (x,y,z) =-E tnS.+A ei+,

t4 1(5)

where 4n represents the time spent by the moving objectat position n. In the limit of a continuous distribution of*

(f) objects in the ensemble, the field at the hologram platewould be

1 r trEgah(ixcy)=- r StdtimAe-i.

tr J

FIG. 4. Reconstructions ofseven holograms of other reso-nant frequencies of the samecan bottom.

field is the function which produces a latent image inthe photographic emulsion. The quasi-instantaneousvalue of the time-varying, latent-image field is expressedas the product of (2) and its conjugate,

EE*= Al+ I St I 2+A e-St±+A eiOSt*. (3)

The photographic record obtained is the time average of(3) for the exposure interval lr,

1 ir 1 pr

- EE*dt=A 2+- I IStI2dttrJ t J 0

1 C tr 1 ptr+AeiO- J Stdt+Aeit J| St*dt. (4)

The third and fourth terms of (4) are the recordedhistory of interference between the object and referencefields. Their sum is the pattern component of the re-cording which diffracts part of a suitable re-illuminatingbeam into images of the hologrammed subject.1 Thesecond term is a history of mutual interference within

(6)

The photographic record for this timne-independent fieldmay be written,

PavEEav*=A +A ft Stdt +AeriO- Stdt

1 for+Aeio- J St*dt. (7)

The second term of (7) is the record of interferencebetween the fields from the ensemble of objects. Itconsists of both interference patterns within each objectfield, alone, and the pattern of mutual interference be-tween the fields of the ensemble. The presence of thelatter terms in (7) is the only difference between (7)and (4). However, again, because there is a sufficientdifference between the directions of propagation of theobject field and the reference field in the region of spaceoccupied by the hologram, diffractions of the re-illuminating beam owing to the second term of (7) are

-PPI

YjLscxs,'), ,,xo

PI

S(Y, J)

FIG. 5. Reference frame for analysis of vibrating object.

I ,1 1 1 ,1 1 Em-

WAVEFRONT RECONSTRUCTION 1595

R. L. POWELL AND K. A. STETSON

separated from image-forming diffractions. The thirdand fourth terms of (7) are the pattern of interferencebetween each field of the ensemble and the referencefield. They are recorded simultaneously into the holo-gram, whereas the third and fourth terms of (4) arerecorded sequentially. Nevertheless, these two pairs ofterms contain precisely the same recorded information.The diffractions of the re-illuminating beam by thethird and fourth terms of (7) produce the same recon-structed images.

Two points, pertinent to the interpretation of holo-grams of vibrating objects, may be drawn from com-parison of (4) and (7): (a) The modification, due tomotion, of terms in the hologram plate recording whichgive rise to the Leith and Upatnieks images,

ie-ij StdI+eiOk SI*dtI

is the same as if the sequence of object fields had existedsimultaneously. The effect, which is perceived in thehologram reconstruction, is to see an image which isthe same as that of an object which had been a compos-ite of a set of optically nonobstructing objects. Thefringes observed in an image obtained from a hologrammade of a sequence of object fields are identical to thefringes formed by the interference between the variousmembers of the set of object fields. The hologram re-cording of a sequence of object fields is independent ofthe order of sequence. (b) The difference between thesequential and simultaneous recordings (4) and (7)results in different diffracted fields propagating at smallangles to the reilluLminating beam. These are separated,in most holography, from the image-forming diffractionfields, which propagate at large angles to the re-illumi-nating beam.

The purpose of establishing the equivalence betweena sequential recording and a simultaneous recording, asdefined in the previous paragraphs, is to make use ofthe physical interpretation of the hologram reconstruc-tion which the hypothetical ensemble set of objectsprovides. Any hologram reconstruction can be con-sidered an interfering superposition of the set of imagesthat correspond to all positions the object occupiedduring the recording interval. The interference fringesare localized wherever the wavefronts from the set ofimages are suitably alike. In the case of simple boundedmotion of the object, reported in this paper, they arelocalized near the recorded location of the object;however, under suitable conditions they may be local-ized near any plane of the system. It is interesting tonote also that a hologram reconstruction of a stationaryobject is a degeneration of the interferometric recon-struction described, to the case where all images coincideand give nothing but constructive interference.

In order to perceive a hologram reconstruction, andany of the interference effects that may occur owing to

motions of the object during the recording, we mustimage the wavefronts diffracted by the hologram re-cording when it is illuminated by a suitable replica ofthe original reference field. This may be done either byusing a lens system such as the eye or by illuminatingthe hologram with a conjugate to the reference field,that is, a field with the same shaped wavefronts butopposite directions of propagation. In this latter casethe wavefront focuses into an image of the holographedobject without the aid of lenses.' In any case, however,the usual behavior associated with imagery of lightapplies, i.e., the hologram must be illuminated with abeam of finite aperature and the lens system must alsohave a finite aperature. Aberrations may also occur inthis imagery. It is suitable, therefore, to describe animage of the hologram reconstruction as the convolutionof the complex impulse response of the imaging systemwith the object of the hologram, in our case the ensembleobject,','

-lSotdt.

The image is

I(x,,yi)= ff {- fSo(xo~yoxzoO)dtl

XK(x,-xo, y,-yo)dxodyo, (8)

where K(xovyo) is the complex impulse response of theimaging system, and

Zo= f(xo,yo)

describes the object surface.The impulse response K is, in general, nonzero for a

range of x0, yo which is large with respect to either therange of object motion or the fine detail of the surfaceof the object. Still, it is nonzero for a small range ofx0, yo, with respect to the total object for any good imag-ing system.

Now let us write the time dependence of So explicitlyas a set of time-dependent translation terms x', y', andz' for the xo, yo, and zo coordinates, respectively,

So(xo,yozoxt)0=SO(Xo-X', yo-y', zo-Z')

Let us also redefine the coordinate system so that thez axis lies along the optical axis of the imaging systemdescribed in Eq. (8). Let us further postulate that z'is so small that the shape of the wavefront So does notchange after propagating a distance z'. Then we maywrite So as

So(x-o-x', yoy, Zo-z')=SS(xo-X', yo-y', zo)eiO(Z').

4 M. Born and E. Wolf, Principales of Optics (Pergamon Press,New York and London, 1964), Chap. 9, p. 480.

6 E. H. Linfoot, Fourier ifetliods in Optical Image Evaluation(The Focal Press, New York and London, 1964).

1596 Vol. 55

December1965 VIBRATION ANALYSIS BY WAVEFRONT RECONSTRUCTION 1597

Next let us also assume that the impulse responsefunction K is sufficiently large with respect to x' and y'and is sufficiently well-behaved with respect to ampli-tude and phase variation, that to a first-order approxi-mation its value is the same at xo-x' and yo-y' as itis at xo and yo. Thus we use the approximate equality

K(xi-xo, y1-yo)-K~x1- (xo-x'), yi- (yo-y')]-

Finally, we assume that x', y', and z' are slowlyvarying functions of x0 and yo, and that over the regionof nonzero K, which defines the practical limits of inte-gration, they may be considered constant. Making useof all of these conditions we may rewrite Eq. (8) as

1 .tr .kIxlyl)= ei¢z)J So(xo- x/ yo-z Z" o)

Ir o -yk*K[xi-(xo-x'), yr-(yo-y)]adxodyodt, (9)

inverting the order of integration. The double integralwith respect to x0 and yo may be recognized, to withinthe limits of our approximations, as the time-invariantimage of the stationary object S0 . What our approxima-tions have done is to limit the discussion to imagingsystems which are unable to resolve the lateral motionsof the object. The above equation may be abbreviated as

I(x.,yi) = {l/tr ei4df)dtlIst(xl,yO), (10)

where IAt is the image of the stationary object.Let us consider in more detail the formation of ck(z').

Consider a vector displacement of the object whosemagnitude is defined as

r n (xo,yo) cos[cv+ 4u(xo,yo)]. (11)

Let 01 be the angle between the axis of observationand the vector displacement, and let 02 be the anglebetween the direction of propagation of the incidentlight upon the object and the vector displacement. Itcan be shown by simple geometry that the expressionfor the phase function g5(z') can be written as

4(z') = (27r/N) (cos01+ COS02)rn(X0,y0)

X cos[cot+ (xo,yo)]. (12)

Once the function k(xo,yo) is expressed in this form,it has been shown8 that the integral of Eq. (10) may beevaluated as the zero-order Bessel function of the magni-tude of the argument ':

I(xi,yi) = Joa(21 r/X) (cos0i+ cos02)m(xo,yo)]

XIst(xl,yl). (13)

It must be appreciated of course that Ol and 02 are inthemselves slowly varying functions of x0 and yo be-cause the object is usually illuminated with a spherical

6 H. Osterberg, J. Opt. Soc. Am. 22, 19 (1932).

wave and the hologram reconstruction images subtenda wide angle with respect to the axis of any imagingsystem.

An important observation as to the form of Eq. (10)has been made by Kozma of this laboratory. If thevector displacement r is a random function of timewhich is stationary and ergodic, the integral of Eq. (10)which is of the form of a time average may be expressedas an ensemble average. The ensemble average is writtenin terms of the probability density function of r,p(r). That is to say

1 rt 271i -M 1=- y exp k(cosoi+coso2 )r dt

r 27riJl exp -(Cosoi+cos02)r]p(r)dr. (14)

Because of the exponential nature of the functionexp[( 2 7ri/X) (cos0+cos0 2)r), however, the ensembleaverage written in Eq. (14) has a definition in itself.It is the characteristic function of the probability den-sity function and is the Fourier transform of p(r) withrespect to (27r/X) (cos0i+ cos02). Thus, we see again thatthe fringe pattern we view in the reconstruction isunique only to the probability density function of themotion of the object and not to the particular motionitself.

IV. APPLICATIONS AND LIMITATIONS

This technique may have widespread applications.Any system which operates by mechanical vibrationsmay be studied with the detailed analysis made possibleby this technique. Examples of such systems are audio-speaker diaphragms, musical instruments such as per-cussion or stringed instruments, or audio transducers ofmany sorts. Also the technique may be applicable tomodels of larger systems where vibrations are undesir-able, such as bridges, architectural structures, aero-dynamic, or hydrofoil structures. The chief advantageis that the structure under analysis need not be modified.No fibers need be attached, no sensing mechanisms needto touch the vibrating structure, measurements may bemade in vacuum or under water or any such transparentmedium; in general, the vibration detection is independ-ent of the physical system under measurement. Thesubject surfaces do not have to be polished or opticallycoated, and no mirrors have to be attached to the struc-ture under analysis. Yet the precision of measurementsmay be to within fractions of a micron.

There are several limitations of the system, however.The detection cannot be carried out in real time usingvisual persistence, and thus the hologram recordinggives only after-the-fact information about what hashappened, rather than an indication of what is happen-ing. The use of faster development processes might helpthis somewhat. Also, the vibrations of the object must

R. L. POWELL AND K. A. STETSON

not be too great in amplitude or the vibration informa-tion becomes impossible to detect. Each successivepeak in the image reconstruction becomes weaker asthe amplitude of the vibration increases. This dif-ficulty may be alleviated by holography with longerwavelengths. The stability of the rest position of theobject is also critical, as mentioned in the opening sec-tion of this paper. This may cause difficulties whenfreely suspended subjects are examined.

It is of historical importance to contrast this methodof vibration analysis with that reported in Ref. 6,cited in the mathematical analysis preceding this sec-tion. In this reference, Osterberg reported the use of aninterferometer to study the vibration modes of piezo-electric quartz crystals. The crystals are polished andthen used as one of the mirrors of the interferometer.Once a simple set of straight line fringes has been ob-tained, the crystal is excited by an alternating electricfield. The resonances of the crystal are detected by thechange in visibility and phase reversal of the fringes.In this case, visual persistence serves to integrate thevibrating fringe pattern and modify its visibility ac-

cording to the same zero-order Bessel function of thevibration amplitude we have described. This techniqueoperates in real time, since visual persistence may beused, at the price of having to optically polish the sub-ject, which, in the case of the peizoelectric crystal, isnot necessarily a disadvantage. It is hoped that thispaper, nonetheless, will stimulate new interest in theOsterberg process.

ACKNOWLEDGMENTS

The authors would like to acknowledge the interestof A. Kozma in the process, which led to his settingforth the interpretation of statistical motions presentedin this paper. Also, we would like to acknowledge con-versations with H. Osterberg which have helped furtheran understanding of the basic similarities of the twoprocesses. Finally, we would like to acknowledge thehelp of E. Leith and J. Upatnieks in clarifying our under-standing of the two-beam hologram process, and par-ticularly J. Upatnieks for proposing in the summer of1964 the use of hologram for analysis of vibrationspofobjects.

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 55, NUMBER 12 DECEMBER 1965

Laser Emission in Ionized Mercury: Isotope Shift, Linewidth, andPrecise Wavelength*

R. L. BYERI W. E. BELL, E. HODGES, AND A. L. BLOOM

Spectra-Physics, Inc., 1255 Terra Bella A venue, Mowntain View, California 94041(Received 29 April 1965)

The 6150-A laser transition in Hg II generated in a pulsed hollow cathode discharge, has been studiedunder high resolution. The line shows structure corresponding to the expected shifts from the even isotopes,with a spacing of approximately 26.5 mK (800 Mc/sec) between adjacent components, in rough agreementwith earlier results. The fullwidth at half-maximum of each component has been measured as 500 Mc/sec.The wavelength has been compared to that of thorium secondary wavelength standards and it has beendetermined that the vacuum wavelength of the "02Hg component is 6151.1650 .. The possibility of themercury hollow cathode laser as a laboratory wavelength standard is pointed out.

I. INTRODUCTION

LASER action from gaseous ions was first observedin a pulsed mercury-helium discharge."2 Although

many other types of ion lasers have been discovered,mercury is still of interest because the large atomicweight implies narrower emission profile linewidthsthan might be obtained from other ions. In this paperwe report the results of interferometric measurementson the hyperfine structure of the visible laser transition7p'P-1 -7s2S1, at 6150 A. We have found that the ex-

* Work supported il part by the U. S. Arny Signal Corps.t Present address: Dept. of Applied Physics, Stanford Univer-

sity, Stanford, California 94305.'W. E. Bell, Appl. Phys. Letters 4, 34 (1964).2 A. L. Bloom, W. E. Bell, and F. 0. Lopez, Phys. Rev. 135,

A578 (1964).

pected isotope structure is clearly resolved and thatthe linewidth of each component of the emission spec-trum is 500 Mc/sec, the width to be expected frommercury near room temperature. It is about one-thirdthe Doppler width in a helium-neon laser and is instriking contrast to the noble gas ion lasers,3 -' whichtypically have widths of the order of 0.1 cm-l (3000Mc/sec).

II. APPARATUS AND TECHNIQUE

The laser that was used in these experiments is shownin Fig. 1. The tube was filled with helium to a pressure

I W. B. Bridges, Appl. Phys. Letters 4, 128 (1964).4 E. I. Gordon, E. F. Labuda, and W. B. Bridges, Appl. Phys.

Letters 4, 178 (1964).5 W. B. Bridges, Proc. IEEE 52, 843 (1964).

1598 Vol. 55