Holographic Techniques for the Study of Dynamic

Particle Fields

J. D. Trolinger, R. A. Belz, and W. M. Farmer

The techniques of small particle holography are extended in an elementary way to show that multipleexposure holography can be used to study the dynamic properties of particle fields. Experimental tech-niques for performing such a study are then described. Finally, as a typical example, a multiple exposurehologram of an aerosol is presented with a portion of the data extracted from it. The hologram exhibitsa vast amount of information including the determination of the velocity and density field, size distribu-tion, flow structure, and diffusion rate.

Introduction

The visual observation of a volumetric array ofdynamic elements, having individual dimensions muchsmaller than the volume dimensions, could rarely beachieved before the advent of holography. It is pos-sible within the limits of holography to study such anarray while, attAhe same time, observing the motion ofall its elemen The extension of holography to thestudy of motion is in its very infancy, but for certainhighly specialized cases, it can already be applied topractical problems. The purpose of this paper is todescribe studies dealing with the determination ofsizes, densities, and velocity fields associated withdynamic particle arrays.

Small Particle Holography

The science of in-line holography of small particlesowes its origin and most of its development to Thomp-son and his co-workers.' A fairly complete treatmentof their work, and that of others, on the subject isincluded in a recent book by De Velis and Reynolds.2

The general approach to the theoretical problem con-cerns itself with the solution of the Helmholtz equationand subsequent fitting of the boundary conditions.To simplify the discussion, Babinet's principle is appliedhere.

The in-line hologram of a small particle is perhaps theeasiest of all hologram systems to describe theoreticallyand to record experimentally. Consider a plane wave-front of amplitude U (Fig. 1), which passes through aplane X7 - in which some object is placed. Theresulting wavefront with amplitude U (iF) is recorded inthe far field of the object in the x-y plane. Babinet'sprinciple states that if U('F) were added to the wave-

The authors are with ARO, Inc., Arnold Air Force Station,Tennessee 37389, and The University of Tennessee SpaceInstitute.

Received 9 September 1968.

front U' (f), produced by the object's complement,then, the resulting amplitude would be Uo.

It is of interest here to determine the intensity oflight U in the x-y plane which results when a disk(representing an opaque particle) is placed in the A-tplane. It is seen that

U, = U - Ua, (1)

where U,, is the amplitude in the observation planewhen the object plane is opaque except for an apertureof the same diameter as the disk. The resulting inten-sity in the recording plane is, therefore,

I = UUp* = U, 2

=+ U0 12 + I Ua - UOU* - Uo*Ua. (2)

It follows that the intensity UpU,* actually forms ahologram of the aperture which is complementary tothe disk. The simplest way to show this isto requirethat the photographic recording medium is relatedlinearly to the intensity so that its amplitude trans-mission coefficient can be expressed as

T = 1 - KI, (3)

where K is a film constant and I is the light intensitystriking the film. Even when the linear relation ofEq. (3) is not obeyed, a hologram is formed, but withless efficiency. This more complicated case is con-sidered later. When the processed photographic re-cording is illuminated by a plane wave Uo, the resultingwavefront emerging from the plate is U', where

U' = Uo(I - KI) = Uo(1 - KUU,*). (4)

From Eqs. (2) and (4),

U' = U0(1 -K U - K U,, 12)

+ UoUoKU.* + I Uo 2KU.. (5)

May 1969 / Vol. 8, No. 5 / APPLIED OPTICS 957

Object Plane Recording Plane

Quasimonochromatic Coherent 1YRadiation

z

Fig. 1. Geometry of the hologram formation process.

The last term in Eq. (5) is a wave exactly like the wavecreated b an aperture which is located behind therecording plane at the place where the particle hadpreviously been. This is the wavefront of a virtualimage. The second term represents a real-image wave-front in front of the photographic recording, while the

a

first term represents a spatially modulated plane wave.The photographic recording has thus been shown to bea hologram of a screen with an aperture the shape ofthe disk. Figure 2(a) is a high contrast, photographicrecording of the real image reconstructed by such ahologram. The object, a glass slide, was covered withopaque paint particles by allowing the aerosol from aspray paint can to fall upon it. The hologram wasformed by He-Ne laser illumination and Kodak SO-243film. When an image is reconstructed from an in-linehologram, a new hologram is formed in addition to theimage, and is superimposed upon the image. Thissecondary hologram is a representation of the dustparticles in the reconstruction system, plus any infor-mation stored on the primary hologram and all out-of-focus images (real or virtual). The reference wave,

b

C

TFig. 2. Reconstructed real image of a particle field. (a) High contrast; (b) low contrast (overexposed); (c) reconstructed real imagewith high pass filtering.

:958 APPLIED OPTICS / Vol. 8, No. 5 / May 1969

acting to form holograms of these objects, is the firstterm in Eq. (5). In fact, if the image represented bythe second term in Eq. (5) is in focus, the secondaryhologram of the third and first terms will be super-imposed upon it.

To illustrate this point, the same reconstructed imageshown in Fig. 2(a) was recorded photographically andoverexposed [Fig. 2(b)] to emphasize the existence ofthe secondary hologram and the spatial modulation ofthe first term. Some interference between the con-jugate and focused waves also exists, but only for largeparticles is this interference noticeable. It is often use-ful to remove most of the light caused by the first term.This can be accomplished by using a high pass (dc)optical filter in the reconstructing process. This willeliminate essentially all of the plane radiation. There-fore, returning to Eq. (5), and eliminating all planewave terms, U' becomes

U' = KUOUoUa* + K I U0 J2 Ua.

Fig. 3. Direction sensing of a multiple image by controlledexposure.

(6)

The term Ua can be further resolved approximately intotwo terms, a scattered wave U and an unscatteredwave. The scattered wave represents a tiny ring oflight scattered at the edge of the aperture, and the un-scattered wave represents light that passed far enoughfrom the edge that it was not scattered. Since theunscattered light is plane radiation, it is removed bythe high pass filter. The only portion of the wave-front that will be imaged, then, is the scattered radia-tion (the ring of light). Figure 2(c) is a photographicrecording of the same reconstructed image shown inFig. 2(a) and (b), except that the wavefront has beenpassed through a high pass optical filter. The second-ary effects have been removed.

High pass filtering does cause new problems, however,since in practice it does remove some low frequencyinformation as well as plane d radiation. Since thefilter removes all paraxial radiation, it does improve thez imaging sensitivity. In fact, as the diameter of thehigh pass filter is made larger, the accuracy of locatingthe exact image plane is improved, and the depth offield of the image is decreased. This is equivalent toremoving the central portion of a lens to decrease itsdepth of field.

The determination of particle sizes, shapes, and den-sities can be made by direct observation of the recon-structed image. In the reconstructed volume, one hasessentially a frozen three-dimensional array character-izing the original subject sitting in free space availablefor study by any optical instrument. Since the limita-tions of the technique are well covered in the references,they are not discussed here.

We have not found the usefulness of the technique tobe specifically limited to the Fraunhofer diffractionregion as suggested by Thompson.' If the particledensity is sufficiently low, high quality holograms canbe formed even when they lie in the Fresnel zone.Figure 2 (a) contains the reconstructed image of a 2-mmdiam particle located 50 cm from the hologram, whichis well within the Fresnel zone.

Fig. 4. Double exposure hologram of an aerosol spray.

Multiple Exposure Holography

The above discussion need be extended only slightlyto encompass one method of velocity determination ofelements in the array. A number of techniques havebeen proposed to accomplish this.A'4 In fact, many ofthe usual photographic techniques for velocity deter-mination can be extended to holography. These in-clude motion picture holography and stroboscopicholography. If the motion is slow, the extension isquite trivial. One simply makes a new hologram atdifferent time intervals and then superimposes thevarious three-dimensional images to study motion ofthe elements during the intervals. As the motionbecomes more dynamic, new requirements becomeevident. Exposure time for each hologram must bereduced, requiring higher optical power density, andthe intervals between exposures must be shortened.The Q-switched ruby laser represents the present limitin high power and short exposure. One quickly findsthat this laser is necessary for all but the slowest motion,particularly for small particles.

May 1969 / Vol. 8, No. 5 / APPLIED OPTICS 959

factors; these values range from about 10-300 usec,while typical flash lamps can be excited sufficiently fortimes in the order of 1000 usec. This means that onecould achieve up to about twenty pulses by multipleQ-switching with present-day systems. In fact, withno Q-switching at all, a laser usually generates, in itsnormal firing mode, twenty or more pulses which are10-200 nsec wide. Each pulse has a substructure also,but this substructure is of no particular interest for thisapplication.

An active Q-switch can be timed to open and closed_ * 25'electronically at the proper intervals. A passive Q-

switch can also be adjusted so that it opens and closesspontaneously at various intervals, but with much lesscontrol and repeatability. In the present experiments,passive Q-switches were used because of their lower costand because they produce better holograms. When thetechniques are carried beyond the research laboratory,active Q-switching would certainly be desirable becauseof their controllability.

(a)

Recording and Reconstruction

Consider the case in which a particle passes before aphotographic plate and two extremely short exposuresof the plates are made. If the emulsion continues toact linearly, Eq. (3), the resulting amplitude trans-mission coefficient will be given by Eqs. (2) and (3).

T = 1 -K(l o0 I + 1 U02 12 + U !2 + I Ua2 12 )

+ KUoiUti* + KUGi*Uai + KU02Ua2* + KUO2*Ua2.

(7)

flol

(b)

Fig. 5. Two planes of the reconstructed image of a doublyexposed hologram (region marked A in Fig. 4); (a) z = 15 cm;

(b) z = 16 cm.

Multiple exposure of an object can be accomplishedby using several lasers synchronized to fire at the desiredinterval. Fortunately, there are less expensive ways toachieve this. Multiple pulsing of a ruby laser can beaccomplished with either active or passive Q-switches.This is possible because the flash lamp which pumps theruby can be actuated considerably longer than the timerequired to energize and de-energize the ruby. The Q-switch is timed to open when the laser gain coefficientexceeds zero to allow lasing and then to close until thegain coefficient once more exceeds zero and so on. Theprocess can continue as long as the flash lamp pumpsthe ruby. The time required between pulses varies Fig. 6. Reconstructed image in a high density region (regionwith pumping rate, the ruby temperature, and other marked B in Fig. 4).

960 APPLIED OPTICS / Vol. 8, No. 5 / May 1969

15

14

13

12

2:

e

2vo

11

Ic

9

8

7'0 1 2 3 4

Position in Spray (cm)

5 6

Fig. 7. Axial speed profile across an aerosol spray.

Now, if the hologram is illuminated by a plane waveUo, a transmitted wave U' will have the form:

U' = Uoi[l - K(l U0o 12 + U0 2 12 + I Uai 12

+ I Ua2 2)] + KUol(UoU.I* + Uo2Ua2*)

+ K(I U 0 1 12 U.i + I U 01 I U 02 1 Ua 2 ). (8)

Equation (8) leads one to the conclusion that the realand virtual wavefronts of the particle are produced inboth exposures without any degradation. This canonly happen when the assumption of linearity [Eq. (3) ]holds true. The transmission vs exposure curve canbe expanded in terms of linear and higher order terms.The linear terms lead once again to the reconstructionof a double image. The higher order terms causeinterference and degradation of the hologram.

In a double exposure hologram it would be conven-ient if images constructed by the second exposure couldbe differentiated from the first to give a sense of direc-tion. That this can be achieved is shown, for example,by the third term in Eq. (8) (virtual image term). Ifthe product I U0 1 I I U0 2 differs from I U 0 1 12, the inten-

sity of the reconstructions will differ according to theratio. Figure 3 is a photograph of the reconstructedimage of a double-exposed hologram which was madewith the second exposure being twice the first. Thisillustrates the potential of the technique for directionsensing. A more accurate determination of the inten-sity ratio and additional discussion of nonlinearities hasbeen presented by Wyant and Givens.5

Figure 4 is the positive of a double-exposed, in-linehologram taken near the nozzle of a high pressureatomizer. A 60-p diam wire with a 300-p disk attachedto its center, was stretched across the field of viewnormal to the direction of flow. The hologram wasmade on an Agfa-Gevaert Scientia plate with a KoradKi ruby laser. The Q-switch was a passive cell filledwith a cryptocyanine dye solution adjusted for doublepulsing. The hologram contains a vast amount of data,a few of which are presented. Figures 5(a) and 5(b)are photographs of the volume reconstructed from thearea marked A in Fig. 4 with the camera focused onplanes which are 1 cm apart. Some of the in-focusparticle pairs are designated. The particles shown inthe reconstruction are water droplets with diametersvarying between 20 A and 150 . The velocities varyup to 15 m per second. Figure 6 shows a reconstructionof the area marked B. This region is difficult to analyzebecause the particle density was high, but even so, aconsiderable amount of information was extracted fromthis region. Figure 7 is the axial speed profile takenacross the center of the flow near the reference wire.Such phenomena as turbulence, droplet oscillation,breakup, collisions, acoustic phenomena, instabilities,and diffusion can be observed in this hologram.

Because of the amount of data contained in a holo-gram of the type under discussion, it would be highlydesirable to develop optical reduction techniques forprocessing the data into its final form. The hologramwould then serve as the input to an optical computerwhich is programmed to extract the desired data.

Other areas in which double exposure holography hasbeen applied successfully include electrolysis, bubbleformation, water jets, and other arrays of scatteringcenters. The technique shows promise in the study ofseed and contaminant materials in various types offlows, in the study of rocket exhausts, chemical reac-tions, impact, and other dynamic phenomena.

The research reported in this paper was sponsored bythe Arnold Engineering Development Center, ArnoldAir Force Station, Tennessee, under a contract.

References

1. B. J. Thompson, J. H. Ward, and W. R. Zinky, Appl. Opt. 6,519 (1967).

2. J. B. De Velis and G. 0. Reynolds, Theory and Applicationsof Holography (Addison-Wesley Publishing Company, Inc.,Reading, Mass., 1967).

3. J. D. Redman, J. Sci. Instrum. 44, 1032 (1967).4. J. D. Trolinger, W. M. Farmer, and R. A. Belz, Appl. Opt.

7, 1640 (1968).5, J. C. Wyant and M. P. Givens, J. Opt. Soc. Amer. 58, 357

(1968).

December 1970 issue will feature Eclipse Optics editor L. Larmore

May 1969 / Vol. 8, No. 5 / APPLIED OPTICS 961

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