Resolution Factors in Edgeline Holography

J. D. Trolinger and T. H. Gee

When an in-line Fresnel hologram of an object such as a projectile in flight is made, the reconstructioncomprises an image of the outside edge of the object superimposed upon a Fresnel diffraction pattern ofthe edge and an unmodulated portion of the reconstruction beam. When the reconstructed image isbandpass filtered, the only remaining significant contribution is that of a diffraction pattern which issymmetrical about an edgeline gaussian image of the object. The present paper discusses the applicationof this type of holography in accurately locating the edge of a large dynamic object, the position of whichis not accurately known in any dimension. A theoretical and experimental analysis was performed tostudy the effects of motion, hologram size, film type, and practical limitations upon the attainable resolu-tion in the reconstructed image. The bandlimiting effect of motion is used to relate the motion effectedresolution limit of holography to that of photography. The study shows that an edgeline can be ac-curately located even at high velocity normal to the edge.

IntroductionOne of the most successful applications of holography

is that of Fraunhofer in-line holography of small par-ticles.' 2 The requirements for high quality holographicreconstructions include satisfaction of the far field con-dition by individual particles and an object field trans-parency that is high enough to allow a considerableportion of the coherent beam of light to pass throughthe field onto the hologram where it serves as a referencebeam. If the far field condition is not satisfied, theinterference between the two conjugate images and thetransmitted reconstruction illumination badly de-teriorates the reconstructed image.' It has beenshown2 '3 that this degrading interference is causedlargely by the interference of the images with thetransmitted reconstruction illumination, and it can beremoved by high pass filtering during reconstruction.It was shown then that high quality holograms of largeobjects could be made with a single beam of light usingthe in-line methods if the reconstructed wave is highpass filtered to remove the dc component from the ob-served light." 4 The reconstruction then comprises anedgeline of the object that was holographed.

This type of holography does not, of course, lead tothe profound visual effects of standard off-axis holog-raphy; however, a certain number of problem areasexist in which the characteristics of edgeline holography

J. D. Trolinger is with ARO, Inc., Arnold Air Force Station,Tennessee 37389; T. H. Gee is with the University of TennesseeSpace Institute, Tullahoma, Tennessee 37388.

Received 29 June 1970.

appear to make its application highly advantageousover other types of holography. For example, if theapplication involves locating precisely the position andoutline of a dynamic object, edgeline holography can beused to perform this task more easily and more ac-curately than off-axis holography. Of course, if oneknows precisely where the object will be located in theplane normal to its direction of flight at the time of therecording, photography could probably outperformeither of the above methods. The specific applicationin mind during these studies involved the accuratelocation and determination of ablation of a projectilein hypersonic flight. The present method for per-forming this measurement is laser photography. Theinability to focus the laser camera exactly on the model(because one does not know the exact location of themodel soon enough to focus the camera) limits theaccuracy of this measurement to a few thousandths ofan inch. This resolution is sufficient in most cases forpositional measurement, but rarely is it high enough forablation observation.

The object of this paper is to report the results ofan investigation to see if edgeline holography could beused to perform these measurements with greater res-olution than existing systems give. The results showthat the application is feasible and that the resolutioncan be made higher with objects traveling up to thou-sands of feet per second. In addition, by using multipleexposure techniques, accurate measurements of thevelocity of fast moving objects can be made.

Theory

Spatial bandpass filtering of an image is performedin a variety of ways. One of the simplest of these is

June 1971 / Vol. 10, No. 6 / APPLIED OPTICS 1319

d,

Variable Bandpass Filter forOne-Dimensional Fourier Transform

dc Spot Variable High-Pass\VI / Filter

Low- ~er ]

2b fg

I Q- __ ,~~-- Image Plane

Filter

f

Object ImagePlane Plane

Fig. 1. Variable bandpass filtering apparatus. For two-dimensional transforms, fixed annular apertures are used.

shown in Fig. 1. An object field is imaged by a lensthrough a filter that is placed in the back focal plane ofthat lens. The transmission of this filter defines thetype of information that is removed from the image ofthe object. Low pass filtering occurs regardless of theexistence of an aperture in the focal plane because of thefinite size of the lenses that are used or because of theeffective system aperture that may arise from someother source. A bandpass filtered image can be ex-plained simply in terms of straight edges. The band-pass filtered image of a straight edge appears as adiffraction pattern which is symmetrical about thegaussian image of the edge. The equation for theamplitude in the bandpass filtered image is easily de-rived.' Assuming unity magnification,

G(x) = 1/7r[Si(2bx) - Si(27rax)], b > a > 0, (1)

where b is the object spatial frequency at the upper bandstop and a is the frequency at the lower band stop. Anumber of useful conclusions can be drawn from thisequation. The equation has been plotted in Fig. 2 forthe frequency a = 1, b = 2, 5, 10, and 50. The di-mensions of a and b are the inverse dimensions of x,and this makes the graph extremely versatile. Thegraph is applicable for any frequency ka and kb (wherek is an arbitrary factor) if the dimension x is replaced byx/k. Since the function G(x) is an odd function, theintensity that is a square of this function will be sym-metrical about the gaussian edge (x = 0). The darkline of symmetry that coincides with the gaussian imageof the edge will henceforth be termed the dark field edge-line.

The quantities that can play an important role inaccurately locating the dark field edgeline are the sharp-ness of the diffraction lines, the distance to the first

maximum, As, and the distance to the first zero,These are given by the following:

dG-G = 2(b-a)

and if b>> a,

Ax 0.4/a,

AT io 1/2b.

Ax.

(2)

(3)

(4)

Typical cases could involve values of b exceeding 104lines cm-' leading to a As less than 1 ,u.

0.6'

- 50 WcmI0.51

0.4.

0 .3

-0.2-

-b -5 cmt

Gil . [Si t2eeb. - Si 2erax)]

b - Upper Bandstop, cm1a - Lower Bandstup Equals One cm-1x - Distance from the Gaussian Edge Image, cm

Gist - Amplitude. arbitrary units

-b 2cm-1

0.2 0.4 0.6 0.8 1.0. Cm

1.2 1.4 1.6 1.8 2.0

Fig. 2. Amplitude in a bandpass filtering image of an edge.

Factors That Reduce Resolution in Practice

The resolution of a photographic image is decreasedeither by defocusing or by motion blur during the re-

1320 APPLIED OPTICS / Vol. 10, No. 6 / June 1971

> l 4S1

cording. Either of these lead to a rounding of sharpedges that would have occurred in the image. Motionblurring affects only edges normal to the motion direc-tion. Defocusing does not apply, of course, to thehologram, but motion has an equivalent effect on thereconstructed image as it would have on the photo-graphic images. High frequency fringes running per-pendicular to the motion direction are smeared togethergiving the effect of low pass filtering (i.e., broadeningand rounding the sharp lines). When collimated lightis used, the fringe frequency on the hologram possessesa one-to-one correspondence with the spatial frequencycontent of the diffraction object radiation. Suchholograms will exhibit a high frequency cutoff for edgesthat have components normal to the motion. The cut-off frequency is given by

b= 1/VAt,

50-micron-diameter Wire

... _.

Fig. 3. Arrangement of cylinder and straight edge.

(5)

where At is the recording time and V is the velocitycomponent normal to the edge. Assuming the fringesmearing to be equivalent to low pass filtering at thecutoff frequency b then the line As, according toEqs. (4) and (5), must be at least as large as

As, = VAt, (6)

which is also the blur that would occur in the photo-graphic image. The significance of this result is that itsuggests that the limitations of photography andholography for this type of application are equivalent.

ExperimentalThe analysis of the last section was based upon a

thin, straight edge object. Many applications areconcerned with spheres, cones, or cylinders and, inparticular, on-axis, back-lighted reconstructions ofsuch. The fourier spectrum is not exactly the same,but the similarity is apparently close enough that theanalysis is applicable to solid bodies such as these.The differences in the two cases that are easily notableare that in the case of the cylinder: (a) one half ofthe fourier spectrum is eliminated because it is blockedby the cylinder, and (b) a reflection component exists.An effect similar to (a) occurs in the hologram of bothcylinders and edges. That portion of light that isdiffracted into the shadow of the object is not holo-graphed, because no reference beams exist there. Forlarge objects, this simply removes half of the fourierspectrum of the object and does not affect the informa-tion content.

To illustrate some of the features of Eq. (1) (Fig. 2)and to provide a comparison of the filtered image of astraight edge and cylinder, a thin straight edge and a2.5-cm diam cylinder, placed end to end, except forseparation by a 50-, diam wire as shown in Fig. 3, wereconsidered (the wire was used throughout the experi-ment for measurement purposes). Employing theelectrooptics configuration depicted by Fig. 1, whichgave an over-all magnification of 150, the resultingband-limited image of the edges of the thin straight edgeand cylinder are shown in Fig. 4(a)-(c). It may benoted that the dark field edgeline appears for both the

Fig. 4. Bandlimited images of an edge. (a) a = 25 cm-', b =1600 cm-'; (b) a = 25 cm-', b = 250 cm-'; (c) a = 25 cm-', b= 50 cm-'. Gap between cylinder and straight edge is 50 g.

Fig. 5. In-line hologram of a .22-caliber bullet in flight.

straight edge and the cylinder. Comparison of thephotographs in Fig. 4 illustrates that the width of thedark field edgeline varies inversely as the upper cutofffrequency. In some cases it is difficult to distinguishthis line because of its extremely narrow dimension[As = 3 in Fig. 4(a) ]. The edges of the reference wireare hardly visible in this figure due to the band-stoppingeffect of the wedge that was used for the high pass por-tion of the filter.

Figure 5 is a copy of an in-line hologram of a .22caliber bullet in flight. The hologram was made with aQ-switched ruby laser having a pulse width of 15 nsec.

Figures 6(a) and 6(b) are the reconstructed unfilteredand filtered images photographed from the closed circuitTV monitor shown in Fig. 1. Figures 6(c) and 6(d)

June 1971 / Vol. 10, No. 6 / APPLIED OPTICS 1321

(a) (c)

(b) (d)

Fig. 6. Reconstructed image of a .22-caliber bullet in flight.(a) Unfiltered. (b) Bandpass filtered. (c) Enlargement

of the nose unfiltered. (d) Enlargement of the nose filtered.

show the enlarged tip of the bullet unfiltered and fil-tered. The edge of the bullet can consistently be lo-cated to better than 15 A. It is interesting to note thatthe unfiltered case is nearly as clear as the filtered case.This is made possible by the filtering action of the TVcontrol unit, which clips all optical energy below acertain level. No effects of motion are detectable inthese reconstructions.

To illustrate the validity of the results at highervelocity, time exposures of a model in motion were madeusing a cw laser. Figure 7 is the reconstructed filteredoutline of the stationary model. The best observedresolution in this reconstruction is approximately 3 .Figure 8 includes an enlarged portion of the recon-structed image of the model moving at an equivalentvelocity of 15,000 m sec'1 (i.e., the motion during theexposure was the same as would have occurred in 15nsec). The motion broadening of the dark field edge-line of the model base predicted by the previous sectionwas within experimental error for the cases tested. Thebandwidths of these holograms were also measured bypassing a knife edge from the center of the fourier trans-form to a position at which reconstructed image edgesparallel to the knife edge were no longer visible on theTV monitor. Measured bandwidths ranged from b =500 cm-' for the stationary cases to b = 35 cm-' for thecase shown in Fig. 8. In the former case, the upperbandstop was caused by the film plate size; in thelatter, it was caused by the motion. Practical applica-

tion of bandwidth measurements for determining pro-jectile velocity is still under study. These data showthat films with medium resolutions are entirely suitablefor many applications. Consequently, the techniquehas been found usable with high speed movie camerasusing standard film allowing motion picture holographyof high speed objects.' 7

Other practical considerations concerning imagequality should be mentioned. The transfer lens usedin these experiments was noted to introduce distortionin the over-all image. A different choice of lens systemwould be required if one chose to study large portionsof the image. In Figs. 6-8, the reconstructions areformed by somewhat broken lines. These are the re-sult of holographic as well as optical system defects.Holographic defects are the result of the failure of holo-grams to store and reconstruct the original informationaccurately. Known causes of these failures includephase and amplitude noise and distortion introduced bythe emulsion onto the reconstructed wave, as well asthe inability to meet certain holographic requirements

Fig. 7. Filtered reconstructed image of a model (model basediameter is 5.0 cm).

Fig. 8. Filtered reconstructed enlargement of a portion of themodel right edge and base (base movement during recording was

0.25 mm; true width of the field shown was 7 mm).

1322 APPLIED OPTICS / Vol. 10, No. 6 / June 1971

exactly. The former can be caused by system noise,holographic nonlinearities, stressing or damaging of theemulsion during processing, or by inhomogeneities.The latter effects are caused by the inability to alignthe hologram exactly in the proper reconstruction wave.Extremely small angular misalignment of the hologramintroduces astigmatism into the image. The idealsystem requires the same wavelength and radius ofcurvature for the formation and reconstruction. Evenin view of all these defects, the data shown here illus-trate that the limitations are not overbearing.

If one has a reconstruction viewing system that em-ploys adjustable clipping of all information below acertain intensity level (such as a CCTV system), and ifthe hologram is optimized, then the unfiltered recon-structed image can be made as sharp as the filteredimage. The bright edge of the coherent image of anedge does not, however, correspond exactly to thegaussian image (Ref. 8).

Conclusions

It can be concluded that edgeline holography doesprovide a method for locating the edge of a fast movingobject that, in some cases, exceeds the accuracy ofphotography. The effect of motion is essentially thesame for holography and photography. Holography isnot limited by problems in focusing. An apparentproblem is the introduction of phase and amplitudenoise introduced in the reconstructed image by theemulsion and the failure of the hologram to reconstruct

the original wave precisely. These studies indicated alimit of resolution of approximately 2 , in locating theedge of a cylinder because of these effects. They alsoindicated that bandpass filtering of the reconstructedoutline image is not always advantageous.

The authors wish to acknowledge the assistance ofK. R. Kneile who provided computer support for thesestudies.

References1. D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).2. B. J. Thompson, J. H. Ward, and W. R. Zinky, Appl. Opt. 6,

519 (1967).3. J. D. Trolinger, W. M. Farmer, and R. A. Belz, Appl. Opt. 8,

957 (1969).4. G. W. Stroke, An Introduction to Coherent Optics and Holog-

raphy (Academic Press, New York, 1966).5. S. Fujumura, Japan. J. Appl. Phys. 7, 60 (1968).6. J. E. O'Hare, unpublished work; ARO, Inc., Arnold AF Sta.,

Tenn.7. J. D. Trolinger and J. E. O'Hare, Aerodynamic Holography,

AEDC-TR-70-44 (1970).8. B. J. Thompson, in Progress in Optics, E. Wolf, Ed. (North-

Holland, Amsterdam, 1969), vol. 7, p. 212.

Information about future meetings should be sent tothe Managing Editor, P. R. WAKELING, WINC,

1613 Nineteenth Street N. W., Washington, D. C. 20009

Centennial Conference on Visual Service,Ohio State University, 10-12 November 1970

Reported by J. M. Enoch, Washington UniversityMedical School

As part of the continuing 100th Anniversary Celebration beingheld at OSU last year, a series of meetings was organized bythe Department of Ophthalmology, the School of Optometry,and the Institute for Research in Vision. The highlight of thisseries was a fine meeting dealing with many aspects of basicvisual science. Glenn Fry, the chairman of the organizing com-mittee, had asked the speakers to review the state of the art intheir respective specialties and to put their work in historicalperspective. Further, each speaker was given adequate time todevelop his argument-and even to indulge in a bit of fantasy-which encouraged lively debate. In short, a stimulating andproductive meeting resulted.

There was no single subject that served as the focus of thegathering-rather, broad coverage was encouraged. Most of thedebate centered about the transmission and processing of spatial(and temporal) information by the visual system. These ques-tions were considered in papers by Fergus Campbell, FloydRatliff, and Don Kelly. Adaptation processes, discussed by

both Matt Alpern and Bob Boynton, also generated consider-able interest.

Westheimer's historical overview of ocular motility proved tobe a high point of the meeting. His presentation was followedby Larry Stark's analysis of several parameters of oculomotormechanisms from an engineering point of view. Fankhauserconsidered models useful in the automation of perimetry, andIngelstam analyzed the speckle patterns that are produced bylaser beams.

Several papers dealt with receptor processes. These includedKen Brown's very nice analysis of the electrical activity of rodsand cones, J.M. Enoch's attempt to initiate quantitative analysisin retinal receptor optics, Yves LeGrand's most interesting treat-ment of microspectrophotometric data, and Martin Bouman'sdiscussion of scotopic cone and photopic rod vision. Riggsfurther developed his careful analysis of objective electrophys-iological indicators of human vision with emphasis on fovealfunction. Herbert Schober considered information capacityof the human eye, and Deane Judd presented an analysis of lociof constant hue from both a theoretical and experimental pointof view.

Unfortunately, no proceedings volume is planned, although thepossibility of developing some of the material into a feature inApplied Optics has not yet been discarded.

June 1971 / Vol. 10, No. 6 / APPLIED OPTICS 1323