Computer-Generated Holographic Imagesrlepci/IMGS322/CGH_Nassar... · 2015. 2. 20. · Computer-Generated Holographic Images Using a PC to Generate Affordable Holograms [Editor’s

Computer-GeneratedHolographic Images

Using a PC to Generate Affordable Holograms

[Editor’s Note: This article is apractical tutorialonmethods togenerateahologram using an MS-DOS computerwith a VGA screen, a 35mm camera, anda laser the hologram).

Dale Nassar has written a largemanuscript covering laser basics, lighttheo y, and the fundamentals of generalholography. This article, while a practicaltutorial, is an excerpt from the largerwork.

If you would like to purchase Dale’scomplete work, send $7.50 to: Computersand Holography, 4 Park St., Vernon, CT06066.1

H olographyisaphotographicprocess which, unlike ordinary pho-tography, does not record an image ofthe scene photographed, but encodesthe emanating light rays themselves.The resulting optical record is called ahologram and can instantly recon-struct the recorded light rays. Holo-grams an illusion of the origi-nal scene in three-dimensional spacethat is remarkably life-like.

The beautiful images created bythis unique recording process aremade possible by the coherent light ofthe laser. However, because a holo-gram can be considered an array ofmany bits of information, I decided toinvestigate the practicality of compu-terized hologram synthesis. In thisarticle I will demonstrate, withoutcomplex analysis, how holographicsynthesis can be accomplished in thecomputer room with no special opti-cal materials or holographic lab. Thelaser’s role in conventional hologramformation is completely emulated by

22

and in some very crucialsituations the computer outperformsthe laser.

The synthesizing method I use isstraightforward and is designed tobe easily understood and inexpen-sively applied with standard photo-graphic and computer equipment. Ona more advanced level, a parallelprocessing environment also lendsitself to the application as the holo-graphic bits are mutually independ-

THE SINUSOIDAL GRATING

A thorough understanding of theprocess of optical interference caneasily be had by assuming light to bemade up of sinusoidal waves of en-ergy (hence the expression “lightwaves”). Figure 1 illustrates a sinu-soidal waveform and the key ele-ments of its structure as defined inphysical optics.

It is important to be aware of thefact that light waves are travelingwaves; that is, the contour of thewaveform of Figure 1 should be con-sidered toward the right atthe speed of light. To get a mentalpicture of what this means, considera particle on the time axis in Figure that is allowed to move only verti-cally in response to the amplitude ofthe passing light wave. Then theeffect the wave has on the particle isa very rapid sinusoidal vertical mo-tion (vibration) about a fixed point onthe horizontal axis. It is obvious thatthe frequency of a light wave is ex-tremely high, visible light has a fre-quency of the order of Hz (100,000

FEATUREARTICLE

Dale Nassar

These important quantities arcrelated by the very simple (and obvi-ous) expression where is thefrequency in Hz, c is the velocity of thewave in m/s (3 x for light) and is the wavelength in meters. The

period of the wave, is the reciprocalof the frequency. represents thetime required for one wavelength topass a given point. Mathematically,the energy of a wave is a measure of i tsintensity, which is proportional to thesquare of itsamplitude. Thisencrgy iswhat does the work responsible forexposing photosensitive film.

When two plane waves atthe surface of a film, as shown in across-sectional view in Figure interference pattern recorded consistsof a series of parallel line fringes (inthe diagram the lines are lar to the page). This is called a photo-graphic grating and appears as inPhoto 1. Figure 2b depicts the samesituation but with a larger angle be-tween theinterferingbeams. As illus-trated, the effect of increasing the anglebetween the two beams causes thefringe spacing to become finer. 3 theamplitude transmis-sion across the surface of the gratingof Photo 1. There are no abruptchanges in the transmission-thevariation is sinusoidal with the fre-quency of the waveform the spatial frequency of the grating.

In Fourier analysis it is shownthat a wave with very sharply chang-ing shape such as a square wave canbe broken down into many sinusoidalcomponents, while a sinusoidal waveis the purest formpossible. In the caseof the abruptly changing amplitudetransmission of the grating, the result

Figure shows basic waveform while wavelengths

of the extremes of the visible spectrum.

is many orders of diffracted beams.Each diffraction order consists of twobeamsdeflected at equal angles meas-ured above and below the zero-order(straight through) beam. The angle ofdeflection of the diffracted beam iscalculated from the standard gratingequation:

where is the fringe spacing and the wavelengthinvolved. On hand, the sinusoidal grating producesonly one diffraction order.

THE ZONE PLATE:HOLOGRAM OF A POINT

From a holographic point of view,an object consists of many tiny surfacepoints or resolution elements. Whenlight is reflected from such an objectonto the film, each resolution elementof the object treated as if it werea point source of light generating acoherent spherical wavefront. Figure4a is is a hologram of a basic a resolution element (smallest resolv-able point) of the object. Let’s definethe axis of the system as the line pass-ing through the object point and cen-ter of the film. Symmetry exists aroundthis axis, and the microscopic patternrecorded on the film will have theform of concentric circles as shown inPhoto 2. Notice that the fringe spacingis relatively coarse at the center of the

Figure plane waves meet,an interference pattern consisting of a se-ries of parallel line fringes is recorded.

system, but becomes finer, approach-ing one wavelength, as the wavesmove radially outward from the cen-ter of the film. This pattern of alter-nately light and dark circular fringesis called a zone plate and is the generalappearance of a hologram of a singlepoint.

Figure 4b shows what happenswhen theprocessed filmisilluminated.The fringes diffract the light waves asif they were coming from the locationof the point source, forming a virtualimage of the point. A set of converg-ing waves forming a real image of thepoint on the opposite side of the holo-gram is also formed. If this were theactual object wave used in the record-ing process in place of the divergingpoint source, exactly the same inter-ference pattern would have resulted.The certain amount of error present inthe system is desired to give the mathe-matical points physical dimension.

The first holograms were made in1948 (12 years before the invention ofthe laser) by Dr. Dennis of theImperial College of London with thelight from a mercury arc lamp whichhad a coherence length of only about0.1 mmand a bandwidth of about 1 which is low coherence by the stan-dards of the laser. Because of the poorsources of coherent light available atthe time, these were on-axis type holo-grams and the object was restricted totwo-dimensional transparencies withopaque lettering. These conditionsgreatly reduced the coherence require-ment. The light was shined directlythrough the transparency onto thefilm. The light passing through theclear areas served as the reference

Figure larger angle theInterfering beams causes the fringe spac-ing to be

Photo 1 -When recorded on film. the inter-ference pattern shown above is aphotographic grating.

beam and the light diffracted by theedges of the lettering served as theobject beam. At this time the conceptof off-axis holography was unknown.Around 1961 Emmett Leith and JurisUpatneiks of the University of Michi-gan, in an attempt to separate the realand virtual images of holo-gram, made off-axis holograms withthe gas laser. The discovery of holog-raphy, or wavefront reconstruction asthe technique was called at the time,earned prize in phys-ics in died in 1979.

THE FASCINATING FRESNEL

A Fresnel zone plate strikingsimilarity to the interference patternof the hologram of a single point. Weuse the properties associated with theFresnel zone plate in many of the cal-culating procedures required to ducecomputer-generated holograms.In deriving the structure of a Fresnelzone plate we make use of Huygen’sprinciple which simply states (and canbe proven) that each point on awavefront may be regarded as a newsource of secondary (of the

24 INK

Figure frequency of the amplitude transmission across the surfaceof the grating in Photo represents the spatial frequency of the grating.

Photo 2-A hologram of a point consists ofconcentric circles on the film.

Figure 4-_(a) A hologram of asingle point consists of alternatelylight and dark circular fringes andis called a zone plate. Whenthe processed hologram is illumi-nated, a origi-nal point is formed.

BEAM

ILLUMINATING \

BEAM

same wavelength) and the interactionof these is responsible forinterference effects observed. Figure5a illustrates the principle. Here aplane wave illuminates an opaquescreen with a pinhole in it. The pin-hole acts as a new source of sphericalwaves as shown by the segments ofcircular arcs. The small circle repre-sents a secondary of thespherical wavefront. The amplitudeof this secondary is not thesame in all directions but varies ac-cording to:

a)

where A is amplitude and a is theangle at which the radiating ampli-tude is to be calculated. This equationis known as the obliquity factor. Theobliquity factor has a maximum valueof 1 which occurs when a = 0, corre-sponding to the direction of travel ofthe source. At 90 degrees the obliq-uity factor gives a value of and at180 degrees the obliquity factor is zeroindicating that, as shown in Figure there is no wave in the backward di-rection. Figure is a polar graph ofthe amplitude and intensity distribu-tion as predicted by the obliquity fac-tor. It follows from Equation 2 that theintensity of the secondary isgiven by

In this respect we can ignore thelight source once the coherentwavefront isdefined at the diffractingaperture(s). The Fresnel zone plate isa patternof concentric transparent andopaqueringsdesigned tofocusabeamof plane wavefronts incident upon it

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much like a magnifying lens. To de-rive some very important propertiesof the Fresnel zone plate we will di-vide a plane wavefront into the vari-ous zones as shown in Figure 6. Wecan assume that the entire flat surfaceof wavefrontconsistsof tinypoint sources of light, each emittingspherical wavelets. Now considersome point illuminated only bylight from the wavefront, located a

past the wavefront as illus-trated. We now will divide a portionof the wavefront into zones such thatthere is a maximum concentration oflight produced at point P. This isdone by allowing only the portions oflight emitted from the wavefront toreach that would interfere construc-tively with any other light from thewavefront reaching P. Referring toFigure 6, consider the perpendicularfrom to the plane wavefront. Theintersection with the wavefront isdenoted by 0. We now divide theplane wavefront into series of circlesof radii r2, r3 . centered at 0such that each circle is a half wave-length further than the preced-ing one. We now can see that thecircles, beginning with the innermostcircle, are at distances

from P. The phases at of any secon-dary from any given circu-lar zone will not differ by more than

(one half wavelength). If we gofrom one zone to the next, the ampli-tude of the wave reaching changessign. Therefore, if we block everyother zone, only constructive inter-ference will occur at (consideringonly the portion of the wavefrontencountering the zone plate). Theresult of thisconstructionist Fresnelzone plate. It does not matter if westart by blackening the center zone or

remember, intensity is proportionalto the square of the amplitude.

This discussion should also sug-gest that the performance of a holo-gram is unaffected if its dark and lightareas are interchanged. This is in-deed the case-a hologram does notproduce negatives. It is informative

Figure As plane wave illuminatesan opaque screen with a pinhole, the pin-hole acts us new source of sphericalwaves. At the obliquity factor iszero. indicating that there is no wave in thebackward direction. The andintensity distribution as predicted by theobliquity factor is plotted on a polar graph.

to look at some actual magnitudes in-volved with the zone plate. For onlythe first zone, it can be shown that theintensity at is increased four times.This is rather surprising for the case ofan opaque center zone since this im-plies that there should be a bright spotin the center of the shadow cast by anopaque circular obstacle. This is in-deed the case and the placement ofsuch a single disk increases the inten-sity at by four. For a zone plate withonly 20 zones, the intensity at isincreased by Looking at the

of Figure 7, we see that a series of

26

right triangles are formed. For thetriangle formed by the radius, wehave, from the Pythagorean theorem:

Eliminating since this value is neg-ligible for light:

= nfh

We now have a simple formulafor constructing a Fresnel zone platewith desired properties. For example,if the above-mentioned zone plate of20 zones were to focus the light from aHe-Ne laser at 10 cm, the entire zoneplate would have a radius of onlyabout 1 mm. If we solve the zone plateequation for f, the focallength,a very convenient and useful formulais obtained:

2

f

We will make much use of thisrelationship. ConsideragainthepointP when the plane wave is encounter-ing a single opaque zone of radius As previously stated, the intensity isfour times as great as that which re-sults from the plane wave alone. Nowif the radius of the opaque disk isexpanded to cover the first two Fresnelzones, theintensityat Pdrops zero. If we continue this process ofincreasing the radius of the opaquedisk, the intensity at goes through aseries of maxima and minima as thenumber of zones included accordingto the formula becomes even or odd.The result is the same if an open aper-ture is used allowing only increasingcircular areas of the plane wavefrontto emerge while blocking the rest.

Because of the abrupt changesbetween transparency and opacity inthe Fresnel zone plate, there will be re-gions of secondary concentrations oflight. A series of secondary foci alongthe axis between the primary focalpoint and the zone plate are readilyobserved if a white card is movedalong the area. These foci are fainter

PLANE WAVEFRONT

Figure Fresnelzone is pattern of concentric transparent and opaque ringsdesigned focus beam wavefronts incident upon it much magnifying/ens.

Photo transmissions of a Gaborzone plate and a Fresnelzone plate are shown

than the point at P and progressivelydiminish as the zone plate is ap-proached. are found at distancesf/3, f/5, f/7,. If you give the abovediscussion a little extra considerationyou will realize that these secondaryfoci are produced by single zonesacting in groups of

There also exist various lightconcentrations at points off the axis ofthe Fresnel zone plate. The

analysis of these off-axismaxima and minima are very com-plex, the results of which verify thepresence of concentric circular fringescentered on the axis. These secondaryfoci do not occur in the holographiczone plate. The sinusoidal variationsin the opacity of fringes causes cancel-lation of these higher order diffrac-tions (by superposition of the secon-dary Photos 3a and 3b

28 CIRCUIT R INK

illustrate respectively the amplitudetransmission of a zone plateand a Fresnel zone plate. A sinusoidalzone plate is also called a zoneplate. It should be interesting tocompute the concentrations of lightproduced by a zone plate byusing as parameters the secondarywavelets, obliquity factor, and sinu-soidal transmission of the film.

To make a distinction betweendiffraction and interference, diffrac-tion refers to the situation verylarge number of tiny of awavefront, such as the Huygen secon-dary wavelets, are summed (inte-grated) to produce a the pattern while,interference refers to the interaction(simple addition) of a smaller numberof beams. Briefly, the hologram inter-ference pattern will be calculated bysumming all of the sinusoidal wavesemitted from each point of the objectand calculating the resultant phaseand amplitude at the hologram sur-face and then assigning either trans-parency or opacity at that point. Thissummation is done for each point.

PRELIMINARY CONSIDERATIONS

The procedure used to producethe computer-generated hologramswillconsistofthefollowingthreesteps:

An optical interference patternof a mathematically represented sceneis computer calculated by digital ap-proximation. This interference pat-tern is of the type produced by an axis holographic recording process.

The pattern is then reducedphotographically thus becoming atransmission hologram designed tobe viewed with laser light.

This pattern is then drawn on a

screen or plotter producing a mono-chromatic output.

Consideration of the standardrecording process of an off-axis trans-mission hologram reveals some diffi-culties that will be encountered inreproducing the process by artificialmeans. We have seen that this proce-dure produces an extremely fine in-terference pattern. Specifically, Equa-tion 1 implies that, when the anglebetween the two recording beams

approaches the fringe spacingapproaches that of the wavelength ofthe light involved, which for a He-Nelaser corresponds to about 1600 pairs/mm. Such large angles arerequired because it is desirable for theoff-axis scene to be near the film, thuspermitting a large angular viewingrange. This necessitates an extremelyhigh resolution film such as the Kodak649F Spectroscopic emulsion which iscapable of resolving a maximum of7000 line-pairs/mm. A x 5-inchhologram with a maximumdeflectionangle of 60” will be required to recordabout 132 billion dots.

The hologram also records a grayscale. If 64 levels of gray are assumed,the effective data content exceeds 8trillion (Thesecalculationsareveryconservative. Toprevent aliasing errors, the resolutionin each direction should be at leastdoubled, and preferably quadrupled.)The size of the hologram is significantbecause the observer moves his headduring viewing to exploit parallax.The dimensions of the hologramshould thus be considerably largerthan the separation of one’s eyes.

The calculation time of the com-puter-generated holograms will bedecreased by reduction of the follow-ing four parameters:

Now consider the time requiredto numerically calculate the data con-tent for a hologram of a typical (small)object consisting of 1 million resolu-tion elements. This means that therewould be calculations re-quired foreachof the 132billionpointsof the hologram. Although the laserwould produce this data instantly ona photographic emulsion (the highestinformation storage material known),the process would take a computer,working at a rate of one million calcu-lations per second, over 4000 years!

size-The hologramswill be no larger than the standardframe size of the popular 35mm film(36 mm x 24 mm).

Quantify of resolution elements ofsubject-The subject will be a simplegeometrical shape such as a circle con-sisting of only a few pixels.

Angles between object and reference maximum angle here will

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be minimized for a given fringe spac-ing of the hologram.

Gray will be no grayscale. The hologram will consist ofonly transparent or opaque areas.There is a very mysterious and littleknown property of holograms that isof great significance in this applica-tion: The gray scale of the subject is in-dependent of the gray scale of thehologram. One may deduce that if thehologram is of binary form, then thereconstructed image must also bebinary in nature. This is not the case.The reconstructed image may have acontinuous gray scale regardless ofthe binary nature of the film. Anylevel of brightness that is assigned toany pixel in the recording process isstored in its relative proportion in thewave summation over the entire holo-gram area.

HARDWARE CONSIDERATIONS

Holographic patterns will bedrawn using the following three typesof output devices:

A standard VGA display of 640 x480 dot resolution and hologram reso-lution (640 x 427).

A pen plotter with an effectiveplot area of 864 mm x 546 mm and mm resolution with a 0.3-mm tip di-ameter pen giving a hologram

A multihead laser plotter with aneffective plot area of

resolution giving a holo-gram resolution of 48000 x 32000 x

The effective plot area of eachdevice is shown in parentheses to obtain a width-to-height ratio ofthat equal to the standard 35mm filmframe This clipping represents asignificant time savings when thereduced pattern is to be of maximumsize (36 mm x 24 mm).

I used technical pan film to photo-graph the holographic interferencepatterns since this able and can resolve up to 400 pairs/mm at various contrast levels.This film is also ideal for applicationsinvolving a He-Ne laser as it has a

high sensitivity to light in the redportion of the spectrum. EktagraphicHC slide film might also be used be-cause of its 750 line-pair/mm resolu-tion.

REDUCTION METHOD

Because we are creating a holo-gram artificially by imitation of theactual process on a large scale andthen reducing it for illumination withthe light from a He-Ne laser, we mustconsider what effect reduction of thepattern as well as reduction of the re-cording wavelength has on the finalimage. Keeping in mind that in theplotting process, a large wavelength(imaginary, of course) can be associ-ated with the recording pattern, wewill look at the effect, on the recon-structed image, of reducing the holo-gram pattern as well as the recon-structing wavelength.

To simplify matters for illustra-tion, suppose that the object consistsof a single point A as illustrated inFigure 7. We know the values of the

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the edges of the plot. Notice how thejagged edges arrange themselves toproduce several families of secondaryzone plates. When this pattern is illu-minated with laser light, each of thesesecondary zone plates has a focusingeffect, and the error emerges as amatrix of concentrations of light (sec-ondary foci) about the primary centerfocal point as shown in Photo 4b.Photos illustrate zone plate for-mation on a VGA screen with antiali-

factors of and 3. As can beseen, the aliasing error decreases (less Figure a hologram is to be formed without exceeding the maximum spatial

secondary zone plate contrast) as thefrequency, then all object points must be confined to the shaded region. Any pointslocated outside this area such that larger angles are produced between the interfering

antialiasing factor increases. From beams produce errors in the

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this data, I decided to use an antiali- factor of 2 for the

generated holograms.

SYNTHESIZING A HOLOGRAM

Let’s start our example by calcu-lating the interference pattern for ahologram of a computer-generatedcurve (a three-leaved polar rose). Here,the situation for the VGA display (640dots horizontally by 480 dots verti-cally) is used. These displays areusually about 10” x 7.5”. Because thewidth/height ratio is lightly differentthan that of a 35mm camera, whenphotographing with a 36-mm x mm viewfinder, simply fill the areavertically. Ideally there will be a smallvertical strip along one edge of a scale (36 mm x 24 mm) hologram, butall pixels will be used. The spatial fre-quency of a VGA display is about 1.26line-pairs/mm. We will divide thisvalue by 2.5 to prevent error: 0.504 line-pairs/mm. I will conser-

vatively round this value to 0.5 inactual calculations.

The hologram plane is defined ina three-dimensional Cartesian coor-dinate system as illustrated in Figure9. The origin of the system has coordi-nates with the coordi-nate signs assigned as implied by thedrawing. The hologram plane coin-cides with the xy plane with its topedge on the axis and upper-leftcorner at the origin. Note that is to

and isdownward to matchthe native coordinate system of thecomputer display. The direction is

32 CELLLAR INKReader 155

Photo A typical zone plate. Themaxtrix of dots are errors caused by secon-dary zone plates. Zone plates with

factors of and 3.

toward the rear looking through thehologram plane. The coordinates ofthe lower right corner of the hologramextent have VGA coordinates(639,479). The screen size photo-graphed will be (0.254 m) by 7.5”(0.192 m). The position, in three-di-mensional space, of the lower-rightcorner of the hologram is located at

Let’s give the rose aradius of 0.1 m and let it consist ofabout 41 (I incremented the full polarrotation of pi by0.075) equally spacedpixels of equal intensity (more on in-tensity assignments shortly).

We will now consider how closethe reconstructed image can form fromthe hologram plane. If the entire screenof 0.254 m x 0.192 m is to be reduced asto just fill a 36-mmx film framethen the reduction factor is 8, as shownin Figure This will result in the0.500 line-pairs/mm of the plot to be-come about 4 line-pairs/mm in thefinal hologram. We also know that thesynthetic wavelength should be eighttimes that of the reconstruction (he-lium-neon laser) wavelength, or 5.0624x m. Equation 4 tells us that themaximum angle that the hologramcan deflect the reconstruction beam isabout 0.146”. This corresponds to aminimum object distance (on the plotscale) of about 63 m if the maximumhologram radius is, using the diago-nal, 6.25”. Notice that an additional39.52 m must be added to the mini-mum object distance if the entire m rose is to be recorded. Since itwould be desirable for the image to besomewhat closer to the hologram, let’slook at a reduction twice as great.

CENTER

Figure hologram plane is a three-dimensional Cartesian coordinatesystem and is shown as the shaded region.

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A reduction of 16 (Figure corresponds to a synthetic wavelengthof 1.0125 x m and a maximumdeflection angle of 0.29“. This bringsthe minimum large-scale image dis-tance down to 31.5 m. However, wenow have only one quarter of thehologram area (when photo-graphed, the image seen through thecamera’s viewfinder should occupyone quarter of thearea). After review-ing the parameters for reductions by24 and 32, I decided to use 16.

When writing the graphics rou-tines, it would be much more conven-ient to work in units of pixels ratherthan meters-since 0.254 m corre-sponds to 640 pixels (for wehave the relationship 2520 pixels/m.Now we can simply multiply anymeter value by 2520 to work directlyin graphics mode.

The polar equation of the leaved rose is:

r =

where r is the dependent variable, f isthe independent variable, and a is theradius of the rose. The transformationfrom polar to Cartesian coordinates isaccomplished with the tions:

x(t) = y(t) =

where is the center of the rose.We could tilt the figure out of the

xy plane by adding a sinusoidalfunction However, because inthisexamplethe hologramisrelativelycourse and the object is small anddistant from the hologram plane, thistilt would not very noticeable in thereconstruction.

Listing 1 is a simple program that will allow the user todefine and edit a holographic imagebefore it is processed by the programof Listing 2, which draws the to-photograph holographic interfer-ence pattern on the high-resolutiongraphics screen. Remember points in the image means longerdrawing time. With a math coproces-sor in a 286 machine running at 12MHz, the rose plot takes about 12

34 INK

VGA plot to be (6328 x (8) 5.0624 x

maximum spatial frequency 1 0.315 line-pairs/mm deflection angle:

= sin-’ line-pair/m) (5.0624 x 1 0.146’

VALID OBJECT

HOLOGRAM

0.161 m

63 m

(OBJECT DISTANCE)

VGA Reduction by 16: 1.0125 x object distance 31.5 m

Figure amount ofreduction used when making the hologram affects how closethe reconstructed image be to the hologram plane. Part shows a reduction factorof 8, while shows a factor of 16.

SCREEN 12: COLOR 4: pi = 3.1416: a = 12.5

FOR = TO 1 * pi STEP 0.02 PLACE ANY FUNCTION HERE **

* y = 240 + * SIN(t)

NEXT t

DO: LOOP WHILE =

listing program allows the user to define and edit a holographicimage before it is processed.

hours to complete. Without thecoprocessor, it takes several days.

In defining the interference pat-tern, each point of the rose is consid-ered to be emitting light, thus illumi-nating the hologram plane with radia-tion of the synthetic wavelength. Eachpoint emits light with a specified ini-tial phase and reaches each point ofthe hologram plane with a specificphase. For each point in the hologramplane, the sum of the waves from eachpoint of the circle is calculated and thepoint is assigned to be either transpar-ent or opaque depending upon theresult of the summation. There is alsoa phase value present at the hologramplane. For simplicity, I assigned aphase value of zero at the hologramplane and let all points on the rose

start emission with a sine wave (0initial phase angle and increasingamplitude) of unit amplitude. At thehologram plane opacity was assignedif the wave summation at that pointwas greater than or equal to zero andtransparency otherwise.pher can choose any trigger leveldesired. Also, different points of anobject can be assigned various ampli-tudes for a proportional intensity inthe reconstructions.

Another consideration in the re-construction of the image of a binaryhologram is the formation of extrane-ous images due to higher order dif-fractions. From the experiments per-formed here, these higher order dif-fractions were so dim that they wereunnoticeable. However, Equation 1

5 A-Z10 SCREEN 12: pi = = 0: CLS20 1 = 0.0255145: = 129528: h = 320: k = 240: a = 25230 FOR = 0 TO 63940 FOR = TO 47950 FOR t = 0 TO 1 * pi STEP 0.07555 r = a * * 60 px = h r * py = k t r * SIN(t)70 d = 2) t 0.580 phase = (2 * pi * d9 0 = + 100 NEXT t110 IF s 0 THEN COLOR 7: PSET GOT0 130120 COLOR 0: PSET 130 = 0135 NEXT y140 NEXT x150 DO: LOOP WHILE =

END

program the ready-to-photograph holographic interfer-ence pattern on the high-resolution graphics screen.

may be used to calculate the of the actual interference tion angles of the second-order tern. Photo illustrates the ages in the reconstruction to of the reconstructed real imagemine an object size (or fringe spacing) projected at the predicted focal to ensure distortion-free images.

VGA-RESOLUTION HOLOGRAMSObservation of the pattern of the

on-axis hologram has a somewhatAND RECONSTRUCTIONS fuzzy outline of the rose, and one may

think that the image is formed by lightPhoto 5a shows the subject of the rays passing straight through the

first hologram and Photo 5b is a film-this is not the case as can be

easily shown in several ways. First,the pattern is not sharp and can’tproduce the observed bright points oflight by projection. Secondly, if theimage is viewed between the holo-gram and the focal point, only a blurappears.

For a more dramatic illustration, Icut the hologram into quarters and thefull image was reproduced in eachpiece. To illustrate this further, I plot-ted a small offset segment of the holo-gram having none of the rose-shapedoutline When illuminated,this (now offset) hologram fully re-produces the rose as illustrated inPhoto Also, the distant virtualimage of the rose can be seen by look-ing through the hologram toward theilluminating laser. The hologramshows off-axis properties with lessthan one-third of VGA resolution!

Photo 6 can be photographed andused as a hologram. If this is done, thefinal image making up the fringesshould be 0.32” x 0.25”. It should bephotographedwithalightbackgroundwhich will become a dark outline for

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Photo a small offset ofthe interference pattern shown inPhoto 5b eliminates any hint of the

outline. When i//u-minuted, the projected real im-age is identical to that in Photo

Photoshappedthe petals

toexpected

blocking extraneous laser light whenviewing the hologram. It appears thatany type of black-and-white film canbe used for the VGA holograms.

I also plotted the negative of thepattern simply by reversing the pixels(turning blank pixels on and turninglighted pixels off). The negative pro-duced results identical to those just il-lustrated.

For the next hologram, I assigneda higher brightness to a few of thepixels making up the rose. Pixelsmaking up three small segments ofthe rose were assigned larger ampli-tudes (all others are normally unity).Photo 7a illustrates the rose with sev-eral brighter pixels. Photo 7b showsthe resulting holographic pattern andPhoto is its reconstruction. The setsof two, three, and four bright pixelswere assigned amplitudes of three,four, and five, respectively.

The experiments presented hereshow that in holography the mentsof coherence, stability, and resolution recording media are not asstrict as many people believe.

THE FUTURE

Holographic display devices mayhave a bright future. The nature ofsuch a device will operate on drasti-cally different principles than the CRT(pixels emit incoherent light), blyusinga sort of supercomputer (par-allel processor) to calculate an inter-ference pattern in a reasonable span oftime. There will a great changein the manner in which graphics im-ages are created on these devices.Obviously, no image is drawn on anysurface-only an interference pattern.The entire image will be present ornot-nowhere in between. A feasibleconstruction would be a display ma-trix consisting of liquid crystal “shut-ters” that could be toggled open orclosed. The matrix would be illumi-nated by spread (no eye hazard) laserlight from behind. The shutters neednot cover the entire display surfacesince their purpose is only to directlight. Several cluster arrangements,each consisting of shutters of closespacing, would be suitable.

Full-color holographic displayscan be produced by use of illuminat-ing lasers of the primary colors. Colorholograms are calculated by assign-ing each pixel three synthetic wave-lengths, each corresponding to one ofthe primary colors. The waves wouldbe of amplitude characteristic of theobject point. The monochrome inter-ference pattern is then capable ofproducing the scene in full color.

like special thanks to ing people for their patience and inthis project: May Lou Nassar, Maria Palmer,Joe Lombardo, and Charles Palmer

Dale Nassar has a B.S. in physicsfrom South-eastern Louisiana University Hishobbies include gymnastics (university team)and springboard diving.

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