image-plane cylindrical holographic stereogram

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Image-plane cylindrical holographic stereogram

Yih-Shyang Cheng and Ray-Cheng Chang

A two-step holographic process is introduced to fabricate a cylindrical multiplex hologram as an image-plane hologram. By adoption of the achromatic angle in the process the hologram is capable of gener-ating an achromatic image. The most important factors, the location as well as the width of the viewingslits, that affect the quality of the observed image are analyzed and discussed. The change of aspectratio for the observed image as a function of the viewing distance is theoretically and numericallyanalyzed. This method can not only eliminate the annoying picket-fence effect but can also increase thevertical viewing range for the observer. Computer simulations as well as experimental results areprovided. © 2000 Optical Society of America

OCIS codes: 090.0090, 090.2870, 090.4220.

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1. Introduction

Holographic stereograms ~HS’s!, or multiplex holo-grams, were invented to overcome the limitations onsubjects, which should be rigid, static, and relativelysmall for conventional holograms.1 Rather than di-rect illumination of the subject with laser light, aseries of two-dimensional ~2D! photographs, or per-spectives, are taken from a three-dimensional ~3D!scene under any convenient light. These 2D photo-graphs are fed into a laser-illuminated system toform a HS. Upon reconstruction a small fraction ofthe diffracted light is directed to each viewing zone,bearing an individual perspective view. The rightand the left eyes of the observer, at two viewingzones, can perceive a pair of 2D images, which arethen fused into a 3D image in the brain. When botheyes sweep horizontally across the viewing zones,stereo pairs with continuously varying binocularparallax are received, giving the impression ofquasi-holographic images.

Since the introduction of the planar-type1 holo-graphic stereogram, various formats of HS have beendeveloped, which include cylindrical-type,2,3 conical-type,4,5 and disk-type6 HS’s. The planar HS benefitsrom its geometrical shape and hence is more suitableor embossing production. However, the planar for-at also greatly limits the horizontal viewable angle

The authors are with the Institute of Optical Sciences, NationalCentral University, Chungli, Taiwan 32054. Y.-S. Cheng’s e-mailaddress is [email protected].

Received 5 October 1999.0003-6935y00y234058-12$15.00y0© 2000 Optical Society of America

4058 APPLIED OPTICS y Vol. 39, No. 23 y 10 August 2000

and depth sensation of the image, which bounds theapplication mainly in entertainment displays. Therecently invented disk-type HS is suitable for massproduction with the well-developed compact-disc~CD! technology. This type of hologram is capable ofdisplaying a rotating image behind the rotating ho-logram disk. The cylindrical and the conical-typeHS’s are quite different from the planar one, sincethey can display images either inside the hologramcylinder or above the open end of the hologram cone,providing a 360° range of view for observers. Owingto this characteristic, the cylindrical HS was appliedto display 3D images reconstructed from tomographicdata7 and to the field of computer-aided design.6 Al-hough the cylindrical HS has the potential to displayedical, scientific, and engineering data, it inher-

ntly possesses some drawbacks. Since 2D data areabricated as a long thin component hologram withn anamorphic optical system, the reconstructed im-ge is inevitably overlaid by a vertical line structure.his notorious phenomenon is called the picket-fenceffect. In addition, the observed image lacks sharp-ess in the horizontal direction, since each 2D image

s not projected onto the hologram plane in the cor-esponding direction. The observed image also ap-ears in rainbow colors, because it is generated fromrainbow-type hologram.In this paper, rather than using the single-step

rocess as is done with traditional multiplex holog-aphy, we propose a two-step procedure to record theylindrical HS as an image-plane hologram. The re-ulting hologram is called the image-plane cylindricalS ~IPCHS!. Comparing the image reconstructed

rom the IPCHS with that from traditional cylindri-al HS, we find that the image is clearer, brighter,

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easier to observe, and without the picket-fence effect.An achromatic image is observed when both eyes areplaced near the middle of the viewing zones. In thispaper we first introduce the two-step recording pro-cess for the IPCHS. A series of 2D perspectives areholographically recorded as closely spaced strips on aphotosensitive film in the first step. These perspec-tives are then retrieved and imaged onto another filmto form a synthesized image-plane hologram. Thefinished hologram, or IPCHS, is then illuminatedwith an incandescent light bulb ~a white-light linesource! after being bent into a cylinder. During ob-servation of the hologram, two eyes of the observerreceive a pair of 2D perspectives, which are thenfused into a 3D image with depth sensation. Theadvantages inherent to the two-step process and thecharacteristics of the IPCHS are discussed followedby detailed analysis and discussion of parametersthat affect the quality of the image. Two of the mostimportant parameters are the location and the widthof the viewing slits. The former determines the best,or correct, viewing distance for the eyes, whereas thelatter affects the parallax resolution and clarity of theobserved image. The effect when the eyes are notsituated at a correct viewing position and the distor-tion that results when the parameters for the holo-graphic process are not properly selected arediscussed. Finally, the above analysis is experimen-tally verified, and an example of our experimentalresults is given.

2. Recording Method and Property of the Image-PlaneCylindrical Holographic Stereogram

To make a cylindrical multiplex hologram, a series of2D perspectives should be prepared. These 2D im-ages can be generated from a real scene containing3D objects or a virtual scene drawn by computer. Inthe former case photographs are taken with the ob-ject placed on a turntable and a camera situated at afixed position ~or, alternatively, with the camera mov-ing along a circular track around the object!. Theptical axis of the camera is perpendicular to the axisf rotation for the object. Successive pictures areaken at equal angular separation. In the later casehe virtual 3D information is transferred into a seriesf 2D perspectives by computer program without thehysical existence of the camera. This is particu-arly useful in dealing with complex 3D informationn the realms of science and engineering. The fin-shed perspectives are then ready to be fed into theecording optical system for multiplex hologram.

Our method for producing the IPCHS is a two-steprocess. In the first step, which is called the mas-ering process, the perspectives are sequentially sentnto a laser-illuminated optical system ~Fig. 1! and

displayed on a spatial light modulator which, in ourcase, is a liquid-crystal display ~LCD! panel. Thelens L2 images and enlarges the 2D object onto adiffusing screen ~DS!. The scattered light from itserves as the object wave for holographic recording.A holographic film H1 ~called the master! in contact

ith a masking slit is situated at some distance be-

ind the DS. This holographic film is tilted at thechromatic angle a, which is defined as the directionf the rainbow spectrum with respect to the normal ofhe hologram plane.8 The masking slit is placed on

the x–z plane with its center aligned on the axis of theoptical system. An off-axis, coherent plane wave,illuminating the masking slit, serves as the referencewave. The resulting interference pattern is re-corded as a long thin component hologram on themaster. After one exposure, the master is trans-lated by a distance equal to the width of the slit alongthe y axis, and the next 2D object is displayed on the

CD. This system is then ready for the recording ofhe next component hologram. This process contin-es until all the 2D objects are recorded as componentolograms side by side.In the second step ~the transferring process! theaster H1 is illuminated with a conjugate reference

wave ~Fig. 2!. Each component hologram recon-tructs a 2D image to the plane where the diffusioncreen was originally situated. Since the masteras at an angle a with respect to the normal of the

diffusion screen, the retrieved images would also in-

Fig. 1. Optical system for recording the master hologram H1,hich is tilted at the achromatic angle a. Input perspectives areisplayed on a LCD and imaged onto the diffusion screen DSequentially. A horizontal slit with proper width in contact withhe master is used to limit the exposure area of the componentologram. M, mirror; L, lens; VBS, variable beam splitter; SF,patial filter.

Fig. 2. Optical system for making the IPCHS. The informationstored in the master H1 is retrieved with a reference wave andecorded on the second holographic film H2 with the help of aylindrical reference wave. M, mirror; L, lens; VBS, variableeam splitter; SF, spatial filter.

10 August 2000 y Vol. 39, No. 23 y APPLIED OPTICS 4059

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cline at the same angle. At this plane another ho-lographic film, H2, is used to record the reconstructedimages as an image-plane hologram ~called the trans-fer!. The reference wave for this transfer is a coher-ent line source along the direction of the y axis, whichis formed by introduction of a cylindrical lens Lc intothe route of a collimated wave. This line source of-fers a collimated wave in the vertical ~y! direction buta divergent one in the horizontal direction at thehologram plane. The position of line source shouldbe carefully calculated to match certain conditionsand thus eliminate some distortion of the final ob-served image. This point is discussed in Section 3.

When the processed transfer H2 is illuminated by aconjugate reference wave, a small fraction of light isdiffracted to each one of many viewing slits. By po-sitioning one eye at one viewing slit, one can see a 2Dimage belonging to that particular component holo-gram. Thus the right and the left eyes of the ob-server would perceive a pair of slightly differentimages, which, after being fused in the brain, give theobserver the impression of a 3D image. Since theIPCHS is bent into a cylinder before observation, theviewing slit would be shifted in the horizontal direc-tion naturally. The effect on the observed imageand method to compensate for the image distortion isalso discussed in section 3.

To avoid the bias build-up or mismatch problemdue to multiple exposure, the information stored inthe master hologram should be transferred to thesecond hologram all at once. However, in reality,the finite dimensions of the optical elements usuallyobstructs this. For example, the length of an IPCHSof diameter 10 cm is more than 30 cm. A lens, whichcan produce a uniform, collimated wave with suffi-ciently good quality to illuminate an area of such adimension, is impractical. An alternative method toachieve the transfer process is to subdivide the mas-ter, as well as the transfer, into several exposuresegments. The whole transferring process can beaccomplished by sequential exposure of these seg-ments. Referring to Fig. 3, assume that the transferis subdivided into several exposure segments, each of

Fig. 3. Schematic figure showing the relationship between thewidths of the exposure segments on the master H1 and the transfer

2.


length L2. The information that will be recorded inone segment ~the shaded area! of the transfer is fromhe shaded area of the master which has a length L1.

The length L1 is the minimum illumination length ofthe reference wave on the master, which can be ex-pressed as L1 5 L2 1 W, where W is the width of themage in one component hologram. After one expo-ure, both the master and the transfer are translatedaterally by a distance L2. The information in the

next segment of the master is then ready to be trans-ferred. This process is repeated until all the 2D per-spectives stored in the master are transferred to thesecond hologram. With careful adjustment, eachsegment on the transfer can correctly capture thedesired images. One point we should note is that,when the first and the final transferring processesare performed, some extra component hologramsshould be added to both ends of the master to offerenough images for the first and the final segments ofthe transfer. Since the information needed for eachsegment of the transfer comes from a wider segmentof the master, these extra component holograms arenecessary to complete the transfer process ~Fig. 4!.

hey can easily be added to the master during theastering process because they contain perspectives

hat are continuous with those in the master. Tovoid the lines of discontinuity among the segmentsn the transfer, another method can be adopted forhe transfer process. Again, the master is spatiallyivided into several segments, each of length L1.

When one of the segments is illuminated by the con-jugate reference wave, a real image occupying alength L2 5 L1 1 W is generated for recording as oneegment on the transfer ~Fig. 5!. Although doublexposures happen at both sides of each exposure seg-ent, the diffraction efficiency is not significantly

egraded for those regions. We note that, as in therevious method, some extra component hologramsre required at both ends of the master.The IPCHS is an image-plane hologram, since all

he 2D perspectives retrieved from the master holo-ram are imaged onto the hologram plane in theransfer step. Each of the 2D perspectives is de-igned to be seen through a vertical slit, called the

Fig. 4. Illustration showing that extra component holograms arerequired at the end of the master hologram.

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viewing slit, which is essentially the real image ofthat particular component hologram. Comparedwith the image observed in a traditional cylindricalmultiplex hologram in which the perspectives are im-aged onto the hologram plane only in the verticaldirection, the image observed in the IPCHS issharper, since the 2D perspectives are imaged di-rectly onto the hologram plane. The image-planenature permits the IPCHS to be replayed with thewhite-light extended source. However, unlike theusual image-plane hologram, a light source with con-siderable horizontal extension is unfavorable, be-cause it would broaden the viewing slits.Consequently, the viewing slits, originally designedto separate the viewing space into many zones, willoverlap. One eye may therefore be covered by morethan one viewing slit at a specified viewing position,which then results in image blur, owing to the seeingof ghost images. Hence the proper light source forviewing the image in IPCHS is a vertically situatedwhite-light line source with small horizontal exten-sion.
Because of the spatial-averaging process of the 2Dperspectives, there are no dark lines on the hologramsurface such as those on the traditional cylindricalmultiplex hologram. The two-step process movesthe plane of component holograms ~master! out of the

lane of the final hologram ~transfer! and makes ithe plane of viewing slits. Therefore the vertical linetructure overlapping the observed image is removed,nd consequently the image quality is improved.The viewing slit is an important factor relating to

he quality of the observed image from the IPCHS.e note that, when the transfer remains in flat for-at and illuminated with the conjugate referenceave, the viewing slits are the real images of the

omponent holograms. However, when the transfers bent into a cylinder and illuminated with an arbi-rary point source, the inconsistency of the position ofhe reference source as well as the curvature of theologram will change the position and the shape ofach viewing slit. Fortunately, with proper selec-

Fig. 5. To eliminate the lines between the exposure segments onthe transfer, the exposure area on the transfer can be taken to belarger than that on the master. However, this results in doubleexposure at the sides of each exposure segment on the transfer.

tion of recording parameters, the viewing slits canstill be separated in space. When the observerplaces his eyes at the plane, which is now curved, ofviewing slits, a pair of slightly different images isreceived. These two images give the impression ofseeing a 3D image. If the observer’s eyes are notsituated at the plane of viewing slits, a 3D image canstill be observed. However, it appears to bestretched or compressed in the horizontal directionaccording to the relative position of the eyes withrespect to the viewing slits. This effect is discussedat the end of Section 3.

Basically the viewing slits are inclined verticalstripes, since the component holograms are tilted,with respect to the final hologram, at the so-calledachromatic angle a, which can be expressed as

a 5 arctan~sin u!, (1)

here u is the incident angle of reference wave.8Upon illumination, the viewing slits corresponding todifferent wavelengths are spread along an inclinedplane. If the viewing slits are long enough, the eyescan see an achromatic image at the central viewingfield where the viewing slits of all wavelengths over-lap. When the master is not tilted at the achromaticangle, the viewing slits of different wavelengthswould be displaced longitudinally rather thanaligned in the same plane. The eyes of the observertherefore cannot be placed at the location of viewingslits for all the wavelengths simultaneously. Sincethe aspect ratio of the observed image changes withthe distance between the eyes and the viewing slits,color blur is more serious toward the two sides of theimage.

A point, less noticed, is that the vertical field ofview is larger for the IPCHS than for the usual mul-tiplex hologram. For a rainbow-type multiplex ho-logram the viewing zone is restricted to a narrowband of horizontal slits. The image-plane nature ofthe IPCHS permits the viewing slit to be extendedvertically without degrading the image quality.Therefore the field of view in the vertical directioncan be extended as long as the height of the originalcomponent hologram ~master! is increased.

3. Factors Affecting Image Quality

As stressed above, the most important factors affect-ing image quality are the location and the width ofthe viewing slits. These two factors are discussedbelow. Although the viewing slits are designed asthe place for viewing, nevertheless, observers aregenerally not at the desired distance, which results insome distortion on the observed image. At the endof this section we discuss the effect on the image asthe distance between the eyes and the viewing slits isvaried.

A. Slit Location

The viewing slit for each individual 2D image wasoriginally a long thin strip inclined at the achromaticangle. Owing to the bending effect of the hologram,


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the location of the viewing slit in the horizontal di-rection is shifted. The use of an inconsistent recon-struction reference wave would also change thelocation of the viewing slit in the vertical direction.Hence the vertical and the horizontal locations ofeach viewing slit are usually different if the recordingparameters are not properly selected. This astig-matic effect would consequently introduces distortionon the observed image. To minimize the distortion,the viewing slit in both the vertical and the horizontaldirections should be located at the pupil plane of theobserver.

By geometrically tracing the rays diffracted fromthe hologram, we can find the location of the viewingslit in both directions. Before we proceed with thecalculation, the basic diffraction equations in holog-raphy are reviewed. Referring to Fig. 6, we see thatan object ray Eo at azimuthal angles uo and fo isnterfering with a reference ray Ec, which is assumed

to lie on the y–z plane and makes an angle uc withespect to the z axis. The directional spatial fre-uency of the interference fringes recorded at theoint Q on the hologram can be expressed as

fxax 1 fyay 5sin uo cos fo

lax

1 Ssin uo sin fo

l1

sin uc

l Day, (2)

here l is the wavelength of both the object and thereference rays. The recorded interference fringes atQ are then illuminated with a reconstruction refer-ence ray Er, and an image ray Ei at azimuthal anglesui and fi is produced. Assume that the ray Er lies inthe y–z plane and makes an angle ur with respect tothe z axis; we then have

sin ui cos fi

l9ax 1 Ssin ui cos fi

l91

sin ur

l9 Day 5 fxax 1 fyay,

(3)

Fig. 6. Object ray Eo and reference ray Ec are interfering at pointQ, and the interference fringes are recorded. These interferencefringes are read by the reconstruction reference ray Er, and theimage ray Ei is generated.


where l9 is the wavelength in the reconstruction pro-cess. Substituting Eq. ~2! into Eq. ~3!, we get

sin ui cos fi

l95

sin uo cos fo

l, (4a)

sin ui sin fi

l95

sin uo sin fo

l1

sin uc

l2

sin ur

l9. (4b)

These two equations are the basic diffraction equa-tions used extensively in the analysis.

In the following we consider how the location of theilluminating source and the radius of the hologramcylinder change the location of the viewing slit in thecorresponding direction. Referring to Fig. 7, we seethat the shaded area is a 2D perspective that wasimaged onto the transfer in the hologram-formingprocess. We take its center to coincide with the or-igin of coordinates O. If this transfer is illuminatedby the conjugate reference wave, which is a one-dimensional ~1D! collimated and one-dimensionalconvergent wave that would focus into a line SL1, thereal image of its corresponding component hologram,shown as the dashed gray line, would be recon-structed. The position of this real image is the loca-tion of the viewing slit of concern. Basically, toobtain a more precise result, more diffracted raysshould be used in the ray-tracing process. However,a satisfactory result can still be obtained even thoughonly a few rays are used. Here, for simplicity of

Fig. 7. 2D perspective ~shaded area! on the transfer is illumi-nated by the conjugate reference wave that would focus to be a lineSL1. The real image of the component hologram, or the viewinglit, is generated at the dashed gray line.

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calculation, we use two diffracted rays at the center ofthe vertical ~horizontal! edges of the perspective ho-logram to determine the horizontal ~vertical! locationf its viewing slit. Moreover, we trace only for theentral point Po of the real image of the component

hologram, which, as we described above, is an in-clined strip.

Ray tracing is performed in the horizontal and thevertical directions separately, since the location ofviewing slit is influenced mainly by the radius of thehologram cylinder in one direction but by the diver-gence of the illuminating reference wave in the otherdirection. In the horizontal direction the point ofintersection for two diffracted rays from O and C isused to estimate the position of the viewing slit ~Fig.7!. When the hologram is under conjugate illumina-tion, the diffracted rays would travel along the oppo-site direction to those of object rays, forming the realimage point Po. Therefore the diffracted ray frompoint O will travel along the z axis to reach point Po.The other ray from point C will travel in the x–z planewith azimuthal angles

uo9 5 arctan~xiyd!,

fo9 5 0, (5)

where xi is the distance between points O and C andd is the distance between point Po and the hologramsurface. Since the illuminating wave is collimatedin the horizontal direction, the incident angle for tworays at points O and C would be the same, which is uc.

If the 1D convergent light is replaced with a diver-gent one SL2 as shown in Fig. 8, the reconstructionreference ray for either point O or point C would be atthe same incident angle ur, which is generally differ-ent from uc, since a white-light line source is used formage reconstruction. ur 5 uc happens only for theentral point of the illuminating source. For sim-licity of calculation we deal with this case only for

Fig. 8. Diverging cylindrical wave from line SL2 is illuminatingthe 2D perspective on the transfer. The vertical location for thecenter of the viewing slit is shifted from Po to Po9.

the central point of the illuminating source. Owingto the fact that the incident rays for both points O andC do not change, the diffracted rays would still travelalong the same routes as those shown in Fig. 7.

When this hologram is bent into a cylinder of ra-dius R, the line SL2 would be collapsed into a point Pr~Fig. 9!. The incident angle at the points on the arcCD is ur, or in our case uc. Although the hologram isnow a cylinder, the incident angles for reconstructionreference rays at both points O and C are still un-changed. However, the diffracted rays, thoughmaintaining original relative directions to the holo-gram surface, will change their directions. For pointO the diffracted ray still travels along the z axis.But, for point C, it would be rotated by an angle~xiyR! outward to reach the new image point Po9.The distance d9 from the origin of coordinates O to thenew image point Po9 can be expressed as

d9 5 RF sin~xiyR!

tan~uo9 2 xiyR!1 cosSxi

RD 2 1G . (6)

Note that d9 may be a negative value according toEq. ~6!, which means that the diffracted rays do notfocus to form a real image, or the viewing slit. Inother words, the diffracted rays from the hologramare divergent rather than convergent. The absolutevalue of d9 is therefore the distance from the diverg-ing center to the origin of coordinates on the holo-gram surface. Generally, in this case, only part ofthe diffracted rays from a 2D perspective can be ob-served, owing to the finite dimension of the eye.However, the observer can still see the complete im-age, since the missing parts would be filled up byother images from the adjacent perspectives. This isnot the case we prefer, because greater image distor-tion would be perceived. Hence the parameter d~master-to-transfer distance! and R ~radius of holo-ram! should be properly chosen to produce the view-ng slit at the desired position. Note that d9 is a

Fig. 9. Owing to the bending effect of the hologram, the horizontallocation of the viewing slit is shifted. When the divergence of theilluminating reference source is properly chosen, the location of theviewing slit in the vertical direction can be shifted by the sameamount.


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function of xi, which means that the diffracted raysfrom different parts of the hologram do not focus atone position. However, the variation of d9 is not toogreat, so that taking xi as one fourth of the width ofthe perspective gives a satisfactory paraxial result.

Now, we consider the case in the vertical directionin which the location of viewing slit is determined bytracing of two vertically separated diffracted raysfrom points O and A ~Fig. 7!. Since the reconstruc-tion reference ray for point O makes an angle uc withrespect to the z axis, the incident angle uc0 of theeference ray for point A would be

uc0 5 arctan@tan uc 1 ~yiydc!#, (7)

where yi is the distance between O and A and dc ishat from Pc to the surface of hologram. The direc-ion of the diffracted ray from point O is along the zxis and that from point A can be described by thezimuthal angles

uo0 5 arctan~yiyd!,

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where d is the distance from Po to the origin of coor-dinates. When the 1D divergent source SL2 is usedfor illumination ~Fig. 8!, the vertical focal point ofdiffracted rays will be moved away from Po. Theincident angle of the reconstruction reference ray atO is ur, which, in the present case, is assumed to beequal to uc. Therefore the incident angle ur0 at point

can be expressed as

ur0 5 arctan@tan uc 2 ~yiyR!#, (9)

where R is the distance between the line SL2 and thehologram surface. The direction of the diffractedray at O is still along the z axis, whereas that at A isrotated upward, since the reference ray is now at theangle ur0, which is different from ur. Assume thathe diffracted rays from points A and O intersect atoint Po9; then the direction for light ray APo9 can beescribed by the azimuthal angles

ui0 5 arctan~yiyd0!,

fi0 5 2py2, (10)

where d0 is the distance from point Po9 to the holo-ram surface. Substituting Eqs. ~7!–~10! into Eqs.

~4!, we have

2sinFarctanSyi

d0DG 5 2sinFarctanSyi

dDG1 sinFarctanStan uc 1

yi

dcDG

2 sinFarctanStan uc 2yi

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(11)


Since a white-light source is used to reconstruct theimage, the broadband nature of white light can haveonly a particular component coincide with that in thehologram-forming process. In arriving at the aboveequation we thus assumed that the wavelength l9 inthe reconstruction process is the same as that in thehologram-forming process, for simplicity of analysis.This equation implies that the distance dc is decidedonce the distance d0 is known. Generally, we firstobtain d9 from Eq. ~6!. Then, by equating d9 to d0,the location of the viewing slit is made equal in thehorizontal and the vertical directions. When thisvalue is substituted into Eq. ~11!, suitable dc is ob-tained. If we choose d9 not equal to d0, that is, thehorizontal and the vertical foci of diffracted rays arenot the same, the observed images will appear to bewider on the top and narrower on the bottom. Fig-ure 10~a! shows an example of the result from com-

uter simulation in which dc is set to be infinite.The result for matched condition, which is dc 5 5.7cm, is shown in Fig. 10~b!. Both figures are drawnor a viewing point at a distance 55 cm from theenter of the hologram. The eyes of the observer aressumed to be on the x–z plane. Other parameterssed for these figures are R 5 10 cm, ur 5 uc 5 45°,

l9 5 l 5 633 nm, and d 5 8 cm. From these figureswe can see that the distortion of the image is cor-rected if the location of the viewing slit is made thesame for both the horizontal and the vertical direc-tions.

B. Slit Width

The slit width is also an important factor for theimage quality of the IPCHS. A slit of sufficientwidth can ensure that an eye receives only one per-spective view without seeing ghost images. How-ever, if the slit is too wide, the parallax resolution willbe degraded, since fewer 2D images can be synthe-sized in one hologram cylinder. A slightly jumpyimage is perceived in this condition especially whenthe eyes sweep across the viewing slits quickly. Thisdiscontinuity usually annoys the observer and istherefore unfavorable. In contrast, a smaller slitwidth not only improves the parallax resolution butalso makes the transition smoother as the eye of theobserver moves from one viewing slit to another.However, the horizontal spatial resolution is de-graded when the slit becomes too narrow. In thiscase the slit width becomes the limiting aperture in-stead of the diameter of the eye pupil. When morethan one viewing slit fills an eye pupil, ghost imagesare observed and, consequently, image blur results.To ensure that the observer sees a clear image withsatisfactory parallax resolution, but without ghostimages, a viewing slit with suitable width should bechosen. We can do this by choosing a simple crite-rion. Figure 11 shows an eye observing two pointsP1 and P2 on the hologram surface, which are thecentral points of two adjacent perspectives. Whenthese two points can be seen simultaneously, an ob-ject point then becomes two points on the image,which consequently result in image blur. To avoid

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this situation, the minimum separation S betweenpoints P1 and P2, which is also the suitable width forthe viewing slit, should be

S 5 pRy~R 1 de!, (12)

where p is the diameter of the eye pupil, R is theradius of the hologram cylinder, and de is the sepa-ration between the hologram surface and the eye. A

Fig. 10. ~a! Computer simulation of the trapezoid distortion onocations in the orthogonal directions. ~b! Trapezoid distortion oirections are made equal.

slit with width wider than S, though ensuring elim-ination of ghost images, would degrade the observedimage, as mentioned above. A narrower slit wouldcause image blur, which is not the preferred case.

C. Aspect Ratio

Although the IPCHS is designed to be viewed fromwhere the viewing slits are located, nevertheless, an

bserved image, which is due to the mismatch of the viewing-slitimage is corrected if the viewing-slit locations in the orthogonal

the on the


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image with satisfactory quality can still be perceivedeven though the eyes of the observer are not properlysituated. However, the observed image would pos-sess more distortion as the eyes are farther awayfrom their correct position. This is because, whenthe eye is out of the plane of the viewing slits, theobserver cannot see a complete image through a sin-gle viewing slit but instead views only part of theimage through the slit. The final observed image isthus composed of many partial images from adjacentperspectives. This effect introduces two kinds of dis-tortion. One of them is called time smearing, whichis quite familiar in multiplex holography.9 Theother, called aspect distortion, occurs when the ob-served image appears to be either stretched or com-pressed in the horizontal direction. The formeroccurs because different parts of the observed imagecome from different perspective views that corre-spond to a slowly moving object taken at a differenttime. Hence the observed image does not show ei-ther correct perspective view or correct movement asexpected. Note that the images bearing differentperspective views are imaged onto the hologram sur-face, with a small separation among one another.Therefore when the final observed image is composedof partial images from the closely spaced perspectiveviews, its aspect ratio is changed, which results in thesecond kind of distortion. Since time smearing isquite familiar in multiplex holography and happensonly with moving objects, we will thus concentrate ondeveloping relationships governing the change of as-pect ratio as the relative position of the eye to theviewing slits is varied.

Figure 12 shows one eye of the observer, at Pe,watching the IPCHS, which is centered at the originof coordinates with its axis coinciding with the y axis.

he radius of the IPCHS is R. We specify a numberfor each 2D image on the hologram surface and itscorresponding viewing slit. The one with its centeron the z axis is numbered 0, and its center is denoteds H0, whereas the corresponding viewing slit is

called P0. Those 2D images with their centers lo-cated at the right- ~left-! hand side of the eye, or at the

ositive ~negative! direction of the x axis, are given a

Fig. 11. Illustration showing the relationship between the diam-eter of eye pupil p and the centers of two adjacent perspectives P1

and P2 for setting proper width for original component holograms.


positive ~negative! number for each image. Sincethe eye is not at the location of the viewing slit P0, itwould see the edge of the image Q through the nthviewing slit Pn. From simple geometrical consider-ation we find

P0 Pn 5 nS~R 1 d9!yR, (13)

here S is the separation between two adjacent 2Dperspectives, which is also the width of the compo-nent hologram in the master, and d9 is the distancefrom the hologram surface to the viewing slit. As-sume that the value nS is much smaller than R,which is often the case in practice; the arc P0Pn canthen be approximated as a straight line. From thesimilarity of two triangles, PeP0Pn and PeAQ, wehave

nS~R 1 d9!

R~de 2 d9!5

R sin C

de 1 R 2 R cos C, (14)

here C is the angle of the arc QH0 subtended frompoint O. The length for the arc QH0 is ~Wy2 1 nS!,where W is the width of a 2D image on the hologramsurface. If the number n is small so that nS is muchmaller than ~Wy2!, then we can obtain

n 5R2~de 2 d9!sin~Wy2R!

S~R 1 d9!$de 1 R@1 2 cos~Wy2R!#%. (15)

When the eye is moved farther than the viewinglit, the number n is increased, which results intretching of the observed image in the horizontalirection. In contrast, when the eye is moved closero the hologram surface than the viewing slit, necreases and the observed image shrinks in theorizontal direction. Figure 13~a! shows the rela-ionship between n and the viewing distance de for

an IPCHS with viewing slits formed at approxi-mately d9 5 45 cm from the hologram surface. Thevalue n is rounded, since it should be an integer in the

Fig. 12. Eye of the observer is at the position Pe, which is off thelocation of the viewing slit Po. The edge of image Q can be seen bythe eye through the nth viewing slit Pn. From this geometricalrelationship, one can estimate the number of perspectives thatcontribute partial images to the final observed image.

T

p

atr

ftpv

practical case. Other parameters used for this plotare radius of hologram cylinder R 5 10 cm, width of2D perspective W 5 4 cm, and slit width S 5 0.5 mm.

he viewing distance de is taken to vary from 30 to120 cm to match the practical case. This then re-sults in the variation of n from 24 to 4. In this

roper viewing range the value nS is much smallerthan both R and ~Wy2!, which satisfies our previousassumptions.

The width of the observed image along the holo-gram surface is

W9 5 2@~Wy2! 1 nS# 5 W 1 2nS. (16)

Note that the actual width ~height! of the observedimage is the projection of this width ~height of 2D

Fig. 13. Number of perspectives, n, on the hologram surface andthe aspect ratio as a function of the viewing distance de, which isrom the eye to the hologram surface. The parameters used forhese plots are radius of hologram cylinder R 5 10 cm, width of 2Derspective W 5 4 cm, slit width S 5 0.5 mm, and distance ofiewing slit to hologram surface d9 5 45 cm.

perspective! onto the x–y plane from the eye. Thespect ratio is defined as the width-to-height ratio ofhe observed image. Figure 13~b! shows the aspectatio for various viewing distance de. From this fig-

ure we note that, when the observer is farther awayfrom the viewing slit, the widening rate of the ob-served image becomes slower. As the observermoves closer to the hologram surface from the view-ing slit, the compression rate of the image becomesfaster, and the image is more distorted.

4. Experimental Results

To demonstrate the above derived theory, we per-formed some experiments. Holographic films of di-mension 4 in. 3 5 in. are used as both the master andthe transfer. The separation d between the centralpoints of the master and the transfer is set at 8 cm,and the radius of the hologram cylinder R is designedat 10 cm. These two parameters determine the lo-cation of the viewing slits to be approximately 45 cmfrom the hologram. In the recording of the masterthe film is tilted at the achromatic angle a, which,according to Eq. ~1!, is 35° when the reference beamis incident at 45° from above ~Fig. 1!. The slit widthfor this hologram is set to be 0.5 mm, which, as pre-scribed by Eq. ~12!, would produce a viewing slit of ;3mm wide. This is about the diameter of the eyepupil. Several films are joined together to obtainsufficient length for both the master and the transfer.

In the viewing process an incandescent light bulb,whose filament length is ;2 cm, is used as the refer-ence source. This light source hangs on the axis ofthe hologram cylinder and produces a reference wavewith a 45° incident angle. Figure 14 shows an ach-romatic image, which originally has a width of 4 cmand a height of 3.5 cm, reconstructed from theIPCHS. The achromatic image is observed in thecentral viewing zone, which has an angular extent of;5°. The overall vertical viewing field is ;45°,which is considerably broader than that of the tradi-tional multiplex holograms. This viewing field canbe further extended if the height of the recordingmaterial is increased. Comparing the observed im-age ~Fig. 14! with that from a traditional multiplex

Fig. 14. Reconstructed achromatic image from IPCHS showingthat the annoying vertical line structure overlaying the image intraditional cylindrical multiplex holography is eliminated.


2htItaoicvtmwalerav

qpmTiocitdivqotcdfiwwefooatpthvsIp

t

4

hologram ~Fig. 15!, we note that the so-called picket-fence effect is eliminated in the IPCHS.

If an IPCHS does not use the achromatic angle, theobserved image will be composed of many images ofdifferent colors overlapping one another. Since thewidth of the red image is the widest and that of theblue is the narrowest, the perceived image will ex-hibit significant blur toward its horizontal edges.This difficulty is eliminated if we incorporate the ach-romatic angle in the recording of the IPCHS as wasdone above.

In viewing the IPCHS, when the eyes are closer tothe hologram than the viewing slits, the observedimage will shrink in the horizontal direction. As theeyes move away from the viewing slits, the observedimage slowly stretches in the horizontal direction.In our observation the variation of the aspect ratio issmall for a viewing distance farther than the properdistance by an amount of 50 cm. This result coin-cides with our previous analysis and ensures that theobserver will see a nondistorted image in a wide view-ing range. Note that, when the eyes of the observermove up and down, not only the color but also thewidth of the image changes. This is because theviewing slits are inclined stripes in the observationspace, which is closer to the hologram in the upperviewing zone and farther in the lower viewing zone.When the eyes of the observer move upward, they arerelatively far from the viewing slits and perceive anarrower image. In contrast, as the eyes of the ob-server move downward, a wider image is observed.This result agrees with the analysis in Section 3.

The slit width was set as 0.5 mm to produce aviewing slit with a width comparable with the diam-eter of the human eye. A slit with width wider thanthis value makes the observed image jumpier as theeyes sweep horizontally across the viewing slits.When a slit with width slightly narrower than 0.5mm is used, according to our previous discussion, theclarity of the image may be somewhat sacrificed.

Fig. 15. Image from traditional cylindrical multiplex hologramshowing the picket-fence effect ~exaggerated for comparison withhe image in Fig. 14!.


However, in our experimental demonstration it is notsignificant.

5. Conclusion

We have demonstrated that, by use of a two-steprecording process, it is possible to fabricate a cylin-drical holographic stereogram as an image-plane ho-logram ~called IPCHS!. In the first step a series ofD perspectives are fabricated as a series of long thinolograms side by side. The holographic film ~mas-er! for recording is tilted at the achromatic angle.n the second step, with conjugate wave illumination,he information stored in the hologram is read out,nd all the 2D perspectives are imaged onto the sec-nd holographic film ~transfer!. A cylindrical waves used as the reference wave for recording. Withareful registration an IPCHS that is capable of pro-iding 360° parallax is obtained by spatial division ofhe master and the transfer into many suitable seg-ents to complete the transfer process. Underhite-light line-source illumination the observed im-ge is brighter than that in traditional multiplex ho-ography and is without the annoying picket-fenceffect. The vertical field of view is broad so that theeconstructed image is easy to observe. Moreover,n achromatic image can be seen around the centraliewing zone.The most important factors, which influence the

uality of the observed image, are analyzed in thisaper. First, the location of the viewing slits deter-ines the best viewing distance for the observer.he horizontal location of the slits is affected primar-

ly by the master-to-transfer distance and the radiusf the hologram cylinder. However, its vertical lo-ation is affected by the divergence of the illuminat-ng reference wave. A ray-tracing method is appliedo analyze the locations of the viewing slits in bothirections. The trapezoid distortion of the observedmage, which occurs when the horizontal and theertical locations of the viewing slits are different, isualitatively analyzed. By changing the divergencef the reference wave, we can move the vertical loca-ion of the viewing slit to coincide with its horizontalounterpart. This technique removes the trapezoidistortion of the image. Second, to maintain satis-actory parallax resolution and clarity of the observedmage, a simple criterion to determine suitable slitidth is provided. This slit width ensures that theidth of the viewing slit is equal to the diameter ofye pupil. Third, the distortion of the aspect ratioor the observed image, which occurs when the eyes ofbserver are not at the plane of viewing slits, is the-retically as well as numerically analyzed. Gener-lly, when the eyes are closer to the hologram thanhe viewing slits, the observed image will be com-ressed rapidly in the horizontal direction. In con-rast, the image is only slightly stretched in theorizontal direction as the eyes move away from theiewing slits. No significant image distortion is ob-erved even when the observer is a few meters away.n our experiment all parameters discussed in thisaper are either qualitatively or quantitatively

Technologies, D. F. McAllister and W. E. Robbins, eds., Proc.
tested. An example of an observed achromatic im-age from the IPCHS is provided for comparison withthat from the traditional cylindrical multiplex holo-gram.
We acknowledge the National Science Council forfinancial support through grant NSC-88-2215-E-008-006. This paper was presented at the 1999 OSAAnnual Meeting, Santa Clara, California.

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image-plane cylindrical holographic stereogram

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