Image Scanning by Rotation of a Hologram

Ivan Cindrich

Reconstructed image scanning can be provided by simple rotation of a hologram. Motion of the image is

described as a function of the rotated angle. Application to the problem of beam forming and deflection isdiscussed.

Introduction

A hologram which is rotated during reconstructioncan serve as a means of image scanning. Motion of theimage is described as a function of the rotated angle,with the illuminating beam maintained fixed in posi-tion. This idea may be used to provide light beamforming and deflection for line scan recording or displaydevices which use a laser, or simply for controlled move-ment of a reconstructed scene to facilitate viewing.

General Properties

Quality of the image obtained when a hologram is ro-tated is the same as in normal viewing without rota-tion, as long as the illuminating beam wavefront isradially symmetric relative to the rotation axis selected.This is true because with such rotational symmetry theilluminating wavefront will have a spatial phase varia-tion at the hologram which is independent of rotationabout this particular axis. However, in describingimage motion in this section, the rotation axis is takennormal to the surface of the hologram record. This is-done because the normally directed axis is usually mostconvenient for implementing rotation provisions.

The position of the reconstructed image' can beestimated from the phase cJ of the light field emergingfrom the hologram when it is illuminated. We con-sider the first order conjugate image (real for examplescited herein) and assume a hologram made as a filmtransparency. However, other images which may oc-cur, and other types of record media, may be consideredsimilarly. The phase '1 can be written as = , +Or - 4)e where 4 c, O'r, and 0, are phase functions giving

The author is in the Radar and Optics Laboratory, The Uni-versity of Michigan, Ann Arbor, Michigan 48107.

Received 27 February 1967.This work was sponsored jointly by United States Air Force

Avionics Laboratory, Wright-Patterson Air Force Base under acontract and United States Army Electronics Command undera contract.

the spatial phase variations, at the record media, of thereconstruction (illuminating) beam, reference beam, andobject beam, respectively. The phase functions aredescribed in the geometry of Fig. 1 where the hologramsurface is defined in the plane of the x and y axes whichare fixed in space. For convenience, light propagationdirection is taken from left to right, i.e., with a z com-ponent for propagation velocity which is positive.Taking each of the beams with a spherical wavefront,the centers of wavefront curvature are the pointsPc(XcZc), Pr(XriyrZr)X and P(xoyozo). With theassumed direction of propagation, the object pointmust always have a negative value for its zo coordinate.The distance from the origin of the xyz coordinate sys-tem to these points is Rc, Rr, and R0 . The distancefrom the point P(PrPO) to a general point P(x,y) inthe xy plane is rc(rrro). With this notation we have

4 = (27r/Ac)rc + (27r/No)rr - (2ir/Xo)rO,

where X, is the wavelength used in hologram construc-tion and X0 in reconstruction. The radial distances inthe above expression are expanded in a form that allowsfor large off-axis positions of the points P0, Pr, and Po,since this situation is of interest in applications to lightbeam deflection. For example, expansion of r is interms of x0, Yo, x, y, and Ro (where Ro is used rather thanthe frequently encountered ze). To allow for a scalechange in the hologram by a factor n prior to recon-struction, we replace x by x/m and y by y/n in theexpansion for rr and ro. Thus, the spatial variations inthe hologram plane after the scale change are spreadover distances m times the distances in the originalrecording. Rotation of the hologram about the z axisby an angle a requires that x,, Yo, Xr, and Yr, which aremeasured relative to the hologram before rotation, betransformed into the xy coordinate system which isalways fixed in space. This is readily done with theresult that we replace xe by x, cosa - yo sina, y byx, sina + Yo cosa, Xr by Xr cosa - y, sina, and Yr byxr sina + Yr cosa.

Using the above described approach to expansion of

September 1967 / Vol. 6, No. 9 / APPLIED OPTICS 1531

y

P (,y 0 ,zo)

ro xP(xy)

0~~~~~~~~~~~~~~X r ~~~~~~~~~~~~~~Z

Fig. 1. Describing the location of the center of curvature P0 fora spherical wave. P(x,y) is a point in the xy plane.

LI), we can write

2 [ + m2R, m ]+YR R l + 70R- m2Rol

2X rx + r(x, cosa - yr sina) P(Xo cosa - o sina)LR, mR, ?nRO

2F ye + u(xr sina + Yrcosa) p(xo sina + Yo cosa)R mR, mRe,

+ (R + R Re) + (. + Ar

where

k = (2 r/X,),p =u

and the A's contain all terms of order greater than one in1/R.

The first-order estimate of the reconstructed pointimage coordinates, denoted here as Xb, yb, Rb, can beidentified from the expanded expression for c as

[Xc p( r o(Yr O\ 1Xb = Rb I + C o csa \r o

Yb = Rb + - ±( - -) sinaa[Re, m R, RRr R)

Rb = lfl2RcRrRoin'RR + pR R- pRrR,

These position coordinates define a circular pathin space as a varies. The point image is rotated byan angle a about the center of its circular path whichis located at (Rb x/R,, Rb y/R,). The square of thepath radius p2 is given by

[(X - X) 2 + 2 o) ] Rb-

This interpretation for the image path must of coursebe restricted to the cases where we perform the re-construction process so as actually to realize a reason-able reproduction of the point object, i.e., we wish toavoid excessive aberrations, which dictates a constrainton the permissible locations for the reference and il-

lumination beams and the values for p and n. Follow-ing the analysis by Meir,1 third-order aberrations canbe eliminated here for any point image with t = ,xclRc = -Xr/Rr, y/R = -Yr/Rr, and with planewavebeams for reference and illumination. We mustalso require that a be fixed, and to be consistent withthe above constraints we need, a = 0. There is anexception to this restriction on a if we require that thebeams be normally directed, i.e., that xc = Xr = Y =Yr = 0. For this case, a may take on any value andthird-order aberrations can still be avoided. The imagenow has a circular path with center on the rotation axisand a radius equal to the off-axis distance of the pointobject.

Using R rather than z components in the expansionsfor 4J makes it inconvenient to interpret the effect of achange in x or y position coordinates of the beams sincethe radial distances which appear would also change buttheir change would not be explicitly evident. For thisreason, an alternative form of the expansions, where theratios x/R, etc., are expressed as sine and cosine func-tions of appropriately selected angles, can be useful.The alternative expression for cJ contains only the radialdistances and angular positions of the beams. Spheri-cal coordinate system angles for Rc, Rr, and R, are usedand denoted as and a with appropriate subscripts.j3 is measured from the z axis to Rc, while a is the anglebetween the plane containing the x and z axes and theplane containing Rc and the z axis. Similar definitionsapply for /r, ar, Bo and ae, but these angles are measuredrelative to the nonrotated hologram. After making thesuggested substitutions the form of xb and Yb changes tothe following:

Xb = Rb [sing3 cosmec + (u/7n)(sin, coScar - sin3e cosaxo)cosa

- (/m) (sing, sina - sin3e, sinae,)sina],

Yb = R [sin,8e, sina + (/m) (sing, cosar sing,, cosae,)sina

+(r/m) (sin3r sina, -sinl,, sinao)cosa].

The center of the circular path of the image now ap-pears as (Rb sino,3 cosa,, Rb sine, sina,) and its radiussquared p2 is:

[(sinlg, Cosa - sinj,8 cosao)'

+ (singr sina, - sinf3e sinao)2] Rb

In this section the position of the image of a singlepoint object, P,(x,,yz,), was reviewed and positionequations were modified to include the effects of holo-gran rotation. The expressions for and for imageposition can of course be extended to a collection ofpoint objects (or a three-dimensional scene) and othertypes of beams such as those with plane or cylindricalwavefronts. Thus, a controlled movement of the re-constructed image of a variety of objects can be en-visioned and some of their basic properties analyzedfrom the equations described above. The quality ofthe image can be more extensively analyzed in terms ofaberrations as in Refs. 1 and 2. Other pertinent topics,

1532 APPLIED OPTICS / Vol. 6, No. 9 / September 1967

such as efficiency,3'4 the effect of finite record mediathickness, 3 and hologram resolution' have also beendiscussed in previous publications, some of which areused here as references.

The following section will be directed to the problemof light beam forming and deflection.

Beam Forming and Deflection

Beam forming is inherent in the hologram recon-struction process. For example, consider the hologramof the single point object which is illuminated with anormally directed plane wave beam. A convergentbeam results in the image space to give a real first-order image. This can be interpreted as a conversionof the incoming collimated beam into the form of theconvergent output beam. Beams corresponding toother image orders which may exist can usually bespatially separated and left unused or blocked with amask if necessary. Deflection of the convergent beamof the first-order image along a circular path is providedby rotation of the hologram as described previously.Thus, scanning of a focused spot of light, as depicted inFig. 2, is readily accomplished. Other forms of beamssuch as collimated or convergent cylindrical beams cansimilarly be formed and scanned.

Experimental results obtained for the case citedabove are typified by the scannable light spot shown atone position along its scan path in Fig. 3. This figureshows the intensity distribution of the real image of asimulated point object which can be moved on a circularpath of about 141 mm radius. The central region of thespot is 6 u across at its widest (the oval shape was dueto poor alignment during reconstruction). The holo-gram was made with an off-axis point object which wassimulated by the crossover of a convergent beam passedthrough a 2.4-,4 diam pinhole. Reference and recon-struction beams with plane wavefronts, directed normalto the hologram plane, were used and a 30-mm diamregion of the hologram was illuminated for reconstruc-tion. Exposure and viewing of the hologram was donewith a He-Ne laser. In terms of the variables in theequations for 1 and image position, an R, = 200 mm,xe = 141.4 mm, y, = 0, Rr = Xr = Yr = 0 Re =c, xc= Y = 0, and i = m= I were used. The re-

y

X i f / ,Focused/ / \ Light

/ \Spot

/ F~~~~~~~~~~~~~~~~~4 ,

-; Illuminating-Ra~-ms. .

Hologram /Section

z

I

Fig. 2. A hologram of a point object is rotated relative to thefixed xyz coordinates giving a reconstructed image which rotates

in the image space.

Fig. 3. Intensity distribution of the scannable reconstructed realimage of a (simulated) point object.

constructed image (spot) of interest appeared at Rb =-Re, Xb = x0 cosa, Yb = x, sina.

Repetitive scanning over a path of finite length orraster scanning over an area, with a broad range of timebetween the start of successive scans to choose from,can also be provided by utilizing a hologram of morethan one point object. The detailed procedure used tomake a hologram for this type of application can varysomewhat. A single point object can be used and therecording plate given multiple exposures with the plateposition changed between successive exposures; or,several point sources properly located can be used withonly a single exposure of the plate being required. Thetime between the start of successive scans can be se-lected from a range of values between the time requiredfor the hologram to be rotated one revolution down tozero, depending on the number of point objects used.An alternative method is possible where superpositionof the recorded object wavefronts is avoided. Forexample, successive exposures with a point object canbe made on adjacent nonoverlapping sections along acircular path on the plate. This may be done by fixingthe object position and rotating the plate between ex-posures while having a suitable mask placed betweenobject and plate to avoid any overlap of successive ex-posures. In reconstruction a hologram made in thisway is rotated such that the sections which had beenexposed separately are passed through the illuminatingbeam individually.

Typically, thus far, the reconstructed image of in-terest has been shown to move on a circular path whenthe hologram is rotated. In the case of scanning afocused spot of light, a straight line scan path or, atleast, a path of large radius may be preferred. Thepath radius can obviously be increased by increasingthe off-axis position, (xO2 + yo)l, of the point objectused in recording. However, an additional means ofchanging the radius of the circular scan path is availablewithout pushing the object further off the hologramrotation axis if we allow some departure from the con-straints used to eliminate third-order aberrations men-tioned previously.

Recalling the aberration discussion involving recon-struction of a point from the object field with rotationaxis normal to the hologram, it was stated that forarbitrary rotation angle a, we need a reference andilluminating beam which are normal to the hologramplane, along with other constraints associated with anunrotated hologram. If we disregard the requirementfor normally directed beams and use off-axis beams (butconform to the other constraints), then for a = 0 third-order aberrations are zero, and as a increases the aber-rations will gradually appear. Thus, a small range of

September 1967 / Vol. 6, No. 9 / APPLIED OPTICS 1533

I-

rotation of the hologram is permissible if some aberra-tion is allowed. Now, however, by use of an off-axisreference and illuminating beam a circular image pathwith radius approaching twice the object to rotationaxis (z axis) distance can be realized. This is seenfrom the image position equations, remembering thatthe constraints needed to eliminate third order aberra-tions for the unrotated hologram of a point object areused.

One approach that might be used to obtain motionalong a straight line, rather than a circular path, whenattempting to deflect a focused spot of light is to il-luminate the hologram so as to use the diffractionproperties of only one dimension. Illumination alonga straight line on the hologram surface will allow beamdeflection in one direction, if indeed the line is of suf-ficiently small width. Such illumination might be pro-vided with the aid of a slit or by use of a convergingcylindrical beam which comes to focus at the hologram

surface. However, translation of the light spot in thefocal plane will not be linearly related to the hologramrotation angle but will be related through sina (or cosa)as can be seen from the equations for image position.In the other, orthogonal, dimension of the reconstructedbeam, spreading may exist and would require a cylindri-cal lens, for example, to focus in this dimension if re-quired for the application.

References1. R. W. Meier, J. Opt. Soc. Am. 55, 987 (1965).2. E. N. Leith, J. Upatnicks, and K. A. Haines, J. Opt. Soc.

Am. 55, 981 (1965).3. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, and N.

Massey, Appl. Opt. 5, 1303 (966).4. N. George and J. W. Matthew, Appl. Phys. Letters 9, 212

(1966).5. G. B. Parrent, Jr. and G. 0. Reynolds, Soc. Phot. Instr.

Engrs. J. 3,219 (1965).

RUSSIAN MASERSTypical examples of Russian maser papers pub-lished in recent issues of OSA's monthly trans-lation journal OPTICS AND SPECTROSCOPY:

CaF2-Sm2+Crystal Optical Generator Excited by a Ruby Laser Konyukhov, Marchenko andProkhorovQ-Switching of a Resonator in Ruby Laser with the Aid of Clearing Substances of the Phthalo-cyanine Series Gryaznov, Lebedev and ChastovShadow Projections of a Spark in Air Occurring When Laser Radiation Is Brought into FocusMalyshev, Ostrovskaya, and ChelidzeInvestigation of the Output Power of a Ne-He Laser as a Function of Various ParametersBasov, Markin, and NikitinTransmission and Reflection of a Plane-Parallel Layer in the Conditions of Amplificationand Generation Stepanov and KhapalyukDetection of the Isotope K40 by Means of Optical Pumping Alexandrov and ChodovoySome Problems of the Nonlinear Theory of the Optical Properties of Plane-Parallel LayersSamson and StepanovCalculation of the Maser Power of a System of Particles with Three Energy Levels Stepanovand SamsonNonstationary Emission of a Three-Level Laser SamsonOn the Emission Spectrum of a Laser on the Base CaF2:Sm2 + Ananyev, Mak, and SedovVariation in the Temperature of Surroundings under the Action of the Laser Pulse V. N.RudenkoQ-Switching of Lasers by KS-19 Filters Lisita, Kulish, Geets, and KovalInvestigations of the Dependence of the Discharge Parameters and Generation Power ofHe-Ne Laser on the Hollow Cathode Diameter Znamenskii, Buinov, and Bursakov

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1534 APPLIED OPTICS / Vol. 6, No. 9 / September 1967