Partitioned-Field Uniaxial Holographic Lenses

Partitioned-field uniaxial holographic lenses

Ana M. Lopez, Jesus Atencia, Jose Tornos, and Manuel Quintanilla

The efficiency and aberration of partitioned-field uniaxial volume holographic compound lenses aretheoretically and experimentally studied. These systems increase the image fields of holographic vol-ume lenses, limited by the angular selectivity that is typical of these elements. At the same time,working with uniaxial systems has led to a decrease in aberration because two recording points �thatbehave as aberration-free points� are used. The extension of the image field is experimentally proved.© 2002 Optical Society of America

OCIS codes: 090.0090, 090.2890, 090.7330, 080.3620, 110.0110.

1. Introduction

A hologram recorded with the interference of beamsgenerated by two point sources behaves as a lens thatoptically conjugates these two points. To obtainhigh-efficiency lenses it is necessary to record volumeholograms with grating periods that are smaller thanthe grating thickness. So high-efficiency holo-graphic lenses are biaxial elements: The input op-tical axis and the output axis do not have the samedirection. This kind of lens has poor aberration per-formance,1 so the quality of the image that is formedis limited. Quintanilla and Arias reported2 a uniax-ial system composed of two single high-efficiency bi-axial elements. With this system, high efficiencyand good aberration performance are compatible.

One of the attributes of volume holograms is angu-lar selectivity. The efficiency of the diffracted orderis maximum when, under illumination with the samewavelength used at recording, the reconstructionwave vector is the same as any of the constructionwave vectors. When the incident wave separatesfrom the construction beams, the diffraction effi-ciency decreases quickly if the illumination wave vec-tor is parallel to the plane defined by the constructionwave vectors. We call this plane the construction-beams plane. This fact limits the directional sensi-

A. M. Lopez is with the Departamento de Ingenierıa Electronicay Comunicaciones, Universidad de Zaragoza, Escuela Politecnicade Teruel, Ciudad Escolar s�n, 44071 Teruel, Spain. J. Atencia�[email protected]�, J. Tornos, and M. Quintanilla are withthe Departamento de Fısica Aplicada, Facultad de Ciencias, Uni-versidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain.

Received 13 July 2001; revised manuscript received 1 October2001.

0003-6935�02�101872-10$15.00�0© 2002 Optical Society of America

1872 APPLIED OPTICS � Vol. 41, No. 10 � 1 April 2002

tivity of a volume holographic lens, which results inthe acceptance of only a portion of the object field andproduces an image with appreciable intensity.3When the projection of the incident direction on theconstruction-beams plane is close to that of the con-struction beams, the diffraction efficiency is hardlymodified. So the intensity does not change appre-ciably when the object coordinate varies orthogonallyto the construction-beams plane, but it decreasesquickly with the coordinate inside this plane. Theevolution of efficiency as a function of this secondcoordinate follows an approximately sinc2 function.4For Agfa 8E75 plates �emulsion thickness, �6.5 �m�and an angle of 30° between the interfering beams,the acceptance angle is of the order of 10°.

As the recording beams define the construction-beams plane, lenses registered by use of differentwaves form images of distinct portions of the sameobject plane. In previous publications this charac-teristic was used to increase the image field of uni-axial compound systems that comprise elements thatwork satisfactoraly for different object zones. Inthis way, systems with two gratings recorded withthe same diverging object wave and different refer-ence plane waves have been designed, built, andcharacterized. These gratings can be recorded ondifferent sections of the system pupil5 or superposedupon a full lens area.6 With these solutions we ex-tend the efficient image field to a cross-shaped areacentered on the image plane’s coordinate axis.

Another type of field extension has been pro-posed7,8 that is produced by overlapping of simplebiaxial lenses, each covering a different parallel por-tion of the object field, onto a single holographicrecord. The field covered by each component can bemade to blend smoothly into the adjacent field cov-ered by another component. It has been proposed

that these systems be used to reduce the aberrationeffect in biaxial holographic systems. In effect, ho-lographic lenses are aberration free when the objectpoint coincides with one of construction points, butaberrations grow quickly when the reconstructionbeams do not match the construction beams. Whenthe off-axis aberrations for one holographic lens com-ponent become excessive, the efficiency of that com-ponent is decreased if the ray directions are otherthan at the Bragg angle, and another componenttakes over, so image points with high degrees of ab-erration can be avoided. By multiplexing variouslenses it is possible to image different zones of theobject plane to minimize aberrations. We call theseelements “partitioned-field holographic lenses,” fol-lowing the designation by Leith and Upatnieks.7

The solution proposed in the paper of Leith andUpatnieks is not feasible because they overlappedholographic lenses with different superficial gratingstructures, such that each lens formed the image ofthe same point on the object plane onto a differentplace at image plane. This procedure results in amultiple-image effect.

Quintanilla and de Frutos8 succeeded in reducingaberrations for points far from the center of the objectplane. However, the image obtained was not goodbecause of the presence of multiple images that couldnot be made to coincide. They recorded three biaxiallenses on the same hologram, so every object pointhad three images that had to match. As the threelenses act on the same position, they must deflectrays that originated in the same point by an equalangle. The surface gratings of the three structuresmust be the same. For this to happen, three couplesof recording points that are related to the same in-terference structure on the plane of the plate must bechosen. A careful interferometric method was usedto prevent multiple images, but this method is highlysensitive to plate movement, which is highly likely tooccur between expositions.

Our aim in the present research is to study apartitioned-field holographic system composed of twooverlapping uniaxial holographic lenses, each of themformed by two biaxial holographic elements. Westate the conditions for correct matching of the im-ages given by the different multiplexed lenses. Wemake a study of the efficiency of the final system andprove that the system produces an extension of theimaged field. We study system aberrations anddemonstrate the ability of the proposed system tocorrect aberrations. We construct one of these sys-tems and experimentally prove the performance offield extension and aberration.

2. Geometrical Analysis of a Partitioned-FieldHolographic Lens

The simplest example of a plane holographic elementis one constructed by the interference of two sphericaland coherent waves that come from real or virtualpoint sources C1 and C2, as shown in Fig. 1. Ingeneral, sources C1 and C2 are not aligned with cen-ter A of the hologram, so a biaxial holographic lens

with optical reference axis C1A, AC2 is obtained.Usually object and image planes are chosen parallelto the hologram plane �as shown in Fig. 1� or perpen-dicular to the optical axis. Gaussian �paraxial� dis-tances are measured on a defined reference axiswhose origin is at A.

If the angle between the two optical axes is largeenough, a volume element is recorded. As we ex-plained above, angular selectivity limits the area ofthe object that corresponds to an image with appre-ciable intensity. The portions of the object planethat are efficiently reconstructed are different forlenses recorded with different recording beams. In apartitioned-field lens two or more different lenses arerecorded on the same plate. The construction geom-etry is chosen to include different zones of the objectplane. Several images of the same object but differ-ent intensity distributions are formed. To producegood global performance these images must coincideexactly. In biaxial systems, as reported by Quin-tanilla and de Frutos,8 this goal suffers from greatcomplexity.

However, working with uniaxial systems in aparaxial approximation produces a match of all im-ages formed. But a simple uniaxial lens does notfulfill the conditions of a volume hologram at thelens’s center, so poor efficiency is obtained. The so-lution is the use of a system of two biaxial holographiclenses that, when they are joined, perform as a uni-axial compound system,5 as shown in Fig. 2. Quin-tanilla and Arias2 have proved that the compoundsystem is totally equivalent, except for efficiency, tothe simple uniaxial system. Figure 3�a� shows theimage formation of a compound uniaxial lens whoseimage field is limited by angular selectivity. Figure3�b� shows a partitioned-field made by the overlap-ping of three uniaxial compound lenses. The raythat goes through the centers of all the lenses is notdeflected; the frequency of the surface grating is thesame for all the compound systems constructed.The frequency value is zero because the systems areuniaxial. The partitioned-field compound systemoperates as a set of uniaxial lenses that extend theimaged field to a greater rectangular area and giveoptimum aberration performance. The intensity

Fig. 1. Construction and reconstruction parameters for a biaxialholographic optical element.

1 April 2002 � Vol. 41, No. 10 � APPLIED OPTICS 1873

distribution at the image plane is schematized in Fig.4 for the � coordinate.

A plane holographic lens works as a thin lens thatgives a Gaussian image at a distance ri,9 which isgiven by

1ri

�1ro

� �� 1r1

�1r2� , (1)

where � � �� is the relation between constructionand reconstruction wavelength and with relations of

magnification

xi

ri�

xo

ro, (2)

yi cos 2

ri�

yo cos 1

ro� �� 1��sin 1 � sin 2� (3)

for the object and the image planes, respectively, per-pendicular to the optical axis. When the object andthe image planes are parallel to the lens, Eq. �3� for ymagnification is transformed into

yi

ri�

yo

ro� �� 1��sin 1 � sin 2�. (4)

It is important to note that, in general, a biaxialholographic lens operates as an anamorphic lens,with a different magnification in each direction. Forthe purpose of this study of partitioned-field lenses,only object and image planes parallel to the lens areconsidered.

When C1, A, and C2 are aligned, we have in generala uniaxial noncentered system. For this case, 1 �2, and, particularizing Eq. �4�, we have

yi

ri�

yo

ro, (5)

so a uniaxial system is isomorphic; i.e., magnifica-tions are the same in both orthogonal directions.Another interesting property of this system is that anobject ray that reaches the center A �the optical cen-ter� of the hologram is not deflected. For the pur-pose of constructing partitioned-field lenses, this isan interesting characteristic. In fact, if we take uni-axial systems constructed with angles � 1 � 2,and we choose distances of C1 and C2 for every ,given by r1�� and r2��, the conditions

r1�� r1�0��cos , (6)

r2�� r2�0��cos (7)

Fig. 2. Recording and reconstruction geometry for the uniaxialcompound system.

Fig. 3. Image fields that result from �a� a compound uniaxialholographic lens limited by angular selectivity and from �b� thepartitioned-field holographic lens.

Fig. 4. Dependence of light intensity on the � coordinate at theimage plane of a partitioned-field lens.


are verified. Then, if we multiplex various of thesesuniaxial lenses to form a partitioned-field lens, for anobject plane parallel to the lens the paraxial imageplanes given by all component lenses coincide, veri-fying that, when

ro�� ro�0��cos , (8)

we have

ri�� ri�0��cos . (9)

With the relationships given by Eqs. �8� and �9� we getthat magnifications are the same for all lenses con-structed under the conditions given by Eqs. �6� and �7�.This fact demonstrate that when one is working withuniaxial systems, in the paraxial approximation all theimages formed match because the surface-grating fre-quency is zero at the center so the ray that goesthrough the centers of all the lenses is not deflected.

We must also study the third-order wave aberra-tion of these systems to characterize its imaging per-formance. Calculation of the third-order waveaberration of the most general biaxial holographiclens can be made with usual wave-front matchingmethod. We consider a holographic element con-structed with two spherical waves with phases 1and 2 at point H of a hologram relative to its center,A �see Fig. 1�. If we illuminate it with a wave ofphase 0, the hologram acts as a phase mask, ��1 2�, and the diffracted wave that exits from the ho-lographic lens �only the minus term is efficient involume holograms� has a phase

�i � o � �1 � 2�. (10)

In general, we can consider that �i � i � W, wherei is the phase of a spherical wave centered at Gauss-ian image O� and W expresses the deformation of thewave front from an ideal wave front, so

W � �o � �1 � 2�� i. (11)

Taking polar coordinates ��, �� normalized to pupilradius �m on the hologram plane �0 � � � 1, 0 � � �2��, we can write W in the known form

where the superscript c indicates a central coeffi-cient, i.e., calculated with the pupil on the holo-graphic optical element’s plane �followingHopkins’s terminology10�. Central coefficients for

object and image planes parallel to the lens aregiven in Appendix A.2

From the equations in Appendix A we can deductsome interesting properties of uniaxial lenses. It iseasy to see that the terms D�

c and D�c are always zero

for uniaxial lenses �1 � 2 � � but not for biaxiallenses. When r1 � r2, another useful property canbe found: If we consider an object point on axis, withyo � 0, and another object point at a symmetricalposition with respect to the normal of the plate, withyo � 2ro sin �as we can see from Fig. 5, on ameridianal plane�, we find that all aberration termshave the same value �except for the sign�. Whenro � r1, both yo � 0 and yo � 2r1 sin , verifying thatall aberration terms are zero. This property will behighly useful for partitioned-field holographic lensesto minimize the effects of aberration on image per-formance, as we shall demonstrate below.

We propose the construction of a partitioned-fieldlens by multiplexing two biaxial elements in thesame plate and joining them in the same form asshown in Fig. 2. The compound system, shown inFig. 6, operates as a set of uniaxial lenses with apartitioned field that provides an increase of theobject plane area that can be imaged with appre-ciable efficiency, owing to angular selectivity ofvolume holograms. An efficiency study and exper-imental results appear below.

3. Theoretical Calculation of the Diffraction Efficiencyof Partitioned-Field Lenses

The image field of a volume holographic lens isstrongly limited for the object coordinate inside the

Fig. 5. Uniaxial lens at a meridianal plane.

W � 2�

��1⁄8Sc�4 spherical

� 1⁄2C�c�3 sin � � 1⁄2C�

c�3 cos � coma

� 1⁄2A�c�2 sin2 � � 1⁄2A�

c�2 cos2 � � A��c �2 sin � cos � astigmatism

� 1⁄4Fc�2 curvature

� 1⁄2D�c� sin � � 1⁄2D�

c� cos ��, distortion, (12)


construction-beams plane and is practically unlim-ited in the orthogonal direction. So this field showsan approximately rectangular shape, as sketched inFig. 6�a�, for the recording geometry in Fig. 2. Fordifferent recording geometries, different image fieldsare obtained. We can see from Fig. 2 and 6�a� that,with the position of focus of the divergent recordingwave fixed, the propagation direction of the planewave determines the orientation of the area of max-imum efficiency with respect to the axis of the objectplane. When the optical axis of the uniaxial lensforms an angle normal to the plate, if the recordingplane wave propagates in a direction parallel to the�� plane the efficient image field will correspond toobject points close to yo � 0. If we multiplex two of

these lenses, we can obtain the image field extensionshown in Fig. 6�b�. The complete system operates asa uniaxial centered lens, except for aberrations.

To study the partitioned-field lens we take a gen-eral coordinate system ��, �, �� with origin at center Aof a hologram. In Fig. 7 the efficiencies as a functionof coordinate � of two different lenses, recorded with � 4.41° and * � 4.41°, are added in an indepen-dent way. We chose the object recording points toextend the image field to a specific range of � wherethe efficiency is maintained nearly constant. In thisway we forced the efficiency curve to reach an effi-ciency equal to half of the maximum at the samevalue of �. This requirement must be compatiblewith a good aberration performance.

This independent superposition permits us tochoose the recording geometries of the two gratings.But these two structures, which are recorded on thesame material, are not independent, so to evaluatethe actual efficiency requires a three coupled-wavetheory.

Two exposures are made on the same plate. Therecording beams are shown in Fig. 8. After thesetwo exposures and adequate processing, the dielectricconstant of the holographic medium shows a periodicvariation that follows the two volume structures thathave been constructed11:

ε�r� � εo � �ε1 cos�K1�r� � r� � �ε2 cos�K2�r� � r�.(13)

The two gratings have been sequentially recordedand are assumed to be superposed in a linear way.On every hologram point two gratings are present,with grating vectors K1�r� and K2�r� determined bythe propagation vectors of the recording waves:

K1�r� � �1�r� � �1�r�, (14)

K2�r� � �2�r� � �2�r�, (15)

where �1�r� and �2�r� are the propagation vectors ofthe object recording waves, �o1 and �o2, respectively,and �1�r� and �2�r� are the wave vectors of the ref-

Fig. 6. �a� Schematic of the limited image field of the compounduniaxial system of Fig. 2. As the average construction-beamsplane of the system is the �� plane, the intensity is limited for they coordinate of the object. �b� Proposed partitioned-field lens withtwo multiplexed uniaxial elements, which produces an extension ofthe field in the y direction.

Fig. 7. Efficiencies of the image formed by two compound sys-tems, � 4.41° and * � 4.41°. The independent addition ofthese two efficiency curves is also shown.

Fig. 8. Construction beams of the two lenses that are recorded ona single element of the compound system.


erence recording waves, �r1 and �r2. The propaga-tion vectors of the object waves vary in a continuousway with the plate coordinates, whereas the direc-tions of the reference beams are the same for allpoints of the plate. The amplitudes of the dielectricconstant modulation, �ε1 and �ε2, are assumed con-stant, as are those for the two gratings.

The hologram is illuminated with an electromag-netic wave with electric field Einc � Eo exp� j�� t � � r��. The incident propagation vector is deter-mined by object point position ro and the impact pointon the plate, r. As volume holograms are recorded,every grating generates a single diffracted beam. Sothree orders are present at the output: an ordergenerated for every lens and the one that correspondsto the undiffracted light. Their propagation vectors,obtained from the K-closure condition,12 are

�, (16)

�1 � � � K1, (17)

�2 � � � K2. (18)

The electric and the magnetic fields inside the holo-grams have three components:

E�r� � R� z�exp� i� � r� � S1� z�� i�1 � r� � S2� z�

� � i�2 � r�, (19)

H�r� � �εo�� U� z�exp� i� � r� � T1� z�� i�1 � r�

� T2� z�� i�2 � r��, (20)

where R�z� and U�z� are the amplitudes of the elec-tric and the magnetic fields, respectively, of the trans-mitted order with propagation vector ��r�, S1�z� andT1�z� are the amplitudes that correspond to diffractedorder �1�r�, and S2�z� and T2�z� are for �2�r�.

These fields must verify the Maxwell equations

� � E � i��oH, (21)

� � H � i�εoε�r�E (22)

in the medium with a periodic dielectric constantgiven by Eq. �13�. The application of these two vec-torial equations produces six scalar equations inwhich the three components of the electric field andthe three components of the magnetic field of thethree orders inside the material are coupled. Thatgives a total of 6 � 3 � 18 amplitude components.Eliminating the z components, we obtain a system of12 equations with 12 unknowns: R�, R�, U�, U�, S1�,S1�, T1�, T1�, S2�, S2�, T2�, and T2�. Following theRunge–Kutta method, we obtain the value of the fieldamplitudes at the output of the plate. Several ap-proximations are made. Index continuity betweenthe external and the internal media is assumed, soreflected orders are not taken into account.13 To in-troduce reflection losses and the ray deflection that isdue to the discontinuities air–emulsion–air, we cal-culate Fresnel coefficients and apply Snell’s law.This approximation is known as approximate

boundary matching.14 We consider the global be-havior of the lens and extend the efficiency calcu-lation to the whole pupil area. From every objectpoint there emerges light that arrives at every platepoint. We take a discrete set of plate points andconsider that all of them are illuminated with anenergetic flux equal to 1. The diffracted energy foreach of them is calculated. The final flux associ-ated with the object point ��, �, ro sin � is obtainedas the sum of the contributions of this set of pointson the plate. The approximation of a locally homo-geneous grating was used.15 Aberrations have notbeen considered here.

The three orders at the output of the first plateilluminate the second one, which is similar to the firstplate but is rotated 180° about the � axis. Its dielec-tric constant is

ε��r� � εo � �ε1 cos�K1��r� � r�

� �ε2 cos�K2��r� � r�, (23)

where Ki� and Ki are related in the form

Kix��, �� Kix� �, ��, (24)

Kiy��, �� Kiy� �, ��, (25)

Kiz��, �� Kiz� �, ��. (26)

Every incident beam is diffracted by the two gratingsthat are present in the second plate. At the output,only two of these orders, viz., those defined by thepropagation vectors, contribute efficiently to the im-age:

� � K1 � K1�, (27)

� � K2 � K2�. (28)

In Fig. 9 are shown the efficiency curves for inci-dence variations inside the selectivity plane, whichcorrespond to the three orders at the output of thefirst plate. The same index modulation has beenassumed for the two lenses. This modulation cor-responds to 100% efficiency for a single lens �whose

Fig. 9. Diffraction efficiencies of the beams generated by the firstplate where two gratings have been recorded. The curves areplotted as a function of the � coordinate of the object plane, with� � 0. The distance of incidence is equal to the focal distance ofthe system.


material thickness is taken equal to 6.5 �m�. InFig. 10�a� the total image efficiency as a function ofobject coordinates is represented. The object planeis 130 mm to the left of the plates. So the systemworks with unity magnification. In Fig. 10�b� thisefficiency is calculated for operation with only grat-ing 1. If we compare the two curves it is easy toverify that the efficient image field has been ex-tended. The maximum efficiency is reached forpoints close to the position of the recording sources�lines � � 10 mm and � � 10 mm�. If the abso-lute value of � is greater than 10, the efficiencydecreases and it disappears for � � �40 mm. Theefficiency also decreases when � tends to zero, butfor this value there is a nonzero minimum. In thiszone both lenses are efficient, and the diffractionproblem is a problem of three coupled waves. Ascan also be seen from Fig. 9, the two diffractedorders exchange energy with each other and withthe transmitted order that transports the energy,which does not appear on the image.

4. Experimental Results

We constructed a system with beams generated froma He–Ne laser, using Agfa-Gevaert holographicplates 8E75 processed by the method of silver halide–sensitized gelatin.16 We recorded all the plates un-der index-matched conditions �with a black plate atthe back� to prevent reflection between the glass oremulsion and air and with a beam ratio of approxi-mately unity. We chose the silver halide–sensitizedgelatin process to create elements with low-levelnoise. However, because of the frequency limitationof this processing method,17 we obtained low-efficiency holograms ��23% diffraction efficiency�.As in other similar systems, precise alignment of thetwo plates was necessary for exact focusing of thedifferent diffracted orders at the same position toavoid double-image effects.

Figure 11 illustrates the setup for real-image for-mation of a diffusing object �square distribution of5-mm period� for this holographic system. The realimage was projected onto a diffusing screen and cap-tured by a CCD camera. Fresnel lenses were usedas a condenser lens to improve illumination and as a

Fig. 10. �a� Theoretical efficiency of the image produced by thecompound system upon which the gratings defined in Fig. 8 havebeen superposed, as a function of the object plane coordinates.The system acts with unity magnification. �b� The same calcula-tion for action of grating 1 only.

Fig. 11. Experimental setup for studying real-image formation bya holographic lens.

Fig. 12. Image of a periodic diffusing object formed by the com-pound system working with unity magnification.


field lens to capture the image. To minimize aber-rations we placed the object plane 130 mm away fromthe system �so this plane included the point sources ofthe recording� and a 632.8-nm filter was used. Withthis setup the system operates with unity magnifica-tion.

Extended but low-efficiency images were obtained.In Fig. 12 we show an image of the diffusing periodicobject.5,6 The way in which the estimations of Fig.10�a� were obtained can be clearly seen. The inten-sity is limited mainly to one direction. Two maximaare present, separated by a zone of lower intensity.

To validate our theoretical estimation in a quanti-tative way we measured the diffraction efficiency ofthe image when the system was illuminated with adivergent wave whose origin was twice the distanceof the focal length of the compound lens.6 The effi-ciency varied as a function of angle �, which formedthe � axis with the line that goes from the illuminat-ing point source to the center of the pupil. We cal-culated this efficiency after having obtained the

values of the parameters, i.e., thickness, amplitudeindex modulation, and absorption coefficient, of therecorded gratings. Theoretical and experimental re-sults are compared in Fig. 13. We can see a goodagreement between them.

To appreciate the image quality of the system wecalculated the spot diagrams obtained by ray tracingthrough the complete system.18 Figure 14 showsspot diagrams at the paraxial image plane of one ofthe simple uniaxial lenses, with � 4.41°, for apupil of A � 14 mm and unity magnification. Wecan see that, when the object point coincides with oneof the construction points, the image at the paraxialimage plane is a perfect point �that corresponds to � �0, � � 10�. Beyond this point the diagrams aretypically astigmatic, as expected, because sphericalaberration and coma cancel out in a symmetrical sys-tem operating with unity magnification. For � � 0and � � 10 we can see that aberration cancels, aspredicted from third-order aberration terms in Sec-tion 2. Although at this point the efficiency of thisuniaxial lens is almost zero, this behavior of aberra-tions leads to a better match between the imagesproduced by the two multiplexed uniaxial lenses thatconstitute the partitioned-field system.

Figure 15 shows the effect of a focal shift on a spotdiagram. We can see that a smaller spot diagram isobtained when � � 0 if we move the image planeaway from the paraxial plane, but, when � � 10 mmor more, the spot diagram grows. With a focal shiftof 0.5 mm we get a good match at the center and theborder of the field. With this defocus the system isable to resolve more than 10 lines�mm at both thecenter and the edge of field.

We evaluated the quality of the image produced bythe system by using the high-resolution 1951 U.S. AirForce test pattern.5,6 From Fig. 16 one can observethe test image when it is placed at the center of theobject plane. To obtain this image, one places theCCD array �without a lens object� directly upon the

Fig. 13. Ratio of energy of the image and total incident energeticflux. Comparison of theoretical and experimental results.

Fig. 14. Spot diagrams at the paraxial image plane for unitymagnification obtained from one of the uniaxial lenses.

Fig. 15. Spot diagram for unity magnification at the paraxialimage plane and at image planes with different focal shifts.


image plane. Rectangular pixel dimensions are 11.5and 27 �m; hence the maximum resolution providedby the CCD is 43 lines�mm horizontal and 18lines�mm vertical. It is possible to see that the sys-tem is even able to resolve as many as 16 lines�mm.This resolution is of the same order as that predictedby the spot diagram.

To complete our study it is interesting to check theimage quality of a lens that does not operate withunity magnification. For comparison with the pre-viously studied system we chose a lens with the samefocal length and aperture, so we got r1 � �, r2 � 65mm, � 4.41°, and a pupil of A � 7 mm. Figure17 shows the spot diagrams at the paraxial imageplane for this uniaxial lens when we placed the objectplane at infinity. We can see again that, when theobject wave coincides with the construction wave, theimage at the paraxial image plane is a perfect point

�which corresponds to � � 0, � � 5�. For � � 0, and� � 5 �which corresponds to the axis of the otheruniaxial lens in the partitioned-field lens� we can seethat aberration does not cancel but has a minimum.This fact means that the proposed partitioned-fieldlens can also be a good choice for magnification thatis different from unity.

5. Conclusions

Uniaxially centered compound holographic systemsprovide efficient images with good aberration perfor-mance. However, the angular selectivity of volumeholograms drastically limits the area on the objectplane that has an efficient image. We have designedand constructed a compound system that overlaps inthe same element two uniaxial noncentered lenses,each of which is efficient for a distinct zone of theobject field. We have theoretically and experimen-tally proved that the overlap of the two systems in-creases the image field of this kind of element. As aresult of the recorded double structure and the fre-quency limitations of silver halide–sensitized gelatinprocessing, we cannot obtain high-efficiency ele-ments, but we are working with a different process-ing method that we hope will lead us to obtain highlyefficient optical systems. Another important advan-tage of the system studied is its better aberrationperformance, as our ray traces prove.

Following the method described here, one can ob-tain a greater extended field by superposing three ormore noncentered uniaxial systems, as described.

Appendix A: Central Coefficients for a BiaxialHolographic Lens with Object and Image PlanesParallel to the Lens

Sc � �m4� 1

ro3 �

1ri

3 � �� 1r1

3 �1

r23�� ,

C�c � �m

3� xo

ro3 �

xi

ri3� ,

C�c � �m

3�� yo � ro sin 1�

ro3 �

� yi � ri sin 2�

ri3

� ��sin 1

r12 �

sin 2

r22 �� ,

A�c � �m

2�xo2

ro3 �

xi2

ri3� ,

A�c � �m

2�� yo � ro sin 1�2

ro3 �

� yi � ri sin 2�2

ri3

� ��sin2 1

r1�

sin2 2

r2�� ,

A�c � �m

2�xo� yo � ro sin 1�

ro3 �

xi� yi � ri sin 2�

ri3 � ,

Fc � �m2�xo

2 � yo2 � 2yo ro sin 1

ro3

�xi

2 � yi2 � 2yiri sin 2

ri3 � ,

Fig. 16. Image of the 1951 U.S. Air Force resolution test, placedat the center of the object plane. The compound system workswith unity magnification.

Fig. 17. Spot diagrams at the paraxial image plane for an objectat infinity obtained from one of the uniaxial lenses.


D�c � �m�xo� xo

2 � yo2 � 2yo ro sin 1�

ro3

�xi� xi

2 � yi2 � 2yiri sin 2�

ri3 � ,

D�c � �m�� yo � ro sin 1�� xo

2 � yo2 � 2yo ro sin 1�

ro3

�� yi � ri sin 2�� xi

2 � yi2 � 2yiri sin 2�

ri3 � .

This research was supported by the DiputacionGeneral de Aragon �Spain� under grant P095�2000.

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Partitioned-Field Uniaxial Holographic Lenses