the present invention multiplexes holograms using a reference beam consisting of a spectrum of plane waves ( similar to phase code multiplexing , for example ). multiplexing is achieved by shifting ( translating ) the recording medium with respect to the signal and reference beams . alternatively the two beams can be translated in tandem with respect to the stationary medium . fig1 illustrates a holographic memory system in accordance with a first embodiment of the invention . a laser 10 furnishes a laser beam 12 to a diffractive element 14 which produces from the laser beam 12 an array of m plane waves . the m plane waves are then fourier transformed by a spherical lens 15 into m beam sources 16 separated by uniform distances d . the light from the array of m beam sources 16 is transformed into a one - dimensional fan of m plane waves 18 by a spherical lens 20 . the fan of plane waves 18 constitutes the reference beam for holographic recording and reproduction . the diffractive element 14 could be a dammann grating of the type disclosed in dammann et al ., &# 34 ; high efficiency in - line multiple imaging by means of multiple phase holograms ,&# 34 ; optics communications , vol . 3 , no . 5 pages 312 - 315 ( july 1971 ). however , in one implementation , the diffractive element 14 was a hologram with similar properties . the spherical lenses 15 and 20 have focal lengths f r and image the reference beam emanating from the diffractive element 14 onto a recording spot in an optical recording medium 22 . as indicated in fig1 the distances between ( a ) the diffractive element 14 and the center of the spherical lens 15 , ( b ) the center of the spherical lens 15 and the plane of beam sources 16 , ( c ) the plane of beam sources 16 and the center of the spherical lens 20 , and ( d ) the center of the spherical lens 20 and the plane of the recording medium 22 are each preferably f r . a beam splitter 24 splits the laser beam 12 between the diffractive element 14 and a mirror 26 . that portion of the laser beam 12 diverted to the mirror 26 is modulated by a spatial light modulator 28 to provide a signal beam 29 for recording a hologram in the optical recording medium 22 . during recording of a hologram in the medium 22 , recording electronics 27 controls the spatial light modulator 28 in accordance with an information signal representing either an image or an array of data . the signal beam 29 is imaged by spherical lenses 30 and 31 , each of focal length f , onto the same recording spot in the medium 22 that is illuminated by the reference beam 18 . the distances between ( a ) the spatial light modulator 28 and the center of the spherical lens 30 , ( b ) the center of the spherical lens 30 and a half - way point p with the center of the next spherical lens 31 , and ( c ) the center of the spherical lens 31 and the recording spot in the medium 22 are each preferably a uniform displacement f . for reconstruction of a hologram previously recorded in the recording medium 22 , the reference beam 18 is applied as illustrated in fig1 while the signal beam 29 is blocked . the diffracted beam 35 emanating from the recording medium 22 is imaged through spherical lenses 32 and 33 , each of focal length f , onto the plane of a detector 34 . the distances between ( a ) the recording spot in the medium 22 and the center of the lens 32 , ( b ) the center of the lens 32 and a midpoint p &# 39 ; with the center of the next lens 33 , ( c ) the midpoint p &# 39 ; and the center of the lens 33 and ( d ) the center of the next lens 33 and the plane of the detector 34 are each preferably the uniform displacement f . the detector 34 may be an array of image pixels such as a charge coupled device image detector . a servo 36 under control of the recording electronics 27 governs the amount of shift between the medium and the optical beams 18 , 29 with the recording of successive holograms . the description of the preceding two paragraphs with reference to fig1 is pertinent to image plane holography using a 4 - f imaging system . alternatively , imaging can be performed using a single lens of focal length f located a distance d i from the slm 28 and a distance d 0 from the recording medium 22 , the distances d i and d 0 satisfying the lens law with respect to the lens focal length f : d 0 - 1 + d i - 1 = f - 1 . the same method of using a single lens can be used for imaging the reconstruction of holograms onto the ccd sensor 34 . alternative holographic methods include fourier holography and fresnel holography . to implement fourier plane holography , we remove the lenses 30 and 32 and align the slm 28 on the mid - point denoted by p in fig1 ( which is a distance f from the lens 31 ) for recording ; and we align the ccd sensor 34 on the plane p &# 39 ; ( which is a distance f from the lens 33 ) for reconstruction . in that case , the fourier transform of the signal beam ( as modulated by the slm 28 ) appears through the lens 31 on the plane perpendicular to the signal beam axis pp &# 39 ; intersecting the center of the recording material 22 , and is recorded as a hologram . on the other hand , the reconstruction undergoes a second fourier transform ( by the lens 32 ) forming the original image ( inverted ) on the ccd sensor 34 . to implement fresnel plane holography , we alter the fourier plane geometry as follows : in fig1 the lenses 30 and 32 are not present , the slm is located at plane p and the ccd sensor 34 is located at plane p &# 39 ;, the recording medium 22 is located a distance d 1 from the lens 31 and a distance d 2 from the lens 33 such that d 1 + d 2 = 2f . ( the special case of d 1 = d 2 = f reduces to fourier plane holography .) the operation in the fresnel holography case is similar to that of the fourier holography case , except that a defocused version of the fourier transform is recorded as a hologram . in the remaining portion of the description of fig1 it will be assumed that fourier transform holograms are being recorded or reconstructed , so that the lenses 30 and 32 depicted in fig1 are not present , the slm 28 is located on the plane p , and the ccd sensor 34 is located on the plane p &# 39 ;. the reference beam 18 originates from the array 16 of m point sources located in the front focal plane of the fourier lens 20 , and centered around the optical axis z . the lens 20 transforms the field into the fan 18 of m plane waves . the angular separation is uniform , given by δθ ≈ d / f r where d is the distance between successive point sources and f r is the focal length . thus the angle of incidence of the m - th component is : ## equ1 ## the signal beam 29 will have more than one angular component because of the modulation imposed by the spatial light modulator 28 . the angle of incidence of the central component of the signal beam with respect to the z - axis is denoted by θ s . because the reference consists of m plane waves , the recorded image may be thought of as consisting of m separate holograms recorded simultaneously with the same signal beam . during reconstruction , each plane wave in the reference fan 18 reads out not only the hologram it recorded , but also all the holograms recorded by the other plane waves of the reference fan . these additional reconstructions , or &# 34 ; ghosts &# 34 ;, produce images that are shifted with respect to the primary reconstruction , due to the change in read - out angle relative to the recording angle . the ghosts are bragg mismatched by an amount roughly proportional to the angular separation between the plane wave component that originally recorded the hologram and the component that is reconstructing it . for the hologram recorded between the central signal component and the m = 0 - th reference component the amount of bragg mismatch is δk z = 2π1tan θ s δθ / λ when read out by the ± 1 - th reference component . the same relation holds approximately for the other holograms . the diffraction efficiency of these bragg mismatched holograms is proportional to ## equ2 ## where sinc ( χ )= sinπχ /( πχ ) and l is the thickness of the recording medium . it follows that by choosing the angular separation δθ between the reference components such that the sinc function of eq . 2 vanishes , the ghosts will be eliminated , leaving a clean reconstruction . from eq . 2 the required separation is : ## equ3 ## this equation constrains the design of the diffractive element 14 in that , for a given focal length f r of the lens 20 , the separation d between adjacent point sources 16 must be ## equ4 ## in the paraxial approximation . having eliminated the ghosts , this specification now examines what happens to the diffracted light if the hologram is shifted by a distance γ in the χ - direction . the diffracted field ε d is obtained by multiplying the illuminating reference ( consisting of m plane waves ) by the expression for the m recorded holograms shifted by γ . for a hologram recorded with a single plane wave signal beam of incidence angle θ s , the diffracted field at the reconstruction plane is : ## equ5 ## the three dimensional nature of the hologram ( i . e . the z dependence and the constraint imposed on the angular separation between plane waves within the fan 18 by equation 4 ) serves to eliminate the cross - terms m ≠ m &# 39 ; ( ghosts ) from the double summation . from eq . 6 , the diffracted field consists of the reconstruction of the signal at angle θ s weighted by a sum leading to the familiar array function . therefore the intensity of the diffracted field as a function of shift is : ## equ6 ## multiplexing is performed by recording each successive hologram ( corresponding to a successive image on the light modulator 28 ) with a spatial shift γ = λ / mδθ with respect to its two neighbor holograms . because of the periodicity of the array function , at maximum m holograms can be superimposed on the same location for multiplexing along the χ direction in the drawing of fig1 . the period is : ## equ8 ## the shift multiplexing method is particularly well - suited for the implementation of holographic 3 - d disks . a 3 - d disk can be readily implemented with this method by simply using the disk rotation ( which is already part of the system intended to allow accessing of information on different locations on the disk surface ) in order to implement the shift . this simplifies the design of the head since no additional components are required for selective readout . fig2 illustrates how the embodiment of fig1 is implemented in accordance with the foregoing using a disk medium rotated by the servo 36 of fig1 under control of the recording electronics 27 of fig1 . in the embodiment of fig2 it may be possible to eliminate the lenses 15 , 20 , depending upon the selected characteristics of the diffractive element 14 . fig3 illustrates the placement of successive holograms within the recording medium 22 . referring to fig3 the medium 22 is an optical disk divided into plural longitudinal circular tracks 40 labelled n , n + 1 , and so forth . in the embodiment of fig3 adjacent tracks do not overlap but may be arbitrarily close together . as shown in track n , a series of m ( in this case m = 5 ) holograms are recorded , the disk being rotated between subsequent recordings by an angle ## equ9 ## where r n is the radius of track n . this corresponds to a spatial shift of ## equ10 ## with the recording of each hologram . then , after m holograms have been recorded within the period length ## equ11 ## recording is temporarily halted while the disk is rotated until the beam spot ( the intersection of the reference and signal beams in the medium 22 ) is beyond the boundary of the first - recorded hologram in the series of m holograms previously recorded in the period length γ m . then , the entire process is repeated , with m new holograms being recorded ( indicated in dashed line in fig3 ). the &# 34 ; skip &# 34 ; length by which the disk is rotated while recording is temporarily halted is the length l h of one hologram ( as measured in the plane of the surface of the disk ) minus m × the shift displacement between successive holograms γ , or l h - m . sub . γ . in this way , no spot along the entire length of the track has more than m holograms multiplexed along the x - direction . preferably , the foregoing recording process is implemented by the shift servo 36 under the control of the recording electronics 27 . specifically , the recording electronics 27 , which may comprise , for example , a programmed microprocessor , causes the servo 36 to rotate the disk media 22 to a first position , at which time the recording electronics 27 causes the spatial light modulator to transmit a first image on the laser beam 29 . the image is then recorded on the media 22 . thereafter , the servo 36 rotates the disk media 22 by the shift displacement γ . thereafter , the recording electronics causes the spatial light modulator to transmit the next image on the laser beam 29 . the foregoing process is repeated until m holograms have been recorded in this manner , at which time the recording electronics 27 causes the servo 36 to rotate the disk by the skip length l h - m . sub . γ before recording the next series of holograms . once an entire track ( for example track n ) has been filled , the recording electronics 27 causes servoes 45 controlling optical elements in the paths of the reference and signal beams ( such as the reference lens 20 and the signal lens 30 ) to move so as to radially move the beam spot at which the signal and reference beams intersect inside the disk media 20 by the width of one track ( i . e ., by the width of one beam spot ). the size of each hologram along one side ( as measured in a plane perpendicular to the signal beam ) may be expressed as the number of pixels n p along each side of the hologram multiplied by the pixel size δ , or n p δ . however , as measured in a plane parallel to the surface of the disk media 22 , the size of one hologram is ltanθ s + n p δ / cosθ s . the first term ltan θ s is attributable to the angle θ s between the reference beam and the z - axis . the factor cosθ s is attributable to fact that the signal beam is slanted with respect to the plane of the disk media 22 . preferably , the projection of the signal beam 29 onto the disk is a straight line which is parallel to the χ - direction or direction of motion of the disk near the hologram being recorded . this optimizes selectivity along the χ - direction . while the embodiment of fig3 has no overlapping of adjacent tracks ( i . e ., multiplexing of holograms along the radial or y - direction ), overlapping of adjacent tracks may be achieved as illustrated in fig4 by selecting a diffractive element 14 which provides a two - dimensional array of beam sources 16 rather than the line of beam sources disclosed hereinabove . the limitation on the spacing γ y between adjacent holograms along the radial or y - direction is governed by the same principles as those discussed above with respect to the spacing γ along the x - direction , with the exception that required angular separation of the fan of reference beams along the radial or y - direction is : ## equ12 ## the output face of the dammann grating or diffractive element 14 as fourier transformed by the lens 15 is illustrated in fig5 in which the spacing between point sources in the along - track or x - direction is ## equ13 ## as stated above , while the spacing in the radial or y - direction is ## equ14 ## in the two - dimensional array diffractive element 14 of fig5 the number of point sources m in the along - track or x - direction may be greater than the number of point sources m y in the radial or y - direction . thus , the incremental shift spacing γ between successive holograms along the x - direction is as given above , namely γ = λ / mδθ , while along the y - direction the incremental shift spacing γ y between successive holograms is γ y = λ / m y δθ y , where δθ y ≈ d y / f r . the storage density d per unit area that the invention can achieve in the embodiment of fig1 is determined by the thickness - dependent angular selectivity ( eq . 10 ), the number of beams m allowed by the optics , the page size n p δ ( δ is the pixel size and n p the number of pixels ) and the periodicity of the array function . an approximate formula for the density is : ## equ15 ## when shift - multiplexing in one dimensional only . for l = 100 μm and signal incidence angle θ s = 30 °, usage of f / 1 optics allows m = 100 holograms . then , for typical page parameters n p = 1000 , δ = 2 μm , eq . 12 yields d = 21 . 1 bits / μm 2 . in the embodiment of fig6 shift multiplexing is implemented using a spherical wave reference instead of the fan of m plane waves . the dammann grating or diffractive element 14 is removed so that the spherical lens 20 provides a single spherical wave as the reference beam 18 . the spherical wave reference beam of this embodiment is a special case of the multiple plane wave embodiment of fig1 in which there are at least nearly an infinite number of plane waves distributed in a two - dimensional plane and separated by infinitely small separation angles . the analysis of the embodiment of fig6 is as follows : consider a spherical reference wave originating a distance z 0 from the center of the recording material and a plane wave signal incident at angle θ s with respect to the optical axis . the displacement z 0 is preferably selected so that the reference beam spot size matches the signal beam spot size at their intersection within the media 22 . an approximate calculation ( under the paraxial and born approximations and neglecting variable modulation depth ) predicts that the shift selectivity is related to the focal distance and the bragg angular selectivity ( eq . 10 ) as γ bragg ≈ z 0 δθ = λz 0 / ltan θ s . the finite numerical aperture ( na ) broadens the selectivity curve by a factor of γ na ≈ λ / 2 ( na ). therefore we have : ## equ16 ## one advantage of the single spherical wave embodiment of fig6 is that the spherical wave reference beam corresponds to a two - dimensional fan of an infinite number of plane waves each of infinitesimal power with an infinitesimally small separation angle between waves , so that there is no periodicity ( i . e ., γ m is infinite ) and two - dimensional multiplexing ( as in fig5 ) may be performed if desired because of the two - dimensional nature of the spherical wave reference beam . fig7 a illustrates an arrangement in which the signal and reference beams in the embodiment of fig1 illuminates opposite sides of the recording medium 22 and are at least nearly coaxial . the beam splitter 24 separates the reference beam from the reconstructed beam during hologram reconstruction . the lenses 20 and 37 image the diffractive element 14 onto the disk media 22 , thus providing the reference beam . the lenses 30 and 36 image the spatial light modulator 28 onto the disk media 22 , thus providing the signal beam . during hologram reconstruction , light to the spatial light modulator 28 is blocked while the lenses 20 , 32 image the reconstruction onto the detector 34 . the embodiment of fig7 a is useful for thin optical media such as polymer films on the disk 22 . fig7 b illustrates a variation of the geometry of the embodiment of fig7 a for using a spherical wave , in which the reference beam is perpendicular to the plane of the disk media 22 while the signal beam and reconstructed beams make an acute angle with the plane of the disk media 22 . the embodiment of fig7 b is suitable for thick recording media such as a lithium niobate crystal . fig8 illustrates an arrangement in which the signal and reference beams of either the embodiments of fig1 or 6 are mutually perpendicular . the embodiment of fig8 is useful for thick optical media such as a lino3 crystal . this embodiment achieves the theoretically smallest required shift distance . experimentally , we have demonstrated shift selectivity of 2 μm in the x direction and 10 μm in the y direction using an 8 mm thick lithium niobate crystal in implementing the embodiment of fig8 . the embodiment of fig8 may be modified so that the reference beam is a single spherical wave ( as in the embodiment of fig6 ) by eliminating the diffusion element 14 and the lenses 15 and 20 . a working example of the embodiment of fig1 - 3 was demonstrated using a reference fan of 20 plane waves angularly separated by 0 . 5 °. the diffractive element 14 was a hologram rather than a dammann grating . the recording material was dupont hrf - 150 polymer of thickness l = 38 μm . the effect of shift on the reconstruction of a single hologram is shown in fig9 . the signal image was a 100 × 100 random bit pattern . for the particular parameters the theoretical shift selectivity is 2 . 8 μm and the period is 55 μm , in good agreement with the experiment . the reason the periodic reconstructios are weaker is the finite transverse size of the hologram of the diffractive element 14 . fig9 is useful for illustrating the recording method of the invention . the peak width γ is inversely proportional to the number m of the plane waves in the fan . the spacing between peaks γ m limits to m the number of other holograms ( indicated in dashed line in fig9 ) that can be stored between peaks of the first hologram . the response at the reconstruction plane of the detector 34 for three holograms obtained with the embodiment of fig1 - 3 are shown in fig1 . each hologram is reconstructed periodically , following its own array factor . because of the very small thickness of the recording medium in this experiment , we used angular separation smaller than that predicted by eq . 10 . therefore the ghosts had to be filtered out in the fourier plane in this example . for the embodiment of fig6 the graph of fig1 shows the experimental shift selectivity curves for z 0 = 9 , 14 , 24 mm and na of 0 . 3 , 0 . 15 , and 0 . 075 , respectively . the angle of incidence of the signal beam was 40 ° outside the 8 mm thick iron doped linbo 3 crystal ( refractive index n ≈ 2 . 24 ). the experimental selectivity agrees with eq . 13 . while the invention has been described with reference to embodiments in which the reference lens 20 is spherical ( so as to provide either a fan of plane waves as in fig1 or a single spherical wave as in fig6 ), the shift multiplexing of the invention described above may also be achieved using different types of reference beams other than a single plane wave . for example , cylindrical waves may be employed ( in which case the reference lens 20 is a cylindrical lens ) or elliptical waves may be employed ( in which case the reference lens 20 is an elliptical lens ). in general , the invention may employ any one of a large variety of non - planar reference waves to attain shift - selective properties in holographic reconstruction . multiple plane waves and a spherical wave are but two examples of non - planar wavefronts , but the invention is not limited thereto . clearly , any generalization of non - planar wavefronts may be employed in carrying out the invention . while the invention has been described in detail by specific reference to preferred embodiments , it is understood that variations and modifications thereof may be made without departing from the true spirit and scope of the invention .