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Effect of lens aberrations on the storage capacity of holographic memories
Volume 20, number 1
OPTICS COMMUNICATIONS
January 1977
EFFECT OF LENS ABERRATIONS ON THE STORAGE CAPACITY OF HOLOGRAPHIC MEMORIES *
Gy. AKOS Hungarian Optical Works, Budapest, Hungary
G. KISS and P. VARGA Central Research Institute for Physics, Budapest, Hungary
Received 16 August 1976
The capacity of a beam addressed holographic memory is strongly affected by the aberrations of the Fourier transform lenses. We studied the field distribution in the hologram and detector planes. The loss due to the lens aberrations can be determined from geometrical aberration data, acquired by the Hartmann method. We give an estimation on the maximum attainable capacity, which seems to be well under 108 bits for real systems.
1. Introduction S
L Z~t4x~
H,L e
?~
The optical arrangements for a holographic readwrite random access memory have been described by various authors [1 ], as have the theoretically obtainable storage density and capacity [2]. We have studied the effect of aberrations of the optical elements and a simplified model has been devised for describing the lose of data capacity due to this effect.
2. Holographic memory arrangements Two possible optical arrangements for a holographic memory are shown in fig. 1. Only the main optical parts are presented; reference beam, lens matrix, etc. were omitted. Addressing is carried out by changing the position of the source in plane S, thus changing the direction from which the data mask (DM) is illuminated. In the hologram plane H, the zeroth order Fourier spectrum of the data mask is recorded. The hologram the constitutes the actual aperture through which the data mask is imaged to the detector plane D. Thus in fig. la, lens L 1 has to perform a 1:1 imaging from plane S to H, while lens L2 images from plane DM to D. In the second arrangement (fig. 1b) the first * This work was supported by the ttungarian National Technical Development Committee (OMFB).
L,
Lz
La
Fig. 1. Optical arrangements of holographic memories.
imaging is done by the combination of lenses L1 and L2, while the combination L2, L 3 performs the second imaging. The storage capacity is maximum if the data mask and total hologram area are of equal size [2]. 63
Volume 20, number 1
OPTICS COMMUNICATIONS
January 1977
3. The effect of lens aberrations 0 T~ =
3.1. Increase in hologram size
I'
We shall now consider the question of the first imaging. The data mask is an array of transparent and opaque bits. In the case of aberration-free imaging the incoming wavefront is a perfect sphere (G) with its centre of curvature located at the image P of the point source (fig. 2), The data mask modulates the wavefront and we obtain the Fourier spectrum centred around point P. This is a highly modulated distribution consisting of overlapping Fourier spectra of individual bits. (The distribution can be smoothed by the use of a random phase mask [3] .) For imperfect imaging the wavefront is distorted (W in fig. 2). Since the bits are relatively small, the imperfect wavefront at one bit can be considered as spherical, but coming from a different direction than the corresponding part of the reference sphere G. This causes the Fourier patterns of the individual bits to shift relatively to one-another. The direction and thus the shift are determined by the wavefront normal. Since for adequate reconstruction of one bit it is necessary to record at least the main lobe of the diffraction pattern, the minimum hologram size has to be enlarged due to this relative shift. Thus it is the amount of transverse displacement, i.e. the transverse geometrical aberrations, that describe the aberration effects. To demonstrate this we have constructed an arrangement corresponding to that of fig. I a. Two 300/4.5 Zeiss Tessar lenses have been used as transformation optics. These lenses were designed for infinity but show relatively good performance at equal conjugate ratios.
w .~
DM
T Fig. 2. Effect of lens aberration in the hologram plane. 64
l "2f
/ rnm
Fig. 3. Shifted Fourier patterns in the hologram plane,
A movable hole of 1.5 mm diameter was placed in front of the first lens. The Fourier diffraction pattern was recorded by a scanning detector in the hologram plane, with 5/am resolution. Fig. 3 shows several diffraction patterns recorded successively, changing the location of the hole in front of the lens. The transverse aberration curves can be obtained by measuring the displacement of the peak of the diffraction pattern when moving the hole across the diameter of the lens at various angles of illumination. This is actually a modified Hartmann method, similar to the one described by Golden [4]. The aberration curves for the Tessar lens are shown in fig. 4, where u represents the aberrations in/am, and x 1 and x 0 represent the coordinates in the planes H and S, respectively. Fig. 5a shows a retransformed alternating bit pattern illuminated on-axis, when no aperture has been placed in the hologram plane; fig. 5b shows the same pattern with a slit limiting the transmitted Fourier spectrum. It can be seen that only those portions of the bit pattern are resolved well for which the aberration curve x 0 = 0 of fig. 4 is close to O. (To enhance the effect, the slit width was chosen smaller than the size corresponding to the double Rayleigh resolution.)
Volume 20, number 1
OPTICS COMMUNICATIONS U
January 1977
[mm]
x,: -30
0.2 0.1
0-0.1 -0.2
/ _
~.~--~'----.~
/
__~_ _ _
...-1 . /
t
~.L
i
/
/,
x,=0
~
/ /
_Iz
-
-
Z~/'
. , _ _ . .
x,= + 30 Fig. 4. Lens aberration curves for the Zeiss Tessar 300/4.5 lens.
In an actual system the aberrations of the storage material substrate [6] should also be considered• (a) 4. Storage capacity considerations
(b)
To calculate the loss of memory capacity we shall begin by considering the bit and hologram sizes db0 and dh0 corresponding to the aberration free system. The equation relating these values is defined by the double Rayleigh criterion:
Fig. 5. Ideal (a) and aberrated (b) transfer of a one-dimensional bit pattern.
db0dh0 = 2 k l f ,
3.2. Detector plane
Let us now consider the aberration effects of the second imaging. Since the aperture corresponding to one sub-hologram is relatively small, the imaging through a sub-hologram can be considered as perfect, except for distortion. Aberrations in the imaging lens cause a shift of the reconstructed bit in the detector plane depending on the location of the subhologram used. This shift can be determined in exactly the same way as in the previous case. Thus, with similar lenses for the two imaging, fig. 4 is also applicable here, u corresponding now to the bit displacement, the coordinate x 1 to hologram location x2, the coordinate x 0 to bit position x 1 . Notice, that in the arrangement of fig. lb, if similar lenses are used, the lenses have to be corrected for both directions to perform the two imaging equally well. This is the same condition as for the transform lenses of coherent optical computers [5].
(1)
where k is the wavelength of light,/)"is the distance from data mask to hologram plane• l = 2 corresponds to the arrangement of fig. 1a and l = 1 for that of fig. lb. The ideally attainable storage capacity of the memory (2), in the case of a squareshaped data mask is C = ( t A l l 6c3J7) 2 ,
(2)
A is the clear aperture of the lens, F is the F-number, c is a duty factor between holograms. Eq. (2) shows, that in the ideal case the capacity increases as the square of the size of a given system. Let us define the maximum aberration ,5 of the len., as
A = 2 max[lu(x 1, x0)l] .
(3)
It may be assumed that the two imaging steps are performed by similar lenses; thus one ,5 value describes the effect of aberrations. To avoid a loss in resolution we have to increase the hologram size and thus the period Ph to Ph = C(dh0 + A).
(4) 65
Volume 20, number 1
OI'TICS COMMUNICATIONS
The factor c is necessary to eliminate hologram crosstalk. To avoid further cross-talk in the detector plane due to the displacement of bits we have to leave a guard band between bits, the bit period Pb thus becomes Pb = db0 + A.
January 1977
to' c.
(5)
It can readily be seen that the capacity which is inversely proportional to the product PbPh is maximum ifdh0 and db0 are chosen to be equal: dh0 = db0 = ~ f .
\
(6)
The capacity can now be written as
c=~
( 5 ~ _ ) 4 I.
Z
2v%Ty+ a
_14
(7) \
The maximum aberration depends on lens design but for a fixed F-number it also increases with the size thereby reducing the obtainable capacity. Though only strictly true for thin lenses, it can be assumed that the geometrical aberrations of a given lens design increase linearly with size: a = ~f,
\ \ \ \ \ \
\
\
(8)
\
\
and the capacity becomes l
/Or
2
2
+ (~/2vgXT),,/7!
C= C~ C~e1 103l 27/4 ~-'-F) (1 ( ~ 6 / 2 X / ) f c o 2
k
6
d
tO
t2
t~
wF
Fig. 6. Plot of C vs. F.
The only parameter relating to the size of the lenses is the focal length f. We can see that by scaling the dimensions of the lens system the capacity slowly approaches a limiting value. It is convenient to write a = co 10 - 3 , where co is a constant of order one for conventional lenses. In this case, for c 2 = 2, the capacity has the form
]4
(10)
The first factor describes the maximum attainable capacity, C~, by increasing the system size infinitely, while the second factor, Crel, contains the size parame t e r s f a n d X. Fig. 6 is a plot of C~ vs. coF for the arrangements shown in fig. la (solid line) and fig. lb (dotted line). The product for commercially obtainable lenses is around 9. Since increasing the F-number also
66
I l l l t l l l
increases the field of view for which the lens has to be corrected, it seems very hard to reduce this factor considerably by proper lens design. For actual sizes the capacity is further reduced. Fig. 7 plots the capacity C vs. focal length for various values of co, in the case when F = 2 and X = 6328/~m. From these calculations it would seem impossible that capacities higher than 107-108 bits could be achieved using conventional optics. This calculation is strictly valid only if the recording beam intensity is uniform. In practice a gaussian reference and reconstruction beam would be used. This tends to complicate somewhat the calculations but the results are similar. No explicit evaluation of the cross-talk between bits or between holograms has been carried out to further simplify calculations.
Volume 20, number 1
OPTICS COMMUNICATIONS
January 1977
5. Conclusions We have shown that lens aberrations considerably affect the storage capacity of a holographic memory if both writing and reading stages are addressed optically. The capacity, instead of varying as the second power of size, as approaching a limit which is determined by the F-number and aberrations o f the transfer lenses. The loss in memory capacity manifests itself in a decrease of storage density. To achieve appreciable capacities and densities it seems necessary to use unconventional (e.g. holographic), aberration-free optics or to address either the writing stage or both write and read cycles nonoptically. The parameter that best describes the capacity loss is the transverse geometrical aberration. This statement appears to be paradoxical for holography, a typically wave-optical method, and results from the simple binary distribution in the data mask.
References
/0~ I0 100
i
500
J
lO00
J
1500
Fig. 7. Plot of C vs..f, with to as parameter.
f[mm]
[1] W.C. Stewart, R.S. Mezrich, L.S. Consentino, E.M. Nagle, F.S. Wendt and R.D. Lohman, RCA Rev. 34 (1973) 5. [2] A. Vander Lugt, Appl. Optics 12 (1973) 1675. [3] Y. Takeda, Japan. J. Appl. Phys. 11 (1972) 656. [4] 1,3. Golden, Appl. Optics 14 (1975) 2391. [5] K. von Bieren, Appl. Optics 10 (1971) 2739. [6] Gy. Akos, G. Kiss and P. Varga, Optica Applicata V. (1975) 15.
67
OPTICS COMMUNICATIONS
January 1977
EFFECT OF LENS ABERRATIONS ON THE STORAGE CAPACITY OF HOLOGRAPHIC MEMORIES *
Gy. AKOS Hungarian Optical Works, Budapest, Hungary
G. KISS and P. VARGA Central Research Institute for Physics, Budapest, Hungary
Received 16 August 1976
The capacity of a beam addressed holographic memory is strongly affected by the aberrations of the Fourier transform lenses. We studied the field distribution in the hologram and detector planes. The loss due to the lens aberrations can be determined from geometrical aberration data, acquired by the Hartmann method. We give an estimation on the maximum attainable capacity, which seems to be well under 108 bits for real systems.
1. Introduction S
L Z~t4x~
H,L e
?~
The optical arrangements for a holographic readwrite random access memory have been described by various authors [1 ], as have the theoretically obtainable storage density and capacity [2]. We have studied the effect of aberrations of the optical elements and a simplified model has been devised for describing the lose of data capacity due to this effect.
2. Holographic memory arrangements Two possible optical arrangements for a holographic memory are shown in fig. 1. Only the main optical parts are presented; reference beam, lens matrix, etc. were omitted. Addressing is carried out by changing the position of the source in plane S, thus changing the direction from which the data mask (DM) is illuminated. In the hologram plane H, the zeroth order Fourier spectrum of the data mask is recorded. The hologram the constitutes the actual aperture through which the data mask is imaged to the detector plane D. Thus in fig. la, lens L 1 has to perform a 1:1 imaging from plane S to H, while lens L2 images from plane DM to D. In the second arrangement (fig. 1b) the first * This work was supported by the ttungarian National Technical Development Committee (OMFB).
L,
Lz
La
Fig. 1. Optical arrangements of holographic memories.
imaging is done by the combination of lenses L1 and L2, while the combination L2, L 3 performs the second imaging. The storage capacity is maximum if the data mask and total hologram area are of equal size [2]. 63
Volume 20, number 1
OPTICS COMMUNICATIONS
January 1977
3. The effect of lens aberrations 0 T~ =
3.1. Increase in hologram size
I'
We shall now consider the question of the first imaging. The data mask is an array of transparent and opaque bits. In the case of aberration-free imaging the incoming wavefront is a perfect sphere (G) with its centre of curvature located at the image P of the point source (fig. 2), The data mask modulates the wavefront and we obtain the Fourier spectrum centred around point P. This is a highly modulated distribution consisting of overlapping Fourier spectra of individual bits. (The distribution can be smoothed by the use of a random phase mask [3] .) For imperfect imaging the wavefront is distorted (W in fig. 2). Since the bits are relatively small, the imperfect wavefront at one bit can be considered as spherical, but coming from a different direction than the corresponding part of the reference sphere G. This causes the Fourier patterns of the individual bits to shift relatively to one-another. The direction and thus the shift are determined by the wavefront normal. Since for adequate reconstruction of one bit it is necessary to record at least the main lobe of the diffraction pattern, the minimum hologram size has to be enlarged due to this relative shift. Thus it is the amount of transverse displacement, i.e. the transverse geometrical aberrations, that describe the aberration effects. To demonstrate this we have constructed an arrangement corresponding to that of fig. I a. Two 300/4.5 Zeiss Tessar lenses have been used as transformation optics. These lenses were designed for infinity but show relatively good performance at equal conjugate ratios.
w .~
DM
T Fig. 2. Effect of lens aberration in the hologram plane. 64
l "2f
/ rnm
Fig. 3. Shifted Fourier patterns in the hologram plane,
A movable hole of 1.5 mm diameter was placed in front of the first lens. The Fourier diffraction pattern was recorded by a scanning detector in the hologram plane, with 5/am resolution. Fig. 3 shows several diffraction patterns recorded successively, changing the location of the hole in front of the lens. The transverse aberration curves can be obtained by measuring the displacement of the peak of the diffraction pattern when moving the hole across the diameter of the lens at various angles of illumination. This is actually a modified Hartmann method, similar to the one described by Golden [4]. The aberration curves for the Tessar lens are shown in fig. 4, where u represents the aberrations in/am, and x 1 and x 0 represent the coordinates in the planes H and S, respectively. Fig. 5a shows a retransformed alternating bit pattern illuminated on-axis, when no aperture has been placed in the hologram plane; fig. 5b shows the same pattern with a slit limiting the transmitted Fourier spectrum. It can be seen that only those portions of the bit pattern are resolved well for which the aberration curve x 0 = 0 of fig. 4 is close to O. (To enhance the effect, the slit width was chosen smaller than the size corresponding to the double Rayleigh resolution.)
Volume 20, number 1
OPTICS COMMUNICATIONS U
January 1977
[mm]
x,: -30
0.2 0.1
0-0.1 -0.2
/ _
~.~--~'----.~
/
__~_ _ _
...-1 . /
t
~.L
i
/
/,
x,=0
~
/ /
_Iz
-
-
Z~/'
. , _ _ . .
x,= + 30 Fig. 4. Lens aberration curves for the Zeiss Tessar 300/4.5 lens.
In an actual system the aberrations of the storage material substrate [6] should also be considered• (a) 4. Storage capacity considerations
(b)
To calculate the loss of memory capacity we shall begin by considering the bit and hologram sizes db0 and dh0 corresponding to the aberration free system. The equation relating these values is defined by the double Rayleigh criterion:
Fig. 5. Ideal (a) and aberrated (b) transfer of a one-dimensional bit pattern.
db0dh0 = 2 k l f ,
3.2. Detector plane
Let us now consider the aberration effects of the second imaging. Since the aperture corresponding to one sub-hologram is relatively small, the imaging through a sub-hologram can be considered as perfect, except for distortion. Aberrations in the imaging lens cause a shift of the reconstructed bit in the detector plane depending on the location of the subhologram used. This shift can be determined in exactly the same way as in the previous case. Thus, with similar lenses for the two imaging, fig. 4 is also applicable here, u corresponding now to the bit displacement, the coordinate x 1 to hologram location x2, the coordinate x 0 to bit position x 1 . Notice, that in the arrangement of fig. lb, if similar lenses are used, the lenses have to be corrected for both directions to perform the two imaging equally well. This is the same condition as for the transform lenses of coherent optical computers [5].
(1)
where k is the wavelength of light,/)"is the distance from data mask to hologram plane• l = 2 corresponds to the arrangement of fig. 1a and l = 1 for that of fig. lb. The ideally attainable storage capacity of the memory (2), in the case of a squareshaped data mask is C = ( t A l l 6c3J7) 2 ,
(2)
A is the clear aperture of the lens, F is the F-number, c is a duty factor between holograms. Eq. (2) shows, that in the ideal case the capacity increases as the square of the size of a given system. Let us define the maximum aberration ,5 of the len., as
A = 2 max[lu(x 1, x0)l] .
(3)
It may be assumed that the two imaging steps are performed by similar lenses; thus one ,5 value describes the effect of aberrations. To avoid a loss in resolution we have to increase the hologram size and thus the period Ph to Ph = C(dh0 + A).
(4) 65
Volume 20, number 1
OI'TICS COMMUNICATIONS
The factor c is necessary to eliminate hologram crosstalk. To avoid further cross-talk in the detector plane due to the displacement of bits we have to leave a guard band between bits, the bit period Pb thus becomes Pb = db0 + A.
January 1977
to' c.
(5)
It can readily be seen that the capacity which is inversely proportional to the product PbPh is maximum ifdh0 and db0 are chosen to be equal: dh0 = db0 = ~ f .
\
(6)
The capacity can now be written as
c=~
( 5 ~ _ ) 4 I.
Z
2v%Ty+ a
_14
(7) \
The maximum aberration depends on lens design but for a fixed F-number it also increases with the size thereby reducing the obtainable capacity. Though only strictly true for thin lenses, it can be assumed that the geometrical aberrations of a given lens design increase linearly with size: a = ~f,
\ \ \ \ \ \
\
\
(8)
\
\
and the capacity becomes l
/Or
2
2
+ (~/2vgXT),,/7!
C= C~ C~e1 103l 27/4 ~-'-F) (1 ( ~ 6 / 2 X / ) f c o 2
k
6
d
tO
t2
t~
wF
Fig. 6. Plot of C vs. F.
The only parameter relating to the size of the lenses is the focal length f. We can see that by scaling the dimensions of the lens system the capacity slowly approaches a limiting value. It is convenient to write a = co 10 - 3 , where co is a constant of order one for conventional lenses. In this case, for c 2 = 2, the capacity has the form
]4
(10)
The first factor describes the maximum attainable capacity, C~, by increasing the system size infinitely, while the second factor, Crel, contains the size parame t e r s f a n d X. Fig. 6 is a plot of C~ vs. coF for the arrangements shown in fig. la (solid line) and fig. lb (dotted line). The product for commercially obtainable lenses is around 9. Since increasing the F-number also
66
I l l l t l l l
increases the field of view for which the lens has to be corrected, it seems very hard to reduce this factor considerably by proper lens design. For actual sizes the capacity is further reduced. Fig. 7 plots the capacity C vs. focal length for various values of co, in the case when F = 2 and X = 6328/~m. From these calculations it would seem impossible that capacities higher than 107-108 bits could be achieved using conventional optics. This calculation is strictly valid only if the recording beam intensity is uniform. In practice a gaussian reference and reconstruction beam would be used. This tends to complicate somewhat the calculations but the results are similar. No explicit evaluation of the cross-talk between bits or between holograms has been carried out to further simplify calculations.
Volume 20, number 1
OPTICS COMMUNICATIONS
January 1977
5. Conclusions We have shown that lens aberrations considerably affect the storage capacity of a holographic memory if both writing and reading stages are addressed optically. The capacity, instead of varying as the second power of size, as approaching a limit which is determined by the F-number and aberrations o f the transfer lenses. The loss in memory capacity manifests itself in a decrease of storage density. To achieve appreciable capacities and densities it seems necessary to use unconventional (e.g. holographic), aberration-free optics or to address either the writing stage or both write and read cycles nonoptically. The parameter that best describes the capacity loss is the transverse geometrical aberration. This statement appears to be paradoxical for holography, a typically wave-optical method, and results from the simple binary distribution in the data mask.
References
/0~ I0 100
i
500
J
lO00
J
1500
Fig. 7. Plot of C vs..f, with to as parameter.
f[mm]
[1] W.C. Stewart, R.S. Mezrich, L.S. Consentino, E.M. Nagle, F.S. Wendt and R.D. Lohman, RCA Rev. 34 (1973) 5. [2] A. Vander Lugt, Appl. Optics 12 (1973) 1675. [3] Y. Takeda, Japan. J. Appl. Phys. 11 (1972) 656. [4] 1,3. Golden, Appl. Optics 14 (1975) 2391. [5] K. von Bieren, Appl. Optics 10 (1971) 2739. [6] Gy. Akos, G. Kiss and P. Varga, Optica Applicata V. (1975) 15.
67