Abstract

Shift multiplexing is a holographic storage method particularly suitable for the implementation of holographic disks. We characterize the performance of shift-multiplexed memories by using a spherical wave as the reference beam. We derive the shift selectivity, the cross talk, the exposure schedule, and the storage density of the method. We give experimental results to verify the theoretical predictions.

© 1996 Optical Society of America

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References

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  1. D. Psaltis, M. Levene, A. Pu, G. Barbastathis, K. Curtis, “Holographic storage using shift multiplexing,” Opt. Lett. 20, 782–784 (1995).
    [CrossRef] [PubMed]
  2. D. Psaltis, “Parallel optical memories,” Byte 17, 179–182 (1992).
  3. H.-Y. S. Li, D. Psaltis, “Three-dimensional holographic disks,” Appl. Opt. 33, 3764–3774 (1994).
    [CrossRef] [PubMed]
  4. K. Wagner, D. Psaltis, “Multilayer optical learning networks,” Appl. Opt. 26, 5061–5076 (1987).
    [CrossRef] [PubMed]
  5. L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1991), pp. 243–253.
  6. H. C. Külich, “A new approach to read volume holograms at different wavelengths,” Opt. Commun. 64, 407–411 (1987).
    [CrossRef]
  7. H. C. Külich, “Reconstructing volume holograms without image field losses,” Appl. Opt. 30, 2850–2857 (1991).
    [CrossRef] [PubMed]
  8. C. Gu, J. Hong, I. McMichael, R. Saxena, F. Mok, “Cross-talk-limited storage capacity of volume holographic memory,” J. Opt. Soc. Am. A 9, 1978–1983 (1992).
    [CrossRef]
  9. A. Yariv, “Interpage and interpixel cross talk in orthogonal (wavelength-multiplexed) holograms,” Opt. Lett. 18, 652–654 (1993).
    [CrossRef] [PubMed]
  10. K. Curtis, C. Gu, D. Psaltis, “Cross talk in wavelength-multiplexed holographic memories,” Opt. Lett. 18, 1001–1003 (1993).
    [CrossRef] [PubMed]
  11. K. Curtis, D. Psaltis, “Cross talk in phase-coded holographic memories,” J. Opt. Soc. Am. A 10, 2547–2550 (1993).
    [CrossRef]
  12. K. Curtis, D. Psaltis, “Cross talk for angle- and wavelength-multiplexed image plane holograms,” Opt. Lett. 19, 1774–1776 (1994).
    [CrossRef] [PubMed]
  13. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), pp. 105–111.
  14. K. Bløtekjaer, “Limitations on holographic storage capacity of photochromic and photorefractive media,” Appl. Opt. 18, 57–67 (1979).
    [CrossRef] [PubMed]
  15. D. Psaltis, D. Brady, K. Wagner, “Adaptive optical networks using photorefractive crystals,” Appl. Opt. 27, 1752–1759 (1988).
    [CrossRef]
  16. E. S. Maniloff, K. M. Johnson, “Maximized photorefrac-tive data storage,” J. Appl. Phys. 70, 4702–4707 (1991).
    [CrossRef]
  17. A. Pu, D. Psaltis, “High-density recording in photopolymer-based holographic three-dimensional disks,” Appl. Opt. 35, 2389–2398 (1996).
    [CrossRef] [PubMed]
  18. K. Curtis, A. Pu, D. Psaltis, “Method for holographic storage using peristrophic multiplexing,” Opt. Lett. 19, 993–994 (1994).
    [CrossRef] [PubMed]
  19. F. H. Mok, G. W. Burr, D. Psaltis, “Angle and space multiplexed random access memory (HRAM),” Opt. Mem. Neural Networks 3, 119–127 (1994).
  20. A. Pu, G. Barbastathis, M. Levene, D. Psaltis, “Shift-multiplexed holographic 3D disk,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 219–221.
  21. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 418–421.
  22. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 33–39.
  23. G. B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, New York, 1973), pp. 371–373.

1996

1995

1994

1993

1992

1991

H. C. Külich, “Reconstructing volume holograms without image field losses,” Appl. Opt. 30, 2850–2857 (1991).
[CrossRef] [PubMed]

E. S. Maniloff, K. M. Johnson, “Maximized photorefrac-tive data storage,” J. Appl. Phys. 70, 4702–4707 (1991).
[CrossRef]

1988

1987

K. Wagner, D. Psaltis, “Multilayer optical learning networks,” Appl. Opt. 26, 5061–5076 (1987).
[CrossRef] [PubMed]

H. C. Külich, “A new approach to read volume holograms at different wavelengths,” Opt. Commun. 64, 407–411 (1987).
[CrossRef]

1979

Barbastathis, G.

D. Psaltis, M. Levene, A. Pu, G. Barbastathis, K. Curtis, “Holographic storage using shift multiplexing,” Opt. Lett. 20, 782–784 (1995).
[CrossRef] [PubMed]

A. Pu, G. Barbastathis, M. Levene, D. Psaltis, “Shift-multiplexed holographic 3D disk,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 219–221.

Bløtekjaer, K.

Brady, D.

Burr, G. W.

F. H. Mok, G. W. Burr, D. Psaltis, “Angle and space multiplexed random access memory (HRAM),” Opt. Mem. Neural Networks 3, 119–127 (1994).

Cooke, D. J.

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1991), pp. 243–253.

Curtis, K.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 33–39.

Gu, C.

Hong, J.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 418–421.

Johnson, K. M.

E. S. Maniloff, K. M. Johnson, “Maximized photorefrac-tive data storage,” J. Appl. Phys. 70, 4702–4707 (1991).
[CrossRef]

Külich, H. C.

H. C. Külich, “Reconstructing volume holograms without image field losses,” Appl. Opt. 30, 2850–2857 (1991).
[CrossRef] [PubMed]

H. C. Külich, “A new approach to read volume holograms at different wavelengths,” Opt. Commun. 64, 407–411 (1987).
[CrossRef]

Levene, M.

D. Psaltis, M. Levene, A. Pu, G. Barbastathis, K. Curtis, “Holographic storage using shift multiplexing,” Opt. Lett. 20, 782–784 (1995).
[CrossRef] [PubMed]

A. Pu, G. Barbastathis, M. Levene, D. Psaltis, “Shift-multiplexed holographic 3D disk,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 219–221.

Li, H.-Y. S.

Maniloff, E. S.

E. S. Maniloff, K. M. Johnson, “Maximized photorefrac-tive data storage,” J. Appl. Phys. 70, 4702–4707 (1991).
[CrossRef]

McMichael, I.

Mok, F.

Mok, F. H.

F. H. Mok, G. W. Burr, D. Psaltis, “Angle and space multiplexed random access memory (HRAM),” Opt. Mem. Neural Networks 3, 119–127 (1994).

Psaltis, D.

A. Pu, D. Psaltis, “High-density recording in photopolymer-based holographic three-dimensional disks,” Appl. Opt. 35, 2389–2398 (1996).
[CrossRef] [PubMed]

D. Psaltis, M. Levene, A. Pu, G. Barbastathis, K. Curtis, “Holographic storage using shift multiplexing,” Opt. Lett. 20, 782–784 (1995).
[CrossRef] [PubMed]

H.-Y. S. Li, D. Psaltis, “Three-dimensional holographic disks,” Appl. Opt. 33, 3764–3774 (1994).
[CrossRef] [PubMed]

K. Curtis, D. Psaltis, “Cross talk for angle- and wavelength-multiplexed image plane holograms,” Opt. Lett. 19, 1774–1776 (1994).
[CrossRef] [PubMed]

K. Curtis, A. Pu, D. Psaltis, “Method for holographic storage using peristrophic multiplexing,” Opt. Lett. 19, 993–994 (1994).
[CrossRef] [PubMed]

F. H. Mok, G. W. Burr, D. Psaltis, “Angle and space multiplexed random access memory (HRAM),” Opt. Mem. Neural Networks 3, 119–127 (1994).

K. Curtis, D. Psaltis, “Cross talk in phase-coded holographic memories,” J. Opt. Soc. Am. A 10, 2547–2550 (1993).
[CrossRef]

K. Curtis, C. Gu, D. Psaltis, “Cross talk in wavelength-multiplexed holographic memories,” Opt. Lett. 18, 1001–1003 (1993).
[CrossRef] [PubMed]

D. Psaltis, “Parallel optical memories,” Byte 17, 179–182 (1992).

D. Psaltis, D. Brady, K. Wagner, “Adaptive optical networks using photorefractive crystals,” Appl. Opt. 27, 1752–1759 (1988).
[CrossRef]

K. Wagner, D. Psaltis, “Multilayer optical learning networks,” Appl. Opt. 26, 5061–5076 (1987).
[CrossRef] [PubMed]

A. Pu, G. Barbastathis, M. Levene, D. Psaltis, “Shift-multiplexed holographic 3D disk,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 219–221.

Pu, A.

Saxena, R.

Solymar, L.

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1991), pp. 243–253.

Wagner, K.

Whitham, G. B.

G. B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, New York, 1973), pp. 371–373.

Yariv, A.

Yeh, P.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), pp. 105–111.

Appl. Opt.

Byte

D. Psaltis, “Parallel optical memories,” Byte 17, 179–182 (1992).

J. Appl. Phys.

E. S. Maniloff, K. M. Johnson, “Maximized photorefrac-tive data storage,” J. Appl. Phys. 70, 4702–4707 (1991).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

H. C. Külich, “A new approach to read volume holograms at different wavelengths,” Opt. Commun. 64, 407–411 (1987).
[CrossRef]

Opt. Lett.

Opt. Mem. Neural Networks

F. H. Mok, G. W. Burr, D. Psaltis, “Angle and space multiplexed random access memory (HRAM),” Opt. Mem. Neural Networks 3, 119–127 (1994).

Other

A. Pu, G. Barbastathis, M. Levene, D. Psaltis, “Shift-multiplexed holographic 3D disk,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 219–221.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 418–421.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 33–39.

G. B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, New York, 1973), pp. 371–373.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), pp. 105–111.

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1991), pp. 243–253.

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Figures (15)

Fig. 1
Fig. 1

Holographic disk implemented with shift multiplexing. SLM, spatial light modulator; NA, numerical aperture.

Fig. 2
Fig. 2

Geometry for shift multiplexing by the use of a spherical reference wave.

Fig. 3
Fig. 3

Experimental setup for the demonstration of shift multiplexing (not drawn to scale).

Fig. 4
Fig. 4

Experimental selectivity curve (diffraction efficiency η versus shift δ). The parameters of the experiment are shown in Fig. 3.

Fig. 5
Fig. 5

Geometry for the theoretical calculation of cross talk in shift multiplexing by the use of a spherical reference wave.

Fig. 6
Fig. 6

Theoretical plots of expected cross-talk power versus pixel location for Fourier-plane shift-multiplexed holograms. The parameters used for the plots were hologram thickness L = 1 mm, angle of incidence of the signal θ S = 20°, wavelength λ = 488 nm, focal length F = 5 cm, and pixel size b = 10 μm.

Fig. 7
Fig. 7

Cross sections of the diffracted pattern at shift location 11 (originally left blank). The upper plot is for multiplexing holograms in the first Bragg null and the lower plot for the second null. The units on both axes are arbitrary, but horizontal and vertical scales are the same in both plots.

Fig. 8
Fig. 8

SNR versus null order p (in multiples of δ = 3.7 μm) for two experiments: single hologram and 21 holograms. Shown also is the theoretical SNR prediction for the maximum number M of allowable shift-multiplexed holograms at the respective null orders.

Fig. 9
Fig. 9

(a) Exposure schedule for sequential recording. Horizontal axis is shift, vertical is recording time. Bars A1, A2,…, A2M denote holograms; the index corresponds to the location on the disk; the horizontal location of a hologram in the graph denotes its shift with respect to the origin (left edge of the first hologram A1), and the vertical location the beginning of its exposure in the schedule. The horizontal separation is equal to the shift selectivity δ; the vertical separation is equal to the constant exposure time t 0 (see text). (b) Nonuniform erasure of hologram A m by its successors A m +1,…, A m + M −1. The diffraction-efficiency curve follows the profile of A m after the recording of all its shift-multiplexed neighbors is complete (see also Appendix B and Fig. 14).

Fig. 10
Fig. 10

Plot of the measured diffraction efficiency (after spatial integration by a single detector) of 50 out of 600 holograms stored with the sequential method. For a shift separation of δshift = 7.4 μm (second null) and an aperture size of s ≈ 3 mm, we have M ≈ 400. Therefore only the first 200 holograms received equal exposure. The exposure time used in this experiment was t 0 = 10 s.

Fig. 11
Fig. 11

Reconstructions of holograms (a) 1, (b) 200, (c) 400, and (d) 600 from the experiment of Fig. 10. The shift direction was from left to right.

Fig. 12
Fig. 12

Geometry for the calculation of storage density in shift-multiplexing geometry (spherical reference incident normally upon the material, signal incident off axis). The case s′ sin θ S ′ < L, ϕ < θ S is shown (see text).

Fig. 13
Fig. 13

Theoretical shift-multiplexing surface storage density in the Fourier plane, with parameters λ = 0.532 nm, n 0 = 1.525, N p = 768, b = 45 μm, amd F = 5.46 cm, the same as those of the experiment reported in Ref. 17, where angle plus peristrophic multiplexing was used. The reference spread used for the shift-multiplexing density calculation is ϕ = 45°, and the angle of incidence of the signal beam is θ S = 60°.

Fig. 14
Fig. 14

Geometry for the calculation of the distortion occuring in shift-multiplexed holograms recorded in photorefractive materials, which is due to partial erasure in the Fourier or the Fresnel regions. The filter is shift variant if the hologram is not centered with respect to the Fourier plane (see also Fig. 9).

Fig. 15
Fig. 15

Numerical results of the effects of shift-induced nonuniformity on Fourier and Fresnel holograms. The horizontal axis is the image coordinate parallel to the shift direction, and the vertical axis is diffracted power, both in arbitrary units. (a) Original chessboard pattern; (b) Nyquist filter (cutoff at ±λF/b) without absorption (τ e = ∞), located at f = F = 5 cm; (c) Nyquist filter with t 0 e = 0.011, f = F = 5 cm (Fourier filter); and (d) Nyquist filter with t 0 e = 0.0092, f = 4 cm (Fresnel filter).

Equations (67)

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R ( x , y , z ) = 1 j λ ( z + z 0 ) exp ( j 2 π z + z 0 λ ) × exp [ j π x 2 + y 2 λ ( z + z 0 ) ] .
S ( x , z ) = exp [ j 2 π u S x λ + j 2 π ( 1 u S 2 2 ) z λ ] ,
R ( x δ , y , z ) R * ( x , y , z ) S ( x , z ) = exp [ j π 2 δ x λ ( z + z 0 ) ] exp [ j π δ 2 λ ( z + z 0 ) ] × exp ( j 2 π u S x λ ) exp [ j 2 π ( 1 u S 2 2 ) z λ ] .
Δ θ S δ ( z + z 0 ) cos θ S .
δ Bragg = λ z 0 L u S .
Δ θ = λ L tan θ S λ L u S .
δ Bragg = z 0 Δ θ .
δ = δ Bragg + Δ x = λ z 0 L tan θ S + λ 2 ( NA ) .
z 0 L 2 = n 0 ( z a L 2 ) .
δ = λ 0 [ z a ( 1 1 n 0 ) L 2 ] L tan θ S + λ 0 2 ( NA ) .
S m ( x , y , z ) = + + dξdη f m ( ξ , η ) × exp [ j 2 π ξ λ F ( sin θ S z + cos θ S x ) j 2 π η y λ F ] × exp [ j 2 π λ ( 1 ξ 2 + η 2 2 F 2 ) ( cos θ S z + sin θ S x ) ] .
E i ( m ) = 1 j λ ( z + z 0 ) exp ( j 2 π z + z 0 λ ) × exp [ j π ( x m δ ) 2 + y 2 λ ( z + z 0 ) ] .
E m ( ξ , η ) m = 0 M 1 f m [ ξ + ( m m ) δ z 0 F , η ] × sinc [ ( m m ) δ L λ z 0 ( u S ξ F ) ] .
P X N m = 0 M 1 sinc 2 [ p ( m m ) ( 1 ξ u S F ) ] ,
p | ξ | u S F 1 , M ,
P X N | ξ | 2 p u S F .
η ( t ) = η 0 [ 1 exp ( t / τ w ) ] 2 ,
η ( t ) = η 1 exp ( 2 t / τ e ) ,
ϕ disk = δ R = λ z 0 R L tan θ S + λ 2 R ( N A ) .
N = 2 π R δ .
η 1 = η 0 [ 1 exp ( t 0 / τ w ) ] 2 .
η l = η 0 [ 1 exp ( t 0 / τ w ) ] 2 exp [ 2 ( l 1 ) t 0 / τ e ] .
t 0 = τ w ln [ 1 + τ e ( M 1 ) τ w ] τ e M
η M = η 0 [ τ e τ e + ( M 1 ) τ w ] 2 [ 1 + τ e ( M 1 ) τ w ] 2 ( M 1 ) τ w / τ e η 0 τ e 2 e 2 M 2 τ w 2 .
η av = η ( x ) d x d x ,
η av = 4 η 0 M exp [ t 0 ( 1 τ w + M 1 τ e ) ] × sinh 2 t 0 2 τ w sinh M t 0 τ e sinh t 0 τ e
η 0 τ e 2 ( 1 e 2 ) 2 M 2 τ w 2 .
η ( q ) = η 0 [ 1 exp ( t q τ w ) ] 2 exp ( q = q + 1 M 1 2 t q τ w ) .
η η 0 τ e 2 M 2 τ w 2 .
D = M N p 2 A .
D image = N p 2 b δ = N p 2 b λ [ z 0 L tan θ S + 1 2 ( NA ) ] .
D Fourier = N p 2 b 4 λ F δ = N p 2 b 4 λ 2 F [ z 0 L tan θ S + 1 2 ( NA ) ] .
z 0 ( L ) = A + B L .
s = { N p b 2 λ 0 F / b , image plane Fourier plane ,
sin χ { λ 0 / b N p b / 2 F , image plane Fourier plane .
sin ϕ = n 0 sin ϕ ,
sin θ S = n 0 sin θ S ,
sin χ = n 0 sin χ ,
s cos θ S = s cos θ S .
A = s 2 cos χ cos θ S cos ( θ S + χ ) tan ϕ cos θ S ,
B = 1 2 tan ( θ S + χ ) tan ϕ .
D image max = n 0 N p sin ϕ λ 0 b [ 1 + tan ( θ S + χ ) cos ϕ tan θ S ]
D Fourier max = n 0 N p 2 b sin ϕ 2 λ 0 2 F [ 1 + tan ( θ S + χ ) cos ϕ tan θ S ]
A = s 2 cos χ tan ϕ cos ( θ S + χ ) ,
B = 1 2 tan ϕ [ 1 tan θ S + 2 tan ( θ S + χ ) cos θ S cos χ sin θ S cos θ S cos ( θ S + χ ) ] .
δ y = z 0 ( 2 λ L ) 1 / 2 + λ 2 ( NA ) .
δ = λ z 0 2 L + λ 2 ( NA )
E d ( r p ) = V E i ( r ) Δ∊ ( r ) G ( r ; r p ) d 3 r ,
G ( r ; r p ) = 1 j λ | r r p | exp ( j 2 π | r r p | λ ) 1 j λ ( z p z ) exp [ j 2 π z p z λ + j π ( x p x ) 2 + ( y p y ) 2 λ ( z p z ) ] ,
R ( r ) = 1 j λ ( z + z 0 ) exp ( j 2 π z + z 0 λ ) × exp [ j π x 2 + y 2 λ ( z + z 0 ) ] ,
S ( r ) = exp ( j 2 π u S x λ ) exp [ j 2 π ( 1 u S 2 2 ) z λ ] .
Δ∊ ( r ) = R * ( x , y , z ) S ( x , z ) ,
E i ( r ) = R ( x δ , y , z ) .
E d ( r p ) exp ( j 2 π z p λ ) λ 2 z 0 2 + d x + d y + d z × rect ( z L ) exp [ j π ( u S 2 z λ δ 2 λ ( z + z 0 ) ) ] × exp [ j 2 π x λ ( u S δ z 0 + z ) ] × 1 λ ( z p z ) exp ( j 2 π z p z λ ) × exp [ j π ( x p x ) 2 + ( y p y ) 2 λ ( z p z ) ] .
+ exp [ j ( a w 2 + 2 b w ) ] d w = ( π | a | ) 1 / 2 exp [ j ( sgn ( a ) π 4 b 2 a ) ] ,
E d ( r P ) exp { j 2 π λ [ u S x p + ( 1 u S 2 2 ) z p ] + j 2 π λ δ ( x p u S z p ) z 0 } λ 2 z 0 2 × sinc [ δ L λ z 0 ( u S x p u S z p z 0 ) ] .
η ( δ ) | E d | 2 | E i | 2 sinc 2 ( δ u S L λ z 0 ) .
δ = m δ Bragg m λ z 0 L u S , m = 1 , 2 ,….
η ( x , x ) = l = 0 m 0 1 exp ( j 2 π x x λ F ) exp [ ( l 0 + l ) t 0 / τ e ] × rect [ x ( l m 0 1 2 ) βδ βδ ] ,
l 0 ( x , f ) = | F f λ F b tan θ S | × { N p b 2 x F δ cos θ S , if f < F λ F b tan θ S N p b 2 + x F δ cos θ S , if f > F λ F b tan θ S ,
β ( x ) = cos θ S + x F sin θ S ,
m 0 ( x ) = 2 λ F b β ( x ) δ .
M ( f ) = { 2 λ F b δ cos θ S ( 1 + | 1 f F | N p b 2 2 λ F ) , if | F f | > λ F b tan θ S 2 λ F b δ cos θ S ( 1 + N p b 2 F tan θ S ) , if | F f | < λ F b tan θ S .
h ( x , x ) = sinc ( κ ) ar ( κ + j ζ; m 0 ) × exp [ 2 π ( l 0 + m 0 1 2 ) ζ ] ,
κ = ( x x ) β ( x ) δ λ F ,
ζ = t 0 2 πτ e ,
ar ( u ; l ) = sin ( l π u ) l sin ( π u ) .

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