Abstract

A solution to the Kukhtarev equations is obtained for a typical holographic memory system in which multiplexed holograms, including the effects owing to a nonuniform beam profile in the focal regions, are used. The various noise mechanisms and storage capacity are analyzed on the basis of this solution. The cross-talk noise of a typical 4f holographic memory configuration with defocus is compared with that of a phase mask. It is shown that the memory capacity and the signal-to-noise can be significantly improved by design of an optimal phase mask. The experimental results with defocus and an eight-level phase mask are presented.

© 1998 Optical Society of America

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References

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  1. Q. Gao, R. Kostuk, “Improvement to holographic digital data-storage systems with random and pseudorandom phase mask,” Appl. Opt. 36, 4853–4861 (1997).
    [CrossRef] [PubMed]
  2. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vineskii, “Holographic storage in electrooptic crystals I. Steady state,” Ferroelectrics 22, 949–960 (1979).
    [CrossRef]
  3. C. Gu, J. Hong, H.-Y. Li, D. Psaltis, P. Yeh, “Dynamics of grating formation in photovotaic media,” J. Appl. Phys. 69, 1167–1172 (1991).
    [CrossRef]
  4. M. G. Moharam, T. K. Gaylord, R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
    [CrossRef]
  5. R. A. Rupp, R. Sommerfeldt, K. H. Ringhofer, E. Kratzig, “Space charge field limitations in photorefractive LiNbO3: Fe crystals,” Appl. Phys. B 51, 362–370 (1990).
    [CrossRef]
  6. E. Ochoa, F. Vachss, L. Hesselink, “Higher-order analysis of the photorefractive effect for large modulation depths,” J. Opt. Soc. Am. A 3, 181–187 (1986).
    [CrossRef]
  7. 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]
  8. E. S. Maniloff, K. M. Johnson, “Maximized photorefractive holographic storage,” J. Appl. Phys. 70, 4702–4707 (1991).
    [CrossRef]
  9. K. Rastani, “Storage capacity and cross talk in angularly multiplexed holograms: two case studies,” Appl. Opt. 32, 3772–3778 (1993).
    [CrossRef] [PubMed]

1997 (1)

1993 (1)

1992 (1)

1991 (2)

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

C. Gu, J. Hong, H.-Y. Li, D. Psaltis, P. Yeh, “Dynamics of grating formation in photovotaic media,” J. Appl. Phys. 69, 1167–1172 (1991).
[CrossRef]

1990 (1)

R. A. Rupp, R. Sommerfeldt, K. H. Ringhofer, E. Kratzig, “Space charge field limitations in photorefractive LiNbO3: Fe crystals,” Appl. Phys. B 51, 362–370 (1990).
[CrossRef]

1986 (1)

1979 (2)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vineskii, “Holographic storage in electrooptic crystals I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

Gao, Q.

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

Gu, C.

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]

C. Gu, J. Hong, H.-Y. Li, D. Psaltis, P. Yeh, “Dynamics of grating formation in photovotaic media,” J. Appl. Phys. 69, 1167–1172 (1991).
[CrossRef]

Hesselink, L.

Hong, J.

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]

C. Gu, J. Hong, H.-Y. Li, D. Psaltis, P. Yeh, “Dynamics of grating formation in photovotaic media,” J. Appl. Phys. 69, 1167–1172 (1991).
[CrossRef]

Johnson, K. M.

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

Kostuk, R.

Kratzig, E.

R. A. Rupp, R. Sommerfeldt, K. H. Ringhofer, E. Kratzig, “Space charge field limitations in photorefractive LiNbO3: Fe crystals,” Appl. Phys. B 51, 362–370 (1990).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vineskii, “Holographic storage in electrooptic crystals I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Li, H.-Y.

C. Gu, J. Hong, H.-Y. Li, D. Psaltis, P. Yeh, “Dynamics of grating formation in photovotaic media,” J. Appl. Phys. 69, 1167–1172 (1991).
[CrossRef]

Magnusson, R.

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

Maniloff, E. S.

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

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vineskii, “Holographic storage in electrooptic crystals I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

McMichael, I.

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

Mok, F.

Ochoa, E.

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vineskii, “Holographic storage in electrooptic crystals I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Psaltis, D.

C. Gu, J. Hong, H.-Y. Li, D. Psaltis, P. Yeh, “Dynamics of grating formation in photovotaic media,” J. Appl. Phys. 69, 1167–1172 (1991).
[CrossRef]

Rastani, K.

Ringhofer, K. H.

R. A. Rupp, R. Sommerfeldt, K. H. Ringhofer, E. Kratzig, “Space charge field limitations in photorefractive LiNbO3: Fe crystals,” Appl. Phys. B 51, 362–370 (1990).
[CrossRef]

Rupp, R. A.

R. A. Rupp, R. Sommerfeldt, K. H. Ringhofer, E. Kratzig, “Space charge field limitations in photorefractive LiNbO3: Fe crystals,” Appl. Phys. B 51, 362–370 (1990).
[CrossRef]

Saxena, R.

Sommerfeldt, R.

R. A. Rupp, R. Sommerfeldt, K. H. Ringhofer, E. Kratzig, “Space charge field limitations in photorefractive LiNbO3: Fe crystals,” Appl. Phys. B 51, 362–370 (1990).
[CrossRef]

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vineskii, “Holographic storage in electrooptic crystals I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Vachss, F.

Vineskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vineskii, “Holographic storage in electrooptic crystals I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Yeh, P.

C. Gu, J. Hong, H.-Y. Li, D. Psaltis, P. Yeh, “Dynamics of grating formation in photovotaic media,” J. Appl. Phys. 69, 1167–1172 (1991).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

R. A. Rupp, R. Sommerfeldt, K. H. Ringhofer, E. Kratzig, “Space charge field limitations in photorefractive LiNbO3: Fe crystals,” Appl. Phys. B 51, 362–370 (1990).
[CrossRef]

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vineskii, “Holographic storage in electrooptic crystals I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

J. Appl. Phys. (3)

C. Gu, J. Hong, H.-Y. Li, D. Psaltis, P. Yeh, “Dynamics of grating formation in photovotaic media,” J. Appl. Phys. 69, 1167–1172 (1991).
[CrossRef]

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

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

J. Opt. Soc. Am. A (2)

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Figures (6)

Fig. 1
Fig. 1

Schematic of a 4f holographic digital memory system.

Fig. 2
Fig. 2

Power spectra of an eight-level phase mask from 4 mm away from focus to the focal plane in 0.5-mm increments.

Fig. 3
Fig. 3

Spot size (1/e 2 points) of the power spectra versus displacement on the z axis. The minimum spot size (z = 8 mm) is at the focal point.

Fig. 4
Fig. 4

Schematic of the experimental configuration used in this paper. The beam ratio is controlled by rotation of the half-wave plate that is closer to the laser. The holograms are multiplexed with a different interbeam angle when the multiplexing lens is translated in the direction that is perpendicular to the reference beam.

Fig. 5
Fig. 5

Average SNR of 35 multiplexed holograms that are equalized with a recording schedule versus the beam ratio for a phase mask and for defocus.

Fig. 6
Fig. 6

Playback images of holograms recorded at the optimal beam ratio for (a) direct imaging, (b) a phase mask, and (c) defocus.

Equations (67)

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N D + t = sI ( x ) ( N D - N D + ) - γ R N D + n ,
n t = N D + t + 1 q J x ,
J = q μ nE + k B T μ   n x - p ( N D - N D + ) I ( x ) ,
  E x = q ( N D + - n - N A ) ,
-   E t = J + J 0 ( t ) = q μ nE + k B T μ   n x - p ( N D - N D + ) I ( x ) + J 0 ( t ) ,
-     E d x t =   V t =   J x d x +   J 0 d x = i + i 0 t ,
i 0 t = 1 L 0 L   J x d x .
U x ,   z =   d x 0 d x 0 exp jxx 0 2 π f λ exp - j   π f λ z f   x 0 2 ,
I x = R 2 + U 2 x + RU * x + R * U x = I 0 x + R 0     d x 0 d x 0 exp jx K + 2 x 0 f λ × exp - j   π f λ z f   x 0 2 + c . c . = I 0 x + R 0 D 0 K + K * ,
N D + x ,   t = N D 0 + x ,   t + N D p + x ,   t K + N D n + x ,   t K * ,
E x ,   t = E 0 x ,   t + E p x ,   t K + E n x ,   t K * ,
n x ,   t = n 0 x ,   t + n p x ,   t K + n n x ,   t K * .
K ¯ K K * ¯ K K ¯ K K * K ¯ = = = = 0 , P x , 0 , 0 ,
N D 0 + t = sI 0 ( N D - N D 0 + ) - sD 0 R 0 P ( N Dp + + N Dn + ) - γ R N D 0 + n 0 - γ R × P ( N Dp + n n + N Dn + n p ) ,
N Dp + t = - sI 0 N Dsp + + sD 0 R 0 ( N D - N D 0 + ) - γ R N D 0 + n p - γ R n 0 N Dp + ,
N Dn + t = - sI 0 N Dn + + sD 0 R 0 ( N D - N D 0 + ) - γ R N D 0 + n n - γ R n 0 N Dn + ,
-   E 0 t = J 0 ( t ) + q μ n 0 E 0 + q μ ( E p n n P + E n n p P ) + kT μ   n 0 x - p [ ( N D - N D 0 + ) × I 0 - D 0 R 0 P ( N Dsp + + N Dn + ) ] ,
-   E p t = q μ E 0 n p + n 0 E sp + kT μ × n p x - jKn p - p [ ( N D - N D 0 + ) × D 0 R 0 - I 0 N Dp + ] ,
-   E n t = q μ E 0 n n + n 0 E n + kT μ n n x - jKn n - p [ ( N D - N D 0 + ) D 0 R 0 - I 0 N Dn + ] ,
  E 0 x = q ( N D 0 + - n 0 - N A ) ,
E p x + jKE p = q ( N Dp + - n p ) ,
E n x - jKE n = q ( N Dn + - n n ) .
N Ds + n s E s = = = 1 2 ( N Dp + + N Dn + ) , N Da + 1 2 ( n p + n n ) ,   n a 1 2 ( E p + E n ) ,   E a = = = 1 2 ( N Dp + - N Dn + ) , 1 2 ( n p - n n ) , 1 2 ( E p - E n ) .
sD 0 R 0 PN Ds + + γ R N D 0 + n 0 + γ R 2 ( N Ds + n s - N Da + n a ) / P = sI 0 ( N D - N D 0 + ) ,
( sI 0 + γ R n 0 ) N Ds + + γ R N D 0 + n s = sD 0 R 0 ( N D - N D 0 + ) ,
sI 0 + γ R n 0 N Da + + γ R N D 0 + n 0 = 0 ,
q μ n 0 E 0 + 2 n s E s - n a E a P + kT μ   n 0 x - p [ ( N D - N D 0 + ) × I 0 - D 0 R 0 PN Ds + ] = 0 ,
q μ n s E 0 + n 0 E s + kT μ n s x + jKn a - p [ ( N D - N D 0 + ) × D 0 R 0 - N Ds + I 0 ] = 0 ,
q μ n a E 0 + n 0 E a + kT μ n a x + jKn s + pN Da + I 0 = 0 ,
-   E 0 x = q ( N D 0 + - n 0 - N A ) ,
E s x + jKE a = q ( N Ds + - n s ) ,
E a x + jKE s = q ( N Da + - n a ) .
E s = ( N D - N D 0 + ) n 0 N D 0 + N D 0 + E m - n m E 0 + n l n m n 0   E D ,
n s = N D - N D 0 + N D 0 + n m + ( sI 0 / γ R + n 0 ) N D 0 + K q × ( E m N D 0 + - n m E 0 ) n l + n m n 0 E D n 0 2 ,
N Ds + = j K q   E a = K q N D - N D 0 + N D 0 + × ( n m E 0 - N D 0 + E m ) n l - n m n 0 E D n 0 2 .
n 0 = n 01 I 0 + δ n ,   E 0 = E 01 + δ E ,   N D 0 + = N A + δ N ,
E s = ( N D - N D 0 + ) n 0 N D 0 + n l n m n 0   E D = E s 01 V 0 ,
n s = N D - N D 0 + N D 0 + n m + sI 0 / γ R + n 0 N D 0 + K q n m E D n 0 = n s 01 D 0 R 0 ,
N Ds + = j K q   E a = - K q N D - N D 0 + N D 0 + n m E D n 0 = N Ds 01 +   V 0 ,
δ n = γ R 2 n s + sD 0 R 0 PN Ds + P q μ   I 0 - γ R n 0 + sI 0 p q μ   D 0 R 0 PN Ds + + n 01 I 0 E + n s E s P E 01 γ R n 0 + sI 0 - γ R N A pI 0 / q μ ,
δ E = F   ( P / I 0 2 ) x ,
F = E 01 ( 2 n s + sD 0 R 0 / γ R ) ( N Ds 01 + / N A ) - p q μ   D 0 R 0 N Ds 01 + + n s E s 01 n 01 ,
δ N = q E x = F   q 2 ( P / I 0 2 ) x 2 ,
N D + x ,   t = N D 0 + x ,   t + N DMp + x ,   t K + N DMn + x ,   t K * + m = 1 M - 1 [ N DMp + K m K m + N DMn + K K m * ] = N D 0 + x ,   t + m = 1 M [ N DMp + K m K m + N DMn + K K m * ] ,
E x ,   t = E 0 x ,   t + m = 1 M   E Mp K m K m E Mn K m K m * ,
n x ,   t = n 0 x ,   t + m = 1 M   n Mp K m K m + n Mn K m K m * ,
N DMp + t = - sI 0 N Dp + W + sD 0 R 0 ( N D - N D 0 + ) - γ R N D 0 + n p × W - γ R n 0 N Dp + W + X 1 - X 2 - sI 0 N DMp + - γ R N D 0 + n Mp - γ R n 0 N DMp + ,
W = 1 M m = 1 M exp jx K xm - K x 0 ¯ ,
X 1 = 1 p K M sI 0 ( N D - N D 0 + ) ¯ ,
X 2 = 1 P K M γ R N D 0 + n 0 ¯ ,
  E Mp t = - q μ W ( E 0 n p + n 0 E p ) - kT μ W n p x + iKn p + p [ ( N D - N D 0 + ) D 0 R 0 - I 0 WN Dp + ] - X 3 + X 4 - q μ ( E 0 n Mp + n 0 E Mp ) - kT μ n Mp x + iKn Mp - pI 0 N DMp + ,
X 3 = 1 P   K M q μ E 0 n 0 ¯ ,
X 4 = 1 P ( K M ) p ( N D - N D 0 + ) I 0 ¯ ,
E Mp x + iKE Mp = q ( N DMp + - n Mp ) ,
j KE Mp = q ( N DMp + - n Mp ) .
E Mp = 1 S 4 S 1 - S 5 S 4 1 - exp - S 4 t + S 5 t ,
S 1 = - q μ W E 0 n p + n 0 E p + p [ ( N D - N D 0 + ) × D 0 R 0 - I 0 WN Dp + ] ,
S 2 = - sI 0 + γ R n 0 N Dp + W + sD 0 R 0 ( N D - N D 0 + ) - γ R N D 0 + n p W - sI 0 + γ R n 0 N DMp + ,
S 3 = q μ E 0 1 γ R N D 0 + ,
S 4 = γ R N D 0 + - j   q μ E 0 K q ,
S 5 = q μ E 0 S 1 S 3 - S 2 .
E Mp = S 1 t = p ( N D - N D 0 + ) D 0 R 0 t - [ q μ W ( E 0 n p + n 0 E p ) + pI 0 WN Dp + + X 3 - X 4 ] t ,
SNR = D 0 R 0     d x ( N D - N D 0 + ) * cx 1 + cx 2 + cx 3 1 / 2 ,
cx 1 =   d xW 2 ( x ) [ ( q μ / p ) ( E 0 n p + n 0 E p ) + I 0 N Dp + ] 2 ,
cx 2 = q μ p     d x K M E 0 n 0 ¯ 2 ,
cx 3 =   d x K M ( N D - N D 0 + ) I 0 ¯ 2 ,
W 2 x = 1 M m = 1 M exp jx K xm - K x 0 ¯ 2 1 / 2 .

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