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)

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]

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]

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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