Abstract

Various channel codes, including binary, gray-scale, threshold, and differential techniques, are compared for digital holographic data storage. The tradeoffs among bit error rate, storage capacity, and system complexity are discussed.

© 1995 Optical Society of America

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  1. J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749–752 (1994).
    [CrossRef] [PubMed]
  2. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  3. M. Schwartz, W. R. Bennett, S. Stein, Communications Systems and Techniques (McGraw-Hill, New York, 1966), Appendix A, pp. 585–589.
  4. V. A. Dombrovskii, S. A. Dombrovskii, E. F. Pen, “Reliability of data readout in a holographic channel with constant characteristics,” Optoelectron. Instrum. Data Process. 6, 69–77 (1988).
  5. J. E. Weaver, T. K. Gaylord, “Evaluation experiments on holographic storage of binary data in electro-optic crystals,” Opt. Eng. 20, 404–411 (1981).
    [CrossRef]
  6. J. I. Marcum, “A statistical theory of target detection by pulsed radar,” IRE Trans. Inf. Theory 6, 59–144 (1960).
    [CrossRef]
  7. D. A. Shnidman, “The calculation of the probability of detection and the generalized Marcum Q-function,” IEEE Trans. Inf. Theory 35, 389–400 (1989).
    [CrossRef]
  8. E. S. Maniloff, K. M. Johnson, “Maximized holographic storage,” J. Appl. Phys. 70, 4702–4707 (1991).
    [CrossRef]
  9. K. S. Shanmugan, A. M. Breipohl, Random Signals: Detection, Estimation, and Data Analysis (Wiley, New York, 1988), Chap. 6, pp. 343–351.
  10. P. W. Hooijmans, Coherent Optical System Design (Wiley, New York, 1994), Appendix B, pp. 339–344.
  11. A. A. Blok, “Effect of data coding methods in holographic memory on characteristics of reconstructed images of data pages,” Optoelectron. Instrum. Data Process. 5, 45–51 (1989).
  12. A. A. Verbovetskii, V. B. Fedorov, “Recording of binary data in paraphase code on phase holograms,” Opt. Spectra 33, 628–630 (1971).
  13. A. B. Marchant, Optical Recording (Addison-Wesley, Reading, Mass., 1990), Chap. 9, pp. 229–242.
  14. C. Gu, J. Hong, I. McMichael, R. Saxena, “Cross-talk-limited storage capacity of volume holographic memory,” J. Opt. Soc. Am. A 9, 1978–1983 (1992).
    [CrossRef]
  15. M. C. Bashaw, J. F. Heanue, A. Aharoni, J. F. Walkup, L. Hesselink, “Crosstalk considerations for angular and phase-encoded multiplexing in volume holography,” J. Opt. Soc. Am. B 11, 1820–1836 (1994).
    [CrossRef]
  16. K. Curtis, C. Gu, D. Psaltis, “Cross talk in wavelength-multiplexed holographic memories,” Opt. Lett. 18, 1001–1003 (1993).
    [CrossRef] [PubMed]
  17. R. Brauer, U. Wojak, F. Wyrowski, O. Bryngdahl, “Digital diffusers for optical holography,” Opt. Lett. 16, 1427–1429 (1991).
    [CrossRef] [PubMed]
  18. X. Yi, P. Yeh, C. Gu, “Statistical analysis of cross-talk noise and storage capacity in volume holographic memory,” Opt. Lett. 19, 1580–1582 (1994).
    [CrossRef] [PubMed]
  19. H. Burkhardt, “Optimal data retrieval for high density storage,” in VLSI and Microelectronic Applications in Intelligent Peripherals and Their Interconnection Networks (IEEE Computer Society, Washington, D.C., 1989), pp. 143–148.
  20. B. Kamali, “Maximum likelihood decoding applied to run length limited codes in a magnetic recording system,” in Proceedings of GLOBECOM Tokyo ’87 (IEEE, New York, 1987), pp. 962–967.
  21. B. Olson, S. Esener, “Multi-dimensional partial response precoding for parallel readout optical memories,” presented at SPIE Symposium on Optics, Imaging, and Instrumentation, San Diego, Calif., July 1994, paper 2297A-39.
  22. M. A. Neifeld, M. McDonald, “Error correction for increasing the usable capacity of photorefractive memories,” Opt. Lett. 19, 1483–1485 (1994).
    [CrossRef] [PubMed]
  23. M. A. Neifeld, J. D. Hayes, “Parallel error correction for optical memories,” Int. J. Opt. Mem. Neural Networks 3, 87–98 (1994)

1994 (5)

1993 (1)

1992 (1)

1991 (2)

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

R. Brauer, U. Wojak, F. Wyrowski, O. Bryngdahl, “Digital diffusers for optical holography,” Opt. Lett. 16, 1427–1429 (1991).
[CrossRef] [PubMed]

1989 (2)

A. A. Blok, “Effect of data coding methods in holographic memory on characteristics of reconstructed images of data pages,” Optoelectron. Instrum. Data Process. 5, 45–51 (1989).

D. A. Shnidman, “The calculation of the probability of detection and the generalized Marcum Q-function,” IEEE Trans. Inf. Theory 35, 389–400 (1989).
[CrossRef]

1988 (1)

V. A. Dombrovskii, S. A. Dombrovskii, E. F. Pen, “Reliability of data readout in a holographic channel with constant characteristics,” Optoelectron. Instrum. Data Process. 6, 69–77 (1988).

1981 (1)

J. E. Weaver, T. K. Gaylord, “Evaluation experiments on holographic storage of binary data in electro-optic crystals,” Opt. Eng. 20, 404–411 (1981).
[CrossRef]

1971 (1)

A. A. Verbovetskii, V. B. Fedorov, “Recording of binary data in paraphase code on phase holograms,” Opt. Spectra 33, 628–630 (1971).

1960 (1)

J. I. Marcum, “A statistical theory of target detection by pulsed radar,” IRE Trans. Inf. Theory 6, 59–144 (1960).
[CrossRef]

Aharoni, A.

Bashaw, M. C.

Bennett, W. R.

M. Schwartz, W. R. Bennett, S. Stein, Communications Systems and Techniques (McGraw-Hill, New York, 1966), Appendix A, pp. 585–589.

Blok, A. A.

A. A. Blok, “Effect of data coding methods in holographic memory on characteristics of reconstructed images of data pages,” Optoelectron. Instrum. Data Process. 5, 45–51 (1989).

Brauer, R.

Breipohl, A. M.

K. S. Shanmugan, A. M. Breipohl, Random Signals: Detection, Estimation, and Data Analysis (Wiley, New York, 1988), Chap. 6, pp. 343–351.

Bryngdahl, O.

Burkhardt, H.

H. Burkhardt, “Optimal data retrieval for high density storage,” in VLSI and Microelectronic Applications in Intelligent Peripherals and Their Interconnection Networks (IEEE Computer Society, Washington, D.C., 1989), pp. 143–148.

Curtis, K.

Dombrovskii, S. A.

V. A. Dombrovskii, S. A. Dombrovskii, E. F. Pen, “Reliability of data readout in a holographic channel with constant characteristics,” Optoelectron. Instrum. Data Process. 6, 69–77 (1988).

Dombrovskii, V. A.

V. A. Dombrovskii, S. A. Dombrovskii, E. F. Pen, “Reliability of data readout in a holographic channel with constant characteristics,” Optoelectron. Instrum. Data Process. 6, 69–77 (1988).

Esener, S.

B. Olson, S. Esener, “Multi-dimensional partial response precoding for parallel readout optical memories,” presented at SPIE Symposium on Optics, Imaging, and Instrumentation, San Diego, Calif., July 1994, paper 2297A-39.

Fedorov, V. B.

A. A. Verbovetskii, V. B. Fedorov, “Recording of binary data in paraphase code on phase holograms,” Opt. Spectra 33, 628–630 (1971).

Gaylord, T. K.

J. E. Weaver, T. K. Gaylord, “Evaluation experiments on holographic storage of binary data in electro-optic crystals,” Opt. Eng. 20, 404–411 (1981).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Gu, C.

Hayes, J. D.

M. A. Neifeld, J. D. Hayes, “Parallel error correction for optical memories,” Int. J. Opt. Mem. Neural Networks 3, 87–98 (1994)

Heanue, J. F.

Hesselink, L.

Hong, J.

Hooijmans, P. W.

P. W. Hooijmans, Coherent Optical System Design (Wiley, New York, 1994), Appendix B, pp. 339–344.

Johnson, K. M.

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

Kamali, B.

B. Kamali, “Maximum likelihood decoding applied to run length limited codes in a magnetic recording system,” in Proceedings of GLOBECOM Tokyo ’87 (IEEE, New York, 1987), pp. 962–967.

Maniloff, E. S.

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

Marchant, A. B.

A. B. Marchant, Optical Recording (Addison-Wesley, Reading, Mass., 1990), Chap. 9, pp. 229–242.

Marcum, J. I.

J. I. Marcum, “A statistical theory of target detection by pulsed radar,” IRE Trans. Inf. Theory 6, 59–144 (1960).
[CrossRef]

McDonald, M.

McMichael, I.

Neifeld, M. A.

M. A. Neifeld, J. D. Hayes, “Parallel error correction for optical memories,” Int. J. Opt. Mem. Neural Networks 3, 87–98 (1994)

M. A. Neifeld, M. McDonald, “Error correction for increasing the usable capacity of photorefractive memories,” Opt. Lett. 19, 1483–1485 (1994).
[CrossRef] [PubMed]

Olson, B.

B. Olson, S. Esener, “Multi-dimensional partial response precoding for parallel readout optical memories,” presented at SPIE Symposium on Optics, Imaging, and Instrumentation, San Diego, Calif., July 1994, paper 2297A-39.

Pen, E. F.

V. A. Dombrovskii, S. A. Dombrovskii, E. F. Pen, “Reliability of data readout in a holographic channel with constant characteristics,” Optoelectron. Instrum. Data Process. 6, 69–77 (1988).

Psaltis, D.

Saxena, R.

Schwartz, M.

M. Schwartz, W. R. Bennett, S. Stein, Communications Systems and Techniques (McGraw-Hill, New York, 1966), Appendix A, pp. 585–589.

Shanmugan, K. S.

K. S. Shanmugan, A. M. Breipohl, Random Signals: Detection, Estimation, and Data Analysis (Wiley, New York, 1988), Chap. 6, pp. 343–351.

Shnidman, D. A.

D. A. Shnidman, “The calculation of the probability of detection and the generalized Marcum Q-function,” IEEE Trans. Inf. Theory 35, 389–400 (1989).
[CrossRef]

Stein, S.

M. Schwartz, W. R. Bennett, S. Stein, Communications Systems and Techniques (McGraw-Hill, New York, 1966), Appendix A, pp. 585–589.

Verbovetskii, A. A.

A. A. Verbovetskii, V. B. Fedorov, “Recording of binary data in paraphase code on phase holograms,” Opt. Spectra 33, 628–630 (1971).

Walkup, J. F.

Weaver, J. E.

J. E. Weaver, T. K. Gaylord, “Evaluation experiments on holographic storage of binary data in electro-optic crystals,” Opt. Eng. 20, 404–411 (1981).
[CrossRef]

Wojak, U.

Wyrowski, F.

Yeh, P.

Yi, X.

IEEE Trans. Inf. Theory (1)

D. A. Shnidman, “The calculation of the probability of detection and the generalized Marcum Q-function,” IEEE Trans. Inf. Theory 35, 389–400 (1989).
[CrossRef]

Int. J. Opt. Mem. Neural Networks (1)

M. A. Neifeld, J. D. Hayes, “Parallel error correction for optical memories,” Int. J. Opt. Mem. Neural Networks 3, 87–98 (1994)

IRE Trans. Inf. Theory (1)

J. I. Marcum, “A statistical theory of target detection by pulsed radar,” IRE Trans. Inf. Theory 6, 59–144 (1960).
[CrossRef]

J. Appl. Phys. (1)

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

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Eng. (1)

J. E. Weaver, T. K. Gaylord, “Evaluation experiments on holographic storage of binary data in electro-optic crystals,” Opt. Eng. 20, 404–411 (1981).
[CrossRef]

Opt. Lett. (4)

Opt. Spectra (1)

A. A. Verbovetskii, V. B. Fedorov, “Recording of binary data in paraphase code on phase holograms,” Opt. Spectra 33, 628–630 (1971).

Optoelectron. Instrum. Data Process. (2)

A. A. Blok, “Effect of data coding methods in holographic memory on characteristics of reconstructed images of data pages,” Optoelectron. Instrum. Data Process. 5, 45–51 (1989).

V. A. Dombrovskii, S. A. Dombrovskii, E. F. Pen, “Reliability of data readout in a holographic channel with constant characteristics,” Optoelectron. Instrum. Data Process. 6, 69–77 (1988).

Science (1)

J. F. Heanue, M. C. Bashaw, L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749–752 (1994).
[CrossRef] [PubMed]

Other (8)

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

M. Schwartz, W. R. Bennett, S. Stein, Communications Systems and Techniques (McGraw-Hill, New York, 1966), Appendix A, pp. 585–589.

K. S. Shanmugan, A. M. Breipohl, Random Signals: Detection, Estimation, and Data Analysis (Wiley, New York, 1988), Chap. 6, pp. 343–351.

P. W. Hooijmans, Coherent Optical System Design (Wiley, New York, 1994), Appendix B, pp. 339–344.

H. Burkhardt, “Optimal data retrieval for high density storage,” in VLSI and Microelectronic Applications in Intelligent Peripherals and Their Interconnection Networks (IEEE Computer Society, Washington, D.C., 1989), pp. 143–148.

B. Kamali, “Maximum likelihood decoding applied to run length limited codes in a magnetic recording system,” in Proceedings of GLOBECOM Tokyo ’87 (IEEE, New York, 1987), pp. 962–967.

B. Olson, S. Esener, “Multi-dimensional partial response precoding for parallel readout optical memories,” presented at SPIE Symposium on Optics, Imaging, and Instrumentation, San Diego, Calif., July 1994, paper 2297A-39.

A. B. Marchant, Optical Recording (Addison-Wesley, Reading, Mass., 1990), Chap. 9, pp. 229–242.

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Figures (15)

Fig. 1
Fig. 1

Typical arrangement of a holographic data storage system.

Fig. 2
Fig. 2

Model of signal transmission, showing a transmitted signal corrupted by random phasor noise.

Fig. 3
Fig. 3

pdf’s for the received intensity for a SNR of 20 dB and (a) C = 20.0, (b) C = 2.0.

Fig. 4
Fig. 4

BER for binary threshold encoding for various values of the contrast C. In all the figures log BER indicates the base-10 logarithm.

Fig. 5
Fig. 5

BER for binary threshold encoding as a function of the deviation from the optimal threshold setting (C = 5.0, SNR = 25 dB).

Fig. 6
Fig. 6

SER for gray-scale threshold encoding with C = 10.0. The numbers next to the curves indicate the number of gray levels. The dashed curve is the BER for binary threshold encoding. The comparison is made at constant capacity. In (a) the levels are equally spaced in intensity. In (b) the levels are equally spaced in amplitude.

Fig. 7
Fig. 7

BER for binary differential encoding (C = 10.0). The dashed curve is the BER for binary threshold encoding.

Fig. 8
Fig. 8

Encoding scheme for generalized differential encoding.

Fig. 9
Fig. 9

BER for generalized differential encoding (C = 10.0). The top curve is the BER when the reference intensity is 0.5(IL + IH). The middle curve corresponds to a reference intensity equal to 0.25 ( I L + I H ) 2. The dashed curve is the BER for binary threshold encoding.

Fig. 10
Fig. 10

Encoding scheme for gray-scale differential encoding. Pixel A encodes the background, pixel B encodes the peak intensity, and pixel C encodes the data.

Fig. 11
Fig. 11

BER for oversampling (C = 10.0). The numbers next to the curves indicate the number of samples. The dashed curve is the BER for binary threshold encoding. In (a) oversampling occurs both at the SLM and at the CCD array. In (b) oversampling occurs only at the CCD array.

Fig. 12
Fig. 12

BER for binary threshold encoding when the peak intensity fluctuates and the threshold setting remains fixed (dashed curve). The peak intensity is assumed to be a normally distributed random variable with a standard deviation equal to 15% of its mean value. The solid curve is the BER for binary differential encoding (C = 10.0).

Fig. 13
Fig. 13

SER for sequence encoding with sequences of length P (C = 10.0). The dashed curve is the BER for binary threshold encoding.

Fig. 14
Fig. 14

BER’s for binary threshold encoding (dashed curve) and binary differential encoding (solid curve) when the SNR exhibits 1/N dependence (C = 10.0). The two curves are seen nearly to coincide.

Fig. 15
Fig. 15

BER’s for binary threshold encoding (dashed curve) and binary differential encoding (solid curve) when the SNR is independent of N (C = 10.0).

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

p R ( r ) = r σ 2 exp ( r 2 + A 2 2 σ 2 ) I 0 ( r A σ 2 ) ,
p I ( I ) = 1 2 σ 2 exp ( I + I ideal 2 σ 2 ) I 0 ( I I ideal σ 2 ) ,
I = I ideal + 2 σ 2 ,
I 2 = I ideal 2 + 8 σ 2 I ideal + 8 σ 4 ,
σ I 2 = 4 σ 2 ( σ 2 + I ideal ) ,
p 1 ( Ĩ ) = S exp [ S ( Ĩ + 1 ) ] I 0 ( 2 S Ĩ ) ,
p 0 ( Ĩ ) = S exp [ S ( Ĩ + 1 C ) ] I 0 ( 2 S Ĩ C ) ,
Q ( α , β ) = β x exp ( α 2 + x 2 2 ) I 0 ( α x ) d x .
Ĩ T 1 4 ( 1 + 1 c ) 2 .
Ĩ T 1 S .
BER = 1 2 0 Ĩ T p 1 ( I ) d I + 1 2 Ĩ T p 0 ( I ) d I
= 1 2 [ Q ( 2 S C , 2 S Ĩ T ) + 1 Q ( 2 S , 2 S Ĩ T ) ] .
I m = I L + m I H I L M 1 .
I m = ( I L + m I H I L M 1 ) 2 .
p m ( Ĩ ) = S exp [ S ( Ĩ + I m I H ) ] I 0 ( 2 S Ĩ I m I H ) .
P error = m = 0 M 1 1 M [ 0 Ĩ T , m p m ( Ĩ ) d Ĩ + Ĩ T , m + 1 p m ( Ĩ ) d Ĩ ] ,
BER = 0 p 1 ( I ) d I I p 0 ( I ) d I
= 1 2 [ Q ( S C , S ) + 1 Q ( S , S C ) ] .
BER = 1 4 [ 2 Q ( S , S Ĩ ref ) + Q ( S Ĩ ref , S ) Q ( S Ĩ ref , S C ) + Q ( S C , S Ĩ ref ) ] ,
I ref = 1 2 ( I L + I H ) ,
I ref = 1 2 ( I L + I H ) .
data = I c I a I b I a D ,
p J 1 ( J ) = S ( J M ) ( M 1 ) / 2 exp [ S ( J + M ) ] × I M 1 ( 2 S M J ) ,
p J 0 ( J ) = S ( J C M ) ( M 1 ) / 2 exp [ S ( J + M C ) ] × I M 1 ( 2 S M J C ) ,
BER = 1 2 [ Q ( 2 M S C , 2 M S Ĩ T ) + 1 Q ( 2 M S , 2 M S Ĩ T ) ] ,
b s = log 2 ( P P / 2 ) .

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