Abstract

We consider a process by which images recorded in high-density holographic storage systems may be degraded. In particular, the recording of many holograms in the same volume results in cross erasure of each hologram by the sum of all the remaining holograms superposed in the same volume. This effect is, in general, spatially nonuniform and varies both transversely across each recorded image and in depth as the image propagates and diffracts throughout the volume of the recording medium. We apply a three-dimensional numerical analysis to assess the bit error rate (BER) resulting from this noise source as a function of various recording parameters. In particular, we use this analysis to determine the optimum exposure schedule for a given BER. We then describe means by which these noise sources can be largely circumvented through appropriate choice of the writing-beam intensities or the recording geometry or the use of random-phase screens.

© 1997 Optical Society of America

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References

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  1. J. Heanue, M. Bashaw, and L. Hesselink, “Volume holographic storage and retrieval of digital data,” Science 265, 749–752 (1994).
    [CrossRef] [PubMed]
  2. J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: techniques and architectures,” Opt. Eng. 34, 2193–2203 (1995).
    [CrossRef]
  3. D. Psaltis and F. Mok, “Holographic memories,” Scientific American, 1995, pp. 70–77.
    [CrossRef]
  4. G. W. Burr, F. H. Mok, and D. Psaltis, “Storage of 10, 000 holograms in LiNbO3:Fe,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CMB7.
  5. K. Blotekjaer, “Limitations on the holographic storage capacity of photochromic and photorefractive media,” Appl. Opt. 18, 57–67 (1979).
    [CrossRef]
  6. D. Psaltis, D. Brady, and K. Wagner, “Adaptive optical neural computers,” Appl. Opt. 27, 1752–1759 (1988).
  7. F. Vachss and P. Yeh, “Image degradation mechanisms in photorefractive amplifiers,” J. Opt. Soc. Am. B 6, 1834–1844 (1989).
    [CrossRef]
  8. J. Goodman, Statistical Optics (Wiley, New York, 1985).

1995 (1)

J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: techniques and architectures,” Opt. Eng. 34, 2193–2203 (1995).
[CrossRef]

1994 (1)

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

1989 (1)

1988 (1)

1979 (1)

Bashaw, M.

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

Blotekjaer, K.

Brady, D.

Chang, T. Y.

J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: techniques and architectures,” Opt. Eng. 34, 2193–2203 (1995).
[CrossRef]

Christian, W.

J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: techniques and architectures,” Opt. Eng. 34, 2193–2203 (1995).
[CrossRef]

Heanue, J.

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

Hesselink, L.

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

Hong, J. H.

J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: techniques and architectures,” Opt. Eng. 34, 2193–2203 (1995).
[CrossRef]

McMichael, I.

J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: techniques and architectures,” Opt. Eng. 34, 2193–2203 (1995).
[CrossRef]

Paek, E. G.

J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: techniques and architectures,” Opt. Eng. 34, 2193–2203 (1995).
[CrossRef]

Psaltis, D.

Vachss, F.

Wagner, K.

Yeh, P.

Appl. Opt. (2)

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

Opt. Eng. (1)

J. H. Hong, I. McMichael, T. Y. Chang, W. Christian, and E. G. Paek, “Volume holographic memory systems: techniques and architectures,” Opt. Eng. 34, 2193–2203 (1995).
[CrossRef]

Science (1)

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

Other (3)

D. Psaltis and F. Mok, “Holographic memories,” Scientific American, 1995, pp. 70–77.
[CrossRef]

G. W. Burr, F. H. Mok, and D. Psaltis, “Storage of 10, 000 holograms in LiNbO3:Fe,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CMB7.

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

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Figures (7)

Fig. 1
Fig. 1

Model hologram recording geometry showing two-dimensional pattern propagated through lens into thick recording medium.

Fig. 2
Fig. 2

Numerical simulation of hologram recording and reconstruction. The input image is numerically propagated to various planes in the recording medium. There the nonuniform intensity effects are modeled by application of the appropriate nonlinear filter function to each propagated image plane. Each filtered plane is then propagated to a common output plane and summed to yield the output amplitude.

Fig. 3
Fig. 3

Input and output error images. The random pattern on the top is processed through the numerical recording and reconstruction model. The output intensity is thresholded to yield a new pattern of 1's and 0's. The image on the bottom is the difference between this thresholded output and the input image. Light squares thus represent error pixels in the reconstruction.

Fig. 4
Fig. 4

Intensity histograms for the hologram 0 to 10 mm from the focal plane. Shown in each plot are histograms for the intensities of the pixels corresponding to input 1's and 0's. Plots are shown for various values of R, the ratio of the reference to input image intensity, and for exposure-schedule parameters of C=1 and C=100, corresponding to the long- and short-exposure regimes, respectively. The vertical dotted or dashed line in each plot is the mean intensity of the reconstructed image before thresholding. Portions of the curve representing the input 0's to the right of this mean thus represent false 1's in the reconstruction. Similarly, portions of the curve representing the 1's that lie to the left of the mean represent false 0's in the output.

Fig. 5
Fig. 5

Intensity histograms for the hologram 5–15 mm from the focal plane. Histograms for the input 0's and 1's are shown along with the output mean. All input parameters are the same as in the corresponding plots in Fig. 4 except for the increased distance from the focal plane. Note the significantly increased separation of the peaks compared with the results in Fig. 4, corresponding to a dramatic reduction in BER.

Fig. 6
Fig. 6

Intensity histograms for a Fourier plane hologram: effects of phase masks. Shown here are histograms for the input 1's and 0's as in Figs. 4 and 5. Here, however, the recording region is taken to be 1 mm thick, centered on the focal plane, and the beam-intensity ratio is unity in each case. Shown are cases with and without a random binary phase mask superimposed on the input image for the exposure schedule parameters of C=1 and C=100. For both exposure parameters the curves representing the input 1's and 0's are nearly indistinguishable in the output when input phase masks are absent, yielding BER rates approaching 50%. Conversely, in both cases when such phase masks are employed the curves are well separated and the error rate greatly reduced.

Fig. 7
Fig. 7

Diffraction efficiency behavior versus BER. Shown here are plots of both average diffraction efficiency (solid curves) and Q calculated from Eq. (13) (dashed lines) versus the exposure-schedule parameter, C, when the BER is held constant. The three plots represent a recording medium 0 to 10 mm from the focal plane with a BER of 0.0002 (one error pixel), 5 to 15 mm from the focal plane with a BER of 0.0002, and 5 to 15 mm from the focal plane with a BER of 0.05, respectively. In each plot the value of C yielding maximum diffraction efficiency calculated from relation (14c) is marked with an arrow. Note that in all cases Q is nearly constant and the location of the efficiency maximum is close to the theoretical prediction.

Equations (29)

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I(r)=I0(r){1+m(r)cos[(kin-kref)·r+ϕ(r)]},
I0(r)=|Ein(r)|2+|Eref(r)|2,
m(r)=2|Ein(r)Eref(r)|I0(r),
Eout(r)=A×1-exp-I0(r)tF0×Ein(r)1+|Ein(r)|2/|Eref(r)|2,
Eout(r)=AI0(r)tF0×Ein(r)1+|Ein(r)|2|Eref(r)|2=AtF0Ein(r)|Eref(r)|2.
Δn(r)=Δn0(r)exp-jIjτjF0Δn0(r)exp-FF0,
Ein(x, y, 0)=Ein+ΔE(x, y, 0)=0.5+ΔE(x, y, 0),
Ein(x, y, z)=0.5×U(x, y, z)+ΔE(x, y, z),
Iin(x, y, z)=|Ein(x, y, z)|2=0.25×|U(x, y, z)|2+|ΔE(x, y, z)|2+Re[U*(x, y, z)ΔE(x, y, z)].
F=j=1NIj(x, y, z)τj=j=1Nτj 1+|U(x, y, z)|24,
σF2=j=1NIj(x, y, z)τj2-F2=j=1Nτj2 1+2|U(x, y, z)|216.
Eout(r)=A×Ein(r)×1-exp[-I0(r)τ0/F0]1+|Ein(r)|2/|Eref(r)|2×exp-F+jN|Erefj(r)|2τjF0,
Eout(r)=A×Ein(r)×1-exp[-I0(r)τ0/F0]1+|Ein(r)|2/|Eref|2×exp-F+|Eref|2jNτjF0.
Eout(r)=A×Ein(r)×1-exp[-I0(r)τ0/F0]1+|Ein(r)|2/|Eref|2×exp-(I¯in+|Eref|2)jNτjF0,
I¯in(r)=1+|U(x, y, z)|24,
σ2I¯=j=1Nτj2j=1Nτj2(1+2|U(x, y, z)|2)16.
p(I¯in)=const.×exp-(I¯in-I¯in(r))22σI2forI¯in0.
τj=F0I¯in+|Eref|2lnj+Cj+C-1,
C=1-exp-I¯in+|Eref|2F0τ0-1=1-exp-0.5+|Eref|2F0τ0-1,
j=1Nτj=F00.5+|Eref|2ln1+NC,
j=1Nτj2F00.5+|Eref|221C-1N+C.
Eout(r)=A×Ein(r)×1-exp[-I0(r)τ0/F0]1+|Ein(r)|2/|Eref|2×exp-I¯in+|Eref|20.5+|Eref|2ln1+NC=A×Ein(r)×1-exp-[(|Ein(r)|2+|Eref|2)/(0.5+|Eref|2)] ln[C/(C-1)]1+|Ein(r)|2/|Eref|2×exp-I¯in+|Eref|20.5+|Eref|2ln1+NC,
Eout(r)A×Ein(r)|Eref|2C(0.5+|Eref|2)×exp-(I¯in+|Eref|2)ln(1+N/C)0.5+|Eref|2.
ln(1+N/C)0.5+|Eref|2=Q(BER),
η¯0.5|A|2|Eref|2(N+C)2(0.5+|Eref|2)2.
η¯|A|2Q2N2×(N/C)2[ln(1+N/C)-Q/2](1+N/C)2 ln2(1+N/C).
η¯opt|A|2N2Q4.98 exp(-Q/2)+4Q,
ln1+NCQ+1.26 exp(-Q/3).
I¯in(r)=12,σ2I¯=j=1Nτj24j=1Nτj2.

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