Abstract

The diffraction efficiency of holographically recorded volume gratings was extensively studied, and it can be accurately predicted as long as the recording wave fronts are simple. The derivation of the diffraction efficiency when complicated wavefronts or images are involved is much more tedious and less explored. In this work we derive operator expressions that can be used to analyze these processes regardless of the shape of the wavefront and the nature of the optical systems through which they propagate. The compact expressions derived are directly applicable to the analysis of volume holographic processes, and the deterioration of the holographic reconstruction quality is derived as a function of the deviations from the recording parameters. The generalized results obtained reduce to the conventional Bragg effect for plane wave recording and reconstruction. Previously unexplored phenomena are discussed and demonstrated through some simple, and practically useful paradigms, including hologram recording and reconstruction in the Fresnel, Fourier transform, and image plane regions, as well as recording with plane and spherical waves. Some prior experimental results are also interpreted mathematically. In subsequent publications the analysis will be explored further to facilitate its application to more complicated architectures.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Solymar, D. J. Cook, Volume Holography and Volume Gratings (Academic, New York, 1981).
  2. R. R. A. Syms, Practical Volume Holography (Clarendon, Oxford, 1990).
  3. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  4. P. C. Sun, K. Oba, Y. T. Mazurenko, S. Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
    [CrossRef]
  5. G. Kriehn, A. Kiruluta, P. E. X. Silveira, S. Weaver, S. Kraut, K. Wagner, R. T. Weverka, L. Griffiths, “Optical BEAMTAP beam-forming and jammer-nulling system for broadband phased-array antennas,” Appl. Opt. 39, 212–230 (2000).
    [CrossRef]
  6. W. L. Wilson, K. Curtis, M. Tackitt, A. Hill, A. Hale, M. Schilling, C. Boyd, S. Campbell, L. Dhar, A. Harris, “High density, high performance optical data storage via volume holography: Viability at last?,” Opt. Quantum Electron. 32, 393–404 (2000).
    [CrossRef]
  7. J. Hong, “Applications of photorefractive crystals for optical neural networks,” Opt. Quantum Electron. 25, S551–S568 (1993).
    [CrossRef]
  8. M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
    [CrossRef]
  9. J. Shamir, Optical Systems and Processes (SPIE Press, Bellingham, Wash., 1999).
  10. M. Nazarathy, J. Shamir, “Wavelength Variation in Fourier Optics and Holography Described by Operator Algebra,” Isr. J. Tech. 18, 224–231 (1980).
  11. M. Nazarathy, J. Shamir, “Holography described by operator algebra,” J. Opt. Soc. Am. 71, 529–535 (1981).
    [CrossRef]
  12. X. M. Yi, P. Yeh, C. Gu, S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE 87, 1912–1930 (1999).
    [CrossRef]
  13. J. D. Kim, S. Lee, H. S. Lee, B. Lee, “Cross talk in holographic memories with lensless phaseconjugate holograms,” J. Opt. Soc. Am. A 17, 2056–2060 (2000).
    [CrossRef]
  14. D. Brady, D. Psaltis, “Control of volume holograms,” J. Opt. Soc. Am. A 9, 1167–1182 (1992).
    [CrossRef]
  15. K. Wagner, D. Psaltis, “Multilayer optical learning networks,” Appl. Opt. 26, 5061–5076 (1987).
    [CrossRef] [PubMed]

2000 (3)

1999 (2)

P. C. Sun, K. Oba, Y. T. Mazurenko, S. Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

X. M. Yi, P. Yeh, C. Gu, S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE 87, 1912–1930 (1999).
[CrossRef]

1993 (1)

J. Hong, “Applications of photorefractive crystals for optical neural networks,” Opt. Quantum Electron. 25, S551–S568 (1993).
[CrossRef]

1992 (1)

1987 (1)

1981 (1)

1980 (2)

M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
[CrossRef]

M. Nazarathy, J. Shamir, “Wavelength Variation in Fourier Optics and Holography Described by Operator Algebra,” Isr. J. Tech. 18, 224–231 (1980).

Boyd, C.

W. L. Wilson, K. Curtis, M. Tackitt, A. Hill, A. Hale, M. Schilling, C. Boyd, S. Campbell, L. Dhar, A. Harris, “High density, high performance optical data storage via volume holography: Viability at last?,” Opt. Quantum Electron. 32, 393–404 (2000).
[CrossRef]

Brady, D.

Campbell, S.

W. L. Wilson, K. Curtis, M. Tackitt, A. Hill, A. Hale, M. Schilling, C. Boyd, S. Campbell, L. Dhar, A. Harris, “High density, high performance optical data storage via volume holography: Viability at last?,” Opt. Quantum Electron. 32, 393–404 (2000).
[CrossRef]

X. M. Yi, P. Yeh, C. Gu, S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE 87, 1912–1930 (1999).
[CrossRef]

Cook, D. J.

L. Solymar, D. J. Cook, Volume Holography and Volume Gratings (Academic, New York, 1981).

Curtis, K.

W. L. Wilson, K. Curtis, M. Tackitt, A. Hill, A. Hale, M. Schilling, C. Boyd, S. Campbell, L. Dhar, A. Harris, “High density, high performance optical data storage via volume holography: Viability at last?,” Opt. Quantum Electron. 32, 393–404 (2000).
[CrossRef]

Dhar, L.

W. L. Wilson, K. Curtis, M. Tackitt, A. Hill, A. Hale, M. Schilling, C. Boyd, S. Campbell, L. Dhar, A. Harris, “High density, high performance optical data storage via volume holography: Viability at last?,” Opt. Quantum Electron. 32, 393–404 (2000).
[CrossRef]

Fainman, S. Y.

P. C. Sun, K. Oba, Y. T. Mazurenko, S. Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Griffiths, L.

Gu, C.

X. M. Yi, P. Yeh, C. Gu, S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE 87, 1912–1930 (1999).
[CrossRef]

Hale, A.

W. L. Wilson, K. Curtis, M. Tackitt, A. Hill, A. Hale, M. Schilling, C. Boyd, S. Campbell, L. Dhar, A. Harris, “High density, high performance optical data storage via volume holography: Viability at last?,” Opt. Quantum Electron. 32, 393–404 (2000).
[CrossRef]

Harris, A.

W. L. Wilson, K. Curtis, M. Tackitt, A. Hill, A. Hale, M. Schilling, C. Boyd, S. Campbell, L. Dhar, A. Harris, “High density, high performance optical data storage via volume holography: Viability at last?,” Opt. Quantum Electron. 32, 393–404 (2000).
[CrossRef]

Hill, A.

W. L. Wilson, K. Curtis, M. Tackitt, A. Hill, A. Hale, M. Schilling, C. Boyd, S. Campbell, L. Dhar, A. Harris, “High density, high performance optical data storage via volume holography: Viability at last?,” Opt. Quantum Electron. 32, 393–404 (2000).
[CrossRef]

Hong, J.

J. Hong, “Applications of photorefractive crystals for optical neural networks,” Opt. Quantum Electron. 25, S551–S568 (1993).
[CrossRef]

Kim, J. D.

Kiruluta, A.

Kraut, S.

Kriehn, G.

Lee, B.

Lee, H. S.

Lee, S.

Mazurenko, Y. T.

P. C. Sun, K. Oba, Y. T. Mazurenko, S. Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

Nazarathy, M.

Oba, K.

P. C. Sun, K. Oba, Y. T. Mazurenko, S. Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

Psaltis, D.

Schilling, M.

W. L. Wilson, K. Curtis, M. Tackitt, A. Hill, A. Hale, M. Schilling, C. Boyd, S. Campbell, L. Dhar, A. Harris, “High density, high performance optical data storage via volume holography: Viability at last?,” Opt. Quantum Electron. 32, 393–404 (2000).
[CrossRef]

Shamir, J.

M. Nazarathy, J. Shamir, “Holography described by operator algebra,” J. Opt. Soc. Am. 71, 529–535 (1981).
[CrossRef]

M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
[CrossRef]

M. Nazarathy, J. Shamir, “Wavelength Variation in Fourier Optics and Holography Described by Operator Algebra,” Isr. J. Tech. 18, 224–231 (1980).

J. Shamir, Optical Systems and Processes (SPIE Press, Bellingham, Wash., 1999).

Silveira, P. E. X.

Solymar, L.

L. Solymar, D. J. Cook, Volume Holography and Volume Gratings (Academic, New York, 1981).

Sun, P. C.

P. C. Sun, K. Oba, Y. T. Mazurenko, S. Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

Syms, R. R. A.

R. R. A. Syms, Practical Volume Holography (Clarendon, Oxford, 1990).

Tackitt, M.

W. L. Wilson, K. Curtis, M. Tackitt, A. Hill, A. Hale, M. Schilling, C. Boyd, S. Campbell, L. Dhar, A. Harris, “High density, high performance optical data storage via volume holography: Viability at last?,” Opt. Quantum Electron. 32, 393–404 (2000).
[CrossRef]

Wagner, K.

Weaver, S.

Weverka, R. T.

Wilson, W. L.

W. L. Wilson, K. Curtis, M. Tackitt, A. Hill, A. Hale, M. Schilling, C. Boyd, S. Campbell, L. Dhar, A. Harris, “High density, high performance optical data storage via volume holography: Viability at last?,” Opt. Quantum Electron. 32, 393–404 (2000).
[CrossRef]

Yeh, P.

X. M. Yi, P. Yeh, C. Gu, S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE 87, 1912–1930 (1999).
[CrossRef]

Yi, X. M.

X. M. Yi, P. Yeh, C. Gu, S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE 87, 1912–1930 (1999).
[CrossRef]

Appl. Opt. (2)

Isr. J. Tech. (1)

M. Nazarathy, J. Shamir, “Wavelength Variation in Fourier Optics and Holography Described by Operator Algebra,” Isr. J. Tech. 18, 224–231 (1980).

J. Opt. Soc. Am. (2)

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

Opt. Quantum Electron. (2)

W. L. Wilson, K. Curtis, M. Tackitt, A. Hill, A. Hale, M. Schilling, C. Boyd, S. Campbell, L. Dhar, A. Harris, “High density, high performance optical data storage via volume holography: Viability at last?,” Opt. Quantum Electron. 32, 393–404 (2000).
[CrossRef]

J. Hong, “Applications of photorefractive crystals for optical neural networks,” Opt. Quantum Electron. 25, S551–S568 (1993).
[CrossRef]

Proc. IEEE (2)

X. M. Yi, P. Yeh, C. Gu, S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE 87, 1912–1930 (1999).
[CrossRef]

P. C. Sun, K. Oba, Y. T. Mazurenko, S. Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

Other (4)

J. Shamir, Optical Systems and Processes (SPIE Press, Bellingham, Wash., 1999).

L. Solymar, D. J. Cook, Volume Holography and Volume Gratings (Academic, New York, 1981).

R. R. A. Syms, Practical Volume Holography (Clarendon, Oxford, 1990).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Schematic diagram of the optical layout. The propagation through the subsystems, OS1, OS2, and OS3 between the respective input planes (output plane for OS3) and z = 0, is described by respective linear operators, T1, T2, and T3.

Fig. 2
Fig. 2

Recording configuration with two spherical waves. One wave is from a point source diverging from A and the other one is a spherical wave converging toward point B positioned symmetrically with A.

Fig. 3
Fig. 3

Numerical evaluation of the predicted diffracted intensity profile in the configuration of Fig. 2 involving off-axis converging and diverging spherical waves. The three-dimensional plot shows the intensity along x, the Bragg plane, for displacements, s x , along x of the readout point source. The expected sinc decrease in diffraction efficiency is clearly visible in the projected total power plot at the front. Sinc sidelobes are also visible along x at the output due to an assumed rectangular transverse aperture.

Fig. 4
Fig. 4

For the same hologram as Fig. 3, the three-dimensional plot shows the intensity profile of the output spot along the out-of-plane, y direction as a function of the reconstructing source displacement in the y direction, s y , with s x = 0. The 3D plot is highlighted by lines representing cross sections of the spot for selected displacements. The reconstructed output falls off much slower than for in-plane displacements and, therefore, the widening of the spot due to the motion blur can be clearly observed. The total power is projected onto the front edge and it shows good agreement with the experimental results presented in Fig. 9 of Ref. 14.

Equations (76)

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

Eout=TEin.
Ga=expjka·ρ,
ρ=xxˆ+yyˆ; ρ=|ρ|,
T1z=zG-xˆ sin θT1 T2z=zGxˆ sin θT2.
Ix, y, t; z  T1zErx, y, t+T2zEox, y, t2.
Hx, y, t; z=TT1zErx, y, t×T2zEox, y, t,
Tft=-t ht-tftdt,
dHax, y, t; z=βHx, y, t; zdz,
dHpx, y, t; z=expjαkHx, y, t; zdz1+jαkHx, y, t; zdz,
dHx, y, t; z=κHx, y, t; zdz,
T3z=T3G-xˆ sin θ-z.
dEdx, y, t; z=T3zdHx, y, t; zux, y, t; z
T1rz=zG-xˆ sin θ+αT1r,
ux, y, t; z=T1rzErx, y, t
dEdx, y, t; z=T3zTT1zErx, y, t*×T2zEox, y, tdz×T1rzErx, y, t
dEdx, y, t; z=TT3zT2zEox, y, t×T1zErx, y, t*×T1rzErx, y, tdz.
dEdx, y, t; z=TT3G-xˆ sin θ-z×zGxˆ sin θT2Eox, y, t×zG-xˆ sin θT1Erx, y, t*×zG-xˆ sin θ+α×T1rErx, y, tdz.
dGm=Qm-dGmSmdd,
Qd=expjkd2x2+y2,
Smfx, y=fx-mx, y-my,
dEdx, y, t; z=TT3G-xˆ sin θ-z×Qxˆ sin θ-zGxˆ sin θSxˆz sin θ×zT2Eox, y, tQ-xˆ sin θ-z×G-xˆ sin θS-xˆz sin θ×zT1Erx, y, t*×Qα-xˆ sin θ-zG-xˆ sin θ+α×S-zxˆ sin θ+α×zT1rErx, y, tdz.
QaQb=Qa+b; GaGb=Ga+b.
dEdx, y, t; z=Qα-xˆ sin θ-zTT3G-xˆ sin θ-z×Gxˆ sin θ+αSxˆz sin θ×zT2Eox, y, t×S-xˆz sin θzT1Erx, y, t*×S-zxˆ sin θ+αz×T1rErx, y, tdz.
Ssa=aSs
dEdx, y, t; z=Qα-xˆ sin θ-zQα+xˆ sin θz×TT3Gα-z×S-zxˆ sin θ+αSxˆz sin θ×zT2Eox, y, t×S-xˆz sin θzT1Erx, y, t*×S-zxˆ sin θ+αz×T1rErx, y, tdz,
Sm1Sm2=Sm1+m2
dEdx, y, t; z=expj2kzαx sin θTT3Gα-z×S-zαzT2Eox, y, t×S-z2xˆ sin θ+αz×T1Erx, y, t*S-z2xˆ sin θ+αzT1rEr(x, y, t)dz.
z=expjkz-1Q-λ2z=expjkzQ-λ2z-1,
fx, y= fx, yexp-2πjux+vydxdy,
z=expjkzjkzQ1zV1λzQ1z,
Vafx, y=fax, ay.
Edx, y, t=z1-l/2z1+l/2expj2kzαx sin θTT3Gα×-1Qλ2z S-zα-1Q-λ2z×T2Eox, y, tS-z2xˆ sin θ+α-1Q-λ2zT1Erx, y, t*×S-z2xˆ sin θ+α-1Q-λ2z×T1rEr(x, y, t)dz,
liml0 Edx, y, t=CTT3GαT2Eox, y, t×T1Erx, y, t*T1rErx, y, t,
T1Erx, y, t=T1rErx, y, t=T2Eox, y, t=1x, y,
μ=2αxλsin θ,
z1-l/2z1+l/2expj2πμzdz=l expj2πμz1sincμl.
T1Erx, y, t=T1rErx, y, t=1x, y,
dEdx, y, t; z=expj2πμzT3GαS-zα×T2Eox, ydz,
ab=a+b,
blur=±αl2,
α1lλsin θ=1α1=λl sin θ
blur1=λ2 sin θ.
T1=T1r=L1; T2=L2; T3=-L2,
dEdx, y, t; z=T-L2-zL2+z×Eox, y, tS-xˆ2z sin θ×L1+zErx, y, t*×L1+zErx, y, tdz.
dEdx, y, t; z=1λ2L1+zL2+z2TQ-1L2+zV-1λL2+zQ-1L2+z×Q1L2+zV1λL2+zQ1L2+zEox, y, t×S-xˆ2z sin θQ1L1+zV1λL1+zQ1L1+zErx, y, t*×Q1L1+zV1λL1+zQ1L+zErx, y, tdz.
dEdx, y, t; z=1λ2L1+zL2+z2TQ-1L2+zV-1λL2+z×V1λL2+zQ1L2+zEox, y, t×S-xˆ2z sin θV1λL1+zQ1L1+zErx, y, t*×V1λL1+zQ1L1+zErx, y, tdz.
VaVb=Vab,
SsVb=VbSbs.
=V-1; -1=V-1
Vb=1|b|2V1b,
dEdx, y, t; z=1λL1+z2TQ-1L2+z-1×Q1L2+zEox, y, t×VL2+zL1+zS-xˆ2z sin θλL1+z×Q1L1+zErx, y, t*×Q1L1+zErx, y, tdz.
dEdx, y, t; z=TQ-1L+zQ1L+z×Eox, y, t * Gxˆ 2z sin θL+z×Q1L+zErx, y, t**Q1L+zErx, y, tdz,
Sm=G-λm
SsQa=QsaG-asQaSs,
dEdx, y, t; z=T  expjkxx-x+yy-yL+z×Eox-x, y-y, t×expjkx2z sin θL+zEr*x-x, y-y, tErx, y, tdxdxdydydz.
Eox, y=Erx, y=δx, y
T1=L; T2=-L; L=L/cos θ,
Erx, y=Ssδx, y
dEdx, y, t; z=Q1L-z-1Q-1L-z×δx, yV-L-zL+z×S-xˆ2z sin θλL+zQ1L+z×δx, y*Q1L+z×Ssδx, ydz.
Qδx, y=1x, y,
dEdx, y, t; z=Q1L-zV-L+zL-z×Gxˆ2z sin θL+zQ1L+z×Ssδx, ydz.
VbGm=GmbVb,
VaQb=Qa2bVa,
VbSm=SmbVb,
QaQb=Qa+b,
dEdx, y, t; z=Q2LL-z2G-xˆ2z sin θL-z×S-sL-zL+zδx, ydz.
dEdx, y; z=expjkLL+z2sx2+sy2×exp-jksx2z sin θL+z×δx-sxL-zL+z, y-syL-zL+zdz.
Ti=V1λfi; i=1, 2, 3.
dEdx, y, t; z=TV1λf3-zzV1λf2×Eox, y, tS-xˆ2z sin θ×zV1λf1Erx, y, t*×zV1λf1Erx, y, tdz.
dEdx, y, t; z=TV1λf3-1Qλ2z×Q-λ2z-1V1λf2×Eox, y, tS-xˆ2z sin θ×Q-λ2z-1V1λf1×Erx, y, t*×Q-λ2z-1V1λf1×Erx, y, tdz.
dEdx, y, t; z=TQzf32Q-zf32Vf2f3×Eox, y, tS-xˆ2zλf3sin θ×Q-zf32Vf1f3Erx, y, t*×Q-zf32Vf1f3Erx, y, tdz.
dEdx, y, t; z=TQzf32Q-zf32V-f2f3×Eox, y, t * Gxˆ2zf3sin θ×Q-zf32Vf1f3Erx, y, t**Q-zf32V-f1f3Erx, y, tdz,
dEdx, y, t; z=T  expjkxx+yyzf32×Eo-f2f3x-x, -f2f3y-y, t×expjkx2zf3sin θexp-jkxx+yyzf32Er*f1f3x-x, f1f3×y-y, tEr-f1f3x, - f1f3y, t×dxdxdydydz,
η1λf3xx-x+yy-yf3+2x sin θ,
Edx, y, t=Tz1-l/2z1+l/2  expj2πηz×Eo-f2f3x-x, -f2f3y-y, t×Er*f1f3x-x, f1f3y-y, t×Er-f1f3x, -f1f3y, tdxdxdydydz.
Edx, y, t=T  expj2πηz1sincηl×Eo-f2f3x-x, -f2f3y-y, t×Er*f1f3x-x, f1f3y-y, t×Er-f1f3x, -f1f3y, tdxdxdydy.

Metrics