Abstract

A simple method for simulating two-dimensional (2-D) distributions of the diffraction lights of a volume-holographic-type VanderLugt correlator is proposed and demonstrated. The simulation results are similar to those of the corresponding experiments, and only a few dozen points are sampled from both the input and the reading patterns. We show that the shifting tolerance to the reading pattern is a result of Bragg degeneracy and is not isotropic. The Bragg degeneracy generates different degrees of cross talk between the horizontal and the vertical directions. With the method we further simulate the 2-D shifting tolerance of the volume-holographic correlator. The simulation results offer a clear picture of the diffraction in a transmission volume hologram used as an optical spatial correlator.

© 1999 Optical Society of America

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  1. J.-P. Huignard, P. Gunter, Photorefractive Materials and Their Applications: I. Fundamental Phenomena (Springer-Verlag, New York, 1988).
  2. J.-P. Huignard, P. Gunter, Photorefractive Materials and Their Applications: II. Applications (Springer-Verlag, New York, 1989).
  3. J. O. White, A. Yariv, “Real-time image processing via four-wave mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5–7 (1980).
    [CrossRef]
  4. J. Yu, F. Mok, D. Psaltis, “Capacity of optical correlators,” in Spatial Light Modulators and Applications II, U. Efron, ed., Proc. SPIE825, 128–135 (1987).
    [CrossRef]
  5. A. E. T. Chiou, P. Yeh, “Symmetry filters using optical correlation and convolution,” Appl. Opt. 29, 1065–1072 (1990).
  6. C. C. Sun, M. W. Chang, S. Yeh, N. Cheng, “Computer-aided photorefractive pattern recognition,” in Optical Information Processing Systems and Architectures III, B. Javidi, ed., Proc. SPIE1564, 199–210 (1991).
    [CrossRef]
  7. C. Gu, J. Hong, S. Campbell, “2-D shift-invariant volume holographic correlator,” Opt. Commun. 88, 309–314 (1991).
    [CrossRef]
  8. K. Y. Hsu, S. Lin, T. Hsieh, L. J. Hu, T. S. Yeh, H. P. Lin, C. H. Lin, S. L. Tu, S. J. Yang, S. E. Hsu, “Photorefractive crystals for real time image processing,” in Photorefractive Materials, Effects, and Applications, Vol. 48 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 204–220.
  9. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  10. A. Chiou, “Optical interconnection method for neural networks using self-pumped phase-conjugate mirrors,” Opt. Lett. 17, 15–17 (1991).
  11. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).
  12. 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]
  13. A. B. VanderLugt, “Signal detection by complex spatial filter,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

1992 (1)

1991 (2)

A. Chiou, “Optical interconnection method for neural networks using self-pumped phase-conjugate mirrors,” Opt. Lett. 17, 15–17 (1991).

C. Gu, J. Hong, S. Campbell, “2-D shift-invariant volume holographic correlator,” Opt. Commun. 88, 309–314 (1991).
[CrossRef]

1990 (1)

A. E. T. Chiou, P. Yeh, “Symmetry filters using optical correlation and convolution,” Appl. Opt. 29, 1065–1072 (1990).

1980 (1)

J. O. White, A. Yariv, “Real-time image processing via four-wave mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5–7 (1980).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

1964 (1)

A. B. VanderLugt, “Signal detection by complex spatial filter,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Campbell, S.

C. Gu, J. Hong, S. Campbell, “2-D shift-invariant volume holographic correlator,” Opt. Commun. 88, 309–314 (1991).
[CrossRef]

Chang, M. W.

C. C. Sun, M. W. Chang, S. Yeh, N. Cheng, “Computer-aided photorefractive pattern recognition,” in Optical Information Processing Systems and Architectures III, B. Javidi, ed., Proc. SPIE1564, 199–210 (1991).
[CrossRef]

Cheng, N.

C. C. Sun, M. W. Chang, S. Yeh, N. Cheng, “Computer-aided photorefractive pattern recognition,” in Optical Information Processing Systems and Architectures III, B. Javidi, ed., Proc. SPIE1564, 199–210 (1991).
[CrossRef]

Chiou, A.

A. Chiou, “Optical interconnection method for neural networks using self-pumped phase-conjugate mirrors,” Opt. Lett. 17, 15–17 (1991).

Chiou, A. E. T.

A. E. T. Chiou, P. Yeh, “Symmetry filters using optical correlation and convolution,” Appl. Opt. 29, 1065–1072 (1990).

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, S. Campbell, “2-D shift-invariant volume holographic correlator,” Opt. Commun. 88, 309–314 (1991).
[CrossRef]

Gunter, P.

J.-P. Huignard, P. Gunter, Photorefractive Materials and Their Applications: I. Fundamental Phenomena (Springer-Verlag, New York, 1988).

J.-P. Huignard, P. Gunter, Photorefractive Materials and Their Applications: II. Applications (Springer-Verlag, New York, 1989).

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, S. Campbell, “2-D shift-invariant volume holographic correlator,” Opt. Commun. 88, 309–314 (1991).
[CrossRef]

Hsieh, T.

K. Y. Hsu, S. Lin, T. Hsieh, L. J. Hu, T. S. Yeh, H. P. Lin, C. H. Lin, S. L. Tu, S. J. Yang, S. E. Hsu, “Photorefractive crystals for real time image processing,” in Photorefractive Materials, Effects, and Applications, Vol. 48 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 204–220.

Hsu, K. Y.

K. Y. Hsu, S. Lin, T. Hsieh, L. J. Hu, T. S. Yeh, H. P. Lin, C. H. Lin, S. L. Tu, S. J. Yang, S. E. Hsu, “Photorefractive crystals for real time image processing,” in Photorefractive Materials, Effects, and Applications, Vol. 48 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 204–220.

Hsu, S. E.

K. Y. Hsu, S. Lin, T. Hsieh, L. J. Hu, T. S. Yeh, H. P. Lin, C. H. Lin, S. L. Tu, S. J. Yang, S. E. Hsu, “Photorefractive crystals for real time image processing,” in Photorefractive Materials, Effects, and Applications, Vol. 48 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 204–220.

Hu, L. J.

K. Y. Hsu, S. Lin, T. Hsieh, L. J. Hu, T. S. Yeh, H. P. Lin, C. H. Lin, S. L. Tu, S. J. Yang, S. E. Hsu, “Photorefractive crystals for real time image processing,” in Photorefractive Materials, Effects, and Applications, Vol. 48 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 204–220.

Huignard, J.-P.

J.-P. Huignard, P. Gunter, Photorefractive Materials and Their Applications: II. Applications (Springer-Verlag, New York, 1989).

J.-P. Huignard, P. Gunter, Photorefractive Materials and Their Applications: I. Fundamental Phenomena (Springer-Verlag, New York, 1988).

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Lin, C. H.

K. Y. Hsu, S. Lin, T. Hsieh, L. J. Hu, T. S. Yeh, H. P. Lin, C. H. Lin, S. L. Tu, S. J. Yang, S. E. Hsu, “Photorefractive crystals for real time image processing,” in Photorefractive Materials, Effects, and Applications, Vol. 48 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 204–220.

Lin, H. P.

K. Y. Hsu, S. Lin, T. Hsieh, L. J. Hu, T. S. Yeh, H. P. Lin, C. H. Lin, S. L. Tu, S. J. Yang, S. E. Hsu, “Photorefractive crystals for real time image processing,” in Photorefractive Materials, Effects, and Applications, Vol. 48 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 204–220.

Lin, S.

K. Y. Hsu, S. Lin, T. Hsieh, L. J. Hu, T. S. Yeh, H. P. Lin, C. H. Lin, S. L. Tu, S. J. Yang, S. E. Hsu, “Photorefractive crystals for real time image processing,” in Photorefractive Materials, Effects, and Applications, Vol. 48 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 204–220.

McMichael, I.

Mok, F.

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]

J. Yu, F. Mok, D. Psaltis, “Capacity of optical correlators,” in Spatial Light Modulators and Applications II, U. Efron, ed., Proc. SPIE825, 128–135 (1987).
[CrossRef]

Psaltis, D.

J. Yu, F. Mok, D. Psaltis, “Capacity of optical correlators,” in Spatial Light Modulators and Applications II, U. Efron, ed., Proc. SPIE825, 128–135 (1987).
[CrossRef]

Saxena, R.

Sun, C. C.

C. C. Sun, M. W. Chang, S. Yeh, N. Cheng, “Computer-aided photorefractive pattern recognition,” in Optical Information Processing Systems and Architectures III, B. Javidi, ed., Proc. SPIE1564, 199–210 (1991).
[CrossRef]

Tu, S. L.

K. Y. Hsu, S. Lin, T. Hsieh, L. J. Hu, T. S. Yeh, H. P. Lin, C. H. Lin, S. L. Tu, S. J. Yang, S. E. Hsu, “Photorefractive crystals for real time image processing,” in Photorefractive Materials, Effects, and Applications, Vol. 48 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 204–220.

VanderLugt, A. B.

A. B. VanderLugt, “Signal detection by complex spatial filter,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

White, J. O.

J. O. White, A. Yariv, “Real-time image processing via four-wave mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5–7 (1980).
[CrossRef]

Yang, S. J.

K. Y. Hsu, S. Lin, T. Hsieh, L. J. Hu, T. S. Yeh, H. P. Lin, C. H. Lin, S. L. Tu, S. J. Yang, S. E. Hsu, “Photorefractive crystals for real time image processing,” in Photorefractive Materials, Effects, and Applications, Vol. 48 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 204–220.

Yariv, A.

J. O. White, A. Yariv, “Real-time image processing via four-wave mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5–7 (1980).
[CrossRef]

Yeh, P.

A. E. T. Chiou, P. Yeh, “Symmetry filters using optical correlation and convolution,” Appl. Opt. 29, 1065–1072 (1990).

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

Yeh, S.

C. C. Sun, M. W. Chang, S. Yeh, N. Cheng, “Computer-aided photorefractive pattern recognition,” in Optical Information Processing Systems and Architectures III, B. Javidi, ed., Proc. SPIE1564, 199–210 (1991).
[CrossRef]

Yeh, T. S.

K. Y. Hsu, S. Lin, T. Hsieh, L. J. Hu, T. S. Yeh, H. P. Lin, C. H. Lin, S. L. Tu, S. J. Yang, S. E. Hsu, “Photorefractive crystals for real time image processing,” in Photorefractive Materials, Effects, and Applications, Vol. 48 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 204–220.

Yu, J.

J. Yu, F. Mok, D. Psaltis, “Capacity of optical correlators,” in Spatial Light Modulators and Applications II, U. Efron, ed., Proc. SPIE825, 128–135 (1987).
[CrossRef]

Appl. Opt. (1)

A. E. T. Chiou, P. Yeh, “Symmetry filters using optical correlation and convolution,” Appl. Opt. 29, 1065–1072 (1990).

Appl. Phys. Lett. (1)

J. O. White, A. Yariv, “Real-time image processing via four-wave mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5–7 (1980).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

IEEE Trans. Inf. Theory (1)

A. B. VanderLugt, “Signal detection by complex spatial filter,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

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

Opt. Commun. (1)

C. Gu, J. Hong, S. Campbell, “2-D shift-invariant volume holographic correlator,” Opt. Commun. 88, 309–314 (1991).
[CrossRef]

Opt. Lett. (1)

A. Chiou, “Optical interconnection method for neural networks using self-pumped phase-conjugate mirrors,” Opt. Lett. 17, 15–17 (1991).

Other (6)

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

K. Y. Hsu, S. Lin, T. Hsieh, L. J. Hu, T. S. Yeh, H. P. Lin, C. H. Lin, S. L. Tu, S. J. Yang, S. E. Hsu, “Photorefractive crystals for real time image processing,” in Photorefractive Materials, Effects, and Applications, Vol. 48 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 204–220.

C. C. Sun, M. W. Chang, S. Yeh, N. Cheng, “Computer-aided photorefractive pattern recognition,” in Optical Information Processing Systems and Architectures III, B. Javidi, ed., Proc. SPIE1564, 199–210 (1991).
[CrossRef]

J. Yu, F. Mok, D. Psaltis, “Capacity of optical correlators,” in Spatial Light Modulators and Applications II, U. Efron, ed., Proc. SPIE825, 128–135 (1987).
[CrossRef]

J.-P. Huignard, P. Gunter, Photorefractive Materials and Their Applications: I. Fundamental Phenomena (Springer-Verlag, New York, 1988).

J.-P. Huignard, P. Gunter, Photorefractive Materials and Their Applications: II. Applications (Springer-Verlag, New York, 1989).

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Figures (13)

Fig. 1
Fig. 1

(a) Incident condition for a transmission volume hologram. (b) Bragg condition in the k space of a volume hologram. (c) Bragg-mismatch condition.

Fig. 2
Fig. 2

Schematic diagram of the Bragg degeneracy associated with the grating vector K g .

Fig. 3
Fig. 3

VanderLugt correlator. FT lens, Fourier-transform lens.

Fig. 4
Fig. 4

Input (reading) pattern is sampled with an elementary plane wave.

Fig. 5
Fig. 5

Coordinate system in k space. Coordinate system 1 is transformed into coordinate system 5 for calculation of the BMF.

Fig. 6
Fig. 6

Calculation results for the diffraction of a grating written by two plane waves.

Fig. 7
Fig. 7

Δα and Δβ are the angular deviations from the Bragg angle in the horizontal and the vertical directions.

Fig. 8
Fig. 8

Input (reading pattern is a rectangle with angular spreading 0.4°.

Fig. 9
Fig. 9

Simulations of the diffraction patterns: (a) Δα = 0°, Δβ = 0°, Δh = 10; (b) Δα = 0.2°, Δβ = 0°, Δh = 5; (c) Δα = 0.4°, Δβ = 0°, Δh = 5; (d) Δα = 0°, Δβ = -1°, Δh = 5; (e) Δα = 0°, Δβ = 2.5°, Δh = 2; (f) Δα = 0°, Δβ = 5°, Δh = 1; (g) without Bragg constraint, Δα = 0°, Δβ = 0°, Δh = 2.5, where Δh is the contour interval. Both of the vertical and horizontal axes of the figures are in the unit of internal angles 0.4°/12. The central peaks corresponding to shift invariance are located at the centers of the figures. The coordinates are (a) (0, 0); (b) (6, 0); (c) (12, 0); (d) (0, -30); (e) (0, 75); (f) (0, 150); and (g) (0, 0), respectively.

Fig. 10
Fig. 10

Experimental measurements: (a) diffraction along the Bragg degeneracy, (b) diffraction in the horizontal direction.

Fig. 11
Fig. 11

Experimental setup. M, mirror; 1/2 λ, half-wave plate; PBS, polarized beam splitter; BS, beam splitter; S, shutter; FT lens, Fourier-transform lens.

Fig. 12
Fig. 12

Experimental results under the conditions corresponding to those in Fig. 9. The contrast and brightness are enhanced in (d)–(f). The vertical patterns in all figure parts expand at a 0.4° angle in the crystal.

Fig. 13
Fig. 13

Simulations of the range of shifting tolerance for crystal thickness (a) 4 mm and (b) 0.4 mm.

Equations (18)

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Kr+Kg=Kd,
ηL=11+U2/4|Γ|2sin2|Γ|2+U241/2L,
Γ=πλ cos θ Δn,
U=λ4πn cos θ|Kd-Kg|2-|Kr|2Kg·Δθ,
I=cos βI cos-θ+αI, cos βI sin-θ+αI, sin βI1,
B=cos θ, sin θ, 01,
R=cos βR cos-θ+αR, cos βR sin-θ+αR, sin βR1,
T12=cos ϕ1sin ϕ10-sin ϕ1cos ϕ10001,
ϕ1=tan-1yE1/xE1.
T23=cos ϕ20sin ϕ2010-sin ϕ20cos ϕ2,
ϕ2=tan-1zF2/xF2.
T34=1000cos ϕ3-sin ϕ30sin ϕ3cos ϕ3,
ϕ3=tan-1-zB3/yB3.
T45=cos ϕ40sin ϕ4010sin ϕ40cos ϕ4,
ϕ4=tan-1zR4/xR4.
BMF=|xG5-xD5| 2πnλ=|xG5-1-yG51/2| 2πnλ,
Idx0, y0=IriMjN ηijx, yδx-x0, y-y0,
IΔα, ΔβI0, 0=12,

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