Abstract

We investigate light diffraction in a direction perpendicular to the surface of a photorefractive crystal by anisotropic or isotropic Bragg diffraction. We estimate the resolution limit by calculating the diffraction pattern as a coherent sum of waves that are diffracted from Fourier-transformed gratings. We show that the resolution limit for anisotropic and isotropic Bragg diffraction depends only on the thickness of the crystal and the ratio of the refractive index to the wavelength of the diffracted wave. We present experimental results for a KNbO3 photorefractive spatial light modulator with a resolution of more than 30 line pairs/mm.

© 1990 Optical Society of America

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References

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  1. A. A. Kamshilin and M. P. Petrov, “Holographic image conversion in a Bi12SiO20crystal,” Sov. Tech. Phys. Lett. 6, 144 (1980).
  2. Y. Shi, D. Psaltis, A. Marrakchi, and A. R. Tanguay, “Photorefractive incoherent-to-coherent optical converter,” Appl. Opt. 22, 3665 (1983).
    [Crossref] [PubMed]
  3. E. Voit and P. Günter, “Photorefractive spatial light modulation by anisotropic self-diffraction in KNbO3crystals,” Opt. Lett. 12, 769 (1987).
    [Crossref] [PubMed]
  4. A. Marrakchi, A. R. Tanguay, J. Yu, and D. Psaltis, “Physical characterization of the photorefractive incoherent-to-coherent optical converter,” Opt. Eng. 24, 124 (1985).
    [Crossref]
  5. P. Amrhein, E. Voit, and P. Günter, “Electro-chemically reduced KNbO3-crystals for photorefractive incoherent to coherent conversion,” Proc. Soc. Photo-Opt. Instrum. Eng. 1018, 28 (1988).
  6. P. Amrhein and P. Günter, “Resolution limit for anisotropic Bragg diffraction from finite holographic gratings,” Opt. Lett. (to be published).
  7. E. Voit, C. Zaldo, and P. Günter, “Optically induced variable light deflection by anisotropic Bragg diffraction in photorefractive KNbO3,” Opt. Lett. 11, 309 (1986).
    [Crossref]
  8. S. I. Bozhelvol’nyi, “Bragg diffraction of light by a set of parallel phase gratings: analysis and applications,” Opt. Quantum Electron. 21, 397 (1989).
    [Crossref]
  9. E. Voit, “Anisotropic Bragg diffraction in photorefractive crystals,” in Electro-Optic and Photorefractive Materials. P. Günter, ed., Vol. 18 of Springer Series in Proceedings in Physics (Springer-Verlag, Berlin, 1987), p. 246.
    [Crossref]
  10. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909 (1969).
    [Crossref]

1989 (1)

S. I. Bozhelvol’nyi, “Bragg diffraction of light by a set of parallel phase gratings: analysis and applications,” Opt. Quantum Electron. 21, 397 (1989).
[Crossref]

1988 (1)

P. Amrhein, E. Voit, and P. Günter, “Electro-chemically reduced KNbO3-crystals for photorefractive incoherent to coherent conversion,” Proc. Soc. Photo-Opt. Instrum. Eng. 1018, 28 (1988).

1987 (1)

1986 (1)

1985 (1)

A. Marrakchi, A. R. Tanguay, J. Yu, and D. Psaltis, “Physical characterization of the photorefractive incoherent-to-coherent optical converter,” Opt. Eng. 24, 124 (1985).
[Crossref]

1983 (1)

1980 (1)

A. A. Kamshilin and M. P. Petrov, “Holographic image conversion in a Bi12SiO20crystal,” Sov. Tech. Phys. Lett. 6, 144 (1980).

1969 (1)

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

Amrhein, P.

P. Amrhein, E. Voit, and P. Günter, “Electro-chemically reduced KNbO3-crystals for photorefractive incoherent to coherent conversion,” Proc. Soc. Photo-Opt. Instrum. Eng. 1018, 28 (1988).

P. Amrhein and P. Günter, “Resolution limit for anisotropic Bragg diffraction from finite holographic gratings,” Opt. Lett. (to be published).

Bozhelvol’nyi, S. I.

S. I. Bozhelvol’nyi, “Bragg diffraction of light by a set of parallel phase gratings: analysis and applications,” Opt. Quantum Electron. 21, 397 (1989).
[Crossref]

Günter, P.

P. Amrhein, E. Voit, and P. Günter, “Electro-chemically reduced KNbO3-crystals for photorefractive incoherent to coherent conversion,” Proc. Soc. Photo-Opt. Instrum. Eng. 1018, 28 (1988).

E. Voit and P. Günter, “Photorefractive spatial light modulation by anisotropic self-diffraction in KNbO3crystals,” Opt. Lett. 12, 769 (1987).
[Crossref] [PubMed]

E. Voit, C. Zaldo, and P. Günter, “Optically induced variable light deflection by anisotropic Bragg diffraction in photorefractive KNbO3,” Opt. Lett. 11, 309 (1986).
[Crossref]

P. Amrhein and P. Günter, “Resolution limit for anisotropic Bragg diffraction from finite holographic gratings,” Opt. Lett. (to be published).

Kamshilin, A. A.

A. A. Kamshilin and M. P. Petrov, “Holographic image conversion in a Bi12SiO20crystal,” Sov. Tech. Phys. Lett. 6, 144 (1980).

Kogelnik, H.

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

Marrakchi, A.

A. Marrakchi, A. R. Tanguay, J. Yu, and D. Psaltis, “Physical characterization of the photorefractive incoherent-to-coherent optical converter,” Opt. Eng. 24, 124 (1985).
[Crossref]

Y. Shi, D. Psaltis, A. Marrakchi, and A. R. Tanguay, “Photorefractive incoherent-to-coherent optical converter,” Appl. Opt. 22, 3665 (1983).
[Crossref] [PubMed]

Petrov, M. P.

A. A. Kamshilin and M. P. Petrov, “Holographic image conversion in a Bi12SiO20crystal,” Sov. Tech. Phys. Lett. 6, 144 (1980).

Psaltis, D.

A. Marrakchi, A. R. Tanguay, J. Yu, and D. Psaltis, “Physical characterization of the photorefractive incoherent-to-coherent optical converter,” Opt. Eng. 24, 124 (1985).
[Crossref]

Y. Shi, D. Psaltis, A. Marrakchi, and A. R. Tanguay, “Photorefractive incoherent-to-coherent optical converter,” Appl. Opt. 22, 3665 (1983).
[Crossref] [PubMed]

Shi, Y.

Tanguay, A. R.

A. Marrakchi, A. R. Tanguay, J. Yu, and D. Psaltis, “Physical characterization of the photorefractive incoherent-to-coherent optical converter,” Opt. Eng. 24, 124 (1985).
[Crossref]

Y. Shi, D. Psaltis, A. Marrakchi, and A. R. Tanguay, “Photorefractive incoherent-to-coherent optical converter,” Appl. Opt. 22, 3665 (1983).
[Crossref] [PubMed]

Voit, E.

P. Amrhein, E. Voit, and P. Günter, “Electro-chemically reduced KNbO3-crystals for photorefractive incoherent to coherent conversion,” Proc. Soc. Photo-Opt. Instrum. Eng. 1018, 28 (1988).

E. Voit and P. Günter, “Photorefractive spatial light modulation by anisotropic self-diffraction in KNbO3crystals,” Opt. Lett. 12, 769 (1987).
[Crossref] [PubMed]

E. Voit, C. Zaldo, and P. Günter, “Optically induced variable light deflection by anisotropic Bragg diffraction in photorefractive KNbO3,” Opt. Lett. 11, 309 (1986).
[Crossref]

E. Voit, “Anisotropic Bragg diffraction in photorefractive crystals,” in Electro-Optic and Photorefractive Materials. P. Günter, ed., Vol. 18 of Springer Series in Proceedings in Physics (Springer-Verlag, Berlin, 1987), p. 246.
[Crossref]

Yu, J.

A. Marrakchi, A. R. Tanguay, J. Yu, and D. Psaltis, “Physical characterization of the photorefractive incoherent-to-coherent optical converter,” Opt. Eng. 24, 124 (1985).
[Crossref]

Zaldo, C.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

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

Opt. Eng. (1)

A. Marrakchi, A. R. Tanguay, J. Yu, and D. Psaltis, “Physical characterization of the photorefractive incoherent-to-coherent optical converter,” Opt. Eng. 24, 124 (1985).
[Crossref]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

S. I. Bozhelvol’nyi, “Bragg diffraction of light by a set of parallel phase gratings: analysis and applications,” Opt. Quantum Electron. 21, 397 (1989).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

P. Amrhein, E. Voit, and P. Günter, “Electro-chemically reduced KNbO3-crystals for photorefractive incoherent to coherent conversion,” Proc. Soc. Photo-Opt. Instrum. Eng. 1018, 28 (1988).

Sov. Tech. Phys. Lett. (1)

A. A. Kamshilin and M. P. Petrov, “Holographic image conversion in a Bi12SiO20crystal,” Sov. Tech. Phys. Lett. 6, 144 (1980).

Other (2)

E. Voit, “Anisotropic Bragg diffraction in photorefractive crystals,” in Electro-Optic and Photorefractive Materials. P. Günter, ed., Vol. 18 of Springer Series in Proceedings in Physics (Springer-Verlag, Berlin, 1987), p. 246.
[Crossref]

P. Amrhein and P. Günter, “Resolution limit for anisotropic Bragg diffraction from finite holographic gratings,” Opt. Lett. (to be published).

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Figures (9)

Fig. 1
Fig. 1

Experimental setup for spatial light modulation. Two interferring Ar+-laser beams write a phase grating in a KNbO3 crystal. A He–Ne laser beam is Bragg diffracted anisotropically from the phase grating. An incoherent beam projects an image into the crystal plate. The coherent diffracted image is contrast reversed.

Fig. 2
Fig. 2

Wave-vector diagrams for isotropic Bragg diffraction with a beam diffracted perpendicular to the crystal surfaces for hologram writing and readout. Kg is the grating vector and k1, k2, ki, and kd are the wave vectors of the two writing beams, the incident readout beam, and the diffracted beam. All polarizations are along the x axis (perpendicular to the plane of the figure).

Fig. 3
Fig. 3

Writing angles Θ1 and Θ2, measured outside the crystal, calculated as a function of readout angle ϕ.

Fig. 4
Fig. 4

Slanted phase grating of two pixels in KNbO3 crystal. The diffracted wave leaves the crystal along the y axis. d is the crystal thickness. Each pixel has a width of 2L.

Fig. 5
Fig. 5

Wave-vector diagram for Bragg diffraction with a diffracted beam along the y axis. δk is the deviation from the Bragg condition with k the grating vector and kd the wave vector of the diffracted beam.

Fig. 6
Fig. 6

Calculated, normalized intensity distribution of the diffracted beam at the exit face of the crystal as a function of position on the z axis. The two pixels are marked with thin vertical lines. A half-maximum intensity point is reached for L = 6.2 μm. [resolution R = (4L)−1 = 40 line pairs/mm]. For L < 6.2 μm (e.g., L = 5.8 μm), other half-maximum intensity points appear that do not correspond to a true image of the original picture.

Fig. 7
Fig. 7

Resolution limit R as a function of the ratio n/λ for KNbO3 for two crystal thicknesses d = 0.5 mm and d = 0.83 mm and for an assumed diffraction efficiency η = 0.01. lp is line pairs.

Fig. 8
Fig. 8

Resolution limit R as a function of crystal thickness for KNbO3 for the case of anisotropic Bragg diffraction. Parameters: n/λ = 3.6; readout wavelength, λ = 632.8 nm; diffraction efficiency, η = 0.01; lp is line pairs.

Fig. 9
Fig. 9

Contrast-reversed image with a resolution of 30 line pairs/mm. The size of the crystal shown is 6.22 × 6.24 mm2.

Equations (13)

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Δ n ( y , z ) = Δ 0 sin ( K g cos χ 0 y + K g sin χ z ) for D < y D , Δ n ( y , z ) = L < z < L otherwise ,
Δ n ( y , z ) Δ n ( k y , k z ) exp [ i ( k y y + k z z ) ] d k y d k z ,
Δ n ( k y , k z ) = 1 2 π Δ n ( y , z ) exp [ i ( k y y + k z z ) ] d y d z ,
Δ n ( k y , k z ) = 2 Δ 0 i { sin [ ( K g cos χ k y ) D ] K g cos χ k y sin [ ( K g sin χ k z ) L ] K g sin χ k z sin [ ( K g cos χ + k y ) D ] K g cos χ + k υ sin [ ( K g sin χ + k z ) L ] K g sin χ + k z } .
Δ n ( k y , k z ) = 2 Δ 0 sin ( k y D ) k y sin ( k z L ) k z .
Δ n ( k y , k z ) = 2 Δ 0 sin ( k y D ) k y 2 sin ( k z L ) cos ( 2 k z L ) k z .
Δ n ( k y , k z ) = 2 Δ 0 sin ( k y D ) k y 4 sin ( k z L ) cos ( 2 k z L ) cos ( 4 k z L ) k z .
E ( k y , k z ) = ν sin [ ( ν 2 + ξ 2 ) 1 / 2 ] ( ν 2 + ξ 2 ) 1 / 2 exp ( i ξ ) ,
δ k ( k y , k z ) = k d [ k z 2 + ( k y k d ) 2 ] 1 / 2 ,
E ( k y , k z ) = ν sin [ ( ν 0 2 + ξ 2 ) 1 / 2 ] ν 0 2 + ξ 2 exp ( i ξ ) ,
ϕ ( k y , k z , z ) = k d ( d cos Θ + z sin Θ d ) , tan Θ = k z k y + k d .
E diff ( z ) = E ( k y , k z ) exp [ i ϕ ( k y , k z , z ) ] d k y , d k z .
R = 1 4 L ,

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