Abstract

We propose and demonstrate a new photorefractive real-time holographic deconvolution technique for adaptive one-way image transmission through aberrating media by means of four-wave mixing. In contrast with earlier methods, which typically required various codings of the exact phase or two-way image transmission for correcting phase distortion, our technique relies on one-way image transmission through the use of exact phase information. Our technique can simultaneously correct both amplitude and phase distortions. We include several forms of image degradation, various test cases, and experimental results. We characterize the performance as a function of the input beam ratios for four metrics: signal-to-noise ratio, normalized root-mean-square error, edge restoration, and peak-to-total energy ratio. In our characterization we use false-color graphic images to display the best beam-intensity ratio two-dimensional region(s) for each of these metrics. Test cases are simulated at the optimal values of the beam-intensity ratios. We demonstrate our results through both experiment and computer simulation.

© 2006 Optical Society of America

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  1. F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, 1999).
    [CrossRef]
  2. V. P. Pauca, B. L. Elerbroek, R. J. Plemmons, and X. Sun, "Structure matrix representations of two parameter Hankle transforms in adaptive optics," Linear Algebr. Appl. 316, 29-43 (2000).
    [CrossRef]
  3. J. Khoury, J. Fu, and C. L. Woods, "Phase-coding technique for one-way image transmission through aberrating media," Opt. Lett. 19, 1645-1647 (1994).
    [CrossRef] [PubMed]
  4. J. Khoury, J. Fu, and C. L. Woods, "Phase coding technique for signal recovery from distortion," Opt. Eng. 34, 840-848 (1995).
    [CrossRef]
  5. D. M. Pepper and A. Yariv, "Compensation for phase distortions in nonlinear media by phase conjugation," Opt. Lett. 5, 59-60 (1980).
    [CrossRef] [PubMed]
  6. G. R. Ayers and J. C. Dainty, "Interative blind deconvolution method and its applications," Opt. Lett. 13, 547-549 (1988).
    [CrossRef] [PubMed]
  7. D. M. Pepper, J. AuYeung, D. Fekete, and A. Yariv, "Spatial convolution and correlation of optical fields via degenerate four-wave mixing," Opt. Lett. 3, 7-9 (1978).
    [CrossRef] [PubMed]
  8. J. O. White and A. Yariv, "Real-time image processing via four-wave mixing in photorefractive materials," Appl. Phys. Lett. 37, 5-7 (1980).
    [CrossRef]
  9. E. Ochoa, L. Hesselink, and J. W. Goodman, "Real-time intensity inversion using two-wave and four-wave mixing in photorefractive Bi12GeO20," Appl. Opt. 24, 1826-1832 (1985).
    [CrossRef] [PubMed]
  10. Y. H. Ja, "Real-time image division in four-wave mixing with photorefractive BGO crystals," Opt. Commun. 44, 24-26 (1982).
    [CrossRef]
  11. Y. H. Ja, "Real-time image deblurring using four-wave mixing," Opt. Quantum Electron. 15, 457-459 (1983).
    [CrossRef]
  12. J. A. Khoury and R. W. Eason, "Photorefractive deconvolution techniques for optical differentiation of images," J. Mod. Opt. 36, 369-376 (1989).
    [CrossRef]
  13. C. De Matos, A. Le Corre, H. L. Haridon, S. Gosselin, and B. Lambert, "Fe-doped InGaAs/InGaAsP photorefractive multiple quantum well devices operating at 1.55 μm," Appl. Phys. Lett. 70, 3591-3593 (1997).
    [CrossRef]
  14. A. Yariv, Optical Electronics in Modern Communications, Oxford Series in Electrical and Computer Engineering (1997).
  15. T. Weyrauch and M. A. Vorontsov, "Mitigation of atmospheric-turbulence-induced phase aberrations over a 2.3 km near-horizontal propagation path with a dual-conjugate adaptive optics system," in Proceedings of the 2004 AMOS Technical Conference (2004), pp. 672-681.
  16. L. Beresnev and M. Vorontsov, "Atmospheric turbulence wave-front phase aberration sensing with differential Zernike filter," in Proceedings of the 2004 AMOS Technical Conference (2004), pp. 289-298.
  17. T. Weyrauch, M. A. Vorontsov, L. Beresnev, and L. Liu, "Atmospheric compensation over a 2.3 km propagation path with a multi-conjugate (piston-MEMS/modal DM) adaptive system," in Target-in-the-Loop: Atmospheric Tracking, Imaging, and Compensation, M.T.Valley and M.A.Vorontsov, eds., Proc. SPIE 5552,73-84 (2004).

2000

V. P. Pauca, B. L. Elerbroek, R. J. Plemmons, and X. Sun, "Structure matrix representations of two parameter Hankle transforms in adaptive optics," Linear Algebr. Appl. 316, 29-43 (2000).
[CrossRef]

1997

C. De Matos, A. Le Corre, H. L. Haridon, S. Gosselin, and B. Lambert, "Fe-doped InGaAs/InGaAsP photorefractive multiple quantum well devices operating at 1.55 μm," Appl. Phys. Lett. 70, 3591-3593 (1997).
[CrossRef]

1995

J. Khoury, J. Fu, and C. L. Woods, "Phase coding technique for signal recovery from distortion," Opt. Eng. 34, 840-848 (1995).
[CrossRef]

1994

1989

J. A. Khoury and R. W. Eason, "Photorefractive deconvolution techniques for optical differentiation of images," J. Mod. Opt. 36, 369-376 (1989).
[CrossRef]

1988

1985

1983

Y. H. Ja, "Real-time image deblurring using four-wave mixing," Opt. Quantum Electron. 15, 457-459 (1983).
[CrossRef]

1982

Y. H. Ja, "Real-time image division in four-wave mixing with photorefractive BGO crystals," Opt. Commun. 44, 24-26 (1982).
[CrossRef]

1980

D. M. Pepper and A. Yariv, "Compensation for phase distortions in nonlinear media by phase conjugation," Opt. Lett. 5, 59-60 (1980).
[CrossRef] [PubMed]

J. O. White and A. Yariv, "Real-time image processing via four-wave mixing in photorefractive materials," Appl. Phys. Lett. 37, 5-7 (1980).
[CrossRef]

1978

AuYeung, J.

Ayers, G. R.

Beresnev, L.

L. Beresnev and M. Vorontsov, "Atmospheric turbulence wave-front phase aberration sensing with differential Zernike filter," in Proceedings of the 2004 AMOS Technical Conference (2004), pp. 289-298.

T. Weyrauch, M. A. Vorontsov, L. Beresnev, and L. Liu, "Atmospheric compensation over a 2.3 km propagation path with a multi-conjugate (piston-MEMS/modal DM) adaptive system," in Target-in-the-Loop: Atmospheric Tracking, Imaging, and Compensation, M.T.Valley and M.A.Vorontsov, eds., Proc. SPIE 5552,73-84 (2004).

Dainty, J. C.

De Matos, C.

C. De Matos, A. Le Corre, H. L. Haridon, S. Gosselin, and B. Lambert, "Fe-doped InGaAs/InGaAsP photorefractive multiple quantum well devices operating at 1.55 μm," Appl. Phys. Lett. 70, 3591-3593 (1997).
[CrossRef]

Eason, R. W.

J. A. Khoury and R. W. Eason, "Photorefractive deconvolution techniques for optical differentiation of images," J. Mod. Opt. 36, 369-376 (1989).
[CrossRef]

Elerbroek, B. L.

V. P. Pauca, B. L. Elerbroek, R. J. Plemmons, and X. Sun, "Structure matrix representations of two parameter Hankle transforms in adaptive optics," Linear Algebr. Appl. 316, 29-43 (2000).
[CrossRef]

Fekete, D.

Fu, J.

J. Khoury, J. Fu, and C. L. Woods, "Phase coding technique for signal recovery from distortion," Opt. Eng. 34, 840-848 (1995).
[CrossRef]

J. Khoury, J. Fu, and C. L. Woods, "Phase-coding technique for one-way image transmission through aberrating media," Opt. Lett. 19, 1645-1647 (1994).
[CrossRef] [PubMed]

Goodman, J. W.

Gosselin, S.

C. De Matos, A. Le Corre, H. L. Haridon, S. Gosselin, and B. Lambert, "Fe-doped InGaAs/InGaAsP photorefractive multiple quantum well devices operating at 1.55 μm," Appl. Phys. Lett. 70, 3591-3593 (1997).
[CrossRef]

Haridon, H. L.

C. De Matos, A. Le Corre, H. L. Haridon, S. Gosselin, and B. Lambert, "Fe-doped InGaAs/InGaAsP photorefractive multiple quantum well devices operating at 1.55 μm," Appl. Phys. Lett. 70, 3591-3593 (1997).
[CrossRef]

Hesselink, L.

Ja, Y. H.

Y. H. Ja, "Real-time image deblurring using four-wave mixing," Opt. Quantum Electron. 15, 457-459 (1983).
[CrossRef]

Y. H. Ja, "Real-time image division in four-wave mixing with photorefractive BGO crystals," Opt. Commun. 44, 24-26 (1982).
[CrossRef]

Khoury, J.

J. Khoury, J. Fu, and C. L. Woods, "Phase coding technique for signal recovery from distortion," Opt. Eng. 34, 840-848 (1995).
[CrossRef]

J. Khoury, J. Fu, and C. L. Woods, "Phase-coding technique for one-way image transmission through aberrating media," Opt. Lett. 19, 1645-1647 (1994).
[CrossRef] [PubMed]

Khoury, J. A.

J. A. Khoury and R. W. Eason, "Photorefractive deconvolution techniques for optical differentiation of images," J. Mod. Opt. 36, 369-376 (1989).
[CrossRef]

Lambert, B.

C. De Matos, A. Le Corre, H. L. Haridon, S. Gosselin, and B. Lambert, "Fe-doped InGaAs/InGaAsP photorefractive multiple quantum well devices operating at 1.55 μm," Appl. Phys. Lett. 70, 3591-3593 (1997).
[CrossRef]

Le Corre, A.

C. De Matos, A. Le Corre, H. L. Haridon, S. Gosselin, and B. Lambert, "Fe-doped InGaAs/InGaAsP photorefractive multiple quantum well devices operating at 1.55 μm," Appl. Phys. Lett. 70, 3591-3593 (1997).
[CrossRef]

Liu, L.

T. Weyrauch, M. A. Vorontsov, L. Beresnev, and L. Liu, "Atmospheric compensation over a 2.3 km propagation path with a multi-conjugate (piston-MEMS/modal DM) adaptive system," in Target-in-the-Loop: Atmospheric Tracking, Imaging, and Compensation, M.T.Valley and M.A.Vorontsov, eds., Proc. SPIE 5552,73-84 (2004).

Ochoa, E.

Pauca, V. P.

V. P. Pauca, B. L. Elerbroek, R. J. Plemmons, and X. Sun, "Structure matrix representations of two parameter Hankle transforms in adaptive optics," Linear Algebr. Appl. 316, 29-43 (2000).
[CrossRef]

Pepper, D. M.

Plemmons, R. J.

V. P. Pauca, B. L. Elerbroek, R. J. Plemmons, and X. Sun, "Structure matrix representations of two parameter Hankle transforms in adaptive optics," Linear Algebr. Appl. 316, 29-43 (2000).
[CrossRef]

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, 1999).
[CrossRef]

Sun, X.

V. P. Pauca, B. L. Elerbroek, R. J. Plemmons, and X. Sun, "Structure matrix representations of two parameter Hankle transforms in adaptive optics," Linear Algebr. Appl. 316, 29-43 (2000).
[CrossRef]

Vorontsov, M.

L. Beresnev and M. Vorontsov, "Atmospheric turbulence wave-front phase aberration sensing with differential Zernike filter," in Proceedings of the 2004 AMOS Technical Conference (2004), pp. 289-298.

Vorontsov, M. A.

T. Weyrauch, M. A. Vorontsov, L. Beresnev, and L. Liu, "Atmospheric compensation over a 2.3 km propagation path with a multi-conjugate (piston-MEMS/modal DM) adaptive system," in Target-in-the-Loop: Atmospheric Tracking, Imaging, and Compensation, M.T.Valley and M.A.Vorontsov, eds., Proc. SPIE 5552,73-84 (2004).

T. Weyrauch and M. A. Vorontsov, "Mitigation of atmospheric-turbulence-induced phase aberrations over a 2.3 km near-horizontal propagation path with a dual-conjugate adaptive optics system," in Proceedings of the 2004 AMOS Technical Conference (2004), pp. 672-681.

Weyrauch, T.

T. Weyrauch and M. A. Vorontsov, "Mitigation of atmospheric-turbulence-induced phase aberrations over a 2.3 km near-horizontal propagation path with a dual-conjugate adaptive optics system," in Proceedings of the 2004 AMOS Technical Conference (2004), pp. 672-681.

T. Weyrauch, M. A. Vorontsov, L. Beresnev, and L. Liu, "Atmospheric compensation over a 2.3 km propagation path with a multi-conjugate (piston-MEMS/modal DM) adaptive system," in Target-in-the-Loop: Atmospheric Tracking, Imaging, and Compensation, M.T.Valley and M.A.Vorontsov, eds., Proc. SPIE 5552,73-84 (2004).

White, J. O.

J. O. White and A. Yariv, "Real-time image processing via four-wave mixing in photorefractive materials," Appl. Phys. Lett. 37, 5-7 (1980).
[CrossRef]

Woods, C. L.

J. Khoury, J. Fu, and C. L. Woods, "Phase coding technique for signal recovery from distortion," Opt. Eng. 34, 840-848 (1995).
[CrossRef]

J. Khoury, J. Fu, and C. L. Woods, "Phase-coding technique for one-way image transmission through aberrating media," Opt. Lett. 19, 1645-1647 (1994).
[CrossRef] [PubMed]

Yariv, A.

D. M. Pepper and A. Yariv, "Compensation for phase distortions in nonlinear media by phase conjugation," Opt. Lett. 5, 59-60 (1980).
[CrossRef] [PubMed]

J. O. White and A. Yariv, "Real-time image processing via four-wave mixing in photorefractive materials," Appl. Phys. Lett. 37, 5-7 (1980).
[CrossRef]

D. M. Pepper, J. AuYeung, D. Fekete, and A. Yariv, "Spatial convolution and correlation of optical fields via degenerate four-wave mixing," Opt. Lett. 3, 7-9 (1978).
[CrossRef] [PubMed]

A. Yariv, Optical Electronics in Modern Communications, Oxford Series in Electrical and Computer Engineering (1997).

Appl. Opt.

Appl. Phys. Lett.

C. De Matos, A. Le Corre, H. L. Haridon, S. Gosselin, and B. Lambert, "Fe-doped InGaAs/InGaAsP photorefractive multiple quantum well devices operating at 1.55 μm," Appl. Phys. Lett. 70, 3591-3593 (1997).
[CrossRef]

J. O. White and A. Yariv, "Real-time image processing via four-wave mixing in photorefractive materials," Appl. Phys. Lett. 37, 5-7 (1980).
[CrossRef]

J. Mod. Opt.

J. A. Khoury and R. W. Eason, "Photorefractive deconvolution techniques for optical differentiation of images," J. Mod. Opt. 36, 369-376 (1989).
[CrossRef]

Linear Algebr. Appl.

V. P. Pauca, B. L. Elerbroek, R. J. Plemmons, and X. Sun, "Structure matrix representations of two parameter Hankle transforms in adaptive optics," Linear Algebr. Appl. 316, 29-43 (2000).
[CrossRef]

Opt. Commun.

Y. H. Ja, "Real-time image division in four-wave mixing with photorefractive BGO crystals," Opt. Commun. 44, 24-26 (1982).
[CrossRef]

Opt. Eng.

J. Khoury, J. Fu, and C. L. Woods, "Phase coding technique for signal recovery from distortion," Opt. Eng. 34, 840-848 (1995).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

Y. H. Ja, "Real-time image deblurring using four-wave mixing," Opt. Quantum Electron. 15, 457-459 (1983).
[CrossRef]

Other

A. Yariv, Optical Electronics in Modern Communications, Oxford Series in Electrical and Computer Engineering (1997).

T. Weyrauch and M. A. Vorontsov, "Mitigation of atmospheric-turbulence-induced phase aberrations over a 2.3 km near-horizontal propagation path with a dual-conjugate adaptive optics system," in Proceedings of the 2004 AMOS Technical Conference (2004), pp. 672-681.

L. Beresnev and M. Vorontsov, "Atmospheric turbulence wave-front phase aberration sensing with differential Zernike filter," in Proceedings of the 2004 AMOS Technical Conference (2004), pp. 289-298.

T. Weyrauch, M. A. Vorontsov, L. Beresnev, and L. Liu, "Atmospheric compensation over a 2.3 km propagation path with a multi-conjugate (piston-MEMS/modal DM) adaptive system," in Target-in-the-Loop: Atmospheric Tracking, Imaging, and Compensation, M.T.Valley and M.A.Vorontsov, eds., Proc. SPIE 5552,73-84 (2004).

F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, 1999).
[CrossRef]

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Figures (16)

Fig. 1
Fig. 1

Four-wave mixing in a photorefractive medium.

Fig. 2
Fig. 2

Aberration correction by four-wave mixing.

Fig. 3
Fig. 3

Schematic diagram of the experimental arrangement.

Fig. 4
Fig. 4

(Color online) Experimental results: A, input; B, blurred input; C, Image corrected by a photorefractive crystal.

Fig. 5
Fig. 5

Test inputs binary: A, point source (single pixel in the center); B, three-bar pattern; C, 2D comb function. Gray scale: D, human face.

Fig. 6
Fig. 6

Aberrators: A, motion (50 × 2 pixel rectangular binary blur); B, phase only (128 × 128 pixel uniformly distributed random phase aberrator); C, misfocusing (circular blur with a radius of 5 pixels); D, combination of the three.

Fig. 7
Fig. 7

Point source image restoration tests: A, blurred point source with rectangular aberrator plus noise; B, corrected image; C, difference between original and corrected images; A′, blurred point source with phase-only aberrator plus noise; B′, corrected image; C′, difference between original and corrected images; A″, blurred point source with circular blur plus noise; B″, corrected image; C″, difference between original and corrected images.

Fig. 8
Fig. 8

Bar image restoration tests: A, three-bar pattern with motion blur plus noise; B, corrected image; C, difference between original and corrected images; A′, blurred three-bar pattern with phase-only aberrator plus noise; B′, corrected image; C′, difference between original and corrected images; A″, blurred three-bar pattern with circular blur plus noise; B″, corrected image; C″, difference between original and corrected images.

Fig. 9
Fig. 9

Image restorations obtained for 2D comb function input.

Fig. 10
Fig. 10

Image restorations obtained for gray-scale human face input.

Fig. 11
Fig. 11

Color-coded metric performance plotted versus beam intensity ratios m and p (for a circular aberrator) for the x axis (10−10 to 1010) and the y axis (1010 to 10−10), respectively. Top, performance of SNR for three noise levels: A, σ = 1; A′, σ = 10; A″ σ = 25. Middle, NMSE for B, σ = 1; B′, σ = 10; B″, σ = 25. Bottom, ER for C, σ = 1; C′, σ = 10; C″, σ = 25.

Fig. 12
Fig. 12

Performance of PTE for three noise levels: A, σ = 1; B, σ = 10; C, σ = 25.

Fig. 13
Fig. 13

Corrections obtained for three-bar input with a circular aberrator at three beam ratios in our optimal region.

Fig. 14
Fig. 14

Color-coded metric performance plotted versus beam intensity ratios m and p for rectangular aberration.

Fig. 15
Fig. 15

Performance of PTE for three noise levels A, σ = 1; B, σ = 10; C, σ = 25.

Fig. 16
Fig. 16

Corrections obtained for three-bar input with a rectangular aberrator at three beam ratios in our optimal region.

Equations (8)

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

A 4 = ( A 1 A 3 * ) A 2 I 1 + I 3 + I 2 ,
A 4 ( v x , v y ) = [ ( A 1 / λ f ) H ( v x , v y ) ( A 3 * / λ f ) H * ( v x , v y ) F * ( v x , v y ) ] A 2 / λ f I 1 | H ( v x , v y ) | 2 / ( λ f ) 2 + I 3 | H ( v x , v y ) | 2 | F ( v x , v y ) | 2 / ( λ f ) 2 + I 2 / ( λ f ) 2 ,
A 4 = m F * ( v x , v y ) H * ( v x , v y ) H ( v x , v y ) m | H ( v x , v y ) | 2 + m p + | F ( v x , v y ) | 2 | H ( v x , v y ) | 2 × ( A 2 / λ f ) .
A 4 ( v x , v y ) = α m F * ( v x , v y ) m p ( A 2 / λ f ) .
A 4 ( v x , v y ) = m F * ( v x , v y ) m ( A 2 / λ f ) .
E R =
| Blurred_image_edge | 2 / | Blurred_image | 2 | Restored_image_edge | 2 / | Restored_image | 2 .
P T E = | max ( FFT_output ) | 2 | FFT_output | 2 .

Metrics