Abstract

A general theory of polarization and spatial information recovery by modal dispersal and phase conjugation is presented by means of a coherency matrix formalism. The theory is applied to a system that consists of a multimode modal-scrambling fiber terminated by a conventional phase-conjugate mirror that reflects only one polarization component. The degree of polarization and the signal-to-noise ratio of the reconstructed field are discussed as a function of input-beam launching conditions. Some experimental results are also shown for comparison with the theory.

© 1988 Optical Society of America

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  1. L. G. Cohen, “Measured attenuation and depolarization of light transmitted along glass fibers,” Bell. Syst. Tech. J. 50, 23–42 (1971).
    [CrossRef]
  2. B. Ya. Zel’dovich, V. V. Shkunov, “Reversal of the wave front of light in the case of depolarized pumping,” Sov. Phys. JETP 48, 214–219 (1978).
  3. B. Ya. Zel’dovich, V. V. Shkunov, “Spatial-polarization wavefront reversal in four-photon interaction,” Sov. J. Quantum Electron. 9, 379–381 (1979).
    [CrossRef]
  4. V. N. Blashchuk, B. Ya. Zel’dovich, A. V. Mamaev, N. T. Pilipetsky, V. V. Shkunov, “Complete wavefront reversal of depolarized radiation under degenerate four-photon interaction conditions (theory and experiment),” Sov. J. Quantum Electron. 10, 356–358 (1980).
    [CrossRef]
  5. G. Martin, L. K. Lam, R. W. Hellwarth, “Generation of time-reversed replica of a nonuniformly polarized image-bearing optical beam,” Opt. Lett. 5, 185–187 (1980).
    [CrossRef]
  6. S. Saikan, M. Kiguchi, “Generation of phase-conjugated vector wave fronts in atomic vapors,” Opt. Lett. 7, 555–557 (1982).
    [CrossRef] [PubMed]
  7. P. Yeh, “Scalar phase conjugator for polarization correction,” Opt. Commun. 51, 195–197 (1984).
    [CrossRef]
  8. I. McMichael, M. Khoshnevisan, P. Yeh, “Polarization-preserving phase conjugator,” Opt. Lett. 11, 525–527 (1986).
    [CrossRef] [PubMed]
  9. K. Kyuma, A. Yariv, S.-K. Kwong, “Polarization recovery in phase conjugation by modal dispersal,” Appl. Phys. Lett. 49, 617–619 (1986).
    [CrossRef]
  10. A. Yariv, Y. Tomita, K. Kyuma, “Theoretical model for modal dispersal of polarization information and its recovery by phase conjugation,” Opt. Lett. 11, 809–811 (1986).
    [CrossRef] [PubMed]
  11. S.-K. Kwong, R. Yahalom, K. Kyuma, A. Yariv, “Optical phase-conjugate correction for propagation distortion in nonreciprocal media,” Opt. Lett. 12, 337–339 (1987).
    [CrossRef] [PubMed]
  12. Y. Tomita, K. Kyuma, R. Yahalom, A. Yariv, “Demonstration of correction of amplitude distortion by modal dispersal and phase conjugation,” Opt. Lett. 12, 1020–1022 (1987).
    [CrossRef] [PubMed]
  13. R. Yahalom, K. Kyuma, A. Yariv, “Phase conjugation of mode scrambled optical beams: application to spatial recovery and interbeam temporal information exchange,” Appl. Phys. Lett. 50, 792–794 (1987).
    [CrossRef]
  14. R. Yahalom, A. Agranat, A. Yariv, “Optical threshold mechanism using fiber coupled phase conjugate mirror,” in Digest of Topical Meeting on Photonic Switching (Optical Society of America, Washington, D.C., 1987), paper FB3.
  15. Y. Tomita, R. Yahalom, A. Yariv, “Fidelity of polarization and spatial information recovery using a fiber-coupled phase conjugate mirror,” Opt. Lett. 12, 1017–1019 (1987).
    [CrossRef] [PubMed]
  16. I. McMichael, P. Yeh, P. Beckwith, “Correction of polarization and modal scrambling in multimode fibers by phase conjugation,” Opt. Lett. 12, 507–509 (1987).
    [CrossRef] [PubMed]
  17. K. Kim, L. Mandel, E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
    [CrossRef]
  18. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 357.
  19. J. Feinberg, “Self-pumped, continuous-wave phase conjugator using internal reflection,” Opt. Lett. 7, 486–488 (1982).
    [CrossRef] [PubMed]
  20. S. Solimento, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), pp. 569–573.
  21. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), p. 292.
  22. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), pp. 544–555.
  23. Since the input-beam N.A. ≈ ϕ/2f(f is a focal length of a lens) and M/Ntotal= (N.A./N.A.fiber)2/2 (M is the number of modes corresponding to ϕ for one polarization; Ntotal= 2N; N.A.fiberis the fiber’s N.A.), we obtain M/Ntotal= (ϕ/ϕ0)2/2. In addition, since ϕ2 is proportional to the solid angle Ω subtended by the beam launched into the fiber, then it also gives (ϕ/ϕ0)2= Ω/Ω0.
  24. J. W. Goodman, “Film-grain noise in wavefront-reconstruction imaging,” J. Opt. Soc. Am. 57, 493–502 (1967).
    [CrossRef] [PubMed]
  25. J. Ohtsubo, T. Asakura, “Statistical properties of the sum of partially developed speckle patterns,” Opt. Lett. 1, 98–100 (1977).
    [CrossRef] [PubMed]
  26. P. F. Steeger, T. Asakura, K. Zocha, A. F. Fercher, “Statistics of the Stokes parameters in speckle fields,” J. Opt. Soc. Am. A 1, 677–682 (1984).
    [CrossRef]
  27. P. F. Steeger, T. Asakura, A. F. Fercher, “Polarization preservation in circular multimode optical fibers and its measurement by a speckle method,” IEEE J. Lightwave Technol. LT-2, 435–441 (1984).
    [CrossRef]
  28. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–75.
  29. In practice, however, the use of the analyzer may improve the SNR better than that in the calculation because of the rejection of unwanted noise that is due to the backreflection from optical components.
  30. A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28, 88–89 (1976).
    [CrossRef]
  31. A. Yariv, “On transmission and recovery of three-dimensional image information in optical fibers,” J. Opt. Soc. Am. 66, 301–306 (1976).
    [CrossRef]
  32. A. Gover, C. P. Lee, A. Yariv, “Direct transmission of pictorial information in multimode optical fibers,” J. Opt. Soc. Am. 66, 306–311 (1976).
    [CrossRef]
  33. V. V. Ivakhnik, V. M. Petnikova, M. S. Solomatin, V. V. Shuvalov, “Compensation of wave front distortions in a thick inhomogeneous medium,” Sov. J. Quantum Electron. 10, 373–375 (1980).
    [CrossRef]
  34. G. J. Dunning, R. C. Lind, “Demonstration of image transmission through fibers by optical phase conjugation,” Opt. Lett. 7, 558–560 (1982).
    [CrossRef] [PubMed]
  35. B. Fisher, S. Sternklar, “Image transmission and interferometry with multimode fibers using self-pumped phase conjugation,” Appl. Phys. Lett. 46, 113–114 (1985).
    [CrossRef]
  36. B. Fisher, D. Peri, “Real-time three-dimensional imaging through fiber bundles by four-wave mixing,” Opt. Lett. 10, 182–183 (1985).
    [CrossRef]
  37. P. H. Beckwith, I. McMichael, P. Yeh, “Image distortion in multimode fibers and restoration by polarization-preserving phase conjugation,” Opt. Lett. 12, 510–512 (1987).
    [CrossRef] [PubMed]
  38. B. Fisher, S. Sternklar, S. Weiss, “Photorefractive oscillation with intracavity image and multimode fiber,” Appl. Phys. Lett. 48, 1567–1569 (1986).
    [CrossRef]
  39. S. Sternklar, S. Weiss, M. Segev, B. Fisher, “Mach–Zehnder interferometer with multimode fibers using the double phase-conjugate mirror,” Appl. Opt. 25, 4518–4520 (1986).
    [CrossRef] [PubMed]
  40. B. Fisher, S. Sternklar, “New optical gyroscope based on the ring passive phase conjugator,” Appl. Phys. Lett. 47, 1–3 (1985).
    [CrossRef]
  41. P. Yeh, I. McMichael, M. Khoshnevisan, “Phase-conjugate fiber-optic gyro,” Appl. Opt. 25, 1029–1030 (1986).
    [CrossRef] [PubMed]
  42. I. McMichael, P. Yeh, “Self-pumped phase-conjugate fiber-optic gyro,” Opt. Lett. 11, 686–688 (1986).
    [CrossRef] [PubMed]
  43. E. McMichael, P. Beckwith, P. Yeh, “Phase-conjugate multimode fiber gyro,” in Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1987), paper FA4.
  44. H. J. Caulfield, J. Shamir, Q. He, “Flexible two-way optical interconnections in layered computers,” Appl. Opt. 26, 2291–2292 (1987).
    [CrossRef] [PubMed]

1987 (8)

1986 (7)

1985 (3)

B. Fisher, S. Sternklar, “New optical gyroscope based on the ring passive phase conjugator,” Appl. Phys. Lett. 47, 1–3 (1985).
[CrossRef]

B. Fisher, S. Sternklar, “Image transmission and interferometry with multimode fibers using self-pumped phase conjugation,” Appl. Phys. Lett. 46, 113–114 (1985).
[CrossRef]

B. Fisher, D. Peri, “Real-time three-dimensional imaging through fiber bundles by four-wave mixing,” Opt. Lett. 10, 182–183 (1985).
[CrossRef]

1984 (3)

P. F. Steeger, T. Asakura, K. Zocha, A. F. Fercher, “Statistics of the Stokes parameters in speckle fields,” J. Opt. Soc. Am. A 1, 677–682 (1984).
[CrossRef]

P. Yeh, “Scalar phase conjugator for polarization correction,” Opt. Commun. 51, 195–197 (1984).
[CrossRef]

P. F. Steeger, T. Asakura, A. F. Fercher, “Polarization preservation in circular multimode optical fibers and its measurement by a speckle method,” IEEE J. Lightwave Technol. LT-2, 435–441 (1984).
[CrossRef]

1982 (3)

1980 (3)

V. V. Ivakhnik, V. M. Petnikova, M. S. Solomatin, V. V. Shuvalov, “Compensation of wave front distortions in a thick inhomogeneous medium,” Sov. J. Quantum Electron. 10, 373–375 (1980).
[CrossRef]

V. N. Blashchuk, B. Ya. Zel’dovich, A. V. Mamaev, N. T. Pilipetsky, V. V. Shkunov, “Complete wavefront reversal of depolarized radiation under degenerate four-photon interaction conditions (theory and experiment),” Sov. J. Quantum Electron. 10, 356–358 (1980).
[CrossRef]

G. Martin, L. K. Lam, R. W. Hellwarth, “Generation of time-reversed replica of a nonuniformly polarized image-bearing optical beam,” Opt. Lett. 5, 185–187 (1980).
[CrossRef]

1979 (1)

B. Ya. Zel’dovich, V. V. Shkunov, “Spatial-polarization wavefront reversal in four-photon interaction,” Sov. J. Quantum Electron. 9, 379–381 (1979).
[CrossRef]

1978 (1)

B. Ya. Zel’dovich, V. V. Shkunov, “Reversal of the wave front of light in the case of depolarized pumping,” Sov. Phys. JETP 48, 214–219 (1978).

1977 (1)

1976 (3)

1971 (1)

L. G. Cohen, “Measured attenuation and depolarization of light transmitted along glass fibers,” Bell. Syst. Tech. J. 50, 23–42 (1971).
[CrossRef]

1967 (1)

Agranat, A.

R. Yahalom, A. Agranat, A. Yariv, “Optical threshold mechanism using fiber coupled phase conjugate mirror,” in Digest of Topical Meeting on Photonic Switching (Optical Society of America, Washington, D.C., 1987), paper FB3.

Asakura, T.

Beckwith, P.

I. McMichael, P. Yeh, P. Beckwith, “Correction of polarization and modal scrambling in multimode fibers by phase conjugation,” Opt. Lett. 12, 507–509 (1987).
[CrossRef] [PubMed]

E. McMichael, P. Beckwith, P. Yeh, “Phase-conjugate multimode fiber gyro,” in Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1987), paper FA4.

Beckwith, P. H.

Blashchuk, V. N.

V. N. Blashchuk, B. Ya. Zel’dovich, A. V. Mamaev, N. T. Pilipetsky, V. V. Shkunov, “Complete wavefront reversal of depolarized radiation under degenerate four-photon interaction conditions (theory and experiment),” Sov. J. Quantum Electron. 10, 356–358 (1980).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), pp. 544–555.

Caulfield, H. J.

Cohen, L. G.

L. G. Cohen, “Measured attenuation and depolarization of light transmitted along glass fibers,” Bell. Syst. Tech. J. 50, 23–42 (1971).
[CrossRef]

Crosignani, B.

S. Solimento, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), pp. 569–573.

Di Porto, P.

S. Solimento, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), pp. 569–573.

Dunning, G. J.

Feinberg, J.

Fercher, A. F.

P. F. Steeger, T. Asakura, K. Zocha, A. F. Fercher, “Statistics of the Stokes parameters in speckle fields,” J. Opt. Soc. Am. A 1, 677–682 (1984).
[CrossRef]

P. F. Steeger, T. Asakura, A. F. Fercher, “Polarization preservation in circular multimode optical fibers and its measurement by a speckle method,” IEEE J. Lightwave Technol. LT-2, 435–441 (1984).
[CrossRef]

Fisher, B.

B. Fisher, S. Sternklar, S. Weiss, “Photorefractive oscillation with intracavity image and multimode fiber,” Appl. Phys. Lett. 48, 1567–1569 (1986).
[CrossRef]

S. Sternklar, S. Weiss, M. Segev, B. Fisher, “Mach–Zehnder interferometer with multimode fibers using the double phase-conjugate mirror,” Appl. Opt. 25, 4518–4520 (1986).
[CrossRef] [PubMed]

B. Fisher, S. Sternklar, “Image transmission and interferometry with multimode fibers using self-pumped phase conjugation,” Appl. Phys. Lett. 46, 113–114 (1985).
[CrossRef]

B. Fisher, S. Sternklar, “New optical gyroscope based on the ring passive phase conjugator,” Appl. Phys. Lett. 47, 1–3 (1985).
[CrossRef]

B. Fisher, D. Peri, “Real-time three-dimensional imaging through fiber bundles by four-wave mixing,” Opt. Lett. 10, 182–183 (1985).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Film-grain noise in wavefront-reconstruction imaging,” J. Opt. Soc. Am. 57, 493–502 (1967).
[CrossRef] [PubMed]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–75.

Gover, A.

He, Q.

Hellwarth, R. W.

Ivakhnik, V. V.

V. V. Ivakhnik, V. M. Petnikova, M. S. Solomatin, V. V. Shuvalov, “Compensation of wave front distortions in a thick inhomogeneous medium,” Sov. J. Quantum Electron. 10, 373–375 (1980).
[CrossRef]

Khoshnevisan, M.

Kiguchi, M.

Kim, K.

Kwong, S.-K.

S.-K. Kwong, R. Yahalom, K. Kyuma, A. Yariv, “Optical phase-conjugate correction for propagation distortion in nonreciprocal media,” Opt. Lett. 12, 337–339 (1987).
[CrossRef] [PubMed]

K. Kyuma, A. Yariv, S.-K. Kwong, “Polarization recovery in phase conjugation by modal dispersal,” Appl. Phys. Lett. 49, 617–619 (1986).
[CrossRef]

Kyuma, K.

Lam, L. K.

Lee, C. P.

Lind, R. C.

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), p. 292.

Mamaev, A. V.

V. N. Blashchuk, B. Ya. Zel’dovich, A. V. Mamaev, N. T. Pilipetsky, V. V. Shkunov, “Complete wavefront reversal of depolarized radiation under degenerate four-photon interaction conditions (theory and experiment),” Sov. J. Quantum Electron. 10, 356–358 (1980).
[CrossRef]

Mandel, L.

Martin, G.

McMichael, E.

E. McMichael, P. Beckwith, P. Yeh, “Phase-conjugate multimode fiber gyro,” in Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1987), paper FA4.

McMichael, I.

Ohtsubo, J.

Peri, D.

Petnikova, V. M.

V. V. Ivakhnik, V. M. Petnikova, M. S. Solomatin, V. V. Shuvalov, “Compensation of wave front distortions in a thick inhomogeneous medium,” Sov. J. Quantum Electron. 10, 373–375 (1980).
[CrossRef]

Pilipetsky, N. T.

V. N. Blashchuk, B. Ya. Zel’dovich, A. V. Mamaev, N. T. Pilipetsky, V. V. Shkunov, “Complete wavefront reversal of depolarized radiation under degenerate four-photon interaction conditions (theory and experiment),” Sov. J. Quantum Electron. 10, 356–358 (1980).
[CrossRef]

Saikan, S.

Segev, M.

Shamir, J.

Shkunov, V. V.

V. N. Blashchuk, B. Ya. Zel’dovich, A. V. Mamaev, N. T. Pilipetsky, V. V. Shkunov, “Complete wavefront reversal of depolarized radiation under degenerate four-photon interaction conditions (theory and experiment),” Sov. J. Quantum Electron. 10, 356–358 (1980).
[CrossRef]

B. Ya. Zel’dovich, V. V. Shkunov, “Spatial-polarization wavefront reversal in four-photon interaction,” Sov. J. Quantum Electron. 9, 379–381 (1979).
[CrossRef]

B. Ya. Zel’dovich, V. V. Shkunov, “Reversal of the wave front of light in the case of depolarized pumping,” Sov. Phys. JETP 48, 214–219 (1978).

Shuvalov, V. V.

V. V. Ivakhnik, V. M. Petnikova, M. S. Solomatin, V. V. Shuvalov, “Compensation of wave front distortions in a thick inhomogeneous medium,” Sov. J. Quantum Electron. 10, 373–375 (1980).
[CrossRef]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), p. 292.

Solimento, S.

S. Solimento, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), pp. 569–573.

Solomatin, M. S.

V. V. Ivakhnik, V. M. Petnikova, M. S. Solomatin, V. V. Shuvalov, “Compensation of wave front distortions in a thick inhomogeneous medium,” Sov. J. Quantum Electron. 10, 373–375 (1980).
[CrossRef]

Steeger, P. F.

P. F. Steeger, T. Asakura, A. F. Fercher, “Polarization preservation in circular multimode optical fibers and its measurement by a speckle method,” IEEE J. Lightwave Technol. LT-2, 435–441 (1984).
[CrossRef]

P. F. Steeger, T. Asakura, K. Zocha, A. F. Fercher, “Statistics of the Stokes parameters in speckle fields,” J. Opt. Soc. Am. A 1, 677–682 (1984).
[CrossRef]

Sternklar, S.

B. Fisher, S. Sternklar, S. Weiss, “Photorefractive oscillation with intracavity image and multimode fiber,” Appl. Phys. Lett. 48, 1567–1569 (1986).
[CrossRef]

S. Sternklar, S. Weiss, M. Segev, B. Fisher, “Mach–Zehnder interferometer with multimode fibers using the double phase-conjugate mirror,” Appl. Opt. 25, 4518–4520 (1986).
[CrossRef] [PubMed]

B. Fisher, S. Sternklar, “Image transmission and interferometry with multimode fibers using self-pumped phase conjugation,” Appl. Phys. Lett. 46, 113–114 (1985).
[CrossRef]

B. Fisher, S. Sternklar, “New optical gyroscope based on the ring passive phase conjugator,” Appl. Phys. Lett. 47, 1–3 (1985).
[CrossRef]

Tomita, Y.

Weiss, S.

S. Sternklar, S. Weiss, M. Segev, B. Fisher, “Mach–Zehnder interferometer with multimode fibers using the double phase-conjugate mirror,” Appl. Opt. 25, 4518–4520 (1986).
[CrossRef] [PubMed]

B. Fisher, S. Sternklar, S. Weiss, “Photorefractive oscillation with intracavity image and multimode fiber,” Appl. Phys. Lett. 48, 1567–1569 (1986).
[CrossRef]

Wolf, E.

Yahalom, R.

R. Yahalom, K. Kyuma, A. Yariv, “Phase conjugation of mode scrambled optical beams: application to spatial recovery and interbeam temporal information exchange,” Appl. Phys. Lett. 50, 792–794 (1987).
[CrossRef]

Y. Tomita, K. Kyuma, R. Yahalom, A. Yariv, “Demonstration of correction of amplitude distortion by modal dispersal and phase conjugation,” Opt. Lett. 12, 1020–1022 (1987).
[CrossRef] [PubMed]

S.-K. Kwong, R. Yahalom, K. Kyuma, A. Yariv, “Optical phase-conjugate correction for propagation distortion in nonreciprocal media,” Opt. Lett. 12, 337–339 (1987).
[CrossRef] [PubMed]

Y. Tomita, R. Yahalom, A. Yariv, “Fidelity of polarization and spatial information recovery using a fiber-coupled phase conjugate mirror,” Opt. Lett. 12, 1017–1019 (1987).
[CrossRef] [PubMed]

R. Yahalom, A. Agranat, A. Yariv, “Optical threshold mechanism using fiber coupled phase conjugate mirror,” in Digest of Topical Meeting on Photonic Switching (Optical Society of America, Washington, D.C., 1987), paper FB3.

Yariv, A.

R. Yahalom, K. Kyuma, A. Yariv, “Phase conjugation of mode scrambled optical beams: application to spatial recovery and interbeam temporal information exchange,” Appl. Phys. Lett. 50, 792–794 (1987).
[CrossRef]

Y. Tomita, R. Yahalom, A. Yariv, “Fidelity of polarization and spatial information recovery using a fiber-coupled phase conjugate mirror,” Opt. Lett. 12, 1017–1019 (1987).
[CrossRef] [PubMed]

S.-K. Kwong, R. Yahalom, K. Kyuma, A. Yariv, “Optical phase-conjugate correction for propagation distortion in nonreciprocal media,” Opt. Lett. 12, 337–339 (1987).
[CrossRef] [PubMed]

Y. Tomita, K. Kyuma, R. Yahalom, A. Yariv, “Demonstration of correction of amplitude distortion by modal dispersal and phase conjugation,” Opt. Lett. 12, 1020–1022 (1987).
[CrossRef] [PubMed]

A. Yariv, Y. Tomita, K. Kyuma, “Theoretical model for modal dispersal of polarization information and its recovery by phase conjugation,” Opt. Lett. 11, 809–811 (1986).
[CrossRef] [PubMed]

K. Kyuma, A. Yariv, S.-K. Kwong, “Polarization recovery in phase conjugation by modal dispersal,” Appl. Phys. Lett. 49, 617–619 (1986).
[CrossRef]

A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28, 88–89 (1976).
[CrossRef]

A. Yariv, “On transmission and recovery of three-dimensional image information in optical fibers,” J. Opt. Soc. Am. 66, 301–306 (1976).
[CrossRef]

A. Gover, C. P. Lee, A. Yariv, “Direct transmission of pictorial information in multimode optical fibers,” J. Opt. Soc. Am. 66, 306–311 (1976).
[CrossRef]

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 357.

R. Yahalom, A. Agranat, A. Yariv, “Optical threshold mechanism using fiber coupled phase conjugate mirror,” in Digest of Topical Meeting on Photonic Switching (Optical Society of America, Washington, D.C., 1987), paper FB3.

Yeh, P.

Zel’dovich, B. Ya.

V. N. Blashchuk, B. Ya. Zel’dovich, A. V. Mamaev, N. T. Pilipetsky, V. V. Shkunov, “Complete wavefront reversal of depolarized radiation under degenerate four-photon interaction conditions (theory and experiment),” Sov. J. Quantum Electron. 10, 356–358 (1980).
[CrossRef]

B. Ya. Zel’dovich, V. V. Shkunov, “Spatial-polarization wavefront reversal in four-photon interaction,” Sov. J. Quantum Electron. 9, 379–381 (1979).
[CrossRef]

B. Ya. Zel’dovich, V. V. Shkunov, “Reversal of the wave front of light in the case of depolarized pumping,” Sov. Phys. JETP 48, 214–219 (1978).

Zocha, K.

Appl. Opt. (3)

Appl. Phys. Lett. (6)

B. Fisher, S. Sternklar, S. Weiss, “Photorefractive oscillation with intracavity image and multimode fiber,” Appl. Phys. Lett. 48, 1567–1569 (1986).
[CrossRef]

B. Fisher, S. Sternklar, “New optical gyroscope based on the ring passive phase conjugator,” Appl. Phys. Lett. 47, 1–3 (1985).
[CrossRef]

B. Fisher, S. Sternklar, “Image transmission and interferometry with multimode fibers using self-pumped phase conjugation,” Appl. Phys. Lett. 46, 113–114 (1985).
[CrossRef]

K. Kyuma, A. Yariv, S.-K. Kwong, “Polarization recovery in phase conjugation by modal dispersal,” Appl. Phys. Lett. 49, 617–619 (1986).
[CrossRef]

R. Yahalom, K. Kyuma, A. Yariv, “Phase conjugation of mode scrambled optical beams: application to spatial recovery and interbeam temporal information exchange,” Appl. Phys. Lett. 50, 792–794 (1987).
[CrossRef]

A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28, 88–89 (1976).
[CrossRef]

Bell. Syst. Tech. J. (1)

L. G. Cohen, “Measured attenuation and depolarization of light transmitted along glass fibers,” Bell. Syst. Tech. J. 50, 23–42 (1971).
[CrossRef]

IEEE J. Lightwave Technol. (1)

P. F. Steeger, T. Asakura, A. F. Fercher, “Polarization preservation in circular multimode optical fibers and its measurement by a speckle method,” IEEE J. Lightwave Technol. LT-2, 435–441 (1984).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Commun. (1)

P. Yeh, “Scalar phase conjugator for polarization correction,” Opt. Commun. 51, 195–197 (1984).
[CrossRef]

Opt. Lett. (14)

I. McMichael, M. Khoshnevisan, P. Yeh, “Polarization-preserving phase conjugator,” Opt. Lett. 11, 525–527 (1986).
[CrossRef] [PubMed]

G. Martin, L. K. Lam, R. W. Hellwarth, “Generation of time-reversed replica of a nonuniformly polarized image-bearing optical beam,” Opt. Lett. 5, 185–187 (1980).
[CrossRef]

S. Saikan, M. Kiguchi, “Generation of phase-conjugated vector wave fronts in atomic vapors,” Opt. Lett. 7, 555–557 (1982).
[CrossRef] [PubMed]

Y. Tomita, R. Yahalom, A. Yariv, “Fidelity of polarization and spatial information recovery using a fiber-coupled phase conjugate mirror,” Opt. Lett. 12, 1017–1019 (1987).
[CrossRef] [PubMed]

I. McMichael, P. Yeh, P. Beckwith, “Correction of polarization and modal scrambling in multimode fibers by phase conjugation,” Opt. Lett. 12, 507–509 (1987).
[CrossRef] [PubMed]

A. Yariv, Y. Tomita, K. Kyuma, “Theoretical model for modal dispersal of polarization information and its recovery by phase conjugation,” Opt. Lett. 11, 809–811 (1986).
[CrossRef] [PubMed]

S.-K. Kwong, R. Yahalom, K. Kyuma, A. Yariv, “Optical phase-conjugate correction for propagation distortion in nonreciprocal media,” Opt. Lett. 12, 337–339 (1987).
[CrossRef] [PubMed]

Y. Tomita, K. Kyuma, R. Yahalom, A. Yariv, “Demonstration of correction of amplitude distortion by modal dispersal and phase conjugation,” Opt. Lett. 12, 1020–1022 (1987).
[CrossRef] [PubMed]

G. J. Dunning, R. C. Lind, “Demonstration of image transmission through fibers by optical phase conjugation,” Opt. Lett. 7, 558–560 (1982).
[CrossRef] [PubMed]

J. Ohtsubo, T. Asakura, “Statistical properties of the sum of partially developed speckle patterns,” Opt. Lett. 1, 98–100 (1977).
[CrossRef] [PubMed]

J. Feinberg, “Self-pumped, continuous-wave phase conjugator using internal reflection,” Opt. Lett. 7, 486–488 (1982).
[CrossRef] [PubMed]

B. Fisher, D. Peri, “Real-time three-dimensional imaging through fiber bundles by four-wave mixing,” Opt. Lett. 10, 182–183 (1985).
[CrossRef]

P. H. Beckwith, I. McMichael, P. Yeh, “Image distortion in multimode fibers and restoration by polarization-preserving phase conjugation,” Opt. Lett. 12, 510–512 (1987).
[CrossRef] [PubMed]

I. McMichael, P. Yeh, “Self-pumped phase-conjugate fiber-optic gyro,” Opt. Lett. 11, 686–688 (1986).
[CrossRef] [PubMed]

Sov. J. Quantum Electron. (3)

V. V. Ivakhnik, V. M. Petnikova, M. S. Solomatin, V. V. Shuvalov, “Compensation of wave front distortions in a thick inhomogeneous medium,” Sov. J. Quantum Electron. 10, 373–375 (1980).
[CrossRef]

B. Ya. Zel’dovich, V. V. Shkunov, “Spatial-polarization wavefront reversal in four-photon interaction,” Sov. J. Quantum Electron. 9, 379–381 (1979).
[CrossRef]

V. N. Blashchuk, B. Ya. Zel’dovich, A. V. Mamaev, N. T. Pilipetsky, V. V. Shkunov, “Complete wavefront reversal of depolarized radiation under degenerate four-photon interaction conditions (theory and experiment),” Sov. J. Quantum Electron. 10, 356–358 (1980).
[CrossRef]

Sov. Phys. JETP (1)

B. Ya. Zel’dovich, V. V. Shkunov, “Reversal of the wave front of light in the case of depolarized pumping,” Sov. Phys. JETP 48, 214–219 (1978).

Other (9)

R. Yahalom, A. Agranat, A. Yariv, “Optical threshold mechanism using fiber coupled phase conjugate mirror,” in Digest of Topical Meeting on Photonic Switching (Optical Society of America, Washington, D.C., 1987), paper FB3.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–75.

In practice, however, the use of the analyzer may improve the SNR better than that in the calculation because of the rejection of unwanted noise that is due to the backreflection from optical components.

S. Solimento, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, Orlando, Fla., 1986), pp. 569–573.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), p. 292.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), pp. 544–555.

Since the input-beam N.A. ≈ ϕ/2f(f is a focal length of a lens) and M/Ntotal= (N.A./N.A.fiber)2/2 (M is the number of modes corresponding to ϕ for one polarization; Ntotal= 2N; N.A.fiberis the fiber’s N.A.), we obtain M/Ntotal= (ϕ/ϕ0)2/2. In addition, since ϕ2 is proportional to the solid angle Ω subtended by the beam launched into the fiber, then it also gives (ϕ/ϕ0)2= Ω/Ω0.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), p. 357.

E. McMichael, P. Beckwith, P. Yeh, “Phase-conjugate multimode fiber gyro,” in Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1987), paper FA4.

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Figures (4)

Fig. 1
Fig. 1

Schematic of the FCPCM for polarization and spatial information recovery. The (polarization- and modal-scrambling) multimode fiber is assumed to be linear with negligible loss.

Fig. 2
Fig. 2

Diagrammatic description of the formation of the field E(4): (a) deterministic phase-conjugate paths that result in true phase conjugation of the input field E(1); (b) randomly scattered phase-conjugate paths that result in the noise.

Fig. 3
Fig. 3

Theoretical curves of (a) the phase-conjugate reflectivity R and (b) the degree of polarization P(4) as a function of (ϕ/ϕ0)2 for (q, η) = (0, 1), (0.035, 1), (0, 0.8), (0.035, 0.8). The Gaussian distribution (ψ/ϕ0 = 0.5) of the depolarized noise intensity is assumed. The experimental data of (a) R (○) and (b) p (○) and P(4) (●) are also shown.

Fig. 4
Fig. 4

Theoretical curves of the two SNR’s, (SNR)xy and (SNR)x, at the center of the signal beam as a function of (ϕ/ϕ0)2. The Gaussian distribution (ψ/ϕ0 = 0.5) of the noise intensity is assumed. The insets are photographs of the x-polarized phase-conjugate images of the letter H for (a) (ϕ/ϕ0)2 = 0.015 and (b) (ϕ/ϕ0)2 = 0.74.

Tables (1)

Tables Icon

Table 1 Experimental Data of the Stokes Parameters and the Degree of Polarization of the Field E(2)a

Equations (100)

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E ( 1 ) = n = 1 N [ a x n ( 1 ) e x n + a y n ( 1 ) e y n ] [ A x ( 1 ) A y ( 1 ) ] ,
E ( 4 ) = r M C M * [ E ( 1 ) ] * ,
M = [ M x x M x y M y x M y y ] ,
C = [ I 0 0 0 ] ,
M M = [ I 0 0 I ] ,
( M x x ) i k ( M x x ) i k * + ( M y x ) i k ( M y x ) i k * = δ k k ,
( M y y ) i k ( M y y ) i k * + ( M x y ) i k ( M x y ) i k * = δ k k ,
( M x y ) i k ( M x x ) i k * + ( M y y ) i k ( M y x ) i k * = 0 ,
( M x x ) i k ( M x y ) i k * + ( M y x ) i k ( M y y ) i k * = 0 ,
( M x x ) k i ( M x x ) k i * + ( M x y ) k i ( M x y ) k i * = δ k k ,
( M y y ) k i ( M y y ) k i * + ( M y x ) k i ( M y x ) k i * = δ k k ,
( M y x ) k i ( M x x ) k i * + ( M y y ) k i ( M x y ) k i * = 0 ,
( M x x ) k i ( M y x ) k i * + ( M x y ) k i ( M y y ) k i * = 0 ,
M M * = [ I 0 0 I ] .
M = M t ,
( M x x ) i j = ( M x x ) j i ,
( M y y ) i j = ( M y y ) j i ,
( M x y ) i j = ( M y x ) j i ,
( M y x ) i j = ( M x y ) j i .
L ( 2 ) E ( 2 ) E ( 2 ) = M E ( 1 ) E ( 1 ) M = [ L x x ( 2 ) L x y ( 2 ) L x y ( 2 ) L y y ( 2 ) ] ,
L x x ( 2 ) = M x x L x x ( 1 ) M x x + M x x L x y ( 1 ) M x y + M x y L x y ( 1 ) M x x + M x y L y y ( 1 ) M x y ,
L y y ( 2 ) = M y x L x x ( 1 ) M y x + M y x L x y ( 1 ) M y y + M y y L x y ( 1 ) M y x + M y y L y y ( 1 ) M y y ,
L x y ( 2 ) = M x x L x x ( 1 ) M y x + M x x L x y ( 1 ) M y y + M x y L x y ( 1 ) M y x + M x y L y y ( 1 ) M y y ,
J ( 2 ) [ J x x ( 2 ) J x y ( 2 ) J x y ( 2 ) * J y y ( 2 ) ] .
J i j ( 2 ) = k = 1 N l = 1 N σ [ L i j ( 2 ) ] k l e i k e * j l d x d y = ( const . ) × k = 1 N [ L i j ( 2 ) ] k k = ( const . ) × Tr [ L i j ( 2 ) ] .
[ L x x ( 2 ) ] i i = ( M x x ) i k ( M x x ) i k * [ L x x ( 1 ) ] k k = ( M x x ) i k 2 [ L x x ( 1 ) ] k k + ( M x x ) i k ( M x x ) i k * [ L x x ( 1 ) ] k k ( k k ) ,
[ L y y ( 2 ) ] i i = ( M y x ) i k 2 [ L x x ( 1 ) ] k k + ( M y x ) i k ( M y x ) i k * [ L x x ( 1 ) ] k k ( k k ) ,
[ L x y ( 2 ) ] i i = ( M x x ) i k ( M y x ) i k * [ L x x ( 1 ) ] k k .
J x x ( 2 ) = a k k [ L x x ( 1 ) ] k k + a k k [ L x x ( 1 ) ] k k ( k k ) ,
J y y ( 2 ) = ( δ k k - a k k ) [ L x x ( 1 ) ] k k - a k k [ L x x ( 1 ) ] k k ( k k ) ,
J x y ( 2 ) = b k k [ L x x ( 1 ) ] k k ,
a k k ( M x x ) i k ( M x x ) i k *
b k k ( M x x ) i k ( M y x ) i k * .
s 0 J x x ( 2 ) + J y y ( 2 ) = k = 1 N [ L x x ( 1 ) ] k k ,
s 1 J x x ( 2 ) - J y y ( 2 ) = 2 a k k [ L x x ( 1 ) ] k k + 2 a k k [ L x x ( 1 ) ] k k ( k k ) - s 0 ,
s 2 J x y ( 2 ) + J y x ( 2 ) = 2 Re { b k k [ L x x ( 1 ) ] k k } ,
s 3 i [ J y x ( 2 ) - J x y ( 2 ) ] = 2 Im { b k k [ L x x ( 1 ) ] k k } ,
P ( 2 ) = ( s 1 2 + s 2 2 + s 3 2 ) 1 / 2 s 0 .
P ( 2 ) ( q 2 + u 2 ) 1 / 2 ,
q 2 a k k [ L x x ( 1 ) ] k k ( k k ) s 0
u 2 b k k [ L x x ( 1 ) ] k k s 0 .
E ( 4 ) = r S [ E ( 1 ) ] * ,
S 1 = 1 2 [ I 0 0 I ]
S 2 = [ D Q Q D ] ,
D i j = { 0 ( i = j ) ( M x x ) k i ( M x x ) k j * ( i j ) ,
D i j = { 0 ( i = j ) - ( M y y ) k i ( M y y ) k j * ( i j ) ,
Q i j = ( M x x ) k i ( M x y ) k j * .
E ( 4 ) = ½ r [ E ( 1 ) ] * + V ,
L ( 4 ) E ( 4 ) E ( 4 ) = r 2 S L ( 1 ) * S = r 2 { ¼ L ( 1 ) * + S 2 L ( 1 ) * S 2 + ½ [ S 2 L ( 1 ) * + L ( 1 ) * S 2 ] } .
S 2 L ( 1 ) * S 2 = [ D L x x ( 1 ) * D D L x x ( 1 ) * Q Q L x x ( 1 ) * D Q L x x ( 1 ) * Q ] ,
1 2 [ S 2 L ( 1 ) * + L ( 1 ) * S 2 ] = 1 2 [ D L x x ( 1 ) * + L x x ( 1 ) * D L x x ( 1 ) * Q Q L x x ( 1 ) * 0 ] .
[ D L x x ( 1 ) * D ] i i = D i l 2 [ L x x ( 1 ) ] l l + D i l D i l * [ L x x ( 1 ) ] * l l ( l l ) ,
[ Q L x x ( 1 ) * Q ] i i = Q l i 2 [ L x x ( 1 ) ] l l + Q l i * Q l i [ L x x ( 1 ) ] * l l ( l l ) ,
[ D L x x ( 1 ) * Q ] i i = D i l Q l i [ L x x ( 1 ) ] l l * ,
[ D L x x ( 1 ) * + L x x ( 1 ) * D ] i i = 2 Re { D i l [ L x x ( 1 ) ] l i * } ,
[ L x x ( 1 ) * Q ] i i = Q l i [ L x x ( 1 ) ] i l * ,
J ( 4 ) [ J x x ( 4 ) J x y ( 4 ) J x y ( 4 ) * J y y ( 4 ) ] = ¼ r 2 J ( 1 ) * + J noise ( 4 ) ,
J ( 1 ) = [ s 0 0 0 0 ]
J nosie ( 4 ) = { Tr [ L x x ( 4 ) ] - ¼ r 2 s 0 Tr [ L x y ( 4 ) ] Tr [ L x y ( 4 ) ] Tr [ L y y ( 4 ) ] } .
[ J noise ( 4 ) ] x x = ( α 1 + ½ q s 0 ) r 2 ,
[ J noise ( 4 ) ] y y = ( ¼ s 0 - α 1 ) r 2 ,
[ J noise ( 4 ) ] x y = ( α 2 + ¼ v s 0 ) r 2 ,
α 1 = Tr [ D L x x ( 1 ) * D ] ,
α 2 = Tr [ D L x x ( 1 ) * Q ] ,
v = 2 Tr [ L x x ( 1 ) * Q ] s 0 ,
P N Tr [ J noise ( 4 ) ] = ¼ r 2 s 0 ( 1 + 2 q ) .
J noise ( 4 ) λ [ 1 + 4 q 2 v 2 v 1 ] ,
P ( 4 ) 1 + 2 q 2 ( 1 + q ) ,
R J x x ( 4 ) s 0 = r 2 ( 3 + 4 q ) ,
J noise ( 4 ) [ λ 0 0 λ ] ,
Θ i true phase - conjugate power in the i th mode 2 λ = [ L x x ( 1 ) ] i i s 0 ,
Λ i polarized noise power in the i th mode 4 q λ = Re [ D L x x ( 1 ) * ] i i ( q s 0 2 ) ,
Δ i depolarized noise power in the i th mode of each polarization λ = [ D L x x ( 1 ) * D ] i i ( s 0 8 ) [ Q L x x ( 1 ) * Q ] i i ( s 0 8 ) .
P pol . = 2 λ ( i = 1 M Θ i + 2 q i = 1 M Λ i )
P M = 2 λ i = 1 M Δ i ,
P ( 4 ) = P pol . P pol . + P M = 1 + 2 q β 1 1 + 2 q β 1 + β 2
R = 1 8 r 2 i = 1 M Θ i ( 2 + 4 q β 1 + β 2 ) ,
β 1 i = 1 M Λ i i = 1 M Θ i
β 2 i = 1 M Δ i i = 1 M Θ i ,
E ( 3 ) = r 0 E x ( 2 ) * + E w ( 3 ) = r 0 E x ( 2 ) * + [ r 1 ] E x ( 2 ) * ,
Tr [ E ( 3 ) E ( 3 ) ] = r 0 2 Tr [ E x ( 2 ) * E x ( 2 ) * ] + Tr [ E w ( 3 ) E w ( 3 ) ] + 2 Re { Tr [ r 0 E x ( 2 ) * E w ( 3 ) ] } r 2 Tr [ E x ( 2 ) * E x ( 2 ) * ] .
η r 0 2 r 2 ,
E ( 4 ) = r 0 S E ( 1 ) * + M E w ( 3 ) E t ( 4 ) + E w ( 4 ) .
J ( 4 ) = J t ( 4 ) + J w ( 4 ) η λ [ 2 + 4 q + 1 2 v 2 v 1 ] + 2 ( 1 - η ) λ [ 1 + q 0 0 1 + q ] ,
P ( 4 ) 1 + 2 q β 1 1 + 2 q β 1 + [ 1 + 2 ( - 1 ) ( 1 + q ) ] β 2
R = 1 8 r 2 i = 1 M Θ i { 2 + 4 q β 1 + [ 1 + 2 ( - 1 ) ( 1 + q ) ] β 2 } ,
I d = I 0 exp ( - 8 r 2 ψ 2 ) ,
i = 1 M Δ i 1 - exp [ - 2 ( ϕ / ϕ 0 ) 2 / ( ψ / ϕ 0 ) 2 ] ,
ϕ 0 the input - beam diameter corresponding to N
ϕ the input - beam diameter corresponding to M ,
Φ I x ( i v ) = 1 ( 1 - i v I ¯ noise ) exp [ - I s I ¯ noise + I s I ¯ noise ( 1 - i v I ¯ noise ) ] ,
Φ I y ( i v ) = 1 ( 1 - i v I ¯ noise ) .
Φ I ( i v ) = 1 ( 1 - i v I ¯ noise ) 2 exp [ - I s I ¯ noise + I s I ¯ noise ( 1 - i v I ¯ noise ) ] .
σ I = [ I ¯ 2 - ( I ¯ ) 2 ] 1 / 2 = [ 1 ( i ) 2 2 v 2 Φ I ( i v ) v = 0 - ( I s + 2 I ¯ noise ) 2 ] 1 / 2 = [ 2 I ¯ noise ( I ¯ noise + I s ) ] 1 / 2 .
( SNR ) x y I s σ I = γ ( 2 1 + 2 γ ) 1 / 2 ,
( SNR ) x = 2 γ ( 1 + 4 γ ) 1 / 2 .
γ = 1 2 ( ψ ϕ 0 ) 2 ( ϕ ϕ 0 ) 2 at the center or the signal beam .
M i j = m i j exp ( i ϕ i j ) ;
k = 1 N ( M i j ) k l 2 1 2 ,
l = 1 N ( M i j ) k l 2 1 2 ,

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