Abstract

The structure and dynamics of the four-wave mixing transmission grating in a photorefractive crystal were investigated. The steady-state grating’s amplitude distribution looks like a motionless soliton. When the value of coupling strength is more than 2, there are three solutions for the steady-state grating: two stable and one unstable. The dynamics of all these gratings are described by the sine-Gordon equation. The four-wave mixing grating’s structure and location in space are determined by the input beams’ ratio, phase difference, or both.

© 1998 Optical Society of America

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References

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  1. M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
    [Crossref]
  2. A. A. Zozulya and V. T. Tikhonchuk, “Solution of nonlinear equations for four-wave interactions in photorefractive media,” Kvantovaya Elektron. (Moscow) 15, 1570–1576 (1988) [Sov. J. Quantum Electron.] [Kvant. Elektron. (Moscow) 18, 981–984 (1988)].
  3. A. A. Zolulya and V. T. Tikhonchuk, “Stability of steady states in four-wave mixing in a photorefractive medium,” Pis’ma Zh. Tekh. Fiz. 15, 35–38 (1989) [Sov. Tech. Phys. Lett. 15, 94–95 (1989)].
  4. S. Odoulov, M. Soskin, and A. Khizhnyak, Optical Oscillators with Degenerate Four-Wave Mixing (Harwood, Chur, Switzerland, 1991).
  5. S. Odoulov, M. Soskin, and A. Khyzhniak, Dynamic Grating Lasers (Nuaka, Moscow, 1990) (in Russian).
  6. Ping Xie, Jian-Hua Dai, and Hong-Jun Zhang, “Multigrating optical phase conjugation with considerations of phase effects,” J. Opt. Soc. Am. B 9, 2240–2247 (1992).
    [Crossref]
  7. W. Krolikowski, K. D. Shaw, M. Cronin-Golomb, and A. Bledowski, “Stability analysis and temporal behavior of four-wave mixing in photorefractive crystals,” J. Opt. Soc. Am. B 6, 1828–1833 (1989).
    [Crossref]
  8. J. M. Heaton and L. Solymar, “Transients effects during dynamic hologram formation in BSO crystals: theory and experiment,” IEEE J. Quantum Electron. 24, 558–567 (1988).
    [Crossref]
  9. A. Bledowski, W. Krolikowski, and A. Kujuwski, “Temporal instabilities in single grating photorefractive four-wave mixing,” J. Opt. Soc. Am. B 6, 1544–1547 (1989).
    [Crossref]
  10. V. V. Eliseev, A. A. Zozulya, G. D. Bacher, and J. Feinberg, “Self-bending of light beams in photorefractive phase conjugators,” J. Opt. Soc. Am. B 9, 398–404 (1992).
    [Crossref]
  11. M. Jeganathan, M. C. Bashaw, and L. Hesseling, “Evolution and propagation of grating envelopes during erasure in bulk photorefractive media,” J. Opt. Soc. Am. B 12, 1370–1383 (1995).
    [Crossref]
  12. M. R. Belic, J. Leonardy, D. Timotijevic, and F. Kaised, “Spatiotemporal effects in double phase conjugation,” J. Opt. Soc. Am. B 12, 1602–1616 (1995).
    [Crossref]
  13. A. I. Khizniak and S. A. Bougaychouk, “Edge dislocation of FWM dynamic grating in a photorefractive medium,” in Conference on Lasers and Electro-Optics Europe, 1996 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), paper CMK6.
  14. J. Leonardy, F. Kaiser, M. R. Belic, and D. Timotijevic, “Oscillation versus amplification in double phase conjugation,” Opt. Commun. 131, 279–284 (1996).
    [Crossref]
  15. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Harcourt Brace Jovanovich, New York, 1982).
  16. S. A. Bugaichuk, A. G. Kutana, and A. I. Khizhyak, “Spatial structure of holographic gratings in photorefractive crystals with a nonlocal response,” Quantum Electron. 27, 727–731 (1997).
    [Crossref]
  17. T. Shimura, A. Nakamura, K. Kuroda, H. Wakimoto, and M. Thukakoshi, “Stabilization of the phase conjugate reflectivity in photorefractive four-wave mixing,” in Conference on Lasers and Electro-Optics Europe, 1996 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), paper CTuK40.

1997 (1)

S. A. Bugaichuk, A. G. Kutana, and A. I. Khizhyak, “Spatial structure of holographic gratings in photorefractive crystals with a nonlocal response,” Quantum Electron. 27, 727–731 (1997).
[Crossref]

1996 (1)

J. Leonardy, F. Kaiser, M. R. Belic, and D. Timotijevic, “Oscillation versus amplification in double phase conjugation,” Opt. Commun. 131, 279–284 (1996).
[Crossref]

1995 (2)

1992 (2)

1989 (3)

1988 (2)

J. M. Heaton and L. Solymar, “Transients effects during dynamic hologram formation in BSO crystals: theory and experiment,” IEEE J. Quantum Electron. 24, 558–567 (1988).
[Crossref]

A. A. Zozulya and V. T. Tikhonchuk, “Solution of nonlinear equations for four-wave interactions in photorefractive media,” Kvantovaya Elektron. (Moscow) 15, 1570–1576 (1988) [Sov. J. Quantum Electron.] [Kvant. Elektron. (Moscow) 18, 981–984 (1988)].

1984 (1)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[Crossref]

Bacher, G. D.

Bashaw, M. C.

Belic, M. R.

J. Leonardy, F. Kaiser, M. R. Belic, and D. Timotijevic, “Oscillation versus amplification in double phase conjugation,” Opt. Commun. 131, 279–284 (1996).
[Crossref]

M. R. Belic, J. Leonardy, D. Timotijevic, and F. Kaised, “Spatiotemporal effects in double phase conjugation,” J. Opt. Soc. Am. B 12, 1602–1616 (1995).
[Crossref]

Bledowski, A.

Bougaychouk, S. A.

A. I. Khizniak and S. A. Bougaychouk, “Edge dislocation of FWM dynamic grating in a photorefractive medium,” in Conference on Lasers and Electro-Optics Europe, 1996 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), paper CMK6.

Bugaichuk, S. A.

S. A. Bugaichuk, A. G. Kutana, and A. I. Khizhyak, “Spatial structure of holographic gratings in photorefractive crystals with a nonlocal response,” Quantum Electron. 27, 727–731 (1997).
[Crossref]

Cronin-Golomb, M.

W. Krolikowski, K. D. Shaw, M. Cronin-Golomb, and A. Bledowski, “Stability analysis and temporal behavior of four-wave mixing in photorefractive crystals,” J. Opt. Soc. Am. B 6, 1828–1833 (1989).
[Crossref]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[Crossref]

Dai, Jian-Hua

Dodd, R. K.

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Harcourt Brace Jovanovich, New York, 1982).

Eilbeck, J. C.

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Harcourt Brace Jovanovich, New York, 1982).

Eliseev, V. V.

Feinberg, J.

Fischer, B.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[Crossref]

Gibbon, J. D.

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Harcourt Brace Jovanovich, New York, 1982).

Heaton, J. M.

J. M. Heaton and L. Solymar, “Transients effects during dynamic hologram formation in BSO crystals: theory and experiment,” IEEE J. Quantum Electron. 24, 558–567 (1988).
[Crossref]

Hesseling, L.

Jeganathan, M.

Kaised, F.

Kaiser, F.

J. Leonardy, F. Kaiser, M. R. Belic, and D. Timotijevic, “Oscillation versus amplification in double phase conjugation,” Opt. Commun. 131, 279–284 (1996).
[Crossref]

Khizhnyak, A.

S. Odoulov, M. Soskin, and A. Khizhnyak, Optical Oscillators with Degenerate Four-Wave Mixing (Harwood, Chur, Switzerland, 1991).

Khizhyak, A. I.

S. A. Bugaichuk, A. G. Kutana, and A. I. Khizhyak, “Spatial structure of holographic gratings in photorefractive crystals with a nonlocal response,” Quantum Electron. 27, 727–731 (1997).
[Crossref]

Khizniak, A. I.

A. I. Khizniak and S. A. Bougaychouk, “Edge dislocation of FWM dynamic grating in a photorefractive medium,” in Conference on Lasers and Electro-Optics Europe, 1996 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), paper CMK6.

Khyzhniak, A.

S. Odoulov, M. Soskin, and A. Khyzhniak, Dynamic Grating Lasers (Nuaka, Moscow, 1990) (in Russian).

Krolikowski, W.

Kujuwski, A.

Kuroda, K.

T. Shimura, A. Nakamura, K. Kuroda, H. Wakimoto, and M. Thukakoshi, “Stabilization of the phase conjugate reflectivity in photorefractive four-wave mixing,” in Conference on Lasers and Electro-Optics Europe, 1996 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), paper CTuK40.

Kutana, A. G.

S. A. Bugaichuk, A. G. Kutana, and A. I. Khizhyak, “Spatial structure of holographic gratings in photorefractive crystals with a nonlocal response,” Quantum Electron. 27, 727–731 (1997).
[Crossref]

Leonardy, J.

J. Leonardy, F. Kaiser, M. R. Belic, and D. Timotijevic, “Oscillation versus amplification in double phase conjugation,” Opt. Commun. 131, 279–284 (1996).
[Crossref]

M. R. Belic, J. Leonardy, D. Timotijevic, and F. Kaised, “Spatiotemporal effects in double phase conjugation,” J. Opt. Soc. Am. B 12, 1602–1616 (1995).
[Crossref]

Morris, H. C.

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Harcourt Brace Jovanovich, New York, 1982).

Nakamura, A.

T. Shimura, A. Nakamura, K. Kuroda, H. Wakimoto, and M. Thukakoshi, “Stabilization of the phase conjugate reflectivity in photorefractive four-wave mixing,” in Conference on Lasers and Electro-Optics Europe, 1996 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), paper CTuK40.

Odoulov, S.

S. Odoulov, M. Soskin, and A. Khyzhniak, Dynamic Grating Lasers (Nuaka, Moscow, 1990) (in Russian).

S. Odoulov, M. Soskin, and A. Khizhnyak, Optical Oscillators with Degenerate Four-Wave Mixing (Harwood, Chur, Switzerland, 1991).

Shaw, K. D.

Shimura, T.

T. Shimura, A. Nakamura, K. Kuroda, H. Wakimoto, and M. Thukakoshi, “Stabilization of the phase conjugate reflectivity in photorefractive four-wave mixing,” in Conference on Lasers and Electro-Optics Europe, 1996 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), paper CTuK40.

Solymar, L.

J. M. Heaton and L. Solymar, “Transients effects during dynamic hologram formation in BSO crystals: theory and experiment,” IEEE J. Quantum Electron. 24, 558–567 (1988).
[Crossref]

Soskin, M.

S. Odoulov, M. Soskin, and A. Khizhnyak, Optical Oscillators with Degenerate Four-Wave Mixing (Harwood, Chur, Switzerland, 1991).

S. Odoulov, M. Soskin, and A. Khyzhniak, Dynamic Grating Lasers (Nuaka, Moscow, 1990) (in Russian).

Thukakoshi, M.

T. Shimura, A. Nakamura, K. Kuroda, H. Wakimoto, and M. Thukakoshi, “Stabilization of the phase conjugate reflectivity in photorefractive four-wave mixing,” in Conference on Lasers and Electro-Optics Europe, 1996 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), paper CTuK40.

Tikhonchuk, V. T.

A. A. Zolulya and V. T. Tikhonchuk, “Stability of steady states in four-wave mixing in a photorefractive medium,” Pis’ma Zh. Tekh. Fiz. 15, 35–38 (1989) [Sov. Tech. Phys. Lett. 15, 94–95 (1989)].

A. A. Zozulya and V. T. Tikhonchuk, “Solution of nonlinear equations for four-wave interactions in photorefractive media,” Kvantovaya Elektron. (Moscow) 15, 1570–1576 (1988) [Sov. J. Quantum Electron.] [Kvant. Elektron. (Moscow) 18, 981–984 (1988)].

Timotijevic, D.

J. Leonardy, F. Kaiser, M. R. Belic, and D. Timotijevic, “Oscillation versus amplification in double phase conjugation,” Opt. Commun. 131, 279–284 (1996).
[Crossref]

M. R. Belic, J. Leonardy, D. Timotijevic, and F. Kaised, “Spatiotemporal effects in double phase conjugation,” J. Opt. Soc. Am. B 12, 1602–1616 (1995).
[Crossref]

Wakimoto, H.

T. Shimura, A. Nakamura, K. Kuroda, H. Wakimoto, and M. Thukakoshi, “Stabilization of the phase conjugate reflectivity in photorefractive four-wave mixing,” in Conference on Lasers and Electro-Optics Europe, 1996 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), paper CTuK40.

White, J. O.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[Crossref]

Xie, Ping

Yariv, A.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[Crossref]

Zhang, Hong-Jun

Zolulya, A. A.

A. A. Zolulya and V. T. Tikhonchuk, “Stability of steady states in four-wave mixing in a photorefractive medium,” Pis’ma Zh. Tekh. Fiz. 15, 35–38 (1989) [Sov. Tech. Phys. Lett. 15, 94–95 (1989)].

Zozulya, A. A.

V. V. Eliseev, A. A. Zozulya, G. D. Bacher, and J. Feinberg, “Self-bending of light beams in photorefractive phase conjugators,” J. Opt. Soc. Am. B 9, 398–404 (1992).
[Crossref]

A. A. Zozulya and V. T. Tikhonchuk, “Solution of nonlinear equations for four-wave interactions in photorefractive media,” Kvantovaya Elektron. (Moscow) 15, 1570–1576 (1988) [Sov. J. Quantum Electron.] [Kvant. Elektron. (Moscow) 18, 981–984 (1988)].

IEEE J. Quantum Electron. (2)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[Crossref]

J. M. Heaton and L. Solymar, “Transients effects during dynamic hologram formation in BSO crystals: theory and experiment,” IEEE J. Quantum Electron. 24, 558–567 (1988).
[Crossref]

J. Opt. Soc. Am. B (6)

Kvantovaya Elektron. (Moscow) (1)

A. A. Zozulya and V. T. Tikhonchuk, “Solution of nonlinear equations for four-wave interactions in photorefractive media,” Kvantovaya Elektron. (Moscow) 15, 1570–1576 (1988) [Sov. J. Quantum Electron.] [Kvant. Elektron. (Moscow) 18, 981–984 (1988)].

Opt. Commun. (1)

J. Leonardy, F. Kaiser, M. R. Belic, and D. Timotijevic, “Oscillation versus amplification in double phase conjugation,” Opt. Commun. 131, 279–284 (1996).
[Crossref]

Pis’ma Zh. Tekh. Fiz. (1)

A. A. Zolulya and V. T. Tikhonchuk, “Stability of steady states in four-wave mixing in a photorefractive medium,” Pis’ma Zh. Tekh. Fiz. 15, 35–38 (1989) [Sov. Tech. Phys. Lett. 15, 94–95 (1989)].

Quantum Electron. (1)

S. A. Bugaichuk, A. G. Kutana, and A. I. Khizhyak, “Spatial structure of holographic gratings in photorefractive crystals with a nonlocal response,” Quantum Electron. 27, 727–731 (1997).
[Crossref]

Other (5)

T. Shimura, A. Nakamura, K. Kuroda, H. Wakimoto, and M. Thukakoshi, “Stabilization of the phase conjugate reflectivity in photorefractive four-wave mixing,” in Conference on Lasers and Electro-Optics Europe, 1996 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), paper CTuK40.

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Harcourt Brace Jovanovich, New York, 1982).

S. Odoulov, M. Soskin, and A. Khizhnyak, Optical Oscillators with Degenerate Four-Wave Mixing (Harwood, Chur, Switzerland, 1991).

S. Odoulov, M. Soskin, and A. Khyzhniak, Dynamic Grating Lasers (Nuaka, Moscow, 1990) (in Russian).

A. I. Khizniak and S. A. Bougaychouk, “Edge dislocation of FWM dynamic grating in a photorefractive medium,” in Conference on Lasers and Electro-Optics Europe, 1996 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), paper CMK6.

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Figures (13)

Fig. 1
Fig. 1

Transmission geometry FWM in a PRC. 1–4, interacted waves. Curve 5 demonstrates the grating amplitude distribution.

Fig. 2
Fig. 2

Grating amplitude distribution for a DPCM: 1, J32=0.01; 2, J32=1; 3, J32=100.

Fig. 3
Fig. 3

(a) Wave intensities and grating amplitude distribution within a PRC for Φ(0)=0. (b) Grating patterns (thinner solid lines) and interference patterns of waves 1 and 3 (thicker solid lines) and 2 and 4 (dashed lines).

Fig. 4
Fig. 4

(a) Wave intensities and grating amplitude distribution within a PRC for Φ(0)=π. (b) Grating indices (thinner solid lines) and interference patterns of waves 1 and 3 (thicker solid lines) and 2 and 4 (dashed lines).

Fig. 5
Fig. 5

Steady-state solutions for FWM grating amplitude: 1, 2, stable solutions; 3, an unstable one. γl=10, J1(0)=0.1, J3(0)=0.4, J2(l)=0.5.

Fig. 6
Fig. 6

Distance to the cross section of the optical dislocation position as a function of the intensity of beam 1 for Φ(0)=π and J2(l)=J1(0)+J3(0).

Fig. 7
Fig. 7

Grating recording dynamics in a pure crystal for γl=10.

Fig. 8
Fig. 8

Dynamics of the unstable solution for γl=10.

Fig. 9
Fig. 9

Grating recording dynamics for Φ(0)=π. γl=10.

Fig. 10
Fig. 10

Dependence of DPCM diffraction efficiency on beam intensity ratio for various values of PRC coupling strength (numbers near curves).

Fig. 11
Fig. 11

Dependence of FWM diffraction efficiency on input beam ratio and phase difference for J1(0)+J3(0)=J2(l). Solid curves, Φ(0)=0; loops, Φ(0)=π; dashed curves, unstable solutions; γl=10. Numbers near curves, intensities of wave 1 [J1(0)].

Fig. 12
Fig. 12

Dependence of phase-conjugation reflection RPC on coupling strength γl for some intensity values of beam 1.

Fig. 13
Fig. 13

Optical control of the probe beam’s space location (5–6). Beams 1–3 are explained in text.

Equations (28)

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

dA1dz=γ[A1A3+A2A4 cos(Φ)]A3,
dA3dz=-γ[A1A3+A2A4 cos(Φ)]A1,
dA2dz=γ[A1A3 cos(Φ)+A2A4]A4,
dA4dz=-γ[A1A3 cos(Φ)+A2A4]A2,
B=A1A3+A2A4 cos(Φ).
d2 ln(B)dz2=-4γ2B2,
B=Ccosh(2γCz+b),
u(z)=γ0z B(z)dz.
A1(u)=A1(0)cos(u)±A3(0)sin(u),
A3(u)=A1(0)cos(u)+A3(0)sin(u),
A2(u)=A2(l)cos(ul-u),
A4(u)=A2(l)sin(ul-u),
η=J4(0)/J2(l)=sin2(ul),
J1(0)/J1=sin2γ-lsh0 Bmdz,
J1(0)/J3(0)=tan2γ0lsh Bmdz,
x-1x+1=ln(x)ln(P)γl,
P=1+J1(0)xJ3(0) x-JJx-11/2/
1-J1(0)J3(0)x Jx-1x-J1/2
lsh=l ln(Pm)ln(xm)ln(Pm),
η=(xm-J)(Jxm-1)(1+xm2).
J1(l)=[J1(0)+J3(0)]η,
J3(l)=[J1(0)+J3(0)](1-η),
J2(0)=J2(l)J3(0)J1 (1-η)1/2±J1(0)J1 η1/22,
J4(0)=J2(l)J3(0)J1 η1/2J1(0)J1 (1-η)1/22.
2vτz+vz=R sin(v),
u(τ, z)=0z B(τ, z)dz,
α=α(τ)=arctan [J3(0)-J1(0)]sin(2ul)±2A1(0)A3(0)cos(2ul)[J3(0)-J1(0)]cos(2ul)2A1(0)A3(0)sin(2ul)-J2(l),
R=R(τ){[J3(0)+J1(0)]2+J22(l)+2J2(l)[J1(0)-J3(0)]cos(2ul)±2A1(0)A3(0)sin(2ul)}1/2.

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