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.

[Optical Society of America ]

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  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. IEJQA7 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) KVEKA3 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. STPLD2 15 , 94 95 ( 1989
  4. Ping Xie , Jian-Hua Dai , and Hong-Jun Zhang , Multigrating optical phase conjugation with considerations of phase effects , J. Opt. Soc. Am. B JOBPDE 9 , 2240 2247 ( 1992
    [CrossRef]
  5. 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 JOBPDE 6 , 1828 1833 ( 1989
    [CrossRef]
  6. J. M. Heaton and L. Solymar , Transients effects during dynamic hologram formation in BSO crystals: theory and experiment , IEEE J. Quantum Electron. IEJQA7 24 , 558 567 ( 1988
    [CrossRef]
  7. A. Bledowski , W. Krolikowski , and A. Kujuwski , Temporal instabilities in single grating photorefractive four-wave mixing , J. Opt. Soc. Am. B JOBPDE 6 , 1544 1547 ( 1989
    [CrossRef]
  8. 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 JOBPDE 9 , 398 404 ( 1992
    [CrossRef]
  9. 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 JOBPDE 12 , 1370 1383 ( 1995
    [CrossRef]
  10. M. R. Belic , J. Leonardy , D. Timotijevic , and F. Kaised , Spatiotemporal effects in double phase conjugation , J. Opt. Soc. Am. B JOBPDE 12 , 1602 1616 ( 1995
    [CrossRef]
  11. J. Leonardy , F. Kaiser , M. R. Belic , and D. Timotijevic , Oscillation versus amplification in double phase conjugation , Opt. Commun. OPCOB8 131 , 279 284 ( 1996
    [CrossRef]
  12. S. A. Bugaichuk , A. G. Kutana , and A. I. Khizhyak , Spatial structure of holographic gratings in photorefractive crystals with a nonlocal response , Quantum Electron. QUELEZ 27 , 727 731 ( 1997
    [CrossRef]

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. QUELEZ 27 , 727 731 ( 1997
[CrossRef]

Hesseling, L

Kaised, F

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. QUELEZ 27 , 727 731 ( 1997
[CrossRef]

Kujuwski, A

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. QUELEZ 27 , 727 731 ( 1997
[CrossRef]

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. STPLD2 15 , 94 95 ( 1989

Other (12)

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. IEJQA7 QE-20 , 12 30 ( 1984
[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) KVEKA3 18 , 981 984 ( 1988

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. STPLD2 15 , 94 95 ( 1989

Ping Xie , Jian-Hua Dai , and Hong-Jun Zhang , Multigrating optical phase conjugation with considerations of phase effects , J. Opt. Soc. Am. B JOBPDE 9 , 2240 2247 ( 1992
[CrossRef]

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 JOBPDE 6 , 1828 1833 ( 1989
[CrossRef]

J. M. Heaton and L. Solymar , Transients effects during dynamic hologram formation in BSO crystals: theory and experiment , IEEE J. Quantum Electron. IEJQA7 24 , 558 567 ( 1988
[CrossRef]

A. Bledowski , W. Krolikowski , and A. Kujuwski , Temporal instabilities in single grating photorefractive four-wave mixing , J. Opt. Soc. Am. B JOBPDE 6 , 1544 1547 ( 1989
[CrossRef]

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 JOBPDE 9 , 398 404 ( 1992
[CrossRef]

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 JOBPDE 12 , 1370 1383 ( 1995
[CrossRef]

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

J. Leonardy , F. Kaiser , M. R. Belic , and D. Timotijevic , Oscillation versus amplification in double phase conjugation , Opt. Commun. OPCOB8 131 , 279 284 ( 1996
[CrossRef]

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


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.

Metrics