Abstract

Reduction of the temporal fluctuation of the signal intensity in an optical wave-mixing system with a reflection grating is studied by use of a feedback of the output pump with appropriate feedback reflectivity. The advantage of the method is that the temporal fluctuation of the output intensity can be significantly reduced, although its mean intensity is not reduced.

© 2002 Optical Society of America

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References

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  1. P. Xie, I. A. Taj, T. Mishima, “Reducing temporal fluctuation of signal intensity in optical wave mixing,” IEEE J. Quantum Electron. 37, 664–671 (2001).
    [CrossRef]
  2. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 4.
  3. Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984).
  4. R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992).
  5. M. Cronin-Golomb, B. Fisher, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–20 (1984).
    [CrossRef]
  6. M. R. Belic, “Exact solution to the degenerate four-wave mixing in reflection geometry in photorefractive media,” Phys. Rev. A 31, 3169–3174 (1985).
    [CrossRef] [PubMed]
  7. P. Xie, I. A. Taj, T. Mishima, “Origin of temporal fluctuation in the photorefractive effect,” J. Opt. Soc. Am. B 18, 479–484 (2001).
    [CrossRef]
  8. P. Xie, T. Mishima, “Temporal fluctuation in photorefractive wave mixing,” IEEE J. Quantum Electron. QE-37, 1248–1255 (2001).
  9. H. Risken, The Fokker-Planck Equation: Method of Solution and Application (Springer-Verlag, Berlin, 1984), pp. 60–62.
  10. R. L. Honeycutt, “Stochastic Runge-Kutta algorithms. I. White noise,” Phys. Rev. A 45, 600–603 (1992).
    [CrossRef] [PubMed]
  11. Throughout this paper we maintain a time delay [defined in Eq. (13)] of Δt = 1014τ. The effect of time delay Δt on reduction of efficiency Vfeedback/Vwithout has been discussed in detail in Ref. 1. In general, the smaller the Δt, the smaller the Vfeedback/Vwithout is. For Δt = 1014τ the reduction efficiency Vfeedback/Vwithout changes only slightly when Δt decreases.
  12. S. I. Stepanov, M. P. Petrov, “Degenerate four-wave mixing via shifted phase holograms in cubic photorefractive crystals,” Opt. Commun. 53, 64–68 (1985).
    [CrossRef]
  13. H. Kong, C. Lin, A. M. Biernacki, M. Cronin-Golomb, “Photorefractive phase conjugation with orthogonally polarized pumping beams,” Opt. Lett. 13, 324–326 (1988).
    [CrossRef] [PubMed]

2001 (3)

P. Xie, I. A. Taj, T. Mishima, “Reducing temporal fluctuation of signal intensity in optical wave mixing,” IEEE J. Quantum Electron. 37, 664–671 (2001).
[CrossRef]

P. Xie, T. Mishima, “Temporal fluctuation in photorefractive wave mixing,” IEEE J. Quantum Electron. QE-37, 1248–1255 (2001).

P. Xie, I. A. Taj, T. Mishima, “Origin of temporal fluctuation in the photorefractive effect,” J. Opt. Soc. Am. B 18, 479–484 (2001).
[CrossRef]

1992 (1)

R. L. Honeycutt, “Stochastic Runge-Kutta algorithms. I. White noise,” Phys. Rev. A 45, 600–603 (1992).
[CrossRef] [PubMed]

1988 (1)

1985 (2)

M. R. Belic, “Exact solution to the degenerate four-wave mixing in reflection geometry in photorefractive media,” Phys. Rev. A 31, 3169–3174 (1985).
[CrossRef] [PubMed]

S. I. Stepanov, M. P. Petrov, “Degenerate four-wave mixing via shifted phase holograms in cubic photorefractive crystals,” Opt. Commun. 53, 64–68 (1985).
[CrossRef]

1984 (1)

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

Belic, M. R.

M. R. Belic, “Exact solution to the degenerate four-wave mixing in reflection geometry in photorefractive media,” Phys. Rev. A 31, 3169–3174 (1985).
[CrossRef] [PubMed]

Biernacki, A. M.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992).

Cronin-Golomb, M.

H. Kong, C. Lin, A. M. Biernacki, M. Cronin-Golomb, “Photorefractive phase conjugation with orthogonally polarized pumping beams,” Opt. Lett. 13, 324–326 (1988).
[CrossRef] [PubMed]

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

Fisher, B.

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

Honeycutt, R. L.

R. L. Honeycutt, “Stochastic Runge-Kutta algorithms. I. White noise,” Phys. Rev. A 45, 600–603 (1992).
[CrossRef] [PubMed]

Kong, H.

Lin, C.

Mishima, T.

P. Xie, I. A. Taj, T. Mishima, “Origin of temporal fluctuation in the photorefractive effect,” J. Opt. Soc. Am. B 18, 479–484 (2001).
[CrossRef]

P. Xie, I. A. Taj, T. Mishima, “Reducing temporal fluctuation of signal intensity in optical wave mixing,” IEEE J. Quantum Electron. 37, 664–671 (2001).
[CrossRef]

P. Xie, T. Mishima, “Temporal fluctuation in photorefractive wave mixing,” IEEE J. Quantum Electron. QE-37, 1248–1255 (2001).

Petrov, M. P.

S. I. Stepanov, M. P. Petrov, “Degenerate four-wave mixing via shifted phase holograms in cubic photorefractive crystals,” Opt. Commun. 53, 64–68 (1985).
[CrossRef]

Risken, H.

H. Risken, The Fokker-Planck Equation: Method of Solution and Application (Springer-Verlag, Berlin, 1984), pp. 60–62.

Shen, Y. R.

Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984).

Stepanov, S. I.

S. I. Stepanov, M. P. Petrov, “Degenerate four-wave mixing via shifted phase holograms in cubic photorefractive crystals,” Opt. Commun. 53, 64–68 (1985).
[CrossRef]

Taj, I. A.

P. Xie, I. A. Taj, T. Mishima, “Reducing temporal fluctuation of signal intensity in optical wave mixing,” IEEE J. Quantum Electron. 37, 664–671 (2001).
[CrossRef]

P. Xie, I. A. Taj, T. Mishima, “Origin of temporal fluctuation in the photorefractive effect,” J. Opt. Soc. Am. B 18, 479–484 (2001).
[CrossRef]

White, J. O.

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

Xie, P.

P. Xie, I. A. Taj, T. Mishima, “Origin of temporal fluctuation in the photorefractive effect,” J. Opt. Soc. Am. B 18, 479–484 (2001).
[CrossRef]

P. Xie, I. A. Taj, T. Mishima, “Reducing temporal fluctuation of signal intensity in optical wave mixing,” IEEE J. Quantum Electron. 37, 664–671 (2001).
[CrossRef]

P. Xie, T. Mishima, “Temporal fluctuation in photorefractive wave mixing,” IEEE J. Quantum Electron. QE-37, 1248–1255 (2001).

Yariv, A.

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

Yeh, P.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 4.

IEEE J. Quantum Electron. (3)

P. Xie, I. A. Taj, T. Mishima, “Reducing temporal fluctuation of signal intensity in optical wave mixing,” IEEE J. Quantum Electron. 37, 664–671 (2001).
[CrossRef]

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

P. Xie, T. Mishima, “Temporal fluctuation in photorefractive wave mixing,” IEEE J. Quantum Electron. QE-37, 1248–1255 (2001).

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

S. I. Stepanov, M. P. Petrov, “Degenerate four-wave mixing via shifted phase holograms in cubic photorefractive crystals,” Opt. Commun. 53, 64–68 (1985).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (2)

R. L. Honeycutt, “Stochastic Runge-Kutta algorithms. I. White noise,” Phys. Rev. A 45, 600–603 (1992).
[CrossRef] [PubMed]

M. R. Belic, “Exact solution to the degenerate four-wave mixing in reflection geometry in photorefractive media,” Phys. Rev. A 31, 3169–3174 (1985).
[CrossRef] [PubMed]

Other (5)

H. Risken, The Fokker-Planck Equation: Method of Solution and Application (Springer-Verlag, Berlin, 1984), pp. 60–62.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 4.

Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984).

R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992).

Throughout this paper we maintain a time delay [defined in Eq. (13)] of Δt = 1014τ. The effect of time delay Δt on reduction of efficiency Vfeedback/Vwithout has been discussed in detail in Ref. 1. In general, the smaller the Δt, the smaller the Vfeedback/Vwithout is. For Δt = 1014τ the reduction efficiency Vfeedback/Vwithout changes only slightly when Δt decreases.

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Figures (9)

Fig. 1
Fig. 1

(a) Two-wave mixing with a reflection grating: 1, the pump beam and 2, the signal beam. (b) Degenerate four-wave mixing with a reflection grating: 1 and 4, pump beams and 2, the phase-conjugate beam that is the signal beam in this paper.

Fig. 2
Fig. 2

I 1(L) (curves 1 and 2) and I 2(0) (curves 3 and 4) as functions of αL in contradirectional two-wave mixing. The solid curves were obtained from the approximate solution in approximations (4a) and (4b), and the dashed curves were obtained from the numerical solution of approximations (21a)–(21c) without noise.

Fig. 3
Fig. 3

Temporal evolution of the output signal intensity I 2(0, t) (a) without and (b) with a feedback cavity.

Fig. 4
Fig. 4

I 1(L)exp(2αL)/I 1(0) as a function of |A in|2 (upper graphs) and V feedback/V without as a function of I 1(L)exp(2αL)/I 1(0) (lower graphs) for D = 10-3 (curve 1) and D = 10-5 (curve 2) with (a) α = 0, (b) αL = 0.15, (c) αL = 0.3. All of the graphs are for γ I L = 0.

Fig. 5
Fig. 5

V feedback/V without as a function of αL for D = 10-4 (curve 1) and D = 10-5 (curve 2), where we used |A in|2 to obtain I 1(L)exp(2αL)/I 1(0) ≈ R.

Fig. 6
Fig. 6

V feedback/V without as a function of γI L for D = 10-3 (curve 1), D = 10-4 (curve 2), and D = 10-5 (curve 3) with γ R L = 2.5 and α = 0, where we used |A in|2 = 35 and R = 0.4, resulting in I 1(0) = 97.2.

Fig. 7
Fig. 7

Temporal evolution of the output pump phase ϕ1(L, t) for γ I L = 2 and D = 10-3.

Fig. 8
Fig. 8

I 1(L) (curves 1 and 2) and I 2(0) (curves 3 and 4) as a function of αL in four-wave mixing with a reflection grating. The solid curves were obtained from the approximate solution in approximations (4a) and (4b), and the dashed curves were obtained from the numerical solution of Eqs. (22a) –(22e) without noise.

Fig. 9
Fig. 9

I 1(L)exp(2αL)/I 1(0) as a function of |A in|2 and (b) V feedback/V without as a function of I 1(L)exp(2αL)/I 1(0) for D = 10-3 (curve 1) and D = 10-5 (curve 2) with αL = 0.2.

Equations (34)

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I1z-I2z=const.
I10+I2L=I20+I1L.
I1αzI1α=0zexp-αz,
I2αzI2α=0zexpαz-L.
I1α=0LI1αLexpαL,
I2α=00I2α0expαL.
I20+I1LI10+I2Lexp-αL,
DE I10=I20,
I20+I1LI10exp-αL.
DE+ΔtI10=I20+t,
I10, t=I10-t
DE+ΔtI10, t=I20,
I20+I1L, tI10exp-αL-texp-αL.
I1L, tI1L-texp-αL.
A10, t=Ain+RexpiδA1L, t-Δt,
A10, tAin+RI1L-texp-αL1/2×expi2nπn=0, 1.
R=I1LI10exp2αL,
I20, t=I20-1I20+14I1Lt2.
A10, t=Ain+RI1L-texp-αL1/2×expi2nπ+ϕtn=0, 1,,
I20, tI201-4 I1LI10expαL1-I1LI10expαLsin2ϕt2+21-I1LI10expαL1-2 I1LI10expαLsin2ϕt2 t
I20, tI201-I1LI10expαL1-I1LI10expαLϕt2+1-I1LI10expαL1-2 I1LI10expαLϕt22 t.
I1LexpαL/I101-I1LexpαL/I10ϕt2
fx=x1-x,
gx=1-x1-2x,
I20, tI201-fxϕt2+gxϕt22 t.
A1z=-QA2-α2 A1,
A2*z=-QA1*+α2 A2*,
τ Qt+Q=γI0 A1A2*,
V=I20, t-I20, t21/2/I20, t
A1z=-Q1A2-α2 A1,
A2*z=-Q1A1*+α2 A2*,
A3z=-Q2A4-α2 A3,
A4*z=-Q2A3*+α2 A4*,
τ Qit+Qi=γim=14 |Am|2A1A2*+A3*A4,

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