Abstract

The analytical solution for the interaction of three diffraction orders in the Kerr medium is obtained by reducing the problem to the completely integrable Hamiltonian task. Intensities of all waves are periodic with propagation length and linearly related, the amplitudes are quasi-periodic and expressed in elliptic functions. Symmetrical four-order interaction also admits an analytical solution.

© 2000 Optical Society of America

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References

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  1. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst.Tech.J 48,2909–2947 (1969).
  2. R. Magnusson and T.K. Gaylord “Analysis of multiwave diffraction of thick gratings,” J.Opt.Soc.Am. 67,1165–1170(1977).
    [CrossRef]
  3. L. Solymar, D.J. Webb, and A. Grunnet-Jepsen, The Physics and applications of photorefractive crystals (Calderon, Oxford, 1996).
  4. A. Apolinar-Iribe, N. Korneev, and J.J. Sanchéz -Mondragon “Beam amplification resulting from non-Bragg wave mixing in photorefractive strontium barium niobate,” Opt.Lett. 23,1877–79 (1998).
    [CrossRef]
  5. N. Korneev, A. Apolinar-Iribe, and J.J. Sanchéz -Mondragon, “Theory of multiple beam interaction in photorefractive media,” J. Opt. Soc. Am. B 16,580–586(1999).
    [CrossRef]
  6. V.I. ArnoldMathematical Methods of Classical Mechanics (2-nd edition, Springer-Verlag1989).
  7. V.E. Zakharov and A.B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov.Physics JETP, 61,62–69(1972).
  8. D. ter HaarElements of Hamiltonian mechanics.(Pergamon Press, 1971).
  9. V.I. Bespalov and V.I. Talanov “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310(1966).

1999 (1)

1998 (1)

A. Apolinar-Iribe, N. Korneev, and J.J. Sanchéz -Mondragon “Beam amplification resulting from non-Bragg wave mixing in photorefractive strontium barium niobate,” Opt.Lett. 23,1877–79 (1998).
[CrossRef]

1977 (1)

R. Magnusson and T.K. Gaylord “Analysis of multiwave diffraction of thick gratings,” J.Opt.Soc.Am. 67,1165–1170(1977).
[CrossRef]

1972 (1)

V.E. Zakharov and A.B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov.Physics JETP, 61,62–69(1972).

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst.Tech.J 48,2909–2947 (1969).

1966 (1)

V.I. Bespalov and V.I. Talanov “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310(1966).

Apolinar-Iribe, A.

N. Korneev, A. Apolinar-Iribe, and J.J. Sanchéz -Mondragon, “Theory of multiple beam interaction in photorefractive media,” J. Opt. Soc. Am. B 16,580–586(1999).
[CrossRef]

A. Apolinar-Iribe, N. Korneev, and J.J. Sanchéz -Mondragon “Beam amplification resulting from non-Bragg wave mixing in photorefractive strontium barium niobate,” Opt.Lett. 23,1877–79 (1998).
[CrossRef]

Arnold, V.I.

V.I. ArnoldMathematical Methods of Classical Mechanics (2-nd edition, Springer-Verlag1989).

Bespalov, V.I.

V.I. Bespalov and V.I. Talanov “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310(1966).

Gaylord, T.K.

R. Magnusson and T.K. Gaylord “Analysis of multiwave diffraction of thick gratings,” J.Opt.Soc.Am. 67,1165–1170(1977).
[CrossRef]

Grunnet-Jepsen, A.

L. Solymar, D.J. Webb, and A. Grunnet-Jepsen, The Physics and applications of photorefractive crystals (Calderon, Oxford, 1996).

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst.Tech.J 48,2909–2947 (1969).

Korneev, N.

N. Korneev, A. Apolinar-Iribe, and J.J. Sanchéz -Mondragon, “Theory of multiple beam interaction in photorefractive media,” J. Opt. Soc. Am. B 16,580–586(1999).
[CrossRef]

A. Apolinar-Iribe, N. Korneev, and J.J. Sanchéz -Mondragon “Beam amplification resulting from non-Bragg wave mixing in photorefractive strontium barium niobate,” Opt.Lett. 23,1877–79 (1998).
[CrossRef]

Magnusson, R.

R. Magnusson and T.K. Gaylord “Analysis of multiwave diffraction of thick gratings,” J.Opt.Soc.Am. 67,1165–1170(1977).
[CrossRef]

Sanchéz -Mondragon, J.J.

N. Korneev, A. Apolinar-Iribe, and J.J. Sanchéz -Mondragon, “Theory of multiple beam interaction in photorefractive media,” J. Opt. Soc. Am. B 16,580–586(1999).
[CrossRef]

A. Apolinar-Iribe, N. Korneev, and J.J. Sanchéz -Mondragon “Beam amplification resulting from non-Bragg wave mixing in photorefractive strontium barium niobate,” Opt.Lett. 23,1877–79 (1998).
[CrossRef]

Shabat, A.B.

V.E. Zakharov and A.B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov.Physics JETP, 61,62–69(1972).

Solymar, L.

L. Solymar, D.J. Webb, and A. Grunnet-Jepsen, The Physics and applications of photorefractive crystals (Calderon, Oxford, 1996).

Talanov, V.I.

V.I. Bespalov and V.I. Talanov “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310(1966).

ter Haar, D.

D. ter HaarElements of Hamiltonian mechanics.(Pergamon Press, 1971).

Webb, D.J.

L. Solymar, D.J. Webb, and A. Grunnet-Jepsen, The Physics and applications of photorefractive crystals (Calderon, Oxford, 1996).

Zakharov, V.E.

V.E. Zakharov and A.B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov.Physics JETP, 61,62–69(1972).

Bell Syst.Tech.J (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst.Tech.J 48,2909–2947 (1969).

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

J.Opt.Soc.Am. (1)

R. Magnusson and T.K. Gaylord “Analysis of multiwave diffraction of thick gratings,” J.Opt.Soc.Am. 67,1165–1170(1977).
[CrossRef]

JETP Lett. (1)

V.I. Bespalov and V.I. Talanov “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310(1966).

Opt.Lett. (1)

A. Apolinar-Iribe, N. Korneev, and J.J. Sanchéz -Mondragon “Beam amplification resulting from non-Bragg wave mixing in photorefractive strontium barium niobate,” Opt.Lett. 23,1877–79 (1998).
[CrossRef]

Sov.Physics JETP (1)

V.E. Zakharov and A.B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov.Physics JETP, 61,62–69(1972).

Other (3)

D. ter HaarElements of Hamiltonian mechanics.(Pergamon Press, 1971).

L. Solymar, D.J. Webb, and A. Grunnet-Jepsen, The Physics and applications of photorefractive crystals (Calderon, Oxford, 1996).

V.I. ArnoldMathematical Methods of Classical Mechanics (2-nd edition, Springer-Verlag1989).

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Figures (3)

Fig. 1.
Fig. 1.

The trajectory in the complex plane for S 1. It is seen that the amplitude gains a constant phase shift after one period of intensity. The numerical solution of equations (3–5) for K=1.8, κ=0.4, and initial conditions S -1=0.3, S 0=1, S 1=0.1+0.24i.

Fig. 2.
Fig. 2.

The maximal intensity transferred to the initially weak side order depending on the nonlinearity parameter q. The central beam intensity I 0=1.

Fig. 3.
Fig. 3.

Phase portraits of trajectories with different initial conditions on the Poincaré sphere for a four-beam interaction. The nonlinearity parameter is q=0.5.

Equations (19)

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i z ψ + 1 2 xx ψ + κ ψ 2 ψ = 0 .
ψ ( x , z ) = k S k ( z ) exp ( i k x ) ,
i z S 1 = 1 2 K 2 S 1 κ ( S 1 2 S 1 + 2 S 0 2 S 1 + 2 S 1 2 S 1 + S 0 2 S 1 * ) ,
i z S 0 = κ ( S 0 2 S 0 + 2 S 1 2 S 0 + 2 S 1 2 S 0 + 2 S 0 * S 1 S 1 ) ,
i z S 1 = 1 2 K 2 S 1 κ ( S 1 2 S 1 + 2 S 0 2 S 1 + 2 S 1 2 S 1 + S 0 2 S 1 * ) .
H ( ψ , ψ * ) = 1 2 ( x ψ 2 κ ψ 4 ) d x .
H ( S , S * ) = 1 2 k k 2 S k S k * 1 2 κ a 1 + a 2 = a 3 + a 4 S a 1 S a 2 S a 3 * S a 4 * ,
i z S k = H S k * .
i z U = { U , H } = k U S k H S k * U S k * H S k
H = ( 1 2 K 2 κ I 0 ) ( I I 0 ) 1 2 κ I 2 1 4 κ ( ( I I 0 ) 2 M 2 K 2 ) 2 κ Re ( S 0 * 2 S 1 S 1 ) .
z I 0 = 4 κ I m ( S 0 * 2 S 1 S 1 ) .
( Im ( S 0 * 2 S 1 S 1 ) ) 2 + ( Re ( S 0 * 2 S 1 S 1 ) ) 2 = I 0 2 I 1 I 1 = 1 4 I 0 2 ( ( I I 0 ) 2 M 2 K 2 ) ,
i z ln ( S 0 ) = κ ( S 0 2 + 2 S 1 2 + 2 S 1 2 + 2 S 0 * 2 S 1 S 1 S 0 2 ) .
i z S 1 = 1 8 K 2 S 1 κ ( ( I + S 1 2 + 2 Re ( S 1 S 2 * ) ) S 1 + ( S 1 2 + 4 Re ( S 1 S 2 * ) ) S 2 ) ,
i z S 2 = 9 8 K 2 S 2 κ ( ( I + S 2 2 ) S 2 + ( S 1 2 + 4 Re ( S 1 S 2 * ) ) S 1 ) .
H = 9 4 K 2 S 2 2 + 1 4 K 2 S 1 2 1 2 κ ( I 2 + 2 S 1 4 + 2 S 2 4 + 8 S 1 2 Re ( S 1 S 2 * ) + 16 ( Re ( S 1 S 2 * ) ) 2 )
z I 2 = 2 κ ( S 1 2 + 4 Re ( S 1 S 2 * ) ) Im ( S 1 S 2 * ) ,
A 2 + B 2 + C 2 = I 2 4 ,
2 K 2 A + κ ( A 2 + IC + 2 AC + 4 C 2 ) = const ,

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