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.

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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. Sanchez-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. Sanchez -Mondragon, "Theory of multiple beam interaction in photorefractive media," J. Opt. Soc. Am. B 16, 80- 86(1999).
    [CrossRef]
  6. V. I. Arnold Mathematical Methods of Classical Mechanics ( 2-nd edition, Springer-Verlag 1989).
  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 Haar, Elements 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).

Other

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

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

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

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

N. Korneev, A. Apolinar-Iribe,and J. J. Sanchez -Mondragon, "Theory of multiple beam interaction in photorefractive media," J. Opt. Soc. Am. B 16, 80- 86(1999).
[CrossRef]

V. I. Arnold Mathematical Methods of Classical Mechanics ( 2-nd edition, Springer-Verlag 1989).

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).

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

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

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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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