## Analytical solutions for three and four diffraction orders interaction in Kerr media

Optics Express, Vol. 7, Issue 9, pp. 299-304 (2000)

http://dx.doi.org/10.1364/OE.7.000299

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### 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.

© Optical Society of America

## 1 Introduction.

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

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]

## 2 The Hamiltonian for interacting diffraction orders.

*ψ*is the amplitude of wave depending on one transversal coordinate

*x*and the propagation coordinate

*z; κ*is the nonlinear coefficient, which is positive for self-focusing media and negative for self-defocusing ones. We are interested in the initial value problem when

*ψ*(

*x, z*) is known for

*z*=

*z*

_{0}, and we need to determine it for

*z*>

*z*

_{0}. The N-order interaction corresponds to the amplitude written as:

*S*

_{k}are the amplitudes of orders and the sum is taken over appropriate wavevectors k. For the odd number of orders we take for

*k*values …-2

*K*,-

*K*, 0,

*K*, 2

*K*…. If the number of orders is even, they are … - 3/2

*K*,-1/2

*K*, 1/2

*K*, 3/2

*K*…. Here

*K*=2

*π*/Λ, where Λ is the interference fringes period. By substituting Eq.(2) into NLSE and combining the coefficients for Fourier harmonics, the infinite system of coupled nonlinear differential equations follows. We can, nevertheless, truncate this infinite system by assuming that higher diffraction orders are small and forcing the corresponding amplitudes to be zero. The coupled differential equations for the general case are written down in [5

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]

*i∂*

_{z}

*ψ*=

*δH/δψ**.

*S*

_{k}and

*S**

_{k}. For soliton problems, the integration in Eq.(6) is done over the whole

*x*axis. In our case we integrate over one period of interference fringes and normalize the result.

*a*can take any possible value of

*k*. The Hamiltonian is real and symmetric with respect to

*S*and its complexconj ugate.

*qk*and momenta

*pk*with

*S*

_{k}=1/√2(

*qk*+

*ip*

_{k}). For the optical task, the complexno tation is more natural.

*C*

_{n}in the original work of Zakharov and Shabat [7]. Of them, apart from the Hamiltonian itself, we will utilize the intensity :

*I*=∑

_{k}

*S*

_{k}

*S**

_{k}, and the momentum:

*M*=∑

_{k}

*kS*

_{k}

*S**

_{k}. Their conservation for our case does not follow automatically from the argument of [7]. It is necessary to prove that truncated sums are integrals of motion for the truncated Hamiltonian. For any function

*U*(

**S, S***) the condition for its conservation is the equality to zero of the Poisson bracket :

## 3 Solution for three orders.

*I*

_{-1},

*I*

_{0},

*I*

_{1}. Then

*I*

_{-1}+

*I*

_{0}+

*I*

_{1}=

*I*, and

*I*

_{1}-

*I*

_{-1}=

*M/K*are conserved. It follows:

*∂*

_{z}

*I*0 is expressed as a square root of the fourth degree polynomial of

*I*

_{0}, and it follows that the intensity is an elliptic function. So, all order intensities are periodic with a propagation length. It is not true for amplitudes, because phases generally do not return to their initial value after a period of intensity. The solutions for amplitudes can be obtained by writing down the equations for

*i∂*

_{z}ln(

*S*

_{k})=

*i∂*

_{z}

*S*

_{k}

*/S*

_{k}. For example, for

*S*

_{0}we have

*S*

_{1}

*S*

_{-1}, and all of them can be expressed as functions of

*I*

_{0}using Eqs.(10–12). It follows that the phases are sums of periodic and linear functions of propagation length. The phase gains over the intensity period for 3 orders are not independent, because the combination

*S*

_{1}

*S*

_{-1}is periodic. In Figure 1 we present the computer calculation for a typical trajectory of the diffraction order. The formal solution in elliptic functions requiers the knowledge of the roots of the general fourth order polynomial, and thus it is quite cumbersome. Practically, the simplest way is the direct computer calculation. The most important property of three orders interaction is the quasi-periodical character of amplitudes and periodicity of intensities. So, it is possible to calculate amplitudes over one period of intensities, and to expand the solution for arbitrary propagation length.

## 4 Symmetrical four orders interaction.

*S*

_{-2}=

*S*

_{2}and

*S*

_{-1}=

*S*

_{1}. These relations are maintained with propagation. Resulting evolution equations include only two orders :

*I*=2|

*S*

_{1}|

^{2}+2|

*S*

_{2}|

^{2}is also the conserved quantity, there are two integrals of motion. This permits us to state that the task is completely integrable. The equation for the second order intensity is:

*H*and

*I*to express the right-hand side of this equation as a function of

*I*

_{2}only, reducing the solution to integration. The integral, nevertheless, is more complicated than the elliptic one. Again, the intensities are periodic with propagation distance and amplitudes are quasi-periodic.

## 5 Representation of solutions on the Poincaré sphere and their properties.

*S*

_{1},

*S*

_{2}we introduce Stokes parameters given by:

*A*=

*I*

_{1}-

*I*

_{2}=

*S*

_{1}

*S**

_{1}-

*S*

_{2}

*S**

_{2},

*B*=-

*i*(

*S*

_{1}

*S**

_{2}-

*S**

_{1}

*S*

_{2}),

*C*=

*S*

_{1}

*S**

_{2}+

*S**

_{1}

*S*

_{2}. The intensity conservation leads to the identity:

*A*(

*z*),

*B*(

*z*),

*C*(

*z*) lie on a sphere. To determine their form one has to rewrite the Hamiltonian in terms of Stokes parameters. This gives the condition:

*A*-

*C*plane are concentric ellipses, and trajectories are intersections of elliptic cylinders parallel to

*B*axis, and the sphere. In Fig.3, we show these ellipces built for illustration purposes with the direct numerical solution of motion equations. The ellipticity and orientation of axes of the ellipses do not depend on

*q*parameter or the ellipse size. Their centers are given by

*q*, the center position moves along the straight line. The A parameter gives the intensity of beams (

*I*

_{1,2}=

*I*/4±

*A*/2), thus the Figure 3 contains a wealth of information about behavior of order intensities in particular situations. For example, trajectories with one maximum per period or with two of them separated by one minimum are possible, and the maximal values can be obtained by solving algebraic ecuations. The self-similar solutions for which which order intensities do not change with propagation are also well seen. For the situation of the Fig.3, there are sixo f them, and two are unstable.

## 6 Discussion and conclusions.

*q*parameter), or limited propagation length when initially the energy was concentrated in a small number of central orders. It seems probable that the approximation to the analytical solution (action-angle variables) can still be obtained by using truncated integrals of motion, though some of these integrals are approximations.

## Acknowledgments

## References and links

1. | H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst.Tech.J |

2. | R. Magnusson and T.K. Gaylord “Analysis of multiwave diffraction of thick gratings,” J.Opt.Soc.Am. |

3. | L. Solymar, D.J. Webb, and A. Grunnet-Jepsen, |

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. |

5. | N. Korneev, A. Apolinar-Iribe, and J.J. Sanchéz -Mondragon, “Theory of multiple beam interaction in photorefractive media,” J. Opt. Soc. Am. B |

6. | V.I. Arnold |

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 , |

8. | D. ter Haar |

9. | V.I. Bespalov and V.I. Talanov “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. |

**OCIS Codes**

(090.7330) Holography : Volume gratings

(190.5330) Nonlinear optics : Photorefractive optics

(190.7070) Nonlinear optics : Two-wave mixing

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 25, 2000

Published: October 23, 2000

**Citation**

N. Korneev, "Analytical solutions for three and four diffraction orders interaction in Kerr media," Opt. Express **7**, 299-304 (2000)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-9-299

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### References

- 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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