## Shaping the optical components of solitary three-wave weakly coupled states in a two-mode waveguide

Optics Express, Vol. 11, Issue 14, pp. 1643-1649 (2003)

http://dx.doi.org/10.1364/OE.11.001643

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### Abstract

Temporal Bragg solitary waves in the form of collinear three-wave weakly coupled states are studied theoretically and experimentally in a two-mode optical waveguide, exhibiting square-law nonlinearity. The dynamics of shaping their optical components, bright and dark, is studied, and the roles of localizing pulse width and phase mismatch are revealed.

© 2003 Optical Society of America

## 1. Introduction

3. A.S. Shcherbakov and A.Aguirre Lopez. “Observation of the optical components inherent in multi-wave non-collinear acousto-optical coupled states.” Opt. Express **10**, 1398–1403 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1398 [CrossRef] [PubMed]

## 2. Fundamental properties of stationary collinear three-wave coupled states

_{k}(k=0, 1, 2) are the normalized complex amplitudes; 2q is the mismatch of wave numbers. Using the substitutions C

_{k}=a

_{k}exp(iφ

_{k}); one can convert Eqs. (1) to the following equations for the real amplitudes a

_{k}and the real phases φ

_{k}

_{k}, E

_{k}, and λ

_{k}are the constants determined by the boundary conditions. Equations (2) for the amplitudes a

_{k}have the following solutions

_{k}, β, η and the modulus κ=β/η of elliptic functions do not depend on the coordinate x. They are all determined by the boundary conditions and the mismatch q. The parameters α

_{k}specify the backgrounds. When α

_{k}=0, we yield ζ

_{k}=0, λ

_{0}=-η

^{2}β

^{2}(1-κ

^{2}), λ

_{1}=η

^{2}β

^{2}, λ

_{2}=-η

^{4}(1-κ

^{2}) and linear dependences of the phases φ

_{k}on the coordinate x. The terms b

_{k}represent the oscillating portions of solutions, evaluating the extent of localization for the coupled state. For the functions b

_{k}one can find

_{j}(j=0,1,2) are the constants; F

_{0}-F

_{1}=F

_{2}and F

_{1}>0. Equations (4) can be converted into three equations, independent of each other, with cubic-law nonlinearity, see Eqs. (5a). These equations can be considered as the motion equations d

^{2}b

_{i}/dx

^{2}=-dU

_{i}/db

_{i}for some particles in the real-valued potentials U

_{i}(b

_{i}), see Eqs. (5b)

_{0}=-1, p

_{0}=F

_{0}, q

_{1}=1, p

_{1}=-F

_{1}, q

_{2}=-1, and p

_{2}=-F

_{2}; H

_{k}=0 for the oscillating portions of solutions. For k=1 with b

_{1}(x

_{0})=0, Eq.(5b) gives the potential that has a local minimum at b

_{1}=0 and two absolute maxima at

_{1}(b

_{1}) and carry the topological charge Q=Δ [b

_{1}(x→+∞)-b

_{1}(x→-∞)], where Δ is a constant [2, 4, 5]. The topological charge Q reflects conservation of the boundary conditions of optical components inherent in the stationary coupled state. Substituting U

_{1}(b

_{1}) in Eq.(5a), we yield the solitary kink solution:

_{1}(x) represents a shock wave of envelope or the dark optical component of the coupled state. The topological charge, associated with the wave b

_{1}, can estimated as Q=±1 with Δ=(2F

_{1})

^{-1/2}.

_{0}and b

_{2}. The potentials U

_{k}(b

_{k}) with k=(0, 2) are given by Eqs. (5b) with F

_{0}>0 and F

_{2}<0. Each of these potentials exhibits only one local maximum at b

_{k}=0. The waves b

_{0}and b

_{2}shape the bright components

_{k}(x→-∞)=b

_{k}(x→+∞)=0 and Q=0. For these waves the even potentials correspond to the even asymptotes b

_{k}(x→-∞)=b

_{k}(x→+∞)=0, but at any finite distance the symmetry in these waves turns to be broken, because the absolute minima of U

_{k}(b

_{k}) are degenerated, and they can be reached in two different points instead of one. The particles oscillate spontaneously only in one direction, towards the right or the left, of the local maxima of U

_{k}(b

_{k}). Since the symmetrical states with the least energy at b

_{k}=0 are unstable, either of two signs can be realized in the relations b

_{k}(x

_{0})=±|b

_{k}(x

_{0})|. This phenomenon is known as the spontaneous breaking of symmetry [5] that is inherent in topologically uncharged bright components of the coupled state.

## 3. Weakly coupled states in the quasi-stationary case; the localization conditions

_{2}and being of non-optical nature. Because the number of interacting photons is several orders less than the number of scattering non-optical slow quanta in a medium, essentially effective Bragg scattering of light can be achieved without any observable influence of the scattering process on that non-optical wave. The velocities of light modes can be approximated by the same value c, because the length of a waveguide does not exceed 10 cm. In this regime, the above-mentioned set of three combined nonlinear partial differential equations [4], describing a three-wave co-directional collinear interaction with mismatched wave numbers, has to be transformed and falls into a homogeneous wave equation for a slow wave, which possesses the traveling-wave solution U(x-vt), v is its velocity, and the pair of combined equations

_{0,1}=a

_{0,1}(x, t)exp(iΦ

_{0,1}[x, t]), γ

_{0,1}=∂Φ

_{0,1}/∂x, Eqs. (6) can be converted into equations

_{0,1}=±q (u/

^{-1}(∂

_{0,1}u/

_{0,1}=0. Now, we focus on the process of localization in the case, when first, two facets of a waveguide at x=0 and x=L

_{0}bound the area of interaction and the spatial length l

_{0}of the non-optical pulse is much less than L

_{0}; and second, the non-optical pulse u(x, t)=U

_{0}(θ[z-vt]-θ[x-l

_{0}-vt]) has a rectangular shape with the amplitude U

_{0}. We analyze Eqs. (7) and (8) with the fixed magnitude of q and the natural boundary conditions a

_{0}(x=0, t)=1, a

_{1}(x=0, t)=0 and trace the dynamics of the phenomenon as far as the localizing pulse of the non-optical wave is incoming through the facet x=0, passing along a waveguide, and issuing through the facet x=L

_{0}with the constant velocity v. There are two possibilities. The first of them is connected with a quasi-stationary description of this effect with the assumption that v≪c, while the second one presupposes a weak inequality v<c. With a quasi-stationary approach, we may put ∂u/∂x≈0 in Eqs. (7), (8) everywhere, excluding the points x={0, l

_{0}}, and yield γ

_{0,1}=±q. Then, we follow three stages in the localization processes.

**Stage 1: Localizing pulse is incoming through the facet x=0:**Exploiting γ

_{0,1}=±q, Eqs. (11) can be solved exactly. The intensities of light waves on x∈(0, l

_{0}) are given by

**Stage 2: Localizing pulse is passing in a medium.**The rectangular pulse as the whole is in a waveguide, so ∂u/∂x=0 and x=l

_{0}in Eqs. (9) for the region (l

_{0}, L

_{0}-l

_{0}).

_{0}|

^{2}in Eqs. (9) exhibits a background, whose level is determined by the mismatch q; the second one represents the oscillating portion of solution, i.e. the localized part of the incident light imposed on a background. The scattered light contains the only oscillating portion of field that gives the localization condition

^{2})=π

^{2}N

^{2}, where x

_{C}is the spatial size of the localization area with v≪c and N=0, 1, 2, …

_{0}x (α is to be found), when the localizing pulse is incoming through the facet x=0. In so doing, we have to take into account the fact that the solution to Eqs. (7) is known only if the last coefficients are proportional to u

^{2}[6], i.e.

_{0,1}=q

^{2}ζ

^{2}x

^{2}with ζ=

*const*. That is why we are forced to exploit the smallness of mismatch, believing that q≪1, and to find approximate solutions to Eqs. (7), (8) at this stage. Resolving this algebraic equation relative to γ

_{0,1}, we yield

_{0,1}, Eqs. (8) can be satisfied with an accuracy of q

^{2}, while Eqs. (7) can be solved exactly. The intensities of light waves with α=ζ on the interval of x∈(0, l

_{0}) are given by

_{0,1}as γ

_{0}≈-qxζ and γ

_{1}=-q[(4/3)+xζ] on the interval of x∈(0, l

_{0}) and then use the conservation law. Stage 2 with v<c is governed by Eqs. (9) as well, because again ∂u/∂x=0; finally, we can invert and apply Eqs. (10) to stage 3. The parameter α makes it possible to join Eqs. (9) and (10) at the point l

_{0}, therefore the localization condition takes the form α

^{2}

^{2})=4π

^{2}N

^{2}, where x

_{S}is the spatial size of localization area with v<c.

## 4. Computer simulation and experimental verification in the quasi-stationary case.

_{1}|

^{2}, when both the amplitude U

_{0}and the mismatch q are fixed, while its width τ

_{0}=l

_{0}/v is increasing plot by plot in the temporal scale of τ

_{C}=x

_{C}/v. Figures 1(b) and (d) illustrate shaping the scattered optical components of one- and two-pulse weakly coupled states.

_{0}=3 cm) and possessed the photoelastic constant p

_{45}=0.06, making possible to couple two optical modes. The schematic arrangement of the experiments was similar to the scheme for acousto-optical filtering [7] and includes a continuous-wave polarized light beam, a crystalline waveguide, output analyzer, and photodetector. During the experiments rather effective (> 10%) Bragg scattering of the light was observed without any effect on the acoustic wave, when their powers were approximately equal to 100 mW each, so the regime of weak coupling had taken place. However, the emphasis was on the dynamics of shaping the coupled states and the quantitative estimation of their temporal characteristics and not on the amplitude parameters of this process. The intensity distributions in both incident and scattered optical components of coupled states as functions of the acoustic power density, the localizing pulse width τ

_{0}, and the frequency mismatch Δf=q v/π has been measured. The oscilloscope traces in Fig. 2 illustrate the particular case, when only the localizing pulse width τ

_{0}is varied. One can see four sequential steps in shaping the optical components of the coupled states in a waveguide, which are in agreement with the analysis performed, see Eqs. (9) and Fig. 1.

_{0}is varied, see Fig. 2 and Figs. 3(a) and (b). However, if the mismatch Δf increases, the amplitude of the coupled state decreases with fixed τ

_{0}and P, compare traces in Figs. 3(a) and (c).

## 5. Conclusion

8. D.E. Pelinovsky and Yu.S. Kivshar, “Stability criterion for multi-component solitary waves,” Phys. Rev. E **62**, 8668–8676 (2000). [CrossRef]

## Acknowledgement

## References & links

1. | A.P. Sukhorukov. |

2. | A.S. Shcherbakov. |

3. | A.S. Shcherbakov and A.Aguirre Lopez. “Observation of the optical components inherent in multi-wave non-collinear acousto-optical coupled states.” Opt. Express |

4. | R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Morris. |

5. | R. Rajaraman. |

6. | E. Kamke. |

7. | F. Yu. |

8. | D.E. Pelinovsky and Yu.S. Kivshar, “Stability criterion for multi-component solitary waves,” Phys. Rev. E |

**OCIS Codes**

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Research Papers

**History**

Original Manuscript: April 14, 2003

Revised Manuscript: June 14, 2003

Published: July 14, 2003

**Citation**

Alexandre Shcherbakov and A. Aguirre-Lopez, "Shaping the optical components of solitary threewave weakly coupled states in a two-mode waveguide," Opt. Express **11**, 1643-1649 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-14-1643

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### References

- A.P.Sukhorukov. Nonlinear Wave Interactions in Optics and Radiophysics. (Nauka Press, Moscow. 1988).
- A. S.Shcherbakov. A three-wave interaction. Stationary coup led states. (St. Petersburg State Technical University Press, St. Petersburg. 1998).
- A.S.Shcherbakov and A.Aguirre Lopez. �??Observation of the optical components inherent in multi-wave non-collinear acousto-optical coupled states.�?? Opt. Express 10, 1398-1403 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1398">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-24-1398</a>. [CrossRef] [PubMed]
- R.K.Dodd, J.C.Eilbeck, J.D.Gibbon, and H.C.Morris. Solitons and Nonlinear Wave Equations. (Academic Press, Orlando. 1984).
- R. Rajaraman. Solitons and Instantons, North-Holland Publishing Company, Amsterdam (1982).
- E.Kamke. Differentialgleichungen. Losungmethoden und Losungen. Part I (Chelsea Co. NY. 1974).
- F.Yu. Introduction to Information Optics. (Academic Press, San Diego. 2001).
- D.E.Pelinovsky and Yu.S.Kivshar, �??Stability criterion for multi-component solitary waves,�?? Phys. Rev. E 62, 8668-8676 (2000). [CrossRef]

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