## 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

In a number of cases the analysis of three-wave processes leads to finding various solitary waves in the form of coupled states, where waves of the same or even different nature become mutually trapped and propagate together [1–3]. In particular, such coupled states can be shaped via stationary co-directional collinear interaction of two optical modes with some non-optical third wave in a dispersive waveguide due to the balancing action of the square-law nonlinearity. The profiles of all the waves are steady at three different current frequencies, because the interaction exhibits itself as a mechanism of stabilizing self-action. Mismatching the wave numbers can be also included in the analysis, giving us an opportunity to follow the process more sequentially. The development of a quasi-stationary model for describing such a phenomenon and its verification are the subjects of this work. The presented approach offers a clear view of this phenomenon and predicts an opportunity of sculpturing multi-pulse coupled states. Moreover, it allows one to determine a certain topological charge, being intrinsic to each component of the coupled state, as well as to reveal the spontaneously broken symmetry. The analysis was examined experimentally using the Bragg acousto-optical interaction in a two-mode optical waveguide and all the predicted effects were observed.

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

A three-wave co-directional collinear interaction with mismatched wave numbers is described by a set of three combined nonlinear partial differential equations [4]. In the particular case of the stationary regime that set can be reduced and written as

where x is the spatial coordinate, C_{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}

Here ζ_{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

The parameters α_{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

where n,m=0,1,2; k≠n≠m≠k; F_{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)

where q_{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
${b}_{1}=\pm \sqrt{\frac{{F}_{1}}{2}}$
. The stationary localized solution connect these two maxima of U_{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:
${b}_{1}\left(x\right)=\pm \sqrt{\frac{{F}_{1}}{2}}\xb7\mathrm{tanh}\lfloor \sqrt{\left(\frac{{F}_{1}}{2}\right)}\left(x-{x}_{0}\right)\rfloor $
. The kink solution b_{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}.

Let us look now at the waves b_{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
${b}_{0}\left(x\right)=\pm \sqrt{{F}_{0}}\phantom{\rule{.2em}{0ex}}\xb7\mathrm{sech}\phantom{\rule{.2em}{0ex}}\left[\sqrt{{F}_{0}}\left(x-{x}_{0}\right)\right]$
and
${b}_{2}\left(x\right)=\pm \sqrt{-{F}_{2}}\xb7\mathrm{sech}\left[\sqrt{-{F}_{2}}\left(x-{x}_{0}\right)\right]$
of the coupled state with the asymptotes b_{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

Here, we consider a regime of weak coupling, when two light modes are scattered from the pulse of a relatively slow wave, taken instead of the wave C_{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

When the non-optical pulse U(x-vt)=u(x-vt)exp(iφ) has the constant phase φ, and C_{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

It follows from Eqs. (8) that γ_{0,1}=±q (u/${\mathrm{a}}_{0,1}^{2}$) ∫u^{-1}(∂${\mathrm{a}}_{0,1}^{2}$/∂x) dx+Γ_{0,1}u/${\mathrm{a}}_{0,1}^{2}$, but here our consideration will be restricted by the simplest choice of Γ_{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

$${\mid {C}_{1}\mid}^{2}=\frac{{U}_{0}^{2}}{{U}_{0}^{2}+{q}^{2}}{\mathrm{sin}}^{2}\left(x\sqrt{{U}_{0}^{2}+{q}^{2}}\right).$$

To find the coefficients in Eqs. (9) we use the conservation law ${\mathrm{a}}_{0}^{2}$+${\mathrm{a}}_{1}^{2}$=1.

**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}).

**Stage3: Localizing pulse is issuing through the facet x=L**_{0}**.** This stage is symmetrical to stage 1, so Eqs. (9), can be inverted and related to the region of x∈(L_{0}-l_{0}, L_{0}).

The first summand in |C_{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 ${\mathrm{x}}_{\mathrm{C}}^{2}$ (${\mathrm{U}}_{0}^{2}$+q^{2})=π^{2} N^{2}, where x_{C} is the spatial size of the localization area with v≪c and N=0, 1, 2, …

On the second possibility (v<c), it is reasonable to put u=αU_{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. ${\mathrm{\gamma}}_{0.1}^{2}$∓2qγ_{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
${\gamma}_{\mathrm{0,1}}=\pm q\left(1\pm \sqrt{1+{\zeta}^{2}{x}^{2}}\right)$
. In terms of these values for γ_{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

$${\mid {C}_{1}\mid}^{2}=\frac{{U}_{0}^{2}}{{U}_{0}^{2}+{q}^{2}}{\mathrm{sin}}^{2}\left(\frac{\alpha {x}^{2}}{2}\sqrt{{U}_{0}^{2}+{q}^{2}}\right).$$

To find the coefficients in Eqs. (10) we approximate γ_{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}${\mathrm{x}}_{\mathrm{S}}^{4}$(${\mathrm{U}}_{0}^{2}$+q^{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.

Shaping the optical components of solitary three-wave weakly coupled states was simulated using Eqs. (9). As an example, Fig.1 shows a set of plots for the scattered light intensity |C_{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.

Verification of the analysis performed has been carried out due to our trial acousto-optical experiments with an optical wavelength of 633 nm in a two-mode crystalline waveguide based on calcium molybdate oriented along the x-axis (exact synchronism acoustic frequency 43.7 MHz, L_{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.

The amplitudes of one- and multi-pulse weakly coupled states are the same, when the acoustic power density P and the frequency mismatch Δf are fixed and only the acoustic pulse width τ_{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

We conclude from the analysis performed and the experimental data obtained that temporal Bragg solitary waves in the form of quasi-stationary three-wave weakly coupled states can be shaped due to collinear scattering of light by the non-optical wave in a two-mode waveguide. Experimentally observed stability of the coupled states gives us a hope that the adequate stability criterion [8] can be elaborated. The data in Fig. 3 show that the regime of multi-wave coupled states can be potentially applied to electronically controlled conversion of 1B/1B binary encoded electronic digital signals into 1B/NB binary encoded trains of optical pulses.

## Acknowledgement

This work has been supported by the CONACyT, Mexico (Project # U 41998-F).

## References & links

**1. **A.P. Sukhorukov. *Nonlinear Wave Interactions in Optics and Radiophysics*. (Nauka Press, Moscow. 1988).

**2. **A.S. Shcherbakov. *A three-wave interaction. Stationary coupled states*. (St.Petersburg State Technical University Press, St.Petersburg. 1998).

**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]

**4. **R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Morris. *Solitons and Nonlinear Wave Equations*. (Academic Press, Orlando. 1984).

**5. **R. Rajaraman. *Solitons and Instantons*. (North-Holland Publishing Company, Amsterdam. 1982)

**6. **E. Kamke. *Differentialgleichungen. Losungmethoden und Losungen. Part I* (Chelsea Co. NY. 1974).

**7. **F. Yu. *Introduction to Information Optics*. (Academic Press, San Diego. 2001).

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