1. Introduction

Bright spatial solitons in optics are light beams that conserve their size and shape through a balance between diffraction and self-focusing [1]. They have been reported and investigated over the years in various media with quadratic, cubic, quintic, saturable, local as well as nonlocal responses [2–5]. Spatially-nonlocal nonlinearities have attracted a great deal of interest because of their ability to prevent the catastrophic collapse of finite size beams under self-focusing, thus supporting the formation of stable scalar [6–8] as well as vector solitons [9–14].

Typically, single-input fundamental bright solitons are bell-shaped with one hump. Multi-peak stationary vector solitons can be formed, e.g., by simultaneously propagating coupled nonlinear beams [15] but, as such, they result from several high-order solitons and exhibit nontrivial phasefronts. Some of us recently reported on the existence of multi-peak scalar fundamental solitons - supermode solitons - in systems with competing nonlinearities of opposite signs [16,17]. Among them are, e. g., reorientational liquid crystals in the presence of absorption [18], photorefractive materials with thermal effects, Lithium Niobate crystals subject to additional photogalvanic or pyroelectric responses [19,20]. At variance with self-guided soleakons [21], such non-diffracting multi-humped wavepackets are supermodes of the self-induced (all-optical) waveguide and may exhibit a bistable power-vs- amplitude characteristic [17,22].

In this Paper we study Kerr-like multi-hump spatial solitons generated in a nonlocal medium when opposite (focusing and defocusing) nonlinearities compete due to either scalar (one-beam) or vectorial (two-beam) excitations. Although both scalar and vector supermode solitons have flat phasefronts, their dynamics and stability show marked differences; vector supermodes, specifically, are more amenable to experimental observations when, for instance, employing two colors in self-focusing materials with a spectrally localized (defocusing) absorption resonance [18,23]. We explore the family of multi-peak supermode solitons and investigate their features and stability, even in the presence of linear absorption. We also find spontaneous spatial symmetry breaking in the trajectory of self-guided two-color supermodes for certain range of parameters.

2. One-dimensional Kerr-like nonlocal model

We consider the nonlinear propagation of one-dimensional ((1+1)D) light beams under the paraxial approximation. Their slowly-varying envelope evolution can be simply described by:

For the sake of simplicity, we assume non-dispersive Kerr nonlinear indices, thus $\alpha _1 = \alpha _2 = \alpha$, $\rho _1 = \rho _2 = \rho$ and $\beta _1 = \beta _2 = \beta$. Substituting (4) in (3) and rescaling with: $X = \dfrac {x}{\sigma _a}$, $Z= \dfrac {-z}{2 n_{02} k_{02} \sigma _a^2}$, $\psi _i = \sqrt {\dfrac {2 n_{02} k_{02}^2 \sigma _a^2 ( \alpha \int \vert E_1 \vert ^2 dx + \rho \int \vert E_2 \vert ^2 dx )}{\int \vert E_i \vert ^2 dx}} E_i$, we obtain the coupled equations

3. Results and discussion

Stationary bright multi-hump solitons of the system Eqs. (6) were investigated using the imaginary-time method [24–26] and confirmed by the modified squared-operator method [27]. Soliton propagation was studied by employing the finite-difference beam-propagation method with a Crank-Nicholson algorithm [28] and evanescent-field boundary conditions [29] in order to eliminate reflections. We injected complex noise into each propagating component in order to assess the soliton stability.

3.1 Supermode solitons in the scalar (one-beam) case

Single-beam supermode solitons stemming from opposite competing nonlinearities correspond to Eqs. (6) for $K=N=1$ and/or $A=0$. In the former case the equations are identical as $\psi _1=\psi _2$. In the latter the second equation is independent of $\psi _1$ and contains neither $N$ nor $K$. In this limit the first of Eqs. (6) describes the linear propagation of $\psi _1$ in the all-optical potential determined by the $\omega _2$ beam. Hence, in the limits $K=N=1$ or $A=0$ Eqs. (6) correspond to the scalar case and, at high enough excitations, support two-hump fundamental spatial solitons [16]. Figures 1(a) and 1(b) show several examples of these supermode solitons, i.e., flat-phase fundamental solitons with profiles evolving from single to multiple humps versus beam intensity and nonlocality range, respectively.

 

Fig. 1. Top: emergence of multi-hump scalar supermode solitons for $A=0, K=1, N=1, B=0.5$. (a) Transverse profile for $\sigma =0.7$ versus input power $P$; (b) transverse profile at constant power $P=500$ versus ratio $\sigma$. Bottom: normalized transverse intensity distribution $|\Psi _n|^2=|\Psi |^2/max{|\Psi |^2}$ of scalar supermode solitons (blue ) and normalized nonlinear potential $F_n=F/max{(F)}$ (magenta) for cases with (c) one hump, (d) two humps and (e) deformed-bell shape; the overhanging panels (dashed boxes) show zoomed-in details of the potential. Here $N=1, K=1, A=0, B=0.5$

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The origin of multiple humps can be understood by examining the nonlinear potential of system (7) in its scalar limit: $F(\psi ) = \left [ R_a \ast \vert \psi \vert ^2 \right ] - \textrm {B} \left [ R_r \ast \vert \psi \vert ^2 \right ]$. When the beam power or the nonlocality range mismatch are small, the overall potential remains bell-shaped, with a width $w_F \simeq \sqrt {w^2 + 1}$ where $w$ is the beam waist. As the competition between focusing and defocusing gets pronounced, the potential $F$ tends to flatten owing to feedback between beam and self-induced waveguide profiles, while additional peaks begin to appear, as apparent in Figs. 1(c)-(e) for three sample values of the nonlocality ratio $\sigma$.

The formation of bright solitons requires a positive overall nonlinearity, which poses a limitation $0<B<1$ on the defocusing strength. In addition, the second derivative $\partial _{XX} F(\psi _1, \psi _2 )$ of the nonlinear potential needs be positive near the beam axis, in order to ensure a minimum in $F$ and the insurgence of two humps. Such minimum requires defocusing to be more localised with $\sigma < 1$.

To evaluate the nonlinear potential we assumed solitons at threshold to have a Gaussian shape with $\psi _1 = \psi _2 = \psi _{max} \exp (- \dfrac {X^2}{w^2})$, with $w$ the beam waist as introduced above. This yields the necessary condition

Since the soliton width cannot be evaluated exactly, we can gain some insight by assuming the nonlocality to be high ($\sigma _{r,a} \gg w$). In this limit we have:

Figure 2 presents a map of supermode soliton states in the scalar case. The number of humps depends on $\sigma$: as the ratio decreases, more and more humps appear until they eventually merge into a smoother profile (see, e. g., the rightmost case in Fig. 2).

 

Fig. 2. Number of humps for scalar supermode solitons. The dashed line plots $\sigma = \sqrt [3]{B}$. Upper graphs: intensity profiles calculated for $B=0.7$ at $\sigma$ values marked by circles in the bottom map, with corresponding color coding. The map was calculated for $N=1, K=1, P=500$.

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3.2 Supermode solitons in the vector (two beam) case

Here we consider two beams of distinct colors collinearly co-propagating and each exciting one of the opposite nonlinear responses, i. e., $A=1$ in Eqs. (6); we assume $N \approx 1$ for $\omega _1$ and $\omega _2$ within the optical (VIS-NIR) spectrum. The model behaves differently for $K$ larger or smaller than one, because the two wavefunctions have unequal nonlinearity-to-diffraction ratio and the latter is proportional to $K^2$ for $\psi _1$ evolution. We found stable supermode solutions for $K \leq 1$ when the defocusing component had a multi-peak structure while the focusing one remained single-humped. Figure 3(a) displays the marked difference between profiles as $K$ varies in the interval $0.4 \leq K \leq 1$.

 

Fig. 3. (a) Profiles of two-hump solitons for various wavevector ratios $K$. Here $N=1, A=1, B=0.5, \sigma =0.7, P=500$. (b) Three-hump soliton profiles for $A=0$ and $A=1$. Here $N=1, K=0.6, B=0.6, \sigma =0.6, P=500$. Note that when $A=0$ the overall nonlinear response depends on the defocusing beam. In this case the focusing component evolves according to linear propagation in the external potential due to defocusing.

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For $K>1$ our numerical experiments yielded collapse of vector supermode solitons, with instabilities ignited in the self-focusing beam; the defocusing component showed a similar behaviour at slightly longer propagation distances. Figure 4(a) illustrates this trend: we added a 0.1% initial amplitude noise and observed collapse of the supermode wavepacket as well as an intriguing bending of its trajectory. As $K$ approached unity, the propagation distance before collapse diverged asymptotically, implying supermode soliton stability for $K \leq 1$. Some insight on the unstable behavior for $K > 1$ can be gained by observing that $K$ does not play a role in the second equation of system (6), whereas $K$ multiplies the nonlinear term in the first equation, thereby reducing its equalizing role for $K>1$ and making the vector soliton vulnerable to collapse. This was confirmed by simulations as, by setting $N K^2=1$ in the first equation of (6), supermode solitons became stable for every $K$. It is worth underlining that such instability does not occur in scalar case, i. e., when $A=0$.

 

Fig. 4. (a) Evolution of an unstable vector supermode soliton in the propagation plane $(x, z)$ for $K=1.2$. Here $N=1, A=1, B=0.5, \sigma = 0.65, P=500$. (b) Trajectory bending and collapse of a vector supermode soliton. Here $N=1, K=0.9, A=1, B=0.5, \sigma = 0.65, P=500$. (c) Evolution of a Vector supermode soliton in the presence of absorption $\alpha = 0.25$. Here $N=1, K=0.9, A=1, B=0.5, \sigma = 0.65, P=500$.

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In general, vector supermode solitons for $K \neq 1$ present a richer domain of configurations than found in the scalar case, as visible in Fig. 5 for $K=0.6$. The self-defocusing component exhibits similar dynamics as described above (Section 3.1) for scalar solitons: the number of peaks increases as the ratio $\sigma$ decreases; meanwhile, the peak to valley contrast decreases, as well, until a distorted, quasi-triangular, profile emerges. Conversely, in the transitions from single to multiple humps (see Fig. 5), the self-focusing component tends directly towards a triangular profile as the nonlocality ratio $\sigma$ decreases, without intermediate states encompassing multiple peaks. We could even observe supermode solitons with two peaks in the focusing component and four in the defocusing one (not shown in Fig. 5).

 

Fig. 5. Number of humps versus defocusing strength $B$ in vector supermode solitons for $A=1, K=0.6, N=1, P=500$. The dashed line represents $\sigma = \sqrt [3]{B}$. Right: intensity profiles for $B=0.5$ at $\sigma$ values marked by circles on the bottom graph. Line colors correspond to those in the map. The solid/dotted lines refer to focusing/defocusing beams, respectively.

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Our simulations of vector supermode solitons revealed some unexpected dynamics as a small amount of (either amplitude or phase) noise resulted in random trajectory bending. The amount of amplitude noise affected the distance over which such instability occurred before soliton collapse for every $K>1$ (see Fig. 4). Such trajectory instability can be attributed to the mutual coupling of Eqs. (6) even in the limit $A=1$ examined above, and is a novel example of purely optical spontaneous symmetry breaking (SSB) with self-localized light beams in space [30]. At variance with earlier reports in liquid crystals where both topological and optical SSB were intertwined [30], here the nonlinear interplay of light beam components in the presence of noise induces an internal imbalance in transverse momentum, with an asymmetric phasefront distortion resulting in off-axis acceleration. When analyzing this intriguing SSB phenomenon, we observed it to disappear for $A$ going to zero (scalar supermode solitons). In addition, trajectory SSB occurred even for $K$ approaching unity and was less pronounced for vector supermode solitons with a small number of humps and thus larger $\sigma$.

In several materials, linear absorption is often linked to thermal effects causing self-defocusing, e.g., in nematic liquid crystals [18,23,31]. Small absorption can also help stabilizing self-localized structured beams over finite propagation distances [32]. In order to verify its role in conjunction with vector supermode solitons, we added a dissipation term $-i \alpha \psi _2$ to the second of Eqs. (6). Typical results for small $\alpha$ are presented in Fig. 4(c). Spontaneous trajectory symmetry breaking is still seen to take place, with additional dynamics as the vector solution oscillates from double to single-peak states, while the power in the self-defocusing component gets attenuated. Eventually, fission and decay of vector supermode solitons are observed. As expected, an increased absorption shortens the propagation distance before soliton collapse.

Finally, while the Gaussian kernel may not match realistic materials or configurations, we found that our results are substantially independent of the detailed functional shape of the nonlocality $R$, as we verified by repeating all simulations with a Lorenzian profile. The corresponding supermode states remained essentially unchanged, even though the vector solutions became slightly less robust that in the Gaussian nonlocal response case discussed above.

4. Conclusions

In conclusion, we theoretically and numerically investigated the formation and most relevant features of scalar and vector bright supermode solitons in a system with competing nonlocal nonlinearities of opposite signs, driven by either a single beam or two co-propagating incoherent beams of distinct wavelengths, respectively. Vector supermode spatial solitons exhibit a wealth of features, including different number of humps in each component, stability when the ratio of the wavelengths exceeds unity, spontaneous symmetry breaking instabilities with trajectory bending and oscillations.

As the use of two distinct light beams governing opposite nonlinear responses allows tailoring the competition between self-focusing and self-defocusing, these results hold great potentials towards the experimental demonstration of supermode solitons in media with spectrally-decoupled competing nonlinearities, including nematic liquid and photorefractive crystals in the presence of a weak absorption resonance.