1. Introduction

Symmetry breaking can be easily found in many physical systems [1, 2]. It takes place when a symmetric system in a meta-stable state is able to transit to an asymmetric state with lower potential. Like a ball sitting on the top of a symmetric hill, it may fall down to the left or right depending on the initial perturbation.

Discrete solitons, that were first discovered by Christodoulides et al. [3], with even parity have a similar behavior. When an even discrete soliton [4] propagates in a nonlinear waveguide array, it is unstable [5, 6] and will gradually become asymmetric due to perturbation. The symmetry breaking instability does not occur in the homogenous medium. The in-phase symmetric beams attract each other and the π -out-of-phase antisymmetric beams repel each other. In both cases, the total intensity profiles remain symmetric. However, for the discrete solitons in the nonlinear waveguide array, whether they are in-phase or π -out-of-phase in the adjacent waveguides, they will undergo the symmetry breaking. This is because the beam tends to self-trap to the position with higher intensity, so an initial intensity difference will be amplified, which results in the devastating symmetry-breaking [5, 6]. This argument about symmetry breaking in the discrete regime is based on the following premise: the nonlinear index change is much smaller than that of the original refractive index wells [Fig. 1(a)]. Under this premise, one can say that the beam never induces a defect at the central potential barrier, but instead, it needs to ‘choose’ a well on its left or right side and therefore becomes asymmetric.

Following this direction, we studied the symmetry breaking of optical solitons in the nonlinear waveguide array and investigated the relation between the nonlinearity and the degree of symmetry breaking. Furthermore, being inspired by the earlier works that focused on the suppression of modulational instability (MI) [7, 8], transverse instability (TI) [9], and azimuthal instability (AI) [10] by partially incoherent light, in this paper, we try to understand whether the other type of instability, say, the symmetry breaking instability, can be suppressed by rendering the discrete solitons partially incoherent.

 

Fig. 1. (a) Refractive index (b) The dimensionless potential function. The waveguide displacement is d=7 µm. Note that the center is the potential barrier. (c) The bandgap structure. (solid line) The diffraction relation with periodically modulated refractive index. (dotted-line) The diffraction relation in a homogeneous medium. The gray area shows the forbidden band. In the first band, normal diffraction occurs at the central region (k_x~0), while anomalous diffraction occurs near the edge (k_x~±π/d).

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For MI and TI, the light wave can become stable if the coherence degree is below a specific threshold value [7, 9]. The azimuthal instability can also be partially suppressed provided that the degree of incoherence of the optical vortices [10] is strong enough. In our study, we found that the symmetry-breaking instability can be suppressed by adding spatial incoherence.

2. Discrete solitons and symmetry breaking

We consider a slowly varying normalized beam envelop A(x, z) propagating along z-direction in a photonic structure with a periodic modulation of the optical refractive index in x-direction. Propagation in such system can be described by the nonlinear Schrödinger equation with a periodic potential function [4],

where $k=\frac{2\pi \overline{n}}{\lambda }$ (λ=0.532µm and n̄=2.35 in this simulation), n′ ₂ represents the strength of the nonlinearity, and ν(x) is the dimensionless periodic potential satisfying the relation n(x)=[n ^̄2(1-ν(x))]^1/2, where n(x) and n̄ are the local and the average refractive index, respectively [Fig. 1(a)]. Note that we assume the potential ν(x) satisfying ν(x)=ν(-x) for convenience, and assume that the magnitude of the nonlinear term n′ ₂|A|² is much smaller than that of the modulated refractive index ν(x) to meet the premise mentioned above.

Equation (1) at least supports two kinds of discrete solitons, staggered and unstaggered ones, when the wave function in the adjacent waveguide [11] is in-phase (π -out-of-phase), the nonlinearity has to be positive (negative), i.e., self-focusing (self-defocusing), in order to form the unstaggered (staggered) discrete solitons. Here, we present the unstaggered discrete solitons first. We assume a real and even input field envelop satisfying A ₀(x,0)=A ₀(-x,0). To make the symmetry breaking occur in a relatively short distance, we put a small random noise, such that A(x,0)=a(x)A ₀(x,0)+ε (x), a(x)=1+0.1×rand(x), ε (x)=0.01×rand(x), and rand(x) ∊ [0,1] is the uniformly distributed pseudo-random variable.

 

Fig. 2. The symmetry breaking of the unstaggered discrete solitons. The input (gray) and output (black) intensity profiles (upper row) and evolution of the beam along z-direction (lower row). (a) n′ ₂=0.000585 and M_out=0.0318. (b) n′ ₂=0.000729 and M_out=0.2107. (c) Higher nonlinearity results in higher M_out, and higher M_out means more unstable discrete solitons.

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Fig. 3. The symmetry breaking of the staggered discrete solitons. The input (gray) and output (black) intensity profiles (upper row) and evolution of the beam along z-direction (lower row). (a) n′ ₂=-0.000744 andM_out=0.0182. (b) n′ ₂=-0.000912 M_out=0.3811. (c) Stronger self-defocusing nonlinear effect leads to more unstable staggered discrete solitons.

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To describe the symmetry breaking phenomenon, we define a normalized intensity contrast that indicates the extent of instability:

where I _pl and I_pr are the peak intensity on the left and right first waveguides, respectively. It is clearly seen that M_out is larger than M_in due to the symmetry breaking instability [Fig. 2]. For a given waveform in the same waveguide array, the higher nonlinear effect is, the more unstable the beam becomes (larger M_out) [Fig. 2(c)]. We then consider the staggered-mode soliton, whose Floquet-Bloch spectrum lies at the bottom of the first band, which requires self-defocusing nonlinearity. We found the similar dependence of symmetry breaking instability on the strength of the nonlinearity [Fig. 3]. Even in an instance of the simulation, it becomes so asymmetric and finally transform into the one-peak discrete soliton [Fig. 3(b)].

3. Partially coherent discrete solitons and symmetry breaking

Most studies about spatial optical solitons mainly focused on coherent entities until some theoretical [12, 13] and experimental [14] studies on the incoherent solitons have been conducted. Even more recently, the interplay of discrete diffraction (normal/anomalous diffraction) in the periodic modulated refractive index and statistical properties, both spatially [15, 16, 17] and temporally [18], has been studied. We therefore would like to study the symmetry breaking phenomena of partially incoherent discrete solitons with even parity. The method we used is the commonly used coherent density approach [12]. Here we simulate only for the noninstantaneous Kerr nonlinearity, which can be obtained by photorefractive nonlinearity at low saturable regime [9, 19] when a suitable background light is added or by the thermal nonlinearity.

 

Fig. 4. The symmetry breaking of coherent and partially incoherent discrete solitons. The input (gray) and output (black) intensity profiles (upper row) and evolution of the beam along z-direction (lower row). (a) The coherent unstaggered discrete soliton, with n′ ₂=0.000639 and M_out=0.0707. (b) The partially incoherent (θ ₀=0.0025) unstaggered discrete soliton, with n′ ₂=0.000639 and M_out=0.0021. (d) The coherent staggered discrete soliton, with n′ ₂=-0.000936 and M_out=0.5048. (e) The partially incoherent (θ ₀=0.0025 rad) staggered discrete soliton, with n′ ₂=-0.000936 and M_out=0.0304. (c)(f) The M_out versus the strength of nonlinearity under different spatial incoherence. (c) for unstaggered and (f) for staggered modes.

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The evolution of the partially coherentwave A(x, z) in the periodic potential can be effectively described by an infinite set of the coupled paraxial nonlinear differential equations in the limit Δθ→0 [4, 12], and with each coherent component satisfying

At the input facet, each component A _j(x,0) is weighted by the angular spectrum G_N(θ)=(π ^1/2 θ ² ₀)^-1 exp(-θ ²/θ ²0), such that A_j(x,0) ∝G ^1/2 _N (jΔθ)A(x,0) with A(x,0) being the same noise-adding input amplitude in Section 2, and θ ₀ represents the degree of the incoherence. In our numerical simulation, we use 201 components (-100≤j≤100) from -3θ ₀ to 3θ ₀. The results of the simulation are only shown by the total intensity I(x, z)=Σ¹⁰⁰ _j=-₁₀₀ |A_j(x, z)|².

The results of the symmetry breaking instability for the unstaggered mode under different degrees of spatial coherence are shown in Figs. 4(a)–4(c). When the light is made partially incoherent (θ ₀=0.0025 rad), the symmetry breaking instability is suppressed [Fig. 4(b)] with M_out=0.002 as compared to the case M_out=0.071 when the light is coherent under the same strength of self-focusing nonlinearity. We also simulate for other degrees of spatial coherence. Figure 4(c) shows that in all different nonlinearity strength, the incoherence can indeed decrease the symmetry breaking instability or even suppress it. We believe the mechanism behind the suppression of symmetry breaking instability resembles that in MI, TI, and AI. The symmetry breaking of an even beam is due to the fact that the intensity difference made by the initial random noise will be amplified by the nonlinear effect. However, when the beam is made partially incoherent, this amplification is compensated by the stronger diffraction of an incoherent light beam. We then proceed to the simulation for the staggered modes. As we compare the results of Fig. 4(d) and Fig. 4(e), it can be seen that the symmetry breaking instability is greatly reduced from M_out=0.51 to M_out=0.03 when the light is made partially incoherent with θ ₀=0.0025 rad. With a series of simulations, Fig. 4(f) shows the result similar to that of the unstaggered mode; the incoherence can suppress the symmetry breaking instability of the staggered mode as well. It is also noted that, when adding incoherence, the staggered mode is spread wider during propagation than the unstaggered mode, making the intensity at the central part much smaller. This may be an indication that the width of the staggered mode is more affected by the incoherence than the unstaggered mode.

4. Conclusion

We have studied the symmetry breaking of even discrete solitons in the 1-D nonlinear waveguide array for both coherent and partially coherent, staggered and unstaggered cases. We have demonstrated, for the first time to our knowledge, that the stable propagation of an even beam in a nonlinear waveguide array can indeed be observed provided that the beam is made partially incoherent. This is true for both staggered and unstaggered discrete solitons. The authors gratefully acknowledge National Science Council for the financial support.