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(2+1)D surface solitons at the interface between a linear medium and a nonlocal nonlinear medium

We address (2+1)D surface solitons occurring at the interface between a linear medium and a nonlocal nonlinear medium whose nonlinear contribution to the refractive index has a initial value at the interface. We find that there exist stable single and dipole surface solitons which do not exhibit a power threshold. The properties of the surface solitons can be affected by the initial value and the degree of nonlocality. When a laser beam is launched away from the interface, the beam will be periodic oscillations.

Surface waves localized at the interface of two different optical materials have many novel properties, which have been studied in nonlocal media recently. The light beam trajectory can be strongly affected by the presence of interfaces, because beams propagating in nonlocal media cause refractive index changes in regions far exceeding the beam width. Under proper conditions, stationary surface waves can propagate along the interface in both local nonlinear media [28][29][30][31] and nonlocal nonlinear media [32][33][34][35][36][37][38][39].
Thermal media [34,35,38] and photorefractive crystals [39] have been also utilized to demonstrate (2+1)D surface solitons. In this paper, we will study (2+1)D surface solitons occurring at the interface between a linear medium and a nonlocal nonlinear medium whose nonlinear contribution to the refractive index has a initial value at the interface. We find that there exist single surface solitons at the edge and the corner of the two 2D media and dipole surface solitons at the edge of the two 2D media. These stable solitons do not exhibit a power threshold. The positions of the peak values and full width at half maximum(FWHM) of the surface solitons can be affected by the degree of nonlocality. However, the initial value can only influence the positions of the peak values of the surface solitons. In addition, when a laser beam is launched away from the interface, the beam will be periodic oscillations, even if the launch position is far away from the interface.

Theoretical Model
Considering a laser beam propagating along the interface between a nonlocal nonlinear medium and a linear medium, the complex amplitude E(X, Y, Z) of the light field satisfies the scalar wave equation [32, 34, 39] where k 0 = 2π/λ is the wave number in vacuum, and n = n L + ∆n for the nonlinear medium(X ≤ 0) and n = n 0 for the linear medium(X ≥ 0). ∆n represents the nonlinear contribution to the refractive index and may originate from any diffusive nonlinear effect, which can be written by [32] where w m is the characteristic length of the nonlinear response and n 2 is the nonlinear index coefficient. For the local case, w m → 0, we have ∆n = n 2 |E| 2 .
We search for stationary soliton solutions of Eqs. (3) and (4) numerically in the form a(x, y, z) = u(x, y) exp(ibz), where u is the real function and b is a real propagation constant of spatial solitons in the normalized system. and

A. Single surface solitons
We firstly talk about new surface-wave soliton solutions at the edge of the two 2D media.
Here, we assume that the normalized nonlinear contribution to the refractive index at the interface(y = 0) has a initial value φ d (φ d > 0). The boundary conditions for the fields at the interface are the continuity of the transverse field(a(y → 0 − ) = a(y → 0 + )) and its derivative(da(y → 0 − )/dy = da(y → 0 + )/dy). Because the width of the surface solitons is much smaller than the sample width, a and φ vanishes at the other boundary.
Three different solitons are separately shown in Fig. 1  where ρ(x, y) is a broadband random perturbation. The fact which is shown in Fig. 2(b) confirms the result of Fig. 2(a). Then we proceed to address the dynamics behavior of the propagation of surface solitons. For convenience, the (1+1)D circumstance is considered. Next, we consider that new soliton solutions at the corner of the two 2D media. Here, we assume that the normalized nonlinear contribution to the refractive index at x = 0 and . The boundary conditions for the fields at the interface meet the continuity conditions.  Fig. 3(b)), but the solitons are farther detached from the interface with φ d = 5 and d = 20( Fig. 3(a)) or φ d = 10 and d = 10( Fig. 3(c)). To further illustrate the influence of φ d and d on the solitons, we display that y max and FWHM versus φ d in Fig. 3(d) or d in Fig. 3(e). The energy flow U monotonically increasing with b shown in Fig. 4(a) and the direct numerical simulations of Eqs. (3) and (4) with noise σ 2 noise = 0.05 shown in Fig. 4(b) explain that the solitons are stable. The results can be similarly illustrated as the solitons at the edge of the interface.

B. Two dimensions dipole surface solitons
In addition to single surface solitons, we also find a 2D stationary dipole surface solitons.
The surface solitons are found numerically by a standard relaxation method which converges to a stationary solution after some iterations provided that a suitable guess for initial field distribution. The boundary conditions are the same as the single surface solitons at the edge of the interface.   Contrarily, in a bulk diffusive medium, it is known that weak input perturbations can cause slow but progressively increasing oscillations of the bright spots forming a dipole, resulting specially in their slow decay into fundamental solitons [35]. So, we think that the presence of a interface possessing a initial value for φ leads to stabilization of dipole solitons. This is further illustrated by the results of Fig. 6(c). when the beam is launched y = −1.91µm away from the interface, the stationary soliton cannot be formed, but it is periodically oscillating.
This are very good description of the boundaries of the role of solitons.

Conclusion
In conclusion, we have addressed (2+1)D surface solitons occurring at the interface between a linear medium and a nonlocal nonlinear medium whose nonlinear contribution to the refractive index has a initial value at the interface. We find that there exist stable single