�2� 1�D surface solitons at the interface between a linearmedium and a nonlocal nonlinear medium

Zhiwei Shi,1,2 Huagang Li,3 and Qi Guo1,*1Key Laboratory of Photonic Information Technology of Guangdong Higher Education Institutes, South China Normal

University, Guangzhou 510631, China2Faculty of Information Engineering, Guangdong University of Technology, Guangzhou 510006, China

3Department of Physics, Guangdong Institute of Education, Guangzhou 510303, China*Corresponding author: guoq@scnu.edu.cn

Received May 24, 2011; revised August 25, 2011; accepted August 25, 2011;posted August 26, 2011 (Doc. ID 147927); published September 23, 2011

We address ð2þ 1ÞD surface solitons occurring at the interface between a linear medium and a nonlocal nonlinearmedium whose nonlinear contribution to the refractive index has an initial value at the interface. We find thatthere exist stable single and dipole surface solitons that do not exhibit a power threshold. The properties of thesurface solitons can be affected by the initial value and the degree of nonlocality. When a laser beam is launchedaway from the interface, the beam will oscillate periodically. © 2011 Optical Society of America

OCIS codes: 190.4350, 190.6135.

1. INTRODUCTIONNonlocal spatial solitons have been investigated for decadesboth theoretically and experimentally. Spatial nonlocality is ageneric property in different materials including the photore-fractive media [1,2], thermal nonlinear media [3–7], and liquidcrystals [8–13]. Nonlocality can lead to new kinds of wavesthat would have been otherwise impossible in local nonlinearmedia [14–29]. In particular, for two-dimensional media withdifferent types of nonlocal response, they can support station-ary multipoles [15–20], stable vortices [21–26], and rotating[27,28] and spiraling [5,29] soliton states.

Surface waves localized at the interface of two different op-tical materials have many novel properties, which have beenstudied in nonlocal media recently. The light beam trajectorycan be strongly affected by the presence of interfaces, be-cause beams propagating in nonlocal media cause refractiveindex changes in regions far exceeding the beamwidth. Underproper conditions, stationary surface waves can propagatealong the interface in both local nonlinear media [30–33] andnonlocal nonlinear media [34–43].

Thermal media [36,37,40], photorefractive crystals [41], andphotonic lattices [44,45] have been utilized to demonstrateð2þ 1ÞD surface solitons. In this paper, we will study ð2þ1ÞD surface solitons occurring at the interface between a lin-ear medium and a nonlocal nonlinear medium whose non-linear contribution to the refractive index has an initialvalue at the interface. We find that there exist single and di-pole surface solitons. These stable solitons do not exhibit apower threshold. The positions of the peak values and FWHMof the surface solitons can be affected by the degree ofnonlocality. However, the initial value can only influencethe positions of the peak values of the surface solitons. In ad-dition, when a laser beam is launched away from the interface,the beamwill oscillate periodically, even if the launch positionis far away from the interface.

2. THEORETICAL MODELFor a laser beam propagating along the interface between anonlocal nonlinear medium and a linear medium, the complexamplitude EðX; Y; ZÞ of the light field satisfies the scalar waveequation [34,36,41]

∇2XYE þ ∂2E

∂Z2 þ k20n2E ¼ 0; ð1Þ

where k0 ¼ 2π=λ is the wavenumber in vacuum, λ is thewavelength in vacuum, n ¼ nL þΔn for the nonlinear med-ium (nL is the refractive index in the absence of light), and n ¼n0 for the linear medium. Δn represents the nonlinear contri-bution to the refractive index and may originate from anydiffusive nonlinear effect, which can be written by [34]

w2m∇

2XYΔn −Δnþ n2jEj2 ¼ 0; ð2Þ

where wm is the characteristic length of the nonlinear re-sponse and n2 is the nonlinear index coefficient. For the localcase, wm → 0, we have Δn ¼ n2jEj2.

Let us put EðX; Y; ZÞ ¼ E0ðX; Y; ZÞ expðiβZÞ and submit itinto Eq. (1) and (2) (β being the wavenumber in the media).Then, using the slowly varying envelope approximation andintroducing the normalized variables x ¼ X=w0, y ¼ Y=w0,z ¼ Z=ðβw2

0Þ, a ¼ k0w0ffiffiffiffiffiffiffiffiffiffiffin2nL

pE0, and ϕ ¼ k20w

20nLΔn, we get

12∇2

⊥aþ i∂zaþ β1aþ ϕa ¼ 0 in a nonlocal medium; ð3aÞ

d2∇2⊥ϕ − ϕþ jaj2 ¼ 0 in a nonlocal medium; ð3bÞ

and

12∇2

⊥aþ i∂zaþ β2a ¼ 0 in a linear medium; ð4Þ

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0740-3224/11/102472-06$15.00/0 © 2011 Optical Society of America

where ∇2⊥¼ ∂2x þ ∂2y, β1 ¼ w2

0ðk20n2L − β2Þ=2, β2 ¼ w2

0ðk20n20−

β2Þ=2, w0 is the beam width, and d ¼ wm=w0 stands for thedegree of nonlocality of the nonlinear response. Equation (3b)describes a local nonlinear response as d → 0 and a stronglynonlocal response as d → ∞.

We search for stationary soliton solutions of Eqs. (3) and(4) numerically in the form aðx; y; zÞ ¼ uðx; yÞ expðibzÞ,where u is the real function and b is a real propagationconstant of spatial solitons in the normalized system:

12∇2

⊥u − buþ β1uþ ϕu ¼ 0 in a nonlocal medium; ð5aÞ

d2∇2⊥ϕ − ϕþ juj2 ¼ 0 in a nonlocal medium; ð5bÞ

and

12∇2

⊥u − buþ β2u ¼ 0 in a linear medium: ð6Þ

We can write Eq. (5b) in the convolution form, namelyϕðx;yÞ¼ R

0−∞

R0−∞Rðx−x0;y−y0Þjuðx0;y0Þj2dx0dy0, where Rðx; yÞ

is a response function of the nonlocal nonlinear medium. Fromthis point, we can say that the nonlinear contribution at the

interface can have a initial value, even if the surface solitonshave vanishing amplitudes at the same time.

3. NUMERICAL RESULTSA. Single Surface SolitonsFirst, we assume the initial value is ϕdðϕd > 0Þ at y ¼ 0. Theboundary conditions for the fields at the interface are the con-tinuity of the transverse field [aðy → 0−Þ ¼ aðy → 0þÞ] and itsderivative [daðy → 0−Þ=dy ¼ daðy → 0þÞ=dy]. Because thewidth of the surface solitons is much smaller than the samplewidth, a and ϕ vanish at the other boundary. Under theseboundary conditions, we find one kind of ð2þ 1ÞD single sur-face solitons.

Three different solitons are separately shown inFigs. 1(a)–1(c). Here, white dashed lines indicate interface po-sitions, and all quantities are plotted in arbitrary dimension-less units in the corresponding figures in this paper. Wecan easily see that the soliton is closely attached to the inter-face with ϕd ¼ 7 and d ¼ 20 in Fig. 1(b), but the solitons arefarther detached from the interfaces with ϕd ¼ 1 and d ¼ 20 inFig. 1(a) and ϕd ¼ 7 and d ¼ 5 in Fig. 1(c). Obviously, ϕd and dinfluence the positions and shapes of the solitons. To furtherexplain this point, we see the change of the positions of thepeak values ymax and FWHM of the surface wave solitons

Fig. 1. (Color online) Sketches of 2D single surface solitons at the edge of the interface with (a) ϕd ¼ 1, d ¼ 20, (b) ϕd ¼ 7, d ¼ 20, and (c) ϕd ¼ 7,d ¼ 5; the normalized nonlinear contribution to the refractive index with (d) ϕd ¼ 7, d ¼ 1, (e) ϕd ¼ 7, d ¼ 20, and (f) ϕd ¼ 7, d ¼ 5; the positions ofthe peak values ymax and FWHM versus (g) the boundary value ϕd and (h) the nonlocal degree d.

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versus the boundary value ϕd [Fig. 1(g)] at the interface andthe degree of nonlocality d [Fig. 1(h)]. From Fig. 1(g), we cansee that under the condition (d ¼ 20), the soliton will be at-tracted to the interface, and more and more significant partsof the optical power reside in the linear meidium as ϕd in-creases. The larger ϕd is, the larger a “surface force” exertedon the beam by the interface becomes. However, ϕd cannotinfluence the beamwidth of solitons. In Fig. 1(h), the changingof the refractive index of the nonlocal nonlinear medium in-duced by d results in the changing of FWHM and ymax of thesolitons at ϕd ¼ 7. FWHM will increase when d increases.

However, ymax changes intricately with d. This point can beexplained by the normalized nonlinear contribution to the re-fractive index ϕ shown in Figs. 1(d)–1(f). In Fig. 1(d), a pullforce exerted on the beam by the nonlocal nonlinearity will begreater than the “surface force” exerted by the interface. Nowthe soliton will depart from the interface [see Fig. 1(a)]. Whend ≈ 5 [Fig. 1(f)], the difference between the two forces will begreatest, namely, the soliton will be farthest detached from theinterface [see Fig. 1(c)]. However, when d further increases[Fig. 1(e)], the “surface force” will be greater than the forceexerted by nonlocal nonlinearty, so the soliton is closely at-tached to the interface [see Fig. 1(b)]. Therefore, the nonlocalnonlinearity introduces a pull force that can make the solitonsdetach from the interface, while the “surface force” exerted bythe interface can attract the solitons toward the interface.

Figure 2(a) shows that the energy flow U ¼R∞−∞

R∞−∞ jaj2dxdy of the single two-dimensional (2D) surface

solitons monotonically increases with b where dU=db > 0.This shows that the solitons are stable [35,36]. Here, to furtherelucidate the stability of the surface solitons, we do direct nu-merical simulations of Eqs. (3) and (4) perturbed by inputnoise with variance σ2noise ¼ 0:05 on both amplitude and phase.In all cases analyzed in this paper, the results were foundnumerically by a finite-difference method. Although the split-step Fourier method is commonly used for analyzing the pro-blems, it is not used because there is a step of the refractiveindex at the interface between the nonlocal nonlinear mediumand the linear medium. Figure 2(b) confirms the result ofFig. 2(a). The soliton distribution looks almost unchangedafter propagation for a distance of 15 diffraction lengths. Thenwe proceed to address the dynamic behavior of the propaga-tion of surface solitons. For convenience, the ð1þ 1ÞD circum-stance is considered [42]. In Ref. [42], when the beam islaunched at the interface, the soliton is very stable, propagat-ing smoothly without distortion or deviations in its trajectory.Moreover, when a narrow beam is launched y ¼ −1:99 μmaway from the interface [see Fig. 2(c)], the beam maintainsa localized shape, but it is attracted to the surface due to a

Fig. 3. (Color online) Sketches of 2D single surface solitons at the corner of the interface with (a) ϕd ¼ 5, d ¼ 20, (b) ϕd ¼ 10, d ¼ 20, and(c) ϕd ¼ 10, d ¼ 10; the positions of the peak values ymax and FWHM versus (d) the boundary value ϕd and (e) the nonlocal degree d.

Fig. 2. (Color online) (a) Energy flow U versus the propagationconstant b with d ¼ 20 and ϕd ¼ 7, (b) stable propagation of surfacesolitons in Fig. 1(b) with noise σ2noise ¼ 0:05 for a distance of 15 dif-fraction lengths, (c) trajectories of the incident beam with the beamcenter coordinates y ¼ −1:99 μm. White dashed line indicates inter-face position. All quantities are plotted in arbitrary dimensionlessunits.

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gravitational force exerted by the interface and departs from itdue to a pull force exerted by the nonlocal nonlinearity, thenis again attracted to the interface, where it departs from theinterface again, in a fully periodic fashion. So, the oscillationsare periodic by virtue of the cooperation of the forces exertedby the boundaries and the nonlocal nonlinearity, even if thelaunch position is far away from the position of the surfacesoliton [36].

Next, we consider the new soliton solutions at the corner ofthe two 2D media. Here, we assume that the normalizednonlinear contributions to the refractive index at x ¼ 0 andy ¼ 0 have an initial value ϕ ¼ ϕdðϕd > 0Þ. The boundary

conditions for the fields at the interface meet the continuityconditions.

Figures 3(a)–3(c) show three different solitons. The solitonis closely attached to the interface with ϕd ¼ 10 and d ¼ 20[see Fig. 3(b)], but the solitons are detached from the interfacewith ϕd ¼ 5 and d ¼ 20 [Fig. 3(a)] or ϕd ¼ 10 and d ¼ 10[Fig. 3(c)]. To further illustrate the influence of ϕd and don the solitons, we display the changing of ymax and FWHMversus ϕd in Fig. 3(d) and d in Fig. 3(e). The energy flow Umonotonically increasing with b shown in Fig. 4(a) and thedirect numerical simulations of Eqs. (3) and (4) with noiseσ2noise ¼ 0:05 shown in Fig. 4(b) explain that the solitonsare stable. The results can be similarly illustrated for thesolitons at the interface.

B. Dipole Surface SolitonsIn addition to single surface solitons, we also find 2D station-ary dipole surface solitons. The surface solitons are foundnumerically by a standard relaxation method that convergesto a stationary solution after some iterations provided that asuitable guess for the initial field distribution is given. Theboundary conditions are the same as for the first kind of singlesurface solitons.

Figure 5(b) depicts the amplitude for a dipole soliton whichis closely attached to the interface with ϕd ¼ 10 and d ¼ 20.However, the solitons are detached from the interface withϕd ¼ 1 and d ¼ 20 in Fig. 5(a) or ϕd ¼ 10 and d ¼ 10 inFig. 5(c). Because the poles of the solitons are almost sym-metric in the x direction, the changing of ymax and FWHMof one pole of the solitons can explain the changing of the po-sition and shape of the dipoles. Figures 5(d) and 5(e) showymax and FWHM as functions of the boundary value ϕd andthe degree of nonlocality d, respectively. The results can alsobe similarly illustrated as the solitons at the interface.

Figure 6(a) shows that the energy flow monotonically in-creases with the propagation constant for dipole surfacesolitons. With increasing energy flow, surface dipoles becomemore localized, i.e., the distance between poles along the x

Fig. 4. (Color online) (a) Energy flow U versus the propagationconstant b with d ¼ 20 and ϕd ¼ 10, (b) stable propagation of surfacesolitons in Fig. 3(b) with noise σ2noise ¼ 0:05 for a distance of 15diffraction lengths.

Fig. 5. (Color online) Sketch of 2D dipole surface solitons with (a) ϕd ¼ 1, d ¼ 20, (b) ϕd ¼ 10, d ¼ 20, and (c) ϕd ¼ 10, d ¼ 10; the positions of thepeak values ymax and FWHM versus (d) the boundary value ϕd and (e) the nonlocal degree d.

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axis and their widths decrease. To further elucidate thestability of the surface dipoles, we do the direct numericalsimulations of Eqs. (3) and (4) with noise σ2noise ¼ 0:05. Parti-cularly, the dipole surface solitons are stable in the entire ex-istence domain. Figure 6(b) depicts a typical evolutiondynamic. The considerable input perturbations cause onlysmall oscillations of amplitudes of the two poles formingthe dipole, but the dipoles keep their internal structures overhuge distances. Contrarily, in a bulk diffusive medium, it isknown that weak input perturbations can cause slow butprogressively increasing oscillations of the bright spots form-ing a dipole, resulting specially in their slow decay into funda-mental solitons [37]. So, we conclude that the presence of aninterface possessing a initial value for ϕ leads to stabilizationof dipole solitons. This is further illustrated by the results ofFig. 6(c). When the beam is launched y ¼ −1:91 μm away fromthe interface, the stationary soliton cannot be formed, but it isperiodically oscillating. This is a very good description of theaction of the boundaries on solitons.

4. CONCLUSIONIn conclusion, ð2þ 1ÞD surface solitons occurring at theinterface between a linear medium and a nonlocal nonlinearmedium whose nonlinear contribution to the refractive indexhas an initial value at the interface are addressed. There existstable single and dipole surface solitons, and such solitons canbe found at any optical power level. The degree of nonlocalityhas influence on the positions of the peak values and FWHMof the surface solitons, but the initial value can only influencethe positions of the peak values of the surface solitons. In ad-dition, when a laser beam is launched away from the interface,the beamwill periodically oscillate, even if the launch positionis far away from the interface.

ACKNOWLEDGMENTSThis research was supported by the National Natural ScienceFoundation of China (NSFC) (Grants No. 11074080 and10904041), the Specialized Research Fund for the DoctoralProgram of Higher Education (Grant No. 20094407110008),

and the Natural Science Foundation of Guangdong Provinceof China (Grant No. 10151063101000017).

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