The refraction of dark solitons at a planar boundary separating two defocusing Kerr media is simulated and analyzed, for the first time (to our knowledge). Analysis is based on the nonlinear Helmholtz equation and is thus valid for any angle of incidence.  A new law, governing refraction of black solitons, is combined with one describing bright soliton refraction to yield a generalized Snell’s law whose validity is verified numerically.  The complexity of gray soliton refraction is also analyzed, and illustrated by a change from external to internal refraction on varying the soliton contrast parameter

Aceves

Andersen

Boardman

Bradley

Cancellieri

Chamorro-Posada

Chen

Christian

Ciattoni

Fibich

Graham S. McDonald

Iturbe-Castillo

Julio Sánchez-Curto

Kartashov

Kivshar

Luther-Davies

Mihalache

Pedro Chamorro-Posada

Scheuer

Shadrivov

Skinner

Stegeman

Swartzlander

Sánchez-Curto

English

Sanchez-Curto, J

Chamorro-Posada, P

McDonald, GS

University of Salford Institutional Repository

Dark solitons at nonlinear interfacesJulio Sánchez-Curto,1,∗ Pedro Chamorro-Posada,1 and Graham S. McDonald21Departamento de Teoŕıa de la Señal y Comunicaciones e Ingenieŕıa Telemática,Universidad de Valladolid, ETSI Telecomunicación, Paseo Belén 15, Valladolid 47011, Spain2Joule Physics Laboratory, School of Computing, Science and Engineering,University of Salford, Salford M5 4WT, United Kingdom and∗Corresponding author: julsan@tel.uva.esThe refraction of dark solitons at a planar boundary separating two defocusingKerr media is simulated and analyzed, for the first time. Analysis is based onthe nonlinear Helmholtz equation, and is thus valid for any angle of incidence.A new law, governing refraction of black solitons, is combined with one describ-ing bright soliton refraction, to yield a generalized Snell’s law whose validity isverified numerically. The complexity of gray soliton refraction is also analyzed,and illustrated by a change from external to internal refraction on varying thesoliton contrast parameter.c© 2010 Optical Society of AmericaOCIS codes: 190.3270, 190.6135.Solitons are universal nonlinear waves and material in-terfaces play a fundamental role as boundary conditions.In particular, spatial solitons are predicted to become keyelements of emerging photonic technologies [1, 2]. Thebehaviour of soliton beams at nonlinear interfaces hasbeen extensively treated in the literature, where Kerr-type, and also saturable-Kerr [3], photorefractive [4], andquadratic soliton [5, 6] refraction properties have been re-viewed and proposed for the design of all-optical devices[7–10]. Most previous works on nonlinear interfaces havetwo features in common. First, analysis has been per-formed assuming the paraxial approximation, and usingthe nonlinear Schrödinger (NLS) equation as the solitonpropagation model [11]. Second, previous studies onlyanalyzed refraction properties of bright spatial solitons.In fact, only a few investigations have studied the be-haviour of dark solitons at nonlinear interfaces in a parax-ial context, where attention was restricted to nonlinearsurface waves at Kerr-type media [12, 13] or to kink soli-tons arising at surfaces of optical lattices imprinted indefocusing media [14]. To the authors’ knowledge, therefraction of dark solitons at nonlinear interfaces has notpreviously been studied.Nonparaxial theory based on the nonlinear Helmholtz(NLH) equation [15, 16] permits one to overcome in-trinsic angular limitations of NLS descriptions. In con-trast to other nonparaxial regimes [17, 18], where effectshave their origin in the strong focusing of high-intensitybeams, we consider broad (compared to the optical wave-length) beams of moderate power. Nonparaxiality thenarises solely from angular effects. Exact analytical solu-tions of NLH equations have been found in the form ofbright Kerr [16], dark Kerr [19], two-component [20], andbistable [21] Helmholtz solitons, and have allowed exten-sion of paraxial soliton theory to arbitrary-angle regimes.Soliton refraction effects have a strong inherent angularcharacter and constitute an excellent testbed for non-paraxial Helmholtz theory [22, 23].Spatial dark solitons present localized intensity dipson modulationally-stable plane waves [24]. In the physi-cal realization of such nonlinear waves, the infinite back-ground is replaced with a beam [25–27]. As with theirbright counterparts, dark solitons have also been pro-posed for all-optical signal processing devices [27, 28].In this Letter, the laws governing the refraction ofHelmholtz dark solitons [19] at interfaces separating twodefocusing Kerr media are presented. Figure 1(a) illus-trates the geometry of soliton refraction, where a blacksoliton is incident at angle θi on an interface separatingtwo defocusing Kerr media, and refracts at angle θt. Thetotal refractive index of medium i is n0i − αiI, whereαi > 0 is the Kerr coefficient and I the optical intensity.Assuming a relatively low value of αi, the approximationn2 ≈ n20i − 2n0iαiI is used.For a TE optical field, the complex envelope u of aforward propagating beam evolves according toκ∂2u∂ζ2+ j∂u∂ζ+12∂2u∂ξ2− |u|2 u =[∆4κ− (1 − α) |u|2]H(ξ)u. (1)A derivation from the Helmholtz equation is detailed in[23]. Here, the focusing Kerr nonlinearity is replacedby a defocusing one. H(ξ) is the Heaviside function,ξ = 21/2x/w0 and ζ = z/LD are the normalized trans-verse and longitudinal coordinates, respectively, and w0is the waist of a reference Gaussian beam with diffractionlength LD = kw20/2. κ = 1/k2w20 is a nonparaxiality pa-rameter, while ∆ = 1 − n202/n201 and α = α2/α1 accountfor the linear and nonlinear refractive index mismatch atthe interface, respectively.The general Helmholtz dark soliton in the second2medium is [19]u(ξ, ζ) = u0 (A tanh Θ + jF ) exp(−jζ2κ)× expj√1 − ∆ − 4κu20α1 + 2κV 2(−V ξ + ζ2κ)(2)whereΘ =u0Aα1/2 (ξ + W ζ)√1 + 2κW 2and W =V − V01 + 2κVV0. (3)V is a transverse velocity arising from an arbitraryrotation angle θ of the laboratory coordinates, V =tan(θ)(2κ)−1/2 [15], and W is the net transverse velocityof the dark soliton resulting from the combination of Vand the intrinsic transverse velocityV0 = u0Fα1/2[1 − ∆ −(2 + F 2)2κu20α]−1/2. (4)F = (1−A2)1/2 is the soliton grayness/contrast parame-ter and u0 is the background amplitude. The dark solitonsolution in medium 1 [19] is recovered from Eqs. (2)-(4)by setting α = 1 and ∆ = 0. Our study only addressesdark soliton refraction, since significant reflection of thesupporting beam may destroy the plane background re-quired for stable dark soliton propagation. Our analy-sis is, thus, necessarily restricted to cases with negligiblebeam reflection at the interface.First, we consider black solitons. The law governingtheir refraction is derived from ensuring continuity of thephase of the supporting beam across the interface. Us-ing Eq. (2), and tan2(θ) = 2κV 2, we find a generalizedSnell’s law for black and bright [22, 23] soliton refractionγ±n01 cos(θi) = n02 cos(θt) (5)where γ± = (1 + 4κβ±)1/2 [1 + 4κβ±α(1 − ∆)−1]−1/2isa nonlinear correction term. The subscript identifies theresult for bright (+) and black (−) Helmholtz solitons.In the case of bright solitons, one has [22, 23] β+ = η20/2whereas for black solitons β− = −u20; the nonlinear roleplayed by bright soliton amplitude (η0) is thus replacedby the amplitude of the black soliton background. Inphysical terms, bright solitons are perfectly collimatednonlinear beams and behave at interfaces in a fashionsimilar to plane waves, but it is the nonlinear plane wavethat determines the refraction of black solitons (whichare linear at their core).The value of γ− is real, from an assumed condition ofHelmholtz type of nonparaxiality: 4κu20 ≪ 1, which inturn implies that ∆ + 4κu20α < 1. While the former con-dition establishes the physical restriction that the totalrefractive index in the first medium must remain positive[19], 2|α1|E20 < n01, the latter determines an analogousrestriction for the second medium, 2|α2|E20 < n02.The total (linear plus nonlinear) refractive index mis-match across the interface is given by δ± = ∆+4κβ±(1−α). In the same way that δ+ determines how bright soli-tons refract [23], δ− can be used to interpret Snell’s lawfor black soliton refraction at self-defocusing interfaces.In Fig. 1, three different cases of a black soliton im-pinging on an interface at 25o are studied. Figure 1(b)shows how a black soliton undergoes internal refractionfor an interface with discontinuity only in the linear re-fractive index (∆ = 0.0024 and δ− > 0). The case ofan interface with exclusively nonlinear index mismatch,where one expects external refraction, is illustrated inFig. 1(d) (α = 0.4 and δ− < 0). With both mismatchcontributions present, they may cancel when the trans-parency/nonrefracting condition (δ− = 0) is met. Thisis verified in Fig. 1(c), where the dip position does notdeviate from the initial propagation direction.(a) (b)xzθtθin01−α1In02−α2Iξζδ−>00 10 20 3050−5ξζδ−=00 10 20 3050−5ξζδ−<00 10 20 3050−5(c) (d)Fig. 1. (a) Refraction geometry. A black soliton experi-encing: (b) internal refraction, (c) transparency, and (d)external refraction. Here, u0 = 1 and κ = 10−3, givingθt = 24.851o, 25o and 25.147o, respectively. In scaledunits, one obtains W = 10.356, 10.427 and 10.497.The value of α not only affects the soliton angle ofrefraction but can also induce significant change in soli-ton width. The input field distribution for the sec-ond medium generally presents a perturbed-soliton ini-tial condition [19, 29]. After crossing the interface, thesoliton narrows (α > 1) or broadens (α < 1), and theemission of 2N0 gray solitons is expected, where N0 isthe largest integer satisfying N0 <√α [19, 29].Attention is now turned to the refraction of gray soli-tons. As for black (F = 0) solitons, the phase slopeof the background beam associated with its transversevelocity V must be continuous across the interface. Inthe gray (F 6= 0) case, an additional independent con-dition arises from assuring the continuity of the phasestructure of the supporting beam: the intrinsic phasejump of the gray soliton which, from Eq. (2), amountsto −2 tan−1 (F/A) must be the same on both sides of theinterface. It depends solely on the grayness parameter F ,and its preservation upon refraction implies the conserva-tion of F . Black (gray) solitons impinging the nonlinearinterface are thus refracted as black (gray) solitons in thesecond medium.3Refraction properties of gray solitons are also depen-dent on the value of F . This effect is shown in Fig.2, where two solitons with different values of F en-counter the same interface (∆ = −0.016 and α = 3) withthe same angle of incidence for the background beamθi = 30o. The left picture of Fig. 2 shows a gray solitonwith F = 0.05 undergoing external refraction. However,internal refraction is experienced by the gray soliton withF = 0.6, as demonstrated in the right side of Fig. 2.Larger F entails a larger intrinsic velocity component,Eq. (4), and a smaller net transverse velocity, W . Thisreduces the angle of refraction until it is less than theangle of incidence. Refraction properties of gray solitonsthus present novel features (not found in bright or blacksoliton refraction).ξζF=0.055 10 15100−10ξζF=0.65 10 155−5−15Fig. 2. External (left) and internal (right) refraction.In this work, simulations have employed a nonparax-ial beam propagation method [30] for the numerical in-tegration of the NLH equation. Rotational symmetry[15] allowed us to consider solitons traveling with zerotransverse velocity encountering an obliquely-orientatedinterface. The background beam, supporting the soli-tons, was a raised cosine h(ξ) = cos2[π/rL(|ξ| − L1)]if L1 < |ξ| < L2, h(ξ) = 1 if |ξ| ≤ L1 and h(ξ) = 0 if|ξ| ≥ L2, with roll-off factor r = 0.5, grid length L = 160,L1 = (1 − r)L/4 and L2 = (1 + r)L/4.In summary, dark soliton refraction at interfaces sepa-rating two defocusing Kerr media has been analyzed forthe first time. Analysis has been undertaken within theframework of Helmholtz theory, which yields valid resultsfor beams propagating at arbitrary angles. We have pro-vided a unified theory in which a compact generalisedSnell’s law describes the refraction of both bright andblack solitons. Numerical results show excellent agree-ment with analytical predictions. Analysis of gray soli-ton refraction revealed a richer complexity that was ex-plained in terms of the properties of the exact solutions.One key finding is that soliton grayness is conserved dur-ing refraction.This work has been supported by the Spanish Minis-terio de Educación y Ciencia and Fondo Europeo de De-sarrollo Regional, project TEC2007-67429-C02-01, andJunta de Castilla y León, project VA001A08.References1. G.I. Stegeman and M. Segev, Science 286, 1518(1999).2. S. Trillo and W. Torruellas, Spatial solitons(Springer Verlag, Berlin, 2000).3. P.J. Bradley and C. De Angelis, Opt. Commun.130, 205 (1996).4. A.D. Boardman, P. Bontemps, W. Ilecki, and A.A.Zharov, J. Mod. Opt. 47, 1941 (2000).5. I.V. Shadrivov and A.A. Zharov, J. Opt. Soc. Am.B 19, 596 (2002).6. I.V. Shadrivov and A.A. Zharov, Opt. Commun.216, 47 (2003).7. D. Mihalache, M. Bertolotti, and C. Sibilia, Progr.Optics 27, 227 (1989).8. A.B. Aceves, J.V. Moloney, and A.C. Newell, Phys.Rev. A 39, 1828 (1989).9. J. Scheuer and M. Orenstein, J. Opt. Soc. Am. B22, 1260 (2005).10. G. Cancellieri, F. Chiaraluce, E. Gambi, andP. Pierleoni, J. Opt. Soc. Am. B 12, 1300 (1995).11. A.B. Aceves, J.V. Moloney, and A.C. Newell, Phys.Rev. A 39, 1809 (1989).12. S.R. Skinner and D.R. Andersen, J. Opt. Soc. Am.B 8, 759 (1991).13. Y. Chen, Phys. Rev. A 45, 4974 (1992).14. Y.V. Kartashov, V.A. Vysloukh, and L. Torner,Opt. Express. 14, 12365 (2006).15. P. Chamorro-Posada, G.S. McDonald, and G.H.C.New, J. Mod. Opt. 45, 1111 (1998).16. P. Chamorro-Posada, G.S. McDonald, and G.H.C.New, J. Opt. Soc. Am. B 19, 1216 (2002).17. G. Fibich, Phys. Rev. Lett. 76, 4356 (1996).18. A. Ciattoni, B. Crosignani, S. Mookherjea, andA. Yariv, Opt. Lett. 30, 516 (2005).19. P. Chamorro-Posada and G.S. McDonald, Opt.Lett. 28, 825 (2003).20. J.M. Christian, G.S. McDonald, and P. Chamorro-Posada, Phys. Rev. E 74, 066612 (2006).21. J.M. Christian, G.S. McDonald, and P. Chamorro-Posada, Phys. Rev. A 76, 033833 (2007).22. J. Sánchez-Curto, P. Chamorro-Posada, and G.S.McDonald, Opt. Lett. 32, 1126 (2007).23. J. Sánchez-Curto, P. Chamorro-Posada, and G.S.McDonald, J. Opt. A: Pure Appl. Opt. 11, 054015(2009).24. Yu.S. Kivshar and B. Luther-Davies, Phys. Rep.298, 81 (1998).25. D.R. Andersen, D.E. Hooton, G.A. Swartzlander,Jr., and A.E. Kaplan, Opt. Lett. 15, 783 (1990).26. G.A. Swartzlander, Jr., D.R. Andersen, J.J. Regan,H. Yin, and A.E. Kaplan, Phys. Rev. Lett. 66, 1583(1991).27. M.D. Iturbe-Castillo, J.J. Sánchez-Mondragón, S.I.Stepanov, M.B. Klein, and B.A. Wechsler, Opt.Commun. 118, 515 (1995).28. B. Luther-Davies and Y. Xiaoping, Opt. Lett. 17,496 (1992).29. Yu.S. Kivshar, IEEE J. Quantum Electron. 29, 250(1993).30. P. Chamorro-Posada, G.S. McDonald, and G.H.C.New, Opt. Commun. 192, 1 (2001).

Dark solitons at nonlinear interfaces

The refraction of dark solitons at a planar boundary separating two defocusing Kerr media is simulated and analyzed, for the first time. Analysis is based on the nonlinear Helmholtz equation, and is thus valid for any angle of incidence. A new law, governing refraction of black solitons, is combined with one describ-ing bright soliton refraction, to yield a generalized Snell’s law whose validity is verified numerically. The complexity of gray soliton refraction is also analyzed, and illustrated by a change from external to internal refraction on varying the soliton contrast parameter

Dark solitons at nonlinear interfaces

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