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OPTICS LETTERS / Vol. 34, No. 5 / March 1, 2009
Localization of light in a parabolically bending waveguide array in thermal nonlinear media Fangwei Ye,1,* Liangwei Dong,2 and Bambi Hu1,3 1
Department of Physics, Centre for Nonlinear Studies and The Beijing-Hong Kong-Singapore Joint Centre for Nonlinear and Complex Systems (Hong Kong), Hong Kong Baptist University, Kowloon Tong, China 2 Institute of Information Optics, Zhejiang Normal University, Jinhua, China, 321004 3 Department of Physics, University of Houston, Houston, Texas 77204-5005, USA *Corresponding author:
[email protected]
Received December 2, 2008; revised January 12, 2009; accepted January 14, 2009; posted January 26, 2009 (Doc. ID 104730); published February 23, 2009 A parabolically longitudinally bending waveguide array imprinted into thermal nonlinear media is found to support the localized stationary solitons. The localization results from the suppression of a curvature effect by the nonlinearity with an infinite-range nonlocality. The localization criterion is given analytically. Solitons propagate stably along a curved trajectory without bending loss, and their locations are significantly influenced by the waveguide curvature. These solitons represent the first example of stationary localized solitons encountered in curved waveguides. © 2009 Optical Society of America OCIS codes: 190.5530, 190.4360, 060.1810.
Light localization in a waveguide array has attracted extensive research interests in past years [1]. Dynamic localization, i.e., a light beam periodically spreading and reshaping itself into its initial state, has been demonstrated in a periodically curved waveguide array [2,3]. In a straight array with a transverse ramp of refractive index, Bloch oscillations were investigated [4,5]. The linear index ramp is realized in different ways; among others, special attention deserves an array of waveguides bending with a constant curvature along the longitudinal direction, which, by means of a conformal transformation, is mathematically equivalent to a straight array with a linear index gradient [5]. Bloch oscillation is closely related to the existence of the linear localized Wannier–Stark states [6]. However, such localized Wannier–Stark states can exist when only the lowest photonic band is considered. Once the full model is taken, Landau–Zener tunneling appears leading to the radiation losses [5]. In the nonlinear region, selftrapping and localized solitons emerge [1,7]. However, all of the localized solitons so far found are encountered in a straight waveguide array. Otherwise the bending would delocalize the soliton by inducing extended oscillating tails [8,9]. Many nonlinear materials are characterized by nonlocal nonlinear responses, including thermal materials; the latter are often described as possessing nonlocality with an infinite range [10]. Light propagation in thermal media changes the refractive index distribution in the whole sample, leading to the formation of specific soliton states and interactions that are absent in local or in finite-range nonlocal response materials [11–19]. In this Letter, we study the guidance of a light beam by a longitudinally parabolically bending lattice imprinted into a thermal media. The lattice is truncated to occupy half of the thermal media [7]. In spite of the curvature effect, localized and stationary solitons are found to exist at the lattice surface, and their locations are determined by the lattice curva0146-9592/09/050584-3/$15.00
ture. They are stable in their whole domain of existence and hence propagate parabolically without bending losses. To the best of our knowledge, such solitons are the first examples that are localized at a longitudinally curved waveguide and propagating along a predefined curved trajectory without bending losses. We start our analysis by considering the propagation of a laser beam in a thermal media that can be described by a nonlinear Schrödinger equation for the light field amplitude q and a Poisson equation for nonlinear refractive index change ␦n: i
q z
2␦ n x2
=−
1 2q 2 x2
− V共x − x0共z兲兲q − q␦n,
= − 兩q兩2 .
共1兲
Here x and z are the dimensionless transverse and longitudinal coordinates, respectively; V共x兲 = 0 for x 艋 0 and V共x兲 = P / 2共1 − cos共⍀x兲兲 for x ⬎ 0 describes the lattice refractive index profile; P stands for lattice amplitude and ⍀ for lattice modulation frequency. For simplicity, we fix P = 4 and ⍀ = 4. The thermal material and the lattice are supposed bending as a whole in the positive direction of the x axis with a parabolic trajectory x0共z兲 = a共z2 / 2兲; a represents the bending rate. The boundary conditions are employed as qx±x0共z兲=−L/2 = ␦nx±x0共z兲=−L/2 = 0, with L being the sample width. It is convenient to apply a conformal transformation on Eq. (1) by writing new variables x⬘ = x − x0共z兲, z⬘ = z, and a new envelope q⬘ = q exp共−iaz⬘x⬘ − 共1 / 6兲a2z⬘3兲 to give, on dropping the primes, i
q z
=−
1 2q 2 x2
− V共x兲q − ␦nq − 共− ax兲q,
© 2009 Optical Society of America
March 1, 2009 / Vol. 34, No. 5 / OPTICS LETTERS
2␦ n x2
= − 兩q兩2 ,
585
共2兲
with the boundary conditions now written as qx=±L/2 = ␦nx=±L/2 = 0. Thus the conformal transformation maps the curved waveguide array into a straight one by projecting the curvature effect into the new term −ax. Note that this linear index ramp achieves infinity at x → −⬁. Thus the whole structure usually does not contain any localized waves. Instead, the solution exhibits oscillatory tails at one side 共x → −⬁兲 that extend to infinity [8,9]. Upon propagation, the initially lattice-concentrated light gradually radiates away. This is the case in nonlinear media with local or finite-range responses, because there the index change occurs only in a limited region surrounding the laser beam. Nevertheless, for thermal media, the index change persists even in the region far away from the laser radiation, which induces a possibility to counteract the linear ramp effect. One thus may still expect the formation of a global index maximum in the laser region and of soliton localization in there. We therefore search for localized and stationary solutions of Eq. (2) in the form q = w共x兲exp共ibz兲, where w is a real function and b is the propagation constant. The nodeless soliton solutions at a = 0 are first found [Fig. 1(a)]. We then increase the bending rate a, which, as aforementioned, might induce extended oscillatory tails at the left. However, in spite of this curvature effect, solitons remain well localized at the lattice surface. Solitons at several bending rates are shown in Fig. 1. Such localization can be understood from the refractive index profiles involved [Fig. 2(a)]. The total-
Fig. 2. (Color online) (a) Refractive index profiles corresponding to the soliton in Fig. 1(d). V stands for lattice, ␦n for nonlinear refractive index, −ax for linear ramp, and F for total index profile. (b) Mass center and (c) energy flow of the solitons versus bending rate. (d) Energy flow versus propagation constant at a = 0 and a = 0.05. Owing to the small difference in energy flow for solitons at different bending rates, the two curves in Fig. 2(d) are indistinguishable.
refractive index F共x兲 can be expressed as a sum of three terms, F共x兲 = V共x兲 + 共−ax兲 + ␦n共x兲. Note that the oscillatory tails of the surface waves can only have a chance to appear in the continuous region 共x ⬍ 0兲. To analytically predict when the elimination of tails could happen, we assume that the surface wave is concentrated at x = x0 with a Delta profile w = 冑U␦共x L/2 兩q兩2dx兲. − x0兲, where U stands for energy flow 共U = 兰−L/2 By a Green-function analysis, one finds ␦n共x兲 ⬇ 1 / L共L / 2 − x0兲共L / 2 + x兲U for x 艋 x0 and ␦n共x兲 ⬇ 1 / L共L / 2 + x0兲共L / 2 − x兲U for x ⬎ x0. Importantly enough, the former expression shows that ␦n共x兲 decays to the left boundary with a rate of 共1 / 2 − x0 / L兲U ⬇ U / 2 (assuming x0 Ⰶ L), an opposite effect to that of −ax [Fig. 2(a)]. When the nonlinearityinduced decay rate dominates over the bendinginduced increasing rate, i.e., U 2
Fig. 1. Profiles of surface solitons at (a) a = 0, (b) a = 0.015, (c) a = 0.025, and (d) a = 0.05. Regions in gray correspond to V共x兲 艌 2 while in white V共x兲 ⬍ 2. P = 4, L = 40, and b = 10.
⬎ a,
共3兲
the otherwise existing oscillatory tails are eliminated and localization happens. One can see that when U / 2 ⬍ a, the “soliton” would still “feel” an increasing index environment at its left side and thus delocalizes. We emphasize that, in the derivation of localization criterion Eq. (3), we have approximated the soliton profile as a Delta function positioned at x0 ⬇ 0. In reality, soliton profile details and locations also significantly depend on the lattice parameters; however, as the thermal nonlinear response does not depend
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OPTICS LETTERS / Vol. 34, No. 5 / March 1, 2009
on the wave profile details but rather on the energy flow, the Delta approximation is justified as long as the sample boundaries are far enough from the laser region 共x0 Ⰶ L兲. This is also consistent with our numeric findings in the framework of the full model [Eq. (2)] that Eq. (3) predicts very well for the occurrence of localization. We also emphasize that, in uniform thermal media, if the soliton is not correctly initially positioned, it oscillates periodically during propagation [16]. In a lattice system as in our study, however, such trajectory oscillations are suppressed thanks to the Peierls–Nabarro barrier. The formation of such surface Bloch solitons results from the balance between nonlinearity, lattice, and the curvature effect. We are interested in the curvature effects on the soliton properties. The influence of a on soliton profiles can be appreciated from Fig. 1. At a = 0 (straight waveguide), the surface soliton features two amplitude peaks residing on the first two channels with the peak on the second channel stronger than the first one [Fig. 1(a)]. As a increases, solitons transport more energy from the right to the left through the continuous drop of the second intensity peak and rise of the first one. Thus, at a moderate value a, two peaks are almost equal [Figs. 1(b) and 1(c)]; with further increasing a, the soliton features one dominated peak on the border channel [Fig. 1(d)]. Figure 2(b) displays the mass cenL/2 ter of the soliton xc = 兰−L/2 兩q兩2xdx / U versus bending rates, being a monotonically decreasing function. Figure 2(c) shows that the energy flow changes very slowly with a, in contrast with its rapid changes with the propagation constant [Fig. 2(d)]. From Figs. 2(b) and 2(c), we conclude that the bending rate mainly determines the location of the soliton and its intrinsic power distribution with a higher value of a corresponding to a more “surface” state.
Fig. 3. (Color online) Propagation of the surface soliton of Fig. 1(c). Regions in gray correspond to V共x兲 艌 2 while in white V共x兲 ⬍ 2. Insets show the intensity of light at z = 0 and z = 36, respectively.
The stability of the solitons is studied by both linear stability analysis and beam propagation simulations of original equations [Eq. (1)] or of the transformed one [Eq. (2)]. We find that they are stable in their whole domain of existence that is defined by Eq. (3). A propagation dynamic is shown in Fig. 3, which shows the soliton following adiabatically a bending trajectory, without any radiation loss. Recalling that a bend waveguide usually induces bending loss in media otherwise, we conjecture that the thermal media with an imprinted curved waveguide might be used for lossless light steering and switching. While a constant curvature is necessary for the formation of localized stationary solitons, it should be unnecessary for loss eliminations. Finally, we mention that dipole and other stable higher-order localized solitons could also exist, and their positions are also greatly controlled by the lattice curvature. F. Ye thanks J. Lin for useful discussions. This work is supported in part by Hong Kong Baptist University and the Hong Kong Research Grants Council. L. Dong acknowledges the support of the National Natural Science Foundation of China (NSFC) (10704067). References 1. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003). 2. G. Lenz, R. Parker, M. C. Wanke, and C. M. de Sterke, Opt. Commun. 218, 87 (2003). 3. S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 96, 243901 (2006). 4. U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998). 5. G. Lenz, I. Talanina, and C. M. de Sterke, Phys. Rev. Lett. 83, 963 (1999). 6. T. Pertsch, U. Peschel, and F. Lederer, Chaos 13, 744 (2003). 7. K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, Opt. Lett. 30, 2466 (2005). 8. W. Krolikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, Phys. Rev. E 54, 5761 (1996). 9. A. V. Gorbach and D. V. Skryabin, Opt. Express 16, 4858 (2008). 10. C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005). 11. I. Kaminer, C. Rotschild, O. Manela, and M. Segev, Opt. Lett. 32, 3209 (2007). 12. F. Ye, Y. V. Kartashov, and L. Torner, Phys. Rev. A 77, 043821 (2008). 13. M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, Opt. Lett. 27, 1460 (2002). 14. A. Dreischuh, D. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, Phys. Rev. Lett. 96, 043901 (2006). 15. Z. Xu, Y. V. Kartashov, and L. Torner, Phys. Rev. Lett. 95, 113901 (2005). 16. B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Opt. Lett. 32, 154 (2007). 17. B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 213901 (2007). 18. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 1774 (2008). 19. N. K. Efremidis, Phys. Rev. A 77, 063824 (2008).
