PHYSICAL REVIEW A 77, 043826 共2008兲
Spinning bearing-shaped solitons in strongly nonlocal nonlinear media Y. J. He State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen University, Guangzhou, 510275, China and School of Electronics and Information, Guangdong Polytechnic Normal University, Guangzhou, 510665, China
Boris A. Malomed Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
Dumitru Mihalache Horia Hulubei National Institute for Physics and Nuclear Engineering (IFIN-HH), 407 Atomistilor, Magurele-Bucharest, 077125, Romania
H. Z. Wang* State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen University, Guangzhou, 510275, China 共Received 15 January 2008; published 18 April 2008兲 Using the model of the so-called “accessible solitons” in nonlocal nonlinear media, we introduce a species of two-dimensional stable spatial solitons featuring the shape of spinning “bearings” 共ring beams periodically modulated in the azimuthal direction兲. They are generated by a superposition of concentric vortices with widely different topological charges. The superposition of two vortices with identical or opposite signs of their charges generates annular chains of quasicircular or oval eddies, respectively, whereas stable multilayered annular solitons are formed by the superposition of more than two vortices. Beams with this structure may form all-optical photonic crystal fibers, and suggest the creation of new matter-wave patterns in Bose-Einstein condensates. DOI: 10.1103/PhysRevA.77.043826
PACS number共s兲: 42.65.Tg, 42.65.Jx
I. INTRODUCTION
Nonlocal nonlinear response is featured by many physical media. It was found that nonlocality can prevent the collapse of self-focusing beams in Kerr-type media 关1,2兴, suppress azimuthal instabilities of vortex solitons 关3,4兴, and stabilize Laguerre and Hermite soliton clusters 关5兴, azimuthons 关6兴, and multipole solitons 关7兴. In particular, a nonlinear medium with strong nonlocality, whose spatial range exceeds the beam’s size, can support solitons with new properties, such as the so-called “accessible solitons,” which are described by quasilinear solutions tantamount to wave functions of the two-dimensional harmonic oscillator 关8兴. Experiments have revealed fundamental 关9兴 and vortex-ring solitons 关10兴 supported by the strong nonlocality, as well as soliton steering 关11兴. In addition, theoretical analyses predict the stabilization of other self-trapped modes 关12兴, partially coherent “accessible” solitons 关13兴, and complex soliton patterns of the Laguerre and Hermite types 关14兴. The topic of solitons in nonlocal optical media was reviewed in Ref. 关15兴. Transversely localized optical vortices, alias vortex solitons, have drawn much attention as objects of fundamental interest, and also due to their potential applications to alloptical information processing, as well as to the guiding and trapping of atoms. Noteworthy properties are also featured by vortex clusters, such as rotation similar to the vortex motion in superfluids 关16兴. Dynamics of vortex clusters in media with competing cubic and quintic nonlinearities has been
*Corresponding author:
[email protected] 1050-2947/2008/77共4兲/043826共5兲
studied too 关17,18兴. However, it was found that, in the general case, multiple vortices nested in the host beam display strong instability, including distortion, drift and annihilation of the vortices 关16兴. In this work, we demonstrate that spinning annular patterns shaped like “bearings” 共see Figs. 2–5 below兲 can be supported by strongly nonlocal nonlinear media. Such patterns are induced by a superposition of two stable concentric vertical solitons with large topological charges, making use of the fact that the higher-order vortices may be stable in nonlocal media. The superposition of the vortices with identical or opposite signs of their topological charges gives rise to solitons in the form of annular chains consisting of a large number of eddies 共small vortices兲 of a circular or oval shape, respectively. The size of the “bearings,” and of eddies of which they are composed, increases with the decrease of the nonlinearity strength. Stable multilayered annular solitons are generated by the superposition of more than two vortices with different topological charges. They may perform stable swinging motion in the external potential.
II. MODEL
To demonstrate the existence of the above-mentioned patterns, we adopt the model of “accessible solitons” 关8兴, which is based on the following version of the two-dimensional 共2D兲 nonlocal nonlinear Schrödinger equation for the evolution of a field amplitude, U共x , y兲, along the transmission distance z, taken in the normalized form,
043826-1
©2008 The American Physical Society
PHYSICAL REVIEW A 77, 043826 共2008兲
HE et al.
FIG. 1. 共Color online兲 Profiles of stable vortex solitons with different topological charges m with widths r0 = 1.025 共a兲 and r0 = 1.017 共b兲.
1 iuz + 共uxx + uyy兲 − Pr2u = 0, P = 2
冕冕
兩u共x,y兲兩2dxdy, 共1兲
where P is the total power of the beam. This equation can be derived, e.g., as a basic propagation equation for thermal nonlinear optical media with a large nonlocality radius; then, factor r2 appears as a result of the expansion of the response function around the beam’s axis 关15兴. First, a family of exact vortices-soliton solutions to Eq. 共1兲 can be easily found in polar coordinates 共r , 兲, u共x,y,z兲 = Ar兩m兩 exp共− r2/2r20兲exp共ikz + im兲, k = − 共1 + 兩m兩兲/r20 ,
共2兲
where integer 共positive or negative兲 m is the topological charge, and the radial width of the beam and its amplitude are related to the total power, r0 = 共2P兲−1/4,
A = 共2共1+兩m兩兲/4/冑兩m兩!兲P共3+兩m兩兲/4 .
共3兲
The maximum of the local power in solution 共2兲 is located at r = 冑兩m兩 · r0. A set of profiles of the vortices solitons with large values of the topological charge, which are used below to construct the “bearing-shaped” annular solitons, are displayed in Fig. 1. All the vortices are stable solutions to Eq. 共1兲, due to the quasilinear character of the equation 共in fact, their stability is tantamount to that of excited states in the quantum-mechanical isotropic harmonic oscillator with two degrees of freedom兲. III. STABLE BEARING-SHAPED SOLITON
A new class of annular solitons can be constructed as a superposition of the vortices with equal widths r0 but different topological charges, m1 and m2. Typical examples of the so obtained “bearing-shaped” solitons 共BSSs兲 are displayed in Fig. 2, where they are generated by pairs of concentric vortices with m1 = 0, m2 = 4, r0 = 1.18 共a兲, m1 = 4, m2 = 14, r0 = 1.025 共b兲, and m1 = −4, m2 = 14, r0 = 1.025 共c兲; note that the two latter cases differ only by the sign of m1. The choice of m1 = 0 in panel 共a兲 makes the core of the soliton filled, while in panels 共b兲 and 共c兲 it is hollow. These BSSs rotate counterclockwise, which corresponds to the sign of the total angular momentum of the vortex pairs that generate these solitons. Their stability was tested in simulations of Eq. 共1兲 by adding 10% noise to the initial conditions.
FIG. 2. 共Color online兲 Stable spinning “bearing-shaped” solitons 共BSSs兲 built as nonlinear superpositions of two concentric vortices 共whose profiles are shown in Fig. 1兲 with equal widths and topological charges m1 = 0, m2 = 4 共a兲, and m1 = ⫾ 4, m2 = 14 共b,c兲. The arrows show the direction of the rotation of each BSS. 共d兲 Phase patterns of the solitons from row 共c兲.
The BSSs are built as regular ring-shaped chains of compact eddies, which are generated by the interference of the two superimposed vortices. It is easy to see that the superposition determines the number of individual elements in the ring as 兩m1 − m2兩, where m1 and m2 are topological charges of the broad vortices which generate it. Each eddy is a quasicircular object if the signs of m1 and m2 are identical, see Figs. 2共a兲 and 2共b兲. In the opposite case, the eddy’s shape is quasielliptic 共oval兲, e.g., for m1 = −4 and m2 = 14 in Fig. 2共c兲. Next, we aim to estimate the size of the eddies nested in the BSS. The point of the intersection of the two vortices, u1 and u2, which build the BSS may be defined as a root of equation 兩u1兩 = 兩u2兩, that is, rS = 共兩m2兩 ! / 兩m1兩!兲1/共2兩m2−m1兩兲 · r0, according to Eqs. 共2兲 and 共3兲 共recall we take a pair of vortices with equal values of r0兲. Then the radius of individual quasicircular eddies which form the BSS is estimated as reddy ⬇ rS / 兩m2 − m1兩, provided that 兩m2 − m1兩 is large enough. With regard to Eq. 共3兲, these estimates demonstrate that, quite naturally, the size of the BSS as a whole, and of individual eddies which constitute it, decreases with the growth of the beam’s power as P−1/4. The angular velocity of the rotation of the “bearing” can also be easily found from the superposition of two expressions 共2兲, ⍀ = r−2 0 共兩m1兩 − 兩m2兩兲 / 共m1 − m2兲, which too is found to be in good agreement with numerical results.
043826-2
SPINNING BEARING-SHAPED SOLITONS IN STRONGLY …
PHYSICAL REVIEW A 77, 043826 共2008兲
FIG. 4. 共Color online兲 Stable swinging motion of the BSS from Fig. 3共a兲, induced by the sudden application of the kick to it in the x direction 共see text兲.
FIG. 3. 共Color online兲 Stable double-layered BSSs built as a superposition of three broad vortices with r1 = r2 = r3 = 1.017, whose profiles are shown in Fig. 1共b兲. The topological charges of the vortices are m1 = 4, m2 = 14, m3 = 30 共a兲; m1 = 4, m2 = 14, m3 = −30 共b兲; m1 = 4, m2 = −14, and m3 = 30 共c兲.
It should be emphasized that, to generate the BSS with a well-defined shape, the superposition zones of the two vortices beams which build it should be chosen appropriately. This actually means that the field amplitude at the intersection point is close to one-half of its maximum value, i.e., 兩u共rS兲兩 ⬇ max共兩u1,2兩兲 / 2. In particular, when the two vortex beams have m1 = −m2, the BBSs amount to known patterns in the form of necklace solitons 关19,20兴. Multilayered BSSs can be constructed by superimposing more than two vortex beams. A straightforward consideration of the interference patterns suggests that the number of layers, NL, and the number of the building vortices, NV, are related by NL = NV − 1. For instance, double-layered BSSs, featuring two ring-shaped chains of compact eddies, are formed by three vortices, see an example for m1 = 4, 兩m2兩 = 14, and 兩m3兩 = 30 in Fig. 3. In the simulations, all these solitons exhibit stable spinning motion upon propagation over indefinitely long distances. A BSS can be set in transverse motion 关in the 共x , y兲 plane兴 by applying a kick to it, i.e., replacing u共x , y兲 by u共x , y兲exp共i␣x兲, with real momentum ␣. In Fig. 4, the resulting stable swinging motion of the soliton is displayed for ␣ = 4. Treating the BSS as an effective particle, we may identify its position as x0共z兲 = 共1 / P兲/兩u共x , y兲兩2xdxdy, where P is the same total power as in Eq. 共1兲. The trajectory of the motion obeys the expected equation for the quasiparticle, d2x0 / dz2 = −2x0, which follows from the effective potential in Eq. 共1兲, V = −r2, cf. the motion of a two-dimensional gap soliton in the parabolic potential analyzed in Ref. 关21兴. The robustness of the BSSs considered above can be further tested by adding strong perturbations to the vortex beams that generate them. For instance, in Fig. 5, we add a sudden displacement, ⌬y = 0.2, to the center of the second
beam building the BSS 共the one with m = 14兲, in addition to the above-mentioned 10% background noise. It is seen that the so perturbed BSS stably propagates over at least 1000 diffraction lengths. The stability of localized states in strongly nonlocal nonlinear media is conduced to by the quasilinear character of Eq. 共1兲 and the presence of the effective global confining potential in it, as discussed above for higher-order vortices. In other words, the stabilization may be explained by the effective diffusion induced by the nonlocal response, which smooths out distortions of the soliton’s shape introduced by perturbations. Generally, vortices and other localized objects 共including the BSSs兲 in nonlocal settings are more stable than their counterparts 共if they exists兲 in local nonlinear media, cf. results for local models collected in Ref. 关22兴. Further details concerning stability aspects of the nonlocality can be found in Refs. 关3,4,10,15兴. While finding the new species of solitons in the present model is facilitated by the quasilinear form of Eq. 共1兲 共which actually was the motivation behind the introduction of the model 关8兴兲, it is very plausible that patterns predicted in this work should be common to isotropic settings featuring nonlocal nonlinearity. Therefore, experimental observation of optical BSSs may be expected in such media as nematic liquid crystals 关9兴, thermal-sensitive glasses 关10兴, and optically active vapors 关23兴. As for the multicharge vortices which are necessary to build the “bearings,” they can be created by means of the technique using dynamic holographic tweezers 关24兴, which makes it possible to generate vortices with the topological charge in the range of many tens and even hundreds 共as said above, higher-order charges are stable in the strongly nonlocal media, on the contrary to
FIG. 5. 共Color online兲 Robust propagation of the BSS from Fig. 3共a兲 in the presence of strong perturbations—the 10% background noise and sudden displacement by ⌬y = 0.2 of the second constituent vortex 共the one with m = 14兲.
043826-3
PHYSICAL REVIEW A 77, 043826 共2008兲
HE et al.
be dealing with an isotropic setting 关26兴. The spatial integration smooths out the singularity of the kernel, making it possible to develop a relatively simple description of the dipolar condensates 关26,31兴, that may be linked to models of nonlocal media. It may also be interesting to extend the present work into the three-dimensional 共3D兲 geometry. Very recently, sable 3D clusters of localized states were predicted in the model of “accessible solitons” 关33兴.
their counterparts in the local models 关22兴兲. It is also relevant to mention that the characteristic structure of the BSSs suggests that they may be used to emulate photonic crystal fibers 关25兴 by means of an all-optical technique. We also aim to discuss a possibility of the creation of similar states in Bose-Einstein condensates 共BECs兲. Because BSSs are generated by the superposition of high-order vortices, it is pertinent to mention that various complex patterns based on vortices were theoretically investigated in the usual BEC models, based on the Gross-Pitaevskii equation共s兲 with the local nonlinearity 共see Ref. 关26兴, and references therein兲. An effective nonlocal nonlinearity appears in condensates made of atoms carrying dipole moments; note that 2D solitons, both isotropic 关27兴 and anisotropic 关28兴 ones, may be essential matter-wave patterns in the latter setting. We expect that a similar nonlocality can support 2D structures of the BSS type. Further analysis of this possibility may be relevant because the strengths of both the dipole-dipole 关29兴 and contact 关30兴 interactions between atoms in the condensate can be controlled by means of external fields, which opens the way to create 2D solitons and other sophisticated states 关27,31兴. In particular, patterns in the form of multilayered BSSs may offer an opportunity to generate highly organized structures composed of a very large number of local vortices: using, for instance, eight broad vortices, with m = 4, 14, 30, 50, 75, 105, 145, and 195, one will arrive at a seven-tier BSS composed of 1037 eddies. Such a possibility is of interest in view of noticeable experimental efforts aimed at creating organized patterns including hundreds of vortices in BEC 关32兴. As concerns the theoretical models adopted for the description of dipolar BEC, the kernel of the dipole-dipole interactions is singular and anisotropic. However, choosing the “pancakelike” configuration, with the dipole moments oriented perpendicular to the plane of the “pancake,” one will
This work was supported by the National Natural Science Foundation of China 共Grant No. 10674183兲, the State Key Research Development Program for Basic Research of China 共Grant No. 2004CB719804兲, and the Ph.D. Degrees Foundation of Ministry of Education of China 共Grant No. 20060558068兲.
关1兴 D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 共1993兲. 关2兴 O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 共2002兲. 关3兴 D. Briedis, D. E. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, Opt. Express 13, 435 共2005兲. 关4兴 A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603共R兲 共2005兲; D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 33, 198 共2008兲. 关5兴 D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 共2007兲. 关6兴 S. Lopez-Aguayo, A. S. Desyatnikov, and Y. S. Kivshar, Opt. Express 14, 7903 共2006兲. 关7兴 Y. V. Kartashov, L. Torner, V. A. Vysloukh, and D. Mihalache, Opt. Lett. 31, 1483 共2006兲. 关8兴 A. Snyder and J. Mitchell, Science 276, 1538 共1997兲. 关9兴 C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 92, 113902 共2004兲. 关10兴 C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 共2005兲. 关11兴 M. Peccianti, K. A. Brzdkiewicz, and G. Assanto, Opt. Lett. 27, 1460 共2002兲; B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, ibid. 32, 154 共2007兲.
关12兴 S. Lopez-Aguayo and J. C. Gutiérrez-Vega, Phys. Rev. A 76, 023814 共2007兲. 关13兴 M. Shen, Q. Wang, J. Shi, P. Hou, and Q. Kong, Phys. Rev. E 73, 056602 共2006兲. 关14兴 W. Zhong and L. Yi, Phys. Rev. A 75, 061801共R兲 共2007兲. 关15兴 W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B: Quantum Semiclassical Opt. 6, S288 共2004兲. 关16兴 D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., Phys. Rev. Lett. 79, 3399 共1997兲; E. B. Sonin, Rev. Mod. Phys. 59, 87 共1987兲. 关17兴 D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, J. Opt. B: Quantum Semiclassical Opt. 6, S333 共2004兲; 6, S341 共2004兲. 关18兴 M. J. Paz-Alonso and H. Michinel, Phys. Rev. Lett. 94, 093901 共2005兲. 关19兴 M. Soljăcić, S. Sears, and M. Segev, Phys. Rev. Lett. 81, 4851 共1998兲; A. S. Desyatnikov and Y. S. Kivshar, Phys. Rev. Lett. 87, 033901 共2001兲; 88, 053901 共2002兲; Y. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, ibid. 89, 273902 共2002兲. 关20兴 Y. J. He, H. H. Fan, J. W. Dong, and H. Z. Wang, Phys. Rev. E
IV. CONCLUSION
In conclusion, we have demonstrated that rotating 2D solitons, built as ring chains of compact eddies, with the shape resembling a “bearing,” exist as a generic class of stable states in strongly nonlocal nonlinear media. These BSSs can be created as a result of the superposition of two or more broad vortices, featuring a multilayer ring structure in the latter case. The sizes of the BSS and individual eddies, as well as the number of eddies in each annular layer, are determined by the topological charges of the underlying vortex beams. The shape of individual eddies nested in BSS built by two broad vortices is either quasicircular or elliptic, if the topological charges of the broad vortices are, respectively, identical or opposite. ACKNOWLEDGMENTS
043826-4
SPINNING BEARING-SHAPED SOLITONS IN STRONGLY …
PHYSICAL REVIEW A 77, 043826 共2008兲
74, 016611 共2006兲; Y. J. He, Boris A. Malomed, and H. Z. Wang, Opt. Express 15, 17502 共2007兲. H. Sakaguchi and B. A. Malomed, J. Phys. B 37, 2225 共2004兲. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B: Quantum Semiclassical Opt. 7, R53 共2005兲. S. Skupin, M. Saffman, and W. Królikowski, Phys. Rev. Lett. 98, 263902 共2007兲. J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901 共2003兲. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, Opt. Lett. 21, 1547 共1996兲; 22, 484 共1997兲. L. C. Crasovan, G. Molina-Terriza, J. P. Torres, L. Torner, V. M. Perez-Garcia, and D. Mihalache, Phys. Rev. E 66, 036612 共2002兲; L.-C. Crasovan, V. Vekslerchik, V. M. Pérez-García, J. P. Torres, D. Mihalache, and L. Torner, Phys. Rev. A 68, 063609 共2003兲; T. Mizushima, N. Kobayashi, and K. Machida, ibid. 70, 043613 共2004兲; M. Möttönen, S. M. M. Virtanen, T. Isoshima, and M. M. Salomaa, ibid. 71, 033626 共2005兲; H. Pu, L. O. Baksmaty, S. Yi, and N. P. Bigelow, Phys. Rev. Lett. 94, 190401 共2005兲. P. Pedri and L. Santos, Phys. Rev. Lett. 95, 200404 共2005兲; V.
M. Lashkin, Phys. Rev. A 75, 043607 共2007兲. 关28兴 I. Tikhonenkov, B. A. Malomed, and A. Vardi, Phys. Rev. Lett. 100, 090406 共2008兲. 关29兴 S. Giovanazzi, A. Görlitz, and T. Pfau, Phys. Rev. Lett. 89, 130401 共2002兲. 关30兴 S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn, and W. Ketterle, Nature 共London兲 392, 151 共1998兲; Ph. Courteille, R. S. Freeland, D. J. Heinzen, F. A. van Abeelen, and B. J. Verhaar, Phys. Rev. Lett. 81, 69 共1998兲; J. L. Roberts, N. R. Claussen, and James P. Burke, Jr., Chris H. Greene, E. A. Cornell, and C. E. Wieman, ibid. 81, 5109 共1998兲; Vladan Vuletic, A. J. Kerman, C. Chin, and S. Chu, ibid. 82, 1406 共1999兲. 关31兴 H. Saito and M. Ueda, Phys. Rev. Lett. 90, 040403 共2003兲; F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, Phys. Rev. A 67, 013605 共2003兲. 关32兴 J. R. Abo-Shaeer et al., Science 292, 476 共2001兲; I. Coddington, P. Engels, V. Schweikhard, and E. A. Cornell, Phys. Rev. Lett. 91, 100402 共2003兲. 关33兴 W.-P. Zhong, L. Yi, R.-H. Xie, M. Belić, and G. Chen, J. Phys. B 41, 025402 共2008兲.
关21兴 关22兴 关23兴 关24兴 关25兴 关26兴
关27兴
043826-5
