Novel Dynamics of a Vortex in Three-Dimensional Dissipative Media with an Umbrella-Shaped Potential *

ISSN: 0256-307X 中国物理快报 Chinese Physics Letters

Volume 31 Number 7 July 2014 A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/0256-307X http://cpl.iphy.ac.cn

C HINESE P HYSICAL S OCIET Y

CHIN. PHYS. LETT. Vol. 31, No. 7 (2014) 074210 Novel Dynamics of a Vortex in Three-Dimensional Dissipative Media with an Umbrella-Shaped Potential *

LIU Yun-Feng(刘云凤), LIU Bin(刘彬)**, HE Xing-Dao(何兴道), LI Shu-Jing(李淑静) Key Laboratory of Nondestructive Test (Ministry of Education), Nanchang Hangkong University, Nanchang 330063

(Received 21 March 2014) We report the novel dynamic of 3D dissipative vortices supported by an umbrella-shaped potential (USP) in the 3D complex Ginzburg–Landau (CGL) equation with the cubic-quintic nonlinearity. The stable solution of vortices with intrinsic vorticity 푆=1 and 2 are obtained in the 3D CGL equation. An appropriate USP forces the vortices continuously to throw out fundamental 3D solitons (light bullets) along the folding umbrella. The dynamic regions of the strength of the potential with the changing number of folding umbrella are analyzed, and the rate of throwing increases with the strength of the potential. A weak potential cannot provide vortices with enough force. Then, the vortices will be stretched into polygons. However, a strong potential will destroy the vortices.

PACS: 42.65.Tg, 05.45.−a DOI: 10.1088/0256-307X/31/7/074210

Solitons in optical media have attracted much In this Letter, we introduce a 3D CGL model with attention.[1−14] A spatiotemporal soliton is referred to an external umbrella-shaped potential. We consider as a ‘light bullet’ localized in all spatial dimensions the nonlinear dynamic on stable dissipative vortices and in the time dimension. The generation of a light initially placed at the apex of the potential. The ex- bullet might be of importance in soliton-based com- tra force of the potential breaks the original dynam- munication systems. An optical vortex soliton is a ical balance of the central vortices. A series of novel self-localized nonlinear wave, which has a point (sin- dynamics are observed. Especially, light bullets are gularity) of zero intensity, and with a phase that twists periodically thrown out from the central vortex. around that point, with a total phase accumulation of We consider the following 3D CQ CGL equation 2휋푆 for a closed circuit around the singularity.[15] The in terms of nonlinear optics, as the evolution equation quantity 푆 is an integer number known as the vorticity for the amplitude of an electromagnetic wave in an or topological charge of the solution. active bulk optical medium,[27,36] Complex Ginzburg–Landau (CGL) equations are (︁퐷 )︁ well known as basic models of the pattern forma- 푖푢 + 푖훿푢 + (1/2 − 푖훽)(푢 + 푢 ) + + 푖훾 푢 푧 푥푥 푦푦 2 푡푡 tion in various nonlinear dissipative media, such + (1 − 푖휀)|푢|2푢 − (휈 − 푖휇)|푢|4푢 = 퐹 (푥, 푦)푢, (1) as in superconductivity and superfluidity, fluid dynamics, reaction-diffusion phenomena, nonlinear op- where (푥, 푦) and 푡 are the transverse coordinates and tics, Bose–Einstein condensates, and quantum field temporal coordinate, 푧 is the propagation distance. theories.[16−18] The CGL equation with the cubic- The coefficients of diffraction and cubic self-focusing quintic nonlinearity has been widely used in nonlin- nonlinearity are scaled respectively, to be 1/2 and 1, ear dissipative optics. Among the important appli- 퐷 is the group-velocity dispersion (GVD) coefficient. cation are passively mode-locked laser systems and In the following we set 퐷 = 1/2 for the anomalous optical transmission lines.[19] A series of nonlinear dispersion propagation regime, 휈 is the quintic self- dynamical reports in this model have been focused defocusing coefficient, 훿 is the coefficient correspond- on the complex stable patterns[20−28] and interac- ing to the linear loss (훿 > 0) or gain (훿 < 0), 휇 > 0 ac- tions of localized pulses.[29−32] Recently, adding ex- counts for the quintic-loss parameter, and 휀 > 0 is the ternal potentials in these models has been a theme cubic-gain coefficient, 훾 > 0 accounts for spectral fil- of extensive studies.[34−36] Desirable patterns of the tering in optics, 훽 is the spatial-diffusion term, which refractive-index modulation in materials described by appears in a model of laser cavities, where it is gener- CGL equation, which may induce the effective poten- ated by the interplay of the dephasing of the local po- tials, can be achieved by means of various techniques, larization in the dielectric medium, cavity loss, and de- such as optics induction[37] and writing patterns by tuning between the cavity’s and atomic frequencies.[39] streams of ultrashort laser pulses.[38] As mentioned above, we will keep 훽 > 0 to secure the

*Supported by the National Natural Science Foundation of China under Grant Nos 61205119, 41066001 and 11104128, the Natural Science Foundation of Jiangxi Province under Grant No 20132BAB212001, and the Natural Science Foundation of Jiangxi Province Office of Education under Grant No GJJ13485. **Corresponding author. Email: [email protected] © 2014 Chinese Physical Society and IOP Publishing Ltd 074210-1 CHIN. PHYS. LETT. Vol. 31, No. 7 (2014) 074210 stability of the vortex solitons in the 3D CGL model. as 퐸(푆 = 1)=415, and 퐸(푆 = 2) = 719. The vortices The last term on the right-hand side of Eq. (1) in- with 푆 > 2 have been reported in Ref. [32], which can- troduces the umbrella-shaped potential in the trans- not steadily propagate. verse plane. The analytical form of 퐹 (푥, 푦) is First, we consider the dynamics of vortices with 푆 = 1, placed at the apex of the USP. A series of novel 퐹 (푥, 푦) = − 푎푟|cos(푚휃/2)|, 푚 ≥ 3 dynamics are studied by performing a large number of √︀ 푟 = 푥2 + 푦2, (2) numerical simulations. The circular vortex is gradually stretched into a where 휃 is the angular coordinate, and 푎 is the strength polygonal one by a weak potential. Figure 2(a) shows of the potential. Integer 푚 stands for the number of that a stable vortex with 푆 = 1 evolves into quadrate folding umbrellas. at 푎 = 0.03. The evolutions of energy at 푎 = 0.02 and 0.03 (shown in Fig. 2(e)) reveal that the size of (a) the polygonal vortex increases with 푎. 15 1.5 2π

t 0 (a) z/ z/ z/ z/ z/ z/ -15 -15 15 0 0 0 0 (b) x 15 -15 y (b) 1.5 2π 15 z/ z/ z/ z/ z/ z/

t 0

-15 (c) -15 15 0 0 0 0 x 15 -15 y z/ z/ z/ z/ z/ z/

Fig. 1. (Color online) The profile (a) and phase (b) of stable 3D vortex solutions with topological charge 푆=1 (d) and 2, respectively. z/ z/ z/ z/ z/ z/

We have solved Eq. (1) using a split-step Fourier 800 (e) 3000 (f) method with typical transverse and longitudinal step 600 2000

sizes ∆푥 = ∆푦 = ∆푡 = 0.3 and ∆푧 = 0.1 as described E 400 1000 in the following. The second-order derivative terms in 200

푥, 푦, and 푧 are solved in Fourier space under the peri- 0 0 0 500 1000 1500 2000 0 500 1000 1500 odic boundary conditions. Other linear and nonlinear z z terms in the equation are solved in real space by using Fig. 2. (a)–(d) Isosurface plots of total intensity a fourth-order Runge–Kutta method. |푢(푥, 푦, 푡)|2, evolutions of the central vortex with 푆 = 1 Generic results may be adequately represented by at (푎 = 0.03, 푚 = 4), (푎 = 0.08, 푚 = 4)(푎 = 0.1, 푚 = 4), setting 훿 = 0.4 휇 = 1, 휀 = 2.43, 훾 = 훽 = 0.5, and and (푎 = 0.16, 푚 = 4). (e) Evolutions of the energy region 휈 = 0.1, corresponding to a physically realistic situ- of 푎 at 푚 = 4 with 푎 = 0.02 (solid line) and 0.03 (dashed line). (f) Evolutions of energy at 푚 = 4 with 푎 = 0.08 ation, and making the evolution relatively fast, thus (dashed line) and 0.1 (solid line). helping to elucidate its salient features.[29,31] Table 1. The relationship between 푎 with 푚 for 푆 = 1. The initial solutions of the vortex in Eq. (1) are set as follows: 푚 Region of 푎 for throwing light bullet from vortices with 푆=1 2 2 [︁ (︁ 푟 푡 )︁]︁ 4 0.038≤ 푎 ≤0.142 푢(푧 = 0, 푟, 푡) = 퐴|푟|푆 exp − + exp(푖푆휃), 푤2 푤2 5 0.041≤ 푎 ≤0.128 (3) 6 0.044≤ 푎 ≤0.116 7 0.045≤ 푎 ≤0.106 where 퐴 is the amplitude, and 푤 is the width. We dis- 8 0.046≤ 푎 ≤0.098 tinguish the vortex solutions with topological charge 푆, also called the angular momentum quantum num- For an appropriate potential, the vortices are ber. The stable vortices with 푆 = 1 and 2 are obtained forced to continuously throw out 푚 jet pulses along in the numerical form by an image-step propagation the slopes of the potential. Then the pulses will self method without potential, as shown in Figs. 1(a) and trap into fundamental solitons. The dynamical evo- 1(b). In this case, the 3D vortices with 푆 = 1 and 2 lutions of continuously throwing out four jets of fun- all are stable, characterized by the following values of damental solitons at 푎 = 0.08 and 0.1 are shown in the energy (alias norm) Figs. 2(b) and 2(c), respectively. Figure 2(f) reveals that the dynamics was obviously periodic. It demon- ∫︁ ∞ ∫︁ +∞ 2 strates that a stronger potential provides for a higher 퐸 = 2휋 푟푑푟 푑푡|푢(푟, 푡)| , (4) rate of throwing. The relationship between 푎 with 0 −∞ 074210-2 CHIN. PHYS. LETT. Vol. 31, No. 7 (2014) 074210

푚 is shown in Table 1 by performing a large numeri- spectively. The evolutions of energy 퐸 are revealed in cal simulation. The stability of vortices with vorticity Fig. 3(f). It reveals that the rate of throwing increase 푆 = 1 is relatively low. It leads to the novel dynamics with the growth of 푎. Thirdly, at larger 푎, the central only appearing at 4≤ 푚 ≤8. Furthermore, the region vortex is broken into five fundamental solitons (shown of 푎 decreases with the growth of 푚. in Fig. 3(d)). For a stronger potential, the central vortices are The relationship between 푎 and 푚 is shown in Ta- broken into 푚 fundamental solitons which slide along ble 2. Compared with 푆 = 1, the vortex with vorticity the slopes of the USP, see a typical example in 푆 = 2 has more stability. It leads to a novel dynamic Fig. 2(d) with 푚 = 4 and 푎 = 0.16. appearing at 3 ≤ 푚 ≤ 10. The region of 푎 also increases with the growth of 푚. (a) In conclusion, we have introduced an umbrella- z/ z/ z/ z/ z/ z/ shaped potential into the 3D CGL equation with the

(b) CQ nonlinearity. For a weak potential, stretching of the circular vortices with 푆 = 1 and 2 into polygonal is z/ z/ z/ z/ z/ z/ observed. The size increases with the strength 푎. If an (c) appropriate strength of potential is used, the setting z/ z/ z/ z/ z/ z/ gives rise to the periodical throwing of light bullets,

(d) which slide along slopes the USP from vortices. The z/ z/ z/ z/ z/ z/ region of 푎 (strength of potential) decreases with the

1200 6000 growth of 푚. The rate of emission increases with the (f) (e) strength of potential 푎. However, a stronger poten- 1000 4000 tial will break the central vortex into 푚 fundamental E 800 2000 3D solitons. The novel dynamic in CGL models will expand the already wide spectrum of relevant appli- 600 0 0 500 1000 1500 2000 0 500 1000 1500 2000 z z cation.

Fig. 3. (a)–(d) Isosurface plots of total intensity |푢(푥, 푦, 푡)|2, evolutions of the central vortex with 푆=2 at (푎=0.015, 푚=5), (푎 = 0.05, 푚 = 5)(푎 = 0.06, 푚 = 5), References and (푎 = 0.11, 푚 = 5). (e) Evolutions of the energy region of 푎 at 푚 = 5 with 푎 = 0.01 (solid line) and 0.015 (dashed [1] Kivshar Y S and Agrawal G P 2003 Optical Solitons: From line). (f) Evolutions of energy at 푚 = 5 with 푎 = 0.05 Fibers to Photonic Crystals (New York: Academic Press) (solid line) and 0.06 (dashed line). [2] Malomed B A, Mihalache D, Wise F and Torner L 2005 J. Opt. B Quantum SemiClass. Opt. 7 R53 Table 2. The relationship between 푎 and 푚 for 푆 = 2. [3] Minardi S, Eilenberger F, Kartashov Y V, Szameit A, Röpke U, Kobelke J, Schuster K, Bartelt H, Nolte S, Torner L, Led- 푚 Region of 푎 for throwing bullet erer F, Tünnermann A and Pertsch T 2010 Phys. Rev. Lett. from vortices with 푆=2 105 263901 3 0.024≤ 푎 ≤0.182 [4] Eilenberger F, Minardi S, Szameit A, Röpke U, Kobelke J, 4 0.026≤ 푎 ≤0.138 Schuster K, Bartelt H, Nolte S, Torner L, Lederer F, Tün- 5 0.028≤ 푎 ≤0.100 nermann A and Pertsch T 2011 Phys. Rev. A 84 013836 6 0.030≤ 푎 ≤0.078 [5] Wu Y and Deng L 2004 Opt. Lett. 29 2064 7 0.032≤ 푎 ≤0.072 [6] Wu Y and Yang X 2007 Appl. Phys. Lett. 91 094104 8 0.034≤ 푎 ≤0.064 [7] Silberberg Y 1990 Opt. Lett. 15 1282 9 0.036≤ 푎 ≤0.046 [8] Aceves A B and Angelis C D 1993 Opt. Lett. 18 110 10 0.038≤ 푎 ≤0.040 [9] Abdollahpour D, Suntsov S, Papazoglou D G and Tzortza- kis S 2010 Phys. Rev. Lett. 105 253901 [10] Chong A, Renninger W H, Christodoulides D N and Wise Secondly, we consider vortices with vorticity 푆 = F W 2010 Nat. Photon. 4 103 2. Three similar dynamics are observed by the varia- [11] Eilenberger F, Prater K, Minardi S, Geiss R, Röpke U, Ko- tion of 푎. We perform the numerical simulation with belke J, Schuster K, Bartelt H, Nolte S, Tünnermann A and Phys. Rev. 3 푚 = 5 as a typical example in Fig. 3. First, at small 푎, Pertsch T 2013 X 041031 [12] Mihalache D 2012 Rom. J. Phys. 57 352 the circular vortex is gradually stretched into a pen- [13] Wu Y and Deng L 2004 Phys. Rev. Lett. 93 143904 tagonal (shown in Fig. 3(a)). The stretching force in- [14] Wu Y 2005 Phys. Rev. A 71 053820 creases with the growth of 푎. The evolutions 퐸 at [15] López-Mariscal C and Gutiérrez-Vega J C 2009 Opt. Pho- ton. News 20 10 푎 = 0.015 and 0.02 (shown in Fig. 3(e)) reveal that the [16] Aranson I S and Kramer L 2002 Rev. Mod. Phys. 74 99 size of pentagonal vortex increases with the growth of [17] Rosanov N, Akhmediev N and Ankievicz A 2005 Dissipative 푎. Secondly, at intermediate 0.028 ≤ 푎 ≤ 0.1, the Solitons (New York: Springer-Verlag) vortices are also forced to periodically throw out five- [18] Malomed B A 2005 Encyclopedia of Nonlinear Science (New York: Routledge) jet fundamental solitons. The dynamical evolutions of [19] Akhmediev N N, Ankiewicz A and Soto-Crespo J M 1998 푎 = 0.05 and 0.06 are shown in Figs. 3(b) and 3(c), re- J. Opt. Soc. Am. B 15 515 074210-3 CHIN. PHYS. LETT. Vol. 31, No. 7 (2014) 074210

[20] Ankiewicz A, Devine N, Akhmediev N and Soto-Crespo J B A 2008 Phys. Rev. A 77 033817 M 2008 Phys. Rev. A 77 033840 [31] Liu B, He X D and Li S J 2011 Phys. Rev. E 84 056607 [21] Crasovan L C, Malomed B A and Mihalache D 2000 Phys. [32] Mihalache D, Mazilu D, Lederer F, Kartashov Y V, L Craso- Rev. E 63 016605 van C, Torner L and Malomed B A 2006 Phys. Rev. Lett. [22] Desyatnikov A, Maimistov A and Malomed B A 2000 Phys. 97 073904 Rev. E 61 3107 [33] Leblond H, Malomed B A and Mihalache D 2009 Phys. Rev. [23] Crasovan L C, Malomed B A and Mihalache D 2001 Phys. A 80 033835 Lett. A 289 59 [34] Sakaguchi H and Malomed B A 2009 Phys. Rev. E 80 [24] Soto-Crespo J M, Akhmediev N, Mejia-Cortés C and 026606 Devine N 2009 Opt. Express 17 4236 [35] He Y J, Malomed B A, Mihalache D, Liu B, Huang H C, [25] Skryabin D V and Vladimirov A G 2002 Phys. Rev. Lett. Yang H and Wang H Z 2009 Opt. Lett. 34 2976 89 044101 [36] Liu B, He Y J, Malomed B A, Wang X S, Kevrekidis P G, [26] Soto-Crespo J M, Grelu P and Akhmediev N 2006 Opt. Wang T B, Leng F C, Qiu Z R and Wang H Z 2010 Opt. Express 14 4013 Lett. 35 1974 [27] Mihalache D, Mazilu D, Lederer F, Leblond H and Malomed [37] Cleff C, Gütlich B and Denz C 2008 Phys. Rev. Lett. 100 B A 2007 Phys. Rev. A 75 033811 233902 [28] Kamagate A, Grelu P, Tchofo-Dinda P, Soto-Crespo J M [38] Szameit A, Burghoff J, Pertsch T, Nolte S, Tünnermann A and Akhmediev N 2009 Phys. Rev. E 79 026609 and Lederer F 2006 Opt. Express 14 6055 [29] Mihalache D, Mazilu D, Lederer F, Leblond H and Malomed [39] Lega J, Moloney J V and Newell A C 1994 Phys. Rev. Lett. B A 2008 Phys. Rev. E 78 056601 73 2978 [30] Mihalache D, Mazilu D, Lederer F, Leblond H and Malomed

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CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES 077201 Current Induced Non-Volatile Resistive Switching Eﬀect in Silicon Devices with Large Magnetoresistance WANG Ji-Min, ZHANG Xiao-Zhong, PIAO Hong-Guang, LUO Zhao-Chu, XIONG Cheng-Yue 077202 Growth of a-Plane InN Film and Its THz Emission WANG Guang-Bing, ZHAO Guo-Zhong, ZHENG Xian-Tong, WANG Ping, CHEN Guang, RONG Xin, WANG Xin-Qiang

077203 Transport and Capacitance Properties of Charge Density Wave in Few-Layer 2H–TaS2 Devices CAO Yu-Fei, CAI Kai-Ming, LI Li-Jun, LU Wen-Jian, SUN Yu-Ping, WANG Kai-You

077401 Two Superconducting Phases and Their Characteristics in Layered BaTi2(Sb1−xBix)2O with x = 0.16 WU Yue, DONG Xiao-Li, MA Ming-Wei, YANG Huai-Xin, ZHANG Chao, ZHOU Fang, ZHOU Xing-Jiang, ZHAO Zhong-Xian

077701 Evolution from Diffuse Ferroelectric to Relaxor Ferroelectric in Pb1−xBax(Fe1/2Nb1/2)O3 Solid Solutions LV Xin, WANG Nan, CHEN Xiang-Ming 077702 Influence of Rapid Thermal Annealing on the Structure and Electrical Properties of Ce-Doped HfO2 Gate Dielectric MENG Yong-Qiang, LIU Zheng-Tang, FENG Li-Ping, CHEN Shuai 077703 Depolarization and Electrical Response of Porous PZT 95/5 Ferroelectric Ceramics under Shock Wave Compression WANG Zhi-Zhu, JIANG Yi-Xuan, ZHANG Pan, WANG Xing-Zhe, HE Hong-Liang 077801 Planar Magnetic Metamaterial Slabs for Magnetic Resonance Imaging Applications LI Chun-Lai, GUO Jie, ZHANG Peng, YU Quan-Qiang, MA Wei-Tao, MIAO Xi-Gen, ZHAO Zhi-Ya, LUAN Lin 077802 Paths for the Non-radiative Recombination Occurring in CdS:CdO/Si Multi-Interface Nanoheterostructure Array LI Yong, WANG Xiao-Bo, ZHAO Jin-Chao, LI Xin-Jian 077803 Silver Nanoparticle Fabrication by Laser Ablation in Polyvinyl Alcohol Solutions Halimah Mohamed. K, Mahmoud Goodarz Naseri, Amir Reza Sadrolhosseini, Arash Dehzangi, Ahmad Kamalianfar, Elias B Saion, Reza Zamiri, Hossein Abastabar Ahangar, Burhanuddin Y. Majlis CROSS-DISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND TECHNOLOGY 078101 Resistive Switching Behavior in Amorphous Aluminum Oxide Film Grown by Chemical Vapor Deposition QUAN Xiao-Tong, ZHU Hui-Chao, CAI Hai-Tao, ZHANG Jia-Qi, WANG Xiao-Jiao 078102 Structural Evolution during the Oxidation Process of Graphite FAN Bing-Bing, GUO Huan-Huan, ZHANG Rui, JIA Yu, SHI Chun-Yan 078103 Structure and Magnetic Properties of (In,Mn)As Based Core-Shell Nanowires Grown on Si(111) by Molecular-Beam Epitaxy PAN Dong, WANG Si-Liang, WANG Hai-Long, YU Xue-Zhe, WANG Xiao-Lei, ZHAO Jian-Hua 078104 Radio-Frequency Performance of Epitaxial Graphene Field-Effect Transistors on Sapphire Substrates LIU Qing-Bin, YU Cui, LI Jia, SONG Xu-Bo, HE Ze-Zhao, LU Wei-Li, GU Guo-Dong, WANG Yuan-Gang, FENG Zhi-Hong 078501 Micrograting Displacement Sensor with Integrated Electrostatic Actuation YAO Bao-Yin, FENG Li-Shuang, WANG Xiao, LIU Wei-Fang, LIU Mei-Hua 078502 Ultralow-Voltage Electric-Double-Layer Oxide-Based Thin-Film Transistors with Faster Switching Response on Flexible Substrates ZHANG Jin, WU Guo-Dong

078503 TixSb2Te Thin Films for Phase Change Memory Applications TANG Shi-Yu, LI Run, OU Xin, XU Han-Ni, XIA Yi-Dong, YIN Jiang, LIU Zhi-Guo 078901 Two Typical Discontinuous Transitions Observed in a Generalized Achlioptas Percolation Process HU Jian-Quan, YANG Hong-Chun, YANG Yu-Ming, FU Chuan-Ji, YANG Chun, SHI Xiao-Hong, JIA Xiao