旋转动力学：径向周期光格子中孤子的研究

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"Rotary dynamics of solitons in radially periodic optical lattices" 这篇研究集中在光学格子中旋转孤子的动力学行为。光学格子是通过光的干涉图案形成的周期性结构，常用于模拟固体物理中的量子效应，并在量子信息处理、光存储和光子晶体等领域有广泛应用。在描述的实验中，研究者关注的是那些在径向周期性光格中被强烈局域化的孤子。孤子是一种非线性波现象，它能保持形状不变地传播，因为非线性效应和波的色散相互平衡。文章指出，这些孤子在远程的槽沟中被陷阱，且具有Kerr型自聚焦非线性。Kerr非线性是指介质的折射率会随着光强的增加而改变，这种效应在光学领域非常常见。研究发现，孤子的旋转稳定性与几个因素有关：非线性波数越大、旋转速度越低以及陷阱槽沟的半径越小，旋转运动就越稳定。这表明非线性效应在维持孤子的旋转状态中起着关键作用。当光格在传播过程中收缩或扩展时，孤子可以继续保持在原来的槽沟中，但其运动方向会改变，向外或向内移动。有趣的是，孤子的旋转线速度与陷阱槽沟的半径成反比，这意味着槽沟半径增大时，旋转速度会减慢，反之则加快。这一发现对于理解和控制光学系统中的孤子动力学有着重要意义。文章引用了Optical Society of America (OSA)的分类代码190.0190（非线性光学）、190.4420（光学孤子）和190.6135（光格子），这些代码代表了研究的核心主题。此外，还提到了solitons在非线性科学多个分支中的重要性，比如生物学、固体物理学以及Bose-Einstein凝聚体（BECs），BECs是量子物理学的一个重要概念，其中原子集体行为类似于单个量子物体。这项工作深化了我们对非线性光学系统中孤子行为的理解，特别是在径向周期性光格中的旋转特性，这可能为未来开发新的光控技术和器件提供理论基础。

August 10, 2010 / Vol. 8, No. 8 / CHINESE OPTICS LETTERS 791

Rotary dynamics of solitons in radially periodic optical

lattices

Keya Zhou (

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), Zhongyi Guo (

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), and Shutian Liu (

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)

∗

Department of Physics, Harbin Institute of Technology, Harbin 150001, China

∗

E-mail: stliu@hit.edu.cn

Received January 4, 2010

We investigate the dynamics of strongly localized solitons trapped in remote troughs of radially periodic

lattices with Kerr-type self-focusing nonlinearity. The rotary motion of solitons is found to be more stable

for larger nonlinear wavenumbers, lower rotating velocity, and shorter radius of the trapping troughs.

When the lattice is shrunk or expanded upon propagation, the solitons can be trapped in the original

trough and move outward or inward, with their rotating linear velocity inversely proportional to the radius

of t he trapp ing troughs.

OCIS codes: 190.0190, 190.4420, 190.6135.

doi: 10.3788/COL20100808.0791.

Solitons in nonlinear s ystems play a crucial role in many

branches of nonlinear science, such as biology, solid state

physics, Bose-Einstein condensates (BECs), and nonlin-

ear optics

[1−6]

. In Kerr medium with a purely cubic non-

linear response, solitons formally exist, but in two- and

three-dimensional (2D and 3 D) cases, they are unstable

against the s patiotemporal collapse induced by the c om-

bined eﬀect of nonlinearity and diﬀraction (anomalous

dispersion)

[6]

. One way to prevent colla pse is to change

the no nlinearity by considering media with qua dratic

(second-harmonic generating) nonlinearity or saturable

nonlinearity. Another way is to introduce transverse

modulation of potentials (optical lattices). In recent

years, many interests have been directed to the latter

case, and many properties of solitons that cannot be ob-

served in bulk uniform nonlinear media have been dis-

covered in transverse modulated media.

In the 2D domain, solitons have be e n investigated in

harmonic lattices

[7,8]

, hexagon lattices

[9]

, axis-symmetric

lattices

[10−16]

, and other kinds of lattices

[17−19]

. Re-

cently, axis-symmetric lattices have been the focus of

attention, where solitons are trapped either at the cen-

ter or in the remote troughs of the lattices. Diﬀerent

forms of solitons have been examined, such as funda-

mental solitons

[10−12]

, dipole solitons

[13]

, ring solitons,

and necklace solitons

[14]

. An interesting phenomenon

takes place in 2D Bessel lattices or radially periodic lat-

tices. When fundamental solitons are pro bed into remote

troughs, they take a rotary motion

[10−12,15]

. However,

recent achievements in this ﬁeld usually employ a funda-

mental soliton as the prob e beam, which is not formally

the exact solution of solitons trapped in remote troughs.

Solitons that are trapped in remote troughs are deter-

mined by the local distribution of the refr active index and

have diﬀerent proﬁles fr om fundamental solitons. Their

corresponding rotary dynamics has never b een reported

befo re.

In this letter, we begin our research with the 2D Gro ss-

Pitaevskii equation (GPE)

[15]

with radially periodic po-

tential. The proﬁles of solitons trapped in remote troughs

of sta tionary lattices are obtained by Fourier iteration

method

[8]

. Using the solved soliton proﬁles as input sig-

nals, their propagation dynamics are numerically sim-

ulated by the split-step Fourier transformation method

(SSFTM). In stationary lattices, the rotating motion o f

a soliton is aﬀected by the potential strength, nonlinear

wavenumber, rotating velocity, and radius of the troug hs.

In particular, we also investigate the rotary motion of a

soliton in dynamic lattices, which are shrunk or expanded

upon propagation. In our simulation, we use a ra ther sta-

ble soliton proﬁle as input signal and ﬁnd that solitons

are trapped in the origina l troughs and rotate around the

axis in a spiral motion upon propagation. The rotating

linear velocity is inversely proportional to the radius of

the troughs, which is determined by the two conserved

quantities of the GPE.

The model follows standard GPE in its no rmalized

form

[15]

∂u

∂t



∂

∂x

∂

∂y



u + V (x, y)u + |u|

u = 0, (1)

written for the wave function u in Cartesian coordinates

(x, y). The equation can describe the evolution of matter-

wave solitons created in BECs with resp ect to time, and

V (x, y) is the corr esponding optical lattice potential. The

dimensionless variables are chosen such that u is mea-

sured in units of the recoil energy E

= ¯h

/2m, where

k = π/ d, and d is a constant re presenting the scale of

the lattices. The temporal coordinate t and spatial co-

ordinates (x, y) a re measured in units of E

/¯h and d/π,

respectively. In our simulation, we consider pe riodic po-

tential in the axial direction as

V (r) = ε c os(2πr/T ), (2)

where ε is the modulation depth of the optical lattice po-

tential, and T = 2π is the dimensionless period along the

radial direction. The solitons trapped in the lattices can

be written as u(x, y, t) = U(x, y)e

jµt

, where µ is the non-

linear wavenumber of the solitons and U(x, y) denotes

the soliton proﬁle. We use the Fourier iteration scheme

used in Ref. [6], choose a Gaussian-like initial condition,

and let the max imum of the trial function centered at

the c e nter of n

be the lattice troug h at po ints (n

T, 0).

1671-7694/2010/080791-04

 2010 Chinese Optics Letters

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