Pis’ma v ZhETF, vol. 111, iss. 5, pp. 303 - 304

March 10

Complex dynamics of optical solitons interacting with nanoparticles

D. A. Dolinina¹⁾, A. S. Shalin, A. V. Yulin

ITMO University, 197101 St.-Petersburg, Russia

Submitted 19 November 2019

Resubmitted 14 February 2020

Accepted 16

February 2020

DOI: 10.31857/S0370274X20050057

(

)

The localized nonlinear patterns in the systems with

∂

∂²

E - iC

E+

1 - iδ + i

gain and loss are refereed as dissipative solitons [1, 2].

∂t

∂x²

1 + |E|²

(

)

The formation of the localized dissipative structures is

∑

)²/ω²

provided not only by the balance of the spreading and

fe-(x-ǫm

(1)

the narrowing of the wave but also by the balance of

∂

the driving force and the losses in the system. A pump

ǫ_m = η

|E(ǫ_m)|²,

(2)

∂t

∂x

of energy is essential and the solitons are defined by the

properties of the system, rather than by the initial con-

where E is a complex amplitude of optical field in the

ditions [3]. That is why the dissipative solitons can be

resonator, C is diffraction coefficient, P is complex am-

easily controlled and are interesting from the practical

plitude of laser pumping, α is coefficient of nonlinearity;

point of view for their potential applications in opto-

δ is laser detuning from resonant frequency, ǫ is coordi-

electronic devices [4, 5].

nate of nanoparticle. Parameter ω is width of a particle

One of the recent proposed applications of the op-

shadow, f is a transparency coefficient of a particle: if

tical solitons is optical trapping [6]. Nowadays optical

f = 0, then particle is transparent and if f = 1 then

trapping [7-9] and transporting [10-14] is actively de-

the particle is opaque. The coefficient η defines the ra-

veloping field and many new effects, that can be used

tio of the dragging force acting on the particle to the

for the trapping, for example such as optical hook [15],

field intensity gradient at the point of particle location.

is presented. In [6] it is proposed to use optical soli-

The numerical simulations of collisions of soliton-

tons for manipulation of nanoparticles placed in or on

particle bound states is performed. To force solitons

the top of the resonator excited by a powerful holding

with trapped particles to move towards each other a

beam. The solution in the form of a bound state of a

phase gradient of the holding beam P = P0e-ikx2 is

soliton and a particle is found, and the stability of the

used. It is shown, that as a result of two-soliton colli-

states is studied. It is shown that the bound states can

sion one soliton is formed, but in dependence of trans-

be dynamically stable and, thus, can be observed exper-

parency of the particles different outcomes are possible.

imentally.

If particles are transparent enough the resulting soliton

In the present Letter we consider the dynamics of the

successfully captures them, see Fig. 1a. If particles are

solitons carrying more than one particle and investigate

too opaque the resulting soliton annihilates and parti-

mutual interaction of the solitons with the trapped par-

cles get released, see Fig. 1d. In the intermediate case

ticles.

the resulting soliton and particles oscillate around some

equilibrium point, see Fig.1c. The result of the collision

The system of interest is a nonlinear Fabry-Perot

can be predicted by the stability analysis of the corre-

resonator pumped by the coherent light with a dielectric

sponding stationary state of single soliton with trapped

particle, located in the surface. The system of this type

particles, see Fig. 1b.

can be described by a generalized nonlinear Schrodinger

Also the interactions of solitons through rescattering

equation for the optical field. The viscous dynamics of

on the particles is considered. The system of interest is

the particle can be obtained via the solution of an or-

two nonlinear wide-aperture resonators separated by a

dinary differential equation for the centre mass of the

relatively thin gap. Each of the resonators is pumped

particle [6]:

by a holding beam, and we assume that the resonators

do not interact with each other directly. However, if a

particle is placed between them, then it feels the evanes-

¹⁾e-mail: dasha.doly@gmail.com

cent fields of the both resonators modes. That is why

Письма в ЖЭТФ том 111 вып. 5 - 6

2020

303

304

D. A. Dolinina, A. S. Shalin, A. V. Yulin

for a flexible and precise control over the particles. Such

accurate many particle manipulation can be used in mi-

crofabrication.

The work has been partially supported by the

Russian Foundation for Basic Research (Projects

#18-02-00414, 18-52-00005) and Ministry of Edu-

cation and Science of the Russian Federation (grant

#2019-220-07-8749, and GOSZADANIE). The calcu-

lations of the soliton dynamics are supported by the

Russian Science Foundation (Project # 18-72-10127).

Full text of the paper is published in JETP Letters

journal. DOI: 10.1134/S002136402005001X

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Fig. 1. (Color online) Collision of solitons with trapped

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= P0e-ikx2 .

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7, 6 (1990).

tons with trapped particles one soliton is formed, which

N. Akhmediev and A. Ankiewicz, Optical Solitons, The-

successfully captures both particles, f = 0.05. A corre-

oretical and Experimental Challenges, Lecture Notes

sponding stationary state is stable, see panel (b). (b) -

in Physics, ed. by K. Porsezian and V.C. Kurakose,

The dependence of the instability increment of the re-

Springer, Berlin (2003), v.613.

sulting soliton with captured particles on the collective

W. J. Firth and C. O. Weiss, Opt. Phot. News 13, 55

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(2002).

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D. A. Dolinina, A.S. Shalin, and A. V. Yulin, Pis’ma v

soliton oscillates because corresponding stationary state

ZhETF 110, 11 (2019).

is unstable, see panel (b). (d) - As a result of the colli-

A. Ashkin, Phys. Rev. Lett. 24, 156 (1970).

sion both solitons annihilate, f = 0.25, because there is

A. Ivinskaya, M. I. Petrov, A. A. Bogdanov, I. Shishkin,

no corresponding stationary state

P. Ginzburg, and A. S. Shalin, Light Sci. Appl. 6, e16258

(2017).

N. Kostina, M. Petrov, A. Ivinskaya, S. Sukhov, A. Bog-

the particle can get attracted to the maximal intensity

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region of the field in each of the cavities. At the same

A. Shalin, Phys. Rev. B 99, 125416 (2019).

time coupling to the particles decreases coupling of the

10.

D. B. Ruffner and D. G. Grier, Phys. Rev. Lett. 109,

holding beam to the guided mode of the cavity. Thus,

163903 (2012).

the modes of the cavities can interact through the par-

11.

M. I. Petrov, S. V. Sukhov, A. A. Bogdanov, A. S. Shalin,

ticles placed between them.

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It is shown that in case of the solitons of same in-

12.

A. S. Shalin and S. V. Sukhov, Plasmonics 8, 2 (2013).

tensity the slower one takes all the particles. The slower

13.

S. Sukhov, N. Kostina, A. Proskurin, M. I. Petrov,

soliton can steadily capture all the particles or just one,

A. A. Bogdanov, S. Sukhov, A. V. Krasavin,

A. Karabchevsky, A. S. Shalin, and P. Ginzburg,

in dependence of transparency of the particles. If both

Opt. Exp. 23, 247 (2015).

solitons move with the similar velocities, in a result of

14.

A. Ivinskaya, N. Kostina, A. Proskurin, M. I. Petrov,

interactions between solitons all trapped particles get

A. A. Bogdanov, S. Sukhov, A. V. Krasavin,

released.

A. Karabchevsky, A. S. Shalin, and P. Ginzburg,

By considered effects it is possible, for example, to

ACS Photonics 5, 4371 (2018).

collect all the particles in the system by two counter-

15.

L. Yue, N. Kostina, A. Proskurin, M. I. Petrov,

propagating solitons and then release the particles at

A. A. Bogdanov, S. Sukhov, A. V. Krasavin,

some desirable point creating a cluster of particles; or

A. Karabchevsky, A. S. Shalin, and P. Ginzburg,

to bring all the particles to the left or to the right edge

Opt. Lett. 43, 4 (2018).

of the cavity. In other words, the solitons can be used

Письма в ЖЭТФ том 111 вып. 5 - 6

2020