Pis’ma v ZhETF, vol. 110, iss. 11, pp. 755 - 756

Dynamics of particles trapped by dissipative solitons

D. A. Dolinina, A. S. Shalin, A. V. Yulin¹⁾

ITMO University, 197101 Saint-Petersburg, Russia

Submitted 22 September 2019

Resubmitted 28 October 2019

Accepted 30

October 2019

DOI: 10.1134/S0370274X19230085

Optical trapping [1-3] and transporting [4-8] of

been studied for many years theoretically [15-17] and

small objects by optical forces has been actively devel-

experimentally [18, 19]. The schematic view of this sys-

oping in the past several decades. Many new effects, that

tem is shown in Fig. 1a. The resonator is pumped by a

can be used for the trapping, for example such as optical

holding beam of coherent light. The nanoparticles par-

hook [9], has been presented. Such optical transport can

tially screen the light exciting the resonator and at the

be widely used in many areas from optics [10], biological

same time spatially nonuniform distribution of the op-

[11] and chemical [12] research, to microfabrication [13].

tical field in the resonator leads to the lateral force that

Optical trapping is based on the balance of two dif-

drags a particle into an area with higher intensity. In

ferent types of optical forces. The first type is scattering

turn, the nanoparticle creates a shadow which locally

forces, which push an object in the direction of light

reduces the intensity of the coherent pumping of the

propagation; and the second type is gradient optical

optical mode which is able to affect the soliton.

forces, which move an object along the gradient of light

As a mathematical model describing the dynamics

intensity [14]. If gradient forces are larger than scat-

of light and motion of the particle in such system, the

tering, then an object is pulled to the area of stronger

Generalized Nonlinear Schrödinger equation with dissi-

light intensity and can be trapped by focused light. Af-

pation and pumping coupled to an ordinary differential

ter trapping an object can be manipulated by moving

equation is used:

light beam in a preferred direction.

∂

∂²

E - iC

E + (γ - iδ + i

)E =

In this letter a new concept of optical manipulation

∂t

∂x²

1 + |E|²

of a small particle by dissipative optical localized waves

= (1 - fe-(x-ǫ)2/ω2 )P,

(1)

is proposed which opens new avenues for flexible con-

∂

trol over nanoparticles dynamics governed by nonlinear

ǫ=η

|E(ǫ)|²,

(2)

∂t

∂x

effects. Further the localized waves are referred as soli-

tons. First, the formation of the solitons can locally en-

where E - is a complex amplitude of optical field in the

hance the intensity of the field and thus allow to use less

resonator, C - diffraction coefficient, P - amplitude of

powerful holding beam. Secondly, changing the phase

laser pumping, γ - decay rate, α - coefficient of nonlin-

gradient of the holding beam (which is a relatively easy

earity; δ - laser detuning from resonant frequency, ǫ -

task) it is possible to move the solitons in a controllable

coordinate of nanoparticle. Parameter ω defines width

way and if the particles are trapped by soliton, then the

of a particle shadow, f defines transparency of a parti-

particles will be moved by it. This mechanism can po-

cle: if f = 0, then the particle is transparent and if f = 1

tentially facilitate the manipulation of nanoobjects by

then the particle is opaque. The coefficient η defines the

light.

ratio of the dragging force acting on the particle to the

To implement this strategy of optical manipulation a

field intensity gradient in the point of the particle loca-

series of problems are to be solved. Primarily, a suitable

tion. For mathematical convenience the dimensionless

physical system providing the existence of the solitons

variables are used.

and a corresponding mathematical model describing the

The stationary localized solutions of the resting soli-

formation of the solitons in the presence of nanoparticles

tons with trapped particles were found and their stabil-

have to be suggested.

ity is analyzed. It was found that in case of the spatially

The physical system considered in the present pa-

uniform pumping field the particle always destabilize

per is a nonlinear Fabry-Perot resonator with dielectric

the soliton, regardless of the transparency of the parti-

nanoparticles placed on top of the resonator. Bistability

cle. Depending on the parameters of the particle such

and formation of the solitons in such resonators have

instability can either destroy the soliton or set the soli-

ton in motion. This motion is affected by two effects,

¹⁾e-mail: alex.v.yulin@gmail.com

the first one is the attraction of the particle to the soli-

Письма в ЖЭТФ том 110 вып. 11 - 12

2019

755

3^∗

756

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

Fig. 1. (Color online) (a) - Schematic view of a Fabry-Perot resonator with a nanoparticle on the upper translucent mirror.

(b) - Instability increment of the resting soliton with a particle placed in the center of soliton, pumping field has phase

dependence on coordinate in the form P = P0e-ikx2 . Parameters: ν = 0.7, E0 = 1, δ = 0.3, γ = 0.2, C = 16, α = 3,

k = 0.005. (c) - Successful capture of a particle by a moving soliton. Pumping field P = P0e^ikx, where k = 0.04, f = 0.3,

other parameters are the same as in (b)

ton and the second one is repelling the soliton from the

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trapped particles can be stabilized by spatially nonuni-

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

form holding beams. In this case stability of the soliton

danov, I. Toftul, M. Nieto-Vesperinas, P. Ginzburg, and

depends on the transparency of a particle. If particle

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

shadow is too deep then soliton collapses. The depen-

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on the transparency of the particle and the velocity of

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ACS Photonics 5, 4371 (2018).

the particle causing its displacement, in the trapping of

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The work has been partially supported by the Rus-

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2019