Nonlinearity-induced optical torque Ivan Toftul1 2Gleb Fedorovich2Denis Kislov2 3 4Kristina Frizyuk2Kirill Koshelev1Yuri Kivshar1and Mihail Petrov2

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Nonlinearity-induced optical torque

Ivan Toftul,1, 2, ∗Gleb Fedorovich,2Denis Kislov,2, 3, 4 Kristina

Frizyuk,2Kirill Koshelev,1Yuri Kivshar,1and Mihail Petrov2

1Nonlinear Physics Center, Research School of Physics, Australia National University, Canberra ACT 2601, Australia

2School of Physics and Engineering, ITMO University, St. Petersburg 197101, Russia

3Riga Technical University, Institute of Telecommunications, Riga 1048, Latvia

4Center for Photonics and 2D Materials, Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia

(Dated: October 11, 2022)

Optically-induced mechanical torque leading to the rotation of small objects requires the presence

of absorption or breaking cylindrical symmetry of a scatterer. A spherical non-absorbing particle

cannot rotate due to the conservation of the angular momentum of light upon scattering. Here, we

suggest a novel physical mechanism for the angular momentum transfer to non-absorbing particles

via nonlinear light scattering. The breaking of symmetry occurs at the microscopic level manifested

in nonlinear negative optical torque due to the excitation of resonant states at the harmonic frequency

with higher projection of angular momentum. The proposed physical mechanism can be veriﬁed

with resonant dielectric nanostructures, and we suggest some speciﬁc realizations.

Introduction. The rotation and spinning of micro-

and nanoscale objects is one of the central goals of op-

tical manipulation since the discovery of optical tweez-

ers [1–8], utilized in controlling biological systems [9–

11], atoms [12,13], and nanoscale objects [14–18]. The

transfer of angular momentum from light to matter re-

sults in a mechanical torque acting on a scatterer [19–

22], being proportional to a diﬀerence between the an-

gular momenta absorbed and re-scattered by the object.

nonzero mechanical torque can appear due to the lack

of rotational symmetry [23–26] or the presence of ab-

sorption [27,28]. The direction and sign of the mechan-

ical torque is deﬁned by the imbalance condition, and

it can be opposite to the projection of the incident an-

gular moment of light leading to negative optical torque

(NOT) [29]. The appearance of linear NOT has recently

been studied both theoretically [29–31] and experimen-

tally [32–35].

Rapid development of all-dielectric nanophotonics [36–

39] brings novel opportunities for optical manipulation.

In contrast to nanoplasmonics, dielectric materials have

lower Ohmic losses [40], which are required for real-

izing optical rotation of cylindrically symmetric struc-

tures [27,28]. However, dielectric structures oﬀer unique

opportunities for observing nonlinear optical processes

such as second harmonic generation (SHG) or third-

harmonic generation (THG) due to large values of bulk

nonlinear susceptibilities. It is also possible to observe ex-

perimentally SHG in trapped particles [41,42]. Recently,

the dramatic enhancement of the SHG eﬃciency for res-

onant all-dielectric nanostructures was reported [43–47].

Here, we suggest utilizing the SHG for a transfer of an-

gular momenta of light to scatterers via nonlinear optical

process. The generated second harmonic (SH) ﬁeld also

may carry the angular momenta and, thus, provides a

contribution to the mechanical torque. We predict that

the angular momentum imbalance between the ﬁelds at

the fundamental and SH frequencies can lead to an op-

FIG. 1. General concept. Circular polarized light at the

frequency ωis launched onto a cylindrical dielectric parti-

cle and generates second-harmonic ﬁelds at the frequency 2ω

that might have diﬀerent angular momentum due to a crys-

talline lattice structure, producing a nonlinearity-induced op-

tical torque enhanced by the Mie resonances.

tical torque even for non-absorbing particles with cylin-

drical symmetry (Fig. 1), and its sign can change from

positive to negative with respect to the incident ﬁeld an-

gular momentum.

Nonlinearity-induced optical torque. We start with

considering circularly polarized plane wave with fre-

quency ωscattered on a dielectric particle possessing the

azimuthal symmetry (see Fig. 1). The plane wave is in-

cident along the axis of the symmetry and carries the

momentum of light of minc~per photon. Due to the

symmetry of the problem the optical torque T(ω)act-

ing on the particle at the fundamental frequency is ex-

actly proportional to the absorption cross section [27,28]

and, in terms of canonical spin angular momenta den-

sity, one can write T(ω)=c/n0·σabsS(ω), where S(ω)=

minc/(2ω)·εε0[E(ω)

0]2ezis the canonical spin angular mo-

menta density [48] with azimuthal number minc =±1 for

right(left) circular polarization and n0=√εµ is the re-

fractive index of the host media; σabs is the total absorp-

arXiv:2210.04021v1 [physics.optics] 8 Oct 2022

tion cross section. For a nonlinear process in dielectric

particles, the energy loss at the fundamental harmonic

(FH) frequency occurs due to the harmonic generation,

so σabs →σSHG. With the consideration of SHG being

a dominant nonlinear process [49], one should also ac-

count for the angular momenta carried out by the SH

ﬁeld (Fig. 1). Hence, there are two components of the

optical nonlinear torque

T=T(ω)+T(2ω),(1)

where T(ω)and T(2ω)are the torques generated by the

ﬁeld at FH and SH. The interference terms with nonzero

frequencies are averaged to zero [50].

For the particles possessing an azimuthal symme-

try with negligible Ohmic losses excited at frequencies

away from two and three photon absorption regions [51],

torque on the FH is deﬁned by the amount of energy

spent on the SHG process. By decomposing SH ﬁeld

into the series of vector spherical harmonics (VSH) it

is possible to express SH generation cross section σSHG

in terms of the radial electromagnetic energy density in

magnetic and electric multipoles in the far ﬁeld WE

mj and

mj [43,50], so torque at the FH is

T(ω)

z=mincT0

σSHG

σgeom

=mincT0

σgeom[k(2ω)]2X

WE

mj +WM

mj ,

(2)

where k(2ω) = n02ω/c,σgeom is the geometric cross sec-

tion, mj are the projection and the total angular mo-

mentum numbers, and T0= 0.5εε0[E(ω)

0]2σgeom/k(ω),

is the maximal torque which can transferred to a plate

of area σgeom once all the momentum of the incident

ﬁeld is absorbed. Coeﬃcients WE

mj and WM

mj depend on

the overlap integral between the nonlinear polarization

P(2ω)=ε0ˆχ(2)E(ω)E(ω)and ﬁeld of the SH mode in the

volume of the particle [50]. Thus WE

mj and WM

mj contains

all the information about nonlinear response.

The torque on the SH frequency can be derived by

the calculating the change of the total angular momenta

ﬂux tensor ˆ

M(2ω)on the SH as T(2ω)=HΣ

M(2ω)·

ndS[19,20,22,52–54]. Surface integration is performed

over arbitrary closed surface Σ which contains the scat-

terer, and nis the outer normal to that surface. This

integral can be taken once the ﬁelds are decomposed into

the VSH series [55–58] and the total torque can be writ-

ten in a compact and elegant way which underpins the

physics behind nonlinearity-induced optical torque [50]:

Tz=T(ω)

z+T(2ω)

z(3)

2T0

σgeom[k(2ω)]2X

(2minc −m)WE

mj +WM

mj .

Eq. (3) is the central result of this work. Generation

of SH photon in the particular VSH state results in (i)

adding a torque corresponding to the spins of two pho-

tons absorbed at the FH and (ii) adding a recoil torque

FIG. 2. Exact solution for a dielectric sphere. (a) SHG ef-

ﬁciency (σSHG/σgeom) of a spherical GaAs nanoparticle with

refractive indices n(ω)

p= 3.28, n(2ω)

p= 3.56 as a function of its

radius. The colored lines show the contribution of diﬀerent

multipolar channels labeled as (m, j). The pump wavelength

is 1550 nm. The multipoles with m= 0 and m= 4 provide

contribution to positive and negative optical torques, respec-

tively. (b) Total optical torque acting on the nanoparticle due

to SHG.

from SH photons emitted with the total angular momen-

tum projection m. This nonlinear optomechanical eﬀect

has not been discussed in the literature and is proposed

for the ﬁrst time, to the best of our knowledge.

Selection rules. For the in-depth analysis of Eq. (3)

we use the symmetry analysis and multipolar decomposi-

tion [43,59–62]. The imbalance between the angular mo-

mentum projections in Eq. (3) immediately shows that

there can appear a nonzero torque induced by nonlinear

optical generation process results. Its sign with respect

to minc strongly depends on the exact multipolar content

of SH ﬁeld. Most of these components are zero due to

the strict selection rules on mduring SHG [43,61,63–65]

imposed by the symmetry of the particle and the crys-

talline lattice, which can be explicitly seen from the over-

lapping integral in WE,H

mj [50]. In order to illustrate the

mechanism of NOT appearance, we consider individual

crystalline structures made of GaAs which is a common

optical material possessing strong second-order nonlinear

optical response. Its zinc-blende lattice structure with

Tdsymmetry provides a single independent component

of the nonlinear tensor χ(2)

xyz [66–69] once the lattice is ori-

ented such that [001] kˆez,[100] kˆex.

The symmetry of GaAs lattice along with the axial

symmetry of the nanoparticle dictates that only m=

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Nonlinearity-inducedopticaltorqueIvanToftul,1,2,GlebFedorovich,2DenisKislov,2,3,4KristinaFrizyuk,2KirillKoshelev,1YuriKivshar,1andMihailPetrov21NonlinearPhysicsCenter,ResearchSchoolofPhysics,AustraliaNationalUniversity,CanberraACT2601,Australia2SchoolofPhysicsandEngineering,ITMOUniversity,St.Peters...

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Nonlinearity-induced optical torque Ivan Toftul1 2Gleb Fedorovich2Denis Kislov2 3 4Kristina Frizyuk2Kirill Koshelev1Yuri Kivshar1and Mihail Petrov2

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