# From far-field to near-field micro- and nanoparticle optical trapping

Theodoros D. Bouloumis  Light-Matter Interactions for Quantum Technologies
Unit, Okinawa Institute of Science and Technology Graduate University, Onna,
Okinawa 904-0495, Japan  Correspondence<EMAIL_ADDRESS><EMAIL_ADDRESS>Síle Nic Chormaic  Light-Matter Interactions for
Quantum Technologies Unit, Okinawa Institute of Science and Technology
Graduate University, Onna, Okinawa 904-0495, Japan  Correspondence:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Optical tweezers is a very well-established technique that has developed into
a standard tool for trapping and manipulating micron and submicron particles
with great success in the last decades. Although the nature of light enforces
restrictions on the minimum particle size that can be efficiently trapped due
to Abbe’s diffraction limit, scientists have managed to overcome this problem
by engineering new devices that exploit near-field effects. Nowadays, metallic
nanostructures can be fabricated which, under laser illumination, produce a
secondary plasmonic field that does not suffer from the diffraction limit.
This advance offers a great improvement in nanoparticle trapping, as it
relaxes the trapping requirements compared to conventional optical tweezers.
In this work, we review the fundamentals of conventional optical tweezers, the
so-called plasmonic tweezers, and related phenomena. Starting from the
conception of the idea by Arthur Ashkin until recent improvements and
applications, we present some of the challenges faced by these techniques as
well as their future perspectives. Emphasis in this review is on the
successive improvements of the techniques and the innovative aspects that have
been devised to overcome some of the main challenges.

###### keywords:

optical tweezers, optical forces, particle trapping, plasmonics, self-induced
back action effect, surface plasmons

##  1 1\. Introduction

Imagine making our fingers one million times smaller and putting them ”inside”
the nanoworld. Now, it is easy to imagine that with these fingers we could
easily grab things of similar size: dielectric nanoparticles, quantum dots
with a DNA strand attached to them, proteins and viruses. More than that, we
could have the ability to move them in space. Well, although we cannot modify
our fingers, we have found a way to manipulate objects of that size using
light!

For more than four centuries, it has been known that light can exert forces on
objects [1]. Much later, in 1873, using Maxwell’s famous electromagnetic
theory [2], the transfer of momentum from light to illuminated objects was
described, resulting in the so-called radiation pressure that leads to objects
moving along the direction of light propagation [3]. There were many
experiments to follow that confirmed Poynting’s calculations, but all of them
were concluding with the fact that these optical forces were so small that it
was difficult even to measure them, let alone utilise them in some meaningful
application. However, as usually happens with science and technology, the
development of techniques to allow this eventually came. Particularly, the
birth of lasers around 1960 [4, 5] opened new possibilities and topics for
research in the field of light-matter interactions.

As lasers became more and more popular in science exploration, Arthur Ashkin,
in 1970, experimentally demonstrated how optical radiation forces exerted by
lasers can be used to change the motion of dielectric microparticles. He even
managed to trap them by creating a stable optical potential well [6], thus
establishing the new research topic that is known today as optical tweezers.

As always, nature follows its own rules, and soon the primary challenge for
optical tweezers became apparent, i.e., the diffraction limit. It seemed to be
impossible to focus light beyond the constraints imposed by this limit and,
consequently, this created a restriction on the smallest size of particle that
could be trapped. Subsequently, the next step was the idea to utilise surface
plasmons excited on metallic nanostructures to confine light into highly
intense optical fields, thus enabling superior trapping performance [7]. The
first experimental demonstration of trapping using plasmonic structures was
reported by Righini et al. [8] in 2007 and, since then, the field of plasmonic
optical tweezers started developing rapidly and opened further scientific
avenues for exploration. Numerous implementations arose from the research on
optical forces and plasmonics which are mentioned elsewhere [9, 10].

##  2 2\. Conventional Optical Tweezers

The Nobel Prize in Physics 2018 was awarded (50%) to Arthur Ashkin ”for the
optical tweezers and their application to biological systems”. The whole
research field started when Ashkin calculated that ”a power $P=1\;W$ of cw
argon laser light at $\lambda=0.5145\;\mu m$ focussed on a lossless dielectric
sphere of radius $r=\lambda$ and density $=1\;gm/cc$ gives a radiation
pressure force $F_{rad}=2qP/c=6.6\cdot 10^{-5}\;dyn$, where $q$, the fraction
of light effectively reflected back, is assumed to be of order 0.1. The
acceleration $=1.2\cdot 10^{8}\;cm/sec^{2}\cong 10^{5}$ times the acceleration
of gravity” [6]. In the same work, he demonstrated the first experimental
approach to test his calculations on transparent, micron-sized latex spheres
in liquids and gas. Indeed, he found that the radiation pressure exerted on
the particles from a focussed laser light beam was able to accelerate them
along the direction of the beam and the measured velocities of the accelerated
particles were in very good agreement with the theoretical predictions.

Ashkin then went even further and demonstrated trapping of particles using one
laser beam and the wall of a glass cell, as well as using two counter-
propagating beams with the same characteristics [6]. However, a few years
later, Ashkin et al. [11], reported trapping of dielectric particles (10
$\mu$m - 25 nm) using a single beam by focussing argon-laser light at 514.5 nm
through a high numerical aperture objective lens ($NA=1.25$). The achievement
of this is attributed to the existence of a force additional to that caused by
the radiation pressure (from now on called the scattering force) which
originates from the axial beam intensity gradient. It then becomes apparent
that, whereas the scattering force depends on the optical intensity and has
the direction of the incident beam, the gradient force depends on the
intensity gradient and is directed along it from low to high intensities. This
allows for optical trapping by balancing these two forces.

The theoretical mechanism that explains this observation, depends on the
relative size of the particle (radius, $r$) in respect to the wavelength of
the laser light ($\lambda$). For $r\gg\lambda$, ray optics can be used and the
reflection and transmission of the beam from the particle can give rise to the
two forces. For $r\ll\lambda$, Rayleigh scattering is assumed and the particle
is treated like a dipole in an external electromagnetic field. The two regimes
are analysed below. Finally, there is the intermediate regime where the
particle size is of the same order of magnitude as the wavelength. In this
case, the approximations mentioned above cannot be used and, in order to
evaluate the forces arising, Maxwell’s stress tensor, which relates the
interactions between electromagnetic forces and mechanical momentum [9, 12],
should be used. To handle this complicated mathematical analysis different
algorithms have been established, such as the transition matrix (T-matrix)
method [13] and the discrete dipole approximation (DDA) [14]. Due to its
complexity this regime is not analysed here.

###  2.1 Ray optics approximation ( $r\gg\lambda$)

We assume spherical particles of higher refractive index than their
surrounding environment, being in a liquid solution and undergoing Brownian
motion. As soon as a particle, randomly moving, enters the light beam, a small
fraction of light is reflected off the surface of the particle and most of it
is refracted on passing through the particle (assuming no absorption). Light
carries momentum and, since refraction is a light-matter interaction
phenomenon, there is a momentum transfer from the photons to the particle. As
is known from geometrical optics, the path of the light changes due to the
refraction, resulting in a change in the momentum of the photons. Obviously,
from conservation of momentum for the light-particle system, there should also
be a change in the momentum of the particle and this creates a force acting on
the particle, $\vec{F}=d\vec{p}/dt$. To get a first insight, we can initially
assume that there is no reflection and part of the beam is refracted inside
the sphere as shown in Figure 1a. According to the work done in [15, 16], the
magnitude of the force on the particle due to the momentum change of a single
ray is given by

$F=\frac{n_{m}}{c}P,$ (1)

where $n_{m}$ is the refractive index of the particle, $c$ is the speed of
light and $P$ is the power of the incident ray. Since the Gaussian beam has a
radial intensity profile, the rays closer to the centre of the beam carry
higher power (i.e., intensity); thus, the resultant force from ray 2 ($F_{2}$)
is stronger than that from ray 1 ($F_{1}$), as shown in Figure 1a.

Figure 1: The scattering and gradient forces acting on a dielectric particle
in the ray optics regime, arising from a free space a and a tightly focussed b
Gaussian beam. In the subplot a the forces push the particle towards the
centre of the beam and along the direction of propagation, whereas in b the
forces drag the particle towards the focus of the beam. The upper panel in
subplot a shows the proposed geometry for calculating the forces using Fresnel
coefficients. The bottom panel shows trapping using counter-propagating beams
with the same characteristics so that the scattering forces cancel each other
to achieve a stable trap. Figure reproduced with permission from [17].

The forces acting on the particle can be reduced into a longitudinal component
parallel to the incident ray and a transversal one perpendicular to it. As
shown in the figure, the longitudinal components of the two forces add up to
create a scattering force, whereas the transversal components subtract leading
to a gradient force towards the beam’s higher intensity, thus moving the
particle to the centre of the Gaussian beam and along its axis. Note that, for
particles with a lower refractive index than the surroundings, the forces
reverse and the particle moves away from the centre of the beam.

If we want to describe the process in a more accurate and mathematically
rigorous way, we have to take into account multiple internal reflections and
refractions of the rays, as shown in the top inset of Figure 1a and Figure 2.
The forces exerted on a particle were first calculated by Roosen [18] by
considering Fresnel’s reflection ($R$) and transmission ($T$) coefficients.
For a detailed derivation see [19, 20]. The resulting forces are:

$F_{scat}=F_{Z}=\sum_{i=1}^{N}\dfrac{n_{m}\cdot
P_{i}}{c}\left[1+R_{i}\cos(2\theta_{i})-\dfrac{T_{i}^{2}[\cos(2\theta_{i}-2r_{i})+R_{i}\cos(2\theta_{i})]}{1+R_{i}^{2}+2R_{i}\cos(2r_{i})}\right]$
(2)

and

$F_{grad}=F_{Y}=\sum_{i}^{N}\dfrac{n_{m}\cdot
P_{i}}{c}\left[R_{i}\sin(2\theta_{i})-\dfrac{T_{i}^{2}[\sin(2\theta_{i}-2r_{i})+R_{i}\sin(2\theta_{i})]}{1+R_{i}^{2}+2R_{i}\cos(2r_{i})}\right],$
(3)

where the sum is over all $N$ rays interacting with the particle, and
$\theta_{i}$ and $r_{i}$ are the incidence and refraction angles,
respectively, as shown in Figure 2. The terms in the square brackets are the
dimensionless trapping efficiencies, $Q_{scat}$ and $Q_{grad}$, and account
for the efficiency of momentum transfer from the light ray to the particle. We
also define the total trapping efficiency of the ray as
$Q_{ray}=\sqrt{Q^{2}_{scat}+Q^{2}_{grad}}$. The Fresnel coefficients, $R$ and
$T$, depend on the polarisation of the incident rays. Therefore, the trapping
efficiencies and the trapping forces will also be polarisation dependent.

Figure 2: Calculation of the scattering and gradient forces acting on a Mie
particle from a single ray, by taking into account geometrical optics and
multiple reflection and transmission events, using Fresnel coefficients $R$
and $T$. Figure reproduced with permission from [19].

In Figure 3, the trapping efficiencies as a function of the ray’s incident
angle are plotted for circularly polarised light hitting a glass $(n_{g}=1.6)$
sphere in water $(n_{s}=1.33)$. We can see that, for incident angles smaller
than $70^{\circ}$, the gradient force dominates, but as the incident angle
increases, the scattering force becomes significant. This means that, for
unfocussed or slightly focussed beams that have a small convergence angle,
inevitably most of the rays (taking into account the Gaussian beam profile)
will hit the surface of the particle with a large incident angle, $\theta$, as
shown in Figure 1a, thereby pushing the particle away. On the contrary, beams
that are tightly focussed under a high NA objective lens, cause the rays to
hit the surface of the particle with small incident angles (Figure 1b). As a
result, the gradient forces strongly dominate over the scattering ones and a
stable trap can be established, as Ashkin et al. experimentally demonstrated
[11]. Note that, in this case, the longitudinal component of the resulting
forces always points towards the beam’s focal point, as shown in Figure 1,
leading to particle trapping close to the focal point. Ashkin’s calculations
[19] confirmed that, in order to create strong, single-beam traps, high
convergence angles are required. For convergence angles smaller than $\sim
30^{\circ}$, single-beam trapping is impossible. Instead, we can use two
counter-propagating beams with the same characteristics, as shown in the
bottom inset of Figure 1a, to cancel out the scattering forces [6].

Figure 3: The calculated Q factors for the gradient, scattering, and total
trapping efficiencies for a single circularly polarised ray acting on a
spherical dielectric particle of effective refractive index $n_{m}=1.2$
($n_{m}=n_{g}/n_{s}$), as a function of the incident angle. Figure reproduced
with permission from [19].

###  2.2 Dipole approximation $(r\ll\lambda)$

In this case, the electric field that the particle experiences is
approximately spatially constant and, assuming a dielectric particle, we can
treat the entire particle as a collection of induced point dipoles in a
homogeneous electric field. Early theoretical work on radiation forces and
scattering effects for subwavelength dielectric media can be found in [15,
21]. Based on this work and the electromagnetic theory for electromagnetically
induced dipoles, we can describe the optical forces and the trapping potential
that arises.

The situation for a homogeneous particle can be briefly described as
follows111The analysis presented is taken from [20].: The oscillating
electromagnetic field from the laser beam causes each of the particle’s point
dipoles to have a dipole moment

$\vec{p}=\alpha_{0}\vec{E},$ (4)

where $\vec{E}$ is the electric field and $\alpha_{0}$ is the polarizability
of the particle, given by the Clausius-Mossotti relation

$\alpha_{0}=4\pi r^{3}\epsilon_{0}\dfrac{\epsilon_{r}-1}{\epsilon_{r}+2}.$ (5)

Here, $r$ is the particle’s radius, $\epsilon_{0}$ is the vacuum dielectric
permittivity, and $\epsilon_{r}$ is the particle’s dielectric permittivity.
The external field causes the dipoles to oscillate and, thus, radiate. Now, we
have to take into account the dipole’s interaction not only with the external
electromagnetic field, but with its own induced scattered field as well. For
that reason, the effective polarizability, $\alpha_{d}$, is introduced as a
radiative reaction correction to the intrinsic polarizability of the particle
[20] and it is given by

$\alpha_{d}=\dfrac{\alpha_{0}}{1-\frac{\epsilon_{r}-1}{\epsilon_{r}+2}[(k_{0}r)^{2}+\frac{2i}{3}(k_{0}r)^{3}]},$
(6)

with $k_{0}$ being the vacuum wavenumber.

Similarly, in order to calculate the forces acting on all the dipoles, we have
to take into account the Lorentz force from the external field and the
radiation forces arising from the dipoles themselves. It is also convenient to
calculate the time-averaged total force since it is the one that is observable
(electromagnetic fields oscillate on the order of $\sim 10^{15}$ Hz which is
very fast). Rigorous calculations have been done in [22, 20, 17]. According to
these works, the resulting, time-averaged force acting on a dipole is given by

$\vec{F}=\frac{1}{4}\alpha^{\prime}_{d}\nabla|\vec{E}|^{2}+\frac{\sigma_{ext,d}}{c}\vec{S}+\frac{c\epsilon_{0}\sigma_{ext,d}}{4\omega
i}\nabla\times(\vec{E}\times\vec{E}^{*}),$ (7)

where $\alpha^{\prime}_{d}$ is the real part of the effective polarizability,
$\omega$ is the angular frequency, $\sigma_{ext,d}$ is the extinction cross-
section, i.e., the active area of the particle that causes part of the energy
of the incident electromagnetic wave to be extinguished due to scattering and
absorption from the particle. It, therefore, indicates the rate of energy loss
from the incident wave. $\vec{S}$ is the time-averaged real part of the
Poynting vector of the incident wave

$\vec{S}=\frac{1}{2}Re\\{\vec{E}\times\vec{H}^{*}\\}.$ (8)

####

In Equation (7), we see that the force acting on a dipole consists of three
terms; the third term is called the spin-curl force and is related to
polarisation gradients in the electromagnetic field that arise when the
polarisation is inhomogeneous. Using a defined polarisation of the incident
beam, this term has a small value compared to the other two terms and that is
why we usually neglect it in optical trapping experiments. The second term is
the scattering force pointing in the direction of the Poynting vector,
$\vec{S}$, and arises from absorption and scattering phenomena that cause
momentum transfer from the field to the particle. The first term is the
gradient force and depends on the particle’s polarizability and the intensity
gradient of the electric field. We know that
$I=\frac{1}{2}c\epsilon_{0}|\vec{E}|^{2}$ and so the gradient force takes the
form

$\vec{F}_{grad}(\vec{r}_{d})=\frac{1}{2}\frac{\alpha^{\prime}_{d}}{c\epsilon_{0}}\nabla
I(\vec{r}_{d}).$ (9)

Equation (9) tells us that, for particles with positive polarizability (i.e.,
a higher refractive index than its surrounding) this force acts towards the
direction of the field’s higher intensity, i.e., the focal point. At the focal
point of a Gaussian beam with a beam waist, $w_{0}$, and radial coordinate,
$\rho$, we can approximate the intensity distribution as

$I(\rho)=I_{0}e^{-2\rho^{2}/w_{0}^{2}},$ (10)

and, for small radial displacements, we can Taylor expand to get

$I(\rho)\approx I_{0}\left(1-2\frac{\rho^{2}}{w_{0}^{2}}\right),$ (11)

and substitute into Equation (9). Thence,

$\vec{F}_{grad}(\vec{r}_{d})=\frac{1}{2}\frac{\alpha^{\prime}_{d}}{c\epsilon_{0}}\frac{\partial}{\partial\rho}\left[I_{0}\left(1-2\frac{\rho^{2}}{w_{0}^{2}}\right)\right]\hat{\rho}=-2\frac{\alpha^{\prime}_{d}}{c\epsilon_{0}}\frac{I_{0}}{w_{0}^{2}}\rho\hat{\rho}.$
(12)

By comparing with the restoring force of the classical harmonic oscillator,
$\vec{F}(x)=-\kappa x\hat{x}$, we get the trapping constant

$\kappa_{\rho}=2\frac{\alpha^{\prime}_{d}}{c\epsilon_{0}}\frac{I_{0}}{w_{0}^{2}},$
(13)

and, by integrating Equation (12), we get the trapping potential

$U(\rho)=\frac{1}{2}\kappa_{\rho}\rho^{2},$ (14)

which is plotted in Figure 4. Note that similar analysis can be used in order
to obtain the potential and the trapping constant along the axial direction.

Let us now present the main limitation of conventional optical tweezers.
Ashkin et al. [11] did some rough calculations and showed that, in order to
create a stable optical trap, resisting the Brownian motion of particles in a
liquid environment, a potential well as deep as $10k_{B}T$ is required, where
$T$ is the temperature in Kelvin. Although in some cases this is easy to
achieve, for subwavelength particles, as they become smaller in size, the
gradient force scales down very quickly, making it impossible to satisfy this
requirement. According to Eqs. (5) and (6), $\alpha^{\prime}_{d}\varpropto
r^{3}$, which means that, when the radius of the particle decreases by a
factor of 10, the polarizability of the particle and, consequently, the
gradient force (Equation (9)) decrease by a factor of 1,000. The trapping
potential is no longer deep and tight enough to hold the particle (Figure 4)
and the trap is inefficient. We can use Equations 5, 6 and 13 to calculate the
change in the trapping stiffness if the particle has a radius of $0.8r$:

$\Delta\kappa=\frac{\kappa_{0.8r}-\kappa_{r}}{\kappa_{r}}=\frac{\alpha^{\prime}_{d0.8r}-\alpha^{\prime}_{dr}}{\alpha^{\prime}_{dr}}.$
(15)

Calculations show that for $r=100$ nm, there is a $53\%$ decrease in the
trapping stiffness when the particle’s radius decreases to $0.8r$. From
Equation (13) we see that, in order to compensate for this effect and increase
the trapping constant and the gradient force, we can either increase the
intensity of the incident field ($I_{0}$) or focus tighter ($w_{0}$). However,
even though in some cases it is experimentally possible to increase the
intensity of the field by a factor of 1,000, the heat accumulation will be
huge and eventually destroy the particle, especially if it is a biological
sample. On the other hand, the diffraction limit allows focussing of the beam
to a certain spot size and this sets a minimum on the particle size that can
be successfully trapped. Additional to these limitations, as the particle
becomes smaller, the viscous drag reduces and the particle undergoes more
intense Brownian motion, making it easier for it to escape from the trap.

Figure 4: Graphical representation of the trapping potential wells for a
polystyrene bead of radius $r$ and 0.8$r$. A small decrease in the particle’s
size creates a significantly shallower and broader trapping potential well,
which offers much weaker confinement and, thus, the particle has a higher
probability to escape from the trap.

##  3 3\. Plasmonic Optical Tweezers

Recent advances in the field of optics and nano-optics have helped to overcome
the diffraction limit problem by using evanescent fields instead of
propagating ones; these have the intrinsic property of confinement beyond the
diffraction limit. A detailed analysis can be found in [23, 24]. The current
trend is to use metallic nanostructures (see [25] for a recent review on
different platforms) in which surface plasmons can be excited at resonant
frequencies and that concentrate the electric field to create highly intense
fields, thereby significantly increasing the trapping potential depth that a
nanoparticle may experience.

In 1992, Kawata et al. were the first to demonstrate ”Movement of micrometer-
sized particles in the evanescent field of a laser beam” [26] and, in 1997,
Novotny et al. were the first to theoretically propose and calculate optical
trapping at the nanoscale, using enhanced evanescent fields from a laser-
illuminated metallic nanotip [7]. Okamoto et al., around the same time, did
similar work, but used a metallic nanoaperture instead of a tip [27]. Figure 5
shows their proposed geometrical model.

Figure 5: The geometrical model, studied by Okamoto et al., for subwavelength
particle trapping, utilising the evanescent field near a metallic
nanoaperture. This is the most commonly studied and reported geometry due to
its simplicity to fabricate using the focussed ion beam (FIB) milling
technique. Figure reproduced with permission from [27].

The advantage of using this kind of configuration comes from the fact that the
incident field is no longer ”responsible” for creating the trapping potential,
but rather for exciting the surface plasmons (SP) on the metal/dielectric
interface. The SPs, in turn, create the strong evanescent field that is
”responsible” for the trapping potential. The main benefit of trapping using
an evanescent field is that, by nature, it has a very high field gradient,
thus exerting a large trapping force (see Equation (9)) with no need to
increase the incident intensity, thereby leading to a reduction of radiation
damage to the sample. In other words, superior trapping conditions can be
achieved with much lower illumination power compared to the conventional
optical tweezers.

It was then just a matter of time for the first experimentally demonstrated
plasmonic optical tweezers to be reported. In 2007, Righini et al., using a
geometry of total internal reflection similar to the one shown in Figure 6a,
and a pattern consisting of 4.8 $\mu$m-diameter gold discs fabricated on
glass, performed multiple trapping of 4.88 $\mu$m polystyrene colloids [8].
Note that the laser beam was unfocussed, with a waist of about 100 $\mu$m and
the intensity was more than 10 times lower than that required for conventional
optical tweezers with similar characteristics. Theoretical work had been done
earlier in order to study the forces arising in such a configuration [28].
Also, in earlier experimental work, the authors used a photonic force
microscope to measure the plasmon radiation forces acting on polystyrene beads
at the localised surface plasmon resonance. They reported forces 40 times
stronger than those obtained in the absence of SP excitation [29].

Figure 6: a The Kretschmann configuration, necessary to excite the surface
plasmon polaritons (SPP). Light is coupled into SPPs under total internal
reflection in order to compensate for the light momentum mismatch. The angle
of incidence, $\theta$, controls both the scattering and the gradient force,
allowing for tuning of the total trapping force. b Excitation of the localised
surface plasmons (LSP) can happen under direct illumination of the metallic
nanostuctures. The geometrical characteristics of the nanostructures on the
metallic thin film define the resonance frequency and the forces that arise
from the plasmonic field. Inset shows the exponential decay of the evanescent
plasmonic field from the surface of the material.

Now, it is important to mention the two distinct types of surface plasmons.
Surface plasmon polaritons (SPP) are propagating electromagnetic surface waves
that appear at the metal/dielectric interface due to the motion of the metal’s
free electrons driven by the incident electromagnetic field. They are
evanescent modes and, thus, they produce localised fields with a high
intensity decaying exponentially away from the metal surface. Due to the very
large intensity gradient, they exert strong gradient forces on the trapped
particles (see Equation (9)), thereby producing stable traps. However, because
SPPs are pure evanescent modes, direct coupling to propagating light is not
possible and, in order to excite them, a different geometrical approach is
required. The most common experimental method is the Kretschmann
configuration, which is shown in Figure 6a. Light is coupled into SPPs under
total internal reflection in order to compensate for the light momentum
mismatch. The crucial parameter in this configuration is the angle of
incidence, $\theta$, which controls both the scattering and the gradient
force, allowing for tuning of the total trapping force.

In contrast, localised surface plasmons (LSP) are related to the bound
electrons that are present near to nanoapertures or nanoparticles much smaller
than the wavelength of the electromagnetic field. Bound electrons are
susceptible to a damping oscillation due to the nucleus attraction and, as a
result, they have a characteristic resonance frequency, unlike SPPs that can
be excited over a wide range of frequencies. The benefits of LSPs are that
they can directly couple to propagating light and their resonance frequency
can be tuned by changing the size and the shape of the
nanoaperture/nanoparticle (Figure 6 b). In a theoretical work done on LSPs,
the dramatic dependence of the strength of the excited evanescent field on the
frequency of the incident electromagnetic field was presented [28]. Detailed
mathematical analysis on the excitation of surface plasmons and the forces
arising can be found in Ref. [20] and [30].

To date, many different configurations have been reported using SPs for
efficient trapping of subwavelength particles, such as plasmonic nanodots
[31], nano-antennas [32], nanocavities [33] and nano-apertures of different
shapes and sizes [34, 35], reporting very low incident power. Note that, in
these cases, rigorous calculations need to be done beforehand, in order to
determine crucial parameters such as resonance wavelength and polarisation in
compliance with the plasmonic field excitation requirements. Additionally, the
fabrication of those nanostructures can also be a challenging task. The most
popular techniques to fabricate structures of these sizes is focussed ion beam
(FIB) milling and electron beam lithography (EBL), with a resolution of about
10 nm.

To conclude, we emphasize that the principle of trapping using surface
plasmons is the same as in the case of conventional optical tweezers, in the
sense that again we have to find the appropriate balance between the gradient
and the scattering force in order to achieve a stable trap. The improvement
comes from the fact that, in plasmonic optical tweezers, the excited plasmonic
field offers stronger gradient forces and a better control over them, as
explained above. However, due to the conductive nature of metals, the
excitation of SPs is connected with heat induction and dissipation to the
surrounding environment and these can increase the destabilisation forces.

###  3.1 Self-Induced Back Action Effect

The diffraction and transmission of light through a single, subwavelength-
sized circular hole on a metallic surface was first theoretically studied in
1944 by Bethe [36]. Assuming a perfectly conducting and infinitely thin
material, Bethe calculated that the transmitted light would scale as
$T\varpropto\left(d/\lambda\right)^{4}$, where $d$ is the radius of the hole
and $\lambda$ the wavelength of the incident light, as illustrated in Figure
7. In 1998, the remarkable phenomenon of the extraordinary transmission of
light was experimentally observed by Ebbesen et al. [37], when they studied
the effects of the geometrical characteristics of an array of tiny holes,
drilled on different metallic films, on the transmission of light. In that
work, UV-Vis-NIR spectrophotometry of the array was performed and it revealed
the existence of maxima in the transmission intensity, see Figure 8a, with
much higher values than predicted from Bethe’s theory. These maxima could not
be explained simply by diffraction theory and they were associated with the
resonant frequencies of the excited surface plasmons of the metal. A detailed
theoretical explanation of this observation was provided a few years later
[38, 39], where the calculated transmission maxima were in very good agreement
with the experiment (Figure 8b).

Figure 7: a Diffraction and transmission of visible light through a
subwavelength circular hole of radius, $d$, on a perfectly conducting and
infinitely thin metallic film. b According to Bethe’s calculation the
transmission of incident light $\lambda$ scales as $(d/\lambda)^{4}$. Figure
inspired by [40]. Figure 8: Experimental (left) and theoretical (right) plots
of the normalised transmittance as a function of the incident light
wavelength, for a square array of holes on a Ag thin film with thickness
$h=$320 nm. The diameter of the holes was 280 nm and the lattice constant 750
nm. Inset in the left plot shows an SEM image of the nano-hole lattice. Image
reproduced with permission from [38].

In addition, in his theoretical work, Abajo investigated the case of filling
the nano-hole with a dielectric material of high refractive index (Si). He
found out that the transmission cross-section at the resonance frequency was
almost three times higher than without the filling, leading to increased
transmission [39]. The increase of the refractive index at the nano-hole
causes the wavelength of the light to decrease:

$n_{fill}=\frac{c}{c_{fill}}>1\Leftrightarrow
n_{fill}=\frac{\lambda}{\lambda_{fill}}>1\Leftrightarrow\lambda_{fill}<\lambda.$
(16)

A consequence of this effect is that, according to Bethe’s theory, the
transmission of light through the subwavelength aperture increases, a
phenomenon also known as dielectric loading. Figure 9 shows how the wavelength
shift causes a significant increase in the transmission of light.

Figure 9: a The presence of a dielectric material of higher refractive index
than the surroundings causes an increase in the transmission of light through
the subwavelength aperture. b Shift of the transmission line due to existence
of the dielectric material, leading to higher transmission. Figure inspired by
[40].

As mentioned in the previous section, at around the same time, the use of
plasmonics for enhanced optical trapping had started to attract attention and,
despite how promising they might seem for trapping subwavelength particles, it
became apparent that their use is limited to particles with a minimum diameter
around 100 nm due to photothermal effects [41].

It was the combination of these two different studies on plasmonics that
brought the realisation that the resonance frequency of the excited plasmonic
field is very sensitive to changes of the local refractive index. Thus, by
proper engineering of the plasmonic structure, the trapped particle itself
could actively contribute to its own trapping potential in a dynamical way
[42]. This plasmonic structure - particle interaction promised high tunability
of the trapping potential, which was no longer a static. This gave rise to the
self-induced back action (SIBA) effect and the first experimental trapping
utilising this effect, where polystyrene spheres of 100 and 50 nm size were
successfully trapped with incident powers as low as 0.7 and 1.9 mW,
respectively [43], pushing further the boundaries of plasmonic nanotweezers.

A comprehensive mathematical analysis of the SIBA effect has been done by
Neumeier et al. [44], for a small dielectric particle trapped in a plasmonic
nanocavity. They demonstrated the additional restoring forces that act on the
particle as it tries to escape from the trap. Below, we present the basic
principle of the SIBA effect following the analysis done in [41].

The gradient force experienced by a nanosphere with radius $r\ll\lambda$ of
the incident light, is given by Equation (9), as mentioned previously. If we
assume small displacements of the particle from the centre of the trap
($|x|\ll\lambda$), then its equation of motion inside the trap is given by

$\gamma\dot{x}(t)+\kappa_{tot}x(t)=\xi(t),$ (17)

where $\gamma$ is the viscous damping [45], assuming that the particle exists
in a liquid environment, $\kappa_{tot}$ is the stiffness of the trap,
indicating how strongly the particle is confined in the trap, and $\xi$
represents thermal fluctuations [46]. Due to the coupling between the cavity
and the particle, the latter causes the plasmon resonance frequency of the
cavity to shift by $\delta\omega_{0}(x_{p})$, where $x_{p}$ indicates the
frequency dependence on the particle’s position. Then, for a cavity with mode
volume, $V_{m}$, and intensity profile, $f(x_{p})$, normalised to 1 for
maximum intensity, the perturbation theory for shifts much smaller than the
cavity eigenfrequency, $\omega_{c}$, yields

$\delta\omega_{0}(x_{p})=\omega_{c}\frac{\alpha_{d}}{2V_{m}\epsilon_{0}}f(x_{p}),$
(18)

with $\alpha_{d}$ being the effective polarizability, from Equation (6). Note
that the magnitude of the shift strongly depends on the relative size of the
particle and the cavity and, as expected from Equation (18), a decrease in the
particle’s size (decrease in $\alpha_{d}$, see Eqs. (5), (6)), decreases the
magnitude of the shift [47].

Now, for incident laser frequency, $\omega$, and
$\Delta\equiv\omega-\omega_{c}$ being the cavity detuning, the intracavity
intensity, $I(\omega)$, is given, on Taylor expansion, as

$I(\omega)=I_{0}\frac{(\Gamma/2)^{2}}{(\Delta-\delta\omega_{0})^{2}+(\Gamma/2)^{2})}\approx
I_{opt}-\frac{2\delta\omega_{0}(x_{p})\Delta}{\Delta^{2}+(\Gamma/2)^{2}}I_{opt}+...$
(19)

where $\Gamma$ is the cavity linewidth and
$I_{opt}=I_{0}\frac{(\Gamma/2)^{2}}{\Delta^{2}+(\Gamma/2)^{2}}$ is the empty
cavity profile.

As can been seen from Equation (19), the intensity of light inside the cavity,
to a first order approximation, is given by the term related to the intensity
of the empty cavity, plus the one related to the frequency shift caused by the
presence of the particle. This second term is the one that causes the SIBA
effect and modifies the optical potential. Following Equation (19), we can
also write the total trapping stiffness, $\kappa_{tot}$, as

$\kappa_{tot}=\kappa_{opt}+\kappa_{SIBA},$ (20)

where $\kappa_{opt}$ is constant and depends on the cavity resonance profile
and $\kappa_{SIBA}$ is a function of the particle’s displacement. According to
Neumeir et al., in order to optimise $\kappa_{SIBA}$, the cavity has to be
constructed such that the back-action parameter,
$\upsilon=\delta\omega_{0}(x_{p})/\Gamma$, is maximised [44]. This means that,
while the particle is trapped in the centre of the trap, the resonance shift
is such that the photon flux from the cavity is less than the maximum
possible. As a consequence, when the particle moves away from the centre of
the trap, the resonance shift causes the photon flux to increase and, thus,
the intensity of the transmitted light increases. From Equation (9) an
increase to the intensity leads to an increase to the gradient force, which
restores the particle back to the centre of the trap. Then, the photon flux
and the intensity decrease and, again, the particle tends to move away from
the trap centre. This kind of feedback is referred to as ”optomechanical
coupling” because there is a continuous response between light and mechanical
motion. The field of optomechanics in plasmonics is rather unexplored and, to
our knowledge, there is only one work that reports an optomechanical coupling
constant [41]. This optomechanical coupling not only relaxes the requirements
for high power trapping, but also prevents the sample from overheating since
most of the time the particle is trapped using a low intensity. It remains
open to exploration to find ways to increase the optomechanical coupling
constant and to achieve even higher particle confinement and motion
transduction.

Figure 10:  a. Three different cases of the resonance profile of a plasmonic
cavity (black line). In case (i), (ii), and (iii) the cavity is designed to be
blue-detuned, slightly blue-detuned and red-detuned, respectively, from the
excitation laser (vertical dashed line). The orange line represents the
frequency shift of the resonance of the cavity due to the existence of the
particle near by it. In case (ii), the SIBA effect has a positive influence on
the trap and, whenever the particle tries to escape, the photon flux
(intensity) increases, which causes, increase to the gradient force. b.
Experimental measurements of the total trapping stiffness as a function of the
incident laser intensity. b(i) corresponds to case a(i), where as the
intensity decreases both the total and the empty cavity trapping stiffness
decrease, making the trap inefficient. b(ii) corresponds to case a(ii), and
clearly shows that, as the laser intensity decreases, the empty cavity
trapping stiffness decreases, but the total stiffness increases due to the
positive contribution from the SIBA effect, thus achieving a stable, self-
adjustable trap. Figures taken from [41], under Creative Commons license: CC
BY-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/.

In Figure 10a, the vertical dashed line represents the excitation laser
wavelength, the black lineshape is the empty cavity mode resonance, and the
orange is the shifted one due to particle trapping. In the first case (Figure
10a(i), blue-shifted) the cavity resonance is set to be blue-detuned from the
excitation laser, such that when the particle is trapped, the resonance red-
shift increases the photon flux to the maximum value and the gradient force
reaches a maximum. However, when the particle moves away from the centre of
the trap, the lineshape blue shifts towards the empty cavity resonance and the
intensity decreases. In order to increase the gradient force, the power of the
laser has to, externally, be increased, thus, it is not the most efficient
scenario for trapping. In the case where the empty mode resonance is slightly
blue-detuned (Figure 10a(ii), resonance), the red-shifting due to a trapping
event creates a symmetrical lineshape around the laser wavelength. As the
particle tries to escape from the trap, the resonance moves towards the empty
mode value and causes an increase in the photon flux and the light intensity
of light, thus increasing the gradient force. Finally, in the case where the
empty cavity resonance is designed to be red-detuned from the excitation laser
(Figure 10a(iii), red-shifted), the trapped object further red-shifts the
resonance, leading to significant reduction in the intensity and the gradient
force. In this configuration, the trapping becomes very inefficient and an
increase to the laser power is necessary to keep the particle in the trap.

The intensity required to keep the particle efficiently in the trap is less in
the second case where the SIBA effect contributes to an increase in the total
trapping stiffness. This was also experimentally observed for the first time
by Mestres et al. [41]. Figure 10b shows the experimental data that confirm
the superior trapping efficiency of a plasmonic cavity, designed to be
slightly blue-detuned from the excitation wavelength (Figure 10a(ii), b(ii)).
The remarkable effect of SIBA is now apparent and, by proper design of the
plasmonic structure, we can have a larger trapping stiffness at a lower laser
power, thereby reducing heat transfer to the specimen.

##  4 4\. Conclusions and Future Perspectives

There are many open questions and challenges to overcome in order to optimise
the plasmonic tweezers and make them able to efficiently trap particles in the
range of less than 10 nm and biological samples such as proteins, viruses and
DNA. In our opinion, one of these important questions is whether it is
possible to enhance the SIBA effect itself so that we achieve even better
optomechanical response of the system and superior trapping performance.

It is amazing to see how a simple idea can be transformed into a powerful
technique for controlling matter, finally leading to the awarding of a Nobel
Prize. But it is even more amazing to see people devoting themselves to
solving a particular problem and pushing the boundaries of science into the
unknown to make it possible. Optical tweezers-based platforms are today,
thanks to all these scientistific efforts, widely used in the fields of
physics [48], biomedicine [49, 50, 51], chemistry [52] and many more. In many
cases, it serves as a tool to manipulate matter while doing other
measurements, for example Raman spectroscopy of biological samples [53, 54,
55]. This gives us a great advantage towards the exploration of the nanoworld
and the advancement of the nanotechnologies. What is next?

#### Acknowledgements

The authors would like to thank Okinawa Institute of Science and Technology
Graduate University for funding, and D. G. Kotsifaki, P. Reece, P. Okada, F.
Pauly, and G. Tkachenko for constructive comments.

## References

  * Kepler [1619] J. Kepler.  _De cometis libelli tres_.  Typis Andreæ Apergeri, sumptibus Sebastiani Mylii bibliopolæ Augustani, 1619.  URL https://books.google.co.jp/books?id=5Z1BQAAACAAJ.
  * Maxwell [1873] J.C. Maxwell.  _A Treatise on Electricity and Magnetism_ , volume 2.  Oxford, 1873.
  * Poynting and William [1884] J. H. Poynting and S. J.XV William.  On the transfer of energy in the electromagnetic field.  _Phil. Trans. R. Soc._ , 175, 1884.
  * Maiman [1960] T. H. Maiman.  Stimulated optical radiation in ruby.  _Nature_ , 187(4736):493–494, 1960.  ISSN 1476-4687.  10.1038/187493a0.  URL https://doi.org/10.1038/187493a0.
  * Schawlow and Townes [1958] A. L. Schawlow and C. H. Townes.  Infrared and optical masers.  _Physical Review_ , 112(6):1940–1949, 1958.  ISSN 0031-899X.  10.1103/PhysRev.112.1940.
  * Ashkin [1970] A. Ashkin.  Acceleration and trapping of particles by radiation pessure.  _Phys. Rev. Lett._ , 24(4):156–159, 1970.  ISSN 0031-9007.  10.1103/PhysRevLett.24.156.
  * Novotny et al. [1997] Lukas Novotny, Randy X. Bian, and X. Sunney Xie.  Theory of nanometric optical tweezers.  _Phys. Rev. Lett._ , 79(4):645–648, 1997.  10.1103/PhysRevLett.79.645.  URL https://link.aps.org/doi/10.1103/PhysRevLett.79.645.
  * Righini et al. [2007] Maurizio Righini, Anna S. Zelenina, Christian Girard, and Romain Quidant.  Parallel and selective trapping in a patterned plasmonic landscape.  _Nat. Photonics_ , 3(7):477–480, 2007.  ISSN 1745-2473 1745-2481.  10.1038/nphys624.
  * Marago et al. [2013] O. M. Marago, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari.  Optical trapping and manipulation of nanostructures.  _Nat. Nanotechnol._ , 8(11):807–19, 2013.  ISSN 1748-3395 (Electronic) 1748-3387 (Linking).  10.1038/nnano.2013.208.  URL https://www.ncbi.nlm.nih.gov/pubmed/24202536.
  * Juan et al. [2011] Mathieu L. Juan, Maurizio Righini, and Romain Quidant.  Plasmon nano-optical tweezers.  _Nature Photon._ , 5(6):349–356, 2011.  ISSN 1749-4885 1749-4893.  10.1038/nphoton.2011.56.
  * Ashkin et al. [1986] A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and Steven Chu.  Observation of a single-beam gradient force optical trap for dielectric particles.  _Opt. Lett._ , 11(5):288–290, 1986.  10.1364/OL.11.000288.  URL http://ol.osa.org/abstract.cfm?URI=ol-11-5-288.
  * Pfeifer et al. [2007] Robert N. C. Pfeifer, Timo A. Nieminen, Norman R. Heckenberg, and Halina Rubinsztein-Dunlop.  Colloquium: Momentum of an electromagnetic wave in dielectric media.  _Rev. Mod. Phys._ , 79:1197–1216, Oct 2007.  10.1103/RevModPhys.79.1197.  URL https://link.aps.org/doi/10.1103/RevModPhys.79.1197.
  * Borghese et al. [2007] Ferdinando Borghese, Paolo Denti, and Rosalba Saija.  _Scattering from model nonspherical particles_.  Springer-Verlag Berlin Heidelberg, 2007.  10.1007/978-3-540-37414-5.
  * Draine and Flatau [1994] Bruce T. Draine and Piotr J. Flatau.  Discrete-dipole approximation for scattering calculations.  _Journal of the Optical Society of America A_ , 11(4):1491–1499, 1994.  10.1364/JOSAA.11.001491.  URL http://josaa.osa.org/abstract.cfm?URI=josaa-11-4-1491.
  * Gordon [1973] James P. Gordon.  Radiation forces and momenta in dielectric media.  _Phys. Rev. A_ , 8(1):14–21, 1973.  ISSN 0556-2791.  10.1103/PhysRevA.8.14.
  * Ashkin and Dziedzic [1973] A. Ashkin and J. M. Dziedzic.  Radiation pressure on a free liquid surface.  _Phys. Rev. Lett._ , 30(4):139–142, 1973.  ISSN 0031-9007.  10.1103/PhysRevLett.30.139.
  * Bradac [2018] Carlo Bradac.  Nanoscale optical trapping: A review.  _Adv. Opt. Mater._ , 6(12), 2018.  ISSN 21951071.  10.1002/adom.201800005.
  * Roosen [1979] Gérald Roosen.  La lévitation optique de sphères.  _Can. J. Phys._ , 57(9):1260–1279, 1979.  ISSN 0008-4204.  10.1139/p79-175.  URL https://doi.org/10.1139/p79-175.
  * Ashkin [1992] A. Ashkin.  Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime.  _Biophys. J._ , 61(2):569–582, 1992.  ISSN 0006-3495.  https://doi.org/10.1016/S0006-3495(92)81860-X.  URL http://www.sciencedirect.com/science/article/pii/S000634959281860X.
  * Jones et al. [2015] Philip Jones, Onofrio M. Marago, and Giovanni Volpe.  _Optical Tweezers: Principles & Applications_.  Cambridge University Press, Cambridge, 2015.  ISBN 9781107051164.
  * M. Purcell and R. Pennypacker [1973] Edward M. Purcell and Carlton R. Pennypacker.  Scattering and absorption of light by nonspherical dielectric grains.  _Astrophys. J._ , 186:705–714, 1973.  10.1086/152538.
  * Spesyvtseva and Dholakia [2016] Susan E. Skelton Spesyvtseva and Kishan Dholakia.  Trapping in a material world.  _ACS Photonics_ , 3(5):719–736, 2016.  10.1021/acsphotonics.6b00023.  URL https://doi.org/10.1021/acsphotonics.6b00023.
  * Vigoureux and Courjon [1992] J. M. Vigoureux and D. Courjon.  Detection of nonradiative fields in light of the heisenberg uncertainty principle and the rayleigh criterion.  _Appl. Opt._ , 31(16):3170–3177, Jun 1992.  10.1364/AO.31.003170.  URL http://ao.osa.org/abstract.cfm?URI=ao-31-16-3170.
  * Novotny and Hecht [2012] Lukas Novotny and Bert Hecht.  _Principles of Nano-Optics_.  Cambridge University Press, Cambridge, 2012.  ISBN 9781107005464 1107005469.
  * Kotsifaki and Nic Chormaic [2019] D Kotsifaki and S. Nic Chormaic.  Plasmonic optical tweezers based on nanostructures: fundamentals, advances and prospect.  _Nanophotonics_ , 8(7):1227–1245, 2019.  10.1515/nanoph-2019-0151.
  * Kawata and Sugiura [1992] Satoshi Kawata and Tadao Sugiura.  Movement of micrometer-sized particles in the evanescent field of a laser beam.  _Opt. Lett._ , 17(11):772–774, Jun 1992.  10.1364/OL.17.000772.  URL http://ol.osa.org/abstract.cfm?URI=ol-17-11-772.
  * Okamoto and Kawata [1999] K. Okamoto and S. Kawata.  Radiation force exerted on subwavelength particles near a nanoaperture.  _Phys. Rev. Lett._ , 83:4534–4537, Nov 1999.  10.1103/PhysRevLett.83.4534.  URL https://link.aps.org/doi/10.1103/PhysRevLett.83.4534.
  * Quidant et al. [2005] Romain Quidant, Dmitri Petrov, and Gonçal Badenes.  Radiation forces on a rayleigh dielectric sphere in a patterned optical near field.  _Opt. Lett._ , 30(9):1009–1011, May 2005.  10.1364/OL.30.001009.  URL http://ol.osa.org/abstract.cfm?URI=ol-30-9-1009.
  * Volpe et al. [2006] G. Volpe, R. Quidant, G. Badenes, and D. Petrov.  Surface plasmon radiation forces.  _Phys. Rev. Lett._ , 96(23):238101, 2006.  ISSN 0031-9007 (Print) 0031-9007 (Linking).  10.1103/PhysRevLett.96.238101.  URL https://www.ncbi.nlm.nih.gov/pubmed/16803408.
  * A. Maier [2007] Stefan A. Maier.  _Plasmonics: Fundamentals and Applications_.  Springer, New York, 01 2007.  ISBN 978-0-387-33150-8.  10.1007/0-387-37825-1.
  * Grigorenko et al. [2008] A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang.  Nanometric optical tweezers based on nanostructured substrates.  _Nat. Photonics_ , 2:365, Nov 2008.  10.1038/nphoton.2008.78.  URL https://www.nature.com/articles/nphoton.2008.78#supplementary-information.
  * Righini et al. [2009] M. Righini, P. Ghenuche, S. Cherukulappurath, V. Myroshnychenko, F. J. García de Abajo, and R. Quidant.  Nano-optical trapping of rayleigh particles and escherichia coli bacteria with resonant optical antennas.  _Nano Lett._ , 9(10):3387–3391, 2009.  10.1021/nl803677x.  URL https://doi.org/10.1021/nl803677x.  PMID: 19159322.
  * Chen et al. [2012] C. Chen, M. L. Juan, Y. Li, G. Maes, G. Borghs, P. Van Dorpe, and R. Quidant.  Enhanced optical trapping and arrangement of nano-objects in a plasmonic nanocavity.  _Nano Lett._ , 12(1):125–32, 2012.  ISSN 1530-6992 (Electronic) 1530-6984 (Linking).  10.1021/nl2031458.  URL https://www.ncbi.nlm.nih.gov/pubmed/22136462.
  * Pang and Gordon [2012] Yuanjie Pang and Reuven Gordon.  Optical trapping of a single protein.  _Nano Lett._ , 12(1):402–406, 2012.  10.1021/nl203719v.  URL https://doi.org/10.1021/nl203719v.  PMID: 22171921.
  * Han et al. [2018] Xue Han, Viet Giang Truong, Prince Sunil Thomas, and Síle Nic Chormaic.  Sequential trapping of single nanoparticles using a gold plasmonic nanohole array.  _Photon. Res._ , 6(10):981–986, Oct 2018.  10.1364/PRJ.6.000981.  URL http://www.osapublishing.org/prj/abstract.cfm?URI=prj-6-10-981.
  * Bethe [1944] H. A. Bethe.  Theory of diffraction by small holes.  _Phys. Rev._ , 66(7-8):163–182, 1944.  ISSN 0031-899X.  10.1103/PhysRev.66.163.
  * Ebbesen et al. [1998] T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff.  Extraordinary optical transmission through sub-wavelength hole arrays.  _Nature_ , 391:667, 1998.  10.1038/35570.  URL https://doi.org/10.1038/35570.
  * Martín-Moreno et al. [2001] L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen.  Theory of extraordinary optical transmission through subwavelength hole arrays.  _Phys. Rev. Lett._ , 86:1114–1117, Feb 2001.  10.1103/PhysRevLett.86.1114.  URL https://link.aps.org/doi/10.1103/PhysRevLett.86.1114.
  * de Abajo [2002] F. J. García de Abajo.  Light transmission through a single cylindrical hole in a metallic film.  _Opt. Express_ , 10(25):1475–1484, Dec 2002.  10.1364/OE.10.001475.  URL http://www.opticsexpress.org/abstract.cfm?URI=oe-10-25-1475.
  * Genet and Ebbesen [2007] C. Genet and T. W. Ebbesen.  Light in tiny holes.  _Nature_ , 445(7123):39–46, 2007.  ISSN 1476-4687 (Electronic) 0028-0836 (Linking).  10.1038/nature05350.  URL https://www.ncbi.nlm.nih.gov/pubmed/17203054.
  * Mestres et al. [2016] P. Mestres, J. Berthelot, S. S. Acimovic, and R. Quidant.  Unraveling the optomechanical nature of plasmonic trapping.  _Light. Sci. Appl._ , 5(7):e16092, 2016.  ISSN 2047-7538 (Electronic) 2047-7538 (Linking).  10.1038/lsa.2016.92.  URL https://www.ncbi.nlm.nih.gov/pubmed/30167173.
  * Sainidou and García de Abajo [2008] R. Sainidou and F. J. García de Abajo.  Optically tunable surfaces with trapped particles in microcavities.  _Phys. Rev. Lett._ , 101:136802, Sep 2008.  10.1103/PhysRevLett.101.136802.  URL https://link.aps.org/doi/10.1103/PhysRevLett.101.136802.
  * Juan et al. [2009] Mathieu L. Juan, Reuven Gordon, Yuanjie Pang, Fatima Eftekhari, and Romain Quidant.  Self-induced back-action optical trapping of dielectric nanoparticles.  _Nat. Phys._ , 5(12):915–919, 2009.  ISSN 1745-2473 1745-2481.  10.1038/nphys1422.
  * Neumeier et al. [2015] Lukas Neumeier, Romain Quidant, and Darrick E. Chang.  Self-induced back-action optical trapping in nanophotonic systems.  _New J. Phys._ , 17(12):123008, 2015.  ISSN 1367-2630.  10.1088/1367-2630/17/12/123008.
  * Svoboda and Block [1994] Karel Svoboda and Steven M. Block.  Biological applications of optical forces.  _Biophys. Biomol. Struct._ , 23:247–285, 1994.
  * Bui et al. [2017] Ann A. M. Bui, Alexander B. Stilgoe, Isaac C. D. Lenton, Lachlan J. Gibson, Anatolii V. Kashchuk, Shu Zhang, Halina Rubinsztein-Dunlop, and Timo A. Nieminen.  Theory and practice of simulation of optical tweezers.  _Journal of Quantitative Spectroscopy and Radiative Transfer_ , 195:66–75, 2017.  ISSN 00224073.  10.1016/j.jqsrt.2016.12.026.
  * Descharmes et al. [2013] N. Descharmes, U. P. Dharanipathy, Z. Diao, M. Tonin, and R. Houdre.  Observation of backaction and self-induced trapping in a planar hollow photonic crystal cavity.  _Phys. Rev. Lett._ , 110(12):123601, 2013.  ISSN 1079-7114 (Electronic) 0031-9007 (Linking).  10.1103/PhysRevLett.110.123601.  URL https://www.ncbi.nlm.nih.gov/pubmed/25166804.
  * Bowman and Padgett [2013] R. W. Bowman and M. J. Padgett.  Optical trapping and binding.  _Rep Prog Phys_ , 76(2):026401, 2013.  ISSN 1361-6633 (Electronic) 0034-4885 (Linking).  10.1088/0034-4885/76/2/026401.  URL https://www.ncbi.nlm.nih.gov/pubmed/23302540.
  * Stevenson et al. [2010] D. J. Stevenson, F. Gunn-Moore, and K. Dholakia.  Light forces the pace: optical manipulation for biophotonics.  _J Biomed Opt_ , 15(4):041503, 2010.  ISSN 1560-2281 (Electronic) 1083-3668 (Linking).  10.1117/1.3475958.  URL https://www.ncbi.nlm.nih.gov/pubmed/20799781.
  * Capitanio and Pavone [2013] M. Capitanio and F. S. Pavone.  Interrogating biology with force: single molecule high-resolution measurements with optical tweezers.  _Biophys J_ , 105(6):1293–303, 2013.  ISSN 1542-0086 (Electronic) 0006-3495 (Linking).  10.1016/j.bpj.2013.08.007.  URL https://www.ncbi.nlm.nih.gov/pubmed/24047980.
  * Nussenzveig [2018] H. M. Nussenzveig.  Cell membrane biophysics with optical tweezers.  _Eur Biophys J_ , 47(5):499–514, 2018.  ISSN 1432-1017 (Electronic) 0175-7571 (Linking).  10.1007/s00249-017-1268-9.  URL https://www.ncbi.nlm.nih.gov/pubmed/29164289.
  * Sugiyama et al. [2012] Teruki Sugiyama, Ken-Ichi Yuyama, and Hiroshi Masuhara.  Laser trapping chemistry : From polymer assembly to amino acid crystallization.  _Acc. of Chem. Res._ , 45(11):1946–1954, 2012\.
  * Snook et al. [2009] R. D. Snook, T. J. Harvey, E. Correia Faria, and P. Gardner.  Raman tweezers and their application to the study of singly trapped eukaryotic cells.  _Integr Biol (Camb)_ , 1(1):43–52, 2009.  ISSN 1757-9708 (Electronic) 1757-9694 (Linking).  10.1039/b815253e.  URL https://www.ncbi.nlm.nih.gov/pubmed/20023790.
  * Chan [2013] J. W. Chan.  Recent advances in laser tweezers raman spectroscopy (ltrs) for label-free analysis of single cells.  _J Biophotonics_ , 6(1):36–48, 2013.  ISSN 1864-0648 (Electronic) 1864-063X (Linking).  10.1002/jbio.201200143.  URL https://www.ncbi.nlm.nih.gov/pubmed/23175434.
  * Gillibert et al. [2019] Raymond Gillibert, Gireeshkumar Balakrishnan, Quentin Deshoules, Morgan Tardivel, Alessandro Magazzù, Maria Grazia Donato, Onofrio M. Maragò, Marc Lamy de La Chapelle, Florent Colas, Fabienne Lagarde, and Pietro G. Gucciardi.  Raman tweezers for small microplastics and nanoplastics identification in seawater.  _Environmental Science & Technology_, 53(15):9003–9013, 2019.  10.1021/acs.est.9b03105.


#  On the Poincaré functional equation

Sergei Kazenas<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, a formula for the solution of the Poincaré functional equation
in algebra of formal power series and its application to continuous iteration
are presented.

The Poincaré functional equation and the associated Schröder’s equation
naturally arise in the analytic iteration theory and have been studied by a
great many authors (see, for instance, [1]). In the following, Poincaré’s
equation is treated algebraically; and the formal solution representation is
derived using finite symbolic manipulations with homogeneous quantum
difference operators.

Here the formal power series treatment is near to that used by Niven [2].

Let $\mathcal{F}$ be an algebra of formal power series over field $\mathbb{C}$
of complex numbers. Any $\mathchorus{f}\in\mathcal{F}$ is determined by the
vector $(\mathchorus{f}_{.j})_{j\geqslant 0}$ of its coefficients, which
belong to $\mathbb{C}$. Recall that for any
$\\{\mathchorus{f},\mathchorus{g}\\}\subset\mathcal{F}$, two basic operations,
addition and multiplication, are defined by
$(\mathchorus{f}+\mathchorus{g})_{.j}=\mathchorus{f}_{.j}+\mathchorus{g}_{.j}$
and $(\mathchorus{f}\mathchorus{g})_{.j}=\sum_{0\leqslant l\leqslant
j}\mathchorus{f}_{.l}\mathchorus{g}_{.j-l}$. If $\mathchorus{f}$ is not a
formal polynomial, then the composition is generally defined by
$(\mathchorus{f}\circ\mathchorus{g})_{.j}=\sum_{0\leqslant l\leqslant
j}\mathchorus{f}_{.l}(\mathchorus{g}^{l})_{.j}$ only when
$\mathchorus{g}_{.0}=0$.

In addition, define a new external binary operation, which can be regarded as
an extension of the composition. Let $\mathcal{A}$ be an algebra over
$\mathbb{C}$ such that there exists isomorphism
$\varphi:\mathcal{A}\rightarrow\mathcal{F}^{\prime}$, where
$\mathcal{F}^{\prime}$ is $\mathcal{F}$ or subalgebra of polynomials. For all
$(\mathchorus{f},a)\in\mathcal{F}^{\prime}\times\mathcal{A}$ such that
$\mathchorus{f}$ is a polynomial or $(\varphi(a))_{.0}=0$, define the
substitution operation by
$\mathchorus{f}[a]:=\varphi^{\circ\scalebox{0.45}[1.0]{$-$}1}(\mathchorus{f}\circ\varphi(a))$.
In all three cases given here, $\mathcal{A}$ naturally is context specific.
The identity $\mathchorus{f}[c]\equiv\mathchorus{f}\circ(c)$, where $0\neq
c\in\mathbb{C}$, implies that $\mathchorus{f}$ is polynomial. The identity
$\mathchorus{f}[\textrm{{x}}\mathchorus{g}]\equiv\mathchorus{f}\circ(\textrm{{x}}\mathchorus{g})$,
where $\textrm{{x}}\in\mathcal{F}$ is such that
$\textrm{{x}}_{.j}=\delta_{1,j}$, implies that
$\mathcal{A}=\mathcal{F}^{\prime}=\mathcal{F}$,
$\varphi=\text{id}_{\mathcal{F}}$. The definition for the polynomial operator
$\mathchorus{p}[\widehat{q\textrm{{x}}}]$, where $\mathchorus{p}$ is a
polynomial, $q\in\mathbb{C}$ and
$\widehat{q\textrm{{x}}}(\,):=(\,)\circ(q\textrm{{x}})$, naturally implies
that $\mathcal{F}^{\prime}$ is the subalgebra of polynomials and
$\varphi(\widehat{q\textrm{{x}}})=\textrm{{x}}$.

Recall that $\mathchorus{f}$ is invertible if and only if
$\mathchorus{f}_{.0}=0$ and $\mathchorus{f}_{.1}\neq 0$. Let $\mathchorus{f}$
be invertible and $q\neq 0$ not be a root of unity. From $\mathchorus{f}$
satisfying Poincaré’s equation

$None$
$\mathchorus{f}\circ(q\mathchorus{x})=\mathchorus{p}\circ\mathchorus{f}\,\text{,}$

immediately follows that $\mathchorus{f}^{\circ\scalebox{0.45}[1.0]{$-$}1}$
satisfies Schröder’s equation
$q\mathchorus{f}^{\circ\scalebox{0.45}[1.0]{$-$}1}=\mathchorus{f}^{\circ\scalebox{0.45}[1.0]{$-$}1}\circ\mathchorus{p}$.

It also follows that an invertible solution exists if and only if
$\mathchorus{p}_{.0}=0$ and $\mathchorus{p}_{.1}=q$. Therefore (1) can be
rewritten in the equivalent form:
$\mathchorus{f}\circ(q^{\scalebox{0.45}[1.0]{$-$}1}\textrm{{x}})=\mathchorus{p}^{\circ\scalebox{0.45}[1.0]{$-$}1}\circ\mathchorus{f}$.
Without loss of generality, the nonzero coefficient $\mathchorus{f}_{.1}$ can
be fixed by $\mathchorus{f}_{.1}=1$ and the other coefficients
$\mathchorus{f}_{.2},\mathchorus{f}_{.3},...$ can be calculated recursively
from

$None$
$\mathchorus{f}_{.j}=\frac{1}{q^{j}-1}\sum_{l=1}^{j-1}\mathchorus{p}_{.l}(\mathchorus{f}^{l})_{.j}\,\text{.}$

However, there are approaches to represent the coefficients of
$\mathchorus{f}$ nonrecursively. One such approach is described below.

For an arbitrary nonzero polynomial $\mathchorus{g}$ define a homogeneous
quantum difference operator in $\mathcal{F}$ by

$D_{\mathchorus{g};q}:=(\textrm{{x}}^{\scalebox{0.45}[1.0]{$-$}n})\mathchorus{g}[\widehat{q\textrm{{x}}}]\,\text{,}$

where operator $(\textrm{{x}}^{\scalebox{0.45}[1.0]{$-$}n})$ is naturally
defined in $\textrm{{x}}^{n}{\mathcal{F}}$, and $n$ is the smallest
nonnegative integer such that $\mathchorus{g}[q^{n}]\neq 0$; let call it the
order of operator $D_{\mathchorus{g};q}$ or $q$-difference order of
$\mathchorus{g}$. For example, $D_{\frac{1-\textrm{{x}}}{1-q};q}$ is the first
order operator, which can be viewed as Jackson derivative operator in
${\mathcal{F}}$.

From
$D_{\mathchorus{g};q}\textrm{{x}}^{m}=(\textrm{{x}}^{\scalebox{0.45}[1.0]{$-$}n})\mathchorus{g}[\widehat{q\textrm{{x}}}]\textrm{{x}}^{m}=(\textrm{{x}}^{\scalebox{0.45}[1.0]{$-$}n})\mathchorus{g}[q^{m}]\textrm{{x}}^{m}=0$
(0 $\leqslant m<n$) follows Macloren-like expansion:

$None$
$\mathchorus{f}_{.j}=\frac{(D_{\mathchorus{g}_{j};q}\mathchorus{f})[0]}{\mathchorus{g}_{j}[q^{j}]}\,\text{,}$

where $D_{\mathchorus{g}_{j};q}$ is of order $j$.

Now the preparation is complete to derive a nonrecursive companion to (2).

######  Theorem 1.

Let $\mathchorus{f}$ satisfy Poincaré’s equation (1). $\mathchorus{f}_{.0}=0$,
$\mathchorus{f}_{.1}=1$; $\mathchorus{g}_{j}$ stands for an arbitrary nonzero
polynomial of $j$-th $q$-difference order, then the coefficients of
$\mathchorus{f}$ can be calculated by

$None$
$\mathchorus{f}_{.j}=\frac{1}{\mathchorus{g}_{j}[q^{j}]}\sum_{i\geqslant
0}\mathchorus{g}_{j.i}(\mathchorus{p}^{\circ i})_{.j}\,\text{.}$

###### Proof.

Calculate $q$-differences:

$D_{\mathchorus{g}_{j};q}\mathchorus{f}=(\textrm{{x}}^{\scalebox{0.45}[1.0]{$-$}j})\mathchorus{g}_{j}[\widehat{q\textrm{{x}}}]\mathchorus{f}=\textrm{{x}}^{\scalebox{0.45}[1.0]{$-$}j}\sum_{i\geqslant
0}\mathchorus{g}_{j.i}\mathchorus{f}[q^{i}\textrm{{x}}]=\textrm{{x}}^{\scalebox{0.45}[1.0]{$-$}j}\sum_{i\geqslant
0}\mathchorus{g}_{j.i}\mathchorus{p}^{\circ j}[\mathchorus{f}]\,\text{.}$

Note that $\frac{\mathchorus{f}-\textrm{{x}}}{\textrm{{x}}}[0]=0$ and
continue:

$\displaystyle(D_{\mathchorus{g}_{j};q}\mathchorus{f})[0]$
$\displaystyle=\left(\textrm{{x}}^{\scalebox{0.45}[1.0]{$-$}j}\sum_{i\geqslant
0}\mathchorus{g}_{j.i}\mathchorus{p}^{\circ
j}\left[\textrm{{x}}\left(1+\frac{\mathchorus{f}-\textrm{{x}}}{\textrm{{x}}}\right)\right]\right)[0]$
$\displaystyle=\left(\textrm{{x}}^{\scalebox{0.45}[1.0]{$-$}j}\sum_{i\geqslant
0}\mathchorus{g}_{j.i}\mathchorus{p}^{\circ
j}[\textrm{{x}}]\right)[0]=\sum_{i\geqslant
0}\mathchorus{g}_{j.i}(\mathchorus{p}^{\circ i})_{.j}\,\text{.}$

Taking in account (3) completes the proof. ∎

As a corollary, one can derive a formula for continuous iteration.

Let $\\{q\\}_{n,j}:=\textrm{{x}}^{n}\prod_{i\leqslant
j-1}(\textrm{{x}}-q^{i})$. It follows from (4) that

$(\mathchorus{p}^{\circ
n})_{.j}=\frac{\\{q^{j}\\}_{n-j,j}}{(\\{q\\}_{n-j,j})_{.n}}\left(\sum_{l=n-j-1}^{n-1}\frac{(\\{q\\}_{n-j-1,j})_{.l}}{\\{q^{j}\\}_{n-j-1,j}}(\mathchorus{p}^{\circ
l})_{.j}-\sum_{l=n-j}^{n-1}\frac{(\\{q\\}_{n-j,j})_{.l}}{\\{q^{j}\\}_{n-j,j}}(\mathchorus{p}^{\circ
l})_{.j}\right)\,\text{,}$

for $j>n$. Assuming coefficients $\mathchorus{p}_{.j}$ are polynomials in $q$
and substituting $q=1$ leads to the following:

$(\mathchorus{p}^{\circ
n})_{.j}=(-1)^{j+1}(\mathchorus{p}^{\circ(n-j)})_{.j}-\sum_{l=1}^{j-1}\binom{j}{l}(-1)^{l}(\mathchorus{p}^{\circ(n-l)})_{.j}=\sum_{l=0}^{j-1}c_{l}n^{l}\,\text{,}$

where coefficients $c_{l}$ can be represented with help of Newton’s
interpolation.

It should be noted that the theory of formal continuous iteration was
developed in detail in [3], for instance, but since $q$ and $\mathchorus{g}$
can be set quite arbitrary, the above approach can be considered for utilizing
in future.

## References

  * [1] Derfel G., Grabner P., Vogl F., _Asymptotics of the Poincaré functions_ , Conf.: Prob. and Math. Ph., 2007
  * [2] Niven I., _Formal power series_ , Amer. Math. Month., 1989, 76(8), 871-889
  * [3] Labelle G., _Sur i’inversion et i’itération continue des séries formalles_ , Europ. J. Combinatorics, 1980, 1, 113-138
  * [4] Kac V., Cheung V., _Quantum calculus_ , Springer, 2002