# Optical chiral sorting forces and their manifestation in evanescent waves
and nanofibres

Sebastian Golat<EMAIL_ADDRESS>Department of Physics, King’s
College London, Strand, London WC2R 2LS, United Kingdom  Jack J. Kingsley-
Smith  Department of Physics, King’s College London, Strand, London WC2R 2LS,
United Kingdom  QinetiQ, Cody Technology Park, Farnborough, GU14 0LX, United
Kingdom  Iago Diez  Josep Martinez-Romeu  Alejandro Martínez  Nanophotonics
Technology Center, Universitat Politècnica de València, Camino de Vera, s/n
Building 8F, 46022, Valencia Spain  Francisco J. Rodríguez-Fortuño
<EMAIL_ADDRESS>Department of Physics, King’s College
London, Strand, London WC2R 2LS, United Kingdom

###### Abstract

Optical fields can exert forces of chiral nature on molecules and
nanoparticles, which would prove extremely valuable in the separation of
enantiomers with pharmaceutical applications, yet it is inherently complex,
and the varied frameworks used in the literature further complicate the
theoretical understanding. This paper unifies existing approaches used to
describe dipolar optical forces and introduces a new symmetry-based ‘force
basis’ consisting of twelve vector fields, each weighted by particle-specific
coefficients, for a streamlined description of force patterns. The approach is
rigorously applied to evanescent waves and dielectric nanofibres, yielding
concise analytical expressions for optical forces. Through this, we identify
optimal strategies for enantiomer separation, offering invaluable guidance for
future experiments.

##  I Introduction

The idea that light may exert a force on matter was proposed by Johannes
Kepler in the 1600s to explain the observation that a comet’s tail always
points away from the sun. Using the phenomenon of optical forces for the
levitation and control of small objects in a designed way had to wait many
centuries. It was first explored by Arthur Ashkin in the 1970s, who proposed
and developed the optical trap [1, 2, 3] (also called optical tweezers),
earning him the 2019 Nobel Prize in Physics. These optical tweezers proved
instrumental in the micromanipulation of molecules, cells, viruses, and atoms
and have revolutionised research fields with the advent of laser cooling and
quantum control of macroscopic objects.

In the optical manipulation of small particles, two main optical forces are
considered: the optical gradient, which tends to attract particles towards the
maxima in electromagnetic energy, and the optical pressure force, which tends
to push particles in the direction of light propagation. Optical tweezers
exploit the gradient force, usually stronger for small particles. By tightly
focusing a laser beam, a particle is trapped in the high-intensity focus of
the beam. The stronger the focusing, the higher the intensity gradient, and
the stronger the trapping force—increasing the stiffness of the trap.

Given the success in the manipulation of matter by optical forces, a question
has arisen in recent years: can light be used to sort chiral particles and
molecules, in other words, to separate two enantiomers (which, apart from
being a mirror-reflection of one another are otherwise identical)? This is a
challenging problem precisely because two enantiomers are identical in many
ways, and because—our body’s building blocks being chiral molecules—the
opposite enantiomers of a given pharmaceutical molecule may have drastically
different effects [4, 5, 6]. The answer to the question is positive—if light
contains some chirality itself.

The expressions for chiral optical forces have been given in many works [7, 8,
9, 10, 11, 12, 13, 14, 15, 16], and different proposals for chiral separation
of enantiomers using light have been put forth [11, 12, 13, 14, 15, 16, 17,
18, 19], with experimental success found only for chiral microparticles that
are large compared to the wavelength [17]. A new researcher in this field
reading the literature is faced with a difficult task due to the many
alternative ways of expressing the forces because of the different notation
and symbols, different unit systems, different definitions, and most
importantly, different ways of grouping terms that make the expressions look
very different from one another. The expressions and theoretical analysis of
these chiral optical forces are not as straightforward as the gradient and
pressure forces for conventional optical manipulation. There are many terms,
and great subtlety is needed. In this work, our aim is threefold: first, to be
as clear as possible, going over the main different forms of the expressions
used in the literature for chiral optical forces in dipolar particles,
pointing out how and why they are equivalent, and making sure they are as
general as possible. Second, in the process of clarifying the expressions, we
uncover and put forward a novel way of expressing the force, which brings
additional clarity, as it highlights the different symmetries involved in the
phenomenon. We write the net force as a linear combination of different vector
force fields, acting as a basis that describes all possible force patterns,
each of them weighted by particle-dependent coefficients. Third, we exemplify
how this simplified notation elegantly applies to two simple example cases: an
evanescent wave and a cylindrical dielectric nanofibre. In the process, we
present generally valid, but surprisingly concise, expressions for diverse
electromagnetic quantities in these modes, and based on this formulation,
describe the best strategies for enantiomer separation available in these
geometries.

This paper is intended to serve as a valuable aid to experimentalists trying
to achieve chiral separation of small (dipolar) enantiomers. As such, and
contrary to a large fraction of the existing literature, we shall adopt SI
units (_Système international d’unités_) together with the convention for
writing the speed of light in vacuum as $c_{0}$, while the phase velocity of
electromagnetic radiation in an arbitrary medium will be denoted
$c={\omega}/{k}={c_{0}}/{n}={1}/{\sqrt{\mu\varepsilon}}$. Throughout the
paper, we further assume a dispersion-free, lossless linear background medium
and monochromatic fields. We shall denote time-dependent real fields with
scripted letters, e.g., $\mathcal{A}(\mathbf{r},t)$, while their time-
independent phasor representation with regular Latin letters, $A(\mathbf{r})$,
such that
$\mathcal{A}(\mathbf{r},t)=\Re[A(\mathbf{r})\text{e}^{-\text{i}\omega t}]$.
One of the advantages of this representation is that any time-averaged
quantity quadratic in the fields can be obtained by simply
$\expectationvalue{\mathcal{A}\mathcal{B}}=\Re[A^{*}B]$ rather than
integrating over time.

We organise this work as follows. In section II we review the most general
expression for a time-averaged optical force on a small particle. In section
III we provide a clear overview of all alternative but equivalent ways of
writing this expression for the case of a linear bi-isotropic dipolar
particle, and we introduce a new classification of the forces based on the
symmetries of the particles and the fields. In the same section, we also
introduce the concept of force basis, which can be used to design the fields
for separating particles which break certain symmetries and in sections IV and
V we apply this concept to two simple yet practical examples of evanescent
waves and dielectric nanofibre, respectively.

##  II Optical forces

The most infallible tool at our disposal to calculate optical forces is to
apply the law of conservation of momentum. Any mechanical momentum that a
particle acquires must come at the expense of the reduced electromagnetic
momentum in the fields. Therefore, in a steady-state time-harmonic field, the
net time-averaged force acting on any piece of matter can be computed as the
total electromagnetic momentum entering, per unit of time, any closed volume
that includes the object. This is achieved by integrating the momentum flux
over a closed surface that surrounds the volume. Derived from first principles
[20, 21, 22], the time-averaged flux density of electromagnetic momentum is
given by the Maxwell stress tensor:

$\\!\\!\expectationvalue{\mathsfbfit{{T}}}=\frac{1}{2}\Re\bigg{[}\varepsilon{\bf\it
E}\otimes{\bf\it E}^{\ast}\\!+\mu{\bf\it H}\otimes{\bf\it
H}^{\ast}\\!-\frac{1}{2}\mathsfbfit{{I}}\big{(}\varepsilon|{\bf\it
E}|^{2}+\mu|{\bf\it H}|^{2}\big{)}\bigg{]},\\!\\!$ (1)

where ${\bf\it E}$ and ${\bf\it H}$ are the total electric and magnetic
fields, the operator $\otimes$ is the outer (or tensor) product of two
vectors, $\mathsfbfit{{I}}$ is the $3\times 3$ identity matrix, an asterisk
indicates complex conjugation, and $\varepsilon$ and $\mu$ are the absolute
permittivity and permeability of the medium, respectively.

Knowing $\expectationvalue{\mathsfbfit{{T}}}$, one can compute the net time-
averaged force acting on any material body via the flux integral:

$\langle{\bf\it
F}\rangle=\oiint_{\mathcal{S}}\expectationvalue{\mathsfbfit{{T}}}\differential{{\bf\it
s}},$ (2)

where $\differential{{\bf\it s}}=\hat{{\bf\it n}}\differential{s}$ is the
surface form, $\hat{{\bf\it n}}$ is the outward normal vector of a closed
surface $\mathcal{S}$ enclosing the body and $\differential{s}$ is the surface
element. We will drop the angular bracket notation for time averages, and
hereafter, each observable will be assumed to be time averaged. This general
method works on any body, regardless of its size. In this work, we are
interested in small particles that undergo Rayleigh scattering and hence
behave as dipoles. The criterion for which technique applies on which domain
is generally described as follows, where $k=\omega/c=2\pi/\lambda$ is the
medium wavenumber and $a$ is the particle size:

$\displaystyle ka$ $\displaystyle\ll{}$ $\displaystyle 1\quad$ — Rayleigh
scattering (3) $\displaystyle ka$ $\displaystyle\approx{}$ $\displaystyle
1\quad$ — Mie resonant regime $\displaystyle ka$ $\displaystyle\gg{}$
$\displaystyle 1\quad$ — geometrical optics

In the Rayleigh limit, any particle under illumination scatters light in an
identical form to an electric ${\bf\it p}$ and/or a magnetic ${\bf\it m}$
dipole. Because the analytical expression of the electric and magnetic fields
of a general combination of an electric and magnetic dipole is well known, one
can substitute such fields into eq. 2 and, analytically performing the
integration [23, 24, 25], arrive at the expression for the force acting on a
dipole in the Rayleigh limit

$\\!{{\bf\it
F}}=\frac{1}{2}\Re\Big{[}\\!\underbrace{\vphantom{\frac{k^{3}}{6\pi}}(\gradient\\!\otimes\\!{\bf\it
E}){\bf\it p}^{\ast}\\!+\\!\mu(\gradient\\!\otimes\\!{\bf\it H}){\bf\it
m}^{\ast}}_{\text{interaction}}\underbrace{-\,\frac{k^{4}\eta}{6\pi}({\bf\it
p}^{\ast}\\!\\!\crossproduct\\!{\bf\it m})}_{\text{recoil}}\\!\Big{]},\\!\\!$
(4)

where ${\bf\it p}$ and ${\bf\it m}$ are electric and magnetic dipole moment
vectors, considered as phasors of time-harmonic dipoles, ${\bf\it E}$ and
${\bf\it H}$ represent some external applied electric and magnetic fields
(excluding those produced by the dipoles) and $\eta=\sqrt{\mu/\varepsilon}$ is
the impedance of the surrounding environment.

Equation 4 constitutes the most general dipole force equation. All the
following equations for the force are derived from eq. 4 by assuming the
linearity of the particle, such that the induced dipoles are proportional to
the applied fields. The linear response of a particle to the applied external
fields can be described using complex polarisabilities. These polarisabilities
($\mathsfbfit{{\alpha}}_{\text{e}},\mathsfbfit{{\alpha}}_{\text{m}},\mathsfbfit{{\alpha}}_{\text{c}},\mathsfbfit{{\alpha}}_{\text{t}}$)
are, in general, second-rank tensors that can be represented by $3\times 3$
square matrices. The linear response of dipole moments to external fields can
be written as follows:

$\pmqty{undef}=\pmqty{undef}\pmqty{undef},$ (5)

where components of these electric-magnetic vectors are chosen so that they
have identical units, e.g., $[\sqrt{\varepsilon}{\bf\it E}]=[\sqrt{\mu}{\bf\it
H}]=\sqrt{$\mathrm{J}\text{\,}{\mathrm{m}}^{-3}$}$ etc. The literature is not
unified on the definitions of these polarisabilities. In particular, different
authors choose different units for these polarisabilities by choosing whether
they are direct responses to external fields ${\bf\it E}$ and ${\bf\it H}$
(used, for example, in [7, 9, 13, 15, 16, 14, 18]) or ${\bf\it E}$ and
${\bf\it B}$ (used in [11, 12, 15]). In this work, following Mun  _et al._
[10], we define all polarisabilities so that they all have dimensions of
volume,
$[\alpha_{\text{e}}]=[\alpha_{\text{m}}]=[\alpha_{\text{c}}]=[\alpha_{\text{t}}]=${\mathrm{m}}^{3}$$,
making the force expressions more elegant. Polarisabilities are a
characteristic of the given particle and typically are proportional to its
volume. In this work, we keep the most general form of polarisability,
allowing the electric-magnetic coupling terms to have both an antisymmetric
part (the chiral polarisability $\alpha_{\text{c}}$) and a symmetric part (the
non-reciprocal polarisability $\alpha_{\text{t}}$). The latter is usually
neglected (Bliokh  _et al._ [8] includes it but neglects the recoil force)
because it only exists for non-reciprocal particles that break time-reversal
symmetry. We keep this response not only for the sake of generality, but also
because it completes a clear framework relating the symmetries of the particle
and the optical fields to the forces that we can expect.

Knowing the incident fields and the polarisability of the particle, the exact
calculation of the optical force is straightforward in two steps: (i)
calculate the induced dipoles using eq. 5, and (ii) substitute them into eq.
4. This substitution may be carried out analytically to arrive at a single-
step equation, but the resulting expression is only manageable under certain
simplifying assumptions, as discussed in the following section. It is also
worth pointing out that if the particle is in an environment that reflects the
dipolar fields back into itself, this must be taken into account in both
steps. In the first step, the dipole moments must be calculated self-
consistently, as the incident fields appearing in eq. 5 have a new term, the
self-reflected fields, which depend on the dipole moments themselves. This can
also be addressed through the concept of an effective polarisability that
depends on the environment [26, 27]. In the second step, the dipolar fields
that are reflected back must be included in the fields in the interaction
terms of eq. 4. This complex interaction allows for a plethora of different
phenomena to arise, such as repulsion from nearby surfaces [28, 29, 30, 31],
pulling forces [32, 26], self torques [33], trapping forces [34], and lateral
force recoils due to excitation of nearby guided modes [13, 35, 36, 37]. These
cases greatly complicate the calculations, but ultimately follow the same
physics as included in eqs. 5 and 4. In the rest of this work, we neglect
back-reflection, as if the particle was in a homogeneous background
environment.

##  III Force on bi-isotropic dipolar particle

In practice, the polarisability tensors will depend on how the particle is
oriented in space relative to the incident fields. However, unless the
particles in question are somehow aligned, for instance, by some external
static field or by nanofabrication, we can assume that for an average
randomly-oriented particle within the sample, the effective polarisability
will be (bi-)isotropic, and therefore can be described by complex
(pseudo)scalars
($\alpha_{\text{e}},\alpha_{\text{m}},\alpha_{\text{c}},\alpha_{\text{t}}$).
In this case eq. 5 becomes

$\begin{split}{\bf\it p}&=\alpha_{\text{e}}\varepsilon{\bf\it
E}+(\alpha_{\text{t}}+{\text{i}}\alpha_{\text{c}}){\bf\it H}/c\,,\\\ {\bf\it
m}&=\alpha_{\text{m}}{\bf\it
H}+(\alpha_{\text{t}}-{\text{i}}\alpha_{\text{c}}){\bf\it
E}/\eta\,.\end{split}$ (6)

Effective polarisabilities for the specific case of a _spherical_ chiral bi-
isotropic particle can be found in appendix A. Equation 6 can be substituted
into eq. 4, and after separating the real and imaginary parts of
polarisabilities and further simplifications, one can write the force in terms
of eight local observables that are quadratic in the fields. These quantities
are: two real scalars

$W_{\text{e}}=\frac{\varepsilon}{4}\absolutevalue{{\bf\it E}}^{2}\,,\quad
W_{\text{m}}=\frac{\mu}{4}\absolutevalue{{\bf\it H}}^{2}\,,$ (7)

which represent electric and magnetic energy densities, respectively; a pair
of real pseudovectors

${\bf\it S}_{\text{e}}=\frac{1}{4\omega}\Im(\varepsilon{{\bf\it
E}^{*}\\!\crossproduct{\bf\it E}})\,,\quad{\bf\it
S}_{\text{m}}=\frac{1}{4\omega}\Im(\mu{{\bf\it H}^{*}\\!\crossproduct{\bf\it
H}})\,,$ (8)

representing electric and magnetic spin angular momentum densities; and a
complex pseudoscalar and vector

$W_{\text{c}}=\frac{\text{i}}{2c}{{\bf\it E}^{*}\\!\dotproduct{\bf\it
H}}\,,\quad{\bf\it\varPi}=\frac{1}{2}{{\bf\it E}\crossproduct{\bf\it
H}^{*}}\,,$ (9)

where $\Re{W_{\text{c}}}/\omega=\mathfrak{S}$ is the time-averaged helicity
density, and $\Re{{\bf\it\varPi}}$ is the time-averaged Poynting vector
representing the flow of active power. Note that quantities
$\Im{W_{\text{c}}}$ and $\Im{{\bf\it\varPi}}$, while often overlooked, are
non-zero in general and also lead to forces felt by particles that break
certain symmetries. The vector $\Im{{\bf\it\varPi}}$ represents the flow of
reactive power [20], while the pseudoscalar $\Im{W_{\text{c}}}$ represents a
static “chirality”111In the sense of handedness. due to $\Re{\bf\it
E}\not\perp\Re{\bf\it H}$ and $\Im{\bf\it E}\not\perp\Im{\bf\it H}$, in this
sense, it can be thought of as a measure of instantaneous _collinearity_ of
the electric with the magnetic field [8].

As mentioned before, the resulting force can be written in more than one way.
One way is to use the complex particle polarisabilities to define the real
scalar quantities that are often for their dimensions called cross-sections:

$\begin{split}\sigma_{\text{ext}}&=\sigma_{\text{e}}+\sigma_{\text{m}}=k\Im(\alpha_{\text{e}}+\alpha_{\text{m}})\,,\\\
\sigma_{\text{rec}}&=\frac{k^{4}}{6\pi}[\Re(\alpha_{\text{e}}^{*}\alpha_{\text{m}})+\left|\alpha_{\text{c}}\right|^{2}-\left|\alpha_{\text{t}}\right|^{2}]\,,\\\
\sigma_{\text{im}}&=\frac{k^{4}}{6\pi}[\Im(\alpha_{\text{e}}^{*}\alpha_{\text{m}})+2\Re(\alpha_{c}^{*}\alpha_{\text{t}})]\,,\\\
\sigma_{\text{re}}&=\sigma_{\text{ext}}-\sigma_{\text{rec}}\,,\end{split}$
(10)

where $\sigma_{\text{ext}}$ is indeed an extinction cross-section. For chiral
forces, we can also define pseudoscalar cross-sections:

$\displaystyle\gamma_{\text{re}}$
$\displaystyle=k\Im\alpha_{\text{c}}\,,\quad\gamma_{\text{im}}=k\Im\alpha_{\text{t}}$
(11) $\displaystyle\gamma^{\text{e}}_{\text{rec}}$
$\displaystyle=\frac{k^{4}}{3\pi}[\Re(\alpha_{\text{e}}^{\ast}\alpha_{\text{c}})+\Im(\alpha_{\text{e}}^{\ast}\alpha_{\text{t}})]$
$\displaystyle\gamma^{\text{m}}_{\text{rec}}$
$\displaystyle=\frac{k^{4}}{3\pi}[\Re(\alpha_{\text{m}}^{\ast}\alpha_{\text{c}})+\Im(\alpha_{\text{m}}^{\ast}\alpha_{\text{t}})]$
$\displaystyle\gamma_{\text{e}}$
$\displaystyle=2\gamma_{\text{re}}-\gamma^{\text{e}}_{\text{rec}}\,,\quad\gamma_{\text{m}}=2\gamma_{\text{re}}-\gamma^{\text{m}}_{\text{rec}}\,.$

These can be used to write the force as in [10, 13, 15, 38]

$\begin{split}\\!\\!\\!\\!{{\bf\it
F}}={}&\gradient{U}+\frac{\sigma_{\text{re}}\Re{\bf\it\varPi}-\sigma_{\text{im}}\Im{\bf\it\varPi}}{c}+\omega\quantity(\gamma_{\text{e}}{\bf\it
S}_{\text{e}}+\gamma_{\text{m}}{\bf\it S}_{\text{m}})\\!\\!\\!\\!\\\
&\\!\\!-\gradient\\!\crossproduct\\!\quantity[\frac{\gamma_{\text{re}}{\Re{\bf\it\varPi}}\\!+\\!\gamma_{\text{im}}{\Im{\bf\it\varPi}}}{\omega}+c(\sigma_{\text{e}}{\bf\it
S}_{\text{e}}\\!+\\!\sigma_{\text{m}}{\bf\it
S}_{\text{m}})]\\!\,,\\!\\!\\!\\!\\!\\!\\!\end{split}$ (12)

where $U$ is a scalar potential defined as

$\\!\\!\\!U=\Re\alpha_{\text{e}}W_{\text{e}}+\Re\alpha_{\text{m}}W_{\text{m}}+\Re\alpha_{\text{c}}\Re{W_{\text{c}}}+\Re\alpha_{\text{t}}\Im{W_{\text{c}}}\,.\\!\\!\\!$
(13)

The chiral terms of this force are those that are a product of pseudoscalar
polarisabilities $\alpha_{\text{c}}$ and cross sections $\gamma$ with
pseudovectors such as $\gradient{W_{\text{c}}}$, ${\bf\it S}_{\text{e}}$,
${\bf\it S}_{\text{m}}$ and $\curl{\bf\it\varPi}$. If we reflect either the
particle or the fields in a mirror, the pseudoscalars or their corresponding
pseudovectors change sign, respectively. The rest of the force is achiral.
Some of the literature that uses eq. 12 gives names to individual terms. The
potential $U$ is sometimes called the free energy and $\gradient U$ are
collectively named the gradient forces. Forces proportional to
$\Re{\bf\it\varPi}$ are referred to as radiation pressure forces222even though
they include the self-recoil of the dipole, while the term with
$\curl\Re{\bf\it\varPi}$ is known as the vortex force. Spin density forces are
the terms directly proportional to ${\bf\it S}_{\text{e}}$ and ${\bf\it
S}_{\text{m}}$, while spin-curl forces are proportional to their curls. The
last term to have a name is $\Im{\bf\it\varPi}$ which is called the flow force
because it is related to the alternating flow of “stored energy” [20]. The
term related to the curl of this vector is only present for nonreciprocal
particles, it is not included in the literature works and hence does not have
a name.

While eq. 12 is an elegant expression relating kinetic properties of light
[eqs. 7, 8 and 9] with observables related to the particle [eqs. 10 and 11],
it has one disadvantage: the expression no longer allows distinguishing the
interaction and recoil parts of the dipole force that appeared in eq. 4. In
order to recover the two terms, we introduce two more vectors:

$\\!\\!{\bf\it
p}_{\text{e}}=\frac{\varepsilon}{4\omega}\Im\bqty{(\gradient\\!\otimes\\!{\bf\it
E}){\bf\it E}^{\ast}}\,,\;{\bf\it
p}_{\text{m}}=\frac{\mu}{4\omega}\Im\bqty{(\gradient\\!\otimes\\!{\bf\it
H}){\bf\it H}^{\ast}}\,,\\!\\!$ (14)

which are often called canonical (or sometimes orbital) linear momentum
densities, on top of which we may define a ‘complex chiral momentum density’
pseudovector:

$\\!\\!{\bf\it p}_{\text{c}}=\frac{1}{4\omega
c}\bqty{(\gradient\\!\otimes\\!{\bf\it H}){\bf\it
E}^{\ast}-(\gradient\\!\otimes\\!{\bf\it E}^{\ast}){\bf\it H}}\,,$ (15)

where $\Re{\bf\it p}_{\text{c}}$ is the chiral and $\Im{\bf\it p}_{\text{c}}$
is the magnetoelectric momentum densities introduced in Bliokh  _et al._ [8].
These canonical momentum densities are directly related to the previously
defined (pseudo)vector observables ${\bf\it\varPi},{\bf\it
S}_{\text{e}},{\bf\it S}_{\text{m}}$ and their curls as follows [39, 40]:

$\begin{split}{\bf\it
p}_{\text{e}}&=\frac{1}{2c^{2}}\Re{\bf\it\varPi}-\frac{1}{2}\curl{{\bf\it
S}_{\text{e}}}\,,\\\ {\bf\it
p}_{\text{m}}&=\frac{1}{2c^{2}}\Re{\bf\it\varPi}-\frac{1}{2}\curl{{\bf\it
S}_{\text{m}}}\,,\\\ \Re{\bf\it p}_{\text{c}}&=k({\bf\it S}_{\text{e}}+{\bf\it
S}_{\text{m}})-\frac{1}{2\omega c}\curl{\Re{\bf\it\varPi}}\,,\\\ \Im{\bf\it
p}_{\text{c}}&=-\frac{1}{2\omega c}\curl{\Im{\bf\it\varPi}}\,,\end{split}$
(16)

which, when substituted into eq. 12, allows us to rewrite the force in a
different way. Confusion may arise due to the fact that a fraction of the
literature uses eq. 12 while another uses eq. 17 below. The two alternative
expressions give the same value for the force, but the terms are grouped
differently (and some terms in one expression are divided into two terms in
the other). While eq. 12 is simpler in terms of the electromagnetic quantities
involved (so we may call it the _EM-centric_ formulation), eq. 17 is simpler
in terms of the particle polarisabilities involved (with no cross sections
needed, so we may call it the _particle-centric_ formulation). This
alternative formulation is used by [9, 14, 16, 18, 23] but missing the terms
involving $\alpha_{\text{t}}$. To our knowledge, no previous publication lists
both non-reciprocal interaction terms and the recoil force. The full force
reads:

$\begin{split}{{\bf\it
F}}={}&\underbrace{\gradient(\Re\alpha_{\text{e}}W_{\text{e}}\\!+\\!\Re\alpha_{\text{m}}W_{\text{m}}\\!+\\!\Re\alpha_{\text{c}}\Re{W_{\text{c}}}\\!+\\!\Re\alpha_{\text{t}}\Im{W_{\text{c}}})}_{\text{gradient
force}}\\\ &\underbrace{+2\omega(\Im\alpha_{\text{e}}{\bf\it
p}_{\text{e}}\\!+\\!\Im\alpha_{\text{m}}{\bf\it
p}_{\text{m}}\\!+\\!\Im\alpha_{\text{c}}\Re{\bf\it
p}_{\text{c}}\\!+\\!\Im\alpha_{\text{t}}\Im{\bf\it
p}_{\text{c}})}_{\text{radiation pressure force}}\\\
&\underbrace{-(\sigma_{\text{rec}}\Re{\bf\it\varPi}\\!+\\!\sigma_{\text{im}}\Im{\bf\it\varPi})/c-\omega\quantity(\gamma^{\text{e}}_{\text{rec}}{\bf\it
S}_{\text{e}}\\!+\\!\gamma^{\text{m}}_{\text{rec}}{\bf\it
S}_{\text{m}})}_{\text{dipole recoil force}}.\\!\\!\\!\end{split}$ (17)

where we can see three distinct terms. The first two are generalisations of
the conservative gradient and radiation pressure forces, while the third term
is purely the recoil force from eq. 4. The advantage of this form is that it
is easier to interpret. Notice that the gradient forces will always point
towards (or away from) the maximal value of _energy densities_ , and they
depend only on real polarisabilities, which means that a particle will feel it
even if there is no absorption of photons, it is a conservative force. The
radiation pressure points along the _canonical momenta_ of the wave and
depends on imaginary polarisabilities, hence extinction cross sections, and it
will require photons to be absorbed or scattered. The recoil force points
along the flow of _active power_ $\Re{{\bf\it\varPi}}$, _reactive power_
$\Im{{\bf\it\varPi}}$ and the electric ${\bf\it S}_{\text{e}}$ and magnetic
${\bf\it S}_{\text{m}}$ _spin_ angular momentum densities and is always
proportional to squares of polarisabilities, while the other two terms were
linearly proportional to the polarisabilities.

One more formulation is worth mentioning for its simplicity and relationship
with symmetries, which was introduced in Bliokh  _et al._ [8] (although
neglecting the recoil force). The key observation is that in free space,
Maxwell’s equations are symmetric under the parity inversion (P):
$(x,y,z)^{\intercal}\mapsto(-x,-y,-z)^{\intercal}$, time reversal (T):
$t\mapsto-t$, and the the duality transformation (D):

$\quantity(\sqrt{\varepsilon}{\bf\it E},\sqrt{\mu}{\bf\it
H})^{\intercal}\mapsto\quantity(\sqrt{\mu}{\bf\it
H},-\sqrt{\varepsilon}{\bf\it E})^{\intercal},$ (18)

which is a symmetry leading to the conservation of optical helicity [41]. One
can then redefine all the quantities involved in the force such that they are
either symmetric or antisymmetric under these transformations. We shall label
these quantities with an index $A\in\\{0,1,2,3\\}$, such that they have the
same behaviour under these transformations as Stokes parameters, i.e., $A=0$
will be symmetric under all these, $A=1$ will be antisymmetric under (D),
$A=2$ will be antisymmetric under all and $A=3$ will be antisymmetric under
parity. The energy densities appearing in the gradient forces can be written
as:

$\displaystyle W_{0}$ $\displaystyle=W_{\text{e}}+W_{\text{m}}\,,\quad$
$\displaystyle W_{2}$ $\displaystyle=-\Im{W_{\text{c}}}\,,$ (19)
$\displaystyle W_{1}$ $\displaystyle=W_{\text{e}}-W_{\text{m}}\,,\quad$
$\displaystyle W_{3}$ $\displaystyle=\phantom{+}\Re{W_{\text{c}}}\,,$

the canonical momenta featured in the pressure force:

$\displaystyle{\bf\it p}_{0}$ $\displaystyle={\bf\it p}_{\text{e}}+{\bf\it
p}_{\text{m}}\,,\quad$ $\displaystyle{\bf\it p}_{2}$
$\displaystyle=-\Im{{\bf\it p}_{\text{c}}}\,,$ (20) $\displaystyle{\bf\it
p}_{1}$ $\displaystyle={\bf\it p}_{\text{e}}-{\bf\it p}_{\text{m}}\,,\quad$
$\displaystyle{\bf\it p}_{3}$ $\displaystyle=\phantom{+}\Re{{\bf\it
p}_{\text{c}}}\,,$

and the _spin-like_ quantities in the recoil force:

$\displaystyle{\bf\it S}_{0}$ $\displaystyle={\bf\it S}_{\text{e}}+{\bf\it
S}_{\text{m}}\,,\quad$ $\displaystyle{\bf\it S}_{2}$
$\displaystyle=-\Im{{\bf\it\varPi}}/(\omega c)\,,$ (21) $\displaystyle{\bf\it
S}_{1}$ $\displaystyle={\bf\it S}_{\text{e}}-{\bf\it S}_{\text{m}}\,,\quad$
$\displaystyle{\bf\it S}_{3}$
$\displaystyle=\phantom{+}\Re{{\bf\it\varPi}}/(\omega c)\,.$

${\bf\it F}=\sum_{i}\lambda_{i}{\bf\it V}\\!_{i}$ | Coefficients $\lambda_{i}$ (dimensions of volume) |  Basis ${\bf\it V}\\!_{i}$ (dimensions of force/volume) | symmetries broken by the particle
---|---|---|---
pseudoscalarpseudo |  pseudoscalar |  pseudovectorpseudo |  pseudovector
Interaction force | Gradient |  $\Re(\alpha_{\text{e}}+\alpha_{\text{m}})/2$ | $\gradient(W_{\text{e}}+W_{\text{m}})$ | –
$\Re(\alpha_{\text{e}}-\alpha_{\text{m}})/2$ | $\gradient(W_{\text{e}}-W_{\text{m}})$ | D
$\Re\alpha_{\text{c}}$ | $\gradient\Re W_{\text{c}}$ | P
$\Re\alpha_{\text{t}}$ | $\gradient\Im W_{\text{c}}$ | T
Pressure | $\Im(\alpha_{\text{e}}+\alpha_{\text{m}})/2$ | $2\omega({\bf\it p}_{\text{e}}+{\bf\it p}_{\text{m}})$ | –
$\Im(\alpha_{\text{e}}-\alpha_{\text{m}})/2$ | $2\omega({\bf\it p}_{\text{e}}-{\bf\it p}_{\text{m}})$ | D
$\Im\alpha_{\text{c}}$ | $2\omega\Re{\bf\it p}_{\text{c}}$ | P
$\Im\alpha_{\text{t}}$ | $2\omega\Im{\bf\it p}_{\text{c}}$ | T
Recoil | Power | $\frac{k^{3}}{6\pi}\\{\Re[\alpha_{\text{e}}^{*}\alpha_{\text{m}}]+\absolutevalue{\alpha_{\text{c}}}^{2}-\absolutevalue{\alpha_{\text{t}}}^{2}\\}$ | $k\Re{{\bf\it\varPi}}/c$ | –
$\frac{k^{3}}{6\pi}\quantity{\Im[\alpha_{\text{e}}^{*}\alpha_{\text{m}}]+2\Re[\alpha_{c}^{*}\alpha_{\text{t}}]}$ | $k\Im{{\bf\it\varPi}}/c$ | D
Spin | $\;\;\frac{k^{3}}{6\pi}\quantity{\Re\quantity[\alpha_{\text{c}}^{*}(\alpha_{\text{e}}+\alpha_{\text{m}})]-\Im\quantity[\alpha_{\text{t}}^{*}(\alpha_{\text{e}}-\alpha_{\text{m}})]}\;\;$ | $k\omega({\bf\it S}_{\text{e}}+{\bf\it S}_{\text{m}})$ | P
$\;\;\frac{k^{3}}{6\pi}\quantity{\Re\quantity[\alpha_{\text{c}}^{*}(\alpha_{\text{e}}-\alpha_{\text{m}})]-\Im\quantity[\alpha_{\text{t}}^{*}(\alpha_{\text{e}}+\alpha_{\text{m}})]}\;\;$ | $k\omega({\bf\it S}_{\text{e}}-{\bf\it S}_{\text{m}})$ | P & D
Table 1: Showing the (pseudo)scalars and (pseudo)vectors involved in the
dipolar force and illustrating the idea of a vector basis for the force. The
last column lists what symmetry (D for duality, P for parity and T for time
reversal) has to be broken by both the particle and the wave for the force
component to be non-zero. When the recoil force is negligible, we can say that
the force forms an eight-dimensional vector space spanned by basis ${\bf\it
V}\\!_{i}$. Formally, the addition of the recoil terms does not promote the
force into a twelve-dimensional vector space because the coefficients of the
recoil terms are fully determined by the products of the coefficients of the
interaction force.

Notably, the quantities in eqs. 19, 20 and 21 are not independent. In
particular, the energy densities eq. 19 are time components of momentum
densities eq. 20 in the relativistic sense, while the _spin-like_ quantities
eq. 21 can be thought of as fluxes of the aforementioned energy densities (for
more details, see appendix B). One can show (see appendix C) that the chiral
energy, momentum, and spin densities are simply differences between these
quantities carried by the positive/negative helicity components of the
electromagnetic field, i.e.,

$W_{3}=W_{+}-W_{-}\,,\quad{\bf\it p}_{3}={\bf\it p}_{+}-{\bf\it
p}_{-}\,,\quad{\bf\it S}_{3}={\bf\it S}_{+}-{\bf\it S}_{-}\,.$

This also justifies why they have dimensions of energy, momentum, and spin
densities, respectively.

The same procedure can be followed for the polarisabilities, which leads to
the following expressions:

$\displaystyle\alpha_{0}$
$\displaystyle=(\alpha_{\text{e}}+\alpha_{\text{m}})/2\,,\quad$
$\displaystyle\alpha_{2}$ $\displaystyle=-\alpha_{\text{t}}\,,$ (22)
$\displaystyle\alpha_{1}$
$\displaystyle=(\alpha_{\text{e}}-\alpha_{\text{m}})/2\,,\quad$
$\displaystyle\alpha_{3}$ $\displaystyle=\phantom{+}\alpha_{\text{c}}\,.$

We can also define square polarisabilities $\beta_{A}$, which are equivalent
to the recoil cross sections eqs. 10 and 11:

$\displaystyle\beta_{0}$
$\displaystyle=\frac{k^{3}}{6\pi}\quantity{\Re\quantity[\alpha_{\text{c}}^{*}(\alpha_{\text{e}}+\alpha_{\text{m}})]-\Im\quantity[\alpha_{\text{t}}^{*}(\alpha_{\text{e}}-\alpha_{\text{m}})]},$
(23) $\displaystyle\beta_{1}$
$\displaystyle=\frac{k^{3}}{6\pi}\quantity{\Re\quantity[\alpha_{\text{c}}^{*}(\alpha_{\text{e}}-\alpha_{\text{m}})]-\Im\quantity[\alpha_{\text{t}}^{*}(\alpha_{\text{e}}+\alpha_{\text{m}})]},$
$\displaystyle-\beta_{2}$
$\displaystyle=\frac{k^{3}}{6\pi}\quantity{\Im[\alpha_{\text{e}}^{*}\alpha_{\text{m}}]+2\Re[\alpha_{c}^{*}\alpha_{\text{t}}]}$
$\displaystyle\beta_{3}$
$\displaystyle=\frac{k^{3}}{6\pi}\\{\Re[\alpha_{\text{e}}^{*}\alpha_{\text{m}}]+\absolutevalue{\alpha_{\text{c}}}^{2}-\absolutevalue{\alpha_{\text{t}}}^{2}\\}\,.$

The minus signs in front of quantities with the label $A=2$ are chosen to
match the signs of Stokes parameters later; however, in eq. 24, they will
cancel out. Expressed in this _basis_ , the optical force takes a surprisingly
simple form

$\\!\\!{{\bf\it F}}=\sum_{A=0}^{3}(\underbrace{\Re{\alpha}_{A}\gradient
W\\!\\!_{A}}_{\text{gradient}}\underbrace{+2\omega\Im{\alpha}_{A}{\bf\it
p}\\!_{A}}_{\text{pressure}}\underbrace{-{\beta}_{A}k\omega{\bf\it
S}\\!_{A}}_{\text{recoil}}),\\!\\!$ (24)

where we have included the recoil terms, which were not previously calculated.
While eqs. 12, 17 and 24 are only true in the dipole approximation an
advantage of the symmetry-based approach is that we can always divide the
force into components, ${\bf\it F}={\bf\it F}_{0}+{\bf\it F}_{1}+{\bf\it
F}_{2}+{\bf\it F}_{3}$, such that ${\bf\it F}_{0}$ is present for every
particle and the remaining components ${\bf\it F}_{A}$ appear only for
particles breaking corresponding symmetries.

We remark that eqs. 12, 17 and 24 are all alternative formulations of the
_same_ force. Ultimately, one may notice that the force on a dipolar particle
can be written as a sum of (pseudo)scalar coefficients multiplied by a
(pseudo)vector basis. In other words, one can write

${\bf\it F}=\sum_{i}\lambda_{i}{\bf\it V}\\!_{i}\,,$ (25)

where eqs. 12, 17 and 24 are simply different choices of the basis vectors and
coefficients. For the representation in eq. 24 we list all the coefficients
and the basis in table 1. Note, however, that this is not formally a vector
basis unless we neglect recoil terms, since the coefficients in eq. 23 are
products of the other coefficients. Despite that, the concept illustrated by
eq. 25 is quite profound: it tells us that we have coefficients $\lambda_{i}$
that carry all the dependence on the particle (and have units of volume),
which multiplied by the basis vectors ${\bf\it V}\\!_{i}$ that depend only on
the incident electromagnetic fields (and whose dimensions are force per volume
or _force density_) gives us the force that a given particle feels. This can
be used in designing optical forces to separate particles with different
symmetries. For example, using this idea, one can plot different basis vectors
${\bf\it V}\\!_{i}({\bf\it r})$ for a given electromagnetic mode (as we will
do later) to get an idea of the types of forces (the directions and
amplitudes) that are possible in a particular waveguide—independently of the
particle.

Looking at table 1, one can see that both the force densities ${\bf\it
V}\\!_{i}$ and associated coefficients $\lambda_{i}$ have the same behaviour
under symmetries of the electromagnetic field, i.e., if one breaks the
symmetry the other does as well such that their product (i.e. the force term)
is a dual symmetric vector (hence parity odd) which is time reversal even/odd
for interaction/recoil force respectively. This gives the formulation in eq.
24 a very useful interpretation. The terms associated with the total energy
$W_{0}$, the total canonical momentum ${\bf\it p}_{0}$, and the active power
$\Re{{\bf\it\varPi}}$ will be symmetric under D, P, and T, and will be felt by
all particles. Other terms will be felt only by particles that break relevant
symmetries (see the last column of table 1). For example, a non-magnetic
dielectric particle breaks the D symmetry, so it will also feel D asymmetric
forces associated with $W_{1}=W_{\text{e}}-W_{\text{m}}$, ${\bf\it
p}_{1}={\bf\it p}_{\text{e}}-{\bf\it p}_{\text{m}}$ and the reactive power
$\Im{{\bf\it\varPi}}$. If we had a particle that breaks duality in the
opposite sense (a dominant magnetic response), then it would feel these forces
in the opposite direction, and one could design an optical field to separate
electric and magnetic particles. In our case, we are interested in chiral
forces that break the P symmetry, i.e., forces associated with
$\Re{W_{\text{c}}}=W_{+}-W_{-}$, $\Re{{\bf\it p}_{\text{c}}}={\bf\it
p}_{+}-{\bf\it p}_{-}$ and spin recoil forces ${\bf\it S}_{\text{e}}$ and
${\bf\it S}_{\text{m}}$ (note that a dual symmetric chiral particle would only
feel ${\bf\it S}_{0}={\bf\it S}_{\text{e}}+{\bf\it S}_{\text{m}}$). Therefore,
these forces are at our disposal if we want to separate enantiomers. As
before, each enantiomer will feel a chiral force of the same magnitude but in
the opposite direction. As a curiosity, a non-reciprocal particle, i.e., a
particle that breaks the T symmetry (as well as D and P), would additionally
feel two new force densities $\Im{W_{\text{c}}}$ and $\Im{{\bf\it
p}_{\text{c}}}$.

##  IV Evanescent wave

We may now apply the formalism to simple though practical examples. Consider a
particle illuminated by an evanescent wave in a linear lossless medium with a
complex wave vector ${\bf\it k}={\bf\it k}^{\prime}+\text{i}{\bf\it
k}^{\prime\prime}=k_{p}\hat{{\bf\it k}}^{\prime}+\text{i}\kappa\hat{{\bf\it
k}}^{\prime\prime}$, where $\kappa>0$ is a real parameter that describes how
evanescent the wave is. Figure 1 illustrates the orthonormal directions of
$\hat{{\bf\it k}}^{\prime}$ (phase advance) and $\hat{{\bf\it
k}}^{\prime\prime}$ (decay direction) in an evanescent wave. Because ${\bf\it
k}\cdot{\bf\it k}=k^{2}$, we know that $k_{p}=\sqrt{k^{2}+\kappa^{2}}$. The
electric field satisfying the transversality condition $\divergence{\bf\it
E}=0$ can be written as:

$\\!\\!\sqrt{\varepsilon}{\bf\it E}({\bf\it r})={\text{e}^{\text{i}{\bf\it
k}\dotproduct{\bf\it r}}}\bqty{A_{s}\quantity(\hat{{\bf\it
k}}^{\prime}\\!\crossproduct\\!\hat{{\bf\it
k}}^{\prime\prime})\\!+\\!A_{p}\quantity(\frac{k_{p}}{k}\hat{{\bf\it
k}}^{\prime\prime}\\!\\!-\text{i}\frac{\kappa}{k}\hat{{\bf\it
k}}^{\prime})\\!},\\!\\!$ (26)

where $A_{s}$ and $A_{p}$ are arbitrary complex amplitudes of two allowed
polarisations, while the magnetic field can be acquired from the
Maxwell–Faraday equation $\curl{{\bf\it E}}=\text{i}\omega\mu{\bf\it H}$. The
duality transformation of eq. 26 can be written as follows:

$\sqrt{\varepsilon}{\bf\it E}\mapsto\sqrt{\mu}{\bf\it
H},\quad(A_{s},A_{p})\mapsto(A_{p},-A_{s})\,.$ (27)

All observables are quadratic in the fields. Hence, the dependence on
polarisation can be written in terms of Stokes parameters, which can be
defined in terms of the polarisation amplitudes

$\displaystyle\mathcal{S}_{0}$
$\displaystyle=\absolutevalue{A_{p}}^{2}+\absolutevalue{A_{s}}^{2},$
$\displaystyle\quad\mathcal{S}_{2}$ $\displaystyle=2\Re(A_{p}^{*}A_{s})\,,$
$\displaystyle\mathcal{S}_{1}$
$\displaystyle=\absolutevalue{A_{p}}^{2}-\absolutevalue{A_{s}}^{2}\,,$
$\displaystyle\quad\mathcal{S}_{3}$ $\displaystyle=2\Im(A_{p}^{*}A_{s})\,.$

Since $A_{p}$ is a complex scalar and $A_{s}$ a pseudoscalar, we can say that
$\mathcal{S}_{0}$ and $\mathcal{S}_{1}$ are scalars, while $\mathcal{S}_{2}$
and $\mathcal{S}_{3}$ are pseudoscalars. Additionally, one can confirm using
eq. 27 that $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ are dual asymmetric while
the rest are symmetric. The energy densities for this field can be calculated
using eqs. 7 and 19, leading to an elegant form, simply proportional to the
Stokes parameters:

$\displaystyle W_{0}$ $\displaystyle=\mathcal{S}_{0}{k_{p}^{2}}f({\bf\it
r})\,,\quad$ $\displaystyle W_{2}$
$\displaystyle=\mathcal{S}_{2}{\kappa^{2}}f({\bf\it r})\,,$ (28)
$\displaystyle W_{1}$ $\displaystyle=\mathcal{S}_{1}{\kappa^{2}}f({\bf\it
r})\,,\quad$ $\displaystyle W_{3}$
$\displaystyle=\mathcal{S}_{3}{k_{p}^{2}}f({\bf\it r})\,,$

where $f({\bf\it r})=\exp(-2{\bf\it k}^{\prime\prime}\\!\\!\dotproduct{\bf\it
r})/{2k^{2}}$. Notice that if a wave is not evanescent $\kappa=0$, then
$W_{1}=W_{2}=0$, which is directly related to the fact that a plane
wave—unlike an evanescent wave—does not break dual symmetry nor time-reversal
symmetry. It is also apparent that the parity can be broken even by a
circularly polarised plane wave since $W_{3}$ is proportional to
$\mathcal{S}_{3}$. Knowing the energy densities, remarkably simple equations
lead to the energy density gradients $\gradient{W}\\!_{A}=-2{\bf\it
k}^{\prime\prime}{W\\!_{A}}$, and to the canonical momenta ${\bf\it
p}_{A}={\bf\it k}^{\prime}{W\\!_{A}}/\omega$, as defined in eq. 20.
Substituting these into eq. 24, we obtain a very simple expression for the
interaction force (gradient and pressure terms):

$\displaystyle{{\bf\it F}}_{\text{int}}$
$\displaystyle=\sum_{A=0}^{3}2W_{A}\quantity(\Im\alpha_{A}{\bf\it
k}^{\prime}-\Re{\alpha}_{A}{\bf\it k}^{\prime\prime})\,,$ (29)

which depends only on the energy densities (Stokes’s parameters) eq. 28 and
relevant polarisabilities. The interaction force has no component in the
lateral ${\bf\it k}^{\prime}\crossproduct{\bf\it k}^{\prime\prime}$ direction;
forces in this direction are coming purely from the recoil part of the force,
which depends on the _spin quantities_ eq. 21. It is common knowledge that
evanescent waves possess transverse spin [42], which is independent of
helicity ($\mathfrak{S}=W_{3}/\omega$). The total spin can be calculated from
eqs. 8 and 21, and has two components, longitudinal and transverse:

$k\omega{\bf\it S}_{0}=\frac{k^{2}}{k_{p}^{2}}W_{3}{\bf\it
k}^{\prime}+\underbrace{\frac{k}{k_{p}^{2}}W_{0}{\bf\it
k}^{\prime}\crossproduct{\bf\it k}^{\prime\prime}}_{\text{transverse spin}}.$
(30)

This transverse spin will lead to a chiral lateral recoil force, as used in
Hayat  _et al._ [14]. It is also known that the real Poynting vector acquires
transverse components in the case of a circularly polarised evanescent wave
(see, e.g. Wei and Rodríguez-Fortuño [43]) explicitly given by

$k\Re({\bf\it\varPi}/c)=k\omega{\bf\it
S}_{3}=\frac{k^{2}}{k_{p}^{2}}W_{0}{\bf\it
k}^{\prime}+\frac{k}{k_{p}^{2}}W_{3}{\bf\it k}^{\prime}\crossproduct{\bf\it
k}^{\prime\prime}.$ (31)

Therefore, even an achiral particle will feel an achiral lateral force if
irradiated by a circularly polarised evanescent wave, as confirmed
experimentally by Antognozzi  _et al._ [44]. In addition to transverse spin
(contained in ${\bf\it S}_{0}$) and transverse Poynting vector (contained in
${\bf\it S}_{3}$), the two remaining spin-like quantities from eq. 21 (which
are the reactive power proportinal to ${\bf\it S}_{2}$ and the difference in
electric and magnetic spins ${\bf\it S}_{1}$) will similarly have transverse
contributions:

$\displaystyle k\omega{\bf\it S}_{2}$
$\displaystyle=-\frac{k^{2}}{\kappa^{2}}W_{1}{\bf\it
k}^{\prime\prime}-\frac{k}{\kappa^{2}}W_{2}{\bf\it
k}^{\prime}\crossproduct{\bf\it k}^{\prime\prime}$ (32) $\displaystyle
k\omega{\bf\it S}_{1}$ $\displaystyle=-\frac{k^{2}}{\kappa^{2}}W_{2}{\bf\it
k}^{\prime\prime}+\frac{k}{\kappa^{2}}W_{1}{\bf\it
k}^{\prime}\crossproduct{\bf\it k}^{\prime\prime}$ (33)

notice that this time, the other contributions point in the direction
$\hat{{\bf\it k}}^{\prime\prime}$ rather than $\hat{{\bf\it k}}^{\prime}$. The
full recoil force (third term in eq. 24) is then

$\begin{split}\\!\\!{{\bf\it
F}}_{\text{rec}}={}&k\Big{(}\frac{{W_{2}}\beta_{2}}{\kappa^{2}}-\frac{{W_{0}}\beta_{0}}{k^{2}_{p}}+\frac{{W_{3}}\beta_{3}}{k^{2}_{p}}-\frac{{W_{1}}\beta_{1}}{\kappa^{2}}\Big{)}({\bf\it
k}^{\prime}\crossproduct{\bf\it k}^{\prime\prime})\\\
&\\!\\!-\frac{k^{2}}{k_{p}^{2}}\Big{(}W_{3}\beta_{0}-W_{0}\beta_{3}\Big{)}{\bf\it
k}^{\prime}+\frac{k^{2}}{\kappa^{2}}\Big{(}W_{1}\beta_{2}+W_{2}\beta_{1}\Big{)}{\bf\it
k}^{\prime\prime}.\\!\\!\end{split}$

The total force ${\bf\it F}={\bf\it F}_{\text{int}}+{\bf\it F}_{\text{rec}}$
exerted by an evanescent wave on a particle can be decomposed into the three
directions given by the evanescent wave propagation, decay, and lateral
directions. This force depends only on the polarisation of the wave (Stokes’
parameters), which controls the energy densities eq. 28, and on the
evanescence factor $\kappa$, since $k_{p}=\sqrt{k^{2}+\kappa^{2}}$, and is
explicitly written as follows:

$\begin{split}{\bf\it F}\dotproduct\hat{{\bf\it
k}}^{\prime}&=2\sum_{A=0}^{3}{k_{p}}W_{A}\Im\alpha_{A}+\frac{k^{2}}{k_{p}}\Big{(}W_{0}\beta_{3}-W_{3}\beta_{0}\Big{)},\\\
{\bf\it F}\dotproduct\hat{{\bf\it
k}}^{\prime\prime}&=-2\sum_{A=0}^{3}{\kappa}W_{A}\Re{\alpha}_{A}+\frac{k^{2}}{\kappa}\Big{(}W_{2}\beta_{1}+W_{1}\beta_{2}\Big{)},\\\
{\bf\it F}\\!\dotproduct\\!\big{(}\hat{{\bf\it
k}}^{\prime}\\!\crossproduct\\!\hat{{\bf\it
k}}^{\prime\prime}\big{)}\\!&=\frac{k{k_{p}}}{\kappa}\Big{(}{{W_{2}}\beta_{2}}\\!-\\!{{W_{1}}\beta_{1}}\Big{)}\\!+\\!\frac{k{\kappa}}{k_{p}}\Big{(}{{W_{3}}\beta_{3}}\\!-\\!{{W_{0}}\beta_{0}}\Big{)}.\end{split}$

The chiral force will only consist of the components that have pseudoscalar
coefficients, which (assuming a reciprocal particle, i.e., $\alpha_{2}=0$) are
$\Re\alpha_{3}=\Re\alpha_{c}$, $\Im\alpha_{3}=\Im\alpha_{c}$,
$\beta_{0}={k^{3}}\Re\left[\alpha_{\text{c}}^{*}\left(\alpha_{\text{e}}+\alpha_{\text{m}}\right)\right]/({6\pi})$
and
$\beta_{1}={k^{3}}\Re\left[\alpha_{\text{c}}^{*}\left(\alpha_{\text{e}}-\alpha_{\text{m}}\right)\right]/({6\pi})$.
Therefore, the chiral force of a reciprocal particle in an evanescent wave is:

$\begin{split}{\bf\it F}_{\text{\\!c}}\dotproduct\hat{{\bf\it
k}}^{\prime}&=2{k_{p}}W_{3}\Im\alpha_{3}-\frac{k^{2}}{k_{p}}W_{3}\beta_{0},\\\
{\bf\it F}_{\text{\\!c}}\dotproduct\hat{{\bf\it
k}}^{\prime\prime}&=-2{\kappa}W_{3}\Re{\alpha}_{3}+\frac{k^{2}}{\kappa}W_{2}\beta_{1},\\\
{\bf\it F}_{\text{\\!c}}\dotproduct\big{(}\hat{{\bf\it
k}}^{\prime}\crossproduct\hat{{\bf\it
k}}^{\prime\prime}\big{)}\\!&=-\frac{k{k_{p}}}{\kappa}{{W_{1}}\beta_{1}}-\frac{k{\kappa}}{k_{p}}{{W_{0}}\beta_{0}}.\end{split}$

From this, one can see that in order to maximise the lateral333Pointing in
direction $\hat{{\bf\it k}}^{\prime}\\!\crossproduct\\!\hat{{\bf\it
k}}^{\prime\prime}$. chiral force, one needs an evanescent wave with a linear
polarisation $\mathcal{S}_{1}=\pm\mathcal{S}_{0}$, which also forces non-
chiral lateral forces to zero and is precisely the configuration proposed by
Hayat  _et al._ [14] and depicted in fig. 1, particle (a). Note that in that
case, the transverse spin-recoil force performs the sorting. Hence, this
method will be more successful for larger particles with non-negligible recoil
force.

Figure 1: Separation of enantiomers using an evanescent wave. An illustrative
figure, forces not to scale. For the larger particle (a) we only show the
lateral recoil force, while for the lossless particle (b) and the lossy
particle (c) we show the interaction forces.

One cannot perform lateral sorting for particles that are too small to feel
recoil forces. If one wants to sort smaller particles that only feel gradient
and pressure forces (${\bf\it F}_{\text{int}}$ from eq. 29), one would have to
choose a circular polarisation $\mathcal{S}_{3}=\sigma\mathcal{S}_{0}$ (where
$\sigma=\pm 1$) and sort the enantiomers in the plane defined by the
evanescent wave. This type of sorting of small enantiomers (suitable for
chiral molecules) in an evanescent field is a novel proposal, to the best of
our knowledge. The circularly polarised evanescent wave has non-zero helicity
$W_{3}\neq 0$, but unlike a plane wave, this helicity is not uniform but
decays exponentially like the energy. This non-uniformity creates a helicity
gradient force that will attract the enantiomer with
$\sigma\Re(\alpha_{\text{c}})>0$ to the region with higher helicity and repel
the other. This is shown in fig. 1, particle (b). The magnitudes of the total
interaction force acting on the positive/negative enantiomer (the positive
enantiomer is the one with $\Re\alpha_{\text{c}}>0$) in terms of the chiral
polarisability of the positive enantiomer $\alpha_{\text{c}}$ will be
$\absolutevalue{{\bf\it F}_{\pm}}=2W_{0}f_{\pm}$, where
$f^{2}_{\pm}={k_{p}^{2}\Im(\alpha_{0}\pm\sigma\alpha_{\text{c}})^{2}+\kappa^{2}\Re(\alpha_{0}\pm\sigma\alpha_{\text{c}})^{2}}$.
For this method, typically the achiral force is stronger than the chiral
forces, and so the particles within the evanescent wave will stick to the
surface due to gradient forces. But one of the enantiomers feels this force
stronger than the other, due to the additive or subtractive chiral force which
reinforces or weakens the achiral force. This allows enantiomer separation
proposals based on the difference in net gradient forces. If the chiral force
is strong enough to overcome the achiral force, then one enantiomer would feel
attraction, and the other repulsion, from the surface. This is illustrated in
fig. 1, particle (b). Particles with non-negligible $\Im\alpha_{\text{c}}$
will also feel a chiral pressure force in the direction of wave propagation
[see fig. 1, particle (c)], which will introduce an angle $\theta$ between the
two forces given by

$\cos{\theta}=\sigma\frac{\;k_{p}^{2}\;\Im\alpha_{\text{c}}\Im\alpha_{0}+\kappa^{2}\;\Re\alpha_{\text{c}}\Re\alpha_{0}\;}{f_{+}(\kappa,\alpha_{0},\alpha_{\text{c}})f_{-}(\kappa,\alpha_{0},\alpha_{\text{c}})}.$
(34)

##  V Cylindrical nanofibre

Intuitively, the modal fields outside a nanofibre are equivalent to ‘radial
evanescent waves’ and we can expect similar behaviour to the previous section,
but with the added richness of having many modes. The eigenmodes of
cylindrical waveguides are well studied, since they can be calculated
analytically: a particularly concise formulation is provided in Picardi  _et
al._ [45]. A dielectric nanofibre aligned with the $z$-axis can be modelled as
a nonmagnetic medium of radius $r_{0}$, characterized by its permittivity and
permeability

$\varepsilon=\left\\{\begin{array}[]{ll}\varepsilon_{1}\varepsilon_{0},&\text{
for }\rho<r_{0}\\\ \varepsilon_{2}\varepsilon_{0},&\text{ for
}\rho>r_{0}\end{array}\quad\text{ and }\quad\mu=\mu_{0}\right..$

The solution to Maxwell’s equations yields electric and magnetic fields that
can exist in and around the dielectric fibre. As Picardi  _et al._ [45]
showed, it is convenient to choose the spin basis $\hat{{\bf\it e}}_{\pm
1}=(\hat{{\bf\it x}}\pm\text{i}\hat{{\bf\it y}})/\sqrt{2}$ and $\hat{{\bf\it
e}}_{0}=\hat{{\bf\it z}}$ to represent the solution. On this basis, any vector
can be expressed as ${\bf\it F}=F_{+1}\hat{{\bf\it e}}_{+1}+F_{0}\hat{{\bf\it
e}}_{0}+F_{-1}\hat{{\bf\it e}}_{-1}$, where the components $F_{s}$ are _spin-
weighted functions_.444spin-weighted functions have wide use in mathematical
physics, particle physics and cosmology. In particular, vector spherical
harmonics and Bessel functions can be generalised and simplified by replacing
them with their spin-weighted counterparts [46]. The advantage of this
representation is that rotation around the $z$-axis by an arbitrary angle
$\theta$ involves just a phase shift in the spin basis by $s\theta$,
$\hat{{\bf\it e}}_{s}\mapsto\text{e}^{\text{i}s\theta}\hat{{\bf\it e}}_{s}$,
and a phase shift by $-s\theta$ in the vector components,
$F_{s}\mapsto\text{e}^{-\text{i}s\theta}F_{s}$ [46]. Picardi  _et al._ [45]
also showed that the total longitudinal angular momentum of the guided modes
in the cylindrical fibre is quantised, that is, ${J}_{z}=\ell W_{0}/\omega$,
where $\ell$ is an integer. The electric field, ${\bf\it E}$, of a single mode
with a definite azimuthal number $\ell$ can be represented by

$\begin{split}\sqrt{\varepsilon}{E}_{\pm 1}({\bf\it
r})&=\frac{kB\mp\text{i}k_{z}A}{\kappa}\operatorname{Z}_{\ell\mp
1}(\kappa\rho)e^{\text{i}(\ell\mp 1)\varphi+\text{i}k_{z}z},\\\
\sqrt{\varepsilon}{E}_{0}({\bf\it
r})&=\sqrt{2}A\operatorname{Z}_{\ell}(\kappa\rho)e^{\text{i}\ell\varphi+\text{i}k_{z}z},\end{split}$
(35)

where $k$ is the wave number in the medium, $k_{z}>k_{0}$ is the mode
propagation constant ($k_{0}$ is the wave number in vacuum),
$\kappa=\sqrt{k^{2}-k_{z}^{2}}$ is the radial wave number. Function
$\operatorname{Z}_{\alpha}(x)$ is defined piecewise as the Bessel/Hankel
function of the first kind inside/outside of the fibre, respectively. The
corresponding magnetic solution, ${\bf\it H}$, can be obtained via the duality
transformation:

$\sqrt{\varepsilon}{\bf\it E}\mapsto\sqrt{\mu}{\bf\it
H},\quad\quad(A,B)\mapsto(B,-A)\,.$ (36)

The dispersion relationship for the propagation constant $k_{z}$ given
parameters ($\omega,r_{0},\varepsilon_{1},\varepsilon_{2},\ell$) is found from
the transcendental characteristic equation (see fig. 2), whereas the complex
constants $A$, $B$ both inside and outside the fibre are determined from the
boundary conditions at $\rho=r_{0}$ (see appendix D). Notice that for a mode
of a given $\ell$ (denoted by each colour in fig. 2),

Figure 2: Numerically calculated dispersion relationships for a dielectric
fibre made of silicon nitride, $\varepsilon_{1}=4.3$, immersed in water,
$\varepsilon_{2}=1.7$ (both values are at
$20\text{\,}\mathrm{\SIUnitSymbolCelsius}$ valid for $\lambda_{0}$ from
$750\text{\,}\mathrm{nm}1750\text{\,}\mathrm{nm}$ [47]). We plot it for the
angular momentum eigenmodes $\ell\in\\{0,\pm 1,\pm 2\\}$ (in blue, orange, and
green). Each $\ell$ has an additional mode number $n$. We plot values
$n\in\\{0,1,2\\}$ (using solid, dashed, and dot-dashed lines). For $\ell\neq
0$ the mode number $n$ is related to the spin number of the dominant mode
$\sigma=(-1)^{n}$ and the number of radial nodes in the energy density
$n_{r}=n/2+(\sigma-1)/4$. The orbital angular momentum of the dominant mode is
then $\ell-\sigma$. The dashed black line represents the optimal radius (see
eq. 56), in this case $r_{0}=$0.18$\,\lambda_{0}$, this value is also used in
figs. 3 and 5.

there is more than one dispersion relationship (solid, dashed, and dash-dotted
lines). We will label these additional modes with a non-negative integer $n$,
which labels the modes from left to right. The solution for $\ell=1$, $n=0$
corresponds to a fundamental circularly polarised eigenmode (CP), whose fields
can be seen in the first row of fig. 3. It turns out that the spin-weighted
components $F_{s}$ (either $\sqrt{\varepsilon}E_{s}$ or $\sqrt{\mu}H_{s}$ eq.
35), although not being individually physical, are eigenfunctions of definite
longitudinal momenta operators with eigenvalues $p_{z}=\hbar k_{z}$,
$J_{z}=\hbar\ell$, $S_{z}=\hbar s$ and $L_{z}=J_{z}-S_{z}=\hbar(\ell-s)$. The
momentum operator acts on these eigenfunctions as

$\hat{{\bf\it p}}F_{s}=-\text{i}\hbar\gradient
F_{s}=\hbar\Bigg{(}{k_{z}}\hat{{\bf\it
z}}+\frac{\ell-s}{\rho}\hat{{\bf\it\varphi}}-\text{i}\kappa\frac{\operatorname{Z}_{\ell-s}^{\prime}}{\operatorname{Z}_{\ell-s}}\hat{{\bf\it\rho}}\Bigg{)}F_{s},$
(37)

where prime represents the derivative of the function with respect to its
argument. Expression in parentheses is a complex wave vector for each
component $F_{s}$

${\bf\it k}_{s}={\bf\it k}_{s}^{\prime}+\text{i}{\bf\it
k}_{s}^{\prime\prime}={k_{z}}\hat{{\bf\it
z}}+\frac{\ell-s}{\rho}\hat{{\bf\it\varphi}}-\text{i}\kappa\frac{\operatorname{Z}_{\ell-s}^{\prime}}{\operatorname{Z}_{\ell-s}}\hat{{\bf\it\rho}}\,,$
(38)

where $\kappa{\operatorname{Z}_{\ell-s}^{\prime}}/{\operatorname{Z}_{\ell-s}}$
is real both inside and outside the fibre. Notice that the simple longitudinal
and azimuthal components of the momentum come from the fact that $F_{s}$ are
eigenfunctions of $p_{z}$ and $L_{z}$. In particular, the longitudinal orbital
angular momentum operator gives

$\hat{L}_{z}F_{s}=\hat{{\bf\it z}}\dotproduct(\hat{{\bf\it
r}}\crossproduct\hat{{\bf\it
p}})F_{s}=-\text{i}\hbar\partialderivative{\varphi}F_{s}=\hbar(\ell-s)F_{s}\,,$
(39)

which means that $\hat{{\bf\it z}}\dotproduct({\bf\it r}\crossproduct{\bf\it
k}_{s})=\ell-s$, which explains the azimuthal component of eq. 38.555Recall
that in cylindrical coordinates ${\bf\it r}=z\hat{{\bf\it
z}}+\rho\hat{{\bf\it\rho}}$.

The orthonormality of the spin basis $\hat{{\bf\it e}}_{s}$ ensures that the
scalar product of any two vectors ${\bf\it F}$ and ${\bf\it G}$ is

${\bf\it F}^{*}\\!\dotproduct{\bf\it
G}=F_{+1}^{*}G^{\vphantom{*}}_{+1}+F_{0}^{*}G^{\vphantom{*}}_{0}+F_{-1}^{*}G^{\vphantom{*}}_{-1}\,,$
(40)

which means that an eigenmode ${\bf\it F}$ (either $\sqrt{\varepsilon}{\bf\it
E}$ or $\sqrt{\mu}{\bf\it H}$) given by $n$ and $\ell$ can be thought of as a
linear combination of independent eigenfunctions $F_{s}\hat{{\bf\it e}}_{s}$
of definite spin

$\hat{S}_{z}(F_{s}\hat{{\bf\it e}}_{s})=\text{i}\hbar\hat{{\bf\it
z}}\crossproduct(F_{s}\hat{{\bf\it e}}_{s})=\hbar s(F_{s}\hat{{\bf\it
e}}_{s})\,.$ (41)

Equation 40 ensures that the energy densities of any eigenmode can be written
as a sum

$W\\!_{A}=\sum_{s=-1}^{1}W\\!_{A}^{s}=W\\!_{A}^{-1}\\!+W\\!_{A}^{0}+W\\!_{A}^{+1},$
(42)

where $W_{A}^{s}$ are energy densities calculated for the individual vector
components $F_{s}\hat{{\bf\it e}}_{s}$ if all other components were zero. To
calculate these, one can define two real and one complex coefficient quadratic
in scalar $A$ and pseudoscalar $B$ such that they have the same symmetries as
$W_{0}$, $W_{3}$ and $W_{\text{c}}$:

$\displaystyle w_{\pm}$
$\displaystyle=\frac{1}{4}\pqty{{\absolutevalue{A}^{2}\pm\absolutevalue{B}^{2}}}\,,\quad$
$\displaystyle w_{\text{c}}$ $\displaystyle=\frac{\text{i}}{2}A^{*}B.$

One can then write the energy densities [eq. 19] associated with each spin-
weighted component. The transverse spin ($s=0$) component has

$\displaystyle W^{0}_{0}$
$\displaystyle=2w_{+}\absolutevalue{\operatorname{Z}_{\ell}}^{2}\,,\quad-$
$\displaystyle W^{0}_{2}$
$\displaystyle=2\Im(w_{\text{c}})\absolutevalue{\operatorname{Z}_{\ell}}^{2}\,,$
(43) $\displaystyle W^{0}_{1}$
$\displaystyle=2w_{-}\absolutevalue{\operatorname{Z}_{\ell}}^{2}\,,\quad$
$\displaystyle W^{0}_{3}$
$\displaystyle=2\Re(w_{\text{c}})\absolutevalue{\operatorname{Z}_{\ell}}^{2}\,,$

while the remaining components will be mixed

$\displaystyle\\!\\!W^{\pm 1}_{0}$
$\displaystyle=\bqty{\frac{k_{z}^{2}+k^{2}}{\absolutevalue{\kappa}^{2}}W_{0}^{0}\pm\frac{2kk_{z}}{\absolutevalue{\kappa}^{2}}W_{3}^{0}}\frac{\absolutevalue{\operatorname{Z}_{\ell\mp
1}}^{2}}{2\absolutevalue{\operatorname{Z}_{\ell}}^{2}},\\!\\!$ (44)
$\displaystyle W^{\pm 1}_{1}$
$\displaystyle=\bqty{{\frac{k_{z}^{2}-k^{2}}{\absolutevalue{\kappa}^{2}}}W^{0}_{1}}\frac{\absolutevalue{\operatorname{Z}_{\ell\mp
1}}^{2}}{2\absolutevalue{\operatorname{Z}_{\ell}}^{2}},\\!\\!$ $\displaystyle
W^{\pm 1}_{2}$
$\displaystyle=\bqty{\frac{k_{z}^{2}-k^{2}}{\absolutevalue{\kappa}^{2}}W^{0}_{2}}\frac{\absolutevalue{\operatorname{Z}_{\ell\mp
1}}^{2}}{2\absolutevalue{\operatorname{Z}_{\ell}}^{2}},$
$\displaystyle\\!\\!W^{\pm 1}_{3}$
$\displaystyle=\bqty{\frac{k_{z}^{2}+k^{2}}{\absolutevalue{\kappa}^{2}}W_{3}^{0}\mp\frac{2kk_{z}}{\absolutevalue{\kappa}^{2}}W_{0}^{0}}\frac{\absolutevalue{\operatorname{Z}_{\ell\mp
1}}^{2}}{2\absolutevalue{\operatorname{Z}_{\ell}}^{2}},$

where $k_{z}$ changes sign under a parity transformation. For a circular
polarisation, from the symmetries, one would expect that $w_{-}$ and $\Im
w_{\text{c}}$ will vanish, but that is not the case for a dielectric fibre due
to the fact that there is electromagnetic asymmetry $A\neq\pm\text{i}B$. This
asymmetry comes from the refractive index being due only to the permittivity
$\varepsilon_{1}$. If instead we have a fibre made of material that has
$\varepsilon_{1}=\mu_{1}=n^{2}$, then $w_{-}$ and $\Im w_{\text{c}}$ will
vanish.

By decomposing the mode into spin-weighted functions, we may write the energy
density gradients identically to a sum of three evanescent waves with
$\gradient{W^{s}_{A}}=-2{\bf\it k}_{s}^{\prime\prime}W^{s}_{A}$, each having
wavevector given by eq. 38 and energy densities given by eqs. 43 and 44:

$\displaystyle\\!\\!\gradient W_{A}$
$\displaystyle=\sum_{s=-1}^{1}\\!\\!2{W^{s}_{A}}\kappa\frac{\operatorname{Z}_{\ell-s}^{\prime}}{\operatorname{Z}_{\ell-s}}\hat{{\bf\it\rho}}.$
(45)

This is valid for all energy density quantities $A=0,1,2,3$ defined in eq. 19.
Remarkably, the same holds for canonical momenta, ${\bf\it p}^{s}_{A}={\bf\it
k}_{s}^{\prime}{W\\!\\!_{A}}/\omega$, leading to

$\displaystyle{\bf\it
p}_{A}=\sum_{s=-1}^{1}\\!\\!\frac{W^{s}_{A}}{\omega}\bqty{{k_{z}}\hat{{\bf\it
z}}+\frac{\ell-s}{\rho}\hat{{\bf\it\varphi}}}.$ (46)

Notice that if we introduced factors of $\hbar$ in both the numerator and
denominator in the above sums, then the canonical momenta (gradients) would
look like a weighted average of real (imaginary) momentum carried by each
spin-weighted function

${\bf\it p}_{A}=\sum_{s=-1}^{1}\\!\\!n^{s}_{A}\,\hbar{\bf\it
k}^{\prime}_{s}\,,\quad\gradient{W\\!\\!_{A}}/(2\omega)=-\sum_{s=-1}^{1}\\!\\!n^{s}_{A}\,\hbar{\bf\it
k}^{\prime\prime}_{s}\,,$

where weights $n^{s}_{A}={W^{s}_{A}}/{(\hbar\omega)}$ for $A=0$ can be
intuitively thought of as the number density of photons with the momentum
given by eq. 37. For $A=3$, it would be the number density of these photons
that break parity, duality for $A=1$, or both and time reversal for $A=2$.
Notice that for $A\neq 0$, these number densities can be positive or negative,
representing whether the photons are more right/left-handed,
electric/magnetic, etc. Using the same logic, eq. 42 is a weighted average of
the energies $\hbar\omega$ carried by each spin component.

Fundamental circular polarisation (CP) mode of the fibre $\ell=1$, $n=0$

Figure 3: Circular force density basis for optical force around a dielectric
fibre guiding total angular momentum eigenmode with $\ell=1$. Arrow maps
represent transverse components, while colour maps represent longitudinal
components of vector quantities plotted in a cross-sectional view of the
fibre. This force basis plot is independent of wavelength. It depends on the
material of the fibre SiN ($\varepsilon_{1}=4.3$), the surrounding medium
water ($\varepsilon_{2}=1.7$) and the radius chosen to be
$r_{0}=$0.18$\,\lambda_{0}$.

The interaction force for a pure fibre eigenmode (later we study mode
combinations) will then be a sum of forces identical to three evanescent waves
eq. 29 with wavevector given by eq. 38 for each of the spin components

${{\bf\it
F}}_{\text{int}}=\sum_{A=0}^{3}\sum_{s=-1}^{1}2W^{s}_{A}\quantity(\Im\alpha_{A}{\bf\it
k}_{s}^{\prime}-\Re{\alpha}_{A}{\bf\it k}_{s}^{\prime\prime})\,.$ (47)

Unfortunately, this is where the similarities with the evanescent waves end.
If a particle is big enough (with respect to the wavelength), it will also
feel the recoil force, which depends on the fluxes of power and helicity.
Those are related to the spin operator, but we can only repeat the trick we
used in eq. 37 for the $z$ component. Therefore, the spin densities have to be
obtained from the definition eq. 21. The total spin density will be

$\\!\\!{\bf\it S}_{0}=\sum_{s=-1}^{1}\frac{W_{0}^{s}}{\omega}s\hat{{\bf\it
z}}-2\quantity(\frac{W_{0}^{0}}{\omega}\frac{k_{z}}{\kappa}\frac{Z_{\ell}^{\prime}}{Z_{\ell}}-\frac{W_{3}^{0}}{\omega}\frac{k\ell}{\kappa^{2}\rho})\hat{{\bf\it\varphi}},\\!\\!$
(48)

where we can see that the longitudinal component is again just the weighted
average of the eigenvalues of each spin component. Notice that there is no
longitudinal contribution from the eigenfunction with $s=0$. Instead, this
function will lead to a transverse spin, which depends only on $W_{A}^{0}$ and
${\bf\it k}_{0}$. Notice that when we talk about longitudinal and transverse
spin in this scenario, we mean with respect to the $z$-direction along the
fibre. The spin quantity related to the real Poynting vector $\omega{\bf\it
S}_{3}=\Re{\bf\it\varPi}/c$ will be very similar:

$\\!\\!{\bf\it S}_{3}=\sum_{s=-1}^{1}\frac{W_{3}^{s}}{\omega}s\hat{{\bf\it
z}}-2\quantity(\frac{W_{3}^{0}}{\omega}\frac{k_{z}}{\kappa}\frac{Z_{\ell}^{\prime}}{Z_{\ell}}+\frac{W_{0}^{0}}{\omega}\frac{k\ell}{\kappa^{2}\rho})\hat{{\bf\it\varphi}},\\!\\!$
(49)

and so will be the spin asymmetry vector

$\\!\\!{\bf\it S}_{1}=\sum_{s=-1}^{1}\frac{W_{1}^{s}}{\omega}s\hat{{\bf\it
z}}-2\quantity(\frac{W_{1}^{0}}{\omega}\frac{k_{z}}{\kappa}\frac{Z_{\ell}^{\prime}}{Z_{\ell}}-\frac{W_{2}^{0}}{\omega}\frac{k\ell}{\kappa^{2}\rho})\hat{{\bf\it\varphi}}.\\!\\!$
(50)

One can see that all of these quantities have only longitudinal and azimuthal
components; the only exception to this will be the spin quantity related to
the imaginary Poynting vector $\omega{\bf\it S}_{2}=-\Im{\bf\it\varPi}/c$

$\\!\\!{\bf\it S}_{2}=\sum_{s=-1}^{1}\frac{W_{2}^{s}}{\omega}s\hat{{\bf\it
z}}+\frac{2}{\kappa}\frac{Z_{\ell}^{\prime}}{Z_{\ell}}\quantity(k_{z}\frac{W_{2}^{0}}{\omega}\hat{{\bf\it\varphi}}-k\frac{W_{1}^{0}}{\omega}\hat{{\bf\it\rho}}),\\!\\!$
(51)

which also has a radial contribution. The analytical form of the total force
${\bf\it F}={\bf\it F}_{\text{int}}+{\bf\it F}_{\text{rec}}$ on a particle
near the fibre is then, in cylindrical basis:

$\begin{split}\\!\\!F_{\rho}=&\sum_{A,s}2\kappa\frac{\operatorname{Z}_{\ell-s}^{\prime}}{\operatorname{Z}_{\ell-s}}\Re\alpha_{A}{W^{s}_{A}}-\frac{2k^{2}}{\kappa}\frac{Z_{\ell}^{\prime}}{Z_{\ell}}\beta_{2}W_{1}^{0},\\!\\!\\\
\\!\\!F_{\varphi}=&\sum_{A,s}2\frac{\ell-s}{\rho}\Im\alpha_{A}{W^{s}_{A}}+\frac{2k^{2}\ell}{\kappa^{2}\rho}\pqty{\beta_{0}{W_{3}^{0}}\\!\\!-\\!\\!\beta_{3}{W_{0}^{0}}\\!+\\!\beta_{2}{W_{2}^{0}}}\\!\\!\\!\\!\\\
&-\frac{2k_{z}k}{\kappa}\frac{Z_{\ell}^{\prime}}{Z_{\ell}}\pqty{\beta_{0}W_{0}^{0}-\beta_{2}W_{2}^{0}+\beta_{3}W_{3}^{0}+\beta_{1}W_{1}^{0}},\\!\\!\\!\\!\\\
\\!\\!F_{z}=&\sum_{A,s}\pqty{2k_{z}\Im\alpha_{A}+{ks}\beta_{A}}{W^{s}_{A}}.\\!\\!\end{split}$

Note that the chiral force (assuming reciprocal particles $\alpha_{2}=0$) are
only the components which feature $\Re(\alpha_{3})$, $\Im(\alpha_{3})$,
$\beta_{0}$ and $\beta_{1}$, i.e.,

$\begin{split}F_{\rho}={}&2\sum_{s}\kappa\frac{\operatorname{Z}_{\ell-s}^{\prime}}{\operatorname{Z}_{\ell-s}}\Re\alpha_{3}{W^{s}_{3}},\\\
F_{\varphi}={}&2\sum_{s}\frac{\ell-s}{\rho}\Im\alpha_{3}{W^{s}_{3}}+\frac{2k^{2}\ell}{\kappa^{2}\rho}\pqty{\beta_{0}{W_{3}^{0}}+\beta_{1}{W_{2}^{0}}}\\\
&-\frac{2k_{z}k}{\kappa}\frac{Z_{\ell}^{\prime}}{Z_{\ell}}\pqty{\beta_{0}W_{0}^{0}+\beta_{1}W_{1}^{0}},\\\
F_{z}={}&2k_{z}\Im\alpha_{3}{W_{3}}+\sum_{s}k{s}(\beta_{0}{W^{s}_{0}}+\beta_{1}{W^{s}_{1}}).\end{split}$

For a visual representation of the different force terms, the total force can
still be expressed using the concept of a basis ${\bf\it
F}=\sum_{i}\lambda_{i}{\bf\it V}\\!_{i}$ as introduced in eq. 25. It is then
very instructive to plot the twelve different basis vector fields ${\bf\it
V}\\!_{i}$ as given in table 1. These are the twelve plots shown in the bottom
rows of fig. 3, and they are entirely determined by the fields of the
circularly polarised mode $(\ell=1,n=0)$ while being independent of the
particle. To supplement this figure, we also include a Python script [48] with
interactive Jupyter notebooks capable of reproducing these twelve plots for
any material parameters and for any mode.

Examining these basis vector fields ${\bf\it V}\\!_{i}$, one can deduce that
small particles (with negligible recoil, and therefore ignoring the last row)
near the fibre will experience a total energy gradient force $\gradient W_{0}$
attracting all particles towards the surface of the fibre, added to a helicity
gradient force $\gradient W_{3}$ which attracts one enantiomer towards the
fibre, while repelling the other one away from it, exactly as was the case of
an evanescent wave described earlier. This is due to the circular polarisation
of the mode having a large helicity density in the fibre, which radially
decays away, creating a gradient in the helicity. Added to these gradient
forces are possible longitudinal ($z$-directed) pressure forces, as shown in
${\bf\it p}_{0}$ and ${\bf\it p}_{3}$ in fig. 3 which are relevant for
particles with $\Im(\alpha_{\text{e}}+\alpha_{\text{m}})\neq 0$ and
$\Im\alpha_{\text{c}}\neq 0$, respectively. This can be easily deduced by
looking at the coefficients $\lambda_{i}$ that multiply the relevant basis
functions in table 1. These gradient and pressure forces are completely
equivalent to the in-plane separation by the evanescent wave (exactly as in
fig. 1 particles b and c), with the only difference being the geometry of the
surface to which the particles will stick. For example, a particle with radius
$300\text{\,}\mathrm{n}\mathrm{m}$, relative permittivity
$\varepsilon_{p}=$2.5$$, permeability $\mu_{p}=$1$$ and chiral parameter
$\absolutevalue{\varkappa}=$0.5$$ (considered in Li  _et al._ [49]) would feel
a chiral force of $205\text{\,}\mathrm{fN}\text{\,}{\mathrm{mW}}^{-1}$ near a
silicon nitride fibre that guides the fundamental circular mode with
wavelength of $1310\text{\,}\mathrm{nm}$ and radius $240\text{\,}\mathrm{nm}$.
In this case, the chiral forces would be strong enough to separate enantiomers
in order of seconds using just $\mathrm{mW}$ of power (taking the same
assumptions as in Martinez-Romeu  _et al._ [50]). Therefore, this would be a
viable configuration for separating small enantiomers. The provided code [48]
can calculate and plot the force fields for these and any other particle
parameters. For bigger particles, the recoil forces become stronger. Looking
at the last row in fig. 3, corresponding to the recoil forces, we can see
there is a strong azimuthal component in both the achiral recoil force
$k\omega{\bf\it S}_{3}=k\Re{{\bf\it\varPi}}/c$ and the chiral spin recoil
force $k\omega{\bf\it S}_{0}=k\omega({\bf\it S}_{\text{e}}+{\bf\it
S}_{\text{m}})$ which would suggest a possible enantiomer sorting mechanism in
this circular mode. In practice, however, the particle-dependent
polarizabilities ($\lambda_{i}$) multiplying the achiral basis are much
stronger than the chiral one, and so the achiral response dominates, making
this circular mode not well suited for recoil force sorting. To achieve strong
chiral recoil forces that are not overtaken by achiral forces, one must rely
on mode combinations to synthesise linearly polarised modes, as shown later.

Figure 4: Maximum force densities outside the fibre normalised by the power in
units of $$\mathrm{fN}\text{\,}{\mathrm{mW}}^{-1}$\lambda_{0}^{-3}$ for the
fundamental circular polarisation modes with $\ell=1$. Solid lines represent
chiral forces, while dashed lines represent achiral forces. Note that these
curves are invariant in these units under a change of wavelength or power. The
black dashed line represents the optimal radius of the fibre which maximises
the chiral gradient force (for CP) and the spin recoil (for LP) (see eq. 56).

Fundamental quasi transverse electric (TE) mode of the fibre $\ell=1$, $n=0$

Figure 5: Linear force density basis for optical force around a dielectric
fibre. Arrow maps represent transverse components, while colour maps represent
longitudinal components of vector quantities plotted in a cross-sectional view
of the fibre. This force basis plot is independent of wavelength. It depends
on the material of the fibre SiN ($\varepsilon_{1}=4.3$), the surrounding
medium water ($\varepsilon_{2}=1.7$) and the radius chosen to be
$r_{0}=$0.18$\,\lambda_{0}$.

The cylindrical fibre can support linear combinations of eigenmodes ${\bf\it
E}_{\ell,n}$ eq. 35 with complex coefficients $a_{\ell,n}$

${\bf\it E}=\sum_{\ell}\sum_{n}a_{\ell,n}{\bf\it E}_{\ell,n}\,,$ (52)

and similarly for the magnetic field. Modes with different $\ell$ and $n$,
unlike the mode components with different $s$, are not linear in the quadratic
observables. Therefore, energy densities will not be simply the sum of
individual energy densities.

$W_{A}\neq\sum_{\ell}\sum_{n}\sum_{s}W^{\ell,n,s}_{A}\,.$

Instead, there will also be interference cross terms. For example, take the
electric energy density

$W_{\text{e}}=\frac{\varepsilon}{4}\sum_{\ell,\ell^{\prime}}\sum_{n,n^{\prime}}a_{\ell^{\prime},n^{\prime}}^{*}a_{\ell,n}{\bf\it
E}^{*}_{\ell^{\prime},n^{\prime}}\dotproduct{\bf\it E}_{\ell,n}\,.$

where the summand is guaranteed to be real only if $\ell^{\prime}=\ell$ and
$n^{\prime}=n$, and in that case it would be the electric energy density of
that particular mode $W_{\text{e}}^{\ell,n}$. Other terms will be generally
complex, and we only get a real energy density once we perform the sum. Let
us, however, label these, in general complex, interference terms
$W_{A}^{\ell,\ell^{\prime},n,n^{\prime},s}$, so that, for instance, if
$A=\text{e}$ (electric energy) we have

$W^{\ell,\ell^{\prime},n,n^{\prime},s}_{\text{e}}=\frac{\varepsilon}{4}a_{\ell^{\prime},n^{\prime}}^{*}a_{\ell,n}{E}^{*}_{\ell^{\prime},n^{\prime},s}{E}_{\ell,n,s}\,,$

where $s$ again labels the component in the spin basis. Similarly, we can
define them for $A=0,1,2,3$. Canonical momenta and gradients of energy
densities will be, in this most general case, as follows:

$\begin{split}{\bf\it
p}_{A}&=\sum_{\ell,n,s}\Re(n^{\ell,n,s}_{A}\,\hbar{\bf\it k}_{\ell,n,s})\,,\\\
{\gradient{W_{A}}}/{(2\omega)}&=-\sum_{\ell,n,s}\Im(n^{\ell,n,s}_{A}\,\hbar{\bf\it
k}_{\ell,n,s})\,,\end{split}$ (53)

where the complex weights in these weighted averages

$n^{\ell,n,s}_{A}=\sum_{\ell^{\prime},n^{\prime}}W_{A}^{\ell,\ell^{\prime},n,n^{\prime},s}/(\hbar\omega)$
(54)

can be thought of as modified number densities of photons with _complex_
momentum $\hbar{\bf\it k}_{\ell,n,s}$ of the species $A$, but taking into
account the interference with all the other eigenmodes. This quantity is, in
general, complex and its phase will redistribute (_rotate_) between how much
of ${\bf\it k}_{\ell,n,s}^{\prime}$ and ${\bf\it k}_{\ell,n,s}^{\prime\prime}$
contributes to the gradient and pressure forces. In theory, one can use these
interferences to design gradient and pressure forces of arbitrary directions
and amplitudes by controlling coefficients $a_{\ell,n}$ in eq. 52. However, in
practice, we don’t have such fine experimental control over these
coefficients.

An example of an experimentally easy-to-obtain mode superposition is that of a
linear polarisation. Two orthogonal linear polarisation modes are usually
called the quasi-transverse electric (TE) and quasi-transverse magnetic (TM)
modes. These modes are obtained in general as the superpositions

${\bf\it E}_{\ell,n}^{\text{TM}}=({\bf\it E}_{\ell,n}+{\bf\it
E}_{-\ell,n})\,,\quad{\bf\it E}_{\ell,n}^{\text{TE}}=\text{i}({\bf\it
E}_{\ell,n}-{\bf\it E}_{-\ell,n}).$ (55)

The fields of the linearly polarised TE mode with $\ell=1$, $n=0$ can be seen
in the top row of fig. 5. In this case, the two modes TE and TM are degenerate
in the sense that their fields are identical, just rotated by an angle
$\pi/2$, and the same is true for all the force densities.

To visualise the possible forces on this TE mode, we once again plot the
twelve different basis vector fields ${\bf\it V}\\!_{i}$ from table 1, shown
in fig. 5. In this linear polarisation, the gradient and pressure forces are
clearly dominated by the non-chiral total energy gradient $\gradient W_{0}$
and momentum ${\bf\it p}_{0}$, hence this mode is not well suited to smaller
molecules in which the gradient and pressure forces dominate. Instead, for
larger particles on which the recoil forces dominate, this linearly polarised
mode shows more promise. Indeed, looking at the recoil terms (last row in fig.
5) one can see an achiral Poynting recoil $k\omega{\bf\it
S}_{3}=k\Re{{\bf\it\varPi}}/c$ that is mostly longitudinal ($z$-directed),
while there is a strong in-plane azimuthal spin angular momentum recoil force
$k\omega{\bf\it S}_{0}=k\omega({\bf\it S}_{\text{e}}+{\bf\it S}_{\text{m}})$
pointing around the fibre. This means that opposite enantiomers will be pushed
in opposite directions around the fibre, resulting in a potential lateral
sorting mechanism. This is caused by the transverse spin, and is exactly
equivalent to the lateral sorting in linearly polarised evanescent waves from
the previous section. Interestingly, this force competes with an achiral force
proportional to the flux of reactive power $-k\omega{\bf\it
S}_{2}=k\Im{\bf\it\varPi}/c$ which depends on the coefficient
$\beta_{2}\sim\Im(\alpha_{\text{e}}^{*}\alpha_{\text{m}})$ and which is
present for any particle that is breaking dual symmetry (i.e. a dominant
electric or magnetic response). For the same particle as before
($a=$300\text{\,}\mathrm{n}\mathrm{m}$$, $\varepsilon_{p}=$2.5$$,
$\mu_{p}=$1$$, and $\absolutevalue{\varkappa}=$0.5$$) and linear polarisation
with $1310\text{\,}\mathrm{nm}$ in a silicon nitride fibre with
$r_{0}=$240\text{\,}\mathrm{nm}$$ the maximal chiral force on the particle
would be $218\text{\,}\mathrm{fN}\text{\,}{\mathrm{mW}}^{-1}$, which would
again make it possible to separate these particles in order of seconds.

Finally, it is interesting to study the effect that the fibre radius has on
the different force terms. To this end, fig. 4 plots the maximum magnitude of
each of the twelve force term basis ${\bf\it V}\\!_{i}$ in both the circularly
polarised and linearly polarised fibre modes as a function of normalised
radius. One can see that there is clearly an optimal fibre radius-to-
wavelength ratio. This optimum is the sweet spot between the radius being too
large (hence mode not interacting with the particles outside) and too small
(hence mode fields very spread outside, with low gradients). In the limit
where the fibre is in a medium for which $n_{2}=0$ this radius coincides with
the point at which the mode with $\ell=0$ starts. The condition for a single-
mode fibre operation is
$k_{0}r_{0}\sqrt{n_{1}^{2}-n_{2}^{2}}<\mathrm{j}_{0,1}$[51], where
$\mathrm{j}_{0,1}$ is the first root of a Bessel function
$\operatorname{J}_{0}$ with the approximate value of $2.40483$. While the
radius at which the fibre can support additional modes depends on $n_{2}$, the
optimal radius does not, leading to a simple expression for the optimal radius
to wavelength ratio for any value of $n_{2}$:

$\frac{r_{0}}{\lambda_{0}}=\frac{\mathrm{j}_{0,1}}{2\pi n_{1}},$ (56)

which only depends on the wavelength and the refractive index of the material
of the fibre.

##  VI Conclusions

The total optical force acting on a general dipolar particle is a relatively
large analytical expression that can be written in many alternative but
equivalent ways. We provided a clear view of the different alternatives used
in the literature and introduced our own alternative that classifies forces
based on the symmetries broken by the particle and the fields. The chiral
forces that separate enantiomers rely on particles and fields that break
parity—but other forces can separate, for instance, electric from magnetic
particles, which break duality symmetry. The concept of a force basis, relying
on twelve force fields, each with its own particle-dependent coefficient, was
introduced—and their manifestation in evanescent waves and nanofibre modes was
exemplified. We also developed very concise analytical expressions for the
optical forces in such modes—including the case of arbitrary nanofibre modes
and their combination. These analytical expressions suggest that the
separation of bigger enantiomers, where the recoil force dominates, should
rely on transverse-spin-based lateral forces requiring linearly polarised
modes, while separation of smaller (molecule-sized) enantiomers should rely on
gradient and pressure forces—so the use of circularly polarised modes provides
a helicity gradient to attract or repel opposite enantiomers towards or away
from the nanofibre. An insight that seems to be also true in the case of a
rectangular waveguide [50]. An optimal radius was found for maximising forces
in nanofibres, which depends only on their material and wavelength. Finally,
we provide an interactive Python script [48] that can reproduce all figs. 3, 5
and 4 for an arbitrary choice of parameters and modes. We hope that our
theoretical work provides guidance and clarity for the design of future
experimental attempts at optical separation of chiral enantiomers near
waveguides, which is of enormous practical importance in the pharmaceutical
domain.

###### Acknowledgements.

We would like to acknowledge the financial support from the European
Innovation Council (HORIZON-EIC) Pathfinder CHIRALFORCE 101046961.

##  Appendix A Dynamic polarisabilities of bi-isotropic dipolar particle

For spherical particles of radius $a$, and relative material properties
($\varepsilon_{p}$, $\mu_{p}$, $\varkappa$), where $\varkappa$ is the material
chirality, in a non-chiral background medium with relative permittivity and
permeability ($\varepsilon_{m}$, $\mu_{m}$), one can use the Clausius-Mossotti
expressions for static polarisabilities

$\begin{split}\alpha_{\text{0e}}&=4\pi
a^{3}\frac{\left(\varepsilon_{p}-\varepsilon_{m}\right)\left(\mu_{p}+2\mu_{m}\right)-\varkappa^{2}}{\left(\varepsilon_{p}+2\varepsilon_{m}\right)\left(\mu_{p}+2\mu_{m}\right)-\varkappa^{2}}\\\
\alpha_{\text{0m}}&=4\pi
a^{3}\frac{\left(\varepsilon_{p}+2\varepsilon_{m}\right)\left(\mu_{p}-\mu_{m}\right)-\varkappa^{2}}{\left(\varepsilon_{p}+2\varepsilon_{m}\right)\left(\mu_{p}+2\mu_{m}\right)-\varkappa^{2}}\\\
\alpha_{\text{0c}}&=12\pi
a^{3}\frac{\varkappa}{\left(\varepsilon_{p}+2\varepsilon_{m}\right)\left(\mu_{p}+2\mu_{m}\right)-\varkappa^{2}}\end{split}$
(57)

As shown by Sersic  _et al._ [27], in order for the polarisabilities to
satisfy energy conservation, one has to apply the tensor radiative correction
by adding radiation damping

$\mathsfbfit{{\alpha}}^{-1}=\mathsfbfit{{\alpha}}^{-1}_{0}-\text{i}\frac{k^{3}}{6\pi}\mathsfbfit{{I}}\,,$
(58)

where $\mathsfbfit{{\alpha}}$ is the full $6\times 6$ square matrices
appearing in eq. 5 and representing all the polarisabilities. While the
importance of radiative corrections is not new, to our knowledge, it has not
been always applied correctly in the chiral case. Sometimes the correction eq.
59 is applied only to the electric and magnetic polarisabilities, assuming the
chiral polarisability to be zero, while the static chiral polarisability from
eq. 57 is uncorrected. For a bi-isotropic dipolar particle, the radiative
corrections coming from eq. 58 are as follows:

$\begin{split}\\!\\!\\!\alpha_{\text{e}}&=\frac{\alpha_{\text{0e}}-\text{i}\frac{k^{3}}{6\pi}(\alpha_{\text{0c}}^{2}-\alpha_{\text{0e}}\alpha_{\text{0m}})}{1+\quantity(\frac{k^{3}}{6\pi})^{2}\quantity(\alpha_{\text{0c}}^{2}-\alpha_{\text{0e}}\alpha_{\text{0m}})-\text{i}\frac{k^{3}}{6\pi}\quantity(\alpha_{\text{0e}}+\alpha_{\text{0m}})},\\!\\!\\!\\\
\\!\\!\\!\alpha_{\text{m}}&=\frac{\alpha_{\text{0m}}-\text{i}\frac{k^{3}}{6\pi}(\alpha_{\text{0c}}^{2}-\alpha_{\text{0e}}\alpha_{\text{0m}})}{1+\quantity(\frac{k^{3}}{6\pi})^{2}\quantity(\alpha_{\text{0c}}^{2}-\alpha_{\text{0e}}\alpha_{\text{0m}})-\text{i}\frac{k^{3}}{6\pi}\quantity(\alpha_{\text{0e}}+\alpha_{\text{0m}})},\\!\\!\\!\\\
\\!\\!\\!\alpha_{\text{c}}&=\frac{\alpha_{\text{0c}}}{1+\quantity(\frac{k^{3}}{6\pi})^{2}\quantity(\alpha_{\text{0c}}^{2}-\alpha_{\text{0e}}\alpha_{\text{0m}})-\text{i}\frac{k^{3}}{6\pi}\quantity(\alpha_{\text{0e}}+\alpha_{\text{0m}})}.\\!\\!\\!\end{split}$
(59)

##  Appendix B Fluxes of energy and helicity densities

Interestingly, the four _spin-like_ quantities, ${\bf\it S}\\!_{A}$, in eq. 21
can be seen as the fluxes of the energy densities $W\\!\\!_{A}$ in eq. 19.
Starting with the real Poynting vector $\Re{{\bf\it\varPi}}=\omega c{\bf\it
S}_{3}$, the Poynting theorem (continuity equation for active power) relates
the time derivative of the total active energy density $W_{0}$ to
$\Re{{\bf\it\varPi}}$ which can be interpreted as the flow of active power
[20]. Somewhat less known is the recently proposed complex Poynting theorem
[20, 52] whose real part gives the usual time-averaged Poynting theorem for a
time-harmonic field,

$-\omega c\divergence{\bf\it S}_{3}=\frac{1}{2}\Re({\bf\it
E}^{\ast}\dotproduct{\bf\it J}),$ (60)

where the term with $W_{0}$ vanishes since the average energy density does not
change with time. The imaginary part relates the reactive or stored energy
density $W_{1}$ and an alternating flow of reactive power
$\Im{{\bf\it\varPi}}=-\omega c{\bf\it S}_{2}$, which in our notation reads:

$2\omega W_{1}-\omega c\divergence{\bf\it S}_{2}=\frac{1}{2}\Im({\bf\it
E}^{\ast}\dotproduct{\bf\it J}).$ (61)

where the term on the right-hand side is the reactive power. Similarly, it is
a well-known fact that the conservation and flow of integrated optical
helicity (the continuity equation) relates helicity density
$\mathfrak{S}=W_{3}/\omega$ to the spin (helicity flux) density ${\bf\it
S}_{0}$ [53, 54]. For time-averaged quantities, the $W_{3}$ will be again
missing like in the case of the real Poynting theorem because the time-
averaged helicity will be conserved and we will get

$-\omega c\divergence{\bf\it S}_{0}=\frac{1}{4}\Im(\eta{\bf\it
H}^{\ast}\dotproduct{\bf\it J})$ (62)

notice the appearance of pseudoscalar quantity in units of power on the right-
hand side, which seems to be a chiral equivalent of active power. A similar
relationship to eq. 61 then exists between the remaining two quantities,
suggesting that ${\bf\it S}_{1}$ is the flow of $W_{2}$

$2\omega W_{2}-\omega c\divergence{\bf\it S}_{1}=\frac{1}{4}\Im(\eta{\bf\it
H}^{\ast}\dotproduct{\bf\it J}).$ (63)

##  Appendix C Chiral energy, canonical momentum and spin angular momentum
densities

One can show that chiral energy density $W_{3}$, chiral momentum ${\bf\it
p}_{3}$ and chiral spin angular momentum density ${\bf\it S}_{3}$ are just
differences between energies, momenta and spins carried by the right- and
left-handed fields. In order to do that one can write the angular spectrum
decomposition of the electric field to separate it into positive and negative
helicity components

$\begin{split}{\bf\it E}({\bf\it r})&=\\!\\!\iiint\tilde{{\bf\it E}}({\bf\it
k})\text{e}^{\text{i}{\bf\it k}\dotproduct{\bf\it r}}\differential[3]k\\\
&=\\!\\!\iiint[\tilde{{\bf\it E}}_{+}({\bf\it k})\hat{{\bf\it e}}_{+}({\bf\it
k})+\tilde{{\bf\it E}}_{-}({\bf\it k})\hat{{\bf\it e}}_{-}({\bf\it
k})]\text{e}^{\text{i}{\bf\it k}\dotproduct{\bf\it r}}\differential[3]k\\\
&={\bf\it E}_{+}({\bf\it r})+{\bf\it E}_{-}({\bf\it r})\,,\end{split}$ (64)

where $\hat{{\bf\it e}}_{\pm}({\bf\it k})$ are the circularly polarised basis
vectors for each plane wave with wave-vector ${\bf\it k}$. We can find the
magnetic fields using the Maxwell-Faraday equation $\curl{{\bf\it
E}}=\text{i}\omega\mu{\bf\it H}$ leading to

$\begin{split}\\!\\!{\bf\it H}({\bf\it
r})&=\frac{1}{\eta}\\!\iiint\hat{{\bf\it k}}\crossproduct\tilde{{\bf\it
E}}({\bf\it k})\text{e}^{\text{i}{\bf\it k}\dotproduct{\bf\it
r}}\differential[3]k\\\ &=\frac{1}{\eta}\\!\iiint[-\text{i}\tilde{{\bf\it
E}}_{+}\hat{{\bf\it e}}_{+}({\bf\it k})+\text{i}\tilde{{\bf\it
E}}_{-}\hat{{\bf\it e}}_{-}({\bf\it k})]\text{e}^{\text{i}{\bf\it
k}\dotproduct{\bf\it r}}\differential[3]k\\\ &={\bf\it H}_{+}({\bf\it
r})+{\bf\it H}_{-}({\bf\it r})=-\frac{\text{i}}{\eta}[{\bf\it E}_{+}({\bf\it
r})-{\bf\it E}_{-}({\bf\it r})]\,.\\!\\!\\!\\!\end{split}$ (65)

Using the definition of $W_{3}$ one can show that

$\displaystyle W_{3}={}$ $\displaystyle\Re
W_{\text{c}}=\frac{1}{4c}\Im\bqty{{\bf\it H}^{\ast}\dotproduct{\bf\it
E}-{\bf\it E}^{\ast}\dotproduct{\bf\it H}}$ $\displaystyle={}$
$\displaystyle\frac{1}{4\omega}\Re\\{\varepsilon[{\bf\it E}_{+}+{\bf\it
E}_{-}]^{\ast}\dotproduct[{\bf\it E}_{+}-{\bf\it E}_{-}]$
$\displaystyle+\varepsilon[{\bf\it E}_{+}-{\bf\it
E}_{-}]^{\ast}\dotproduct[{\bf\it E}_{+}+{\bf\it E}_{-}]\\}$
$\displaystyle={}$
$\displaystyle\frac{1}{4\omega}\pqty{\varepsilon\absolutevalue{{\bf\it
E}_{+}}^{2}-\varepsilon\absolutevalue{{\bf\it E}_{-}}^{2}}=W_{+}-W_{-}\,,$

where $W_{\pm}$ is the energy density carried by the right- or left-handed
component of the field. The same can be done for the canonical momentum
density

$\displaystyle{\bf\it p}_{3}={}$ $\displaystyle\Re{\bf\it
p}_{\text{c}}=\frac{1}{4\omega c}\Re\bqty{{\bf\it
E}^{\ast}\dotproduct(\gradient){\bf\it H}-{\bf\it
H}^{\ast}\dotproduct(\gradient){\bf\it E}}$ $\displaystyle={}$
$\displaystyle\frac{1}{4\omega}\Im\\{\varepsilon[{\bf\it E}_{+}+{\bf\it
E}_{-}]^{\ast}\dotproduct(\gradient)[{\bf\it E}_{+}-{\bf\it E}_{-}]$
$\displaystyle+\varepsilon[{\bf\it E}_{+}-{\bf\it
E}_{-}]^{\ast}\dotproduct(\gradient)[{\bf\it E}_{+}+{\bf\it E}_{-}]\\}$
$\displaystyle={}$ $\displaystyle\frac{1}{4\omega}\Im\bqty{\varepsilon{\bf\it
E}_{+}^{*}\dotproduct(\gradient){\bf\it E}_{+}-\varepsilon{\bf\it
E}_{-}^{\ast}\dotproduct(\gradient){\bf\it E}_{-}}={\bf\it p}_{+}-{\bf\it
p}_{-}\,,$

and for the spin angular momentum density that is related to the Poynting
vector

$\displaystyle{\bf\it S}_{3}={}$ $\displaystyle\frac{1}{\omega
c}\Re{\bf\it\varPi}=\frac{1}{4\omega c}\Re\bqty{{\bf\it
E}^{\ast}\crossproduct{\bf\it H}-{\bf\it H}^{\ast}\crossproduct{\bf\it E}}$
$\displaystyle={}$ $\displaystyle\frac{1}{4\omega}\Im\\{\varepsilon[{\bf\it
E}_{+}+{\bf\it E}_{-}]^{\ast}\crossproduct[{\bf\it E}_{+}-{\bf\it E}_{-}]$
$\displaystyle+\varepsilon[{\bf\it E}_{+}-{\bf\it
E}_{-}]^{\ast}\crossproduct[{\bf\it E}_{+}+{\bf\it E}_{-}]\\}$
$\displaystyle={}$ $\displaystyle\frac{1}{4\omega}\Im\bqty{\varepsilon{\bf\it
E}_{+}^{*}\crossproduct{\bf\it E}_{+}-\varepsilon{\bf\it
E}_{-}^{\ast}\crossproduct{\bf\it E}_{-}}={\bf\it S}_{+}-{\bf\it S}_{-}\,.$

##  Appendix D Boundary conditions for a cylindrical dielectric fibre

At the interface between the dielectric fibre and the surrounding material,
the tangent components of the electric field $E_{z}$, $E_{\varphi}$ and
magnetic field $H_{z}$, $H_{\varphi}$ have to be continuous. These boundary
conditions can be written as an eigenvalue problem

${\bf\it A}_{\ell,n}{\bf\it v}_{\ell,n}=0,$ (66)

where ${\bf\it v}_{\ell,n}=(A_{1},B_{1},A_{2},B_{2})^{\intercal}$ represents
the amplitudes both outside and inside the fibre and the boundary condition
matrix as a function of $(k_{z},\omega,r_{0},\varepsilon_{1},\varepsilon_{2})$
is as follows:

${\bf\it A}_{\ell,n}=\\!\\!\pmqty{undef},$

where $n_{i}=\sqrt{\varepsilon_{i}}$ is the refractive index of medium $i$,
$\operatorname{J}_{\ell}=\operatorname{J}_{\ell}(\kappa_{1}\rho)$,
$\operatorname{H}^{(1)}_{\ell}=\operatorname{H}^{(1)}_{\ell}(\kappa_{2}\rho)$
are Bessel and Hankel functions of the first kind respectively and the prime
represents derivative of the whole argument of a function. Given the
characteristic parameters $(\omega,r_{0},\varepsilon_{1},\varepsilon_{2})$,
one can numerically calculate the longitudinal dispersion relation
$k_{z}/k_{0}$ as a function of $k_{0}r_{0}$ for any mode from the
transcendental equation
$\det\boldsymbol{A}_{\ell,n}(k_{z},\omega,r_{0},\varepsilon_{1},\varepsilon_{2})=0$.
The solutions of this equation for $\ell\in\\{0,\pm 1,\pm 2\\}$ and
$n\in\\{0,1,2\\}$ can be seen in fig. 2 using standard normalised units
popular in the literature.

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