# A decomposition of light’s spin angular momentum density

Alex J. Vernon  Department of Physics, King’s College London, Strand, London
WC2R 2LS, UK  London Centre for Nanotechnology  Sebastian Golat  Department of
Physics, King’s College London, Strand, London WC2R 2LS, UK  London Centre for
Nanotechnology  Claire Rigouzzo  Department of Physics, King’s College London,
Strand, London WC2R 2LS, UK  Eugene A. Lim  Department of Physics, King’s
College London, Strand, London WC2R 2LS, UK  Francisco J. Rodríguez-Fortuño
<EMAIL_ADDRESS>Department of Physics, King’s College
London, Strand, London WC2R 2LS, UK  London Centre for Nanotechnology

###### Abstract

Light carries intrinsic spin angular momentum (SAM) when the electric or
magnetic field vector rotates over time. A familiar vector equation calculates
the direction of light’s SAM density using the right hand rule with reference
to the electric and magnetic polarisation ellipses. Using Maxwell’s equations,
this vector equation can be decomposed into a sum of two distinct terms, akin
to the well-known Poynting vector decomposition into orbital and spin
currents. We present the first general study of this spin decomposition,
showing that the two terms, which we call canonical and Poynting spin, are
chiral analogies to the canonical and spin momenta of light in its interaction
with matter. Both canonical and Poynting spin incorporate spatial variation of
the electric and magnetic fields and are influenced by optical orbital angular
momentum (OAM). The decomposition allows us to show that the OAM of a linearly
polarised vortex beam can impart a first-order preferential force to chiral
matter in the absence of spin.

##  1 Introduction

Optical angular momentum is contributed to by physical orbital and spin parts
[1, 2, 3, 4]. The orbital component can be either extrinsic or intrinsic and
is associated with the spatial structure of the light field. Orbital angular
momentum (OAM) is most famously stirred into light by optical vortices, which
have helical phase fronts twisted around a one-dimensional line singularity
[5, 6, 7, 8]. Spin angular momentum (SAM), meanwhile, develops as the electric
and magnetic field vectors rotate during oscillation [9], and is an intrinsic
quantity, independent of co-ordinate origin. Both OAM and SAM arise from
chiral structures in light, and as such, can access the chirality of matter,
enhancing, pushing, twisting, and torquing with or without a parity bias [10,
11]. Many commonly observed chiroptical interactions, such as circular
dichroism, arise from the coupling of chiral matter to a photon’s spin state.
Recent attention has also turned to OAM-dependent photon absorption and
scattering, with a range of effects theorised [12, 13, 14, 15]. Chiroptical
effects involving OAM often hinge on the strong longitudinal field component
introduced by beam focussing [16, 17]. Unlike for photons with opposite spins,
however, preferential absorption of photons with different OAM handedness does
not occur in the dipole approximation for paraxial light [18, 19].

A well-known orbital and spin separation can be performed locally in the
Poynting vector [20] (kinetic momentum density, when divided by $c^{2}$). The
Poynting vector may be expressed as a sum of the orbital current, pointing in
the direction of canonical momentum, and spin current, proportional to the
virtual spin momentum. This is a physically meaningful decomposition, tying in
to the famous Abraham-Minkowski dilemma [21, 22].

Surprisingly to most readers, a similar vector decomposition can be performed
on the electromagnetic spin density $\mathbf{S}$, which is split into, perhaps
confusingly, orbital-like and spin-like contributions to the total
electromagnetic spin. We name these two terms canonical and Poynting spin.
This little-known decomposition is the focus of our work—amidst discussion by
Shi et al who, without giving the complete decomposition explicitly, linked
the two terms to longitudinal and transverse spin in some specific cases [23,
24, 25], no clear picture exists in the literature on the physical
significance of the two terms in general, in the same way as for the
decomposed Poynting vector. The decomposition as we present has been expressed
previously (e.g. [26]), but with almost no discussion on the function of terms
as component parts of $\mathbf{S}$.

We therefore have two objectives: to clarify and understand the meaning of the
orbit-like and spin-like components of $\mathbf{S}$, and to emphasise their
value for the research community in interpreting electromagnetic SAM density.
For example, by use of the Maxwell stress tensor, one understands that the
canonical and Poynting spin vectors are responsible for different interactions
between light and chiral particles. While the two decomposed spin vectors are
organised independently in general monochromatic light, a remarkable
implication lies in beams carrying OAM, such as vortex beams. OAM imparts a
longitudinal component to both canonical and Poynting spins, even in the
absence of longitudinal total spin. By applying the decomposition to a
linearly polarised vortex beam, we show that a total spin-free longitudinal
chiral force exists—a force which can act even under the dipole approximation,
despite originating from the beam’s OAM.

We generalise the spin decomposition to time-dependent fields and four-vector
representation, unlocking a deeper understanding of the behaviour of the two
terms. We also show that the ability to split spin into two terms appears to
be a general feature of wave fields, similarly to the decomposition of the
Poynting vector, including those of the linearised theories of gravity [27]
and acoustics, which have distinctly different vector structures. Throughout,
we use bold latin letters $\mathbf{E}$ and $\mathbf{H}$ to represent time-
harmonic complex electromagnetic field phasors, which are a function of
position only [e.g. $\mathbf{E(\mathbf{r})}=\mathbf{E}_{0}\exp(i\mathbf{k\cdot
r})$], and scripted characters $\boldsymbol{\mathcal{E}}$ and
$\boldsymbol{\mathcal{H}}$ denoting real, time-varying field vectors (e.g.
$\boldsymbol{\mathcal{E}}(\mathbf{r},t)=\textrm{Re}\\{\mathbf{E(\mathbf{r})}\exp(-i\omega
t)\\}$).

Figure 1:  The component parts of light’s total spin angular momentum density,
visualised at a single point (red circle) in the 3D interference of
monochromatic plane waves. Enlarged electric and magnetic field ellipses and
coloured vector arrows, representing the total spin $\mathbf{S}$, electric and
magnetic spins $\mathbf{S}_{e}$ and $\mathbf{S}_{m}$, and the canonical
$\mathbf{s}_{c}$ and Poynting spins $\mathbf{s}_{p}$ of the decomposition Eq.
(4), are plotted. The electric and magnetic spin vectors are normal to the
$\mathbf{E}$ and $\mathbf{H}$ polarisation ellipses, according to the right-
hand rule and the sense of rotation of the instantaneous field vectors (see
separated diagrams below the main combined image). Projected on the $xy$ plane
are the Poynting vector streamlines, which in this case organise a clear
curling structure. This provides a visual aid for Poynting spin (large black
arrow), which is informed by three-dimensional curling of the Poynting vector.
In structured light where the $\mathbf{E}$ and $\mathbf{H}$ ellipses are de-
coupled, the canonical spin and therefore chiral pressure (Eq. (5)) can point
in a completely different direction to the any of the total spin or its
electric and magnetic contributions.

##  2 Electromagnetic spin decomposition

We will first contextualise the decomposition of electromagnetic SAM density
by recalling a well-known orbital and spin decomposition which exists in free-
space and time-harmonic fields for the time-averaged Poynting vector,

$\mathbf{P}=\frac{1}{2}\textrm{Re}\\{\mathbf{E^{*}\times H}\\},$ (1)

which represents the flux of active power in the light field. The Poynting
vector is separated into the orbital and spin currents [20, 22],

$\mathbf{P}=\underbrace{\frac{c^{2}}{4\omega}\textrm{Im}\\{\epsilon_{0}\mathbf{E^{*}}\cdot(\nabla)\mathbf{E}+\mu_{0}\mathbf{H^{*}}\cdot(\nabla)\mathbf{H}\\}}_{\mathbf{p}_{o}}+\underbrace{\frac{c^{2}}{2}\nabla\times\overbrace{\frac{1}{4\omega}\textrm{Im}\\{\epsilon_{0}\mathbf{E^{*}}\times\mathbf{E}+\mu_{0}\mathbf{H^{*}}\times\mathbf{H}\\}}^{\mathbf{S}}}_{\mathbf{p}_{s}},$
(2)

defined using the complex phasors $\mathbf{E}$ and $\mathbf{H}$. The inner
product notation is $\mathbf{a}\cdot(\nabla)\mathbf{b}=a_{x}\nabla
b_{x}+a_{y}\nabla b_{y}+a_{z}\nabla b_{z}$. When divided by $c^{2}$,
$\mathbf{P}$ has units of a momentum density and is termed the kinetic
momentum density, while the orbital current $\mathbf{p}_{o}$ becomes the
canonical momentum density, corresponding to the expectation value of linear
momentum carried by photons at each point in space (unlike kinetic momentum,
the canonical momentum is a directly measurable quantity). The net flow of the
instantaneous Poynting vector into or out of a volume relates to the change in
electromagnetic energy density over time, according to a continuity equation.
An adjacent conserved quantity of light is its chirality, which has its own
continuity equation and an associated chirality flux [28, 29, 30]. Chirality
$\chi$ is one of an infinite hierarchy of conserved quantities in linear, non-
dispersive media, to which helicity $h$ also belongs [30, 9]—helicity and
chirality are often conflated because they (and their fluxes) are proportional
by a factor of $\omega^{2}$ in monochromatic waves, though in general they are
distinct. In monochromatic light the flux of helicity corresponds to the time-
averaged SAM density (often referred to simply as ‘spin’), given by,

$\mathbf{S}=\frac{1}{4\omega}\textrm{Im}\\{\epsilon_{0}\mathbf{E^{*}\times
E}+\mu_{0}\mathbf{H^{*}\times H}\\}\equiv\mathbf{S}_{e}+\mathbf{S}_{m}\ .$ (3)

Spin’s individual electric and magnetic contributions point in the normal
direction to the electric and magnetic polarisation ellipses, drawn over time
by the instantaneous vectors $\boldsymbol{\mathcal{E}}(\mathbf{r},t)$ and
$\boldsymbol{\mathcal{H}}(\mathbf{r},t)$. Like the Poynting vector, and using
the same procedure, spin $\mathbf{S}$ may be split into a sum of two vectors:
the canonical spin $\mathbf{s}_{c}$ and the Poynting spin $\mathbf{s}_{p}$.
This decomposition is our main focus and is, explicitly,

$\mathbf{S}=\underbrace{\frac{1}{4\omega^{2}}\textrm{Re}\\{\mathbf{E^{*}}\cdot(\nabla)\mathbf{H}-\mathbf{H^{*}}\cdot(\nabla)\mathbf{E}\\}}_{\mathbf{s}_{c}}+\underbrace{\frac{1}{2\omega^{2}}\nabla\times\overbrace{\frac{1}{2}\textrm{Re}\\{\mathbf{E^{*}\times
H}\\}}^{\mathbf{P}}}_{\mathbf{s}_{p}}.$ (4)

Any subsequent mention of ‘canonical spin’ in this work always refers to the
first term,
$\mathbf{s}_{c}=\frac{1}{4\omega^{2}}\textrm{Re}\\{\mathbf{E^{*}}\cdot(\nabla)\mathbf{H}-\mathbf{H^{*}}\cdot(\nabla)\mathbf{E}\\}$,
and ‘Poynting spin’ to the second,
$\mathbf{s}_{p}=\frac{1}{2\omega^{2}}\nabla\times\mathbf{P}$. Their sum
$\mathbf{S}$ (Eq. (3)) we refer to as total spin. This naming scheme reflects
that of the decomposed kinetic momentum, split into canonical momentum and
spin momentum.

##  3 Physical interpretation

Previous works [23, 24] have linked the canonical $\mathbf{s}_{c}$ and
Poynting spin $\mathbf{s}_{p}$ vectors to the longitudinal and transverse spin
of light, respectively. However, this is only true in a very limited number of
cases such as linearly polarised evanescent waves. It cannot be a general
result because well-defined longitudinal and transverse directions do not
exist for many-plane wave or multiple beam interference. In fact, it does not
hold even when there is a well-defined longitudinal direction: a circularly
polarised evanescent wave has both longitudinal and transverse spin components
contained in the Poynting spin vector.

One way to gain an understanding of the physical significance of the canonical
and Poynting spins is to study the interactions between light and chiral
matter. Chiral light can exert preferential forces on enantiomers with
opposite handedness, with direct proportionality to the light’s SAM. When
light shines on a particle much smaller than the wavelength (Rayleigh regime)
it gets polarised and acquires an electric $\mathbf{p}$ and a magnetic
$\mathbf{m}$ dipole. In the linear regime, the dipole moments are proportional
to the incident fields,
$\mathbf{p}=\alpha_{\text{e}}\varepsilon\mathbf{E}+i\alpha_{\text{c}}\mathbf{H}/c$
and
$\mathbf{m}=\alpha_{\text{m}}\mathbf{H}-i\alpha_{\text{c}}\mathbf{E}/\eta$,
where $\alpha_{\text{e}}$, $\alpha_{\text{m}}$ and $\alpha_{\text{c}}$ are the
electric, magnetic, and chiral polarisabilities of the particle. The chiral
polarisability $\alpha_{\text{c}}$ is a pseudoscalar so it changes sign
between the two enantiomers (mirror-reflected versions), and vanishes unless
the matter is chiral. Under these assumptions, the illuminating light exerts a
chiral optical force on the particle (which changes sign for the different
enantiomers) given by [31, 32, 33, 34, 35]:

${\mathbf{F}_{\text{chiral}}}=\underbrace{\omega\nabla[\mathrm{Re}(\alpha_{\text{c}})h]}_{\text{helicity
gradient}}+\underbrace{2\omega
k[\mathrm{Im}(\alpha_{\text{c}})\mathbf{s}_{\text{c}}]}_{\text{chiral
pressure}}-\underbrace{\,\omega\frac{k^{4}}{3\pi}[\mathrm{Re}(\alpha_{\text{e}}^{\ast}\alpha_{\text{c}})\mathbf{S}_{\text{e}}\\!+\\!\mathrm{Re}(\alpha_{\text{m}}^{\ast}\alpha_{\text{c}})\mathbf{S}_{\text{m}}]}_{\text{spin
recoil}},$ (5)

where $k=\omega/c$ is the wavenumber,
$h=-\mathrm{Im}(\mathbf{E}^{*}\cdot\mathbf{H})/(2\omega c)$ is the cycle-
averaged optical helicity density [17, 36], and $\mathbf{S}_{e}$ and
$\mathbf{S}_{m}$ are the electric and magnetic parts of the total spin
$\mathbf{S}$ from Eq. (3). One immediately sees that the canonical spin
$\mathbf{s}_{\text{c}}$, one of the two terms in the spin decomposition [Eq.
(4)], appears in this force equation and is directly responsible for the
chiral pressure (in this context, the product $k\mathbf{s}_{\text{c}}$ is
sometimes called the chiral momentum [26]), while the _total_ spin appears in
the spin recoil term. Poynting spin contributes (together with canoncial spin)
only to the higher-order recoil force caused by the unbalanced radiation
pattern of an electric-magnetic dipole. This also implies that in an
electromagnetic field whose SAM is pure Poynting spin (i.e., zero canonical
spin), the particle will experience no chiral pressure, only helicity
gradients and relatively weaker spin recoil terms of the force.

An alternative route to the physical meaning of the canonical and Poynting
spins is found purely in the two terms’ mathematical expressions as shown in
Eq. (4). The Poynting spin, being proportional to the curl of the Poynting
vector, is easily interpreted as the vorticity in energy flow. The expression
for the canonical spin is much harder to interpret initially, but begins to
unravel if we decompose a general electromagnetic field into circularly
polarised plane waves. Any arbitrary electromagnetic field, no matter how
complicated, can be expressed in momentum space by an angular spectrum (an
infinite sum of plane waves of different wavevectors, weighted by an amplitude
function). Using this property, we can further probe the canonical spin term
in monochromatic light by separating electric and magnetic fields into two
component fields of opposite helicities, indicated by the $+$ and $-$
subscripts,

$\mathbf{E}(\mathbf{r})=\\!\\!\iiint\tilde{\mathbf{E}}(\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{r}}\mathrm{d}^{3}k=\\!\\!\iiint[\tilde{{E}}_{+}(\mathbf{k})\hat{\mathbf{e}}_{+}(\mathbf{k})+\tilde{{E}}_{-}(\mathbf{k})\hat{\mathbf{e}}_{-}(\mathbf{k})]e^{i\mathbf{k}\cdot\mathbf{r}}\mathrm{d}^{3}k=\mathbf{E}_{+}(\mathbf{r})+\mathbf{E}_{-}(\mathbf{r}),$
(6)

where $\hat{\mathbf{e}}_{\pm}(\mathbf{k})$ represent the circularly polarised
unit vectors for each plane wave with wave-vector $\mathbf{k}$ (see, e.g.,
[37]). The magnetic field’s associated helicity components are obtained from
Faraday’s law $\nabla\times{\mathbf{E}}=i\omega\mu\mathbf{H}$ as,

$\mathbf{H}(\mathbf{r})=\frac{1}{\eta}\\!\iiint\frac{\mathbf{k}}{k}\times\tilde{\mathbf{E}}(\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{r}}\mathrm{d}^{3}k=\frac{1}{\eta}\\!\iiint[-i\tilde{{E}}_{+}\hat{\mathbf{e}}_{+}(\mathbf{k})+i\tilde{{E}}_{-}\hat{\mathbf{e}}_{-}(\mathbf{k})]e^{i\mathbf{k}\cdot\mathbf{r}}\mathrm{d}^{3}k=\mathbf{H}_{+}(\mathbf{r})+\mathbf{H}_{-}(\mathbf{r}).$
(7)

where we used the property $(\mathbf{k}/k)\times\hat{\mathbf{e}}_{\pm}=\mp
i\hat{\mathbf{e}}_{\pm}$. Equations (6) and (7) show that the helicity-
separated electric and magnetic fields are related by
$\mathbf{H}_{\pm}(\mathbf{r})=\mp i\mathbf{E}_{\pm}(\mathbf{r})/\eta$,
characteristic of a circularly polarised plane wave—this is true not only in
the spectral representation, but also in the spatial representation, for any
arbitrary field (a property explored in depth in [38]). This allows us to
substitute helicity-separated fields into many dual quantities, including
canonical momentum and canonical spin, gaining further insight. Simply
substituting $\mathbf{E}=\mathbf{E}_{+}+\mathbf{E}_{-}$ and
$\mathbf{H}=-i\left(\mathbf{E}_{+}-\mathbf{E}_{-}\right)/\eta$ into the
expression for orbital current [$\mathbf{p}_{o}$ of Eq. (2)], we find, after
some algebra,

$\mathbf{p}_{o}=\frac{c^{2}}{2\omega}\epsilon_{0}\textrm{Im}\\{\mathbf{E}^{*}_{+}\cdot(\nabla)\mathbf{E}_{+}+\mathbf{E}^{*}_{-}\cdot(\nabla)\mathbf{E}_{-}\\}=c^{2}(\mathbf{p}_{+}+\mathbf{p}_{-}),$
(8)

showing that the helicity segregation of the $\mathbf{E}$ and $\mathbf{H}$
fields translates to a separation of contributions to the field’s momentum
$\mathbf{p}=\mathbf{p}_{+}+\mathbf{p}_{-}$ by photons of positive and negative
helicity. Note that orbital current is proportional to canonical momentum by
$\mathbf{p}_{o}=c^{2}\mathbf{p}$; here, $\mathbf{p}_{+}$ and $\mathbf{p}_{-}$
are helicity-separated momentum densities. Making the same substitution in the
expression for canonical spin $\mathbf{s}_{c}$ [first term of Eq. (4)]
illuminates its meaning,

$\begin{split}\mathbf{s}_{c}&=\frac{1}{4\omega^{2}}\textrm{Re}\big{\\{}\\!-\frac{i}{\eta}[\mathbf{E^{*}_{+}}+\mathbf{E^{*}_{-}}]\cdot(\nabla)[\mathbf{E_{+}}-\mathbf{E_{-}}]+\frac{i}{\eta}[\mathbf{E^{*}_{+}}-\mathbf{E^{*}_{-}}]\cdot(\nabla)[\mathbf{E_{+}}+\mathbf{E_{-}}]\big{\\}}\\\
&=\frac{1}{2k\omega}\epsilon_{0}\textrm{Im}\\{\mathbf{E}^{*}_{+}\cdot(\nabla)\mathbf{E}_{+}-\mathbf{E}^{*}_{-}\cdot(\nabla)\mathbf{E}_{-}\\}=\frac{1}{k}\left(\mathbf{p}_{+}-\mathbf{p}_{-}\right).\end{split}$
(9)

Canonical spin is proportional to the difference in linear momentum densities
carried by photons of oppositely signed helicity. This brings a clear physical
interpretation of the canonical spin, and also helps explain why the chiral
pressure force acts in the direction of $\mathbf{s}_{c}$, as photons of
opposite helicities are absorbed or scattered in different amounts by chiral
particles. In a general structured field,
$\mathbf{p}_{+}-\mathbf{p}_{-}\neq\mathbf{p}_{+}+\mathbf{p}_{-}$ and hence
canonical spin is decoupled from any local longitudinal direction defined by
canonical momentum.

##  4 Mathematical derivation

In this section, we briefly lay out the mathematical steps taken to arrive at
Eq. (4), before developing the expression into a more fundamental 4-vector
description. For each of the canonical and Poynting spin terms, the 4-vector
decomposition incorporates a time component which characterises how the two
terms transform differently between reference frames.

###  4.1 3-vector decomposition

Maxwell’s equations enable both decompositions of the Poynting vector Eq. (2)
and spin Eq. (4) by bringing the relevant vector from its usual representation
into a form which can be separated into two terms using a vector identity,

$\mathbf{a}\times(\nabla\times\mathbf{b})=\mathbf{a}\cdot(\nabla)\mathbf{b}-(\mathbf{a}\cdot\nabla)\mathbf{b},$
(10)

where $\mathbf{a}$ and $\mathbf{b}$ are arbitrary vectors, and the previously
unseen notation on the right hand side means
$(\mathbf{a}\cdot\nabla)\mathbf{b}=a_{x}\partial_{x}\mathbf{b}+a_{y}\partial_{y}\mathbf{b}+a_{z}\partial_{z}\mathbf{b}$.
In time-harmonic fields, the phasors $\mathbf{E}$ and $\mathbf{H}$ can be
substituted for curls of their counterpart via Faraday’s and Ampere’s laws.
Substituting for the un-conjugated phasors in the spin vector definition Eq.
(3) gives an expression in the form of Eq. (10),

$\mathbf{S}=\frac{1}{4\omega}\textrm{Im}\big{\\{}\epsilon_{0}\mathbf{E^{*}}\times\Big{(}\frac{i}{\omega\epsilon_{0}}\nabla\times\mathbf{H}\Big{)}-\mu_{0}\mathbf{H^{*}}\times\Big{(}\frac{i}{\omega\mu_{0}}\nabla\times\mathbf{E}\Big{)}\big{\\}}.$
(11)

Applying the vector identity Eq. (10), we have,

$\mathbf{S}=\frac{1}{4\omega^{2}}\textrm{Re}\\{\mathbf{E^{*}}\cdot(\nabla)\mathbf{H}-\mathbf{H^{*}}\cdot(\nabla)\mathbf{E}\\}+\frac{1}{4\omega^{2}}\textrm{Re}\\{(\mathbf{H^{*}}\cdot\nabla)\mathbf{E}-(\mathbf{E^{*}}\cdot\nabla)\mathbf{H}\\}.$
(12)

Gauss’ law in free space ($\nabla\cdot\mathbf{E}=0$ and
$\nabla\cdot\mathbf{H}=0$), combined with a second vector identity
$\nabla\times(\mathbf{a}\times\mathbf{b})=\mathbf{a}(\nabla\cdot\mathbf{b})-\mathbf{b}(\nabla\cdot\mathbf{a})+(\mathbf{b}\cdot\nabla)\mathbf{a}-(\mathbf{a}\cdot\nabla)\mathbf{b}$,
extracts the curl of the Poynting vector from the second $\textrm{Re}\\{\\}$
term of Eq. (12),

$\frac{1}{4\omega^{2}}\textrm{Re}\\{(\mathbf{H^{*}}\cdot\nabla)\mathbf{E}-(\mathbf{E^{*}}\cdot\nabla)\mathbf{H}\\}=\frac{1}{2\omega^{2}}\nabla\times\frac{1}{2}\textrm{Re}\\{\mathbf{E^{*}\times
H}\\}=\mathbf{s}_{p},$ (13)

completing the final step in obtaining the electromagnetic spin decomposition
as we present in Eq. (4).

In addition to the monochromatic case, canonical and Poynting spin analogies
for the flow of chirality may also be defined in polychromatic fields,
expressed using the full time-dependent fields
$\boldsymbol{\mathcal{E}}(\mathbf{r},t)$ and
$\boldsymbol{\mathcal{H}}(\mathbf{r},t)$ and stemming from the instantaneous
flow of chirality $\boldsymbol{\mathcal{F}}(\mathbf{r},t)$ [28, 29, 39, 30].

$\boldsymbol{\mathcal{F}}=\frac{1}{2}\left(\boldsymbol{\mathcal{E}}\times(\nabla\times\boldsymbol{\mathcal{H}})-\boldsymbol{\mathcal{H}}\times(\nabla\times\boldsymbol{\mathcal{E}})\right).$
(14)

Chiral flow $\boldsymbol{\mathcal{F}}$, unlike the flux of helicity in time-
dependent fields, is defined without the vector potentials
$\boldsymbol{\mathcal{A}}$ and $\boldsymbol{\mathcal{C}}$, which avoids gauge
complications and contains a direct relation to the instantaneous Poynting
vector $\boldsymbol{\mathcal{P=E\times H}}$. For monochromatic light,
$\boldsymbol{\mathcal{F}}$ is not time-dependent and becomes proportional to
the field’s SAM density $\mathbf{S}$. Equation (14) is immediately presented
in a form which can be broken into two vectors using the two identities
detailed in this section, giving the extension of the time-harmonic spin
decomposition to time-dependent fields,

$\boldsymbol{\mathcal{F}}=\frac{1}{2}\left[\left(\boldsymbol{\mathcal{E}}\cdot(\nabla)\boldsymbol{\mathcal{H}}-\boldsymbol{\mathcal{H}}\cdot(\nabla)\boldsymbol{\mathcal{E}}\right)+\nabla\times(\boldsymbol{\mathcal{E\times
H}})\right].$ (15)

The above is a more general decomposition of chirality flow than Eq. (4), and
is valid for polychromatic aperiodic light, at every instant in time. It is
worth stressing again that the flow of chirality $\boldsymbol{\mathcal{F}}$ is
a different quantity to the flow of helicity (SAM density) for general time-
dependent fields. To obtain the helicity flow equivalent of Eq. (15) in the
Coulomb gauge, one simply substitutes
$\boldsymbol{\mathcal{E}}\rightarrow\boldsymbol{\mathcal{A}}$ and
$\boldsymbol{\mathcal{H}}\rightarrow\boldsymbol{\mathcal{C}}$. Equation (15)
assumed a source-free medium, such that
$\nabla\cdot\boldsymbol{\mathcal{E}}=\rho/\epsilon=0$. Dropping this
assumption, Eq. (15) can be further generalised by adding a third term,
$\frac{1}{2}\left[\frac{\rho}{\epsilon}\boldsymbol{\mathcal{H}}\right]$, to
the decomposition.

###  4.2 4-vector representation

In an attempt to extract new physical insight, we reproduce the above spin
decomposition Eq. (4) in 4-vector notation. This notation is particularly
elegant and useful to verify the Lorentz covariance of quantities and
equations. We will begin by introducing some common definitions.
Electrodynamics is intrinsically relativistic, and several related physical
quantities can be expressed via well-known 4-vectors, such as the 4-potential
(grouping the scalar and vector potentials) and the 4-current (grouping the
charge and current densities). 4-vector notation is well known to provide an
efficient way to formulate Maxwell’s equations. Indeed, the time-dependent
Maxwell’s equations (using scripted vectors) in vacuum are a combination of
four equations, two scalar equations
$\mathbf{\nabla}\cdot\boldsymbol{\mathcal{E}}=0,\mathbf{\nabla}\cdot\boldsymbol{\mathcal{H}}=0$
and two vector equations
$\mathbf{\nabla}\times\boldsymbol{\mathcal{E}}=-\mu_{0}\frac{\partial\boldsymbol{\mathcal{H}}}{\partial
t},\mathbf{\nabla}\times\boldsymbol{\mathcal{H}}=\epsilon_{0}\frac{\partial\boldsymbol{\mathcal{E}}}{\partial
t}$. In 4-vector notation, however, we can write Maxwell’s equations as:

$\partial_{\alpha}\mathcal{F}^{\alpha\beta}=0,\qquad\epsilon^{\alpha\beta\gamma\lambda}\partial_{\alpha}\mathcal{F}_{\beta\gamma}=0.$
(16)

Throughout, the greek indices $\mu,\nu,\alpha,\beta$… run from $0$ to $3$,
where the $0^{th}$ component labels the time direction. Repeated indices are
summed over. Note that the distinction between subscript/superscript placement
of indices are important, as $A_{\mu}=\eta_{\mu\nu}A^{\nu}$ where the so-
called Minkowski metric $\eta_{\mu\nu}=\mathrm{diag}(-1,1,1,1)$. We also use
Roman indices $i=1,2,3$ to denote the three-dimensional space. In Eq. (16),
$\epsilon^{\alpha\beta\gamma\lambda}$ is the Levi-Civita symbol [40], while
$\mathcal{F}_{\alpha\beta}$ is the field strength tensor that conveniently
packages the electric and magnetic fields:

$\mathcal{F}_{\mu\nu}=\begin{pmatrix}0&-\mathcal{E}_{x}/c&-\mathcal{E}_{y}/c&-\mathcal{E}_{z}/c\\\
\mathcal{E}_{x}/c&0&-\mu_{0}\mathcal{H}_{z}&\mu_{0}\mathcal{H}_{y}\\\
\mathcal{E}_{y}/c&\mu_{0}\mathcal{H}_{z}&0&-\mu_{0}\mathcal{H}_{x}\\\
\mathcal{E}_{z}/c&-\mu_{0}\mathcal{H}_{y}&\mu_{0}\mathcal{H}_{x}&0\end{pmatrix}.$
(17)

The field strength tensor can be expressed as
$\mathcal{F}_{\alpha\beta}=\partial_{\alpha}\mathcal{A}_{\beta}-\partial_{\beta}\mathcal{A}_{\alpha}$,
where $\mathcal{A}_{\alpha}$ is the 4-vector potential, that encases the
scalar potential $\varphi$ and the vector potential $\boldsymbol{\mathcal{A}}$
and $\partial_{\mu}$ is a generalisation of gradient:

$\mathcal{A}^{\mu}=\begin{pmatrix}\mathcal{A}^{0}\\\
\mathcal{A}^{i}\end{pmatrix}=\begin{pmatrix}\varphi/c\\\
\boldsymbol{\mathcal{A}}\end{pmatrix},\quad\partial_{\mu}=\begin{pmatrix}\frac{\partial}{c\partial
t}\\\ \partial_{i}\end{pmatrix}=\begin{pmatrix}\frac{\partial}{c\partial t}\\\
\nabla\end{pmatrix}.$ (18)

Equation (16) is a striking example of the neatness of the tensorial notation:
the physical content is the same, but the tensorial notation is clearer and
exhibits Lorentz covariance. For time-harmonic fields, these tensors will also
have a phasor representation (e.g.
${\mathcal{F}}_{\mu\nu}(\mathbf{r},t)=\textrm{Re}\\{{F_{\mu\nu}(\mathbf{r})}\exp(-i\omega
t)\\}$), including the time components of the gradient, which for phasors
becomes $\partial_{0}=-i\omega/c$.

We now have the tools to write, using phasors, the time-averaged spin
decomposition Eq. (4) in tensorial notation. The first step is to equivalently
express the spin, defined in Eq. (3), using spatial indices to replicate the
vector operations,

$S^{i}=\frac{1}{4\omega}\epsilon^{ijk}\textrm{Im}\\{\epsilon_{0}E^{*}_{j}E_{k}+\mu_{0}H^{*}_{j}H_{k}\\}.$
(19)

Then, the equivalent of the proposed decomposition in Eq. (4) is:

$S^{i}=\underbrace{\frac{1}{4\omega^{2}}\textrm{Re}\\{E^{*}_{j}\partial_{i}H^{j}-H^{*}_{j}\partial_{i}E^{j}\\}}_{s^{i}_{c}}+\underbrace{\frac{1}{2\omega^{2}}\epsilon^{ijk}\partial_{j}\frac{1}{2}\textrm{Re}\\{\epsilon_{klm}E^{*l}H^{m}\\}}_{s^{i}_{p}}.$
(20)

The next step is to find a 4-vector, for which the spatial part would reduce
to Eq. (20). It turns out that this 4-vector quantity is exactly the helicity
density and flux, a 4-current density associated with conserved helicity [41]

$S^{\mu}=\frac{1}{4}\operatorname{Re}\left\\{A_{\nu}^{*}G^{\nu\mu}+C_{\nu}^{*}F^{\nu\mu}\right\\},$
(21)

where
$C^{\mu}=(\psi,-i\omega\mu_{0}^{-1}(\mathbf{\nabla}\times\mathbf{A}))^{\intercal}$
is the magnetic equivalent to the 4-potential $A_{\mu}$, and
$G_{\mu\nu}=\partial_{\nu}C_{\mu}-\partial_{\mu}C_{\nu}$ is the corresponding
field strength tensor. Note that Eq. (21) is time-averaged and the fields are
in their phasor representation. We include the instantaneous variant in the
supplementary information. Working in the Coulomb gauge, i.e., the scalar and
vector potentials are chosen such that $\nabla\cdot\mathbf{A}=\phi=0$ and
$\nabla\cdot\mathbf{C}=\psi=0$, and after some algebra (see supplementary
Mathematica file [42]), we find that the spatial part of the 4-vector
$S^{\mu}$ reproduces Eq. (20), whilst the temporal part is the cycle-averaged
helicity density $h=-\textrm{Im}\\{\mathbf{E^{*}\cdot H}\\}/(2\omega c)$,

$S^{\mu}=\frac{1}{4\omega}\operatorname{Im}\left\\{\left(\begin{array}[]{c}-\frac{2}{c}\mathbf{E}^{*}\cdot\mathbf{H}\\\
\epsilon_{0}\mathbf{E}^{*}\times\mathbf{E}+\mu_{0}\mathbf{H}^{*}\times\mathbf{H}\end{array}\right)\right\\}=\left(\begin{array}[]{c}h\\\
\mathbf{S}\end{array}\right)~{}.$ (22)

This is consistent with the fact that $h$ and $\mathbf{S}$ are density and
flux associated with integrated helicity, the conserved quantity associated
with the dual symmetry [9, 30] (hence $\nabla\cdot\mathbf{S}=0$ for free-space
monochromatic fields). This four-vector current $S^{\mu}$ has a decomposition
equivalent to Eq. (4),

$S^{\mu}=S^{\mu}_{C}+S^{\mu}_{P},$ (23)

where the individual components are given by,

$\displaystyle
S_{C}^{\mu}=\frac{1}{4}\operatorname{Re}\left\\{A_{\nu}^{*}\left(\partial^{\mu}C^{\nu}\right)-C_{\nu}^{*}\left(\partial^{\mu}A^{\nu}\right)\right\\}=\frac{1}{4\omega^{2}}\left(\begin{array}[]{c}-2(\omega/c)\operatorname{Im}\left\\{\mathbf{E}^{*}\cdot\mathbf{H}\right\\}\\\
\operatorname{Re}\left\\{\mathbf{E}^{*}\cdot(\mathbf{\nabla})\mathbf{H}-\mathbf{H}^{*}\cdot(\mathbf{\nabla})\mathbf{E}\right\\}\end{array}\right)=\left(\begin{array}[]{c}h\\\
\mathbf{s}_{c}\end{array}\right),$ (24) $\displaystyle
S_{P}^{\mu}=\frac{1}{4}\operatorname{Re}\left\\{C_{\nu}^{*}\left(\partial^{\nu}A^{\mu}\right)-A_{\nu}^{*}\left(\partial^{\nu}C^{\mu}\right)\right\\}=\frac{1}{4\omega^{2}}\left(\begin{array}[]{c}0\\\
\operatorname{Re}\left\\{\left(\mathbf{H}^{*}\cdot\nabla\right)\mathbf{E}-\left(\mathbf{E}^{*}\cdot\mathbf{\nabla}\right)\mathbf{H}\right\\}\end{array}\right)=\left(\begin{array}[]{c}0\\\
\mathbf{s}_{p}\end{array}\right),$

reproducing the spin decomposition in 4-vector notation. Note that the
temporal part of the total spin, the helicity density, is carried only by the
canonical 4-spin term while the time component of the Poynting 4-spin is zero.

Let us briefly comment on the choice of the Coulomb gauge. As can be seen from
the supplementary information, the decomposition of $S^{\mu}$ is independent
of the choice of frame and gauge. As already discussed in previous work (e.g.
[9, 41]), while the local density $S^{\mu}$ is not gauge independent, upon
integration one gets integrated helicity, which is a gauge-independent
quantity that is conserved in any scenario where there is electromagnetic dual
symmetry. Furthermore, the integral depends only on the transverse components
of the potentials, which makes the Coulomb gauge the most convenient gauge to
work with, and in this gauge, the helicity flux (spatial component of
$S^{\mu}$) also coincides with the optical spin density. Note, however, that
the gauge fixed version of this decomposition is not invariant under Lorentz
boosts, unless we simultaneously change our potentials to the Coulomb gauge
associated with the new boosted frame.

##  5 Spin decomposition examples

It is instructive to apply the spin decomposition in Eq. (4) to some simple
examples of light: a general evanescent wave and three commonly known focused
beams, namely a Gaussian beam, a radial/azimuthal beam ($l=0$), and an $l=1$
vortex beam (all linearly polarised in the transverse plane). What
differentiates each of these fields are their energy density and phase
structures, due to one-dimensionally monotonic, doughnut or Gaussian real-
space amplitude profiles, and the presence of OAM, two characteristics which
reorient the Poynting vector throughout space and generate different amounts
of Poynting spin. Poynting spin can emerge even when the instantaneous vectors
$\boldsymbol{\mathcal{E}}$ and $\boldsymbol{\mathcal{H}}$ do not actually
rotate, because its counterpart (canonical spin) is able to counterbalance the
total SAM density of the field as necessary. Interplay between spin’s
canonical and Poynting components creates counter-intuitive effects, such as
spin-free chiral interaction forces.

###  5.1 Evanescent wave

The electric and magnetic field phasors of an evanescent wave of arbitrary
polarisation, propagating in the $z$ direction ($k_{z}>k$) and decaying along
the $x$ axis with decay constant $\gamma=\sqrt{k_{z}^{2}-k^{2}}$, are,

$\mathbf{E}=\begin{pmatrix}A_{p}\frac{k_{z}}{k}\\\ A_{s}\\\
-iA_{p}\frac{\gamma}{k}\end{pmatrix}e^{ik_{z}z-\gamma
x}\quad\mathbf{H}=\frac{1}{\eta}\begin{pmatrix}-A_{s}\frac{k_{z}}{k}\\\
A_{p}\\\ iA_{s}\frac{\gamma}{k}\end{pmatrix}e^{ik_{z}z-\gamma x},$ (25)

where $\eta=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}$. Choosing complex values for
$A_{s}$ and $A_{p}$, which are the amplitudes of the evanescent wave’s TE and
TM modes, controls the wave’s polarisation—a circularly polarised wave, for
instance, has $A_{p}=\pm iA_{s}$. The energy density of the wave decays in the
$x$ direction and is given by,

$W=\epsilon_{0}\frac{k_{z}^{2}}{k^{2}}e^{-2\gamma
x}\frac{1}{2}\left(|A_{s}|^{2}+|A_{p}|^{2}\right).$ (26)

Using our formulae, we can calculate the energy-normalised total spin of the
evanescent wave, as well as its canonical and Poynting spins,

$\frac{\mathbf{S}}{W}=\frac{1}{\omega
k_{z}}\left[\gamma\mathbf{\hat{y}}+k\sigma\mathbf{\hat{z}}\right],$ (27)
$\frac{\mathbf{s}_{c}}{W}=\frac{1}{\omega
k_{z}}\left[\frac{k_{z}^{2}}{k}\sigma\mathbf{\hat{z}}\right],$ (28)
$\frac{\mathbf{s}_{p}}{W}=\frac{1}{\omega
k_{z}}\left[\gamma\mathbf{\hat{y}}-\frac{\gamma^{2}}{k}\sigma\mathbf{\hat{z}}\right].$
(29)

The parameter
$\sigma=2\textrm{Im}\\{A_{s}A_{p}^{*}\\}/(|A_{s}|^{2}+|A_{p}|^{2})$ is the
degree of circular polarisation in the sense of a plane wave ($\sigma=\pm 1$
for circular polarisation, $\sigma=0$ for linear polarisation). It is well-
known that evanescent waves carry transverse spin independently of
polarisation [26]. This property is accounted for by the $\mathbf{\hat{y}}$
component of $\mathbf{S}$, which is unaffected by the relationship between
$A_{s}$ and $A_{p}$ and, interestingly, is a product solely of Poynting spin
as was discovered in [23]. Meanwhile, the evanescent wave acquires a
longitudinal spin component if $\sigma\neq 0$, a component which is
contributed to by both decomposed spins, and in different amounts. The over-
generous canonical spin develops a larger $\mathbf{\hat{z}}$ component than is
physical for the total spin of the wave
($\mathbf{s}_{c}\cdot\mathbf{\hat{z}}>\mathbf{S}\cdot\mathbf{\hat{z}}$).
Compensating, the Poynting spin’s $\mathbf{\hat{z}}$ component points
backwards to ensure
$(\mathbf{s}_{c}+\mathbf{s}_{p})\cdot\mathbf{\hat{z}}=\mathbf{S}\cdot\mathbf{\hat{z}}$
(this becomes clear after substituting $\gamma^{2}=k_{z}^{2}-k^{2}$).

Decomposing an evanescent wave’s spin reveals a physical distinction between
its transverse and longitudinal spin components. Since Poynting spin is
responsible for the transverse component of $\mathbf{S}$, transverse chiral
forces felt by an enantiomer arise as it recoils from its own radiation,
rather than from a direct field interaction. This is consistent with the fact
that perpendicular to the wavevector, there is no phase advance to twist the
rotating field vectors into helices—the evanescent wave only acquires helicity
if it carries longitudinal spin ($h=W\sigma/\omega$). Canonical spin, purely
longitudinal on the other hand, couples directly to chiral matter to produce a
significantly stronger (relatively) preferential force.

###  5.2 Beams

Let us first consider a tightly focussed, linearly polarised Gaussian beam,
whose energy density and polarisation in the focal plane is plotted in the top
row of Fig. 2(a). In the transverse $xy$ plane, the projection of the electric
and magnetic field vectors are linear ($x$ and $y$ polarised respectively),
though due to the tight focusing both $\mathbf{E}$ and $\mathbf{H}$ have
significant longitudinal $z$ components, such that they are elliptically
polarised in 3D and contribute a circulation of transverse spin. The Poynting
vector magnitude is not constant across the face of the beam and therefore has
a non-zero curl in the transverse plane (except in the local maximum at the
beam centre). This, the total spin and the decomposed canonical and Poynting
spin of the Gaussian beam are plotted in Fig. 2(a)’s column (descending). Both
$\mathbf{E}$ and $\mathbf{H}$ fields have $z$ components and discretely
symmetric polarisation ellipses in 3D (after a rotation of one field by
$\pi/2$), which appears to reduce the canonical spin of the beam to zero,
$\mathbf{s}_{c}=\mathbf{0}$. All transverse spin developed by focussing of a
linearly polarised Gaussian beam stems from Poynting spin,
$\mathbf{S}=\mathbf{s}_{p}$. A similar conclusion could be made for an
evanescent wave, making it tempting to argue for a general association between
transverse spin and Poynting spin. This notion is not supported by the
following example, however.

An $l=0$ azimuthally polarised (in the electric field) beam and its spin
decomposition is plotted in Fig 2(b). Considering dual spin (contributions
from both $\mathbf{E}$ and $\mathbf{H}$), the decomposition properties of this
beam are identical for a radially polarised beam, which has an azimuthal
magnetic field. Despite tight focussing, the azimuthal electric field vector
does not carry a longitudinal component, as is well-known, and is completely
linearly polarised in 3D. This means that the beam’s wholly transverse spin is
supplied by the magnetic field alone. Combining to give the total spin of the
beam, both canonical and Poynting spins are non-zero—this contrasts with the
Gaussian beam, whose electric and magnetic fields are both elliptical in 3D.
Neither beam considered so far has twisted wavefronts carrying OAM, though due
to their azimuthal transverse spin components, both the Gaussian and radial
beam possess a certain chiral OAM. A chiral particle could access the spatial
structure of the beam’s spin field $\mathbf{S}$, feeling an azimuthal force
(clockwise or anticlockwise depending on the enantiomer) which would cause the
particle to orbit the beam centre with no need for helical phasefronts.

Finally, we treat the $x$-polarised $l=1$ vortex beam [Fig. 2(c)], whose OAM
leads to the most surprising spin decomposition features of the three.
Numerically simulating such a non-paraxial, focussed beam is a challenging
task because there is no unanimous 3D definition of a vector vortex beam (only
scalar and paraxial beams are well-defined). The beams treated in this section
are all electric linearly polarised and generated numerically using an angular
spectrum integration technique [43] which, by convention, forces the electric
field to exactly match the paraxial description of the beam in the focal
plane. All unenforced field components (i.e., all magnetic field components
and the longitudinal electric component throughout space, and the transverse
electric components outside of the focal plane) are subsequently calculated
via Maxwell’s equations. While the beams are, therefore, physically valid,
some magnetic-biased polarisation features develop in the $x$-polarised vortex
beam considered here. In particular, although the electric field remains
perfectly linear, the magnetic field alone gains a slight ellipticity in the
transverse plane, contributing to a small longitudinal total spin outside of
the beam centre as has been shown to exist in focussed linearly polarised
beams [16]. Longitudinal spin remains zero in the singularity, however.
Breaking continuous rotational symmetry via $x$ polarisation, both the vortex
beam’s electric and magnetic fields possess a $z$ component from focussing, in
contrast to the azimuthal beam for which the electric field is perfectly
linearly polarised in 3D. The beam has helical wavefronts so that the average
phase gradient (local wavevector) has an azimuthal component, also inherited
by the Poynting vector (shown in the second row of Fig. 2(c); there is a small
transverse component circulating in the $xy$ plane). The Poynting spin
$\mathbf{s}_{p}\propto\nabla\times\mathbf{P}$, therefore, acquires a
longitudinal component (even at the centre of the beam), in outright defiance
of the fact that both transverse electric and magnetic fields are nearly
linearly polarised, and zero in the centre, as is visible in the lowest plot
of Fig. 2(c). To neutralise the longitudinal Poynting spin and suppress the
total longitudinal spin, the $z$ component of canonical spin $\mathbf{s}_{c}$
must be non-zero too, satisfying $s_{cz}=-s_{pz}$ for $S_{z}=s_{cz}+s_{pz}=0$
in the centre of the beam, and $S_{z}=s_{cz}+s_{pz}\approx 0$ elsewhere (see
the plot second from bottom). In the middle of the beam, this counteracting
canonical spin points in the negative $z$ direction as a consequence of the
beam’s vortex handedness (right, topological charge $+1$), and would switch
sign in a left-handed vortex. Returning now to the chiral force equation Eq.
(5), we can infer that the beam’s non-zero canonical spin, proportional to
chiral momentum, produces a longitudinal chiral force, even though near to the
vortex centre the $\mathbf{E}$ and $\mathbf{H}$ fields have negligible
ellipticity in the transverse plane. Perhaps counter-intuitively, the
longitudinal chiral force is strongest in the centre of the beam where the
electromagnetic field is virtually zero (due, of course, to the longitudinal
curl of the Poynting vector being maximal there). This is another example of
spatially chiral light coupling to chiral matter which, even under the dipole
approximation and in a linearly polarised field, can feel a discriminatory
force by absorbing or scattering photons carrying OAM of a certain handedness.
Only recently has OAM-dependent chirality, able to couple even to chiral
dipoles, been demonstrated in focussed beams, resulting from strong
longitudinal field components [17, 44]. We point out, however, that the chiral
pressure force which we theorise at the centre of a linearly polarised vortex
appears only to require that the Poynting vector has an azimuthal component
(i.e., no requirement for longitudinal field components), and should still be
present in a paraxial beam.

Figure 2:  Spin decomposition of non-paraxial beams. The beams, each of waist
$1.5\lambda$ and separated in columns, are (a) a linearly polarised Gaussian
beam ($\mathbf{E}^{T}||\mathbf{\hat{x}}$), (b) an azimuthally polarised
($\mathbf{E}||\boldsymbol{\hat{\phi}}$), $l=0$ doughnut beam, and (c) a
linearly polarised ($\mathbf{E}^{T}||\mathbf{\hat{x}}$) vortex beam with
topological charge $l=1$. The top row of (isometric) plots across each sub
figure column shows the beam energy density in colour, as well as the Poynting
vector and electric (blue) and magnetic (green) polarisation ellipses, which
are elliptical due to a significant $z$ field component. Subsequent rows are
vector plots of each beam’s Poynting vector $\mathbf{P}$, total spin
$\mathbf{S}$ Eq. (3), canonical spin $\mathbf{s}_{c}$ and Poynting spin
$\mathbf{s}_{p}$ from Eq. (4), respectively. White arrows are projections of
the corresponding vector into the $xy$ plane, while the red arrows are
projections of the vector onto longitudinal $yz$ and $xz$ cut planes. Within
each beam, arrows in the three spin decomposition plots are drawn to a
consistent scale. Each non-paraxial beam is generated using an angular
spectrum integration method [43]. Defining the 3D vector vortex beam (c) is a
difficult problem and the method we used produces a small and physical
longitudinal spin (third row of (c)), contributed by the magnetic field, which
would not be present in a (non-physical) perfect paraxial beam.

##  6 Analogies in other wave fields

Kinetic, canonical and spin momenta analogous to the terms in Eq. (2) can be
identified more generally in other wave fields [45, 46, 47]. This, and the
ability to split the SAM density of light into two distinct terms poses
another curiosity: what does a spin decomposition of another wave field look
like, particularly if, unlike the electromagnetic field, its quanta are not
spin-1? Treated in this section are acoustic and gravitational waves, which
both depart from light’s spin-1 structure. Despite their increased complexity,
linearised theories exist for each of these fields under certain conditions.
In a perfect fluid, acoustic waves are linear and oscillate longitudinally
(spin-0), while gravitational waves are tensorial in nature (spin-2) and
combine linearly when their amplitudes are sufficiently low.

In formulating equivalent expressions for total SAM, canonical and Poynting
spins in linearised gravity, we use the Maxwellian representation of gravity
in the weak field limit [27], i.e., waves propagating over flat spacetime at a
large distance from their source. Gravitational waves detected on earth [48,
49] arrive within this limit, and share some of light’s characteristics; both
fields have two polarisation degrees of freedom (in gravity these are the
$h_{+}$ and $h_{\times}$ polarisations) and are massless. Some of light’s
strangest behaviour, such as in evanescent fields, have also been predicted
with additional properties in gravitational waves [50]. Note that we take the
expressions for the helicity and SAM density that are dual-symmetric (i.e.,
$\mathbf{S}=\mathbf{S}_{e}+\mathbf{S}_{m}$ rather than
$\mathbf{S}=2\mathbf{S}_{e}$), the same as in the case of electromagnetism,
despite the fact that in the case of gravity we have no experimental evidence
favouring its physical relevance over the asymmetric version [51]. However, if
we take the point of view of [9] that only the integrated helicity is a
physically meaningful quantity, the two will be equivalent.

Table 1 summarises differences and similarities between linearised acoustics,
electromagnetism, and linearised gravity with a focus on the spin-related
quantities. An acoustic wave field can be described by a scalar pressure field
$P$ and a vector velocity field $\mathbf{v}$ which, in linearised acoustic
theory, share a Maxwell-like relation [46, 47]. The derivation of the
decomposition for the acoustic field is given in the supplementary material. A
gravitational wave can be described using a metric perturbation $h_{ij}$,
which can be thought of as components of a three-by-three symmetric matrix. If
we consider only $i$-th row or column of this matrix as a vector potential,
${\boldsymbol{\mathbf{A}}}^{i}=h^{ij}\hat{\mathbf{e}}_{j}$, then the
Maxwellian representation of gravity can be written in vector notation; the
derivation of the spin decomposition then becomes simply Eq. (12). The
acoustic helicity density is zero, which is a feature of spin-0 fields, but
interestingly the canonical spin, which we showed in Eq. (41) to be related to
helicity, is also zero. One can see that electromagnetic and gravitational
waves are spin-1 and spin-2 respectively by taking a circularly polarised
wave. The helicity density of a circularly polarised wave will be one(two)
times the energy density and canonical spin will be one(two) times the
canonical momentum for an electromagnetic(gravitational) wave.

Linearised Acoustics Electromagnetism Linearised gravity Field phasors
$P=-i\omega\rho\varphi$ $\mathbf{v}=\mathbf{\nabla}\varphi$
$\mathbf{E}=i\omega\mathbf{A}$
$\mathbf{H}=\frac{1}{\mu_{0}}\mathbf{\nabla}\times\mathbf{A}$
$\mathbf{E}^{i}=i\omega(h^{ij}\hat{\mathbf{e}}_{j})$
$\mathbf{H}^{i}=\frac{1}{\mu_{0}}\nabla\times(h^{ij}\hat{\mathbf{e}}_{j})$
Energy density $\frac{1}{4}\\!\left(\beta|P|^{2}+\rho|\mathbf{v}|^{2}\right)$
$\frac{1}{4}\\!\left(\epsilon_{0}|\mathbf{E}|^{2}+\mu_{0}|\mathbf{H}|^{2}\right)$
$\frac{1}{4}\\!\left(\epsilon_{0}\mathbf{E}^{*}_{i}\cdot\mathbf{E}^{i}+\mu_{0}\mathbf{H}^{*}_{i}\cdot\mathbf{H}^{i}\right)$
Helicity density $0$ $-\frac{1}{2\omega c}\textrm{Im}\\{\mathbf{E^{*}\cdot
H}\\}$ $-\frac{1}{\omega
c}\textrm{Im}\\{\mathbf{E}_{i}^{*}\cdot\mathbf{H}^{i}\\}$ Poynting vector
$\frac{1}{2}\textrm{Re}\\{P^{*}\mathbf{v}\\}$
$\frac{1}{2}\textrm{Re}\\{\mathbf{E^{*}\times H}\\}$
$\frac{1}{2}\textrm{Re}\\{\mathbf{E}_{i}^{*}\times\mathbf{H}^{i}\\}$ SAM
density $\frac{1}{2\omega}\textrm{Im}\\{\rho\mathbf{v^{*}}\times\mathbf{v}\\}$
$\frac{1}{4\omega}\textrm{Im}\\{\epsilon_{0}\mathbf{E^{*}}\times\mathbf{E}+\mu_{0}\mathbf{H^{*}}\times\mathbf{H}\\}$
$\frac{1}{2\omega}\textrm{Im}\\{\epsilon_{0}\mathbf{E}^{*}_{i}\times\mathbf{E}^{i}+\mu_{0}\mathbf{H}^{*}_{i}\times\mathbf{H}^{i}\\}$
Canonical spin $0$
$\frac{1}{4\omega^{2}}\textrm{Re}\\{\mathbf{E^{*}}\cdot(\nabla)\mathbf{H}-\mathbf{H^{*}}\cdot(\nabla)\mathbf{E}\\}$
$\frac{1}{2\omega^{2}}\textrm{Re}\\{\mathbf{E}^{*}_{i}\cdot(\nabla)\mathbf{H}^{i}-\mathbf{H}^{*}_{i}\cdot(\nabla)\mathbf{E}^{i}\\}$
Poynting spin
$\frac{1}{2\omega^{2}}\nabla\times\frac{1}{2}\textrm{Re}\\{P^{*}\mathbf{v}\\}$
$\frac{1}{2\omega^{2}}\nabla\times\frac{1}{2}\textrm{Re}\\{\mathbf{E^{*}\times
H}\\}$
$\frac{1}{\omega^{2}}\nabla\times\frac{1}{2}\textrm{Re}\\{\mathbf{E}^{*}_{i}\times\mathbf{H}^{i}\\}$

Table 1:  Comparison of time-averaged energy, momentum and spin densities
between time-harmonic waves in theories of linearised acoustics,
electromagnetism and linearised gravity. Table inspired by [46, 45, 50]. For
electromagnetism, the potential is considered to be in the Coulomb gauge. For
linearised gravity, $h_{ij}$ are spatial components of the metric perturbation
in the transverse-traceless gauge, $\hat{\mathbf{e}}_{i}$ are basis vectors,
and any repeated indices are summed over (Einstein’s convention). Parameters
$\epsilon_{0}=1/(c^{2}\mu_{0})=c^{2}/(32\pi G)$ were chosen such that the
time-averaged energy density takes the same form as the expression for the
electromagnetic field. A larger version of this table using the index notation
introduced in Sec. 4.2 is given in the supplementary material.

##  7 Conclusions

We have discussed a known but overlooked decomposition which exists for the
spin angular momentum density of light, in a similar way to the Poynting
vector, which can be split into orbital and spin currents. Spin is decomposed
into two terms, which we call the canonical spin $\mathbf{s}_{c}$ and Poynting
spin $\mathbf{s}_{p}$. We have further expressed the decomposition for time-
varying fields, as well as in four-vector notation. The canonical spin is
proportional to chiral linear momentum which can be transferred to chiral
enantiomers in a positive or negative direction, depending on the enantionmer
handedness. The resulting force is a chiral analogy to the (achiral) force due
to radiation pressure—both are relatively strong first order forces and can
act on matter in the dipole approximation.

The mechanism for preferential photon absorption or scattering by chiral
matter is normally associated with the photon’s spin state (i.e., circular
dichroism), and less so to the sign of its OAM (although these interactions
have received recent attention). We emphasised, however, that light is capable
of exerting a first-order chiral force which does not directly depend on the
total SAM of light, rather, its canonical spin, which can take a non-zero
value even in linearly polarised fields. This is due to the second term in the
spin decomposition, the Poynting spin, which by definition (being the curl of
the Poynting vector) depends strongly on light’s OAM: optical vortices
twisting energy flow around the beam axis naturally generate non-zero Poynting
spin, regardless of polarisation. We showed that under linear polarisation, an
optical vortex’s Poynting spin must be compensated by canonical spin, to
produce a longitudinal chiral force in the absence of longitudinal SAM.
Interestingly, this OAM-dependent chiral force is strongest in the centre of
the vortex, where the electromagnetic energy density is minimal, along with
achiral forces such as radiation pressure and gradient force. It is possible
to trap atoms and molecules (using a blue-detuned wavelength with respect to a
strong resonance) in dark spots [52, 53, 54].

The ability to decompose spin appears to be a general property of wave fields,
and as we demonstrated, is straightforward to perform in linearised acoustics
and gravity, whose quanta are considered spin-0 and spin-2 respectively. This
fact could be of significant interest to a broader community, beyond optics.

##  8 Acknowledgements

C. R. acknowledges support from a Science and Technology Facilities Council
(STFC) Doctoral Training Grant. F. J. R-F. and S. G. are supported by EIC-
Pathfinder-CHIRALFORCE (101046961) funded by Innovate UK Horizon Europe
Guarantee (UKRI project 10045438). A. J. V. is supported by EPSRC Grant
EP/R513064/1.

##  9 Supplementary Information

This section supports the main text with four main components. In section 9.1,
we provide an alternative version of Table I of the manuscript, expressed
completely using index notation, which highlights the similarities and
differences between vector and tensor waves. Secondly, section 9.2 is a
discussion of the equivalent SAM density decomposition in linerised acoustics,
which informs the first column of Table I of the manuscript. In section 9.3,
details of the units of the quantities handled in the manuscript’s 4-vector
decomposition are given. Finally in section 9.4, the electromagnetic spin
decomposition is presented in a general gauge, supporting comments made in the
in 4-vector section of the main text.

###  9.1 Alternative table (index notation)

An alternative version of Table 1 from the manuscript is given below, where
all quantities are expressed using index notation.

Linearised acoustics Electromagnetism Linearised gravity Potential field
Scalar field $\varphi$ Vector field $A^{\mu}$ Tensor field $h_{\mu\nu}$ Choice
of gauge n/a Coulomb gauge: $A^{0}=0$ $\partial_{i}A^{i}=0$ Transverse
traceless gauge: $h_{0\mu}=h^{i}{}_{i}=0$ $\partial_{i}h^{ij}=0$ Helmholtz
equation $\nabla^{2}\varphi=-k^{2}\varphi$ $\nabla^{2}A^{i}=-k^{2}A^{i}$
$\nabla^{2}h^{ij}=-k^{2}h^{ij}$ Fields $P=-i\omega\rho\varphi$
$v^{i}=\partial^{i}\varphi$ $E^{i}=i\omega A^{i}$
$\mu_{0}H^{i}=\epsilon^{ijk}\partial_{j}A_{k}$ $E^{ij}=i\omega h^{ij}$
$\mu_{0}H^{ij}=\epsilon^{ikm}\partial_{k}h^{j}_{~{}m}$ Spin
$\frac{\rho}{2\omega}\epsilon^{ijk}\textrm{Im}\\{v^{*}_{j}v_{k}\\}$
$\frac{1}{4\omega}\epsilon^{ijk}\textrm{Im}\\{\epsilon_{0}E^{*}_{j}E_{k}+\mu_{0}H^{*}_{j}H_{k}\\}$
$\frac{1}{2\omega}\epsilon^{ijk}\textrm{Im}\\{\epsilon_{0}E^{*}_{jl}E_{k}{}^{l}+\mu_{0}H^{*}_{jl}H_{k}{}^{l}\\}$
Canonical spin $0$
$\frac{1}{4\omega^{2}}\textrm{Re}\\{E^{*}_{j}\partial_{i}H^{j}-H^{*}_{j}\partial_{i}E^{j}\\}$
$\frac{1}{2\omega^{2}}\textrm{Re}\\{E^{*}_{jn}\partial_{i}H^{jn}-H^{*}_{jn}\partial_{i}E^{jn}\\}$
Poynting spin
$\frac{1}{2\omega^{2}}\epsilon^{ijk}\partial_{j}\frac{1}{2}\textrm{Re}\\{P^{*}v_{k}\\}$
$\frac{1}{2\omega^{2}}\epsilon^{ijk}\partial_{j}\frac{1}{2}\textrm{Re}\\{\epsilon_{klm}E^{*l}H^{m}\\}$
$\frac{1}{\omega^{2}}\varepsilon^{ijk}\partial_{j}\frac{1}{2}\textrm{Re}\\{\varepsilon_{klm}E^{*l}{}_{n}H^{mn}\\}$
Energy density $\frac{1}{4}\\{\beta p^{*}p+\rho v_{i}^{*}v^{i}\\}$
$\frac{1}{4}\\{\epsilon_{0}E^{*}_{j}E^{j}+\mu_{0}H^{*}_{j}H^{j}\\}$
$\frac{1}{4}\\{\epsilon_{0}E^{*}_{jk}E^{jk}+\mu_{0}H^{*}_{jk}H^{jk}\\}$
Maxwell(-like) equations $\partial_{i}P=i\omega\rho v^{i}$
$\partial_{i}v^{i}=i\omega\beta P$ $\epsilon^{ijk}\partial_{j}v_{k}=0$
$\partial_{i}E^{i}=0$ $\partial_{i}H^{i}=0$
$\epsilon^{ijk}\partial_{j}E_{k}=i\omega\mu_{0}H^{i}$
$\epsilon^{ijk}\partial_{j}H_{k}=-i\omega\epsilon_{0}E^{i}$
$\partial_{i}E^{ij}=0$ $\partial_{i}H^{ij}=0$
$\epsilon^{ijk}\partial_{j}E_{k}{}^{l}=i\omega\mu_{0}H^{il}$
$\epsilon^{ijk}\partial_{j}H_{k}{}^{l}=-i\omega\epsilon_{0}E^{il}$

Table 2:  Comparison between acoustics, electromagnetism and linearised
gravity. Table inspired by [46, 45, 50]. The parameter
$\epsilon_{0}=1/(c^{2}\mu_{0})=c^{2}/(32\pi G)$ for linearised gravity was
chosen such that the time-averaged energy density takes the same form as for
electromagnetism [27].

###  9.2 Decomposition in Linearised acoustics

An acoustic wave field can be described by a scalar pressure field $P$ and a
vector velocity field $\mathbf{v}$ which, in linearised acoustic theory, share
a Maxwell-like relation [46, 47]. The time-harmonic equations are,

$\nabla\cdot\mathbf{v}=i\beta\omega P,$ (30) $\nabla P=i\rho\omega\mathbf{v}.$
(31)

where the constants $\beta$ and $\rho$ are the acoustic medium’s
compressibility and mass density respectively. Compared to photons, acoustic
phonons, which are spin-0 quanta in this regime, do not give acoustic fields
as rich a vector structure as light. Constraining the velocity vector is the
longidutinality condition, that is $\nabla\times\mathbf{v}=\mathbf{0}$, a more
restrictive analogy to light’s transversality condition due to Gauss’ law.
Yet, $\mathbf{v}$ can still rotate and generate acoustic SAM, which is
expressed (time-averaged) by,

$\mathbf{S}_{ac}=\frac{\rho}{2\omega}\textrm{Im}\\{\mathbf{v^{*}\times v}\\}.$
(32)

In time-harmonic acoustic fields, where the velocity $\mathbf{v}$ and
displacement field $\mathbf{r}$ vectors are related by
$\mathbf{v}=-i\omega\mathbf{r}$, particles in the acoustic medium have an
elliptical motion. Meanwhile, an acoustic analogy to the Poynting vector can
be defined by mixing the acoustic pressure and velocity fields,

$\mathbf{P}_{ac}=\frac{1}{2}\textrm{Re}\\{P^{*}\mathbf{v}\\}.$ (33)

Taking the curl of $\mathbf{P}_{ac}$ and following with use of Eq. (31), the
acoustic SAM emerges in what appears to be the acoustic analogy to the
presented electromagnetic spin decomposition:

$\mathbf{S}_{ac}=-\frac{1}{2\omega^{2}}\textrm{Re}\\{P^{*}(\nabla\times\mathbf{v})\\}+\frac{1}{2\omega^{2}}\nabla\times\mathbf{P}_{ac}.$
(34)

Due to the longitudinality condition $\nabla\times\mathbf{v=0}$, the first
term in Eq. (34) vanishes leaving the acoustic spin entirely proportional to
the curl of the acoustic Poynting vector. Transverse phonons for which
$\nabla\times\mathbf{v\neq 0}$ can occur in viscous fluids or solids, although
in these media, the acoustic field could no longer be described by a
linearised theory. Compared to the electromagnetic spin decompostion, however,
Eq. (34) highlights the structural distinction between spin-0 and spin-1
fields as only vorticity in the flow of energy can generate SAM in a linear
acoustic field.

###  9.3 On units

Discussions in this section are related to the 4-vector extension of the
decomposition presented in section 4.2 of the manuscript. The field strength
tensor $F_{\mu\nu}$ and conjugate field strength tensor $G_{\mu\nu}$ are not
of the same units. The former has SI units
$\mathrm{k}\mathrm{g}\,\mathrm{s}^{-4}\,\mathrm{A}^{-1}$ whilst the latter has
SI units
$\mathrm{m}^{-1}\,\mathrm{k}\mathrm{g}\,\mathrm{s}^{-5}\,\mathrm{A}^{-1}$.
This comes from the form of the conjugate field strength tensor that can be
derived using the companion Mathematica notebook [42]:

$G_{\mu\nu}=\left[\begin{array}[]{cccc}0&H_{x}/c&H_{y}/c&H_{z}/c\\\
-H_{x}/c&0&-\epsilon_{0}E_{z}&\epsilon_{0}E_{y}\\\
-H_{y}/c&\epsilon_{0}E_{z}&0&-\epsilon_{0}E_{x}\\\
-H_{z}/c&-\epsilon_{0}E_{y}&H_{x}/c^{2}&0\end{array}\right].$ (35)

Finally, we summarize all the SI units of the mentioned tensor and constants:

| E | H | $\epsilon_{0}$ | $\mu_{0}$ | A | $\phi$ | C
---|---|---|---|---|---|---|---
Units | $\mathrm{m}\,\mathrm{k}\mathrm{g}\,\mathrm{s}^{-3}\,\mathrm{A}^{-1}$ | $\mathrm{A}\,\mathrm{m}^{-1}$ | $\mathrm{k}\mathrm{g}^{-1}\,\mathrm{m}^{-3}\,\mathrm{s}^{4}\,\mathrm{A}^{-2}$ | $\mathrm{k}\mathrm{g}\,\mathrm{m}\,\mathrm{s}^{-2}\,\mathrm{A}^{-2}$ | $\mathrm{k}\mathrm{g}\,\mathrm{m}\,\mathrm{s}^{-2}\,\mathrm{A}^{-1}$ | $\mathrm{k}\mathrm{g}\,\mathrm{m}^{2}\,\mathrm{s}^{-3}\,\mathrm{A}^{-1}$ | $\mathrm{A}\,\mathrm{m}^{-1}\,\mathrm{s}$
Table 3: Summary of the SI units of every tensors and constants mentioned in
this work

###  9.4 On the choice of gauge

The electric field and magnetic field, respectively $\mathbf{E}$ and
$\mathbf{H}$, can be expressed in terms of a scalar potential $\phi$ and a
vector potential A,

$\begin{split}&\boldsymbol{\mathcal{E}}=-\frac{\partial\boldsymbol{\mathcal{A}}}{\partial
t}-\mathbf{\nabla}\varphi\\\
&\mu_{0}\boldsymbol{\mathcal{H}}=\mathbf{\nabla}\times\boldsymbol{\mathcal{A}}\end{split}$
(36)

and respectively for the conjugate vector potential:

$\begin{split}\epsilon_{0}\boldsymbol{\mathcal{E}}&=-\mathbf{\nabla}\times\boldsymbol{\mathcal{C}}\\\
\boldsymbol{\mathcal{H}}&=-\frac{\partial\boldsymbol{\mathcal{C}}}{\partial
t}-\mathbf{\nabla}\psi\end{split}$ (37)

When doing computations in the Coulomb gauge, where the scalar potentials
$\varphi=\psi=0$, we can easily express the electric and magnetic field in
terms of the vector potentials through:

$E^{j}=i\omega A^{j}(\omega,r),\qquad H^{j}=i\omega C^{j}(\omega,r),$ (38)

which implies that we can express the four vector potential as:

$A^{\mu}=\frac{1}{i\omega}\begin{pmatrix}0\\\ \mathbf{E}\end{pmatrix},\qquad
C^{\mu}=\frac{1}{i\omega}\begin{pmatrix}0\\\ \mathbf{H}\end{pmatrix}.$ (39)

Working in the Coulomb gauge simplifies greatly the form of the spin
decomposition, giving a clearer physics intuition. However, it is possible to
derive all the results presented in the main work without choosing a gauge. In
a general gauge, we found that the total spin is given by:

$\begin{split}S^{\mu}&=\frac{1}{4}\operatorname{Re}\left\\{A_{\nu}^{*}G^{\nu\mu}+C_{\nu}^{*}F^{\nu\mu}\right\\}=\frac{1}{4}\operatorname{Re}\left\\{\left(\begin{array}[]{c}\frac{1}{c}\left(\mathbf{A}\cdot\mathbf{H}^{*}-\mathbf{C}\cdot\mathbf{E}^{*}\right)\\\
\mu\mathbf{H}^{*}\times\mathbf{C}+\varepsilon\mathbf{E}^{*}\times\mathbf{A}\end{array}\right)+\frac{1}{c^{2}}\left(\begin{array}[]{c}0\\\
\phi\mathbf{H}^{*}-\psi\mathbf{E}^{*}\end{array}\right)\right\\}\\\
&=\frac{1}{4\omega}\operatorname{Im}\left\\{\left(\begin{array}[]{c}-\frac{2}{c}\mathbf{E}^{*}\cdot\mathbf{H}\\\
\epsilon_{0}\mathbf{E}^{*}\times\mathbf{E}+\mu_{0}\mathbf{H}^{*}\times\mathbf{H}\end{array}\right)+\left(\begin{array}[]{c}\frac{1}{c}{\nabla}\cdot\left(\psi\mathbf{E}^{*}-\phi\mathbf{H}^{*}\right)\\\
{\nabla}\times\left(\varepsilon\phi\mathbf{E}^{*}+\mu\psi\mathbf{H}^{*}\right)\end{array}\right)\right\\}.\end{split}$
(40)

Interestingly, the extra terms for a general gauge are total derivatives,
meaning that the integrated helicity is gauge invariant up to a boundary term.
This is because generically, total derivatives vanish under suitable boundary
conditions. The canonical and Poynting spin now read as:

$\displaystyle
S_{C}^{\mu}=\frac{1}{4}\operatorname{Re}\left\\{A_{\nu}^{*}\left(\partial^{\mu}C^{\nu}\right)-C_{\nu}^{*}\left(\partial^{\mu}A^{\nu}\right)\right\\}=\frac{1}{4}\operatorname{Re}\left\\{\begin{pmatrix}-{2i\omega}(\phi^{*}\psi)/c^{3}+{i\omega}(\mathbf{A}^{*}\cdot\mathbf{C}-\mathbf{C}^{*}\cdot\mathbf{A})/c\\\
\mathbf{A}^{*}\cdot(\nabla)\mathbf{C}-\mathbf{C}^{*}\cdot(\nabla)\mathbf{A}+\frac{1}{c^{2}}(\psi^{*}\nabla\phi-\phi^{*}\nabla\psi)\par\end{pmatrix}\right\\},$
(41) $\displaystyle
S_{P}^{\mu}=\frac{1}{4}\operatorname{Re}\left\\{C_{\nu}^{*}\left(\partial^{\nu}A^{\mu}\right)-A_{\nu}^{*}\left(\partial^{\nu}C^{\mu}\right)\right\\}=\frac{1}{4}\operatorname{Re}\left\\{\begin{pmatrix}{2i\omega}(\phi^{*}\psi)/c^{3}+\left[(\mathbf{C}\cdot\nabla)^{*}\phi-(\mathbf{A}\cdot\nabla)^{*}\psi\right]/c\\\
\nabla\times(\mathbf{A}\times\mathbf{C}^{*})-\mathbf{A}^{*}(\partial_{\mu}C^{\mu})+\mathbf{C}^{*}(\partial_{\mu}A^{\mu})\par\end{pmatrix}\right\\},$

where $\partial_{\mu}A^{\mu}=-i\omega\phi/c^{2}+\nabla\cdot\mathbf{A}$ and
same for $C$, which is quantity that is zero in both Lorentz and Coulomb
gauge. An instantaneous dual symmetric gauge independent helicity current
density four-vector is [41]

$\begin{split}S^{\mu}&=\frac{1}{2}\left(\mathcal{A}_{\nu}\mathcal{G}^{\nu\mu}+\mathcal{C}_{\nu}\mathcal{F}^{\nu\mu}\right)=\frac{1}{2}\left(\begin{array}[]{c}\frac{1}{c}\left(\boldsymbol{\mathcal{A}}\cdot\boldsymbol{\mathcal{H}}-\boldsymbol{\mathcal{C}}\cdot\boldsymbol{\mathcal{E}}\right)\\\
\mu\boldsymbol{\mathcal{H}}\times\boldsymbol{\mathcal{C}}+\varepsilon\boldsymbol{\mathcal{E}}\times\boldsymbol{\mathcal{A}}\end{array}\right)+\frac{1}{2c^{2}}\left(\begin{array}[]{c}0\\\
\varphi\boldsymbol{\mathcal{H}}-\psi\boldsymbol{\mathcal{E}}\end{array}\right).\end{split}$
(42)

which can be decomposed into the canonical and Poynting spins in an arbitrary
gauge as follows:

$\displaystyle
S_{C}^{\mu}=\frac{1}{2}[\mathcal{A}_{\nu}\left(\partial^{\mu}\mathcal{C}^{\nu}\right)-\mathcal{C}_{\nu}\left(\partial^{\mu}\mathcal{A}^{\nu}\right)]=\frac{1}{2}\begin{pmatrix}\frac{1}{c}\left(\boldsymbol{\mathcal{A}}\cdot\boldsymbol{\mathcal{H}}-\boldsymbol{\mathcal{C}}\cdot\boldsymbol{\mathcal{E}}\right)\\\
\boldsymbol{\mathcal{A}}\cdot(\nabla)\boldsymbol{\mathcal{C}}-\boldsymbol{\mathcal{C}}\cdot(\nabla)\boldsymbol{\mathcal{A}}\end{pmatrix}+\frac{1}{2c}\begin{pmatrix}{\mathcal{A}}^{\mu}\partial_{\mu}\psi-\mathcal{C}^{\mu}\partial_{\mu}\varphi\\\
\frac{1}{c}\left(\psi\nabla\varphi-\varphi\nabla\psi\right)\end{pmatrix},$
(43) $\displaystyle
S_{P}^{\mu}=\frac{1}{2}[\mathcal{C}_{\nu}\left(\partial^{\nu}\mathcal{A}^{\mu}\right)-\mathcal{A}_{\nu}\left(\partial^{\nu}\mathcal{C}^{\mu}\right)]=\frac{1}{2}\begin{pmatrix}0\\\
(\boldsymbol{\mathcal{C}}\cdot\nabla)\boldsymbol{\mathcal{A}}-(\boldsymbol{\mathcal{A}}\cdot\nabla)\boldsymbol{\mathcal{C}}\par\end{pmatrix}-\frac{1}{2c}\begin{pmatrix}{\mathcal{A}}^{\mu}\partial_{\mu}\psi-\mathcal{C}^{\mu}\partial_{\mu}\varphi\\\
\frac{1}{c}\left(\psi\nabla\varphi-\varphi\nabla\psi\right)\end{pmatrix}+\frac{1}{2c^{2}}\begin{pmatrix}0\\\
\varphi\boldsymbol{\mathcal{H}}-\psi\boldsymbol{\mathcal{E}}\end{pmatrix},$

Note that
$(\boldsymbol{\mathcal{C}}\cdot\nabla)\boldsymbol{\mathcal{A}}-(\boldsymbol{\mathcal{A}}\cdot\nabla)\boldsymbol{\mathcal{C}}=\nabla\times(\boldsymbol{\mathcal{A}}\times\boldsymbol{\mathcal{C}})+\boldsymbol{\mathcal{C}}(\nabla\cdot\boldsymbol{\mathcal{A}})-\boldsymbol{\mathcal{A}}(\nabla\cdot\boldsymbol{\mathcal{C}})$
and ${\mathcal{A}}^{\mu}\partial_{\mu}\psi=\varphi(\partial\psi/\partial
t)/c^{2}+\boldsymbol{\mathcal{A}}\cdot\nabla\psi$ and similarly
$\mathcal{C}^{\mu}\partial_{\mu}\varphi=\psi(\partial\varphi/\partial
t)/c^{2}+\boldsymbol{\mathcal{C}}\cdot\nabla\varphi$.

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