# Reactive helicity and reactive power in nanoscale optics: Evanescent waves.
Kerker conditions. Optical theorems and reactive dichroism

Manuel Nieto-Vesperinas  Instituto de Ciencia de Materiales de Madrid, Consejo
Superior de Investigaciones Científicas.
Campus de Cantoblanco, Madrid 28049, Spain. www.icmm.csic.es/mnv  Xiaohao Xu
Institute of Nanophotonics, Jinan University, Guangzhou 511443, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Considering time-harmonic optical fields, we put forward the complex helicity
and its alternating flow, together with their conservation equation: the
complex helicity theorem. Its imaginary part constitutes a novel law that
rules the build-up of what we establish as the reactive helicity through its
zero time-average flow. Its associated reactive flow, and the imaginary
Poynting momentum that accounts for the accretion of reactive power, are
illustrated in two paradigmatic systems: evanescent waves and fields scattered
from magnetodielectric dipolar nanoparticles. As for the former, we show that
its reactive helicity may be experimentally observed as we  introduce a
reactive spin momentum and a reactive orbital momentum in terms of which we
express the imaginary field momentum, whose transversal component produces an
optical force on a magnetoelectric particle that, as we illustrate, may
surpass and can be discriminated from, the known force due to the so-called
extraordinary momentum. We also uncover a non-conservative force on such a
magnetoelectric particle, acting in the decay direction of the evanescent
wave, and that may also be discriminated from the standard gradient force;
thus making the reactive power of the wavefield also observable.  Concerning
the light scattered by magnetoelectric nanoparticles, we establish two optical
theorems that govern the accretion of reactive helicity and reactive power on
extinction of incident wave helicity and energy. Like a nule total - i.e.
internal plus external - reactive power is at the root of a resonant scattered
power, we show that a zero total reactive helicity underlies a resonant
scattered helicity. These reactive quantities are shown to yield a novel
interpretation of the two Kerker conditions which we demonstrate to be linked
to an absence, or minimum, of the overall scattered reactive energy. Further,
the first Kerker condition, under which the particle becomes dual on
illumination with circularly polarized light, we demonstrate to amount to a
nule overall scattered reactive helicity. Therefore, these two reactive
quantities are shown to underly the directivity of the particle scattering and
emission. In addition, we discover a discriminatory property of the reactive
helicity of chiral light incident on a chiral nanoparticle by excitation of
the external reactive power. This should be useful for optical near-field
enantiomeric separation, an effect that we call reactive dichroism.

##  I  Introduction

Reactive quantities of electromagnetic fields, such as reactive and stored
energy or the imaginary Poynting vector (IPV), are associated to physical
entities that do not propagate in the environment, like evanescent waves in
their varied forms as surface waves raether ; kolokolov ; harrington ,
standing waves harrington , along with near-fields of RF-antennas harrington ;
stratton ; wheeler ; chu ; collin ; jackson ; mcLean ; harrington ; balanis ;
alu . In recent years, advances in photonics and nano-optics led to developing
concepts such as chirality, helicity tang ; barnett2 ; corbato ; nietoheli ;
banzer ; barnett1 ; yan , magnetoelectric effects associated with the
imaginary Poynting vector (IPV) nieto1 , and its consequent transverse spin
momentum and magnetoelectric energy density bliokh1 ; bliokh_rep ; bliokh2 ,
as well as related to the azimuthal imaginary Poynting momentum xu and the
Kerker-type non-conservative intensity gradient force xiao .

In spite of progress in the analysis of optical antennas, mainly addressing
quantum emitters and plasmonic nanoparticles novotny1 ; norris1 ; barnes ,
whose radiative and feeding characteristics is studied from the point of view
of RF-antennas and ciecuit theory hecht ; ziolkowski ; engheta , we have found
few detailed studies ziolkowski on their reactive quantities; although the
effects of reactive power in antenna functionality are well-known harrington ;
wheeler ; chu ; collin ; mcLean ; harrington ; geyi ; balanis ; alu ;
ziolkowski ; e.g. the radiative and reactive energies of an oscillating dipole
are intertwined. In this way, a major task in RF-antenna design has been the
study of its reactive power and $Q$-factor, seeking a minimization of both
quantities in order to match the input energy with its radiative performance,
since a large $Q$ and reactive energy in the antenna convey high ohmic losses,
and a decrease of its operative bandwidth.

By contrast, at optical wavelengths a high reactive power outside the
nanoemitter or scatterer, external stored energy, and $Q$-factor of a low-loss
nanoparticle, or nanoantenna, enhances both its scattered (or radiated) power
and frequency sensitivity, also narrowing its operational bandwith, this being
desirable for its role as e.g. a nano-source or a biosensor, as well as to
reinforce light-matter interactions at the nanoscale, both in plasmonics saad
and Mie-tronics won ; bonod . The reason is that, as we shall see, although
such a large external reactive power and stored energy occur at wavelengths
near those of resonant scattered power, the interior of the particle acts as a
compensating (capacitive or inductive) element so that its reactive power and
stored energy cancel out the external ones close to these resonant wavelengths
at which each of these quantities have near extreme values. The result is
that, in analogy with RF-antenna design ziolkowski , the total (i.e. internal
plus external) reactive power and stored energy are zero close to resonances.

We also show that the same effect occurs for the interior, external, and total
reactive helicities in connection with a maximum efficiency in the helicity
scattered up to the far zone.

Since surface plasmons work at optical frequencies where metals exhibit high
losses, there has been an increasing interest in high refractive index
nanoantennas, on which light exerts a magnetoelectric response with large
electric and magnetic resonances won ; bonod ; nietoSi ; nieto2011 ; mlight ;
staude ; kivshar ; nietolibrev . We will study the key role of reactive
quantities in the scattering from these magnetodielectric particles. The other
archetypical configuration in which we shall address these reactive quantities
is an evanescent wave.

Using time-harmonic wavefields, we will start addressing the flow of reactive
power of the complex Poynting theorem, and since an analogous law for the
helicity flow has never been established, as far as we know, we will introduce
the concepts of complex helicity density comment0 and complex helicity flow,
for which we put forward a conservation equation that we coin as complex
helicity theorem. Its real part is the well-known continuity equation for the
conservation of optical helicity tang ; barnett2 ; corbato ; nietoheli ;
barnett1 , while its imaginary part is a novel law that describes the build-up
of reactive helicity, whose flow has zero time-average and hence it does not
propagate in free-space.

We show that this reactive helicity density and its flow exist, like the
reactive energy and the imaginary Poynting vector, in wavefields that do not
propagate into the environment, as e.g. evanescent  and elliptically (and
circularly in particular) polarized standing waves;  the former being
identical to the so-called magnetoelectric energy density, introduced in
bliokh2 following symmetry arguments but without providing its undelying
physical law.

As for an evanescent wave, whose imaginary Poynting vector and its associated
transversal spin have been studied bliokh1 , while its time-averaged energy
flow, spin angular momentum, Belinfante momentum, and orbital momentum have to
do with the time-averaged energy and helicity densities bliokh1 , we show that
the reactive (i.e. imaginary) Poynting vector, and reactive helicity flow, are
linked with the reactive energy and reactive helicity of the wave. These
relationships provide a physical law for our introduced concept of reactive
helicity, in which the aforementioned magnetoelectric energy bliokh2 , and the
so-called ”real helicity” discussed in a different research kamenetskii , are
unified. Further, we discuss how the reactive power and reactive helicity of
this wavefield, are generated close to the interface through their
corresponding alternate flows along the wave decay direction.

Moreover, we show that the imaginary Poynting momentum of the evanescent
wavefield is  the sum of densities of a reactive spin momentum and a reactive
orbital momentum, which in turn are expressed as differences of the imaginary
magnetic and electric corresponding momenta. This leads us to uncover a non-
conservative optical force on a magnetodielectric particle, directed along the
wave decay direction -i.e. different from the gradient force and larger than
it at certain wavelengths, thus being experimentally detectable-  due to the
IPV.

We also illustrate the presence of a transversal force on such a
magnetodielectric particle, stemming from the corresponding component of the
IPV, which may be of opposite sign and much larger than the known lateral
force  bliokh1 due to the time-averaged Poynting vector; thus being detectable
and making observable the reactive helicity.

As regards fields emitted or scattered by a magnetodielectric dipolar
nanoparticle, we shall show that its reactive energy and angular distribution
of scattered radiation, being intimately interrelated, provide a novel
interpretation of the two Kerker conditions: K1 and K2 kerker ; nieto2011 ;
nietoJNano ; geffrin ; lapin ; banzer1 ; olmos , and hence of the particle
emission directivity. Namely, under plane wave illumination, and at
wavelengths where such a particle fulfils either K1 of zero backscattering, or
K2 of minimum forward scattering, the overall external reactive power around
this body is either zero or almost zero, while the internal and total (i.e.
external plus internal) reactive powers are near zero. Moreover, the overall
external reactive helicity vanishes at K1 wavelengths on illumination with
circularly polarized light.

Concerning feeding the magnetodielectric nanoantenna, we put forward the
reactive power and the reactive helicity optical theorems which quantify the
accretion of external stored reactive energy and reactive helicity in the near
and intermediate-field zones, in terms of the particle excitation and
extinction of energy and helicity of the supplied illumination. The effects
here shown maximizing both the external and internal values of these reactive
quantities, (while minimizing their overall amounts, i,.e. external plus
internal), which tune to resonance their radiation efficiency, establishes an
analogy with the same well-known pursuit in RF antenna design. Therefore, this
work puts forward the importance of reactive quantities which underly some
previously studied concepts in the analysis of optical antennas novotny1 ;
norris1 ; barnes ; hecht .

Finally, we show that the reactive power theorem yields an interpretation of
how by illuminating a chiral particle with a non-free propagating wavefield,
like an elliptically polarized standing wave, evanescent wave, or near field
from a nearby emitter, the reactive helicity of the incident wavefield appears
as a consequence of the generation of reactive energy on interaction with the
sample particle. It is intriguing that, as we find, this incident reactive
helicity emerges analogously as the incident helicity does in a standard
dichroism far-field observation tang . Therefore, we show that the so-called
”magnetoelectric response” of a chiral particle, quoted in bliokh2 , arises as
a consequence of the accretion of its reactive energy from near-field chiral
illumination, e.g. with incident evanescent waves, (or, similarly, with
incident circularly polarized standing waves as proposed in bliokh2 ). As
such, we name reactive dichroism this magnetoelectric phenomenon. It
underlines the observability of such incident reactive helicity, and its
discriminatory property for enantiomer separation by near-field optical
techniques using structured illumination.

##  II The reactive Poynting vector and the reactive power

To fix some concepts to deal with, we first outline the main quantities
involved in the complex Poynting theorem.  We shall assume an arbitrary body
immersed in a lossless homogeneous medium note1 .  Let $\bm{\mathcal{E}}({\bf
r},t)={\bf E}({\bf r})\exp(-i\omega t)$, $\bm{\mathcal{B}}({\bf r},t)={\bf
B}({\bf r})\exp(-i\omega t)$ be a time-harmonic electromagnetic field. It is
well-known that in a body with charges and free electric currents of density
${\bf j}$ contained in a volume $V$ with permitivity $\epsilon$ and
permeability $\mu$, the complex work density, given by the scalar product
${\bf j}^{*}\cdot{\bf E}$, leads after using Maxwell’s equations,
$\nabla\times\bm{\mathcal{E}}=-(1/c)\,\partial\bm{\mathcal{B}}/\partial t$,
$\nabla\times\bm{\mathcal{H}}=(4\pi/c)\,\bm{\mathcal{J}}+(1/c)\,\partial\bm{\mathcal{D}}/\partial
t$,
$[\bm{\mathcal{D}}=\epsilon\bm{\mathcal{E}},\bm{\mathcal{B}}=\mu\bm{\mathcal{H}},\bm{\mathcal{J}}({\bf
r},t)={\bf j}({\bf r})\exp(-i\omega t)$], to the complex Poynting theorem
harrington ; stratton ; jackson ; balanis :

$\displaystyle\int_{V}[\frac{1}{2}\,{\bf j}^{*}\cdot{\bf E}+\nabla\cdot{\bf
S}]d^{3}r=i2\omega\int_{V}(<w_{m}>-<w_{e}>)d^{3}r\,;$
$\displaystyle(\omega=\frac{kc}{\sqrt{\epsilon\mu}}=\frac{2\pi}{\lambda}\frac{c}{\sqrt{\epsilon\mu}}).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(1)

Where * and $<.>$ mean complex conjugated and time-average, respectively. The
complex Poynting vector (CPV) ${\bf S}$, and time-averaged electric and
magnetic energy densities, $<w_{e}>$ and $<w_{m}>$, are

$\displaystyle{\bf S}({\bf r})=\frac{c}{8\pi\mu}{\bf E}({\bf r})\times{\bf
B}^{*}({\bf r}),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle<w_{e}({\bf r})>=\frac{\epsilon}{16\pi}|{\bf E}({\bf
r})|^{2},\,\,\,\,<w_{m}({\bf r})>=\frac{1}{16\pi\mu}|{\bf B}({\bf
r})|^{2}\,.\,\,\,\,\,\,\,\,\,\,\,\,$ (2)

Under our above assumptions, the right side of (1) is purely imaginary, and
the real part of this equation constitutes the well-known Poynting theorem
describing the variation of energy in the body due to the work rate of the
field upon its charges, given by the volume integral of $\frac{1}{2}Re\\{{\bf
j}^{*}\cdot{\bf E}\\}$. This variation is characterized by the flow of the
time-averaged Poynting vector, $<{\bf S}>=Re\\{{\bf S}\\}$, across the surface
$\partial V$ of $V$: $\int_{\partial V}Re\\{{\bf S}\\}\cdot\hat{\bf r}d^{2}r$.
Where $\hat{\bf r}$ is the outward unit normal to $\partial V$.

On the other hand, the imaginary part of (1) formulates that in the steady
state the integral of the imaginary work $\frac{1}{2}Im\\{{\bf j}^{*}\cdot{\bf
E}\\}$, plus the reactive power flux $\int_{\partial V}Im\\{{\bf
S}\\}\cdot{\bf n}d^{2}r$, (which has zero time-average since it is associated
with instantaneous energy flow that alternates back and forth at twice
frequency across $\partial V$), accounts for this reactive power build-up in
and around the body, given by the right side of (1); $2(<w_{m}>-<w_{e}>)$
being the reactive energy density. Unless otherwise stated, we shall
henceforth assume $\epsilon=\mu=1$ for the embedding space.

At this point it is convenient to introduce the instantaneous Poynting vector
harrington ; balanis built by the fields ${\cal E}({\bf
r},t)=\Re\\{\bm{\mathcal{E}}({\bf r},t)\\}$ and ${\cal B}({\bf
r},t)=\Re\\{\bm{\mathcal{B}}({\bf r},t)\\}$,

$\displaystyle\bm{\mathcal{S}}({\bf r},t)=\frac{c}{4\pi}{\cal E}({\bf
r},t)\times{\cal B}({\bf r},t)=\,\,\,\,\,\,$ $\displaystyle<{\bf
S}>+\frac{c}{8\pi}Re\\{{\bf E}({\bf r})\times{\bf B}({\bf r})\exp(-2i\omega
t)\\},\,\,\,\,\,\,$

This is the standard expression of $\bm{\mathcal{S}}({\bf r},t)$. However we
find it more instructive to write it as:

$\displaystyle\bm{\mathcal{S}}({\bf r},t)=\,<{\bf S}>(1+\cos 2\omega
t)+Im\\{{\bf S}\\}\sin 2\omega t\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle+\frac{c}{4\pi}{\bf E}^{R}({\bf r})\times{\bf B}^{I}({\bf
r})\sin 2\omega t-\frac{c}{4\pi}{\bf E}^{I}({\bf r})\times{\bf B}^{I}({\bf
r})\cos 2\omega t.\,\,\,\,\,\ \,\,\,.$ (3)

Where the superscripts $R$ and $I$ denote real and imaginary parts,
respectively. Note that while the term with $<{\bf S}>$ does not change sign,
as expected from that part of the instantaneous Poynting vector associated
with the time-averaged energy flow, the term that contains the IPV or reactive
Poynting vector, $Im\\{{\bf S}\\}$, alternates its sign at frequency $2\omega$
following the variation of $\sin 2\omega t$. This is in accordance with the
above interpretation of the imaginary part of (1). We also see that there is a
generally non-zero contribution to this alternating flow in the last two terms
of (3). Obviously only the $<{\bf S}>$ term remains on time-averaging in (3).

###  II.1 The reactive Poynting vector and the angular spectrum of plane waves

To distinguish the structure of the real and imaginary parts of ${\bf S}$, it
helps to employ the angular spectrum representation of the electromagnetic
field, so as to map these quantities into their spectra in Fourier space. To
this end, we calculate the flow of $\frac{c}{8\pi}{\bf E}({\bf r})\times{\bf
B}^{*}({\bf r})$ across a plane $z=constant$. We use for simplicity a
framework such that the souces are on $z<0$ and thus the integration is done
on the $z=0$ plane. But one may equally choose any other constant value
$z=z_{0}$, providing the sources lie in $z<z_{0}$. The electric field ${\bf
E}({\bf r})$ propagating into the half-space $z\geq 0$ is represented by its
angular spectrum of plane wave components as nietolibro ; mandel :

$\displaystyle{\bf E}({\bf r})=\int_{-\infty}^{\infty}d^{2}{\bf K}\,\,{\bf
e}({\bf K})\exp[i({\bf K}\cdot{\bf R}+k_{z}z)].$ (4)

${\bf r}=({\bf R},z)$, ${\bf R}=(x,y)$,

${\bf k}=({\bf K},k_{z})$, ${\bf K}=(K_{x},K_{y})$, $|{\bf k}|^{2}=k^{2}$. And

$\displaystyle k_{z}=\sqrt{k^{2}-K^{2}}=q_{h}\,,\,K\leq k,\,\,\mbox{for
homogeneous }$ (propagating) plane wave components. $\displaystyle
k_{z}=i\sqrt{K^{2}-k^{2}}=iq_{e}\,,\,K>k,\,\,\mbox{for evanescent }$ plane
wave components. (5)

Using the subscripts $h$ and $e$ for homogeneous and evanescent  components,
respectively, the CPV flux, $\Phi^{Poynt}$, across the plane $z=0$ has real
and imaginary parts given by, (see the proof in Appendix A):

$\displaystyle\Phi^{RPoynt}=\frac{\pi c}{2k}\int_{K\leq k}d^{2}{\bf
K}\,q_{h}|{\bf e}_{h}({\bf K})|^{2},$ (6)

and

$\displaystyle\Phi^{IPoynt}=-\frac{\pi c}{2k}\int_{K>k}d^{2}{\bf
K}\,[q_{e}|{\bf e}_{e}({\bf K})|^{2}-2q_{e}|{e}_{e\,z}({\bf
K})|^{2}].\,\,\,\,\,\,\,\,$ (7)

Eq.(6) is well-known, it expresses the flux $\Phi^{RPoynt}$ of the real part
of the CPV as the momentum power carried on by the propagating components,
($K\leq k$). However, Eq. (7) shows that the flux $\Phi^{IPoynt}$ of the
reactive CPV is momentum associated with power contained in the evanescent
components, $(K>k)$, and as such it does not propagate into $z\geq 0$. Notice
the special role played by the longitudinal component ${e}_{e\,z}({\bf K})$ in
(7).

##  III The reactive helicity and its reactive flow theorem

The optical helicity density of the electromagnetic field in a medium with
constitutive parameters $\epsilon$ and $\mu$ is well-known to be barnett2 ;
nietoheli ; barnett1

$\displaystyle{\mathscr{H}}({\bf
r})=(1/2k)\sqrt{\frac{\epsilon}{\mu}}Im\\{{\bf E}({\bf r})\cdot{\bf
B}^{*}({\bf r})\\}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (8)

We now introduce the quantity

$\displaystyle{\mathscr{H_{R}}}({\bf
r})=(1/2k)\sqrt{\frac{\epsilon}{\mu}}Re\\{{\bf E}({\bf r})\cdot{\bf
B}^{*}({\bf r})\\}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (9)

Like $Im\\{{\bf S}\\}$, this quantity $\mathscr{H_{R}}$ may appear from
${\mathscr{H}}$ when ${\bf E}$ and ${\bf B}$ are chosen $\pi/2$ out of phase.

We shall later see that $\mathscr{H_{R}}$ is exclusive of $Im\\{{\bf S}\\}$,
(and of course of a reactive helicity flow, to be introduced next). Thus we
call ${\mathscr{H_{R}}}$ the reactive helicity density of the field. Some
authors have recently addressed this quantity in different works, and call it
magnetoelectric energy bliokh2 , or just real helicity kamenetskii . However
we keep our denomination by showing that, like the reactive energy, it
fulfills a conservation law.

We consider a body with charges and free currents, embedded in a volume $V$.
From the two Maxwell equations for the spatial vectors, $\nabla\times{\bf
E}=ik{\bf B}$ and $\nabla\times{\bf H}=(4\pi/c)\,{\bf j}-ik{\bf D}$, we derive
the conservation equation for the reactive helicity. First, we employ the
second of these equations and address the scalar product $(4\pi/c){\bf
j}^{*}\cdot{\bf B}$ in $V$. On using the identity: ${\bf
B}\cdot(\nabla\times{\bf H^{*}})={\bf H}^{*}\cdot(\nabla\times{\bf
B})-\nabla\cdot({\bf B}\times{\bf H}^{*})$, one gets

$\displaystyle\int_{V}[-\frac{2\pi}{kn}Im{\\{\bf j}^{*}\cdot{\bf
B}\\}+\,\nabla\cdot\bm{\mathcal{F}}_{B}]\,d^{3}r=\omega\int_{V}\,d^{3}r\,{\mathscr{H_{R}}}.\,\,\,\,\,\,\,\,\,.$
(10)

In (10) $\bm{\mathcal{F}}_{B}=(c/4kn)\Im\\{{\bf H}^{*}\times{\bf B}\\}$ is the
density of magnetic flow of helicity barnett2 ; nietoheli ; bliokh2 ,
($n=\sqrt{\epsilon\mu}$); and it is also proportional to the magnetic part of
the spin angular momentum density barnett2 ; nietoheli ; bliokh2 , (see also
Eq. (C-2) below).

Similarly, we may obtain a conservation equation for $\mathscr{H_{R}}$ with
the electric helicity flow density, $\bm{\mathcal{F}}_{E}=(c/4kn)Im\\{{\bf
E}^{*}\times{\bf D}\\}$. This is done by taking the scalar product: ${\bf
D}\cdot(\nabla\times{\bf E}^{*})=-ik\,{\bf D}\cdot{\bf B}^{*}$ , and
proceeding in an identical way as with the derivation of (10), the result is

$\displaystyle\int_{V}\nabla\cdot\bm{\mathcal{F}}_{E}\,d^{3}r=-\omega\int_{V}\mathscr{H_{R}}\,d^{3}r\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(11)

Adding (11) and (10) one obtains the well-known continuity equation for the
conservation of helicity in the steady state:

$\displaystyle\int_{V}\nabla\cdot\bm{\mathcal{F}}\,d^{3}r\equiv\int_{\partial
V}\bm{\mathcal{F}}\,\cdot\hat{\bf r}\,d^{2}r=\frac{2\pi}{kn}\int_{V}Im{\\{\bf
j}^{*}\cdot{\bf B}\\}\,d^{3}r,\,\,\,\,\,\,\,\,\,\,\,\,$ (12)

which shows that the flow of helicity, or dual-symmetric spin bliokh4 ;
nietoheli :

$\displaystyle\bm{\mathcal{F}}=\bm{\mathcal{F}}_{E}+\bm{\mathcal{F}}_{B}=(c/4kn)Im\\{{\bf
E}^{*}\times{\bf D}+{\bf H}^{*}\times{\bf B}\\},\,\,\,\,\,$ (13)

across the boundary $\partial V$ of $V$ equals the radiated field helicity,
including its dissipation and conversion given by the right side of (12)
poulikakos1 ; poulikakos2 ; gutsche .

However, substracting (11) from (10) leads to the reactive helicity flow
theorem,

$\displaystyle\int_{V}[-\frac{2\pi}{kn}Im{\\{\bf j}^{*}\cdot{\bf
B}\\}+\nabla\cdot\bm{\mathcal{F}}_{{\mathscr{H_{R}}}}]\,d^{3}r=2\omega\int_{V}\mathscr{H_{R}}\,d^{3}r\,;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(14)

where we have introduced the reactive helicity flow associated to a flow of
helicity that vanishes on time-average, although not instantly.

$\displaystyle\bm{\mathcal{F}}_{{\mathscr{H_{R}}}}=\bm{\mathcal{F}}_{B}-\bm{\mathcal{F}}_{E}=(c/4kn)\Im\\{{\bf
H}^{*}\times{\bf B}-{\bf E}^{*}\times{\bf D}\\}=$ $\displaystyle\frac{2\pi
c^{2}}{n}(<\bm{\mathcal{S}}_{m}>-<\bm{\mathcal{S}}_{e}>).\,\,\,\,\,\,\,\,\,$
(15)

If $\partial V$ is in air or vacuum, $n=1$, ${\bf B}={\bf H}$ and ${\bf
D}={\bf E}$ in (13) and (15). $<\bm{\mathcal{S}}_{e}>$ and
$<\bm{\mathcal{S}}_{m}>$ stand for the time-averages of the density of
electric and magnetic spin angular momentum, (cf. Appendix C).

The quantity $2\omega\mathscr{H_{R}}$ is a reactive helicity per unit half-
period,  or just the reactive helicity power,  in analogy with the reactive
power of the complex Poynting theorem Eq. (1). Hence (14) expresses the
conservation of $2\omega\mathscr{H_{R}}$. Indeed (12) and (14) suggest us to
formulate a complex helicity theorem norris_use :

$\displaystyle\int_{V}[-(1+i)\frac{2\pi}{kn}Im{\\{\bf j}^{*}\cdot{\bf
B}\\}+\,\nabla\cdot\bm{\mathcal{F}}_{C}]\,d^{3}r=2i\omega\int_{V}\mathscr{H_{R}}\,d^{3}r,$
$\displaystyle\bm{\mathcal{F}}_{C}=\bm{\mathcal{F}}+i\bm{\mathcal{F}}_{{\mathscr{H_{R}}}}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(16)

where the complex helicity flow , or complex spin angular momentum, is
$\bm{\mathcal{F}}_{C}$. Evidently (16) has a real part which is the standard
helicity conservation equation (12), whereas its imaginary part is Eq. (14)
for the reactive helicity flow and governs the variation of reactive helicity
in $V$, given by the decrease of the integrated source density
$-\frac{2\pi}{kn}Im{\\{\bf j}^{*}\cdot{\bf B}\\}$. This variation is expressed
in terms of the reactive helicity flux $\int_{\partial
V}\bm{\mathcal{F}}_{{\mathscr{H_{R}}}}\cdot\hat{\bf r}\,d^{2}r$, (which has
zero time-average since it comes from $\Im\\{\bm{\mathcal{F}}_{C}\\}$, and
hence represents helicity flowing back and forth to the body across $\partial
V$ in its near-field region, without net propagation), and of integrated
reactive helicity density $2\omega\mathscr{H_{R}}$.

Unless otherwise stated, we shall not drag the factor $2\omega$ when we refer
to the reactive helicity, thus we shall just write $\mathscr{H_{R}}$ for this
quantity. In Section VII we show that $\mathscr{H_{R}}$ is built-up on chiral
light-matter interaction. This gives rise to the phenomenon of reactive
dichroism in the near-field of the body,  addressed in Section VIII.

Equations (10) and (11) suggest that $\mathscr{H_{R}}$ is observable, for
example by detecting the torque exerted by a circularly polarized plane wave
on a dipolar particle on which the field induces a purely electric (e) or a
purely magnetic (m) dipole. In fact, in vacuum $\bm{\mathcal{F}}_{E}$ and
$\bm{\mathcal{F}}_{B}$ are proportional to the optical electric and magnetic
torque, respectively, $\bm{\Gamma}_{E}=(1/8\pi k)\sigma_{e}^{(a)}Im\\{{\bf
E}^{*}\times{\bf E}\\}=c\sigma_{e}^{(a)}<\bm{\mathcal{S}}_{e}>$ and
$\bm{\Gamma}_{B}=(1/8\pi k)\sigma_{m}^{(a)}Im\\{{\bf B}^{*}\times{\bf
B}\\}=c\sigma_{m}^{(a)}<\bm{\mathcal{S}}_{m}>$; $\sigma_{m}^{(a)}$ and
$\sigma_{m}^{(a)}$ being the particle electric and magnetic absorption cross
sections, (see nieto_torque , Section X). Otherwise, if the particle is
magnetodielectric, $\mathscr{H_{R}}$ becomes observable through the e-m
interaction force, Eq. (31) below, [cf. nieto1 Eq. (44)]. This latter
situation is detailed with an evanescent wave in Section IV.B, cf. Eq. (23).

Concerning the $P$, $T$, $D$ symmetries of these novel quantities, namely,

parity, $P:{\bf r}\rightarrow-{\bf r}$, $P:{\bf E}\rightarrow-{\bf E}$,
$P:{\bf B}\rightarrow{\bf B}$,

time-reversal, $T:t\rightarrow-t$, $T:{\bf E}\rightarrow{\bf E}^{*}$, $T:{\bf
B}\rightarrow-{\bf B}^{*}$,

and duality, $D:{\bf E}\rightarrow{\bf B}$, $D:{\bf B}\rightarrow-{\bf E}$,

while it is well-known that

$P:<{\bf S}>\rightarrow-<{\bf S}>$,
$P:\bm{\mathcal{F}}\rightarrow\bm{\mathcal{F}}$,
$P:{\mathscr{H}}\rightarrow-{\mathscr{H}}$;

$T:<{\bf S}>\rightarrow-<{\bf S}>$,
$T:\bm{\mathcal{F}}\rightarrow-\bm{\mathcal{F}}$,
$T:{\mathscr{H}}\rightarrow{\mathscr{H}}$;

$D:<{\bf S}>\rightarrow<{\bf S}>$,
$D:\bm{\mathcal{F}}\rightarrow\bm{\mathcal{F}}$,
$D:{\mathscr{H}}\rightarrow{\mathscr{H}}$;

and we obtain the symmetries for the reactive quantities:

$P:Im\\{{\bf S}\\}\rightarrow-Im\\{{\bf S}\\}$,
$P:\bm{\mathcal{F}}_{\mathscr{H_{R}}}\rightarrow\bm{\mathcal{F}}_{\mathscr{H_{R}}}$,
$P:{\mathscr{H_{R}}}\rightarrow-{\mathscr{H_{R}}}$;

$T:Im\\{{\bf S}\\}\rightarrow Im\\{{\bf S}\\}$,
$T:\bm{\mathcal{F}}_{\mathscr{H_{R}}}\rightarrow-\bm{\mathcal{F}}_{\mathscr{H_{R}}}$,
$T:{\mathscr{H_{R}}}\rightarrow-{\mathscr{H_{R}}}$;

$D:Im\\{{\bf S}\\}\rightarrow-Im\\{{\bf S}\\}$,
$D:\bm{\mathcal{F}}_{\mathscr{H_{R}}}\rightarrow-\bm{\mathcal{F}}_{\mathscr{H_{R}}}$,
$D:{\mathscr{H_{R}}}\rightarrow-{\mathscr{H_{R}}}$.

It is interesting that while under parity the three reactive quantities behave
like their corresponding non-reactive counterparts, they invert their symmetry
under duality applications, in contrast with their non-reactive analogues that
remain $D$-invariant. Under time-reversal, only
$\bm{\mathcal{F}}_{\mathscr{H_{R}}}$ has the same symmetry as its non-reactive
correspondant $\bm{\mathcal{F}}$.

###  III.1 The reactive helicity and the angular spectrum

In order to gain more insight into the different nature of ${\mathscr{H}}$ and
${\mathscr{H_{R}}}$, we shall employ once again the angular spectrum of the
electromagnetic wave. We evaluate the total ${\mathscr{H}}$ and
${\mathscr{H_{R}}}$ per unit z-length by integration of ${\bf E}({\bf
r})\cdot{\bf B}^{*}({\bf r})$ on a plane $z=constant$. Again, we choose
coordinates such that the souces are on $z<0$ and thus the integration is done
on the $z=0$ plane.

Using, as before, the subindex $h$ and $e$ for homogeneous and evanescent
components, respectively, the integral on $z=0$ of ${\bf E}({\bf r})\cdot{\bf
B}^{*}({\bf r})$ is shown in appendix B to yield

$\displaystyle\int_{-\infty}^{\infty}d^{2}{\bf R}\,{\mathscr{H}}({\bf R},0)$
$\displaystyle=(1/2k)\sqrt{\frac{\epsilon}{\mu}}\int_{-\infty}^{\infty}d^{2}{\bf
R}\,Im\\{{\bf E}({\bf R})\cdot{\bf B}^{*}({\bf R})\\}$
$\displaystyle=2\frac{(2\pi)^{2}}{kc}\\{\int_{K\leq k}d^{2}{\bf K}\,{\bf
k}_{h}\cdot\bm{\mathcal{F}}_{E\,h}({\bf
K})\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ $
$\displaystyle+\int_{K>k}d^{2}{\bf K}\,{\bf
K}\cdot\bm{\mathcal{F}}_{E\,e\,\perp}({\bf
K})\\},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (17)

and

$\displaystyle\int_{-\infty}^{\infty}d^{2}{\bf R}\,{\mathscr{H_{R}}}({\bf
R},0)$
$\displaystyle=(1/2k)\sqrt{\frac{\epsilon}{\mu}}\int_{-\infty}^{\infty}d^{2}{\bf
R}\,Re\\{{\bf E}({\bf R})\cdot{\bf B}^{*}({\bf R})\\}$
$\displaystyle=2\frac{(2\pi)^{2}}{kc}\int_{K>k}d^{2}{\bf K}\,q_{e}\,{\bf\cal
F}_{E\,e\,z}({\bf K}).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (18)

In (17) $\bm{\mathcal{F}}_{E\,e}=(\bm{\mathcal{F}}_{E\,e\,\perp},{\bf\cal
F}_{E\,e\,z})$, $\bm{\mathcal{F}}_{E\,e\,\perp}=({\bf\cal
F}_{E\,e\,x}\,,{\bf\cal F}_{E\,e\,y})$. Therefore Eq. (17) shows that in the
domain of propagating components the helicity density of the field in $z=0$
maps into the projection of the electric spin angular momentum of the ${\bf
K}$-plane wave component onto the propagation wavevectors; namely, onto the
real wavevector ${\bf k}_{h}$, while in the evanescent region, it is given by
the projection of the electric spin onto the transversal (propagating)
component ${\bf K}$. (Note that this is in agreement with the standard
definition corbato , but here generalized to include evanescent waves).

Of special interest is, however, Eq. (18), which shows that the reactive
helicity density ${\mathscr{H_{R}}}$ maps in ${\bf K}$-space as the projection
of the electric spin of the ${\bf K}$th-evanescent component onto the
$z$-component $i{\bf k}_{z}^{*}=i(-iq_{e})\hat{\bm{z}}$. Where ${\bf k}$ is
the complex wavevector of this ${\bf K}$th-evanescent wave, [cf. Eq. (5)].
Again, this justifies that we call reactive to the real part of ${\bf E}({\bf
r})\cdot{\bf B}^{*}({\bf r})$.

In this connection, it should be remarked that like the reactive power and the
IPV are linked with non-propagating waves, e.g. evanescent and standing waves
harrington , the reactive helicity ${{\mathscr{H_{R}}}}$ (and hence its flow
$\bm{\mathcal{F}}_{{\mathscr{H_{R}}}}$) exist in evanescent waves as shown in
Eq.(18), as well as in elliptically polarized (and circularly polarized in
particular, CPL) standing wavefields note2 . For CPL waves  the authors of
bliokh2 used the term ”magnetoelectric energy” which, as seen above, is a
reactive quantity since it is the same as $\mathscr{H_{R}}$.

Figure 1: The evanescent wave (yellow) of Eq. (19), with reactive power
density given by (20), propagates in the air, parallel to the interface $z=0$
separating it from a dielectric in $z<0$, with $K$-vector along $OX$. Its
amplitude decays as $\exp(-qz)$. The $<{\bf S}>$-vector and spin momentum,
$<{\bf P}^{S}>$, contained in $OXY$, have $y$-components proportional to the
wave helicity ${\mathscr{H}}$, (see Appendix C). The orbital momentum, $<{\bf
P}^{0}>$, proportional to the energy density $w$ points along $OX$. The
reactive momentum $Im\\{{\bf S}\\}$, contained in $OYZ$, and whose
$z$-component is associated to energy flux bouncing up and down along $OZ$,
[cf. Eq.(28)], may be observed through the time-averaged force ${\bf F}_{em}$,
proportional to $Im\\{{\bf S}\\}$, on a dipolar magnetodielectric particle
placed on the interface. This force is due to the interference of the particle
electric and magnetic induced dipoles. The lateral $y$-force, $({\bf
F}_{em})_{y}$, and transversal $z$-component, $({\bf F}_{em})_{z}$ , due to
$Im\\{{S}\\}_{y}$ and $Im\\{{S}\\}_{z}$, are proportional to the wave reactive
helicity, $\mathscr{H_{R}}$, and reactive power density, $\nabla\cdot
Im\\{{\bf S}\\}$, respectively. The imaginary orbital and spin momenta, ${\bf
P}^{O\,I}$ and ${\bf P}^{S\,I}$, point along the $+z$ and $-z$-direction,
respectively. The flow $\bm{\mathcal{F}}_{\mathscr{H_{R}}}$ of reactive
helicity is, like $Im\\{\bf{S}\\}$, in the $OYZ$-plane with $y$-component
proportional to both the reactive power density, $w_{react}$, and the
magnitude of the $K$-vector; while its $z$-component is proportional to the
reactive helicity density ${\mathscr{H_{R}}}$, so that across the $OXY$-plane
$({\mathcal{F}}_{\mathscr{H_{R}}})_{z}$ holds the imaginary part of the
complex helicity theorem, as shown by Eq. (30).

##  IV Case 1: Reactive power, reactive helicity, and reactive momenta in an
evanescent wave

As illustrated in Fig. 1, we consider a generic time-harmonic evanescent wave
in air, $z\geq 0$, generated by total internal reflection (TIR), at a plane
interface $z=0$ separating air ($\epsilon=\mu=1$) from a dielectric in the
half-space $z\leq 0$. The plane of incidence being $OXZ$. Then the complex
spatial parts of the electric and magnetic vectors in $z>0$, are expressed in
a Cartesian coordinate basis $\\{\hat{\bf x},\hat{\bf y},\hat{\bf z}\\}$ as
($n=1$) born ; nietoOL :

$\displaystyle{\bf
E}=\left(-\frac{iq}{k}T_{\parallel},T_{\perp},\frac{K}{k}T_{\parallel}\right)\exp(iKx-
qz),$ $\displaystyle{\bf
B}=n\left(-\frac{iq}{k}T_{\perp},-T_{\parallel},\frac{K}{k}T_{\perp}\right)\times\exp(iKx-
qz).\,\,\,\,$ (19)

For TE or $s$ (TM or $p$) - polarization , i.e. ${\bf E}^{(i)}$ (${\bf
B}^{(i)}$) perpendicular to the plane of incidence $OXZ$, only those
components with the transmission coefficient $T_{\perp}$, ($T_{\parallel}$)
would be chosen in the incident fields born . $K$ denotes the component,
parallel to the interface, of the wavevector ${\bf k}$: $k({\bf
s}_{xy},s_{z})=(K,0,iq)$, $q=\sqrt{K^{2}-k^{2}}$, $k^{2}=K^{2}-q^{2}$.

###  IV.1 Reactive power and reactive helicity densities

The densities of energy, $w=w_{e}+w_{m}$, and reactive power,
$w_{react}=2\omega(w_{m}-w_{e})$, ($\epsilon=\mu=1$), of this wave are
according to (2)

$\displaystyle\,w=\frac{1}{8\pi}\frac{K^{2}}{k^{2}}(|T_{\perp}|^{2}+|T_{\parallel}|^{2})\exp(-2qz),\,\,\,\,\,\,\,\,\,\,\,\
$ $\displaystyle
w_{react}=\frac{c}{4\pi}\frac{q^{2}}{k}(|T_{\perp}|^{2}-|T_{\parallel}|^{2})\exp(-2qz).\,\,\,\,\,\,\,\,\,\,\,\,$
(20)

And the densities of helicity, $\mathscr{H}$, and reactive helicity,
$\mathscr{H_{R}}$, ($\epsilon=\mu=1$), of this evanescent wave are

$\displaystyle{\mathscr{H}}=-(K^{2}/k^{3})Im\\{T_{\perp}T_{\parallel}^{*}\\}\exp(-2qz);\,\,$
$\displaystyle\mathscr{H_{R}}=(q^{2}/k^{3})Re\\{T_{\perp}T_{\parallel}^{*}\\}\exp(-2qz).\,\,\,\,\,\,\,\,\,\,\,\,$
(21)

###  IV.2 The reactive energy flow: Reactive momentum, imaginary spin and
imaginary orbital momenta

The CPV is written as

$\displaystyle{\bf
S}=\frac{c}{8\pi\mu}[\frac{K}{k}(|T_{\perp}|^{2}+|T_{\parallel}|^{2}),$
$\displaystyle
i2\frac{Kq}{k^{2}}T_{\perp}^{*}T_{\parallel}\,,-i\frac{q}{k}(|T_{\perp}|^{2}-|T_{\parallel}|^{2})]\exp(-2qz)\,\,\,\,\,\,\,\,\,\,$
$\displaystyle=[\frac{ck}{K}w\,,\frac{c}{4\pi}(-\frac{kq}{K}\mathscr{H}+i\frac{kK}{q}\mathscr{H_{R}})\,,-i\frac{1}{2\mu
q}w_{react}].\,\,\,\,\,\,\,\,\,\,\,$ (22)

For the sake of comprehensiveness, in Appendix C we present the well-known
main time-averaged quantities. Here we concentrate on those reactive less-
known and their interralations.

The reactive, or imaginary, part of the CPV is

$\displaystyle Im\\{{\bf
S}\\}=\frac{c}{8\pi\mu}[\,0\,,2\frac{Kq}{k^{2}}Re\\{T_{\perp}T_{\parallel}^{*}\\}\,,\frac{q}{k}(|T_{\parallel}|^{2}-|T_{\perp}|^{2})]$
$\displaystyle\times\exp(-2qz)=\frac{1}{2q}[0\,,\frac{ckK}{2\pi}{\mathscr{H_{R}}}\,,-w_{react}].\,\,\,\,\,\,\,$
(23)

Which yields the reactive or  imaginary momentum of the field, (sometimes
called imaginary Poynting momentum bliokh1 ; xu ), ${\bf g}^{I}=Im\\{{\bf
g}\\}=Im\\{{\bf S}\\}/c^{2}$:

$\displaystyle{\bf g}^{I}=\frac{1}{8\pi
c}[\,0\,,2\frac{Kq}{k^{2}}Re\\{T_{\perp}T_{\parallel}^{*}\\}\,,\frac{q}{k}(|T_{\parallel}|^{2}-|T_{\perp}|^{2})]$
$\displaystyle\times\exp(-2qz)=\frac{1}{2qc^{2}}[0\,,\frac{ckK}{2\pi}{\mathscr{H_{R}}}\,,-w_{react}],\,\,\,\,\,\,\,\,\,\,\,\,$
(24)

whose components come from two vectors that we put forward next: the density
of both reactive spin momentum  $\bm{\mathcal{P}}^{S}$ and reactive orbital
momentum $\bm{\mathcal{P}}^{O}$:

$\displaystyle\bm{\mathcal{P}}^{S}=\frac{1}{2}({\bf P}_{m}^{S\,I}-{\bf
P}_{e}^{S\,I})=\frac{K}{2cq}(0,\frac{k}{2\pi}\mathscr{H_{R}},-\frac{K}{ck^{2}}w_{react}),$
$\displaystyle\bm{\mathcal{P}}^{O}=\frac{1}{2}({\bf P}_{m}^{O\,I}-{\bf
P}_{e}^{O\,I})=\frac{q}{2cq}(0,0,\frac{q}{ck^{2}}w_{react}).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(25)

Namely,

$\displaystyle{\bf g}^{I}=\bm{\mathcal{P}}^{S}+\bm{\mathcal{P}}^{O}.\,\,\,$
(26)

The electric and magnetic imaginary spin and orbital momenta of (25) are given
in Appendix C, Eqs. (C-10)-(C-14).

We emphasize that, as seen in (24)-(26), $g_{y}^{I}$ is fully due to
${\mathcal{P}}_{y}^{S}$, while $g_{z}^{I}$ comes from
${\mathcal{P}}_{z}^{S}+{\mathcal{P}}_{z}^{O}$.

Note that, interestingly and in contrast with Eq.(C-6), denoting ${\bf
P}^{S\,I}=(1/2)({\bf P}_{e}^{S\,I}+{\bf P}_{m}^{S\,I})=(0,0,-\frac{q}{kc}w)$,
${\bf P}^{O\,I}=(1/2)({\bf P}_{e}^{O\,I}+{\bf
P}_{m}^{O\,I})=(0,0,\frac{q}{kc}w)$, one obtains: ${\bf P}^{S\,I}+{\bf
P}^{O\,I}=0$.

###  IV.3 The reactive helicity flow

In turn, the reactive helicity flow [cf. Eq. (15)] is

$\displaystyle\bm{\mathcal{F}}_{\mathscr{H_{R}}}=\frac{c}{4k}[0,\frac{2Kq}{k^{2}}(|T_{\parallel}|^{2}-|T_{\perp}|^{2}),-\frac{4q}{k}Re\\{T_{\perp}T^{*}_{\parallel}\\}]\exp(-2qz)$
$\displaystyle=-\frac{1}{q}\,[0\,,\frac{2\pi
K}{k^{2}}w_{react}\,,ck{\mathscr{H_{R}}}].\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (27)

It should be emphasized that although both $Im\\{{\bf S}\\}$ and
$\bm{\mathcal{F}}_{\mathscr{H_{R}}}$ have an $y$-component proportional to
$K>k$, (as well as proportional to $\mathscr{H_{R}}$ in the former and to
$w_{react}$ in the latter), there is no case of superluminal propagation for
these quantities since both are alternating flows with zero time-average. We
show below that ${\cal F}_{z}$ represents up and down flow of reactive
helicity in the $OZ$ direction, matching with the imaginary part of the
complex helicity theorem, Eq. (16).

From the above equations it is important to remark that while the time-
averaged energy and helicity densities are linked to $<{\bf S}>$, $<{\bf g}>$,
$<\bm{\mathcal{S}}>$, $<{\bf P}^{S}>$ and $<{\bf P}^{O}>$, the densities of
reactive power, $w_{react}$, and reactive helicity, $\mathscr{H_{R}}$, are
exclusive of $Im\\{{\bf S}\\}$, ${\bf g}^{I}$ and
$\bm{\mathcal{F}}_{\mathscr{H_{R}}}$.

We show below that the $z$-component of $Im\\{{\bf S}\\}$, that matches with
the imaginary part of the complex Poynting theorem, is associated with an up
and down flow of reactive power in the decay $z$-direction of the evanescent
wave, and hence it is not a net flow of energy. Analogously happens with the
$z$-component of $\bm{\mathcal{F}}_{\mathscr{H_{R}}}$. However, as seen below,
both the $y$ and $z$-components of $Im\\{{\bf S}\\}$ produce detectable
optical forces and, hence, make the reactive quantities $w_{react}$ and
${\mathscr{H_{R}}}$ observable.

###  IV.4 Ractive power conservation law

The $z$-component of $Im\\{{\bf S}\\}$, [cf. Eq. (23)], depends on the
reactive power density, $\nabla\cdot Im\\{{\bf S}\\}$, of the evanescent wave
in the half-space $z>0$, (which actually concentrates in the near field region
above the interface $z=0$, namely at $z<<\lambda$), flowing back and forth
along $OZ$ at twice the frequency $\omega$, without contributing to a net
energy flow since its time-average is zero. I.e. one has from the CPV theorem:

$\displaystyle\nabla\cdot Im\\{{\bf S}\\}\equiv\partial_{z}Im\\{{\bf
S}\\}=w_{react}$ $\displaystyle\equiv-2qIm\\{{\bf
S}\\}_{z}\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (28)

Which obviously agrees with (23). Therefore taking (28) into account, the
total reactance, balanis of the dielectric-air interface system associated to
the evanescent wave is

$\displaystyle\int_{V}d^{3}r\nabla\cdot Im\\{{\bf
S}\\}=\Sigma\int_{0}^{\infty}dz\,\nabla\cdot Im\\{{\bf S}\\}=-Im\\{{\bf
S}_{z}(z=0)\\}.\,\,\,\,\,\,\,\,\,\,\,$ (29)

$\Sigma$ being the area of the $XY$-plane resulting from the volume
integration.

###  IV.5 Reactive helicity conservation law

Analogously, the $z$-component of $\bm{\mathcal{F}}_{\mathscr{H_{R}}}$, Eq.
(27), depends on the reactive helicity density,
$\nabla\cdot\bm{\mathcal{F}}_{\mathscr{H_{R}}}$, of the evanescent wave in the
half-space $z>0$, concentrated in the near field on $z=0$, flowing up and down
in the the $z$-direction without yielding a net flow since its time-average is
zero. I.e. one has in agreement with Eq.(14), (${\bf j}=0$),

$\displaystyle\nabla\cdot\bm{\mathcal{F}}_{\mathscr{H_{R}}}\equiv\partial_{z}\bm{\mathcal{F}}=\frac{2cq^{2}}{k^{2}}Re\\{T_{\perp}T_{\parallel}^{*}\\}$
$\displaystyle=-2q({\cal F}_{\mathscr{H_{R}}})_{z}\equiv
2\omega{\mathscr{H_{R}}}\,.\,\,\,\,\,\,\,\,\,\,\,\,$ (30)

Hence, the $z$-component of $\bm{\mathcal{F}}_{\mathscr{H_{R}}}$ is
proportional to the reactive power density.

It is evident that while reactive power and reactive helicity exist in
evanescent waves, (see also Eqs. (7) of Section II.A and (18) of Section
III.A), and in standing waves harrington ; bliokh2 , they do not exist in
plane propagating waves, whatever their polarization be. Therefore, reactive
helicity, like reactive energy, exists in the near-field region of scattering
or emitting objects.

Figure 2:  An evanescent wave, (see in Fig.1), is created in $z\geq 0$ by TIR
at the interface $z=0$ of a linearly polarized plane propagating wave of
complex amplitudes $A_{\perp}$ and $A_{\parallel}$ in the dielectric of
refractive index $n=1.5$ with angle of incidence: $60^{o}$. A Si spherical
particle of radius $a=75$ nm is deposited on the interface. Its electric and
magnetic dipole resonances are at $\lambda_{e}=492$ nm and $\lambda_{m}=668$
nm, respectively, [cf. Fig.4(left)]. This wave exerts on the particle the
following transversal and perpendicular optical forces: (Left) With
$A_{\perp}=A_{\parallel}$, force transversal components, $<{\bf F}_{e-m}^{{\bf
g}^{I}}>_{y}$ due to ${g}_{y}^{I}$, $<{\bf F}_{e-m}^{<{\bf g}>}>_{y}$ from
$<{g}_{y}>$, and ratio: $<{\bf F}_{e-m}^{{\bf g}^{I}}>_{y}/<{\bf
F}_{e-m}^{<{\bf g}>}>_{y}$ whose values (in green) are shown in the right
ordinate axis. (Right) Choosing $A_{\parallel}=0$, $A_{\perp}\neq 0$: Normal
component $<{\bf F}_{e-m}^{{\bf g}^{I}}>_{z}$ due to ${g}_{z}^{I}$, gradient
force $<{\bf F}^{grad}>_{z}$, and ratio: $<{\bf F}_{e-m}^{{\bf
g}^{I}}>_{z}/<{\bf F}^{grad}>_{z}$ whose values (in green) are shown in the
right ordinate axis. Notice that the positive values of the gradient force,
repelling the particle from the interface, are due to negative values of
$\alpha_{e}^{R}$ and/or $\alpha_{m}^{R}$. The bump of $<{\bf F}^{grad}>_{z}$
near $\lambda_{e}$ and its steep change of sign close to $\lambda_{m}$, are
due to the gradient force felt by the induced electric and magnetic dipoles,
respectively. The wavelength at which the second Kerker condition holds is
$\lambda_{K2}=608$ nm. The force (normalized to area) units are $fN/\mu m^{2}$
for an incident power of $1$ $mW/mm^{2}$.

###  IV.6 Observability of the transversal and perpendicular components of the
imaginary momentum ${\bf g}^{I}$: Optical forces on a magnetodielectric
dipolar particle due to the reactive helicity and reactive power

Let a magnetodielectric dipolar particle be placed on the $z=0$ interface. An
illuminating wavefield, and in particular the evanescent wave, exerts an
optical force on it due to the interaction between its induced electric (e)
and magnetic (m) dipoles nieto1 , viz.,

$\displaystyle<{\bf F}_{e-m}>=\frac{8\pi
k^{4}c}{3}[-Re(\alpha_{e}\alpha_{m}^{*})<{\bf
g}>+Im(\alpha_{e}\alpha_{m}^{*}){\bf g}^{I}].\,\,\,\,\,\,\,\,\,\ $ (31)

The e and m polarizabilities $\alpha_{e}$ and $\alpha_{m}$ are related with
the $a_{1}$ and $b_{1}$ Mie coefficients of the field scattered by the
particle by nieto1 : $\alpha_{e}=i3a_{1}/2k^{3}$, $\alpha_{m}=i3b_{1}/2k^{3}$.

The first term of (31), proportional to $<{\bf g}>$, has been studied, (see
its main features in Appendix C). Here we are interested in the second term
that contains the reactive momentum ${\bf g}^{I}$.

The $y$-component $<{\bf F}_{e-m}^{{\bf g}^{I}}>_{y}$ due to ${g}_{y}^{I}$ of
(31) was obtained in bliokh1 , being considered by the authors ”a quite
intriguing result” characterized through the second Stokes parameter with a
rather small contribution in measurements of $<{\bf F}_{e-m}>_{y}$. We have
shown above that this force has a reactive origin since it is fully due to the
reactive spin $y$-component, being characterized by the reactive helicity
$\mathscr{H_{R}}$. Furthermore, we establish here that $<{\bf F}_{e-m}^{{\bf
g}^{I}}>_{y}$ is detectable since it may widely exceed the known component
$<{\bf F}_{e-m}^{<{\bf g}>}>_{y}$ due to $<g>_{y}$.

For instance, Fig.2(Left) shows forces on a Si spherical particle placed on
the interface $z=0$: $<{\bf F}_{e-m}^{{\bf g}^{I}}>_{y}$ compared with $<{\bf
F}_{e-m}^{<{\bf g}>}>_{y}$. The particle electric and magnetic dipole
resonances are at $\lambda_{e}=492$ nm and $\lambda_{m}=668$ nm, respectively,
[see Fig.4(left)], and the second Kerker condition (K2) wavelength (at which
the particle scatters minimum forward intensity, see Section V.B) is
$\lambda_{K2}=608$ nm. As seen, the magnitude of $<{\bf F}_{e-m}^{{\bf
g}^{I}}>_{y}$ is much greter than $<{\bf F}_{e-m}^{<{\bf g}>}>_{y}$ near
$\lambda_{e}$ and $\lambda_{m}$ where the latter changes sign, (see the sharp
asymptotic values of the ratio between both forces). Thus $<{\bf
F}_{e-m}^{{\bf g}^{I}}>_{y}$, which keeps negative, should be detectable near
these resonances. Besides, in the proximities of K2, i.e. of $\lambda_{K2}$,
$<{\bf F}_{e-m}^{{\bf g}^{I}}>_{y}=-<{\bf F}_{e-m}^{<{\bf g}>}>_{y}$.

While $<{\bf F}_{e-m}^{<{\bf g}>}>_{z}=0$, one observes in Fig.2(Right)
features of the perpendicular force $<{\bf F}_{e-m}^{{\bf g}^{I}}>_{z}$
similar to those of its $y$-component comparing it with the gradient force
nietoOL , which choosing $T_{\parallel}=0$ reads: $<{\bf
F}^{grad}>_{z}=-(1/2)q[\alpha_{e}^{R}|T_{\perp}|^{2}+\alpha_{m}^{R}|T_{\perp}|^{2}(2K^{2}/k^{2}-1)](0,0,1)$.
The superscript $R$ standing for real part. Again the sharp ratio between both
near $\lambda=665$ nm indicates the wavelenght zone where $<{\bf
F}_{e-m}^{{\bf g}^{I}}>_{z}$, which remains positive, may be detected.
Furthermore, the bump of this ratio in the proximities of $\lambda=520$ nm
shows that this force is over twice the gradient force.

We conclude thereby that there exists a measurable transverse $y$-component of
$<{\bf F}_{e-m}>$ due to the reactive spin momentum density and hence to
${\mathscr{H_{R}}}$, which may be dominant upon the transversal component of
$<{\bf F}_{e-m}>$ stemming from the field (Poynting) momentum, namely from
${\mathscr{H}}$. Hence, ${\mathscr{H_{R}}}$ is observable. Besides, the normal
force $<{\bf F}_{e-m}>_{z}$which is exclusively due ${g}_{z}^{I}$, (since
$<g_{z}>=0$), characterized by the reactive power density $w_{react}$ of the
evanescent wave has not yet been addressed as far as we know, and may be
detected at wavelengths at which, as seen above, clearly exceeds the gradient
force, making $w_{react}$ also an observable quantity

##  V Case 2: Reactive power and reactive helicity from a magnetodielectric
dipolar sphere

We consider a magnetodielectric spherical particle of radius $a$ and volume
$V_{0}$, dipolar in the wide sense, namely whose electric and magnetic
polarizabilities are given by the first electric and magnetic Mie
coefficients, respectively nieto1 ; nietoSi , in air. We first address the
reactive power and stored energy of this magnetoelectric dipole with electric
and magnetic moments ${\bf p}$ and ${\bf m}$, respectively. For a wave, ${\bf
E}^{(i)}$, ${\bf B}^{(i)}$, incident on the particle centered at ${\bf r}={\bf
0}$, the dipolar moments are: ${\bf p}=\alpha_{e}{\bf E}^{(i)}{\bf(0)}$ and
${\bf m}=\alpha_{m}{\bf B}^{(i)}{\bf(0)}$.

###  V.1 Reactive power and stored energy

Concerning the CPV, ${\bf S^{(s)}}=\frac{c}{8\pi}{\bf E}^{(s)}\times{\bf
H}^{(s)\,*}$, of the emitted fields, (cf. Appendix D), we are interested in
its radial component, ${\bf S}^{(s)}\cdot\hat{\bf r}$, across a spherical
surface $\partial V$ of radius $r$ concentric with the particle and enclosing
it. Using the fields of Appendix E, this is straightforwardly integrated,
yielding

$\displaystyle\int_{\partial V}{\bf S}^{(s)}\cdot\hat{\bf
r}\,d^{2}r=\frac{ck^{4}}{3}(|{\bf p}|^{2}+|{\bf
m}|^{2})+\frac{ick}{3r^{3}}(|{\bf p}|^{2}-|{\bf
m}|^{2}).\,\,\,\,\,\,\,\,\,\,\,\,$ (32)

Whose real part, $\int_{\partial V}<{\bf S^{(s)}}>\cdot\hat{\bf
r}\,d^{2}r=\frac{ck^{4}}{3}(|{\bf p}|^{2}+|{\bf m}|^{2})$ is the well-known
radiated (scattered) total power, $W^{(s)}$, independent of the distance $r$
to the center $r=0$, and corresponds to the wevefield re-radiated up to the
far-zone, i.e. that with the $r^{-1}$ dependence, [cf. Appendix D, Eqs. (D-1)
and (D-2)].

Figure 3: 3-D graphs of reactive powers as functions of wavelength $\lambda$
and radius $a$ of a magnetodielelectric sphere of Si in air, illuminated by a
(linearly or circularly polarized) plane wave. (Left): External
$W_{react}^{(s)}$. (Center): Interior $W_{react,int}^{(s)}$. (Right): Total
$W_{react,T}^{(s)}=W_{react}^{(s)}+W_{react,int}^{(s)}$. The sign of
$W_{react,int}^{(s)}$ is opposite to that of $W_{react}^{(s)}$ for all
$\lambda$ and $a$. The dip and peak of $W_{react}^{(s)}$ correspond to the
electric and magnetic resonance, respectively, and are close to these
resonances in $W_{react,int}^{(s)}$. Like the resonances exhibited by the
scattered power $W^{(s)}$ geffrin , these peaks and dips are redshifted as $a$
increases. $W_{react,T}^{(s)}$ vanishes in the proximities of these
resonances, irrespective of $a$. Figure 4:  (Left) Si sphere of radius
$a=75nm$. (Left): Square moduli of the electric and magnetic Mie coefficients,
$a_{1}$ and $b_{1}$, and $d_{1}$ and $c_{1}$ of the external and interior
scattered field, respectively. (Center): Scattered power $W^{(s)}$ and
reactive powers: external $W_{react}^{(s)}$, interior $W_{react,int}^{(s)}$,
and total $W_{react,T}^{(s)}=W_{react}^{(s)}+W_{react,int}^{(s)}$. Notice that
$W_{react,T}^{(s)}$ vanishes at wavelenghts close to those of the electric and
magnetic Mie resonances: $\lambda_{e}=492nm$ and $\lambda_{m}=668nm$ where
$W^{(s)}$ is maximum . The lines $|a_{1}|^{2}$ and $|b_{1}|^{2}$ cross each
other at the Kerker wavelengths: $\lambda_{K1}=738.5nm$, (at which
$W_{react}^{(s)}=0$ and $W_{react,int}^{(s)}\simeq 0$), and
$\lambda_{K2}=608nm$, (where both $W_{react}^{(s)}$ and $W_{react,int}^{(s)}$
are near $0$). (Right): Stored energies: external $W^{(sto)}$, interior
$W^{(sto,int)}$, total $W^{(sto,T)}=W^{(sto)}+W^{(sto,int)}$, and difference
$W^{(sto,Dif)}=W^{(sto)}-W^{(sto,int)}$..

Here we are particularly interested in the imaginary part,

$\displaystyle\int_{\partial V}Im\\{{\bf S}^{(s)}\\}\cdot\hat{\bf
r}\,d^{2}r=\frac{ck}{3r^{3}}(|{\bf p}|^{2}-|{\bf
m}|^{2})]=-W_{react}^{(s)}.\,\,\,\,\,\,\,\,\,\,\,\,$ (33)

Where $W_{react}^{(s)}$ is the reactive power outside $V$. It arises from the
near and intermediate fields: ${\bf E}^{(s)}({\bf r})=\\{\frac{1}{\epsilon
r^{3}}[3{\bf n}({\bf n}\cdot{\bf p})-{\bf p}]-\sqrt{\frac{\mu}{\epsilon}}({\bf
n}\times{\bf m})\frac{ik}{r^{2}}\\}e^{ikr}$, and ${\bf B}^{(s)}({\bf
r})=\\{\frac{\mu}{r^{3}}[3{\bf n}({\bf n}\cdot{\bf m})-{\bf
m}]+\sqrt{\frac{\mu}{\epsilon}}({\bf n}\times{\bf
p})\frac{ik}{r^{2}}\\}e^{ikr}$. (We recall that the expression:
$\frac{ick}{3r^{3}}|{\bf p}|^{2}$ for a purely electric dipolar emitter is
well-known in antenna theory harrington ; stratton ; balanis ). We note that
the $r^{-3}$ dependence of $W_{react}^{(s)}$ makes it to acquire much larger
values than $W^{(s)}$ in the near-field region.

The reactive power $W_{react}^{(s)}$ is the difference between averaged stored
magnetic and electric powers which dominate in the near and intermediate-field
regions around the particle, and that do not propagate. To see it, we write
the mean stored electric and magnetic energy densities as collin ; mcLean ;
geyi $<\tilde{w}_{e}^{(s)}>=<{w}_{e}^{(s)}>-<{w}_{e}^{(FF)}>$,
$<\tilde{w}_{m}^{(s)}>=<{w}_{m}^{(s)}>-<{w}_{m}^{(FF)}>$. While
$<{w}_{e}^{(s)}>$ and $<{w}_{m}^{(s)}>$ are the energy densities of the full
fields ${\bf E}^{(s)}$ and ${\bf B}^{(s)}$, i.e with all terms of their
expressions (D-1)-(D-4) of Appendix D; the supescript $(FF)$ stands for far-
zone electric and magnetic energy densities, i.e corresponding to the fields
(D-1)-(D-4) with only terms of $r^{-1}$ dependence. Since evidently
$<{w}_{m}^{(FF)}>=<{w}_{e}^{(FF)}>$, we have that
$<{w}_{m}^{(s)}>-<{w}_{e}^{(s)}>=<\tilde{w}_{m}^{(s)}>-<\tilde{w}_{e}^{(s)}>$
noteinterf .

Then we write $W_{react}^{(s)}$ in terms of the volume integral outside the
volume $V$ in the above mentioned sphere $\partial V$ of radius $r$, centered
in ${\bf r}={\bf 0}$:

$\displaystyle W_{react}^{(s)}=\frac{ck}{3r^{3}}(|{\bf m}|^{2}-|{\bf
p}|^{2})]$
$\displaystyle=2\omega\int_{V_{\infty}-V}d^{3}r\,(<{w}_{m}^{(s)}>-<{w}_{e}^{(s)}>)$
$\displaystyle=2\omega\int_{V_{\infty}-V}d^{3}r\,(<\tilde{w}_{m}^{(s)}>-<\tilde{w}_{e}^{(s)}>).$
(34)

Where $V_{\infty}$ is the volume of a large sphere ($kr\rightarrow\infty$).
Making $V=V_{0}$ and $r=a$, (34) yields the reactive power outside the
particle. The proof of (34) is given in Appendix E, [cf. Eq. (E-3)]. We
anticipate that Eq. (34) is a consequence of the optical theorem for reactive
power which we put forward in Section VI, [cf. Eq. (47)].

As shown in Eq. (3), $\int_{\partial V}Im\\{{\bf S}^{(s)}\\}\cdot\hat{\bf
r}\,d^{2}r$ is associated to an instantaneous energy flow alternating back and
forth from the scatterer without losses, at frequency $2\omega$, with zero net
energy transport in the embedding vacuum. Nonetheless, this alternating flow
builds $W_{react}^{(s)}$ according to Eq. (33), also deduced from Eq. (34). As
a consequence, there is an accretion of time-averaged non-propagating reactive
power and stored energy, ${W}^{(sto)}$, outside $V$. (We note a concept
analogous to ${W}^{(sto)}$ for purely electric dipolar RF-antennas, cf. e.g.
harrington ; collin ; mcLean ; geyi ). Taking $V=V_{0}$, the total energy
stored outside the particle is obtained by

$\displaystyle{W}^{(sto)}=\int_{V_{\infty}-V_{0}}(<\tilde{w}_{m}^{(s)}>+<\tilde{w}_{e}^{(s)}>)d^{3}r$
$\displaystyle=\int_{V_{\infty}-V_{0}}(<{w}_{m}^{(s)}>+<{w}_{e}^{(s)}>)d^{3}r$
$\displaystyle-\int_{V_{\infty}-V_{0}}(<\tilde{w}_{m}^{(FF)}>+<\tilde{w}_{e}^{(FF)}>)d^{3}r$
$\displaystyle=\frac{1}{6}(|{\bf p}|^{2}+|{\bf
m}|^{2})(\frac{1}{a^{3}}+\frac{2k^{2}}{a}).\,\,\,\,\,\,\,\,\,\,\,\,\,\ $ (35)

Appendix E provides the proof of (35), [cf. Eq. (E-4)].

The quality factor associated with ${W}^{(sto)}$ is
$Q=\frac{2\omega{W}^{(sto)}}{W^{(s)}}=\frac{1}{(ka)^{3}}+\frac{2}{ka}.\,$
Hence being independent of the strength of the electric and/or magnetic dipole
moments.

Since in general a dipolar particle in the wide sense cannot be abstracted as
a point dipole, the overall interior reactive power, $W_{react,int}^{(s)}$ and
the interior stored energy, $W^{(sto,int)}$, obtained analogously to (34) and
(35), but integrating in $V_{0}$ the mean energies of the interior field, are
also of interest. For a linearly, or circularly, polarized incident plane wave
of unit intensity, a straightforward calculation yields
$(kc/3a^{3})(9/4k^{6})(0.055|d_{1}|^{2}\mp 0.018|c_{1}|^{2})$. The upper and
lower sign in $\mp$ apply to $W_{react,int}^{(s)}$ and $W^{(sto,int)}$,
respectively, while $d_{1}$ and $c_{1}$ are the first electric and magnetic
Mie coefficients of the fields inside the particle, (cf. e.g. Eq.(4.45) of
bohren ). Notice the appearance of $|d_{1}|^{2}$ and $|c_{1}|^{2}$ in
$W_{react,int}^{(s)}$ with sign opposite to that of $|{\bf p}|^{2}$ and
$|\bf{m}|^{2}$ in (34), i.e. of the first electric and magnetic external Mie
coefficient squared moduli, $|a_{1}|^{2}$ and $|b_{1}|^{2}$, in
$W_{react}^{(s)}$; we shall see that this has consequences for the total (i.e.
interior plus external) reactive power at resonant wavelengths.

###  V.2 Reactive power, resonances, and Kerker conditions

Fig.3 depicts $W_{react}^{(s)}(\lambda,a)$, $W_{react,int}^{(s)}(\lambda,a)$,
and
$W_{react,T}^{(s)}(\lambda,a)=W_{react}^{(s)}(\lambda,a)+W_{react,int}^{(s)}(\lambda,a)$
for Si spheres within the range of wavelengths where they become dipolar
magnetodielectric, and for different size radii $a$, taking advantage of their
scaling property with their impact parameter, (see Fig.2 of nietoSi , and
geffrin ). The redshift of the electric and magnetic dipole resonances
$\lambda_{e}$ and $\lambda_{m}$ as $a$ grows, observed in the scattering
cross-section geffrin ; kivshar , [see also $W^{(s)}$ in Fig.3(Center)], is
observed in the peaks or dips of these reactive powers as they are much
influenced by these resonances. An important feature of these surfaces is that
the total reactive power vanishes, changing its sign, close to the resonance
wavelengths where $W^{(s)}$ is maximum, irrespective of $a$. This is detailed
in Fig. 4(Center) on a cross sectional plane $a=75nm$ of these surfaces,
depicting the scattered power, along with the external, interior, and total
reactive powers. Also Fig.4(Right) shows the stored energies, while in
Fig.4(Left) one sees the electric and magnetic external first Mie coefficients
$a_{1}$ and $b_{1}$ with resonant maxima at $\lambda_{e}=492nm$ and
$\lambda_{m}=668nm$, respectively, and the internal coefficients, $d_{1}$ and
$c_{1}$.

The vanishing of the total reactive power, $W_{react,T}^{(s)}$, observed in
Figs.3 (Right) and Fig.4(Center) close to the electric and magnetic resonances
$\lambda_{e}$ and $\lambda_{m}$ where the scattered power, $W^{(s)}$, is
produced with maximum efficiency, is quite relevant because it manifests a
cancellation between the internal and external reactive power peaks; the
stored energies [cf. Fig. 4(Right)] being also resonant near these
wavelengths. In fact this matches ziolkowski with knowledge from RF antenna
theory according to which a maximum radiation efficiency is sought by
minimizing their reactive power and $Q$-factor harrington ; wheeler ; chu ;
collin ; mcLean ; geyi ; balanis , even though in these works the capacitive -
electric dipole wavelength $\lambda_{e}$: ${\bf E}$ dominates in $r>a$,
$r<\lambda_{e}$ \- (inductive - magnetic dipole wavelength $\lambda_{m}$:
${\bf B}$ dominates in $r>a$, $r<\lambda_{m}$ -) nature of $W_{react}^{(s)}$
is compensated and tuned to resonance by adding an inductive (capacitive)
storage element.

Such element, here in the optics domain, is provided by the particle interior
through the emergence of a dominant ${\bf B}$ at $\lambda_{e}$ (dominant ${\bf
E}$ at $\lambda_{m}$) of $W_{react,int}^{(s)}$ inside the particle, ($r<a$).
By the same token the total stored energies have peaks in these resonant
wavelengths $\lambda_{e}$ and $\lambda_{m}$, like the scattered power
$W^{(s)}$ [cf. Fig.4(Right)], being evident that the difference of external
and internal stored energies: $W^{(sto,Dif)}=W^{(sto)}-W^{(sto,int)}$
vanishes, like $W_{react,T}^{(s)}$ , close to $\lambda_{e}$ and $\lambda_{m}$.

These results illustrate the concepts of reactive and stored power in and
around a magnetodielectric, or Huygens, particle when choosing configurations
and wavelengths such that the accretion of external reactive power and stored
energy in the particle near-field, through the IPV alternating flow, be as
large as possible; thus scattering with maximum efficiency $W^{(s)}$, and
possessing the highest possible $Q$-factor for applications in light-matter
interactions, while its total reactive power $W_{react,T}^{(s)}$ vanishes or
is near zero at resonant wavelengths. Since, however, these magnetodielectric
nanoresonators have rather low $Q$’s ($Q\leq 7$), external (and internal)
stored power enhancements may be achieved either by sets of such
magnetodielectric particles, even metal coated, using them as building blocks
of photonic molecules, metasurfaces kivsharmeta , or in regimes of bound
states in the continuum bonod ; kivsharBIC .

Returning to Fig. 4(Left), we observe the lines $|a_{1}|^{2}$ and
$|b_{1}|^{2}$ crossing each other at the two Kerker wavelengths:
$\lambda_{K1}=738.5nm$ and $\lambda_{K2}=608nm$, which correspond to the first
Kerker condition (K1), (zero backscattering, $\alpha_{m}=\alpha_{e}$, $|{\bf
p}|=|{\bf m}|$), and second Kerker condition (K2), (minimum forward
scattering, $\alpha_{m}=-\alpha^{*}_{e}$, $|{\bf p}|\simeq|{\bf m}|$), kerker
; nieto2011 ; nietoJNano ; geffrin ; lapin ; staude ; kivshar . One sees in
Fig. 4(Middle) that at $\lambda_{K1}$, $W_{react}^{(s)}=0$ in accordance with
Eqs. (33) and (34), $W_{react,int}^{(s)}\simeq 0$ ; while both
$W_{react}^{(s)}$ and $W_{react,int}^{(s)}\simeq 0$ at $\lambda_{K2}$. Also
notice in Fig.4(Right) that the stored energies, $W^{(sto)}$ and
$W^{(sto,int)}$, and scattered power, $W^{(s)}$, are near minimum in the
proximities of $\lambda_{K1}$ and $\lambda_{K2}$. These features happen in
Fig.3 for any $a$.

Therefore, the analysis based on the particle reactive power allows to
envisage the Kerker conditions, K1 and K2, from a new standpoint:

The two Kerker conditions for a magnetodielectric dipolar particle are those
at which the total reactive power is near zero. Namely, the external reactive
power $W_{react}^{(s)}$ is either zero (in K1), or close to zero (in K2);
while the internal reactive power is near zero both in K1 and K2. In
consequence, the reactive power underlies the angular distribution of
scattered (or radiated) intensity and, hence, the directivity of the
magnetoelectric particle in a way complementary to that formerly addressed in
Mie-tronics and RF-antennas balanis ; geyi .

Figure 5: 3-D graphs of reactive helicities as functions of wavelength
$\lambda$ and radius $a$ of a magnetodielelectric sphere of Si in air,
illuminated by a left circularly polarized incident plane wave, CPL(+).
(Left): External ${\cal H}_{react}^{(s)}(\lambda,a)$. (Center): Interior
${\cal H}_{react,int}^{(s)}(\lambda,a)$. (Right): Total ${\cal
H}_{react,T}^{(s)}(\lambda,a)={\cal H}_{react}^{(s)}(\lambda,a)+{\cal
H}_{react,int}^{(s)}(\lambda,a)$. The two dips of ${\cal H}_{react}^{(s)}$,
and zero crossings of ${\cal H}_{react,int}^{(s)}$, are close to the electric
and magnetic resonant wavelengths $\lambda_{e}$ and $\lambda_{m}$, and are
redshifted with increasing $a$. Irrespective of the value of $a$, ${\cal
H}_{react}^{(s)}=0$ at $\lambda_{K1}$ and ${\cal H}_{react,T}^{(s)}$=0 close
to $\lambda_{e}$ and $\lambda_{m}$. Figure 6: (Left) 3-D graph of the
scattered helicity ${\cal H}^{(s)}(\lambda,a)$ generated by Si spheres in air,
illuminated by a left circularly polarized incident plane wave, CPL(+). It has
two peaks, at $475nm$ and $679nm$, influenced by the electric and magnetic
resonant wavelengths $\lambda_{e}=492nm$ and $\lambda_{m}=668nm$. (Right):
Making $a=75nm$, ${\cal H}^{(s)}(\lambda)$, ${\cal H}_{react}^{(s)}(\lambda)$,
${\cal H}_{react,int}^{(s)}(\lambda)$ and ${\cal H}_{react,T}^{(s)}(\lambda)$.
At the Kerker wavelength $\lambda_{K1}=738.5nm$, ${\cal H}_{react}^{(s)}$
vanishes. Also ${\cal H}_{react,T}^{(s)}(\lambda)$ is zero close to the
wavelengths where the scattered helicity ${\cal H}^{(s)}$ is maximum, and in
the vicinity of extrema of both ${\cal H}_{react}^{(s)}$ and ${\cal
H}_{react,int}^{(s)}$.

###  V.3 The reactive helicity

Concerning the complex helicity flow of the fields scattered by the dipolar
particle, [cf. Eq.(16)],
$\bm{\mathcal{F}}_{C}^{(s)}=\bm{\mathcal{F}}^{(s)}+i\bm{\mathcal{F}}_{{\mathscr{H_{R}}}}^{(s)}=(c/4k)Im\\{({\bf
B}^{(s)\,*}\times{\bf B}^{(s)}+{\bf E}^{(s)\,*}\times{\bf E}^{(s)})+i({\bf
B}^{(s)\,*}\times{\bf B}^{(s)}-{\bf E}^{(s)\,*}\times{\bf E}^{(s)})\\}$, we
obtain the flux $\bm{\mathcal{F}}_{C}^{(s)}\,\cdot\hat{\bf r}$ across the
spherical surface $\partial V$ of radius $r\geq a$ centered in the particle.
From Eqs. (D-1) and (D-2) for the fields in Appendix D, the terms that do not
vanish on integration are

$\displaystyle\int_{\partial V}\bm{\mathcal{F}}_{C}^{(s)}\cdot\hat{\bf
r}\,d^{2}r=\frac{c}{4k}r^{2}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\,\hat{\bf
r}\cdot Im\\{[\hat{\bf r}\times({\bf m}^{*}\times\hat{\bf r})]$
$\displaystyle\times(\hat{\bf r}\times{\bf
p})\frac{k^{2}}{r}(\frac{k^{2}}{r}+\frac{ik}{r^{2}})-C.C.\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle+[\hat{\bf r}\times({\bf p}^{*}\times\hat{\bf
r})]\times(\hat{\bf r}\times{\bf
m})\frac{k^{2}}{r}(\frac{k^{2}}{r}+\frac{ik}{r^{2}})-C.C.\\}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(36)

Where $C.C.$ denotes complex-conjugated of the previous term. A
straightforward calculation of (36) yields

$\displaystyle\int_{\partial V}\bm{\mathcal{F}}_{C}^{(s)}\cdot\hat{\bf
r}\,d^{2}r=\frac{8\pi}{3}ck^{2}\\{k\,Im[{\bf p}\cdot{\bf
m}^{*}]+\frac{i}{r}Re[{\bf p}\cdot{\bf
m}^{*}]\\},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (37)

whose real part, ${\cal H}^{(s)}=\int_{\partial
V}Re\\{\bm{\mathcal{F}}_{C}^{(s)}\\}\cdot\hat{\bf r}\,d^{2}r=\int_{\partial
V}\bm{\mathcal{F}}^{(s)}\cdot\hat{\bf r}\,d^{2}r=\frac{8\pi}{3}ck^{3}Im[{\bf
p}\cdot{\bf m}^{*}]$, is the total helicity of the scattered field, (as such,
it coincides with Eq. (25) of nietoheli ). Like the scattered power, this
helicity does not depend on the distance $r$.

However the imaginary part of (37),

$\displaystyle-{\cal H}_{react}^{(s)}=\int_{\partial
V}Im\\{\bm{\mathcal{F}}_{C}^{(s)}\\}\cdot\hat{\bf r}\,d^{2}r=\int_{\partial
V}\bm{\mathcal{F}}_{{\mathscr{H_{R}}}}^{(s)}\cdot\hat{\bf r}\,d^{2}r$
$\displaystyle=\frac{8\pi}{3r}ck^{2}Re[{\bf p}\cdot{\bf m}^{*}],\,\,\,\,\,\,$
(38)

comes from the interference of the intermediate field with $r^{-2}$ dependence
and the far field. ${\cal H}_{react}^{(s)}$ represents the external reactive
helicity,

$\displaystyle{\cal
H}_{react}^{(s)}=2\omega\int_{V_{\infty}-V}\mathscr{H_{R}}^{(s)}\,d^{3}r$ (39)

outside $V$, as detailed in its optical theorem discussed later, [cf. Eq.
(52)]. It decreases as $r^{-1}$ as $r$ grows. Therefore, in analogy with the
reactive power, in the near-field $(r<<\lambda$) the reactive helicity
dominates upon ${\cal H}^{(s)}$. On making $V=V_{0}$, $r=a$, Eq.(39) becomes
the overall reactive helicity outside the particle.

It is of interest to specify the overall reactive helicity of the field inside
the particle, ${\cal H}_{react,int}^{(s)}$, which is obtained integrating in
$V_{0}$ the helicity of the internal field:
$2\omega\int_{V_{0}}\mathscr{H_{R}}^{(s,int)}\,d^{3}r$. For a left circularly
polarized incident plane wave of unit intensity, this quantity is equal to:
$(8\pi ck^{2}/3a)(9/4k^{6})0.234\times 0.276Re\\{d_{1}c^{*}_{1}\\}$.

Figures 5 illustrate ${\cal H}_{react}^{(s)}(\lambda,a)$, ${\cal
H}_{react,int}^{(s)}(\lambda,a)$, and ${\cal
H}_{react,T}^{(s)}(\lambda,a)={\cal H}_{react}^{(s)}(\lambda,a)+{\cal
H}_{react,int}^{(s)}(\lambda,a)$, generated by Si spheres illuminated by a
left-circularly polarized, CPL(+), plane wave. Again, the redshift with
increasing radius $a$ is observed in the electric and magnetic dipole resonant
dips of ${\cal H}_{react}^{(s)}$. An interesting feature of ${\cal
H}_{react,int}^{(s)}$ is its similitude with ${\cal H}^{(s)}$, [cf.
Fig.6(Center) and Fig.6(Left), as well as Fig.6(Right)], and specially their
similarity with the converted helicity in the range between $\lambda_{e}$ and
$\lambda_{m}$, (compare with Fig. 1(c) of gutsche ). This remarks the
contribution of the interior reactive helicity to the scattered helicity
lineshape, and lays down an intriguing connection with previous studies
poulikakos1 ; poulikakos2 ; gutsche which attribute the conversion of helicity
to contributions of the scatterer volume and surface.

The total reactive helicity ${\cal H}_{react,T}^{(s)}$ plays on the scattered
helicity a role analogous to that of the total reactive power on the scattered
power. As seen in Fig. 6(Left) ${\cal H}^{(s)}$ has resonant dips influenced
by the electric and magnetic resonant wavelengths, $\lambda_{e}$ and
$\lambda_{m}$, of the scattered power. For $a=75$, [cf. Fig.6(Right)], these
dips are at $\lambda=475nm$ and $679nm$ and stem from the resonances at
$\lambda_{e}=492nm$ and $\lambda_{m}=668nm$, respectively. On the other hand,
${\cal H}_{react,T}^{(s)}$ becomes zero very close to these resonant
wavelengths of ${\cal H}^{(s)}$. We conclude, therefore, that the interior
reactive helicity ${\cal H}_{react,int}^{(s)}$ counteracts on the external
reactive helicity ${\cal H}_{react}^{(s)}$ close the resonant wavelengths of
${\cal H}^{(s)}$, (where also ${\cal H}_{react}^{(s)}$ and ${\cal
H}_{react;int}^{(s)}$ are near extreme values), yielding a zero total reactive
helicity ${\cal H}_{react,T}^{(s)}$. This property may be observed in
Fig.5(Right), independently of $a$, and it is clearly seen in Fig.6(Right). We
do not know, however, of any analogy of these effects in RF-antenna heory.

Notice that these high index magnetoelectric particles have the interesting
property of emitting a wavefield in which there are not very large peaks of
the total reactive power [cf. Fig.4(Center)] versus those of the scattered
(radiated) power; although, certainly, where this reactive power has extrema
the scattered (radiated) power is well aside its peaks. However, under chiral
ilumination these fields present high peaks of total reactive helicity versus
its radiated one. For instance, [see in Fig.6(Right)], the $75$ nm particle]
yields a the total reactive helicity with a large dip at 658$nm$ where the
scattered (radiated) helicity is near its minimum value. This occurs at
shifted positions as $a$ varies, [cf. Fig.6(Left) and Fig.5(Right)].

Therefore, the picture that emerges in these illustrations of such optical
nanoantennas, considered as either primary or secondary sources, is that the
reactive power and the reactive helicity, which are concentrated both inside
and in the near and intermediate regions of the source, have a hampering
effect in their far-field scattering (or radiation) efficiency. This is an
analogous effect to that due to the presence of reactive power in RF-antennas.
In consequence, if these nanoantennas emit chiral light, the total reactive
helicity hinders the efficiency of far-field scattered (or radiated) helicity,
so that less of this helicity is emitted due to a build-up of reactive
helicity in and around the nanantenna bornseries .

Under incident CPL these particles are dual at $\lambda_{K1}$, and then ${\bf
p}_{\pm}=\pm i{\bf m}_{\pm}$ corbato ; nietoheli , the upper and lower sign
applying to left circular, CPL(+) and right circular, CPL(-), respectively.
Then ${\bf p}\cdot{\bf m}^{*}$ is purely imaginary at $\lambda_{K1}$
comment_CPL , and one sees from (37) that ${\cal H}_{react}^{(s)}=0$. This is
observed in Fig.5(Left) for any $a$ and in Fig.6(Right) at
$\lambda_{K1}=738.5nm.$

As $a$ varies, ${\cal H}_{react}^{(s)}(\lambda,a)$ vanishes at the
corresponding Kerker wavelength $\lambda_{K1}$; this is seen in detail in
Fig.6(Right) for $a=75nm$.

Therefore, the first Kerker condition, K1, also has the novel property that
under CPL illumination, the magnetodielectric particle, which then becomes
dual and hence emits a wavefield of well-defined helicity equal to the
incident one, corbato ; nietoheli ; gutsche , does not generate external
reactive helicity.

##  VI The reactive power optical theorem

Consider a wavefield, ${\bf E}^{(i)}$, ${\bf H}^{(i)}$ incident on a
magnetodielectric body of volume $V_{0}$. The field at any point of the
embedding medium, (assumed to be vacuum or air), is represented as ${\bf
E}={\bf E}^{(i)}+{\bf E}^{(s)}$, ${\bf H}={\bf H}^{(i)}+{\bf H}^{(s)}$, where
the superscript $(s)$ denotes the scattered field. Maxwell’s equations are
written as:

$\displaystyle\nabla\times{\bf H}^{(i)}=-ik{\bf
D}^{(i)}\,,\,\,\,\,\,\,\,\,\,\,\,\,\nabla\times{\bf E}^{(i)}=ik{\bf
B}^{(i)};\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle\nabla\times{\bf H}^{(s)}=-ik{\bf E}^{(s)}-4\pi ik{\bf
P}+\frac{4\pi}{c}{\bf j}\,,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle\nabla\times{\bf E}^{(s)}=ik{\bf H}^{(s)}+4\pi ik{\bf
M}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (40)

Where ${\bf P}$, ${\bf M}$ and ${\bf j}$ are the polarization, magnetization,
and free current densities in $V_{0}$, respectively. ${\bf D}^{(i)}={\bf
E}^{(i)}$, ${\bf B}^{(i)}={\bf H}^{(i)}$. We insert Eqs. (40) into Eq.(1) for
the total fields ${\bf E}$ and ${\bf H}$, and use the identity:
$\nabla\cdot({\bf E}\times{\bf H}^{*})={\bf H}^{*}\cdot\nabla\times{\bf
E}-{\bf E}\cdot\nabla\times{\bf H}^{*}$, integrating in a volume $V$ that
contains $V_{0}$,

$\displaystyle\int_{V}d^{3}r\,\\{2i\omega(<w^{(i)}_{m}>-<w^{(i)}_{e}>)+i\frac{\omega}{4\pi}Re\\{{\bf
H}^{(i)}\cdot{\bf H}^{(s)\,*}$ $\displaystyle-{\bf E}^{(i)}\cdot{\bf
E}^{(s\,)*}\\}+i\frac{\omega}{2}[{\bf H}^{(i)\,*}\cdot{\bf M}-{\bf
E}^{(i)}\cdot{\bf P}^{*}]$ $\displaystyle-\frac{1}{2}\,{\bf j}^{*}\cdot{\bf
E}^{(i)}+\nabla\cdot{\bf
S}^{(s)}\\}d^{3}r=-\frac{1}{2}\int_{V_{0}}d^{3}r\,{\bf j}^{*}\cdot{\bf
E}\,\,\,\,\,\,\,\,\,$
$\displaystyle+2i\omega\int_{V}(<w_{m}>-<w_{e}>)d^{3}r\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(41)

If the incident field has no evanescent components, i.e. it is source-free and
propagating, it does not store energy, so that the first term of (41) is
identically zero. However if it is evanescent , or it has evanescent
components, we shaw in (28) that it stores reactive power and it is given by
this term. Therefore we shall keep it in the above equation, which is the
complex optical theorem in presence of the scatterer.

If there are no scattering induced sources other than ${\bf P}$ and ${\bf M}$
in the body, the free current ${\bf j}$ conveys the conversion of incident
power into mechanical and/or thermal energy through the work done on the
charges $\frac{1}{2}Re\,\int_{V_{0}}d^{3}r\,{\bf j}^{*}\cdot({\bf E}-{\bf
E}^{(i)})$, which accounts for the decrease of energy ${\cal W}^{(a)}$ from
the wave as power absorbed by the body. Then taking the real part of (41) we
obtain

$\displaystyle\frac{\omega}{2}\int_{V_{0}}d^{3}r\,Im[{\bf H}^{(i)\,*}\cdot{\bf
M}+{\bf E}^{(i)\,*}\cdot{\bf P}]$ $\displaystyle={\cal W}^{(a)}+\int_{\partial
V}d^{2}rRe\\{{\bf S}^{(s)}\\}\cdot\hat{\bm{r}}.$ (42)

Which is the standard optical theorem (OT) for energy born ; nieto1 ,
describing the extinction of incident energy, [left side of (42)], and
consequent absorption and radiation of the total scattered energy. It reduces
to its well-known expression nieto1 for dipolar particles on making ${\bf
P}({\bf r})={\bf p}\,\delta({\bf r})$ and ${\bf M}({\bf r})={\bf
m}\,\delta({\bf r})$, and then $\int_{\partial V}d^{2}rRe\\{{\bf
S}^{(s)}\\}\cdot\hat{\bm{r}}=W^{(s)}$ in accordance with (32).

Here we are, however, interested in the imaginary part of (41),

$\displaystyle\int_{V}d^{3}r\,[2\omega(<w^{(i)}_{m}>-<w^{(i)}_{e}>)+\frac{\omega}{4\pi}Re\\{{\bf
H}^{(i)}\cdot{\bf H}^{(s\,)*}$ $\displaystyle-{\bf E}^{(i)}\cdot{\bf
E}^{(s\,)*}\\}+\frac{\omega}{2}Re\\{{\bf H}^{(i)\,*}\cdot{\bf M}-{\bf
E}^{(i)\,*}\cdot{\bf P}\\}]$
$\displaystyle-\frac{1}{2}\,Im\\{\int_{V_{0}}d^{3}r\,{\bf j}^{*}\cdot{\bf
E}^{(i)}\\}+\int_{\partial V}Im\\{{\bf S}^{(s)}\\}\cdot\hat{\bm{r}}d^{2}r$
$\displaystyle=-\frac{1}{2}\int_{V_{0}}d^{3}r\,Im\\{{\bf j}^{*}\cdot{\bf
E}\\}$
$\displaystyle+2\omega\int_{V}(<w_{m}>-<w_{e}>)d^{3}r\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(43)

Which becomes

$\displaystyle\frac{\omega}{2}\int_{V_{0}}d^{3}r\,Re\\{{\bf
H}^{(i)\,*}\cdot{\bf M}-{\bf E}^{(i)\,*}\cdot{\bf P}\\}=$
$\displaystyle-\frac{1}{2}\int_{V_{0}}d^{3}r\,Im\\{{\bf j}^{*}\cdot{\bf
E}^{(s)}\\}-\int_{\partial V}Im\\{{\bf
S}^{(s)}\\}\cdot\hat{\bm{r}}d^{2}r+\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle+2\omega\int_{V}(<\tilde{w}^{(s)}_{m}>-<\tilde{w}^{(s)}_{e}>)d^{3}r\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(44)

Where we have made use of the fact that
$<{w}_{m}^{(s)\,(FF)}>=<{w}_{e}^{(s)\,(FF)}>$ and hence
$<w^{(s)}_{m}>-<w^{(s)}_{e}>=<\tilde{w}_{m}^{(s)}>-<\tilde{w}_{e}^{(s)}>$,
[cf. paragraph prior to Eq. (34)].

Equation (44) is our formulation of the reactive power optical theorem (ROT)
for a generic scatterer whose response to illumination induces densities of
polarization ${\bf P}$, magnetization ${\bf M}$, and free current ${\bf j}$,
comment ROT . The left side constitutes the extinction of incident energy
which produces the build-up of external reactive power on scattering in the
right side of (44). Thus Eq. (44) describes how, in addition to being radiated
into the far zone as Eq. (42) illustrates, scattering gives rise to non-
radiated energy, stored in $V$ in the form of (external) reactive power,
flowing out from the scattering object and returning to it.

For a dipolar particle the ROT reduces to

$\displaystyle\frac{\omega}{2}Re\\{{\bf H}^{(i)\,*}(0)\cdot{\bf m}-{\bf
E}^{(i)\,*}(0)\cdot{\bf p}\\}=\,\,\,\,\,\,\,\,\,\,$
$\displaystyle-\frac{1}{2}\int_{V_{0}}d^{3}r\,Im\\{{\bf j}^{*}\cdot{\bf
E}^{(s)}\\}-\int_{\partial V}Im\\{{\bf
S}^{(s)}\\}\cdot\hat{\bm{r}}d^{2}r\,\,\,\,\,\,\,\,\,\,$
$\displaystyle+2\omega\int_{V}(<\tilde{w}^{(s)}_{m}>-<\tilde{w}^{(s)}_{e}>)d^{3}r\,.\,\,\,\,\,\,\,\,\,\,\,\,\,$
(45)

The argument $0$ indicates that the fields are evaluated at the particle
center $r=0$.

Concerning a dipolar particle, the process described by the complex optical
theorem (41) is analogous to that in which the feeding energy from an
alternate current $I$, induces an oscillating dipole in a small antenna, which
emits radiated and stored power through the extinction, $\frac{1}{2}ZI^{2}$,
of the driving energy. $Z$ being the antenna input impedance,
$Z=R_{l}+R_{r}-iX$. The dipole loss resistance $R_{l}$ and radiation
resistance $R_{r}$ balanis , generated in accordance with the optical theorem
(42), are $2W^{(a)}/|I|^{2}$ and $2W^{(s)}/|I|^{2}$, respectively. On the
other hand, the dipole reactance $X$ (which for a magnetoelectic dipole is
either capacitive or inductive jackson , depending on the wavelength
$\lambda_{e}$ or $\lambda_{m}$) stems from its external reactive power,
$W_{react}^{(s)}$, balanis whose generation is ruled by the ROT (45). Hence,
in this context the extinction term in the left sides of (42) and (45) may be
associated to $\frac{1}{2}(R_{l}+R_{r})|I|^{2}$ and $-\frac{1}{2}X|I|^{2}$,
respectively.

On taking $\partial V$ and $V$ as $\partial V_{\infty}$ and $V_{\infty}$,
respectively, corresponding to a large sphere, ($kr\rightarrow\infty$), the
flux of scattered CPV across $\partial V_{\infty}$ is real and equals the
total scattered energy $W^{(s)}$. Therefore, Eq. (45) yields

$\displaystyle\frac{\omega}{2}Re\\{{\bf H}^{(i)\,*}(0)\cdot{\bf m}-{\bf
E}^{(i)\,*}(0)\cdot{\bf p}\\}=-\frac{1}{2}\int_{V_{0}}d^{3}r\,Im\\{{\bf
j}^{*}\cdot{\bf E}^{(s)}\\}$
$\displaystyle+2\omega\int_{V_{\infty}}(<\tilde{w}^{(s)}_{m}>-<\tilde{w}^{(s)}_{e}>)d^{3}r\,.\,\,\,\,\,\,\,\,\,\,\,\,$
(46)

Which introduced into (45) leads to

$\displaystyle-\int_{\partial V}Im\\{{\bf
S}^{(s)}\\}\cdot\hat{\bm{r}}d^{2}r=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle
2\omega\int_{V_{\infty}-V}(<\tilde{w}^{(s)}_{m}>-<\tilde{w}^{(s)}_{e}>)d^{3}r=W_{react}^{(s)}\,.\,$
(47)

In contrast with its real part, the flow (47) depends on the integration
domains, $V$ and $\partial V$. Notice that (47) conincides with Eq. (34).

##  VII The reactive helicity optical theorem

Next, we put forward the law which rules the formation of external reactive
helicity by scattering in the near and intermediate-field regions of the
particle through extinction of helicity of the incident wave. Let us introduce
Eqs. (40) into the identities: $\nabla\cdot({\bf H}^{*}\times{\bf H})={\bf
H}\cdot\nabla\times{\bf H}^{*}-{\bf H}^{*}\cdot\nabla\times{\bf H}$ and
$\nabla\cdot({\bf E}^{*}\times{\bf E})={\bf E}\cdot\nabla\times{\bf
E}^{*}-{\bf E}^{*}\cdot\nabla\times{\bf E}$, integrating in a volume $V$ that
contains the scattering volume $V_{0}$. With ${\bf E}={\bf E}^{(i)}+{\bf
E}^{(s)}$, ${\bf H}={\bf H}^{(i)}+{\bf H}^{(s)}$, adding the respective
expressions, employing the conservation equation (12) and using the
definitions (8) and (13), ($n$=1 and ${\bf H}={\bf B}$ outside $V_{0}$), we
arrive at

$\displaystyle 2\pi c\int_{V_{0}}d^{3}rRe\\{{\bf E}^{(i)\,*}\cdot{\bf M}-{\bf
H}^{(i)\,*}\cdot{\bf P}\\}=$ $\displaystyle\int_{\partial
V}d^{2}r\bm{\mathcal{F}}^{(s)}\cdot{\bf
n}+\frac{2\pi}{k}\int_{V_{0}}d^{3}rIm\\{{\bf H}^{(s)\,*}\cdot{\bf j}\\},$ (48)

which is the known optical theorem for the electromagnetic helicity nietoheli
applying to magnetodielectric arbitrary scattering bodies.  Notice that for a
dipolar particle, since ${\bf P}({\bf r})={\bf p}\,\delta({\bf r})$, ${\bf
M}({\bf r})={\bf m}\,\delta({\bf r})$ and $\int_{\partial
V}d^{2}r\bm{\mathcal{F}}^{(s)}\cdot{\bf n}=(8\pi ck^{3}/3)Im[{\bf p}\cdot{\bf
m}^{*}]$, Eq. (48) becomes like Eq. (28) of nietoheli .

However our focus is the conservation law of the reactive helicity. To
formulate it in the form of an optical theorem we substract, rather than add,
the above vector identities, and make use of the conservation law (14) along
with definitions (9) and (15). Then, proceeding as before, it is
straightforward to obtain

$\displaystyle 2\pi c\int_{V_{0}}d^{3}rRe\\{{\bf E}^{(i)\,*}\cdot{\bf M}+{\bf
H}^{(i)\,*}\cdot{\bf P}\\}=$
$\displaystyle-\frac{2\pi}{k}\int_{V_{0}}d^{3}rIm\\{{\bf H}^{(s)\,*}\cdot{\bf
j}\\}-\int_{\partial
V}d^{2}r\bm{\mathcal{F}}_{{\mathscr{H_{R}}}}^{(s)}\cdot{\bf n}$
$\displaystyle+2\omega\int_{V}d^{3}r\mathscr{H_{R}}^{(s)},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(49)

Equation (49) is the reactive helicity optical theorem and applies to a
generic magnetodielectric scatterer re opteheli . The left side represents the
extinction of helicity of the incident wave on build-up outside the body of a
reactive helicity by scattering, given by the right side of (49). Thus, like
the energy, the incident helicity gives rise to a reactive one associated to
the scattered field, which, in addition to the internal reactive helicity, is
stored around the particle, dominating in the near and intermediate-field
regions where it flows back and forth from the scatterer.

This storage is seen by first considering $V$ to be $V_{\infty}$ in (49). Then
[cf. comment Vinfty ]

$\displaystyle\int_{\partial
V_{\infty}}d^{2}r\bm{\mathcal{F}}_{{\mathscr{H_{R}}}}^{(s)}\cdot{\bf n}=0.$
(50)

Therefore,

$\displaystyle 2\pi c\int_{V_{0}}d^{3}rRe\\{{\bf E}^{(i)\,*}\cdot{\bf M}+{\bf
H}^{(i)\,*}\cdot{\bf P}\\}=$
$\displaystyle-\frac{2\pi}{k}\int_{V_{0}}d^{3}rIm\\{{\bf H}^{(s)\,*}\cdot{\bf
j}\\}+2\omega\int_{V_{\infty}}d^{3}r\mathscr{H_{R}}^{(s)},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(51)

which substituted in (49) leads to

$\displaystyle-\int_{\partial
V}d^{2}r\bm{\mathcal{F}}_{{\mathscr{H_{R}}}}^{(s)}\cdot{\bf
n}=2\omega\int_{V_{\infty-V}}d^{3}r\mathscr{H_{R}}^{(s)}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(52)

Notice that if $V=V_{0}$, Eq. (52) accounts for the reactive helicity stored
outside the scattering body.

Likewise, if the scatterer is a dipolar particle, the extinction term in the
left side of (51) becomes $2\pi cRe\\{{\bf E}^{(i)\,*}(0)\cdot{\bf m}+{\bf
H}^{(i)\,*}(0)\cdot{\bf p}\\}$. Then (52) coincides with ${\cal
H}_{react}^{(s)}$, Eq. (39) according to (38), thus proving it; and
illustrates how the external reactive helicity is stored around the particle
without being scattered into the far-zone.

##  VIII Consequence of the reactive power optical theorem: Significance of
the reactive helicity in reactive dichroism

In dichroism, chiral light illuminates a chiral particle, molecule, or
nanostructure. Assuming it dipolar, its constitutive relations for the induced
dipole moments, ${\bf p}$ and ${\bf m}$, and the incident field are

$\displaystyle{\bf p}=\alpha_{e}{\bf E}^{(i)}+\alpha_{em}{\bf
B}^{(i)},\,\,\,\,\,{\bf m}=\alpha_{me}{\bf E}^{(i)}+\alpha_{m}{\bf
B}^{(i)}.\,\,\,\,\,\,\,\,\,\,$ (53)

The electric, magnetic, and magnetoelectric polarizabilities being
$\alpha_{e}$, $\alpha_{m}$, $\alpha_{em}$, and $\alpha_{me}$; and fulfilling
$\alpha_{em}=-\alpha_{me}$ since the object is chiral tang ; barron .

The signal re-emitted (or scattered) by the excitation of this dipolar body
discriminates enantiomers (i.e. particles with either $\alpha_{me}$ or
$-\alpha_{me}$) tang ; barron ; schellman by using as rate of excitation:
${\cal W}^{(s)}=\frac{\omega}{2}Im\\{{\bf E}^{(i)\,*}(0)\cdot{\bf p}+{\bf
B}^{(i)\,*}(0)\cdot{\bf m}\\}$; [cf. left side of Eq. (42)].

For instance, consider the pair of illuminating fields tang : ${\bf\cal
E}^{(i)}({\bf r},t)=Re[\pm{\bf E}^{(i)}({\bf r})\exp(-i\omega t)]$ and
${\bf\cal H}^{(i)}({\bf r},t)=Re[{\bf H}^{(i)}({\bf r})\exp(-i\omega t)]$,
whose respective helicites are: ${\mathscr{H}}_{i}^{+}$ and
${\mathscr{H}}_{i}^{-}$, with
${\mathscr{H}}_{i}^{+}={\mathscr{H}}_{i}=-{\mathscr{H}}_{i}^{-}$. On employing
(42) and (53), the particle excitation rate becomes: ${\cal
W}^{(s)\,\pm}=\frac{\omega}{2}\\{\alpha_{e}^{I}|{\bf
E}^{(i)}(0)|^{2}+\alpha_{m}^{I}|{\bf B}^{(i)}(0)|^{2}\pm
2\alpha_{me}^{R}\,Im[{\bf E}^{(i)}\cdot{\bf
B}^{(i)*\,}]\\}$$=\frac{\omega}{2}\\{\alpha_{e}^{I}|{\bf
E}^{(i)}(0)|^{2}+\alpha_{m}^{I}|{\bf B}^{(i)}(0)|^{2}\pm
4k\alpha_{me}^{R}\,\mathscr{H}_{i}\\}.$ The superscripts $I$ and $R$ denote
imaginary and real part, respectively. Clearly, the sign $+$ or $-$ appears
according to whether the helicity of the illumination is positive:
${\mathscr{H}}_{i}^{+}={\mathscr{H}}_{i}$, or negative:
${\mathscr{H}}_{i}^{-}=-{\mathscr{H}}_{i}$.

Then the above expression of ${\cal W}^{(s)\,\pm}$ yields the well-known
dissymmetry factor $g=({\cal W}^{(s)\,+}-{\cal W}^{(s)\,-})/({\cal
W}^{(s)\,+}+{\cal W}^{(s)\,-})$ barron ; schellman proportional to
$\alpha_{me}^{R}\,\mathscr{H}_{i}$ tang .

However, the ROT establishes that rather than the power radiated in the far-
zone (42), one may address the excitation of stored reactive power, which
dominates in the near and intermediate-field regions of the particle, which is
given by the left side of Eq. (45), viz. ${\cal
W}_{react}^{(s)}=\frac{\omega}{6}\Re\\{{\bf E}^{(i)\,*}(0)\cdot{\bf p}-{\bf
B}^{(i)\,*}(0)\cdot{\bf m}\\}$. Then using the above pair of illumination
fields and Eq. (53), one obtains the discriminatory reactive power

$\displaystyle{\cal W}_{react}^{(s)\,\pm}=\frac{\omega}{2}\Re\\{\pm{\bf
E}^{(i)\,*}(0)\cdot{\bf p}-{\bf B}^{(i)\,*}(0)\cdot{\bf
m}\\}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle=\frac{\omega}{2}\\{\alpha_{e}^{R}|{\bf
E}^{(i)}|^{2}-\alpha_{m}^{R}|{\bf B}^{(i)}|^{2}\mp 2\alpha_{me}^{R}\,\Re[{\bf
E}^{(i)}\cdot{\bf B}^{(i)*\,}]\\}\ \,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle=\frac{\omega}{2}\\{\alpha_{e}^{R}|{\bf
E}^{(i)}|^{2}-\alpha_{m}^{R}|{\bf B}^{(i)}|^{2}\mp
4k\alpha_{me}^{R}\,\mathscr{H_{R}}_{i}\\}.\,\,\,\,\,\,\,\,\,\,\,$ (54)

So that now the sign $-$ or $+$ applies according to whether the reactive
helicity of the illumination is positive:
${\mathscr{H_{R}}}_{i}^{+}={\mathscr{H_{R}}}_{i}$, or negative:
${\mathscr{H_{R}}}_{i}^{-}=-{\mathscr{H_{R}}}_{i}$, respectively; and (54)
yields a dissymmetry factor proportional to
$-\alpha_{me}^{R}\,\mathscr{H_{R}}_{i}$ comment2 .

We propose Eq. (54) as the basis of reactive dichroism observations. At
difference with standard dichroism, it involves a chiral incident field with
non-zero reactive helicity, and constitutes a near-field optics technique.

Therefore,  in an analogous way as detecting the radiated energy ${\cal
W}^{(s)\,\pm}$, (or absorption/extinction energy), in standard dichroism
involves the helicity $\mathscr{H}_{i}$ of the incident wave, in reactive
dichroism, observing the excitation of reactive power ${\cal
W}_{react}^{(s)\,\pm}$ in the chiral particle conveys the incident field
reactive helicity $\mathscr{H_{R}}_{i}$ , which evidently comes out on using
(54) in a dissymmetry factor defined as $g_{react}=2({\cal
W}_{react}^{(s)\,+}-{\cal W}_{react}^{(s)\,-})/({\cal
W}_{react}^{(s)\,+}+{\cal W}_{react}^{(s)\,-})$. Notice that being
$\mathscr{H_{R}}_{i}$ measurable, so is $g_{react}$ in proportion to
$\mathscr{H_{R}}_{i}$.

There is a variety of pairs of illumination wavefields that, like in the above
illustration, are interchangeable by parity, and that one may employ in
experiments. As discussed in previous sections, non-propagating fields in
free-space possess a non-zero $\mathscr{H_{R}}_{i}$; e.g. elliptically (or
circularly, in particular) polarized standing waves, near fields from an
emitter, or evanescent and other surface waves, fulfill Eq. (54).

##  IX Conclusions

Given the broad interest of evanescent waves at the nanoscale, and of small
particles as light emitting nanoantennas,  couplers, and metasurface elements,
the contributions of this paper on its reactive quantities is summarized in
the following main conclusions:

(1) We have established the concepts of complex helicity density and its
complex helicity flow, together with their conservation law that we name
complex helicity theorem. Its real part is the well-known conservation
equation of optical helicity, while its imaginary is a novel law that governs
the build-up of reactive helicity through its imaginary, and thus zero time-
average, flow. The concept of reactive helicity density unifies that of
magnetoelectric energy density, previously introduced by symmetry arguments,
and the so-called real helicity. In this way, we put forward its conservation
law and observability, thus completing the fundamentals of this quantity.

(2) The conservation of reactive helicity and reactive power, and their zero
time-averaged flow, has been illustrated in two paradigmatic systems: an
evanescent wave and a wavefield scattered (or emitted) by a dipolar
magnetodielectric particle. For the former we have put forward reactive
orbital and spin momenta that characterize its imaginary (reactive) field
(Poynting) momentum; showing that the wave density of reactive helicity is
observable from an experimentally detectectable mixed electric-magnetic
transversal optical force  exerted by the evanescent wave on a small high
refractive index particle (which behaves as magnetodielectric) through this
reactive Poynting momentum. On the other hand, we have uncovered a novel non-
conservative force in the decay direction of the evanescent wave, which can be
discriminated from the gradient one and thus detected, making observable the
wavefield reactive power density.

(3) Concerning the stored energy, reactive power, and reactive helicity of the
field scattered by a dipolar magnetodielectric particle, we have shown that
they provide a novel framework to study the particle emission directivity.
This has been illustrated on addressing the two Kerker conditions, K1 of zero
backscattering, and K2 of minimum forward scattering. We have established that
under CPL incident light, the external reactive power is zero in K1 or close
to zero in K2; while the internal reactive power, and hence the total reactive
power, is near zero both in K1 and K2. Also, we have proven an additional
novel property of the particle at K1 wavelengths, namely, it produces a
scattered field with nule overall external reactive helicity.

(4) We have established a reactive helicity optical theorem that governs the
build-up and storage of near-field reactive helicity, on extinction of the
incident helicity as light interacts with a generally magnetodielectric
nanoantenna. Also we have shown that the emission of resonant scattered power
and of resonant scattered helicity coincides with a nule, or near zero, total
reactive power and helicity, respectively, i.e. those given by the sum of the
overall interior and external reactive powers and helicities. Conversely,
peaks of total reactive helicity (reactive power) are associated with poorer
efficiency in the emission of radiated (or scattered) helicity (power).

(5) A reactive power optical theorem has been put forward. It rules the
formation of external reactive power and stored electric and magnetic
energies, which dominate in the near and intermediate-field zones of a
magnetodielectric particle by extinction of the illuminating energy.

(6) This latter theorem provides a framework to studying the near-field
response of chiral nanoparticles to illumination with chiral complex fields.
It is remarkable that in the phenomenon of dichroism on illumination with
chiral light, the incident reactive helicity arises in the near-field region,
as we have shown, while it is well-known that the incident optical helicity
appears from the determination of power emitted in the far-zone. Because of
this, we call reactive dichroism the phenomenon by which this incident
reactive helicity becomes discriminatory for enantiomeric separation. We
propose near-field observation experiments of this reactive phenomenon.

Given the interplay between reactive and radiative quantities of
electromagnetic fields, we believe that the concepts studied in this work
enrich the landscape of photonics as regards nanoantennas and nanoparticle
interactions with light. We expect that the observability of these reactive
effects and quantities should form the basis of future experiments and
techniques. In this respect, addressing reactive quantities on higher order
multipole resonance excitation will be a subject of interest for future
studies. This is of special interest for e.g. nanosensing advances calda and
all-dielectric thermonanophotonics zograf , to be used in effective biomedical
diagnosis and therapies.

Although our analysis has been focused on the nanoscale, these results are
equally valid in the microwave range due the scaling property of high index
particles as Huygens sources, which remain with the same characteristics of
generating large electric and magnetic dipole and multipole resonances as the
illumination wavelength increases.

## Acknowledgments

MN-V work was supported by Ministerio de Ciencia e Innovación of Spain, grant
PGC2018-095777-B-C21. X.X acknowledges the National Natural Science Foundation
of China (11804119). Helpful comments from two anonymous referees are
appreciated.

##  Appendix A PROOF OF EQS. (6) and (7) FOR THE ENERGY FLOW

Since ${\bf B}({\bf r})=(1/ik)\nabla\times{\bf E}({\bf r})$, using Eqs.
(4)-(5) we have

$\displaystyle{\bf B}({\bf r})=\frac{1}{k}\int_{-\infty}^{\infty}d^{2}{\bf
K}\,\,[{\bf k}\times{\bf e}({\bf K})]\exp[i({\bf K}\cdot{\bf
R}+k_{z}z)].\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (B-1)

Then the CPV flux across the plane $z=0$ is

$\displaystyle\Phi^{Poynt}=\frac{c}{8\pi}\int_{-\infty}^{\infty}d^{2}{\bf
R}\,[{\bf E}({\bf r})\times{\bf B}^{*}({\bf
r})]\cdot\hat{\bm{z}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle=\frac{c}{8\pi}\frac{1}{k}\int_{-\infty}^{\infty}d^{2}{\bf
R}\,\int_{-\infty}^{\infty}d^{2}{\bf K}\,d^{2}{\bf K}^{\prime}\,[{\bf e}({\bf
K^{\prime}})\times({\bf k}^{*}\times{\bf e}^{*}({\bf
K}))]\cdot\hat{\bm{z}}\exp[i({\bf K}^{\prime}-{\bf K})\cdot{\bf
R}+(k^{\prime}_{z}-k_{z}^{*})z]\\}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(B-2)

The ${\bf R}$-integral yields $\delta({\bf K}^{\prime}-{\bf
K})=(1/2\pi)^{2}\int_{-\infty}^{\infty}d^{2}{\bf R}\exp[i({\bf
K}^{\prime}-{\bf K})\cdot{\bf R}]$, and subsequent integration in ${\bf
K}^{\prime}$ yields for the right side of (B-2):

$\displaystyle\Phi^{Poynt}=\frac{\pi c}{2k}\int_{\infty}^{\infty}d^{2}{\bf
K}\,[{\bf k}^{*}|{\bf e}({\bf K})|^{2}-{\bf e}^{*}({\bf K})({\bf
k}^{*}\cdot{\bf e}({\bf
K}))]\cdot\hat{\bm{z}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle=\frac{\pi c}{2k}\int_{K\leq k}d^{2}{\bf K}\,q_{h}|{\bf
e}_{h}({\bf K})|^{2}-i\frac{\pi c}{2k}\int_{K>k}d^{2}{\bf K}\,[q_{e}|{\bf
e}_{e}({\bf K})|^{2}-2q_{e}|{e}_{e\,z}({\bf
K})|^{2}].\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (B-3)

Where the ${\bf K}$-integral is split into its homogeneous and evanescent
parts with subscripts $h$ and $e$, respectively. Then ${\bf k}_{h}$ is the
real wavevector $({\bf K},q_{h})$ of the homogeneous propagating plane wave
component of complex amplitude ${\bf e}_{h}({\bf K})$, whereas the complex
wavevector ${\bf k}_{e}=({\bf K},iq_{e})$ corresponds to each evanescent plane
wave component of complex amplitude ${\bf e}_{h}({\bf K})$. Also we have taken
into account that ${\bf k}_{h}^{*}\cdot{\bf e}_{h}({\bf K})=0$ and ${\bf
k}_{e}\cdot{\bf e}_{e}({\bf K})=0$, therefore ${\bf k}_{e}^{*}\cdot{\bf
e}_{e}({\bf K})=2{\bf K}\cdot{\bf e}_{e,\perp}({\bf
K})=-2iq_{e}{e}_{e\,z}({\bf K})$. [${\bf e}_{e}({\bf K})=({\bf
e}_{e\,\perp}({\bf K}),e_{e\,z}({\bf K})$].

Taking real and imaginary parts in (B-3), one has

$\displaystyle\Phi^{RPoynt}=\frac{\pi c}{2k}\int_{K\leq k}d^{2}{\bf
K}\,q_{h}|{\bf e}_{h}({\bf K})|^{2},$ (B-4)

and

$\displaystyle\Phi^{IPoynt}=-\frac{\pi c}{2k}\int_{K>k}d^{2}{\bf
K}\,[q_{e}|{\bf e}_{e}({\bf K})|^{2}-2q_{e}|{e}_{e\,z}({\bf
K})|^{2}].\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (B-5)

Which are Eqs.(6) and (7) of the main text.

##  Appendix B PROOF OF EQS. (17) AND (18)

Using Eqs. (4)-(5), the integral on $z=0$ of ${\bf E}({\bf r})\cdot{\bf
B}^{*}({\bf r})$ is

$\displaystyle\int_{-\infty}^{\infty}d^{2}{\bf R}\,[{\bf E}({\bf r})\cdot{\bf
B}^{*}({\bf r})]=\frac{1}{k}\int_{-\infty}^{\infty}d^{2}{\bf
R}\,\int_{-\infty}^{\infty}d^{2}{\bf K}\,d^{2}{\bf K}^{\prime}[{\bf e}({\bf
K^{\prime}})\cdot({\bf k}^{*}\times{\bf e}^{*}({\bf K}))]\exp[i({\bf
K}^{\prime}-{\bf K})\cdot{\bf R}]\\}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (B-1)

And following the same procedure as in Appendix A, we are led to

$\displaystyle\int_{-\infty}^{\infty}d^{2}{\bf R}\,[{\bf E}({\bf R})\cdot{\bf
B}^{*}({\bf R})]=\frac{(2\pi)^{2}}{k}\int_{-\infty}^{\infty}d^{2}{\bf
K}\,\,{\bf k}^{*}\cdot[{\bf e}^{*}({\bf K})\times{\bf e}({\bf
K})]\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle=\frac{(2\pi)^{2}}{k}\\{\int_{K\leq k}d^{2}{\bf K}\,{\bf
k}_{h}\cdot[{\bf e}_{h}^{*}({\bf K})\times{\bf e}_{h}({\bf
K})]+\int_{K>k}d^{2}{\bf K}\,{\bf K}\cdot[{\bf e}^{*}_{e}({\bf K})\times{\bf
e}_{e}({\bf K})]+\int_{K>k}d^{2}{\bf K}\ (-iq_{e})[{\bf e}_{e}^{*}({\bf
K})\times{\bf e}({\bf K})]_{z}\\}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (B-2)

We recall that $[{\bf e}^{*}({\bf K})\times{\bf e}({\bf K})]=i\Im[{\bf
e}^{*}({\bf K})\times{\bf e}({\bf K})]=(4ki/c)\bm{\mathcal{F}}_{E}({\bf K})$;
where $\bm{\mathcal{F}}_{E}({\bf K})$ is the density of electric spin of each
angular plane wave component in ${\bf K}$-space. Then, taking imaginary and
real parts in (B-2), and using the subindex $h$ and $e$ for homogeneous and
evanescent components, we arrive at

$\displaystyle\int_{-\infty}^{\infty}d^{2}{\bf R}\,{\mathscr{H}}({\bf
R},0)=(1/2k)\sqrt{\frac{\epsilon}{\mu}}\int_{-\infty}^{\infty}d^{2}{\bf
R}\,Im\\{{\bf E}({\bf R})\cdot{\bf B}^{*}({\bf R})\\}$
$\displaystyle=2\frac{(2\pi)^{2}}{kc}\\{\int_{K\leq k}d^{2}{\bf K}\,{\bf
k}_{h}\cdot\bm{\mathcal{F}}_{E\,h}({\bf K})+\int_{K>k}d^{2}{\bf K}\,{\bf
K}\cdot\bm{\mathcal{F}}_{E\,e\,\perp}({\bf
K})\\},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (B-3)

and

$\displaystyle\int_{-\infty}^{\infty}d^{2}{\bf R}\,{\mathscr{H_{R}}}({\bf
R},0)=(1/2k)\sqrt{\frac{\epsilon}{\mu}}\int_{-\infty}^{\infty}d^{2}{\bf
R}\,Re\\{{\bf E}({\bf R})\cdot{\bf B}^{*}({\bf
R})\\}=2\frac{(2\pi)^{2}}{kc}\int_{K>k}d^{2}{\bf
K}\,q_{e}\,\bm{\mathcal{F}}_{E\,e\,z}({\bf
K}).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (B-4)

In Eq. (B-3) $\bm{\mathcal{F}}_{E\,e}=(\bm{\mathcal{F}}_{E\,e\,\perp},{\cal
F}_{E\,e\,z})$, $\bm{\mathcal{F}}_{E\,e\,\perp}=({\cal F}_{E\,e\,x}\,,{\cal
F}_{E\,e\,y})$. Equations (B-3) and (B-4) are (17) and (18) of the main text.

##  Appendix C EVANESCENT WAVE: TIME-AVERAGED QUANTITIES, ELECTRIC AND
MAGNETIC IMAGINARY SPIN AND ORBITAL MOMENTA

The real part of ${\bf S}$ in (22) is

$\displaystyle Re\\{{\bf S}\\}=<{\bf
S}>=\frac{c}{8\pi\mu}[\frac{K}{k}(|T_{\perp}|^{2}+|T_{\parallel}|^{2})\,,2\frac{Kq}{k^{2}}Im\\{T_{\perp}T_{\parallel}^{*}\\}\,,0\,]\exp(-2qz)=\frac{ck}{K}[w\,,-\frac{q}{4\pi}{\mathscr{H}},0].\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(C-1)

${\bf S}$ is well-known to be associated to the optical force on a body.
Considering a magnetodielectric dipolar particle placed in the air on the
$z=0$ interface, the real part, or energy flow density $<{\bf S}>$, of the CPV
is known to constitute a momentum of the radiation whose $x$-component,
proportional to $w$, gives rise to an $x$-force on the particle, separately
acting on the particle electric (e) and magnetic (m) induced dipoles nieto1 .
The component $<{\bf S}>_{y}$ is known to produce a lateral force proportional
to $-\mathscr{H}$ along $OY$ bliokh1 , due to the interference of its e and m
dipoles nieto1 .

On the other hand, the time-averages of the density of spin angular momentum,
$<\bm{\mathcal{S}}>=(1/4\pi
c^{2})\bm{\mathcal{F}}=(1/2)(<\bm{\mathcal{S}}_{e}>+<\bm{\mathcal{S}}_{m}>)$,
where $<\bm{\mathcal{S}}_{e}>=(1/8\pi kc)\Im\\{{\bf E}^{*}\times{\bf E}\\}$
and $<\bm{\mathcal{S}}_{m}>=(1/8\pi kc)Im\\{{\bf B}^{*}\times{\bf B}\\}$, and
of spin curl, or Belinfante spin momentum, $<{\bf P}^{S}>=(1/2)[<{\bf
P}_{e}^{S}>+<{\bf P}_{m}^{S}>]=(1/2)\nabla\times<\bm{\mathcal{S}}>$, are

$\displaystyle<\bm{\mathcal{S}}>=\frac{1}{8\pi
c}[\frac{-2K}{k^{2}}Im\\{T_{\perp}T^{*}_{\parallel}\\},-\frac{Kq}{k^{3}}(|T_{\perp}|^{2}+|T_{\parallel}|^{2})\,,0]\exp(-2qz)=\frac{1}{4\pi
cK}\,[k\mathscr{H}\,,-\frac{4\pi
q}{k}w,\,0].\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (C-2)

With $<{\bf P}_{e}^{S}>=(1/2)\nabla\times<\bm{\mathcal{S}}_{e}>$, $<{\bf
P}_{m}^{S}>=(1/2)\nabla\times<\bm{\mathcal{S}}_{m}>$, and $<{\bf P}^{S}>$
being:

$\displaystyle<{\bf P}_{e}^{S}>=\frac{1}{8\pi
c}[-\frac{2Kq^{2}}{k^{3}}|T_{\parallel}|^{2}\,\frac{2Kq}{k^{2}}Im\\{T_{\perp}T^{*}_{\parallel}\\},\,0]\exp(-2qz).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(C-3) $\displaystyle<{\bf P}_{m}^{S}>=\frac{1}{8\pi
c}[-\frac{2Kq^{2}}{k^{3}}|T_{\perp}|^{2}\,,\frac{2Kq}{k^{2}}Im\\{T_{\perp}T^{*}_{\parallel}\\},\,0]\exp(-2qz).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(C-4) $\displaystyle<{\bf P}^{S}>=\frac{1}{8\pi
c}[-\frac{Kq^{2}}{k^{3}}(|T_{\perp}|^{2}+|T_{\parallel}|^{2})\frac{2Kq}{k^{2}}\Im\\{T_{\perp}T^{*}_{\parallel}\\},\,0]\exp(-2qz)=-\frac{q}{cK}\,[\frac{q}{k}w,\,\frac{k}{4\pi}\mathscr{H},\,0].\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(C-5)

Equation (C-2) remarks that, in agreement with the second term of the right
side of (17), $\mathscr{H}$ appears in the projection of the density of spin
momentum $<\bm{\mathcal{S}}>$ on the $x$-direction, which is that of
propagation of the evanescent wave comment . On the other hand, (C-5)
highlights the transversal $y$-component of the spin momentum proportional to
$\mathscr{H}$, as well as its longitudinal component, along $OX$, proportional
to $w$.

Since the time-average electromagnetic field momentum density $<{\bf g}>=<{\bf
S}>/c^{2}$ holds

$\displaystyle<{\bf g}>=<{\bf P}^{O}>+<{\bf P}^{S}>;$ (C-6)

$<{\bf P}^{O}>$ being the density of time-averaged orbital momentum, one sees
from (C-1) and (C-5) that the transverse $y$-component of $<{\bf g}>$ comes
from $<{P}_{y}^{S}>$, which is characterized by $\mathscr{H}$. Both
$<{g}_{y}>$ and $<{P}^{S}_{y}>$ are $(-1/4\pi
c)(kq/K){\mathscr{H}}\exp(-2qz)$. Therefore the trensverse $y$-component
$<{g}_{y}>$ of the field momentum is provided by the transverse $y$-component
$<{P}_{y}^{S}>$ of Belinfante’s momentum, in agreement with bliokh1 .

In addition, from (C-6), (C-5) and (C-1) we have for the $i$th Cartesian
component of the time-averaged orbital momentum density,

$\displaystyle<{P}_{i}^{O}>=(1/2)(<{P}_{e\,i}^{O}>+<{P}_{m\,i}^{O}>)=(1/2)(1/8\pi
kc)[Im\\{E_{j}^{*}\partial_{i}E_{j}\\}+Im\\{B_{j}^{*}\partial_{i}B_{j}\\}],\,\,\,(i,j=x,y,z),$
$\displaystyle<{\bf P}^{O}>=\frac{1}{8\pi
c}[\frac{K^{3}}{k^{3}}(|T_{\perp}|^{2}+|T_{\parallel}|^{2})\,,\,0\,,\,0]\exp(-2qz)=\frac{w}{c}\,[\frac{K}{k}\,,0\,,0].\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(C-7)

With the electric and magnetic orbital momenta:

$\displaystyle<{\bf P}_{e}^{O}>=\frac{1}{8\pi
c}[\frac{K(2K^{2}-k^{2})}{k^{3}}|T_{\parallel}|^{2}+\frac{K}{k}|T_{\perp}|^{2}\,,\,0\,,\,0]\times\exp(-2qz),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(C-8)

and

$\displaystyle<{\bf P}_{m}^{O}>=\frac{1}{8\pi
c}[\frac{K(2K^{2}-k^{2})}{k^{3}}|T_{\perp}|^{2}+\frac{K}{k}|T_{\parallel}|^{2}\,,\,0\,,\,0]\times\exp(-2qz),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(C-9)

respectively. $<{\bf P}^{O}>$ contains the energy density $w$ of the wave and,
as such, points in the propagation direction $OX$ of the wave, like both
$<{g}_{x}>$ and the $x$-component, $K>k$, of its propagation wavevector. This,
in agreement with bliokh1 , confers to the evanescent wave a superluminal
group velocity; although of course the time-average energy flow $<{S}_{x}>$
propagates with speed less than $c$. Hence $<{P}_{x}^{O}>$ pushes the
aforementioned small particle, placed in the air on the interface, as
radiation pressure along the $x$\- propagation direction.

On the other hand, the electric and magnetic imaginary momenta of (25) are:

$\displaystyle({\bf P}_{e}^{S\,I})_{i}=\frac{1}{8\pi
kc}Re\\{\partial_{j}({E}_{i}^{*}{E}_{j})\\}\,,\,\,\,$ $\displaystyle({\bf
P}_{m}^{S\,I})_{i}=\frac{1}{8\pi
kc}Re\\{\partial_{j}({B}_{i}^{*}{B}_{j})\\}\,,$ $\displaystyle({\bf
P}_{e}^{O\,I})_{i}=-\frac{1}{8\pi kc}\partial_{j}\frac{1}{2}\delta_{ij}|{\bf
E}|^{2}=-\frac{1}{8\pi kc}\frac{1}{2}\partial_{i}|{\bf
E}|^{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ $\displaystyle({\bf
P}_{m}^{O\,I})_{i}=-\frac{1}{8\pi kc}\frac{1}{2}\partial_{i}|{\bf
B}|^{2}\,.\,\,\,\,\,\,(i,j=x,y,z).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (C-10)
$\displaystyle{\bf P}_{e}^{S\,I}=\frac{1}{4\pi
k^{2}c}(0,-Kq\,Re\\{T_{\perp}T_{\parallel}^{*}\\},-\frac{K^{2}q}{k}|T_{\parallel}|^{2})\exp(-2qz).\,\,\,\,\,\,\,\,\,\,$
(C-11) $\displaystyle{\bf P}_{m}^{S\,I}=\frac{1}{4\pi
k^{2}c}(0,\,Kq\,Re\\{T_{\perp}T_{\parallel}^{*}\\},-\frac{K^{2}q}{k}|T_{\perp}|^{2})\exp(-2qz).\,\,\,\,\,\,\,\,\,\,$
(C-12) $\displaystyle{\bf P}_{e}^{O\,I}=\frac{q}{8\pi
kc}(0,0,\frac{K^{2}+q^{2}}{k^{2}}|T_{\parallel}|^{2}+|T_{\perp}|^{2})\exp(-2qz).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(C-13) $\displaystyle{\bf P}_{m}^{O\,I}=\frac{q}{8\pi
kc}(0,0,\frac{K^{2}+q^{2}}{k^{2}}|T_{\perp}|^{2}+|T_{\parallel}|^{2})\exp(-2qz).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(C-14)

##  Appendix D DIPOLE FIELDS

We express the fields emitted by a dipole with electric and magnetic moments
${\bf p}$ and ${\bf m}$ as jackson ,

$\displaystyle{\bf E}^{(s)}({\bf r})=\\{\frac{k^{2}}{\epsilon
r}[\hat{\bm{r}}\times({\bf
p}\times\hat{\bm{r}})]+\frac{1}{\epsilon}[3\hat{\bm{r}}(\hat{\bm{r}}\cdot{\bf
p})-{\bf
p}](\frac{1}{r^{3}}-\frac{ik}{r^{2}})-\sqrt{\frac{\mu}{\epsilon}}(\hat{\bm{r}}\times{\bf
m})(\frac{k^{2}}{r}+\frac{ik}{r^{2}})\\}e^{ikr}.\,\,\,\,({\bf
r}=r\hat{\bm{r}}).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(D-1) $\displaystyle{\bf B}^{(s)}({\bf r})=\\{\frac{\mu
k^{2}}{r}[\hat{\bm{r}}\times({\bf
m}\times\hat{\bm{r}})]+{\mu}[3\hat{\bm{r}}(\hat{\bm{r}}\cdot{\bf m})-{\bf
m}](\frac{1}{r^{3}}-\frac{ik}{r^{2}})+\sqrt{\frac{\mu}{\epsilon}}(\hat{\bm{r}}\times{\bf
p})(\frac{k^{2}}{r}+\frac{ik}{r^{2}})e^{ikr}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
(D-2)

##  Appendix E PROOF OF EQS. (34) AND (35)

The reactive power $W_{react}^{(s)}=\frac{ck}{3r^{3}}(|{\bf p}|^{2}-|{\bf
m}|^{2})$ may also be obtained by performing the volume integration of the
right side of (34) for the magnetoelectric dipole:

$\displaystyle<\tilde{W}_{e}^{(s)}>=\int_{V_{\infty}-V}d^{3}r<\tilde{w}_{e}^{(s)}>=\frac{1}{16\pi}\int_{r}^{\infty}drr^{2}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta[|{\bf
E}^{(s)}|^{2}-|{\bf E}_{FF}^{(s)}|^{2}]=\frac{k^{3}}{6}[|{\bf
p}|^{2}(\frac{1}{(kr)^{3}}+\frac{1}{kr})+\frac{|{\bf
m}|^{2}}{kr}],\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (E-1)

and

$\displaystyle<\tilde{W}_{m}^{(s)}>=\int_{V_{\infty}-V}d^{3}r<\tilde{w}_{m}^{(s)}>=\frac{1}{16\pi}\int_{r}^{\infty}drr^{2}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta[|{\bf
B}^{(s)}|^{2}-|{\bf B}_{FF}^{(s)}|^{2}]=\frac{k^{3}}{6}[|{\bf
m}|^{2}(\frac{1}{(kr)^{3}}+\frac{1}{kr})+\frac{|{\bf
p}|^{2}}{kr}].\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (E-2)

Having used Eqs. (D-1) and (D-2) of Appendix D, and ${\bf E}_{FF}^{(s)}$ and
${\bf B}_{FF}^{(s)}$ being the radiated far-fields with $r^{-1}$ dependence.
Thus the time-averaged reactive power is

$\displaystyle
2\omega(<\tilde{W}_{m}^{(s)}>-<\tilde{W}_{e}^{(s)}>)=W_{react}^{(s)}=\frac{ck}{3r^{3}}(|{\bf
m}|^{2}-|{\bf p}|^{2}).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ (E-3)

Which is (34).

The total time-averaged energy outside the volume $V$ is after Eqs. (E-1) and
(E-2),

$\displaystyle<\tilde{W}_{T}^{(s)}>=<\tilde{W}_{e}^{(s)}>+<\tilde{W}_{m}^{(s)}>=\frac{1}{6}(|{\bf
p}|^{2}+|{\bf m}|^{2})(\frac{1}{r^{3}}+\frac{2k^{2}}{r}).$ (E-4)

Which is (35) when $r\rightarrow a$ .

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  * (56) Notice the difference of  Eqs. (14)-(16) with the conservation equation and quantities of poulikakos1 , (cf. Eqs. (S.I.12)-(S.I.15) of the Supporting Information of poulikakos1 ). In homogeneous lossless media addressed here, (S.I.12)-(S.I.15) describe purely real quantities and $\chi_{e}-\chi_{m}=0$, so that the conservation equation (S.I.12) of poulikakos1 does not provide a law for reactive quantities, although both the optical chirality flux (S.I.15) and the real conservation equation (S.I.12) are $k^{2}$ times the flow of helicity (13) and the conservation of helicity (12), respectively. In order to get complex expresions (S.I.12)-(S.I.15), and $Re[\chi_{e}-\chi_{m}]\neq 0$, $Im[\chi_{e}-\chi_{m}]\neq 0$), lossy media should be considered, but then the imaginary parts of (S.I.12)-(S.I.15) are different to (14)-(16). Hence, the imaginary part (14) of the complex conservation equation (16) is a novel law for the reactive helicity ${\mathscr{H_{R}}}$ and its flow ${\cal F}_{\mathscr{H_{R}}}$ .
  * (57) Let ${\bf E}=(e^{+}{\bm{\epsilon}}^{+}+e^{-}{\bm{\epsilon}}^{-})\cos kz$, ${\bf B}=(e^{+}{\bm{\epsilon}}^{+}-e^{-}{\bm{\epsilon}}^{-})\sin kz$, be an elliptically polarized standing wavefield, expressed in the helicity basis $\hat{\bm{\epsilon}}^{\pm}=\frac{1}{\sqrt{2}}(\hat{\bm{x}}\pm i\hat{\bm{y}})$, with amplitudes CPL(+) (left circularly polarized) $e^{+}$ and CPL(-) (right circularly polarized) $e^{-}$. One has ${\bf E}\times{\bf B}^{*}=-(i/2)(|(e^{+}|^{2}+|e^{-}|^{2})\sin 2kz\,\hat{\bm{z}}$, $|{\bf B}|^{2}-|{\bf E}|^{2}=-(|(e^{+}|^{2}+|e^{-}|^{2})\cos 2kz$, and hence $\nabla\cdot\Im\\{{\bf S}\\}=2\omega(<w_{m}>-<w_{e}>)$. In addition, ${\bf E}\cdot{\bf B}^{*}=(1/2)(|(e^{+}|^{2}-|e^{-}|^{2})\sin 2kz$, ${\bf B}^{*}\times{\bf B}-{\bf E}^{*}\times{\bf E}=-i(|(e^{+}|^{2}-|e^{-}|^{2})\cos 2kz\,\hat{\bm{z}}$, so that $\nabla\cdot{\cal F}_{\mathscr{H_{R}}}=2\omega\mathscr{H_{R}}$. Therefore this kind of wave has no time-averaged energy transport $<{\bf S}>$, nor helicity density ${\mathscr{H}}$, but their densities of reactive power $2\omega(<w_{m}>-<w_{e}>)$, IPV, reactive helicity $\mathscr{H_{R}}$ and its flow ${\cal F}_{\mathscr{H_{R}}}$ , are non-zero. Note that if the standing wave is linearly polarized, (e.g. $E_{y}=B_{x}=0$), it has IPV and stores reactive power, but $\mathscr{H_{R}}=0$.
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  * (62) Notice that Eqs.(19) correspond to a wavefield ${\bf E}({\bf r})$ with angular spectrum ${\bf e}({\bf K^{\prime}})=\left(-\frac{k^{\prime}_{z}}{k}T_{\parallel},T_{\perp},\frac{K^{\prime}}{k}T_{\parallel}\right)\delta({\bf K}^{\prime}-{\bf K}),({\bf k}=({\bf K},k_{z}),{\bf K}=(K_{x},0),k_{z}=iq,K_{x}>k)$.
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  * (64) Note that $<\tilde{w}_{e}^{(s)}>$ and $<\tilde{w}_{m}^{(s)}>$ are not the electric and magnetic energy densities of only the near plus intermediate fields, say ${\bf E}^{(NF,IF)}$, ${\bf B}^{(NF,IF)}$, since they also contain an interference term of these fields with the far fields: $2Re\\{{\bf E}^{(NF,IF)}\cdot{\bf E}^{(FF)}\\}$ and $2Re\\{{\bf B}^{(NF,-F)}\cdot{\bf B}^{(FF)}\\}$, respectively.
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  * (68) Note that one could address a medium composed of particles, in which transport of light occurs. The existence of multiple scattering may be associated with a slow convergence of the Born series due to some coupling between the particles, stringent conditions for convergence under excitation of their resonances, (see e.g. N. A. Ustimenko, D. F. Kornovan, K. V. Baryshnikova, A. B. Evlyukhin, and M. I. Petrov, Multipole Born series approach to light scattering by Mie-resonant nanoparticle structures, arXiv:2108.11920v1 26 Aug 2021), or even with no convergence at all if there is either strong coupling between the induced dipoles (or multipoles). The latter two effects should be associated with higher Q-factors, and thus with larger amounts of stored and reactive powers. The same we would expect to rule the behavior of the scattered helicity and a quality factor which might be introduced for this quantity; this being an area of possible future exploration.
  * (69) For a circularly polarized incident plane wave of unit amplitude, the induced dipoles are ${\bf p}=\alpha_{e}{\bf E}^{(i)}$ and ${\bf m}=\alpha_{m}{\bf B}^{(i)}$ with ${\bf E}^{(i)}=\hat{\bm{\epsilon}}^{\pm}$, ${\bf B}^{(i)}=\mp i\hat{\bm{\epsilon}}^{\pm}$, $\hat{\bm{\epsilon}}^{\pm}=\frac{1}{\sqrt{2}}(\hat{\bm{x}}\pm i\hat{\bm{y}})$, the upper and lower sign applies according to whether it is left circular, CPL(+), or right circular, CPL(-), respectively. Since at K1: $\alpha_{m}=\alpha_{e}$, then ${\bf p}=\alpha_{e}\hat{\bm{\epsilon}}^{\pm}$ and ${\bf m}=\mp i\alpha_{e}\hat{\bm{\epsilon}}^{\pm}$, and thus ${\bf p}=\pm i{\bf m}$ nietoheli . Therefore at K1: ${\bf p}\cdot{\bf m}^{*}=\pm i|\alpha_{e}|^{2}$. However at K2: $\alpha_{m}=-\alpha^{*}_{e}$ and hence ${\bf p}\cdot{\bf m}^{*}=\mp i\alpha_{e}^{2}$.