Reactive helicity and reactive power in nanoscale optics: Evanescent waves. Kerkerconditions. Optical theorems and reactive dichroism

Manuel Nieto-Vesperinas1 and Xiaohao Xu2

1Instituto de Ciencia de Materiales de Madrid, Consejo Superior de Investigaciones Cientıficas.Campus de Cantoblanco, Madrid 28049, Spain. www.icmm.csic.es/mnv

2Institute of Nanophotonics, Jinan University, Guangzhou 511443, China ∗

Considering time-harmonic optical fields, we put forward the complex helicity and its alternatingflow, together with their conservation equation: the complex helicity theorem. Its imaginary partconstitutes a novel law that rules the build-up of what we establish as the reactive helicity throughits zero time-average flow. Its associated reactive flow, and the imaginary Poynting momentum thataccounts for the accretion of reactive power, are illustrated in two paradigmatic systems: evanescentwaves and fields scattered from magnetodielectric dipolar nanoparticles. As for the former, we showthat its reactive helicity may be experimentally observed as we introduce a reactive spin momentumand a reactive orbital momentum in terms of which we express the imaginary field momentum, whosetransversal 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 extraordinarymomentum. We also uncover a non-conservative force on such a magnetoelectric particle, acting inthe decay direction of the evanescent wave, and that may also be discriminated from the standardgradient force; thus making the reactive power of the wavefield also observable. Concerning thelight scattered by magnetoelectric nanoparticles, we establish two optical theorems that govern theaccretion 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 scatteredpower, we show that a zero total reactive helicity underlies a resonant scattered helicity. Thesereactive quantities are shown to yield a novel interpretation of the two Kerker conditions whichwe 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 withcircularly 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 scatteringand emission. In addition, we discover a discriminatory property of the reactive helicity of chirallight incident on a chiral nanoparticle by excitation of the external reactive power. This should beuseful for optical near-field enantiomeric separation, an effect that we call reactive dichroism.

I. INTRODUCTION

Reactive quantities of electromagnetic fields, such asreactive and stored energy or the imaginary Poyntingvector (IPV), are associated to physical entities thatdo not propagate in the environment, like evanescentwaves in their varied forms as surface waves [1–3], stand-ing waves [3], along with near-fields of RF-antennas [3–9, 11, 12]. In recent years, advances in photonics andnano-optics led to developing concepts such as chirality,helicity [13–19], magnetoelectric effects associated withthe imaginary Poynting vector (IPV) [20], and its con-sequent transverse spin momentum and magnetoelectricenergy density [21–23], as well as related to the azimuthalimaginary Poynting momentum [24] and the Kerker-typenon-conservative intensity gradient force [25].

In spite of progress in the analysis of optical anten-nas, mainly addressing quantum emitters and plasmonicnanoparticles [26–28], whose radiative and feeding char-acteristics is studied from the point of view of RF-antennas and ciecuit theory [29–31], we have found fewdetailed studies [30] on their reactive quantities; although

∗ mnieto@icmm.csic.es , xuxhao dakuren@163.com

the effects of reactive power in antenna functionality arewell-known [3, 5–7, 9–12, 30]; e.g. the radiative and reac-tive energies of an oscillating dipole are intertwined. Inthis way, a major task in RF-antenna design has beenthe study of its reactive power and Q-factor, seeking aminimization of both quantities in order to match the in-put energy with its radiative performance, since a largeQ and reactive energy in the antenna convey high ohmiclosses, and a decrease of its operative bandwidth.

By contrast, at optical wavelengths a high reactivepower outside the nanoemitter or scatterer, externalstored energy, and Q-factor of a low-loss nanoparticle, ornanoantenna, enhances both its scattered (or radiated)power and frequency sensitivity, also narrowing its oper-ational 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[32] and Mie-tronics [33, 34]. The reason is that, as weshall see, although such a large external reactive powerand stored energy occur at wavelengths near those of res-onant scattered power, the interior of the particle acts asa compensating (capacitive or inductive) element so thatits reactive power and stored energy cancel out the ex-ternal ones close to these resonant wavelengths at whicheach of these quantities have near extreme values. Theresult is that, in analogy with RF-antenna design [30],

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the total (i.e. internal plus external) reactive power andstored energy are zero close to resonances.

We also show that the same effect occurs for the inte-rior, external, and total reactive helicities in connectionwith a maximum efficiency in the helicity scattered up tothe far zone.

Since surface plasmons work at optical frequencieswhere metals exhibit high losses, there has been an in-creasing interest in high refractive index nanoantennas,on which light exerts a magnetoelectric response withlarge electric and magnetic resonances [33–40]. We willstudy the key role of reactive quantities in the scatter-ing from these magnetodielectric particles. The otherarchetypical configuration in which we shall address thesereactive quantities is an evanescent wave.

Using time-harmonic wavefields, we will start address-ing the flow of reactive power of the complex Poyntingtheorem, and since an analogous law for the helicity flowhas never been established, as far as we know, we will in-troduce the concepts of complex helicity density [41] andcomplex helicity flow, for which we put forward a conser-vation equation that we coin as complex helicity theorem.Its real part is the well-known continuity equation for theconservation of optical helicity [13–16, 18], while its imag-inary part is a novel law that describes the build-up ofreactive helicity, whose flow has zero time-average andhence it does not propagate in free-space.

We show that this reactive helicity density and its flowexist, like the reactive energy and the imaginary Poyntingvector, in wavefields that do not propagate into the en-vironment, as e.g. evanescent and elliptically (and circu-larly in particular) polarized standing waves; the formerbeing identical to the so-called magnetoelectric energydensity, introduced in [23] following symmetry argumentsbut without providing its undelying physical law.

As for an evanescent wave, whose imaginary Poynt-ing vector and its associated transversal spin have beenstudied [21], while its time-averaged energy flow, spinangular momentum, Belinfante momentum, and orbitalmomentum have to do with the time-averaged energyand helicity densities [21], we show that the reactive (i.e.imaginary) Poynting vector, and reactive helicity flow,are linked with the reactive energy and reactive helicityof the wave. These relationships provide a physical lawfor our introduced concept of reactive helicity, in whichthe aforementioned magnetoelectric energy [23], and theso-called ”real helicity” discussed in a different research[42], are unified. Further, we discuss how the reactivepower and reactive helicity of this wavefield, are gener-ated close to the interface through their correspondingalternate flows along the wave decay direction.

Moreover, we show that the imaginary Poynting mo-mentum of the evanescent wavefield is the sum of densi-ties of a reactive spin momentum and a reactive orbitalmomentum, which in turn are expressed as differences ofthe imaginary magnetic and electric corresponding mo-menta. This leads us to uncover a non-conservative op-tical force on a magnetodielectric particle, directed along

the wave decay direction -i.e. different from the gradientforce and larger than it at certain wavelengths, thus beingexperimentally detectable- due to the IPV.

We also illustrate the presence of a transversal forceon such a magnetodielectric particle, stemming from thecorresponding component of the IPV, which may be of op-posite sign and much larger than the known lateral force[21] due to the time-averaged Poynting vector; thus beingdetectable and making observable the reactive helicity.

As regards fields emitted or scattered by a magnetodi-electric dipolar nanoparticle, we shall show that its re-active energy and angular distribution of scattered ra-diation, being intimately interrelated, provide a novelinterpretation of the two Kerker conditions: K1 andK2 [36, 43–48], and hence of the particle emission di-rectivity. Namely, under plane wave illumination, andat wavelengths where such a particle fulfils either K1 ofzero backscattering, or K2 of minimum forward scatter-ing, the overall external reactive power around this bodyis either zero or almost zero, while the internal and to-tal (i.e. external plus internal) reactive powers are nearzero. Moreover, the overall external reactive helicity van-ishes at K1 wavelengths on illumination with circularlypolarized light.

Concerning feeding the magnetodielectric nanoan-tenna, we put forward the reactive power and the re-active helicity optical theorems which quantify the accre-tion of external stored reactive energy and reactive he-licity in the near and intermediate-field zones, in termsof the particle excitation and extinction of energy andhelicity of the supplied illumination. The effects hereshown maximizing both the external and internal valuesof these reactive quantities, (while minimizing their over-all amounts, i,.e. external plus internal), which tune toresonance their radiation efficiency, establishes an anal-ogy with the same well-known pursuit in RF antenna de-sign. Therefore, this work puts forward the importance ofreactive quantities which underly some previously stud-ied concepts in the analysis of optical antennas [26–29].

Finally, we show that the reactive power theorem yieldsan interpretation of how by illuminating a chiral parti-cle with a non-free propagating wavefield, like an ellipti-cally polarized standing wave, evanescent wave, or nearfield from a nearby emitter, the reactive helicity of theincident wavefield appears as a consequence of the gen-eration of reactive energy on interaction with the sam-ple particle. It is intriguing that, as we find, this inci-dent reactive helicity emerges analogously as the incidenthelicity does in a standard dichroism far-field observa-tion [13]. Therefore, we show that the so-called ”mag-netoelectric response” of a chiral particle, quoted in [23],arises as a consequence of the accretion of its reactiveenergy from near-field chiral illumination, e.g. with in-cident evanescent waves, (or, similarly, with incident cir-cularly polarized standing waves as proposed in [23]). Assuch, we name reactive dichroism this magnetoelectricphenomenon. It underlines the observability of such in-

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cident reactive helicity, and its discriminatory propertyfor enantiomer separation by near-field optical techniquesusing structured illumination.

II. THE REACTIVE POYNTING VECTOR ANDTHE REACTIVE POWER

To fix some concepts to deal with, we first outline themain quantities involved in the complex Poynting the-orem. We shall assume an arbitrary body immersedin a lossless homogeneous medium [49]. Let E(r, t) =E(r) exp(−iωt), B(r, t) = B(r) exp(−iωt) be a time-harmonic electromagnetic field. It is well-known that ina body with charges and free electric currents of densityj contained in a volume V with permitivity ε and perme-ability µ, the complex work density, given by the scalarproduct j∗ ·E, leads after using Maxwell’s equations, ∇×E = −(1/c) ∂B/∂t, ∇×H = (4π/c)J + (1/c) ∂D/∂t,[D = εE,B = µH,J (r, t) = j(r) exp(−iωt)], to the com-plex Poynting theorem [3, 4, 8, 11]:∫V

[1

2j∗ ·E +∇ · S]d3r = i2ω

∫V

(< wm > − < we >)d3r ;

(ω =kc√εµ

=2π

λ

c√εµ

). (1)

Where * and < . > mean complex conjugated andtime-average, respectively. The complex Poynting vec-tor (CPV) S, and time-averaged electric and magneticenergy densities, < we > and < wm >, are

S(r) =c

8πµE(r)×B∗(r),

< we(r) >=ε

16π|E(r)|2, < wm(r) >=

1

16πµ|B(r)|2 . (2)

Under our above assumptions, the right side of (1) ispurely imaginary, and the real part of this equation con-stitutes the well-known Poynting theorem describing thevariation of energy in the body due to the work rate ofthe field upon its charges, given by the volume integral of12Re{j

∗ · E}. This variation is characterized by the flowof the time-averaged Poynting vector, < S >= Re{S},across the surface ∂V of V :

∫∂V

Re{S} · rd2r. Where ris the outward unit normal to ∂V .

On the other hand, the imaginary part of (1) formu-lates that in the steady state the integral of the imag-inary work 1

2Im{j∗ · E}, plus the reactive power flux∫

∂VIm{S}·nd2r, (which has zero time-average since it is

associated with instantaneous energy flow that alternatesback and forth at twice frequency across ∂V ), accountsfor this reactive power build-up in and around the body,given by the right side of (1); 2(< wm > − < we >) be-ing the reactive energy density. Unless otherwise stated,we shall henceforth assume ε = µ = 1 for the embeddingspace.

At this point it is convenient to introduce the in-stantaneous Poynting vector [3, 11] built by the fields

E(r, t) = <{E(r, t)} and B(r, t) = <{B(r, t)},

S(r, t) =c

4πE(r, t)× B(r, t) =

< S > +c

8πRe{E(r)×B(r) exp(−2iωt)},

This is the standard expression of S(r, t). However wefind it more instructive to write it as:

S(r, t) =< S > (1 + cos 2ωt) + Im{S} sin 2ωt

+c

4πER(r)×BI(r) sin 2ωt− c

4πEI(r)×BI(r) cos 2ωt. .(3)

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

A. The reactive Poynting vector and the angularspectrum of plane waves

To distinguish the structure of the real and imaginaryparts of S, it helps to employ the angular spectrum repre-sentation of the electromagnetic field, so as to map thesequantities into their spectra in Fourier space. To this end,we calculate the flow of c

8πE(r) × B∗(r) across a planez = constant. We use for simplicity a framework suchthat the souces are on z < 0 and thus the integration isdone on the z = 0 plane. But one may equally chooseany other constant value z = z0, providing the sourceslie in z < z0. The electric field E(r) propagating into thehalf-space z ≥ 0 is represented by its angular spectrumof plane wave components as [50, 51]:

E(r) =

∫ ∞−∞

d2K e(K) exp[i(K ·R + kzz)]. (4)

r = (R, z), R = (x, y),k = (K, kz), K = (Kx,Ky), |k|2 = k2. And

kz =√k2 −K2 = qh , K ≤ k, for homogeneous

(propagating) plane wave components.

kz = i√K2 − k2 = iqe , K > k, for evanescent

plane wave components. (5)

Using the subscripts h and e for homogeneous andevanescent components, respectively, the CPV flux,ΦPoynt, across the plane z = 0 has real and imaginaryparts given by, (see the proof in Appendix A):

4

ΦRPoynt =πc

2k

∫K≤k

d2K qh|eh(K)|2, (6)

and

ΦIPoynt = −πc2k

∫K>k

d2K [qe|ee(K)|2 − 2qe|ee z(K)|2].(7)

Eq.(6) is well-known, it expresses the flux ΦRPoynt ofthe real part of the CPV as the momentum power car-ried on by the propagating components, (K ≤ k). How-ever, Eq. (7) shows that the flux ΦIPoynt of the reactiveCPV is momentum associated with power contained inthe evanescent components, (K > k), and as such it doesnot propagate into z ≥ 0. Notice the special role playedby the longitudinal component ee z(K) in (7).

III. THE REACTIVE HELICITY AND ITSREACTIVE FLOW THEOREM

The optical helicity density of the electromagnetic fieldin a medium with constitutive parameters ε and µ is well-known to be [14, 16, 18]

H (r) = (1/2k)

√ε

µIm{E(r) ·B∗(r)}. (8)

We now introduce the quantity

HR(r) = (1/2k)

√ε

µRe{E(r) ·B∗(r)}. (9)

Like Im{S}, this quantity HR may appear from H whenE and B are chosen π/2 out of phase.

We shall later see that HR is exclusive of Im{S},(and of course of a reactive helicity flow, to be intro-duced next). Thus we call HR the reactive helicity den-sity of the field. Some authors have recently addressedthis quantity in different works, and call it magnetoelec-tric energy [23], or just real helicity [42]. However wekeep our denomination by showing that, like the reactiveenergy, it fulfills a conservation law.

We consider a body with charges and free currents,embedded in a volume V . From the two Maxwellequations for the spatial vectors, ∇ × E = ikB and∇ × H = (4π/c) j − ikD, we derive the conservationequation for the reactive helicity. First, we employ thesecond of these equations and address the scalar product(4π/c)j∗ ·B in V . On using the identity: B · (∇×H∗) =H∗ · (∇×B)−∇ · (B×H∗), one gets∫V

[−2π

knIm{j∗ ·B}+ ∇ ·FB ] d3r = ω

∫V

d3rHR. .(10)

In (10) FB = (c/4kn)={H∗ ×B} is the density of mag-netic flow of helicity [14, 16, 23], (n =

√εµ); and it is

also proportional to the magnetic part of the spin angu-lar momentum density [14, 16, 23], (see also Eq. (C-2)below).

Similarly, we may obtain a conservation equationfor HR with the electric helicity flow density, FE =(c/4kn)Im{E∗ ×D}. This is done by taking the scalarproduct: D · (∇×E∗) = −ikD ·B∗ , and proceeding inan identical way as with the derivation of (10), the resultis ∫

V

∇ ·FE d3r = −ω

∫V

HR d3r . (11)

Adding (11) and (10) one obtains the well-known con-tinuity equation for the conservation of helicity in thesteady state:∫V

∇ ·F d3r ≡∫∂V

F · r d2r =2π

kn

∫V

Im{j∗ ·B} d3r, (12)

which shows that the flow of helicity, or dual-symmetricspin [16, 52]:

F = FE + FB = (c/4kn)Im{E∗ ×D + H∗ ×B},(13)

across the boundary ∂V of V equals the radiated fieldhelicity, including its dissipation and conversion given bythe right side of (12) [53–55].

However, substracting (11) from (10) leads to the re-active helicity flow theorem,∫V

[−2π

knIm{j∗ ·B}+∇ ·FHR ] d3r = 2ω

∫V

HR d3r ; (14)

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

FHR = FB −FE = (c/4kn)={H∗ ×B−E∗ ×D} =

2πc2

n(< Sm > − < Se >).(15)

If ∂V is in air or vacuum, n = 1, B = H and D = Ein (13) and (15). < Se > and < Sm > stand for thetime-averages of the density of electric and magnetic spinangular momentum, (cf. Appendix C).

The quantity 2ωHR is a reactive helicity per unit half-period, or just the reactive helicity power, in analogy withthe reactive power of the complex Poynting theorem Eq.(1). Hence (14) expresses the conservation of 2ωHR.Indeed (12) and (14) suggest us to formulate a complexhelicity theorem [56]:∫V

[−(1 + i)2π

knIm{j∗ ·B}+ ∇ ·FC ] d3r = 2iω

∫V

HR d3r,

FC = F + iFHR . (16)

where the complex helicity flow , or complex spin an-gular momentum, is FC . Evidently (16) has a real partwhich is the standard helicity conservation equation (12),whereas its imaginary part is Eq. (14) for the reactive he-licity flow and governs the variation of reactive helicity inV , given by the decrease of the integrated source density− 2πknIm{j

∗ · B}. This variation is expressed in terms of

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the reactive helicity flux∫∂V

FHR · r d2r, (which has zerotime-average since it comes from ={FC}, and hence rep-resents helicity flowing back and forth to the body across∂V in its near-field region, without net propagation), andof integrated reactive helicity density 2ωHR.

Unless otherwise stated, we shall not drag the factor2ω when we refer to the reactive helicity, thus we shalljust write HR for this quantity. In Section VII we showthat HR is built-up on chiral light-matter interaction.This gives rise to the phenomenon of reactive dichroismin the near-field of the body, addressed in Section VIII.

Equations (10) and (11) suggest that HR is observable,for example by detecting the torque exerted by a circu-larly polarized plane wave on a dipolar particle on whichthe field induces a purely electric (e) or a purely magnetic(m) dipole. In fact, in vacuum FE and FB are propor-tional to the optical electric and magnetic torque, respec-

tively, ΓE = (1/8πk)σ(a)e Im{E∗ × E} = cσ

(a)e < Se >

and ΓB = (1/8πk)σ(a)m Im{B∗ × B} = cσ

(a)m < Sm >;

σ(a)m and σ

(a)m being the particle electric and magnetic ab-

sorption cross sections, (see [58], Section X). Otherwise,if the particle is magnetodielectric, HR becomes observ-able through the e-m interaction force, Eq. (31) below,[cf. [20] Eq. (44)]. This latter situation is detailed withan evanescent wave in Section IV.B, cf. Eq. (23).

Concerning the P , T , D symmetries of these novelquantities, namely,

parity, P : r→ −r, P : E→ −E, P : B→ B,

time-reversal, T : t→ −t, T : E→ E∗, T : B→ −B∗,

and duality, D : E→ B, D : B→ −E,

while it is well-known that

P :< S >→ − < S >, P : F → F , P : H → −H ;

T :< S >→ − < S >, T : F → −F , T : H →H ;

D :< S >→< S >, D : F → F , D : H →H ;

and we obtain the symmetries for the reactive quantities:

P : Im{S} → −Im{S}, P : FHR → FHR ,P : HR → −HR;

T : Im{S} → Im{S}, T : FHR → −FHR ,T : HR → −HR;

D : Im{S} → −Im{S}, D : FHR → −FHR ,D : HR → −HR.

It is interesting that while under parity the three re-active quantities behave like their corresponding non-reactive counterparts, they invert their symmetry underduality applications, in contrast with their non-reactiveanalogues that remain D-invariant. Under time-reversal,only FHR has the same symmetry as its non-reactivecorrespondant F .

A. The reactive helicity and the angular spectrum

In order to gain more insight into the different natureof H and HR, we shall employ once again the angu-lar spectrum of the electromagnetic wave. We evaluatethe total H and HR per unit z-length by integration ofE(r) ·B∗(r) on a plane z = constant. Again, we choosecoordinates such that the souces are on z < 0 and thusthe integration is done on the z = 0 plane.

Using, as before, the subindex h and e for homogeneousand evanescent components, respectively, the integral onz = 0 of E(r) ·B∗(r) is shown in appendix B to yield∫ ∞

−∞d2R H (R, 0)

= (1/2k)

√ε

µ

∫ ∞−∞

d2R Im{E(R) ·B∗(R)}

= 2(2π)2

kc{∫K≤k

d2K kh ·FE h(K)

+

∫K>k

d2K K ·FE e⊥(K)}, (17)

and ∫ ∞−∞

d2R HR(R, 0)

= (1/2k)

√ε

µ

∫ ∞−∞

d2RRe{E(R) ·B∗(R)}

= 2(2π)2

kc

∫K>k

d2K qe FE e z(K). (18)

In (17) FE e = (FE e⊥,FE e z), FE e⊥ = (FE ex ,FE e y).Therefore Eq. (17) shows that in the domain of prop-agating components the helicity density of the field inz = 0 maps into the projection of the electric spin angu-lar momentum of the K-plane wave component onto thepropagation wavevectors; namely, onto the real wavevec-tor kh, while in the evanescent region, it is given by theprojection of the electric spin onto the transversal (prop-agating) component K. (Note that this is in agreementwith the standard definition [15], but here generalized toinclude evanescent waves).

Of special interest is, however, Eq. (18), which showsthat the reactive helicity density HR maps in K-space asthe projection of the electric spin of the Kth-evanescentcomponent onto the z-component ik∗z = i(−iqe)z. Wherek is the complex wavevector of this Kth-evanescent wave,[cf. Eq. (5)]. Again, this justifies that we call reactive tothe real part of E(r) ·B∗(r).

In this connection, it should be remarked that likethe reactive power and the IPV are linked with non-propagating waves, e.g. evanescent and standing waves[3], the reactive helicity HR (and hence its flow FHR)exist in evanescent waves as shown in Eq.(18), as well asin elliptically polarized (and circularly polarized in par-ticular, CPL) standing wavefields [57]. For CPL waves

6

the authors of [23] used the term ”magnetoelectric en-ergy” which, as seen above, is a reactive quantity sinceit is the same as HR.

IV. CASE 1: REACTIVE POWER, REACTIVEHELICITY, AND REACTIVE MOMENTA IN AN

EVANESCENT WAVE

As illustrated in Fig. 1, we consider a generic time-harmonic evanescent wave in air, z ≥ 0, generated bytotal internal reflection (TIR), at a plane interface z = 0separating air (ε = µ = 1) from a dielectric in the half-space z ≤ 0. The plane of incidence being OXZ. Thenthe complex spatial parts of the electric and magneticvectors in z > 0, are expressed in a Cartesian coordinatebasis {x, y, z} as (n = 1) [59, 60]:

E =

(− iqkT‖, T⊥,

K

kT‖

)exp(iKx− qz),

B = n

(− iqkT⊥,−T‖,

K

kT⊥

)× exp(iKx− qz). (19)

For TE or s (TM or p) - polarization , i.e. E(i) (B(i))perpendicular to the plane of incidence OXZ, only thosecomponents with the transmission coefficient T⊥, (T‖)would be chosen in the incident fields [59]. K denotes thecomponent, parallel to the interface, of the wavevector k:k(sxy, sz) = (K, 0, iq), q =

√K2 − k2, k2 = K2 − q2.

A. Reactive power and reactive helicity densities

The densities of energy, w = we + wm, and reactivepower, wreact = 2ω(wm − we), (ε = µ = 1), of this waveare according to (2)

w =1

8π

K2

k2(|T⊥|2 + |T‖|2) exp(−2qz),

wreact =c

4π

q2

k(|T⊥|2 − |T‖|2) exp(−2qz). (20)

And the densities of helicity, H , and reactive helicity,HR, (ε = µ = 1), of this evanescent wave are

H = −(K2/k3)Im{T⊥T ∗‖ } exp(−2qz);

HR = (q2/k3)Re{T⊥T ∗‖ } exp(−2qz). (21)

B. The reactive energy flow: Reactive momentum,imaginary spin and imaginary orbital momenta

The CPV is written as

S =c

8πµ[K

k(|T⊥|2 + |T‖|2),

i2Kq

k2T ∗⊥T‖ ,−i

q

k(|T⊥|2 − |T‖|2)] exp(−2qz)

= [ck

Kw ,

c

4π(−kq

KH + i

kK

qHR) ,−i 1

2µqwreact]. (22)

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

The reactive, or imaginary, part of the CPV is

Im{S} =c

8πµ[ 0 , 2

Kq

k2Re{T⊥T ∗‖ } ,

q

k(|T‖|2 − |T⊥|2)]

× exp(−2qz) =1

2q[0 ,

ckK

2πHR ,−wreact]. (23)

Which yields the reactive or imaginary momentum of thefield, (sometimes called imaginary Poynting momentum[21, 24]), gI = Im{g} = Im{S}/c2:

gI =1

8πc[ 0 , 2

Kq

k2Re{T⊥T ∗‖ } ,

q

k(|T‖|2 − |T⊥|2)]

× exp(−2qz) =1

2qc2[0 ,

ckK

2πHR ,−wreact], (24)

whose components come from two vectors that we putforward next: the density of both reactive spin momentumPS and reactive orbital momentum PO:

PS =1

2(PS I

m −PS Ie ) =

K

2cq(0,

k

2πHR,−

K

ck2wreact),

PO =1

2(PO I

m −PO Ie ) =

q

2cq(0, 0,

q

ck2wreact). (25)

Namely,

gI = PS + PO. (26)

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

We emphasize that, as seen in (24)-(26), gIy is fully due

to PSy , while gIz comes from PSz + POz .Note that, interestingly and in contrast with Eq.(C-

6), denoting PS I = (1/2)(PS Ie + PS I

m ) = (0, 0,− qkcw),

PO I = (1/2)(PO Ie + PO I

m ) = (0, 0, qkcw), one obtains:

PS I + PO I = 0.

C. The reactive helicity flow

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

FHR =c

4k[0,

2Kq

k2(|T‖|2 − |T⊥|2),−4q

kRe{T⊥T ∗‖ }] exp(−2qz)

= −1

q[0 ,

2πK

k2wreact , ckHR]. (27)

It should be emphasized that although both Im{S} andFHR have an y-component proportional to K > k, (aswell as proportional to HR in the former and to wreact inthe latter), there is no case of superluminal propagationfor these quantities since both are alternating flows withzero time-average. We show below that Fz represents upand down flow of reactive helicity in the OZ direction,

7

FIG. 1. The evanescent wave (yellow) of Eq. (19), with reactive power density given by (20), propagates in the air, parallelto the interface z = 0 separating it from a dielectric in z < 0, with K-vector along OX. Its amplitude decays as exp(−qz).The < S >-vector and spin momentum, < PS >, contained in OXY , have y-components proportional to the wave helicityH , (see Appendix C). The orbital momentum, < P0 >, proportional to the energy density w points along OX. The reactivemomentum Im{S}, contained in OY Z, 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 Fem, proportional to Im{S}, on a dipolar magnetodielectricparticle placed on the interface. This force is due to the interference of the particle electric and magnetic induced dipoles.The lateral y-force, (Fem)y, and transversal z-component, (Fem)z , due to Im{S}y and Im{S}z, are proportional to the wavereactive helicity, HR, and reactive power density, ∇ · Im{S}, respectively. The imaginary orbital and spin momenta, PO I

and PS I , point along the +z and −z-direction, respectively. The flow FHR of reactive helicity is, like Im{S}, in the OY Z-plane with y-component proportional to both the reactive power density, wreact, and the magnitude of the K-vector; while itsz-component is proportional to the reactive helicity density HR, so that across the OXY -plane (FHR)z holds the imaginarypart of the complex helicity theorem, as shown by Eq. (30).

matching with the imaginary part of the complex helicitytheorem, Eq. (16).

From the above equations it is important to remarkthat while the time-averaged energy and helicity densitiesare linked to < S >, < g >, < S >, < PS > and< PO >, the densities of reactive power, wreact, andreactive helicity, HR, are exclusive of Im{S}, gI andFHR .

We show below that the z-component of Im{S}, thatmatches with the imaginary part of the complex Poynt-ing theorem, is associated with an up and down flow ofreactive power in the decay z-direction of the evanescentwave, and hence it is not a net flow of energy. Analo-gously happens with the z-component of FHR . However,as seen below, both the y and z-components of Im{S}produce detectable optical forces and, hence, make thereactive quantities wreact and HR observable.

D. Ractive power conservation law

The z-component of Im{S}, [cf. Eq. (23)], depends onthe reactive power density, ∇ · Im{S}, of the evanescentwave in the half-space z > 0, (which actually concen-trates in the near field region above the interface z = 0,namely at z << λ), flowing back and forth along OZat twice the frequency ω, without contributing to a netenergy flow since its time-average is zero. I.e. one hasfrom the CPV theorem:

∇ · Im{S} ≡ ∂zIm{S} = wreact

≡ −2qIm{S}z . (28)

Which obviously agrees with (23). Therefore taking (28)into account, the total reactance, [11] of the dielectric-airinterface system associated to the evanescent wave is∫V

d3r∇ · Im{S} = Σ

∫ ∞0

dz∇ · Im{S} = −Im{Sz(z = 0)}.(29)

8

Σ being the area of the XY -plane resulting from the vol-ume integration.

E. Reactive helicity conservation law

Analogously, the z-component of FHR , Eq. (27), de-pends on the reactive helicity density, ∇ · FHR , of theevanescent wave in the half-space z > 0, concentratedin the near field on z = 0, flowing up and down in thethe z-direction without yielding a net flow since its time-average is zero. I.e. one has in agreement with Eq.(14),(j = 0),

∇ ·FHR ≡ ∂zF =2cq2

k2Re{T⊥T ∗‖ }

= −2q(FHR)z ≡ 2ωHR . (30)

Hence, the z-component of FHR is proportional to thereactive power density.

It is evident that while reactive power and reactivehelicity exist in evanescent waves, (see also Eqs. (7) ofSection II.A and (18) of Section III.A), and in stand-ing waves [3, 23], they do not exist in plane propagatingwaves, whatever their polarization be. Therefore, reac-tive helicity, like reactive energy, exists in the near-fieldregion of scattering or emitting objects.

F. Observability of the transversal andperpendicular components of the imaginary

momentum gI : Optical forces on amagnetodielectric dipolar particle due to the

reactive helicity and reactive power

Let a magnetodielectric dipolar particle be placed onthe z = 0 interface. An illuminating wavefield, and inparticular the evanescent wave, exerts an optical forceon it due to the interaction between its induced electric(e) and magnetic (m) dipoles [20], viz.,

< Fe−m >=8πk4c

3[−Re(αeα∗m) < g > +Im(αeα

∗m)gI ].(31)

The e and m polarizabilities αe and αm are related withthe a1 and b1 Mie coefficients of the field scattered by theparticle by [20]: αe = i3a1/2k

3, αm = i3b1/2k3.

The first term of (31), proportional to < g >, hasbeen studied, (see its main features in Appendix C). Herewe are interested in the second term that contains thereactive momentum gI .

The y-component < FgI

e−m >y due to gIy of (31) wasobtained in [21], being considered by the authors ”aquite intriguing result” characterized through the sec-ond Stokes parameter with a rather small contributionin measurements of < Fe−m >y. We have shown abovethat this force has a reactive origin since it is fully dueto the reactive spin y-component, being characterized bythe reactive helicity HR. Furthermore, we establish here

that < FgI

e−m >y is detectable since it may widely exceed

the known component < F<g>e−m >y due to < g >y.

For instance, Fig.2(Left) shows forces on a Si spheri-

cal particle placed on the interface z = 0: < FgI

e−m >ycompared with < F<g>

e−m >y. The particle electric andmagnetic dipole resonances are at λe = 492 nm andλm = 668 nm, respectively, [see Fig.4(left)], and thesecond Kerker condition (K2) wavelength (at which theparticle scatters minimum forward intensity, see SectionV.B) is λK2 = 608 nm. As seen, the magnitude of

< FgI

e−m >y is much greter than < F<g>e−m >y near λe

and λm where the latter changes sign, (see the sharpasymptotic values of the ratio between both forces). Thus

< FgI

e−m >y, which keeps negative, should be detectablenear these resonances. Besides, in the proximities of K2,

i.e. of λK2, < FgI

e−m >y= − < F<g>e−m >y.

While < F<g>e−m >z= 0, one observes in Fig.2(Right)

features of the perpendicular force < FgI

e−m >z similar tothose of its y-component comparing it with the gradientforce [60], which choosing T‖ = 0 reads: < Fgrad >z=

−(1/2)q[αRe |T⊥|2 + αRm|T⊥|2(2K2/k2 − 1)](0, 0, 1). Thesuperscript R standing for real part. Again the sharpratio between both near λ = 665 nm indicates the wave-lenght zone where < FgI

e−m >z, which remains positive,may be detected. Furthermore, the bump of this ratio inthe proximities of λ = 520 nm shows that this force isover twice the gradient force.

We conclude thereby that there exists a measurabletransverse y-component of < Fe−m > due to the reactivespin momentum density and hence to HR, which may bedominant upon the transversal component of < Fe−m >stemming from the field (Poynting) momentum, namelyfrom H . Hence, HR is observable. Besides, the nor-mal force < Fe−m >zwhich is exclusively due gIz , (since< gz >= 0), characterized by the reactive power densitywreact of the evanescent wave has not yet been addressedas far as we know, and may be detected at wavelengths atwhich, as seen above, clearly exceeds the gradient force,making wreact also an observable quantity

V. CASE 2: REACTIVE POWER ANDREACTIVE HELICITY FROM A

MAGNETODIELECTRIC DIPOLAR SPHERE

We consider a magnetodielectric spherical particle ofradius a and volume V0, dipolar in the wide sense, namelywhose electric and magnetic polarizabilities are given bythe first electric and magnetic Mie coefficients, respec-tively [20, 35], in air. We first address the reactive powerand stored energy of this magnetoelectric dipole withelectric and magnetic moments p and m, respectively.For a wave, E(i), B(i), incident on the particle centeredat r = 0, the dipolar moments are: p = αeE

(i)(0) andm = αmB(i)(0).

9

c

λ (nm)

λ (nm)λ (nm)

CccCcccCcccCccccc

400 440 480 520 560 600 640 680 720 760 8007 103

5.6 103

4.2 103

2.8 103

1.4 103

0

1.4 103

2.8 103

4.2 103

5.6 103

7 103

FgI_y z( )

FgR_y z( )

quo1 z( )

z

λ (nm)

Fy<g>

FygI

Ratio:Fy

gI

Fy<g>

400 440 480 520 560 600 640 680 720 760 8002 104

1.6 104

1.2 104

8 103

4 103

0

4 103

8 103

1.2 104

1.6 104

2 104

FgI_z z( )

Fgrad_z z( )

quo2 z( )

z

CccCcCcccc

0

4

8Ratio:Fz

gI

Fzgrad

Fzgrad

FzgI

0

-4

-8

400 480 560 640 720 800 400 480 560 640 720 800

λ (nm)

5.6x103

2.8x103

0

-2.8x103

-5.6x103

1.6x103

8x103

0

-8x103

-1.6x103

CccCcCccccCcc

CcCcccc -2.4

-1.2

1.2

2.4

λ (nm)

FIG. 2. An evanescent wave, (see in Fig.1), is created in z ≥ 0 by TIR at the interface z = 0 of a linearly polarized planepropagating wave of complex amplitudes A⊥ and A‖ in the dielectric of refractive index n = 1.5 with angle of incidence: 60o.A Si spherical particle of radius a = 75 nm is deposited on the interface. Its electric and magnetic dipole resonances are atλe = 492 nm and λm = 668 nm, respectively, [cf. Fig.4(left)]. This wave exerts on the particle the following transversal and

perpendicular optical forces: (Left) With A⊥ = A‖, force transversal components, < FgI

e−m >y due to gIy , < F<g>e−m >y from

< gy >, and ratio: < FgI

e−m >y / < F<g>e−m >y whose values (in green) are shown in the right ordinate axis. (Right) Choosing

A‖ = 0, A⊥ 6= 0: Normal component < FgI

e−m >z due to gIz , gradient force < Fgrad >z, and ratio: < FgI

e−m >z / < Fgrad >z

whose values (in green) are shown in the right ordinate axis. Notice that the positive values of the gradient force, repellingthe particle from the interface, are due to negative values of αR

e and/or αRm. The bump of < Fgrad >z near λe and its steep

change of sign close to λm, are due to the gradient force felt by the induced electric and magnetic dipoles, respectively. Thewavelength at which the second Kerker condition holds is λK2 = 608 nm. The force (normalized to area) units are fN/µm2

for an incident power of 1 mW/mm2.

A. Reactive power and stored energy

Concerning the CPV, S(s) = c8πE(s) × H(s) ∗, of the

emitted fields, (cf. Appendix D), we are interested in itsradial component, S(s) · r, across a spherical surface ∂Vof radius r concentric with the particle and enclosing it.Using the fields of Appendix E, this is straightforwardlyintegrated, yielding∫∂V

S(s) · r d2r =ck4

3(|p|2 + |m|2) +

ick

3r3(|p|2 − |m|2). (32)

Whose real part,∫∂V

< S(s) > ·r d2r = ck4

3 (|p|2 + |m|2)

is the well-known radiated (scattered) total power, W (s),independent of the distance r to the center r = 0, andcorresponds to the wevefield re-radiated up to the far-zone, i.e. that with the r−1 dependence, [cf. AppendixD, Eqs. (D-1) and (D-2)].

Here we are particularly interested in the imaginarypart,∫∂V

Im{S(s)} · r d2r =ck

3r3(|p|2 − |m|2)] = −W (s)

react. (33)

Where W(s)react is the reactive power outside V . It

arises from the near and intermediate fields: E(s)(r) =

{ 1εr3 [3n(n · p)− p]−

√µε (n×m) ikr2 }e

ikr, and B(s)(r) =

{ µr3 [3n(n ·m)−m] +√

µε (n×p) ikr2 }e

ikr. (We recall that

the expression: ick3r3 |p|

2 for a purely electric dipolar emit-ter is well-known in antenna theory [3, 4, 11]). We note

that the r−3 dependence of W(s)react makes it to acquire

much larger values than W (s) in the near-field region.

The reactive power W(s)react is the difference between av-

eraged stored magnetic and electric powers which dom-inate in the near and intermediate-field regions aroundthe particle, and that do not propagate. To see it, wewrite the mean stored electric and magnetic energy den-

sities as [7, 9, 10] < w(s)e >=< w

(s)e > − < w

(FF )e >,

< w(s)m >=< w

(s)m > − < w

(FF )m >. While < w

(s)e >

and < w(s)m > are the energy densities of the full fields

E(s) and B(s), i.e with all terms of their expressions (D-1)-(D-4) of Appendix D; the supescript (FF ) stands forfar-zone electric and magnetic energy densities, i.e corre-sponding to the fields (D-1)-(D-4) with only terms of r−1

dependence. Since evidently < w(FF )m >=< w

(FF )e >, we

have that < w(s)m > − < w

(s)e >=< w

(s)m > − < w

(s)e >

[64].

Then we write W(s)react in terms of the volume integral

outside the volume V in the above mentioned sphere ∂V

10

p_react pint_react p_retotW(s)react x10-6 W(s)

react,int x10-6 W(s)react,T x10-5

76

68

60

76

68

60

76

68

60

76

68

60

76

68

60

76

68

60

460

680

9

00

460

680

9

00

λ (nm)

λ (nm)λ (nm)a (nm)

a

7.63

5.71

3.9

2.05

1.84

-1.68

1.79

-1.02

-1.99

-3.88

-5.76

-7.65

8.46

4.84

1.22

-2.40

-6.02

-9.64

460

6

80

900

a (nm) a (nm)

FIG. 3. 3-D graphs of reactive powers as functions of wavelength λ and radius a of a magnetodielelectric sphere of Si in air,

illuminated by a (linearly or circularly polarized) plane wave. (Left): External W(s)react. (Center): Interior W

(s)react,int. (Right):

Total W(s)react,T = W

(s)react +W

(s)react,int. The sign of W

(s)react,int is opposite to that of W

(s)react for all λ and a. The dip and peak of

W(s)react correspond to the electric and magnetic resonance, respectively, and are close to these resonances in W

(s)react,int. Like the

resonances exhibited by the scattered power W (s) [45], these peaks and dips are redshifted as a increases. W(s)react,T vanishes in

the proximities of these resonances, irrespective of a.

W(s)react

400 440 480 520 560 600 640 680 720 760 8000

0.27

0.54

0.81

1.08

1.35

1.62

1.89

2.16

2.43

2.7

Ammod2 xx( )

Bmmod2 xx( )

Cmmod2 xx( ) 0.05

Dmmod2 xx( ) 0.05

xx

400 440 480 520 560 600 640 680 720 760 8007.3 10

6

5.8 106

4.4 106

2.9 106

1.5 106

0

1.5 106

2.9 106

4.4 106

5.8 106

7.3 106

P_rad z( )

P_react z( )

Pint_react z( )

P_retot z( )

z

W(s)

400 440 480 520 560 600 640 680 720 760 8006 10

5

0

6 105

1.2 106

1.8 106

2.4 106

3 106

3.6 106

4.2 106

4.8 106

5.4 106

P_sto z( )

P_stoin z( )

P_stotot z( )

P_stototm z( )

z

W(sto)

W(sto,int)

W(sto,T)

W(sto,Dif)

400 480 560 640 720 800

400 480 560 640 720 800

400 480 560 640 720 800

λ (nm)λ (nm) λ (nm)

4.8x106

3.6x106

2.4x106

1.2x106

0

W(sto)

W(sto,int)

W(sto,T)

W(sto,Dif)

5.8x106

2.9x106

0

-2.9x106

-5.8x106

W(s)react

W(s)react,int

W(s)react,T

cccccc

4.8x106

3.6x106

2.4x106

1.2x106

0

2.16

1.62

1.08

0.54

0

|a1|2

0.05 x |d1|2

0.05 x |c1|2

|b1|2

ccccc

FIG. 4. (Left) Si sphere of radius a = 75nm. (Left): Square moduli of the electric and magnetic Mie coefficients, a1 and b1,

and d1 and c1 of the external and interior scattered field, respectively. (Center): Scattered power W (s) and reactive powers:

external W(s)react, interior W

(s)react,int, and total W

(s)react,T = W

(s)react + W

(s)react,int. Notice that W

(s)react,T vanishes at wavelenghts

close to those of the electric and magnetic Mie resonances: λe = 492nm and λm = 668nm where W (s) is maximum . The lines

|a1|2 and |b1|2 cross each other at the Kerker wavelengths: λK1 = 738.5nm, (at which W(s)react = 0 and W

(s)react,int ' 0), and

λK2 = 608nm, (where both W(s)react and W

(s)react,int 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)..

of radius r, centered in r = 0:

W(s)react =

ck

3r3(|m|2 − |p|2)]

= 2ω

∫V∞−V

d3r (< w(s)m > − < w(s)

e >)

= 2ω

∫V∞−V

d3r (< w(s)m > − < w(s)

e >). (34)

Where V∞ is the volume of a large sphere (kr → ∞).Making V = V0 and r = a, (34) yields the reactive poweroutside the particle. The proof of (34) is given in Ap-pendix E, [cf. Eq. (E-3)]. We anticipate that Eq. (34) isa consequence of the optical theorem for reactive powerwhich we put forward in Section VI, [cf. Eq. (47)].

11

As shown in Eq. (3),∫∂V

Im{S(s)} · r d2r is associ-ated to an instantaneous energy flow alternating backand forth from the scatterer without losses, at frequency2ω, with zero net energy transport in the embedding vac-

uum. Nonetheless, this alternating flow builds W(s)react ac-

cording to Eq. (33), also deduced from Eq. (34). As aconsequence, 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) forpurely electric dipolar RF-antennas, cf. e.g. [3, 7, 9, 10]).Taking V = V0, the total energy stored outside the parti-cle is obtained by

W (sto) =

∫V∞−V0

(< w(s)m > + < w(s)

e >)d3r

=

∫V∞−V0

(< w(s)m > + < w(s)

e >)d3r

−∫V∞−V0

(< w(FF )m > + < w(FF )

e >)d3r

=1

6(|p|2 + |m|2)(

1

a3+

2k2

a). (35)

Appendix E provides the proof of (35), [cf. Eq. (E-4)].The quality factor associated with W (sto) is Q =

2ωW (sto)

W (s) = 1(ka)3 + 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 inte-

rior reactive power, W(s)react,int and the interior stored en-

ergy, W (sto,int), obtained analogously to (34) and (35),but integrating in V0 the mean energies of the interiorfield, are also of interest. For a linearly, or circularly, po-larized incident plane wave of unit intensity, a straight-forward calculation yields (kc/3a3)(9/4k6)(0.055|d1|2 ∓0.018|c1|2). The upper and lower sign in ∓ apply to

W(s)react,int and W (sto,int), respectively, while d1 and c1

are the first electric and magnetic Mie coefficients of thefields inside the particle, (cf. e.g. Eq.(4.45) of [65]). No-

tice the appearance of |d1|2 and |c1|2 in W(s)react,int with

sign opposite to that of |p|2 and |m|2 in (34), i.e. ofthe first electric and magnetic external Mie coefficient

squared moduli, |a1|2 and |b1|2, in W(s)react; we shall see

that this has consequences for the total (i.e. interior plusexternal) reactive power at resonant wavelengths.

B. Reactive power, resonances, and Kerkerconditions

Fig.3 depicts W(s)react(λ, a), W

(s)react,int(λ, a), and

W(s)react,T (λ, a) = W

(s)react(λ, a) + W

(s)react,int(λ, a) for Si

spheres within the range of wavelengths where they be-come dipolar magnetodielectric, and for different sizeradii a, taking advantage of their scaling property withtheir impact parameter, (see Fig.2 of [35], and [45]). Theredshift of the electric and magnetic dipole resonances

λe and λm as a grows, observed in the scattering cross-section [39, 45], [see also W (s) in Fig.3(Center)], is ob-served in the peaks or dips of these reactive powers asthey are much influenced by these resonances. An im-portant feature of these surfaces is that the total reactivepower vanishes, changing its sign, close to the resonancewavelengths where W (s) is maximum, irrespective of a.This is detailed in Fig. 4(Center) on a cross sectionalplane a = 75nm of these surfaces, depicting the scatteredpower, along with the external, interior, and total reac-tive powers. Also Fig.4(Right) shows the stored energies,while in Fig.4(Left) one sees the electric and magneticexternal first Mie coefficients a1 and b1 with resonantmaxima at λe = 492nm and λm = 668nm, respectively,and the internal coefficients, d1 and c1.

The vanishing of the total reactive power, W(s)react,T , ob-

served in Figs.3 (Right) and Fig.4(Center) close to theelectric and magnetic resonances λe and λm where thescattered power, W (s), is produced with maximum effi-ciency, is quite relevant because it manifests a cancel-lation between the internal and external reactive powerpeaks; the stored energies [cf. Fig. 4(Right)] being alsoresonant near these wavelengths. In fact this matches[30] with knowledge from RF antenna theory accordingto which a maximum radiation efficiency is sought byminimizing their reactive power and Q-factor [3, 5–7, 9–11], even though in these works the capacitive - electricdipole wavelength λe: E dominates in r > a, r < λe - (in-ductive - magnetic dipole wavelength λm: B dominates

in r > a, r < λm -) nature of W(s)react is compensated and

tuned to resonance by adding an inductive (capacitive)storage element.

Such element, here in the optics domain, is providedby the particle interior through the emergence of a dom-

inant B at λe (dominant E at λm) of W(s)react,int inside

the particle, (r < a). By the same token the total storedenergies have peaks in these resonant wavelengths λe andλm, like the scattered power W (s) [cf. Fig.4(Right)], be-ing evident that the difference of external and internalstored energies: W (sto,Dif) = W (sto) − W (sto,int) van-

ishes, like W(s)react,T , close to λe and λm.

These results illustrate the concepts of reactive andstored power in and around a magnetodielectric, or Huy-gens, particle when choosing configurations and wave-lengths such that the accretion of external reactive powerand stored energy in the particle near-field, through theIPV alternating flow, be as large as possible; thus scat-tering with maximum efficiency W (s), and possessing thehighest possible Q-factor for applications in light-matter

interactions, while its total reactive power W(s)react,T van-

ishes or is near zero at resonant wavelengths. Since, how-ever, these magnetodielectric nanoresonators have ratherlow Q’s (Q ≤ 7), external (and internal) stored power en-hancements may be achieved either by sets of such mag-netodielectric particles, even metal coated, using them asbuilding blocks of photonic molecules, metasurfaces [66],or in regimes of bound states in the continuum [34, 67].

12

Returning to Fig. 4(Left), we observe the lines |a1|2and |b1|2 crossing each other at the two Kerker wave-lengths: λK1 = 738.5nm and λK2 = 608nm, which corre-spond to the first Kerker condition (K1), (zero backscat-tering, αm = αe, |p| = |m|), and second Kerker con-dition (K2), (minimum forward scattering, αm = −α∗e ,|p| ' |m|), [36, 38, 39, 43–46]. One sees in Fig. 4(Mid-

dle) that at λK1, W(s)react = 0 in accordance with Eqs.

(33) and (34), W(s)react,int ' 0 ; while both W

(s)react and

W(s)react,int ' 0 at λK2. Also notice in Fig.4(Right) that

the stored energies, W (sto) and W (sto,int), and scatteredpower, W (s), are near minimum in the proximities of λK1

and λK2. 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 andK2, from a new standpoint:

The two Kerker conditions for a magnetodielectricdipolar particle are those at which the total reactive power

is near zero. Namely, the external reactive power W(s)react

is either zero (in K1), or close to zero (in K2); while theinternal reactive power is near zero both in K1 and K2.In consequence, the reactive power underlies the angu-lar distribution of scattered (or radiated) intensity and,hence, the directivity of the magnetoelectric particle in away complementary to that formerly addressed in Mie-tronics and RF-antennas [10, 11].

C. The reactive helicity

Concerning the complex helicity flow of the fields scat-

tered by the dipolar particle, [cf. Eq.(16)], F (s)C =

F (s)+iF (s)HR

= (c/4k)Im{(B(s) ∗×B(s)+E(s) ∗×E(s))+

i(B(s) ∗×B(s)−E(s) ∗×E(s))}, we obtain the flux F (s)C · r

across the spherical surface ∂V of radius r ≥ a centeredin the particle. From Eqs. (D-1) and (D-2) for the fieldsin Appendix D, the terms that do not vanish on integra-tion are∫∂V

F (s)C · r d

2r =c

4kr2∫ 2π

0

dφ

∫ π

0

dθ r · Im{[r× (m∗ × r)]

×(r× p)k2

r(k2

r+ik

r2)− C.C.

+[r× (p∗ × r)]× (r×m)k2

r(k2

r+ik

r2)− C.C.}. (36)

Where C.C. denotes complex-conjugated of the previousterm. A straightforward calculation of (36) yields∫∂V

F (s)C · r d

2r =8π

3ck2{k Im[p ·m∗] +

i

rRe[p ·m∗]}, (37)

whose real part, H(s) =∫∂V

Re{F (s)C } · r d2r =

∫∂V

F (s) ·r d2r = 8π

3 ck3Im[p ·m∗], is the total helicity of the scat-

tered field, (as such, it coincides with Eq. (25) of [16]).Like the scattered power, this helicity does not dependon the distance r.

However the imaginary part of (37),

−H(s)react =

∫∂V

Im{F (s)C } · r d

2r =

∫∂V

F (s)HR· r d2r

=8π

3rck2Re[p ·m∗], (38)

comes from the interference of the intermediate field withr−2 dependence and the far field. H(s)

react represents theexternal reactive helicity,

H(s)react = 2ω

∫V∞−V

HR(s) d3r (39)

outside V , as detailed in its optical theorem discussedlater, [cf. Eq. (52)]. It decreases as r−1 as r grows.Therefore, in analogy with the reactive power, in thenear-field (r << λ) the reactive helicity dominates uponH(s). On making V = V0, r = a, Eq.(39) becomes theoverall reactive helicity outside the particle.

It is of interest to specify the overall reactive helicity

of the field inside the particle, H(s)react,int, which is ob-

tained integrating in V0 the helicity of the internal field:

2ω∫V0

HR(s,int) d3r. For a left circularly polarized inci-

dent plane wave of unit intensity, this quantity is equalto: (8πck2/3a)(9/4k6)0.234× 0.276Re{d1c∗1}.

Figures 5 illustrate H(s)react(λ, a), H(s)

react,int(λ, a), and

H(s)react,T (λ, a) = H(s)

react(λ, a)+H(s)react,int(λ, a), generated

by Si spheres illuminated by a left-circularly polarized,CPL(+), plane wave. Again, the redshift with increas-ing radius a is observed in the electric and magnetic

dipole resonant dips of H(s)react. An interesting feature of

H(s)react,int is its similitude with H(s), [cf. Fig.6(Center)

and Fig.6(Left), as well as Fig.6(Right)], and speciallytheir similarity with the converted helicity in the rangebetween λe and λm, (compare with Fig. 1(c) of [55]).This remarks the contribution of the interior reactive he-licity to the scattered helicity lineshape, and lays down anintriguing connection with previous studies [53–55] whichattribute the conversion of helicity to contributions of thescatterer volume and surface.

The total reactive helicity H(s)react,T plays on the scat-

tered helicity a role analogous to that of the total reactivepower on the scattered power. As seen in Fig. 6(Left)H(s) has resonant dips influenced by the electric and mag-netic resonant wavelengths, λe and λm, of the scatteredpower. For a = 75, [cf. Fig.6(Right)], these dips are atλ = 475nm and 679nm and stem from the resonancesat λe = 492nm and λm = 668nm, respectively. On the

other hand, H(s)react,T becomes zero very close to these

resonant wavelengths of H(s). We conclude, therefore,

that the interior reactive helicity H(s)react,int counteracts

on the external reactive helicity H(s)react close the resonant

wavelengths of H(s), (where also H(s)react and H(s)

react;int

are near extreme values), yielding a zero total reactive

helicity H(s)react,T . This property may be observed in

Fig.5(Right), independently of a, and it is clearly seen

13

H (s)

react,int x10-9

hel_react

76

68

60a (nm)

460

680

9

00

λ (nm)

H (s)

react x10-9 hel_reactint

76

68

60

460

6

80

900

2.16

1.54

9.11

0.29

-0.34

-0.96

λ (nm)

a (nm)

helretot

0.07

-0.55

-1.16

-1.77

-2.38

-2.99

λ (nm)

460

6

80

900

H (s)

react,T x10-9

76

68

60

a (nm)

1.37

4.65

4.39

-1.34

-2.25

-3.15

FIG. 5. 3-D graphs of reactive helicities as functions of wavelength λ and radius a of a magnetodielelectric sphere of Si in

air, illuminated by a left circularly polarized incident plane wave, CPL(+). (Left): External H(s)react(λ, a). (Center): Interior

H(s)react,int(λ, a). (Right): Total H(s)

react,T (λ, a) = H(s)react(λ, a) + H(s)

react,int(λ, a). The two dips of H(s)react, and zero crossings

of H(s)react,int, are close to the electric and magnetic resonant wavelengths λe and λm, and are redshifted with increasing a.

Irrespective of the value of a, H(s)react = 0 at λK1 and H(s)

react,T =0 close to λe and λm.

hel_rad

460

680

9

00

λ (nm)

H (s) x 10-8

76

68

60

13.10

8.97

4.81

0.65

-3.51

-7.68

a (nm) 400 440 480 520 560 600 640 680 720 760 8003 10

9

2.4 109

1.8 109

1.2 109

6 108

0

6 108

1.2 109

1.8 109

2.4 109

3 109

Hel_rad xx( )

Hel_react xx( )

Hel_reactint xx( )

Helretot xx( )

xx

400 480 560 640 720 800 λ (nm)

3x109

1.8x109

6x108

-6x108

-1.8x109

-3x109

H (s)

H (s)

react

H (s)

react,int

H (s)

react,T

FIG. 6. (Left) 3-D graph of the scattered helicity H(s)(λ, a) generated by Si spheres in air, illuminated by a left circularlypolarized incident plane wave, CPL(+). It has two peaks, at 475nm and 679nm, influenced by the electric and magnetic

resonant wavelengths λe = 492nm and λm = 668nm. (Right): Making a = 75nm, H(s)(λ), H(s)react(λ), H(s)

react,int(λ) and

H(s)react,T (λ). At the Kerker wavelength λK1 = 738.5nm, H(s)

react vanishes. Also H(s)react,T (λ) is zero close to the wavelengths

where the scattered helicity H(s) is maximum, and in the vicinity of extrema of both H(s)react and H(s)

react,int.

in Fig.6(Right). We do not know, however, of any anal-ogy of these effects in RF-antenna heory.

Notice that these high index magnetoelectric particleshave the interesting property of emitting a wavefield inwhich there are not very large peaks of the total reactivepower [cf. Fig.4(Center)] versus those of the scattered

(radiated) power; although, certainly, where this reactivepower has extrema the scattered (radiated) power is wellaside its peaks. However, under chiral ilumination thesefields present high peaks of total reactive helicity versusits radiated one. For instance, [see in Fig.6(Right)], the75 nm particle] yields a the total reactive helicity with

14

a large dip at 658nm where the scattered (radiated) he-licity is near its minimum value. This occurs at shiftedpositions as a varies, [cf. Fig.6(Left) and Fig.5(Right)].

Therefore, the picture that emerges in these illustra-tions of such optical nanoantennas, considered as eitherprimary or secondary sources, is that the reactive powerand the reactive helicity, which are concentrated bothinside and in the near and intermediate regions of thesource, have a hampering effect in their far-field scatter-ing (or radiation) efficiency. This is an analogous effectto that due to the presence of reactive power in RF-antennas. In consequence, if these nanoantennas emitchiral light, the total reactive helicity hinders the effi-ciency of far-field scattered (or radiated) helicity, so thatless of this helicity is emitted due to a build-up of reactivehelicity in and around the nanantenna [68].

Under incident CPL these particles are dual at λK1,and then p± = ±im± [15, 16], the upper and lowersign applying to left circular, CPL(+) and right circular,CPL(-), respectively. Then p ·m∗ is purely imaginary at

λK1 [69], and one sees from (37) that H(s)react = 0. This

is observed in Fig.5(Left) for any a and in Fig.6(Right)at λK1 = 738.5nm.

As a varies, H(s)react(λ, a) vanishes at the correspond-

ing Kerker wavelength λK1; this is seen in detail inFig.6(Right) for a = 75nm.

Therefore, the first Kerker condition, K1, also has thenovel property that under CPL illumination, the magne-todielectric particle, which then becomes dual and henceemits a wavefield of well-defined helicity equal to the inci-dent one, [15, 16, 55], does not generate external reactivehelicity.

VI. THE REACTIVE POWER OPTICALTHEOREM

Consider a wavefield, E(i), H(i) incident on a magne-todielectric body of volume V0. The field at any point ofthe embedding medium, (assumed to be vacuum or air),is represented as E = E(i)+E(s), H = H(i)+H(s), wherethe superscript (s) denotes the scattered field. Maxwell’sequations are written as:

∇×H(i) = −ikD(i) , ∇×E(i) = ikB(i);

∇×H(s) = −ikE(s) − 4πikP +4π

cj ,

∇×E(s) = ikH(s) + 4πikM. (40)

Where P, M and j are the polarization, magnetization,and free current densities in V0, respectively. D(i) = E(i),B(i) = H(i). We insert Eqs. (40) into Eq.(1) for the totalfields E and H, and use the identity: ∇ · (E × H∗) =H∗ · ∇×E−E · ∇×H∗, integrating in a volume V that

contains V0,

∫V

d3r {2iω(< w(i)m > − < w(i)

e >) + iω

4πRe{H(i) ·H(s) ∗

−E(i) ·E(s )∗}+ iω

2[H(i) ∗ ·M−E(i) ·P∗]

−1

2j∗ ·E(i) +∇ · S(s)}d3r = −1

2

∫V0

d3r j∗ ·E

+2iω

∫V

(< wm > − < we >)d3r . (41)

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

If there are no scattering induced sources other thanP and M in the body, the free current j conveys theconversion of incident power into mechanical and/orthermal energy through the work done on the charges12Re

∫V0d3r j∗ · (E − E(i)), which accounts for the de-

crease of energy W(a) from the wave as power absorbedby the body. Then taking the real part of (41) we obtain

ω

2

∫V0

d3r Im[H(i) ∗ ·M + E(i) ∗ ·P]

=W(a) +

∫∂V

d2rRe{S(s)} · r. (42)

Which is the standard optical theorem (OT) for energy[20, 59], describing the extinction of incident energy, [leftside of (42)], and consequent absorption and radiation ofthe total scattered energy. It reduces to its well-knownexpression [20] for dipolar particles on making P(r) =p δ(r) and M(r) = m δ(r), and then

∫∂V

d2rRe{S(s)} ·r = W (s) in accordance with (32).

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

∫V

d3r [2ω(< w(i)m > − < w(i)

e >) +ω

4πRe{H(i) ·H(s )∗

−E(i) ·E(s )∗}+ω

2Re{H(i) ∗ ·M−E(i) ∗ ·P}]

−1

2Im{

∫V0

d3r j∗ ·E(i)}+

∫∂V

Im{S(s)} · rd2r

= −1

2

∫V0

d3r Im{j∗ ·E}

+2ω

∫V

(< wm > − < we >)d3r . (43)

15

Which becomes

ω

2

∫V0

d3r Re{H(i) ∗ ·M−E(i) ∗ ·P} =

−1

2

∫V0

d3r Im{j∗ ·E(s)} −∫∂V

Im{S(s)} · rd2r +

+2ω

∫V

(< w(s)m > − < w(s)

e >)d3r . (44)

Where we have made use of the fact that <w

(s) (FF )m >=< w

(s) (FF )e > and hence < w

(s)m > − <

w(s)e >=< w

(s)m > − < w

(s)e >, [cf. paragraph prior to

Eq. (34)].

Equation (44) is our formulation of the reactive poweroptical theorem (ROT) for a generic scatterer whose re-sponse to illumination induces densities of polarizationP, magnetization M, and free current j, [70]. The leftside constitutes the extinction of incident energy whichproduces the build-up of external reactive power on scat-tering in the right side of (44). Thus Eq. (44) describeshow, in addition to being radiated into the far zone as Eq.(42) illustrates, scattering gives rise to non-radiated en-ergy, stored in V in the form of (external) reactive power,flowing out from the scattering object and returning toit.

For a dipolar particle the ROT reduces to

ω

2Re{H(i) ∗(0) ·m−E(i) ∗(0) · p} =

−1

2

∫V0

d3r Im{j∗ ·E(s)} −∫∂V

Im{S(s)} · rd2r

+2ω

∫V

(< w(s)m > − < w(s)

e >)d3r . (45)

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

Concerning a dipolar particle, the process describedby the complex optical theorem (41) is analogous to thatin which the feeding energy from an alternate current I,induces an oscillating dipole in a small antenna, whichemits radiated and stored power through the extinction,12ZI

2, of the driving energy. Z being the antenna inputimpedance, Z = Rl+Rr− iX. The dipole loss resistanceRl and radiation resistance Rr [11], generated in accor-dance with the optical theorem (42), are 2W (a)/|I|2 and2W (s)/|I|2, respectively. On the other hand, the dipolereactance X (which for a magnetoelectic dipole is eithercapacitive or inductive [8], depending on the wavelength

λe or λm) stems from its external reactive power, W(s)react,

[11] 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 1

2 (Rl + Rr)|I|2 and

− 12X|I|

2, respectively.

On taking ∂V and V as ∂V∞ and V∞, respectively,corresponding to a large sphere, (kr → ∞), the flux ofscattered CPV across ∂V∞ is real and equals the total

scattered energy W (s). Therefore, Eq. (45) yields

ω

2Re{H(i) ∗(0) ·m−E(i) ∗(0) · p} = −1

2

∫V0

d3r Im{j∗ ·E(s)}

+2ω

∫V∞

(< w(s)m > − < w(s)

e >)d3r . (46)

Which introduced into (45) leads to

−∫∂V

Im{S(s)} · rd2r =

2ω

∫V∞−V

(< w(s)m > − < w(s)

e >)d3r = W(s)react . (47)

In contrast with its real part, the flow (47) depends onthe integration domains, V and ∂V . Notice that (47)conincides with Eq. (34).

VII. THE REACTIVE HELICITY OPTICALTHEOREM

Next, we put forward the law which rules the forma-tion of external reactive helicity by scattering in the nearand intermediate-field regions of the particle through ex-tinction of helicity of the incident wave. Let us introduceEqs. (40) into the identities: ∇·(H∗×H) = H ·∇×H∗−H∗ · ∇×H and ∇ · (E∗ ×E) = E · ∇×E∗ −E∗ · ∇×E,integrating in a volume V that contains the scatteringvolume V0. With E = E(i) + E(s), H = H(i) + H(s),adding the respective expressions, employing the conser-vation equation (12) and using the definitions (8) and(13), (n=1 and H = B outside V0), we arrive at

2πc

∫V0

d3rRe{E(i) ∗ ·M−H(i) ∗ ·P} =∫∂V

d2rF (s) · n +2π

k

∫V0

d3rIm{H(s) ∗ · j}, (48)

which is the known optical theorem for the electromag-netic helicity [16] applying to magnetodielectric arbitraryscattering bodies. Notice that for a dipolar particle, since

P(r) = p δ(r), M(r) = m δ(r) and∫∂V

d2rF (s) · n =

(8πck3/3)Im[p ·m∗], Eq. (48) becomes like Eq. (28) of[16].

However our focus is the conservation law of the reac-tive helicity. To formulate it in the form of an opticaltheorem we substract, rather than add, the above vec-tor identities, and make use of the conservation law (14)along with definitions (9) and (15). Then, proceeding asbefore, it is straightforward to obtain

2πc

∫V0

d3rRe{E(i) ∗ ·M + H(i) ∗ ·P} =

−2π

k

∫V0

d3rIm{H(s) ∗ · j} −∫∂V

d2rF (s)HR· n

+2ω

∫V

d3rHR(s), (49)

16

Equation (49) is the reactive helicity optical theorem andapplies to a generic magnetodielectric scatterer [71]. Theleft side represents the extinction of helicity of the inci-dent wave on build-up outside the body of a reactive he-licity by scattering, given by the right side of (49). Thus,like the energy, the incident helicity gives rise to a reac-tive one associated to the scattered field, which, in addi-tion to the internal reactive helicity, is stored around theparticle, dominating in the near and intermediate-fieldregions where it flows back and forth from the scatterer.

This storage is seen by first considering V to be V∞ in(49). Then [cf. [72]]∫

∂V∞

d2rF (s)HR· n = 0. (50)

Therefore,

2πc

∫V0

d3rRe{E(i) ∗ ·M + H(i) ∗ ·P} =

−2π

k

∫V0

d3rIm{H(s) ∗ · j}+ 2ω

∫V∞

d3rHR(s), (51)

which substituted in (49) leads to

−∫∂V

d2rF (s)HR· n = 2ω

∫V∞−V

d3rHR(s). (52)

Notice that if V = V0, Eq. (52) accounts for the reactivehelicity stored outside the scattering body.

Likewise, if the scatterer is a dipolar particle, theextinction term in the left side of (51) becomes2πcRe{E(i) ∗(0) ·m + H(i) ∗(0) · p}. Then (52) coincides

with H(s)react, Eq. (39) according to (38), thus proving

it; and illustrates how the external reactive helicity isstored around the particle without being scattered intothe far-zone.

VIII. CONSEQUENCE OF THE REACTIVEPOWER OPTICAL THEOREM: SIGNIFICANCEOF THE REACTIVE HELICITY IN REACTIVE

DICHROISM

In dichroism, chiral light illuminates a chiral particle,molecule, or nanostructure. Assuming it dipolar, its con-stitutive relations for the induced dipole moments, p andm, and the incident field are

p = αeE(i) + αemB(i), m = αmeE

(i) + αmB(i). (53)

The electric, magnetic, and magnetoelectric polarizabil-ities being αe, αm, αem, and αme; and fulfilling αem =−αme since the object is chiral [13, 73].

The signal re-emitted (or scattered) by the excitationof this dipolar body discriminates enantiomers (i.e. parti-cles with either αme or −αme) [13, 73, 74] by using as rateof excitation: W(s) = ω

2 Im{E(i) ∗(0) · p + B(i) ∗(0) ·m};

[cf. left side of Eq. (42)].For instance, consider the pair of illuminating fields

[13]: E(i)(r, t) = Re[±E(i)(r) exp(−iωt)] and H(i)(r, t) =

Re[H(i)(r) exp(−iωt)], whose respective helicites are:H +i and H −

i , with H +i = Hi = −H −

i . On employ-ing (42) and (53), the particle excitation rate becomes:W(s)± = ω

2 {αIe|E(i)(0)|2 +αIm|B(i)(0)|2± 2αRme Im[E(i) ·

B(i)∗ ]}= ω2 {α

Ie|E(i)(0)|2+αIm|B(i)(0)|2±4kαRme Hi}. The

superscripts I and R denote imaginary and real part, re-spectively. Clearly, the sign + or − appears accordingto whether the helicity of the illumination is positive:H +i = Hi, or negative: H −

i = −Hi.

Then the above expression of W(s)± yields thewell-known dissymmetry factor g = (W(s)+ −W(s)−)/(W(s)++W(s)−) [73, 74] proportional to αRme Hi

[13].

However, the ROT establishes that rather than thepower radiated in the far-zone (42), one may addressthe excitation of stored reactive power, which dominatesin the near and intermediate-field regions of the parti-cle, which is given by the left side of Eq. (45), viz.

W(s)react = ω

6<{E(i) ∗(0) · p − B(i) ∗(0) ·m}. Then using

the above pair of illumination fields and Eq. (53), oneobtains the discriminatory reactive power

W(s)±react =

ω

2<{±E(i) ∗(0) · p−B(i) ∗(0) ·m}

=ω

2{αRe |E(i)|2 − αRm|B(i)|2 ∓ 2αRme <[E(i) ·B(i)∗ ]}

=ω

2{αRe |E(i)|2 − αRm|B(i)|2 ∓ 4kαRme HRi}. (54)

So that now the sign − or + applies according towhether the reactive helicity of the illumination is posi-tive: HR

+i = HRi, or negative: HR

−i = −HRi, respec-

tively; and (54) yields a dissymmetry factor proportionalto −αRme HRi [75].

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

Therefore, in an analogous way as detecting the radi-ated energyW(s)±, (or absorption/extinction energy), instandard dichroism involves the helicity Hi of the inci-dent wave, in reactive dichroism, observing the excitation

of reactive power W(s)±react in the chiral particle conveys

the incident field reactive helicity HRi , which evidentlycomes out on using (54) in a dissymmetry factor defined

as greact = 2(W(s)+react−W

(s)−react)/(W

(s)+react+W

(s)−react). Notice

that being HRi measurable, so is greact in proportion toHRi.

There is a variety of pairs of illumination wavefieldsthat, like in the above illustration, are interchangeableby parity, and that one may employ in experiments. Asdiscussed in previous sections, non-propagating fields infree-space possess a non-zero HRi; e.g. elliptically (orcircularly, in particular) polarized standing waves, nearfields from an emitter, or evanescent and other surfacewaves, fulfill Eq. (54).

17

IX. CONCLUSIONS

Given the broad interest of evanescent waves atthe nanoscale, and of small particles as light emittingnanoantennas, couplers, and metasurface elements, thecontributions of this paper on its reactive quantities issummarized in the following main conclusions:

(1) We have established the concepts of complex he-licity density and its complex helicity flow, together withtheir conservation law that we name complex helicity the-orem. Its real part is the well-known conservation equa-tion of optical helicity, while its imaginary is a novel lawthat governs the build-up of reactive helicity through itsimaginary, and thus zero time-average, flow. The conceptof reactive helicity density unifies that of magnetoelectricenergy density, previously introduced by symmetry argu-ments, and the so-called real helicity. In this way, we putforward its conservation law and observability, thus com-pleting the fundamentals of this quantity.

(2) The conservation of reactive helicity and reactivepower, and their zero time-averaged flow, has been il-lustrated in two paradigmatic systems: an evanescentwave and a wavefield scattered (or emitted) by a dipolarmagnetodielectric particle. For the former we have putforward reactive orbital and spin momenta that charac-terize its imaginary (reactive) field (Poynting) momen-tum; showing that the wave density of reactive helicityis observable from an experimentally detectectable mixedelectric-magnetic transversal optical force exerted by theevanescent wave on a small high refractive index particle(which behaves as magnetodielectric) through this reac-tive Poynting momentum. On the other hand, we haveuncovered a novel non-conservative force in the decay di-rection of the evanescent wave, which can be discrimi-nated from the gradient one and thus detected, makingobservable the wavefield reactive power density.

(3) Concerning the stored energy, reactive power, andreactive helicity of the field scattered by a dipolar mag-netodielectric particle, we have shown that they providea novel framework to study the particle emission direc-tivity. This has been illustrated on addressing the twoKerker conditions, K1 of zero backscattering, and K2 ofminimum forward scattering. We have established thatunder CPL incident light, the external reactive power iszero in K1 or close to zero in K2; while the internal re-active power, and hence the total reactive power, is nearzero both in K1 and K2. Also, we have proven an addi-tional novel property of the particle at K1 wavelengths,namely, it produces a scattered field with nule overallexternal reactive helicity.

(4) We have established a reactive helicity optical the-orem that governs the build-up and storage of near-field reactive helicity, on extinction of the incident he-licity as light interacts with a generally magnetodielec-

tric nanoantenna. Also we have shown that the emissionof resonant scattered power and of resonant scattered he-licity coincides with a nule, or near zero, total reactivepower and helicity, respectively, i.e. those given by thesum of the overall interior and external reactive powersand helicities. Conversely, peaks of total reactive helicity(reactive power) are associated with poorer efficiency inthe emission of radiated (or scattered) helicity (power).

(5) A reactive power optical theorem has been put for-ward. It rules the formation of external reactive powerand stored electric and magnetic energies, which domi-nate in the near and intermediate-field zones of a mag-netodielectric particle by extinction of the illuminatingenergy.

(6) This latter theorem provides a framework to study-ing the near-field response of chiral nanoparticles to illu-mination with chiral complex fields. It is remarkable thatin the phenomenon of dichroism on illumination with chi-ral light, the incident reactive helicity arises in the near-field region, as we have shown, while it is well-knownthat the incident optical helicity appears from the de-termination of power emitted in the far-zone. Becauseof this, we call reactive dichroism the phenomenon bywhich this incident reactive helicity becomes discrimina-tory for enantiomeric separation. We propose near-fieldobservation experiments of this reactive phenomenon.

Given the interplay between reactive and radiativequantities of electromagnetic fields, we believe that theconcepts studied in this work enrich the landscape of pho-tonics as regards nanoantennas and nanoparticle inter-actions with light. We expect that the observability ofthese reactive effects and quantities should form the ba-sis of future experiments and techniques. In this respect,addressing reactive quantities on higher order multipoleresonance excitation will be a subject of interest for fu-ture studies. This is of special interest for e.g. nanosens-ing advances [77] and all-dielectric thermonanophotonics[78], to be used in effective biomedical diagnosis and ther-apies.

Although our analysis has been focused on thenanoscale, these results are equally valid in the mi-crowave range due the scaling property of high index par-ticles as Huygens sources, which remain with the samecharacteristics of generating large electric and magneticdipole and multipole resonances as the illumination wave-length increases.

ACKNOWLEDGMENTS

MN-V work was supported by Ministerio de Ciencia eInnovacion of Spain, grant PGC2018-095777-B-C21. X.Xacknowledges the National Natural Science Foundation ofChina (11804119). Helpful comments from two anony-mous referees are appreciated.

18

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

Since B(r) = (1/ik)∇×E(r), using Eqs. (4)-(5) we have

B(r) =1

k

∫ ∞−∞

d2K [k× e(K)] exp[i(K ·R + kzz)]. (B-1)

Then the CPV flux across the plane z = 0 is

ΦPoynt =c

8π

∫ ∞−∞

d2R [E(r)×B∗(r)] · z

=c

8π

1

k

∫ ∞−∞

d2R

∫ ∞−∞

d2K d2K′ [e(K′)× (k∗ × e∗(K))] · z exp[i(K′ −K) ·R + (k′z − k∗z)z]}. (B-2)

The R-integral yields δ(K′ −K) = (1/2π)2∫∞−∞ d2R exp[i(K′ −K) ·R], and subsequent integration in K′ yields for

the right side of (B-2):

ΦPoynt =πc

2k

∫ ∞∞

d2K [k∗|e(K)|2 − e∗(K)(k∗ · e(K))] · z

=πc

2k

∫K≤k

d2K qh|eh(K)|2 − iπc2k

∫K>k

d2K [qe|ee(K)|2 − 2qe|ee z(K)|2]. (B-3)

Where the K-integral is split into its homogeneous and evanescent parts with subscripts h and e, respectively. Thenkh is the real wavevector (K, qh) of the homogeneous propagating plane wave component of complex amplitudeeh(K), whereas the complex wavevector ke = (K, iqe) corresponds to each evanescent plane wave component ofcomplex amplitude eh(K). Also we have taken into account that k∗h · eh(K) = 0 and ke · ee(K) = 0, thereforek∗e · ee(K) = 2K · ee,⊥(K) = −2iqeee z(K). [ee(K) = (ee⊥(K), ee z(K)].

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

ΦRPoynt =πc

2k

∫K≤k

d2K qh|eh(K)|2, (B-4)

and

ΦIPoynt = −πc2k

∫K>k

d2K [qe|ee(K)|2 − 2qe|ee z(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 E(r) ·B∗(r) is∫ ∞−∞

d2R [E(r) ·B∗(r)] =1

k

∫ ∞−∞

d2R

∫ ∞−∞

d2K d2K′[e(K′) · (k∗ × e∗(K))] exp[i(K′ −K) ·R]}. (B-1)

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

d2R [E(R) ·B∗(R)] =(2π)2

k

∫ ∞−∞

d2K k∗ · [e∗(K)× e(K)]

=(2π)2

k{∫K≤k

d2K kh · [e∗h(K)× eh(K)] +

∫K>k

d2K K · [e∗e(K)× ee(K)] +

∫K>k

d2K (−iqe)[e∗e(K)× e(K)]z}. (B-2)

We recall that [e∗(K) × e(K)] = i=[e∗(K) × e(K)] = (4ki/c)FE(K); where FE(K) is the density of electric spinof each angular plane wave component in K-space. Then, taking imaginary and real parts in (B-2), and using thesubindex h and e for homogeneous and evanescent components, we arrive at∫ ∞

−∞d2R H (R, 0) = (1/2k)

√ε

µ

∫ ∞−∞

d2R Im{E(R) ·B∗(R)}

= 2(2π)2

kc{∫K≤k

d2K kh ·FE h(K) +

∫K>k

d2K K ·FE e⊥(K)}, (B-3)

19

and ∫ ∞−∞

d2R HR(R, 0) = (1/2k)

√ε

µ

∫ ∞−∞

d2RRe{E(R) ·B∗(R)} = 2(2π)2

kc

∫K>k

d2K qeFE e z(K). (B-4)

In Eq. (B-3) FE e = (FE e⊥,FE e z), FE e⊥ = (FE ex ,FE e y). Equations (B-3) and (B-4) are (17) and (18) of themain text.

Appendix C: EVANESCENT WAVE: TIME-AVERAGED QUANTITIES, ELECTRIC AND MAGNETICIMAGINARY SPIN AND ORBITAL MOMENTA

The real part of S in (22) is

Re{S} =< S >=c

8πµ[K

k(|T⊥|2 + |T‖|2) , 2

Kq

k2Im{T⊥T ∗‖ } , 0 ] exp(−2qz) =

ck

K[w ,− q

4πH , 0]. (C-1)

S is well-known to be associated to the optical force on a body. Considering a magnetodielectric dipolar particle placedin the air on the z = 0 interface, the real part, or energy flow density < S >, of the CPV is known to constitute amomentum of the radiation whose x-component, proportional to w, gives rise to an x-force on the particle, separatelyacting on the particle electric (e) and magnetic (m) induced dipoles [20]. The component < S >y is known to producea lateral force proportional to −H along OY [21], due to the interference of its e and m dipoles [20].

On the other hand, the time-averages of the density of spin angular momentum, < S >= (1/4πc2)F = (1/2)(<Se > + < Sm >), where < Se >= (1/8πkc)={E∗ × E} and < Sm >= (1/8πkc)Im{B∗ × B}, and of spin curl, orBelinfante spin momentum, < PS >= (1/2)[< PS

e > + < PSm >] = (1/2)∇× < S >, are

< S >=1

8πc[−2K

k2Im{T⊥T ∗‖ },−

Kq

k3(|T⊥|2 + |T‖|2) , 0] exp(−2qz) =

1

4πcK[kH ,−4πq

kw, 0]. (C-2)

With < PSe >= (1/2)∇× < Se >, < PS

m >= (1/2)∇× < Sm >, and < PS > being:

< PSe >=

1

8πc[−2Kq2

k3|T‖|2

2Kq

k2Im{T⊥T ∗‖ }, 0] exp(−2qz). (C-3)

< PSm >=

1

8πc[−2Kq2

k3|T⊥|2 ,

2Kq

k2Im{T⊥T ∗‖ }, 0] exp(−2qz). (C-4)

< PS >=1

8πc[−Kq

2

k3(|T⊥|2 + |T‖|2)

2Kq

k2={T⊥T ∗‖ }, 0] exp(−2qz) = − q

cK[q

kw,

k

4πH , 0]. (C-5)

Equation (C-2) remarks that, in agreement with the second term of the right side of (17), H appears in the projectionof the density of spin momentum < S > on the x-direction, which is that of propagation of the evanescent wave [62].On the other hand, (C-5) highlights the transversal y-component of the spin momentum proportional to H , as wellas its longitudinal component, along OX, proportional to w.

Since the time-average electromagnetic field momentum density < g >=< S > /c2 holds

< g >=< PO > + < PS >; (C-6)

< PO > being the density of time-averaged orbital momentum, one sees from (C-1) and (C-5) that the trans-verse y-component of < g > comes from < PSy >, which is characterized by H . Both < gy > and < PSy > are(−1/4πc)(kq/K)H exp(−2qz). Therefore the trensverse y-component < gy > of the field momentum is provided bythe transverse y-component < PSy > of Belinfante’s momentum, in agreement with [21].

In addition, from (C-6), (C-5) and (C-1) we have for the ith Cartesian component of the time-averaged orbitalmomentum density,

< POi >= (1/2)(< POe i > + < POmi >) = (1/2)(1/8πkc)[Im{E∗j ∂iEj}+ Im{B∗j ∂iBj}], (i, j = x, y, z),

< PO >=1

8πc[K3

k3(|T⊥|2 + |T‖|2) , 0 , 0] exp(−2qz) =

w

c[K

k, 0 , 0]. (C-7)

With the electric and magnetic orbital momenta:

< POe >=

1

8πc[K(2K2 − k2)

k3|T‖|2 +

K

k|T⊥|2 , 0 , 0]× exp(−2qz), (C-8)

20

and

< POm >=

1

8πc[K(2K2 − k2)

k3|T⊥|2 +

K

k|T‖|2 , 0 , 0]× exp(−2qz), (C-9)

respectively. < PO > contains the energy density w of the wave and, as such, points in the propagation directionOX of the wave, like both < gx > and the x-component, K > k, of its propagation wavevector. This, in agreementwith [21], confers to the evanescent wave a superluminal group velocity; although of course the time-average energyflow < Sx > propagates with speed less than c. Hence < POx > pushes the aforementioned small particle, placed inthe 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:

(PS Ie )i =

1

8πkcRe{∂j(E∗i Ej)} ,

(PS Im )i =

1

8πkcRe{∂j(B∗iBj)} ,

(PO Ie )i = − 1

8πkc∂j

1

2δij |E|2 = − 1

8πkc

1

2∂i|E|2

(PO Im )i = − 1

8πkc

1

2∂i|B|2 . (i, j = x, y, z). (C-10)

PS Ie =

1

4πk2c(0,−KqRe{T⊥T ∗‖ },−

K2q

k|T‖|2) exp(−2qz). (C-11)

PS Im =

1

4πk2c(0, Kq Re{T⊥T ∗‖ },−

K2q

k|T⊥|2) exp(−2qz). (C-12)

PO Ie =

q

8πkc(0, 0,

K2 + q2

k2|T‖|2 + |T⊥|2) exp(−2qz). (C-13)

PO Im =

q

8πkc(0, 0,

K2 + q2

k2|T⊥|2 + |T‖|2) exp(−2qz). (C-14)

Appendix D: DIPOLE FIELDS

We express the fields emitted by a dipole with electric and magnetic moments p and m as [8],

E(s)(r) = {k2

εr[r × (p× r)] +

1

ε[3r(r · p)− p](

1

r3− ik

r2)−

õ

ε(r ×m)(

k2

r+ik

r2)}eikr. (r = rr). (D-1)

B(s)(r) = {µk2

r[r × (m× r)] + µ[3r(r ·m)−m](

1

r3− ik

r2) +

õ

ε(r × p)(

k2

r+ik

r2)eikr. (D-2)

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

The reactive power W(s)react = ck

3r3 (|p|2 − |m|2) may also be obtained by performing the volume integration of theright side of (34) for the magnetoelectric dipole:

< W (s)e >=

∫V∞−V

d3r < w(s)e >=

1

16π

∫ ∞r

drr2∫ 2π

0

dφ

∫ π

0

dθ sin θ[|E(s)|2 − |E(s)FF |

2] =k3

6[|p|2(

1

(kr)3+

1

kr) +|m|2

kr], (E-1)

and

< W (s)m >=

∫V∞−V

d3r < w(s)m >=

1

16π

∫ ∞r

drr2∫ 2π

0

dφ

∫ π

0

dθ sin θ[|B(s)|2 − |B(s)FF |

2] =k3

6[|m|2(

1

(kr)3+

1

kr) +|p|2

kr]. (E-2)

21

Having used Eqs. (D-1) and (D-2) of Appendix D, and E(s)FF and B

(s)FF being the radiated far-fields with r−1 depen-

dence. Thus the time-averaged reactive power is

2ω(< W (s)m > − < W (s)

e >) = W(s)react =

ck

3r3(|m|2 − |p|2). (E-3)

Which is (34).The total time-averaged energy outside the volume V is after Eqs. (E-1) and (E-2),

< W(s)T >=< W (s)

e > + < W (s)m >=

1

6(|p|2 + |m|2)(

1

r3+

2k2

r). (E-4)

Which is (35) when r → a .

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in forward- and back-scattered electromagnetic waves bya single dielectric subwavelength sphere, Nat. Comm. 3,1171 (2012).

[46] S. Person, M. Jain, Z. Lapin, J. J. Saenz, G. Wicks andL. Novotny, Demonstration of zero optical backscatteringfrom single nanoparticles, Nano Lett. 13, 1806 (2013).

[47] A. Bag, M. Neugebauer, P. Wozniak, G. Leuchs, and P.Banzer, Transverse Kerker Scattering for Angstrom Lo-calization of Nanoparticles, Phys. Rev. Lett. 121, 193902(2018).

[48] J. Olmos-Trigo, C. Sanz-Fernandez, D. R. Abujetas,J. Lasa-Alonso, N. de Sousa, A. Garcıa-Etxarri, J. A.Sanchez-Gil, G. Molina-Terriza and J. J. Saenz, KerkerConditions upon Lossless, Absorption, and Optical GainRegimes, Phys. Rev. Lett. 125, 073205 (2020); J. Olmos-Trigo, D. R. Abujetas, C. Sanz-Fernandez, J. A. Sanchez-Gil and J. J. Saenz, Optimal backward light scattering bydipolar particles, Phys. Rev. Research 2, 013225 (2020);J. Olmos-Trigo, D. R. Abujetas, C. Sanz-Fernandez, X.Zambrana-Puyalto, N. de Sousa, J. A. Sanchez-Gil, andJ. J. Saenz, Unveiling dipolar spectral regimes of largedielectric Mie spheres from helicity conservation, Phys.Rev. Research 2, 043021 (2020).

[49] If the embedding medium is lossy and hence it alsopresents a certain dispersion, or complex, there willbe terms associated with effects additional to thoseadressed in this research. See e.g. Section 1.8 of [3] forthe complex Poynting theorem and [53] for the conser-vation of optical chiraliity. For instance, for lossy dis-persive media the electric and magnetic energy termsshould be of the form (1/16π)∂ω[ε′(ω) + iε′′(ω)]|E|2 and(1/16π)∂ω[µ′(ω) + iµ′′(ω)]|H|2, [cf. e.g. Section 80 of L.D. Landau and E.M. Lifshitz, Electrodynamics of Con-tinuous Media, Pergamon Press, (Oxford, 1984)], whichis in fact an approximation for time slowly-varying fields.Such generalizations, and their related problems [10, 30]are outside the aims of this work.

[50] L. Mandel and E. Wolf,Optical Coherence and QuantumOptics, Cambridge U.P., (Cambridge, 1995).

[51] M. Nieto-Vesperinas, Scattering and Diffraction in Phys-ical Optics, 2nd edition, World Scientific, (Singapore,2006).

[52] K. Y. Bliokh, A. Y. Bekshaev and F. Nori, Dual electro-magnetism: helicity, spin, momentum and angular mo-mentum, New J. Phys. 15, 033026 (2013).

[53] L. V. Poulikakos, P. Gutsche, K. M. McPeak, S. Burger,J. Niegemann, C. Hafner and D. J. Norris, Optical chi-rality flux as a useful far-field probe of chiral near fields,ACS Photonics 3, 1619–1625 (2016).

[54] P. Gutsche, L. V. Poulikakos, M. Hammerschmidt, S.Burger and F. Schmidt, Time-harmonic optical chiral-ity in inhomogeneous, space. Proc. SPIE 9756, 97560XarXiv:1603.05011 (2016).

[55] P. Gutsche and M. Nieto-Vesperinas, Optical Chiralityof Time-Harmonic Wavefields for Classification of Scat-terers. Sci. Reps. 8, 9416 (2018).

[56] Notice the difference of Eqs. (14)-(16) with the conser-vation equation and quantities of [53], (cf. Eqs. (S.I.12)-(S.I.15) of the Supporting Information of [53]). In homo-geneous lossless media addressed here, (S.I.12)-(S.I.15)describe purely real quantities and χe − χm = 0, so thatthe conservation equation (S.I.12) of [53] does not providea law for reactive quantities, although both the opticalchirality flux (S.I.15) and the real conservation equation

23

(S.I.12) are k2 times the flow of helicity (13) and theconservation of helicity (12), respectively.In order to get complex expresions (S.I.12)-(S.I.15), andRe[χe − χm] 6= 0, Im[χe − χm] 6= 0), lossy media shouldbe considered, but then the imaginary parts of (S.I.12)-(S.I.15) are different to (14)-(16). Hence, the imaginarypart (14) of the complex conservation equation (16) is anovel law for the reactive helicity HR and its flow FHR

.[57] Let E = (e+ε++e−ε−) cos kz, B = (e+ε+−e−ε−) sin kz,

be an elliptically polarized standing wavefield, expressedin the helicity basis ε± = 1√

2(x ± iy), with ampli-

tudes CPL(+) (left circularly polarized) e+ and CPL(-) (right circularly polarized) e−. One has E × B∗ =−(i/2)(|(e+|2+ |e−|2) sin 2kz z, |B|2−|E|2 = −(|(e+|2+|e−|2) cos 2kz, and hence ∇ · ={S} = 2ω(< wm > − <we >). In addition, E ·B∗ = (1/2)(|(e+|2−|e−|2) sin 2kz,B∗ ×B−E∗ ×E = −i(|(e+|2 − |e−|2) cos 2kz z, so that∇ · FHR = 2ωHR.Therefore this kind of wave has no time-averaged energytransport < S >, nor helicity density H , but their den-sities of reactive power 2ω(< wm > − < we >), IPV,reactive helicity HR and its flow FHR , are non-zero.Note that if the standing wave is linearly polarized, (e.g.Ey = Bx = 0), it has IPV and stores reactive power, butHR = 0.

[58] M. Nieto-Vesperinas, Optical torque: Electromagneticspin and orbital-angular-momentum conservation lawsand their significance.Phys. Rev. A 92, 043843 (2015).

[59] M. Born and E. Wolf,Principles of Optics, CambridgeUniversity Press, Cambridge (1995).

[60] M. Nieto-Vesperinas and J. J. Saenz, “Optical forces froman evanescent wave on a magnetodielectric small parti-cle,” Opt. Lett. 35, 4078–4080 (2010).

[61] M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, Mea-suring the Transverse Spin Density of Light, Phys.Rev.Lett. 114, 063901 (2015).

[62] Notice that Eqs.(19) correspond to a wavefield E(r) with

angular spectrum e(K′) =(− k′z

kT‖, T⊥,

K′

kT‖

)δ(K′ −

K), (k = (K, kz),K = (Kx, 0), kz = iq,Kx > k).[63] P. C. Chaumet and M. Nieto-Vesperinas, Coupled dipole

method determination of the electromagnetic force on aparticle over a flat dielectric substrate. Phys. Rev. B 61,14119-14127 (2000).

[64] Note that < w(s)e > and < w

(s)m > are not the electric and

magnetic energy densities of only the near plus interme-diate fields, say E(NF,IF ), B(NF,IF ), since they also con-tain an interference term of these fields with the far fields:2Re{E(NF,IF ) · E(FF )} and 2Re{B(NF,−F ) ·B(FF )}, re-spectively.

[65] C. F. Bohren and D. R. Huffman, Absorption and Scat-tering of Light by Small Particles, (John Wiley and Sons,1998).

[66] K. Koshelev and Y. Kivshar, DielectricResonant Metaphotonics, ACS Photonics.https://pubs.acs.org/doi/pdf/10.1021/acsphotonics.0c01315.

[67] M. V. Rybin, K. L. Koshelev, Z. F. Sadrieva, K. B. Samu-sev, A. A. Bogdanov, M. F. Limonov and Y. S. Kivshar,High-Q Supercavity Modes in Subwavelength DielectricResonators. Phys Rev. Lett. 119, 243901 (2017).

[68] Note that one could address a medium composed of par-ticles, in which transport of light occurs. The existence

of multiple scattering may be associated with a slow con-vergence of the Born series due to some coupling betweenthe particles, stringent conditions for convergence underexcitation 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 tolight scattering by Mie-resonant nanoparticle structures,arXiv:2108.11920v1 26 Aug 2021), or even with no con-vergence at all if there is either strong coupling betweenthe induced dipoles (or multipoles). The latter two effectsshould be associated with higher Q-factors, and thus withlarger amounts of stored and reactive powers. The samewe would expect to rule the behavior of the scatteredhelicity and a quality factor which might be introducedfor this quantity; this being an area of possible futureexploration.

[69] For a circularly polarized incident plane wave of unit am-

plitude, the induced dipoles are p = αeE(i) and m =

αmB(i) with E(i) = ε±, B(i) = ∓iε±, ε± = 1√2(x± iy),

the upper and lower sign applies according to whetherit is left circular, CPL(+), or right circular, CPL(-), re-spectively. Since at K1: αm = αe, then p = αeε

± andm = ∓iαeε

±, and thus p = ±im [16]. Therefore at K1:p ·m∗ = ±i|αe|2. However at K2: αm = −α∗e and hencep ·m∗ = ∓iα2

e.[70] In this regard we note that a reactive optical theorem

in anisotropic media was reported by E. A. Marengo,A New Theory of the Generalized Optical Theorem inAnisotropic Media, IEEE Trans. Antenn. Propag. 61,2164-2179 (2013).

[71] In contrast with the energy OT and ROT, both helicityOT and ROT contain a real part in the extinction term.This is due to the different functional form in the realand imaginary parts, F and FHR , of the complex flowof helicity FC , as seen in Eqs. (13)-(16).Eq. (16) shows, at difference with those of the CPV.

[72] To prove that∫∂V∞

d2rF (s)HR· n = 0, we write the scat-

tered field at points in the far-zone characterized by thedirection unit vector n as: E(s)(n) = e(n) exp(ikr)/r and

H(s)(n) = h(n) exp(ikr)/r, h = n×e. Then it is straight-forward to see that

∫∂V∞

d2rIm{h∗(n) × h(n)} · n =∫∂V∞

d2rIm{e∗(n) × e(n)} · n. This latter equality and

the definition (15) of F (s)HR

constitute the proof.

[73] L. D. Barron, Molecular light scattering and optical ac-tivity. Cambridge University Press, (Cambridge, 2004).University Press

[74] J. A. Schellman, Circular dichroism and optical rotation.Chem. Rev. 75, 323–331 (1975).

[75] In this connection, it should be noted that an expres-sion akin to such dissymmetry factor, was written in [23]as what the authors called relative magnetoelectric re-sponse of the particle to the illumination. This was doneon employing what they named magnetoelectric absorp-tion rate, determined from <[E(i) · B(i)∗ ], which theycalled magnetoelectric energy; even though the physicalprocess that produces it was not reported. Here we havedemonstrated that the left side of Eqs. (54) and (51), aswell as (52), describe the mechanism through which thisquantity appears and may be observed.

[76] M. F. Picardi, A. V. Zayats, and F. J. Rodrıguez-Fortuno, Janus and Huygens Dipoles: Near-Field Direc-tionality Beyond Spin-Momentum Locking, Phys. Rev.

24

Lett. 120, 117402 (2018).[77] A. Krasnok, M. Caldarola, N. Bonod and Andrea Alu,

Spectroscopy and Biosensing with Optically ResonantDielectric Nanostructures, Advanced Optical Materials

1701094 (2018). DOI: 10.1002/adom.201701094.[78] G. P. Zograf, M. I. Petrov, S. V. Makarov, and Y. S.

Kivshar, All-dielectric thermonanophotonics, Advancesin Optics and Photonics 13, 643 (2021).