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

We put forward the complex helicity theorem. It constitutes a novel law that rules the build-up of the reactive helicity through its zero time-average flow. Its imaginary Poynting momentum accounts for the accretion of reactive power, are illustrated in: evanescent waves and fields scattered from magnetodielectric dipolar nanoparticles. As for the former, we show that its reactive helicity may be experimentally observed as we introduce a reactive spin momentum and a reactive orbital momentum in terms of which we express the imaginary field momentum, whose transversal component produces an optical force on a magnetoelectric particle that, as we illustrate, may surpass and can be discriminated from, the known force due to the so-called extraordinary momentum. We also uncover a non-conservative force on such a magnetoelectric particle, acting in the decay direction of the evanescent wave, and that may also be discriminated from the standard gradient force; thus making the reactive power of the wavefield also observable. Concerning the light scattered by magnetoelectric nanoparticles, we establish two optical theorems that govern the accretion of reactive helicity and reactive power on extinction of incident wave helicity and energy. Like a nule total -- internal plus external -- reactive power is at the root of a resonant scattered power, we show that a zero total reactive helicity underlies a resonant scattered helicity. The first Kerker condition, under which the particle becomes dual on illumination with circularly polarized light, we demonstrate to amount to a nule overall scattered reactive helicity. and emission. We discover a discriminatory property of the reactive helicity of chiral light incident on a chiral nanoparticle by excitation of the external reactive power. This for optical near-field enantiomeric separation, we call reactive dichroism.

In spite of progress in the analysis of optical antennas, mainly addressing quantum emitters and plasmonic nanoparticles [26][27][28], whose radiative and feeding characteristics is studied from the point of view of RFantennas and ciecuit theory [29][30][31], we have found few detailed studies [30] on their reactive quantities; although * mnieto@icmm.csic.es , xuxhao dakuren@163.com the effects of reactive power in antenna functionality are well-known [3, 5-7, 9-12, 30]; e.g. the radiative and reactive energies of an oscillating dipole are intertwined. In this way, a major task in RF-antenna design has been the study of its reactive power and Q-factor, seeking a minimization of both quantities in order to match the input energy with its radiative performance, since a large Q and reactive energy in the antenna convey high ohmic losses, and a decrease of its operative bandwidth.
By contrast, at optical wavelengths a high reactive power outside the nanoemitter or scatterer, external stored energy, and Q-factor of a low-loss nanoparticle, or nanoantenna, enhances both its scattered (or radiated) power and frequency sensitivity, also narrowing its operational bandwith, this being desirable for its role as e.g. a nano-source or a biosensor, as well as to reinforce lightmatter interactions at the nanoscale, both in plasmonics [32] and Mie-tronics [33,34]. The reason is that, as we shall see, although such a large external reactive power and stored energy occur at wavelengths near those of resonant scattered power, the interior of the particle acts as a compensating (capacitive or inductive) element so that its reactive power and stored energy cancel out the external ones close to these resonant wavelengths at which each of these quantities have near extreme values. The result is that, in analogy with RF-antenna design [30], the total (i.e. internal plus external) reactive power and stored energy are zero close to resonances.
We also show that the same effect occurs for the interior, external, and total reactive helicities in connection with a maximum efficiency in the helicity scattered up to the far zone.
Since surface plasmons work at optical frequencies where metals exhibit high losses, there has been an increasing interest in high refractive index nanoantennas, on which light exerts a magnetoelectric response with large electric and magnetic resonances [33][34][35][36][37][38][39][40]. We will study the key role of reactive quantities in the scattering from these magnetodielectric particles. The other archetypical configuration in which we shall address these reactive quantities is an evanescent wave.
Using time-harmonic wavefields, we will start addressing the flow of reactive power of the complex Poynting theorem, and since an analogous law for the helicity flow has never been established, as far as we know, we will introduce the concepts of complex helicity density [41] and complex helicity flow, for which we put forward a conservation equation that we coin as complex helicity theorem. Its real part is the well-known continuity equation for the conservation of optical helicity [13][14][15][16]18], while its imaginary part is a novel law that describes the build-up of reactive helicity, whose flow has zero time-average and hence it does not propagate in free-space.
We show that this reactive helicity density and its flow exist, like the reactive energy and the imaginary Poynting vector, in wavefields that do not propagate into the environment, as e.g. evanescent and elliptically (and circularly in particular) polarized standing waves; the former being identical to the so-called magnetoelectric energy density, introduced in [23] following symmetry arguments but without providing its undelying physical law.
As for an evanescent wave, whose imaginary Poynting vector and its associated transversal spin have been studied [21], while its time-averaged energy flow, spin angular momentum, Belinfante momentum, and orbital momentum have to do with the time-averaged energy and helicity densities [21], we show that the reactive (i.e. imaginary) Poynting vector, and reactive helicity flow, are linked with the reactive energy and reactive helicity of the wave. These relationships provide a physical law for our introduced concept of reactive helicity, in which the aforementioned magnetoelectric energy [23], and the so-called "real helicity" discussed in a different research [42], are unified. Further, we discuss how the reactive power and reactive helicity of this wavefield, are generated close to the interface through their corresponding alternate flows along the wave decay direction.
Moreover, we show that the imaginary Poynting momentum of the evanescent wavefield is the sum of densities of a reactive spin momentum and a reactive orbital momentum, which in turn are expressed as differences of the imaginary magnetic and electric corresponding momenta. This leads us to uncover a non-conservative optical force on a magnetodielectric particle, directed along the wave decay direction -i.e. different from the gradient force and larger than it at certain wavelengths, thus being experimentally detectable-due to the IPV.
We also illustrate the presence of a transversal force on such a magnetodielectric particle, stemming from the corresponding component of the IPV, which may be of opposite sign and much larger than the known lateral force [21] due to the time-averaged Poynting vector; thus being detectable and making observable the reactive helicity.
As regards fields emitted or scattered by a magnetodielectric dipolar nanoparticle, we shall show that its reactive energy and angular distribution of scattered radiation, being intimately interrelated, provide a novel interpretation of the two Kerker conditions: K1 and K2 [36,[43][44][45][46][47][48], and hence of the particle emission directivity. Namely, under plane wave illumination, and at wavelengths where such a particle fulfils either K1 of zero backscattering, or K2 of minimum forward scattering, the overall external reactive power around this body is either zero or almost zero, while the internal and total (i.e. external plus internal) reactive powers are near zero. Moreover, the overall external reactive helicity vanishes at K1 wavelengths on illumination with circularly polarized light.
Concerning feeding the magnetodielectric nanoantenna, we put forward the reactive power and the reactive helicity optical theorems which quantify the accretion of external stored reactive energy and reactive helicity in the near and intermediate-field zones, in terms of the particle excitation and extinction of energy and helicity of the supplied illumination. The effects here shown maximizing both the external and internal values of these reactive quantities, (while minimizing their overall amounts, i,.e. external plus internal), which tune to resonance their radiation efficiency, establishes an analogy with the same well-known pursuit in RF antenna design. Therefore, this work puts forward the importance of reactive quantities which underly some previously studied concepts in the analysis of optical antennas [26][27][28][29].
Finally, we show that the reactive power theorem yields an interpretation of how by illuminating a chiral particle with a non-free propagating wavefield, like an elliptically polarized standing wave, evanescent wave, or near field from a nearby emitter, the reactive helicity of the incident wavefield appears as a consequence of the generation of reactive energy on interaction with the sample particle. It is intriguing that, as we find, this incident reactive helicity emerges analogously as the incident helicity does in a standard dichroism far-field observation [13]. Therefore, we show that the so-called "magnetoelectric response" of a chiral particle, quoted in [23], arises as a consequence of the accretion of its reactive energy from near-field chiral illumination, e.g. with incident evanescent waves, (or, similarly, with incident circularly polarized standing waves as proposed in [23]). As such, we name reactive dichroism this magnetoelectric phenomenon. It underlines the observability of such in-cident reactive helicity, and its discriminatory property for enantiomer separation by near-field optical techniques using structured illumination.

II. THE REACTIVE POYNTING VECTOR AND THE REACTIVE POWER
To fix some concepts to deal with, we first outline the main quantities involved in the complex Poynting theorem. We shall assume an arbitrary body immersed in a lossless homogeneous medium [49]. Let E(r, t) = E(r) exp(−iωt), B(r, t) = B(r) exp(−iωt) be a timeharmonic electromagnetic field. It is well-known that in a body with charges and free electric currents of density j contained in a volume V with permitivity and permeability µ, the complex work density, given by the scalar product j * ·E, leads after using Maxwell's equations, ∇ × , to the complex Poynting theorem [3,4,8,11]: Where * and < . > mean complex conjugated and time-average, respectively. The complex Poynting vector (CPV) S, and time-averaged electric and magnetic energy densities, < w e > and < w m >, are Under our above assumptions, the right side of (1) is purely imaginary, and the real part of this equation constitutes the well-known Poynting theorem describing the variation of energy in the body due to the work rate of the field upon its charges, given by the volume integral of 1 2 Re{j * · E}. This variation is characterized by the flow of the time-averaged Poynting vector, < S >= Re{S}, across the surface ∂V of V : ∂V Re{S} ·rd 2 r. Wherer is the outward unit normal to ∂V . On the other hand, the imaginary part of (1) formulates that in the steady state the integral of the imaginary work 1 2 Im{j * · E}, plus the reactive power flux ∂V Im{S}·nd 2 r, (which has zero time-average since it is associated with instantaneous energy flow that alternates back and forth at twice frequency across ∂V ), accounts for this reactive power build-up in and around the body, given by the right side of (1); 2(< w m > − < w e >) being the reactive energy density. Unless otherwise stated, we shall henceforth assume = µ = 1 for the embedding space.
At this point it is convenient to introduce the instantaneous Poynting vector [3,11] built by the fields E(r, t) = {E(r, t)} and B(r, t) = {B(r, t)}, This is the standard expression of S(r, t). However we find it more instructive to write it as: Where the superscripts R and I denote real and imaginary parts, respectively. Note that while the term with < S > does not change sign, as expected from that part of the instantaneous Poynting vector associated with the time-averaged energy flow, the term that contains the IPV or reactive Poynting vector, Im{S}, alternates its sign at frequency 2ω following the variation of sin 2ωt. This is in accordance with the above interpretation of the imaginary part of (1). We also see that there is a generally non-zero contribution to this alternating flow in the last two terms of (3). Obviously only the < S > term remains on time-averaging in (3).
A. The reactive Poynting vector and the angular spectrum of plane waves To distinguish the structure of the real and imaginary parts of S, it helps to employ the angular spectrum representation of the electromagnetic field, so as to map these quantities into their spectra in Fourier space. To this end, we calculate the flow of c 8π E(r) × B * (r) across a plane z = constant. We use for simplicity a framework such that the souces are on z < 0 and thus the integration is done on the z = 0 plane. But one may equally choose any other constant value z = z 0 , providing the sources lie in z < z 0 . The electric field E(r) propagating into the half-space z ≥ 0 is represented by its angular spectrum of plane wave components as [50,51]: Using the subscripts h and e for homogeneous and evanescent components, respectively, the CPV flux, Φ P oynt , across the plane z = 0 has real and imaginary parts given by, (see the proof in Appendix A): and Eq.(6) is well-known, it expresses the flux Φ RP oynt of the real part of the CPV as the momentum power carried on by the propagating components, (K ≤ k). However, Eq. (7) shows that the flux Φ IP oynt of the reactive CPV is momentum associated with power contained in the evanescent components, (K > k), and as such it does not propagate into z ≥ 0. Notice the special role played by the longitudinal component e e z (K) in (7).

III. THE REACTIVE HELICITY AND ITS REACTIVE FLOW THEOREM
The optical helicity density of the electromagnetic field in a medium with constitutive parameters and µ is wellknown to be [14,16,18] We now introduce the quantity Like Im{S}, this quantity H R may appear from H when E and B are chosen π/2 out of phase. We shall later see that H R is exclusive of Im{S}, (and of course of a reactive helicity flow, to be introduced next). Thus we call H R the reactive helicity density of the field. Some authors have recently addressed this quantity in different works, and call it magnetoelectric energy [23], or just real helicity [42]. However we keep our denomination by showing that, like the reactive energy, it fulfills a conservation law.
We consider a body with charges and free currents, embedded in a volume V . From the two Maxwell equations for the spatial vectors, ∇ × E = ikB and ∇ × H = (4π/c) j − ikD, we derive the conservation equation for the reactive helicity. First, we employ the second of these equations and address the scalar product (4π/c)j * · B in V . On using the identity: In (10) F B = (c/4kn) {H * × B} is the density of magnetic flow of helicity [14,16,23], (n = √ µ); and it is also proportional to the magnetic part of the spin angular momentum density [14,16,23], (see also Eq. (C-2) below).
Similarly, we may obtain a conservation equation for H R with the electric helicity flow density, F E = (c/4kn)Im{E * × D}. This is done by taking the scalar product: D · (∇ × E * ) = −ik D · B * , and proceeding in an identical way as with the derivation of (10), the result is Adding (11) and (10) one obtains the well-known continuity equation for the conservation of helicity in the steady state: which shows that the flow of helicity, or dual-symmetric spin [16,52]: (13) across the boundary ∂V of V equals the radiated field helicity, including its dissipation and conversion given by the right side of (12) [53][54][55]. However, substracting (11) from (10) leads to the reactive helicity flow theorem, where we have introduced the reactive helicity flow associated to a flow of helicity that vanishes on time-average, although not instantly.
If ∂V is in air or vacuum, n = 1, B = H and D = E in (13) and (15). < S e > and < S m > stand for the time-averages of the density of electric and magnetic spin angular momentum, (cf. Appendix C). The quantity 2ωH R is a reactive helicity per unit halfperiod, or just the reactive helicity power, in analogy with the reactive power of the complex Poynting theorem Eq. (1). Hence (14) expresses the conservation of 2ωH R . Indeed (12) and (14) suggest us to formulate a complex helicity theorem [56]: where the complex helicity flow , or complex spin angular momentum, is F C . Evidently (16) has a real part which is the standard helicity conservation equation (12), whereas its imaginary part is Eq. (14) for the reactive helicity flow and governs the variation of reactive helicity in V , given by the decrease of the integrated source density − 2π kn Im{j * · B}. This variation is expressed in terms of the reactive helicity flux ∂V F H R ·r d 2 r, (which has zero time-average since it comes from {F C }, and hence represents helicity flowing back and forth to the body across ∂V in its near-field region, without net propagation), and of integrated reactive helicity density 2ωH R . Unless otherwise stated, we shall not drag the factor 2ω when we refer to the reactive helicity, thus we shall just write H R for this quantity. In Section VII we show that H R is built-up on chiral light-matter interaction. This gives rise to the phenomenon of reactive dichroism in the near-field of the body, addressed in Section VIII.
Equations (10) and (11) suggest that H R is observable, for example by detecting the torque exerted by a circularly polarized plane wave on a dipolar particle on which the field induces a purely electric (e) or a purely magnetic (m) dipole. In fact, in vacuum F E and F B are proportional to the optical electric and magnetic torque, respectively, m and σ (a) m being the particle electric and magnetic absorption cross sections, (see [58], Section X). Otherwise, if the particle is magnetodielectric, H R becomes observable through the e-m interaction force, Eq. (31) below, [cf. [20] Eq. (44)]. This latter situation is detailed with an evanescent wave in Section IV.B, cf. Eq. (23).
Concerning the P , T , D symmetries of these novel quantities, namely, and we obtain the symmetries for the reactive quantities: It is interesting that while under parity the three reactive quantities behave like their corresponding nonreactive counterparts, they invert their symmetry under duality applications, in contrast with their non-reactive analogues that remain D-invariant. Under time-reversal, only F H R has the same symmetry as its non-reactive correspondant F .

A. The reactive helicity and the angular spectrum
In order to gain more insight into the different nature of H and H R , we shall employ once again the angular spectrum of the electromagnetic wave. We evaluate the total H and H R per unit z-length by integration of E(r) · B * (r) on a plane z = constant. Again, we choose coordinates such that the souces are on z < 0 and thus the integration is done on the z = 0 plane.
Using, as before, the subindex h and e for homogeneous and evanescent components, respectively, the integral on and In (17) Therefore Eq. (17) shows that in the domain of propagating components the helicity density of the field in z = 0 maps into the projection of the electric spin angular momentum of the K-plane wave component onto the propagation wavevectors; namely, onto the real wavevector k h , while in the evanescent region, it is given by the projection of the electric spin onto the transversal (propagating) component K. (Note that this is in agreement with the standard definition [15], but here generalized to include evanescent waves). Of special interest is, however, Eq. (18), which shows that the reactive helicity density H R maps in K-space as the projection of the electric spin of the Kth-evanescent Where k is the complex wavevector of this Kth-evanescent wave, [cf. Eq. (5)]. Again, this justifies that we call reactive to the real part of E(r) · B * (r).
In this connection, it should be remarked that like the reactive power and the IPV are linked with nonpropagating waves, e.g. evanescent and standing waves [3], the reactive helicity H R (and hence its flow F H R ) exist in evanescent waves as shown in Eq. (18), as well as in elliptically polarized (and circularly polarized in particular, CPL) standing wavefields [57]. For CPL waves the authors of [23] used the term "magnetoelectric energy" which, as seen above, is a reactive quantity since it is the same as H R .

IV. CASE 1: REACTIVE POWER, REACTIVE HELICITY, AND REACTIVE MOMENTA IN AN EVANESCENT WAVE
As illustrated in Fig. 1, we consider a generic timeharmonic evanescent wave in air, z ≥ 0, generated by total internal reflection (TIR), at a plane interface z = 0 separating air ( = µ = 1) from a dielectric in the halfspace z ≤ 0. The plane of incidence being OXZ. Then the complex spatial parts of the electric and magnetic vectors in z > 0, are expressed in a Cartesian coordinate basis {x,ŷ,ẑ} as (n = 1) [59,60]: For TE or s (TM or p) -polarization , i.e. E (i) (B (i) ) perpendicular to the plane of incidence OXZ, only those components with the transmission coefficient T ⊥ , (T ) would be chosen in the incident fields [59]. K denotes the component, parallel to the interface, of the wavevector k: A. Reactive power and reactive helicity densities The densities of energy, w = w e + w m , and reactive power, w react = 2ω(w m − w e ), ( = µ = 1), of this wave are according to (2) And the densities of helicity, H , and reactive helicity, B. The reactive energy flow: Reactive momentum, imaginary spin and imaginary orbital momenta The CPV is written as w react ]. (22) For the sake of comprehensiveness, in Appendix C we present the well-known main time-averaged quantities.
Here we concentrate on those reactive less-known and their interralations. The reactive, or imaginary, part of the CPV is Which yields the reactive or imaginary momentum of the field, (sometimes called imaginary Poynting momentum [21,24]), g I = Im{g} = Im{S}/c 2 : whose components come from two vectors that we put forward next: the density of both reactive spin momentum P S and reactive orbital momentum P O : Namely, The electric and magnetic imaginary spin and orbital momenta of (25) are given in Appendix C, Eqs. (C-10)-(C- 14). We emphasize that, as seen in (24)- (26), g I y is fully due to P S y , while g I z comes from P S z + P O z . Note that, interestingly and in contrast with Eq.(C-6), denoting P S I = (1/2)(P S I e + P S I m ) = (0, 0, − q kc w), It should be emphasized that although both Im{S} and F H R have an y-component proportional to K > k, (as well as proportional to H R in the former and to w react in the latter), there is no case of superluminal propagation for these quantities since both are alternating flows with zero time-average. We show below that F z represents up and down flow of reactive helicity in the OZ direction,  (19), with reactive power density given by (20), propagates in the air, parallel to the interface z = 0 separating it from a dielectric in z < 0, with K-vector along OX. Its amplitude decays as exp(−qz). The < S >-vector and spin momentum, < P S >, contained in OXY , have y-components proportional to the wave helicity H , (see Appendix C). The orbital momentum, < P 0 >, proportional to the energy density w points along OX. The reactive momentum 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 magnetodielectric particle placed on the interface. This force is due to the interference of the particle electric and magnetic induced dipoles. The lateral y-force, (Fem)y, and transversal z-component, (Fem)z , due to Im{S}y and Im{S}z, are proportional to the wave reactive helicity, H R , and reactive power density, ∇ · Im{S}, respectively. The imaginary orbital and spin momenta, P O I and P S I , point along the +z and −z-direction, respectively. The flow F H R of reactive helicity is, like Im{S}, in the OY Zplane with y-component proportional to both the reactive power density, wreact, and the magnitude of the K-vector; while its z-component is proportional to the reactive helicity density H R , so that across the OXY -plane (F H R )z holds the imaginary part of the complex helicity theorem, as shown by Eq. (30).
matching with the imaginary part of the complex helicity theorem, Eq. (16).
From the above equations it is important to remark that while the time-averaged energy and helicity densities are linked to < S >, < g >, < S >, < P S > and < P O >, the densities of reactive power, w react , and reactive helicity, H R , are exclusive of Im{S}, g I and F H R .
We show below that the z-component of Im{S}, that matches with the imaginary part of the complex Poynting theorem, is associated with an up and down flow of reactive power in the decay z-direction of the evanescent wave, and hence it is not a net flow of energy. Analogously happens with the z-component of F H R . However, as seen below, both the y and z-components of Im{S} produce detectable optical forces and, hence, make the reactive quantities w react and H R observable.

D. Ractive power conservation law
The z-component of Im{S}, [cf. Eq. (23)], depends on the reactive power density, ∇ · Im{S}, of the evanescent wave in the half-space z > 0, (which actually concentrates in the near field region above the interface z = 0, namely at z << λ), flowing back and forth along OZ at twice the frequency ω, without contributing to a net energy flow since its time-average is zero. I.e. one has from the CPV theorem: Which obviously agrees with (23). Therefore taking (28) into account, the total reactance, [11] of the dielectric-air interface system associated to the evanescent wave is Σ being the area of the XY -plane resulting from the volume integration.

E. Reactive helicity conservation law
Analogously, the z-component of F H R , Eq. (27), depends on the reactive helicity density, ∇ · F H R , of the evanescent wave in the half-space z > 0, concentrated in the near field on z = 0, flowing up and down in the the z-direction without yielding a net flow since its timeaverage is zero. I.e. one has in agreement with Eq. (14), Hence, the z-component of F H R is proportional to the reactive power density.
It is evident that while reactive power and reactive helicity exist in evanescent waves, (see also Eqs. (7) of Section II.A and (18) of Section III.A), and in standing waves [3,23], they do not exist in plane propagating waves, whatever their polarization be. Therefore, reactive helicity, like reactive energy, exists in the near-field region of scattering or emitting objects.
F. Observability of the transversal and perpendicular components of the imaginary momentum g I : Optical forces on a magnetodielectric dipolar particle due to the reactive helicity and reactive power Let a magnetodielectric dipolar particle be placed on the z = 0 interface. An illuminating wavefield, and in particular the evanescent wave, exerts an optical force on it due to the interaction between its induced electric (e) and magnetic (m) dipoles [20], viz., The e and m polarizabilities α e and α m are related with the a 1 and b 1 Mie coefficients of the field scattered by the particle by [20]: The first term of (31), proportional to < g >, has been studied, (see its main features in Appendix C). Here we are interested in the second term that contains the reactive momentum g I .
The y-component < F g I e−m > y due to g I y of (31) was obtained in [21], being considered by the authors "a quite intriguing result" characterized through the second Stokes parameter with a rather small contribution in measurements of < F e−m > y . We have shown above that this force has a reactive origin since it is fully due to the reactive spin y-component, being characterized by the reactive helicity H R . Furthermore, we establish here that < F g I 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 spherical particle placed on the interface z = 0: < F g I e−m > y compared with < F <g> e−m > y . The particle electric and magnetic dipole resonances are at λ e = 492 nm and λ m = 668 nm, respectively, [see Fig.4(left)], and the second Kerker condition (K2) wavelength (at which the particle scatters minimum forward intensity, see Section V.B) is λ K2 = 608 nm. As seen, the magnitude of < F g I e−m > y is much greter than < F <g> e−m > y near λ e and λ m where the latter changes sign, (see the sharp asymptotic values of the ratio between both forces). Thus < F g I e−m > y , which keeps negative, should be detectable near these resonances. Besides, in the proximities of K2, Fig.2(Right) features of the perpendicular force < F g I e−m > z similar to those of its y-component comparing it with the gradient force [60], which choosing T = 0 reads: The superscript R standing for real part. Again the sharp ratio between both near λ = 665 nm indicates the wavelenght zone where < F g I e−m > z , which remains positive, may be detected. Furthermore, the bump of this ratio in the proximities of λ = 520 nm shows that this force is over twice the gradient force.
We conclude thereby that there exists a measurable transverse y-component of < F e−m > due to the reactive spin momentum density and hence to H R , which may be dominant upon the transversal component of < F e−m > stemming from the field (Poynting) momentum, namely from H . Hence, H R is observable. Besides, the normal force < F e−m > z which is exclusively due g I z , (since < g z >= 0), characterized by the reactive power density w react of the evanescent wave has not yet been addressed as far as we know, and may be detected at wavelengths at which, as seen above, clearly exceeds the gradient force, making w react also an observable quantity

V. CASE 2: REACTIVE POWER AND REACTIVE HELICITY FROM A MAGNETODIELECTRIC DIPOLAR SPHERE
We consider a magnetodielectric spherical particle of radius a and volume V 0 , dipolar in the wide sense, namely whose electric and magnetic polarizabilities are given by the first electric and magnetic Mie coefficients, respectively [20,35], in air. We first address the reactive power and stored energy of this magnetoelectric dipole with electric and magnetic moments p and m, respectively. For a wave, E (i) , B (i) , incident on the particle centered at r = 0, the dipolar moments are: p = α e E (i) (0) and m = α m B (i) (0).  Fig.1), is created in z ≥ 0 by TIR at the interface z = 0 of a linearly polarized plane propagating wave of complex amplitudes A ⊥ and A in the dielectric of refractive index n = 1.5 with angle of incidence: 60 o . A Si spherical particle of radius a = 75 nm is deposited on the interface. Its electric and magnetic dipole resonances are at λe = 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, < F g I e−m >y due to g I y , < F <g> e−m >y from < gy >, and ratio: < F g I e−m >y / < F <g> e−m >y whose values (in green) are shown in the right ordinate axis. (Right) Choosing A = 0, A ⊥ = 0: Normal component < F g I e−m >z due to g I z , gradient force < F grad >z, and ratio: < F g I e−m >z / < F grad >z whose values (in green) are shown in the right ordinate axis. Notice that the positive values of the gradient force, repelling the particle from the interface, are due to negative values of α R e and/or α R m . The bump of < F grad >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. The wavelength at which the second Kerker condition holds is λK2 = 608 nm. The force (normalized to area) units are f N/µm 2 for an incident power of 1 mW/mm 2 .

A. Reactive power and stored energy
Concerning the CPV, S (s) = c 8π E (s) × H (s) * , of the emitted fields, (cf. Appendix D), we are interested in its radial component, S (s) ·r, across a spherical surface ∂V of radius r concentric with the particle and enclosing it. Using the fields of Appendix E, this is straightforwardly integrated, yielding Whose real part, ∂V < S (s) > ·r d 2 r = ck 4 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, and corresponds to the wevefield re-radiated up to the farzone, i.e. that with the r −1 dependence, [cf. Appendix D, Eqs. (D-1) and (D-2)].
Here we are particularly interested in the imaginary part, Where W (s) react is the reactive power outside V . It arises from the near and intermediate fields: (We recall that the expression: ick 3r 3 |p| 2 for a purely electric dipolar emitter 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 averaged stored magnetic and electric powers which dominate in the near and intermediate-field regions around the particle, and that do not propagate. To see it, we write the mean stored electric and magnetic energy densities as [7,9,10] Where V ∞ is the volume of a large sphere (kr → ∞).
Making V = V 0 and r = a, (34) yields the reactive power outside the particle. The proof of (34)  As shown in Eq. (3), ∂V Im{S (s) } ·r d 2 r is associated to an instantaneous energy flow alternating back and forth from the scatterer without losses, at frequency 2ω, with zero net energy transport in the embedding vacuum. Nonetheless, this alternating flow builds W (s) react according to Eq. (33), also deduced from Eq. (34). As a consequence, there is an accretion of time-averaged nonpropagating reactive power and stored energy, W (sto) , outside V . (We note a concept analogous to W (sto) for purely electric dipolar RF-antennas, cf. e.g. [3,7,9,10]). Taking V = V 0 , the total energy stored outside the particle is obtained by Appendix E provides the proof of (35), [cf. Eq. (E-4)].
The quality factor associated with Hence being independent of the strength of the electric and/or magnetic dipole moments.
Since in general a dipolar particle in the wide sense cannot be abstracted as a point dipole, the overall interior reactive power, W (s) react,int and the interior stored energy, W (sto,int) , obtained analogously to (34) and (35), but integrating in V 0 the mean energies of the interior field, are also of interest. For a linearly, or circularly, polarized incident plane wave of unit intensity, a straightforward calculation yields (kc/3a 3 )(9/4k 6 )(0.055|d 1 | 2 ∓ 0.018|c 1 | 2 ). The upper and lower sign in ∓ apply to W (s) react,int and W (sto,int) , respectively, while d 1 and c 1 are the first electric and magnetic Mie coefficients of the fields inside the particle, (cf. e.g. Eq.(4.45) of [65]). Notice the appearance of |d 1 | 2 and |c 1 | 2 in W (s) react,int with sign opposite to that of |p| 2 and |m| 2 in (34), i.e. of the first electric and magnetic external Mie coefficient squared moduli, |a 1 | 2 and |b 1 | 2 , in W (s) react ; we shall see that this has consequences for the total (i.e. interior plus external) reactive power at resonant wavelengths.   react,int (λ, a) for Si spheres within the range of wavelengths where they become dipolar magnetodielectric, and for different size radii a, taking advantage of their scaling property with their impact parameter, (see Fig.2 of [35], and [45]). The redshift of the electric and magnetic dipole resonances λ e and λ m as a grows, observed in the scattering crosssection [39,45], [see also W (s) in Fig.3(Center)], is observed in the peaks or dips of these reactive powers as they are much influenced by these resonances. An important feature of these surfaces is that the total reactive power vanishes, changing its sign, close to the resonance wavelengths where W (s) is maximum, irrespective of a. This is detailed in Fig. 4(Center) on a cross sectional plane a = 75nm of these surfaces, depicting the scattered power, along with the external, interior, and total reactive powers. Also Fig.4(Right) shows the stored energies, while in Fig.4(Left) one sees the electric and magnetic external first Mie coefficients a 1 and b 1 with resonant maxima at λ e = 492nm and λ m = 668nm, respectively, and the internal coefficients, d 1 and c 1 .
The vanishing of the total reactive power, W  Fig. 4(Right)] being also resonant near these wavelengths. In fact this matches [30] with knowledge from RF antenna theory according to which a maximum radiation efficiency is sought by minimizing their reactive power and Q-factor [3,[5][6][7][9][10][11], even though in these works the capacitive -electric dipole wavelength λ e : E dominates in r > a, r < λ e -(inductive -magnetic dipole wavelength λ m : B dominates in r > a, r < λ m -) 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 provided by the particle interior through the emergence of a dominant B at λ e (dominant E at λ m ) of W (s) react,int inside the particle, (r < a). By the same token the total stored energies have peaks in these resonant wavelengths λ e and λ m , like the scattered power W (s) [cf. Fig.4(Right)], being evident that the difference of external and internal stored energies: W (sto,Dif ) = W (sto) − W (sto,int) vanishes, like W (s) react,T , close to λ e and λ m . These results illustrate the concepts of reactive and stored power in and around a magnetodielectric, or Huygens, particle when choosing configurations and wavelengths such that the accretion of external reactive power and stored energy in the particle near-field, through the IPV alternating flow, be as large as possible; thus scattering with maximum efficiency W (s) , and possessing the highest possible Q-factor for applications in light-matter interactions, while its total reactive power W (s) react,T vanishes or is near zero at resonant wavelengths. Since, however, these magnetodielectric nanoresonators have rather low Q's (Q ≤ 7), external (and internal) stored power enhancements may be achieved either by sets of such magnetodielectric particles, even metal coated, using them as building blocks of photonic molecules, metasurfaces [66], or in regimes of bound states in the continuum [34,67]. Fig. 4(Left), we observe the lines |a 1 | 2 and |b 1 | 2 crossing each other at the two Kerker wavelengths: λ K1 = 738.5nm and λ K2 = 608nm, which correspond to the first Kerker condition (K1), (zero backscattering, α m = α e , |p| = |m|), and second Kerker condition (K2), (minimum forward scattering, α m = −α * e , |p| |m|), [36,38,39,[43][44][45][46]. One sees in Fig. 4(Middle) that at λ K1 , W (s) react = 0 in accordance with Eqs. (33) and (34) Fig.4(Right) that the stored energies, W (sto) and W (sto,int) , and scattered power, W (s) , are near minimum in the proximities of λ K1 and λ K2 . These features happen in Fig.3 for any a. Therefore, the analysis based on the particle reactive power allows to envisage the Kerker conditions, K1 and K2, from a new standpoint:

Returning to
The two Kerker conditions for a magnetodielectric dipolar particle are those at which the total reactive power is near zero. Namely, the external reactive power W (s) react is either zero (in K1), or close to zero (in K2); while the internal reactive power is near zero both in K1 and K2. In consequence, the reactive power underlies the angular distribution of scattered (or radiated) intensity and, hence, the directivity of the magnetoelectric particle in a way complementary to that formerly addressed in Mietronics and RF-antennas [10,11].
Where C.C. denotes complex-conjugated of the previous term. A straightforward calculation of (36) yields (37) whose real part, H (s) = ∂V Re{F (s) , is the total helicity of the scattered field, (as such, it coincides with Eq. (25) of [16]). Like the scattered power, this helicity does not depend on the distance r.
However the imaginary part of (37), (38) comes from the interference of the intermediate field with r −2 dependence and the far field. H (s) react represents the external reactive helicity, outside V , as detailed in its optical theorem discussed later, [cf. Eq. (52)]. It decreases as r −1 as r grows. Therefore, in analogy with the reactive power, in the near-field (r << λ) the reactive helicity dominates upon H (s) . On making V = V 0 , r = a, Eq.(39) becomes the overall reactive helicity outside the particle. It is of interest to specify the overall reactive helicity of the field inside the particle, H (s) react,int , which is obtained integrating in V 0 the helicity of the internal field: For a left circularly polarized incident plane wave of unit intensity, this quantity is equal to: (8πck 2 /3a)(9/4k 6 )0.234 × 0.276Re{d 1 c * 1 }.  Fig. 1(c) of [55]). This remarks the contribution of the interior reactive helicity to the scattered helicity lineshape, and lays down an intriguing connection with previous studies [53][54][55] which attribute the conversion of helicity to contributions of the scatterer volume and surface.

Figures 5 illustrate H
The total reactive helicity H (s) react,T plays on the scattered helicity a role analogous to that of the total reactive power on the scattered power. As seen in Fig. 6  in Fig.6(Right). We do not know, however, of any analogy of these effects in RF-antenna heory.
Notice that these high index magnetoelectric particles have the interesting property of emitting a wavefield in which there are not very large peaks of the total reactive power [cf. Fig.4(Center)] versus those of the scattered (radiated) power; although, certainly, where this reactive power has extrema the scattered (radiated) power is well aside its peaks. However, under chiral ilumination these fields present high peaks of total reactive helicity versus its radiated one. For instance, [see in Fig.6(Right)], the 75 nm particle] yields a the total reactive helicity with a large dip at 658nm where the scattered (radiated) helicity is near its minimum value. This occurs at shifted positions as a varies, [cf. Fig.6(Left) and Fig.5(Right)].
Therefore, the picture that emerges in these illustrations of such optical nanoantennas, considered as either primary or secondary sources, is that the reactive power and the reactive helicity, which are concentrated both inside and in the near and intermediate regions of the source, have a hampering effect in their far-field scattering (or radiation) efficiency. This is an analogous effect to that due to the presence of reactive power in RFantennas. In consequence, if these nanoantennas emit chiral light, the total reactive helicity hinders the efficiency of far-field scattered (or radiated) helicity, so that less of this helicity is emitted due to a build-up of reactive helicity in and around the nanantenna [68].
As a varies, H (s) react (λ, a) vanishes at the corresponding Kerker wavelength λ K1 ; this is seen in detail in Fig.6(Right) for a = 75nm.
Therefore, the first Kerker condition, K1, also has the novel property that under CPL illumination, the magnetodielectric particle, which then becomes dual and hence emits a wavefield of well-defined helicity equal to the incident one, [15,16,55], does not generate external reactive helicity.

VI. THE REACTIVE POWER OPTICAL THEOREM
Consider a wavefield, E (i) , H (i) incident on a magnetodielectric body of volume V 0 . The field at any point of the embedding medium, (assumed to be vacuum or air), is represented as E = E (i) +E (s) , H = H (i) +H (s) , where the superscript (s) denotes the scattered field. Maxwell's equations are written as: Where P, M and j are the polarization, magnetization, and free current densities in V 0 , respectively. D (i) = E (i) , B (i) = H (i) . We insert Eqs. (40) into Eq.(1) for the total fields E and H, and use the identity: ∇ · (E × H * ) = H * · ∇ × E − E · ∇ × H * , integrating in a volume V that If the incident field has no evanescent components, i.e. it is source-free and propagating, it does not store energy, so that the first term of (41) is identically zero. However if it is evanescent , or it has evanescent components, we shaw in (28) that it stores reactive power and it is given by this term. Therefore we shall keep it in the above equation, which is the complex optical theorem in presence of the scatterer.
If there are no scattering induced sources other than P and M in the body, the free current j conveys the conversion of incident power into mechanical and/or thermal energy through the work done on the charges 1 2 Re V0 d 3 r j * · (E − E (i) ), which accounts for the decrease of energy W (a) from the wave as power absorbed by the body. Then taking the real part of (41) we obtain Which is the standard optical theorem (OT) for energy [20,59], describing the extinction of incident energy, [left side of (42)], and consequent absorption and radiation of the total scattered energy. It reduces to its well-known expression [20] for dipolar particles on making P(r) = p δ(r) and M(r) = m δ(r), and then ∂V d 2 rRe{S (s) } · r = W (s) in accordance with (32).
Here we are, however, interested in the imaginary part of (41), Which becomes Where we have made use of the fact that < w Equation (44) is our formulation of the reactive power optical theorem (ROT) for a generic scatterer whose response to illumination induces densities of polarization P, magnetization M, and free current j, [70]. The left side constitutes the extinction of incident energy which produces the build-up of external reactive power on scattering in the right side of (44). Thus Eq. (44) describes how, in addition to being radiated into the far zone as Eq. (42) illustrates, scattering gives rise to non-radiated energy, stored in V in the form of (external) reactive power, flowing out from the scattering object and returning to it.
For a dipolar particle the ROT reduces to The argument 0 indicates that the fields are evaluated at the particle center r = 0. Concerning a dipolar particle, the process described by the complex optical theorem (41) is analogous to that in which the feeding energy from an alternate current I, induces an oscillating dipole in a small antenna, which emits radiated and stored power through the extinction, 1 2 ZI 2 , of the driving energy. Z being the antenna input impedance, Z = R l + R r − iX. The dipole loss resistance R l and radiation resistance R r [11], generated in accordance with the optical theorem (42), are 2W (a) /|I| 2 and 2W (s) /|I| 2 , respectively. On the other hand, the dipole reactance X (which for a magnetoelectic dipole is either capacitive or inductive [8], depending on the wavelength λ e or λ m ) 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 (R l + R r )|I| 2 and − 1 2 X|I| 2 , respectively. On taking ∂V and V as ∂V ∞ and V ∞ , respectively, corresponding to a large sphere, (kr → ∞), the flux of scattered CPV across ∂V ∞ is real and equals the total scattered energy W (s) . Therefore, Eq. (45) yields In contrast with its real part, the flow (47) depends on the integration domains, V and ∂V . Notice that (47) conincides with Eq. (34).

VII. THE REACTIVE HELICITY OPTICAL THEOREM
Next, we put forward the law which rules the formation of external reactive helicity by scattering in the near and intermediate-field regions of the particle through extinction of helicity of the incident wave. Let us introduce Eqs. (40) into the identities: adding the respective expressions, employing the conservation equation (12) and using the definitions (8) and (13), (n=1 and H = B outside V 0 ), we arrive at which is the known optical theorem for the electromagnetic helicity [16] applying to magnetodielectric arbitrary scattering bodies. Notice that for a dipolar particle, since P(r) = p δ(r), M(r) = m δ(r) and ∂V d 2 rF (s) · n = (8πck 3 /3)Im[p · m * ], Eq. (48) becomes like Eq. (28) of [16]. However our focus is the conservation law of the reactive helicity. To formulate it in the form of an optical theorem we substract, rather than add, the above vector identities, and make use of the conservation law (14) along with definitions (9) and (15). Then, proceeding as before, it is straightforward to obtain Equation (49) is the reactive helicity optical theorem and applies to a generic magnetodielectric scatterer [71]. The left side represents the extinction of helicity of the incident wave on build-up outside the body of a reactive helicity by scattering, given by the right side of (49). Thus, like the energy, the incident helicity gives rise to a reactive one associated to the scattered field, which, in addition to the internal reactive helicity, is stored around the particle, dominating in the near and intermediate-field regions where it flows back and forth from the scatterer. This storage is seen by first considering V to be V ∞ in (49) Therefore, which substituted in (49) Notice that if V = V 0 , Eq. (52) accounts for the reactive helicity stored outside the scattering body. Likewise, if the scatterer is a dipolar particle, the extinction term in the left side of (51) becomes 2πcRe{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 is stored around the particle without being scattered into the far-zone.

VIII. CONSEQUENCE OF THE REACTIVE POWER OPTICAL THEOREM: SIGNIFICANCE OF THE REACTIVE HELICITY IN REACTIVE DICHROISM
In dichroism, chiral light illuminates a chiral particle, molecule, or nanostructure. Assuming it dipolar, its constitutive relations for the induced dipole moments, p and m, and the incident field are The electric, magnetic, and magnetoelectric polarizabilities being α e , α m , α em , and α me ; and fulfilling α em = −α me since the object is chiral [13,73].
However, the ROT establishes that rather than the power radiated in the far-zone (42), one may address the excitation of stored reactive power, which dominates in the near and intermediate-field regions of the particle, which is given by the left side of Eq. (45), viz. W (s) Then using the above pair of illumination fields and Eq. (53), one obtains the discriminatory reactive power So that now the sign − or + applies according to whether the reactive helicity of the illumination is positive: H R + i = H Ri , or negative: H R − i = −H Ri , respectively; and (54) yields a dissymmetry factor proportional to −α R me H Ri [75]. We propose Eq. (54) as the basis of reactive dichroism observations. At difference with standard dichroism, it involves a chiral incident field with non-zero reactive helicity, and constitutes a near-field optics technique.
Therefore, in an analogous way as detecting the radiated energy W (s) ± , (or absorption/extinction energy), in standard dichroism involves the helicity H i of the incident wave, in reactive dichroism, observing the excitation of reactive power W (s) ± react in the chiral particle conveys the incident field reactive helicity H Ri , which evidently comes out on using (54) in a dissymmetry factor defined as g react = 2(W There is a variety of pairs of illumination wavefields that, like in the above illustration, are interchangeable by parity, and that one may employ in experiments. As discussed in previous sections, non-propagating fields in free-space possess a non-zero H Ri ; e.g. elliptically (or circularly, in particular) polarized standing waves, near fields from an emitter, or evanescent and other surface waves, fulfill Eq. (54).

IX. CONCLUSIONS
Given the broad interest of evanescent waves at the nanoscale, and of small particles as light emitting nanoantennas, couplers, and metasurface elements, the contributions of this paper on its reactive quantities is summarized in the following main conclusions: (1) We have established the concepts of complex helicity density and its complex helicity flow, together with their conservation law that we name complex helicity theorem. Its real part is the well-known conservation equation of optical helicity, while its imaginary is a novel law that governs the build-up of reactive helicity through its imaginary, and thus zero time-average, flow. The concept of reactive helicity density unifies that of magnetoelectric energy density, previously introduced by symmetry arguments, and the so-called real helicity. In this way, we put forward its conservation law and observability, thus completing the fundamentals of this quantity.
(2) The conservation of reactive helicity and reactive power, and their zero time-averaged flow, has been illustrated in two paradigmatic systems: an evanescent wave and a wavefield scattered (or emitted) by a dipolar magnetodielectric particle. For the former we have put forward reactive orbital and spin momenta that characterize its imaginary (reactive) field (Poynting) momentum; showing that the wave density of reactive helicity is observable from an experimentally detectectable mixed electric-magnetic transversal optical force exerted by the evanescent wave on a small high refractive index particle (which behaves as magnetodielectric) through this reactive Poynting momentum. On the other hand, we have uncovered a novel non-conservative force in the decay direction of the evanescent wave, which can be discriminated from the gradient one and thus detected, making observable the wavefield reactive power density.
(3) Concerning the stored energy, reactive power, and reactive helicity of the field scattered by a dipolar magnetodielectric particle, we have shown that they provide a novel framework to study the particle emission directivity. This has been illustrated on addressing the two Kerker conditions, K1 of zero backscattering, and K2 of minimum forward scattering. We have established that under CPL incident light, the external reactive power is zero in K1 or close to zero in K2; while the internal reactive power, and hence the total reactive power, is near zero both in K1 and K2. Also, we have proven an additional novel property of the particle at K1 wavelengths, namely, it produces a scattered field with nule overall external reactive helicity.
(4) We have established a reactive helicity optical theorem that governs the build-up and storage of nearfield reactive helicity, on extinction of the incident helicity as light interacts with a generally magnetodielec-tric nanoantenna. Also we have shown that the emission of resonant scattered power and of resonant scattered helicity coincides with a nule, or near zero, total reactive power and helicity, respectively, i.e. those given by the sum of the overall interior and external reactive powers and helicities. Conversely, peaks of total reactive helicity (reactive power) are associated with poorer efficiency in the emission of radiated (or scattered) helicity (power).
(5) A reactive power optical theorem has been put forward. It rules the formation of external reactive power and stored electric and magnetic energies, which dominate in the near and intermediate-field zones of a magnetodielectric particle by extinction of the illuminating energy.
(6) This latter theorem provides a framework to studying the near-field response of chiral nanoparticles to illumination with chiral complex fields. It is remarkable that in the phenomenon of dichroism on illumination with chiral light, the incident reactive helicity arises in the nearfield region, as we have shown, while it is well-known that the incident optical helicity appears from the determination of power emitted in the far-zone. Because of this, we call reactive dichroism the phenomenon by which this incident reactive helicity becomes discriminatory for enantiomeric separation. We propose near-field observation experiments of this reactive phenomenon.
Given the interplay between reactive and radiative quantities of electromagnetic fields, we believe that the concepts studied in this work enrich the landscape of photonics as regards nanoantennas and nanoparticle interactions with light. We expect that the observability of these reactive effects and quantities should form the basis of future experiments and techniques. In this respect, addressing reactive quantities on higher order multipole resonance excitation will be a subject of interest for future studies. This is of special interest for e.g. nanosensing advances [77] and all-dielectric thermonanophotonics [78], to be used in effective biomedical diagnosis and therapies.
Although our analysis has been focused on the nanoscale, these results are equally valid in the microwave range due the scaling property of high index particles as Huygens sources, which remain with the same characteristics of generating large electric and magnetic dipole and multipole resonances as the illumination wavelength increases. The real part of S in (22) is S is well-known to be associated to the optical force on a body. Considering a magnetodielectric dipolar particle placed in the air on the z = 0 interface, the real part, or energy flow density < S >, of the CPV is known to constitute a momentum of the radiation whose x-component, proportional to w, gives rise to an x-force on the particle, separately acting on the particle electric (e) and magnetic (m) induced dipoles [20]. The component < S > y is known to produce a lateral force proportional to −H along OY [21], due to the interference of its e and m dipoles [20].
Since the time-average electromagnetic field momentum density < g >=< S > /c 2 holds < g >=< P O > + < P S >; (C-6) < P O > being the density of time-averaged orbital momentum, one sees from (C-1) and (C-5) that the transverse y-component of < g > comes from < P S y >, which is characterized by H . Both < g y > and < P S y > are (−1/4πc)(kq/K)H exp(−2qz). Therefore the trensverse y-component < g y > of the field momentum is provided by the transverse y-component < P S y > 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 orbital momentum density, < P O i >= (1/2)(< P O e i > + < P O m i >) = (1/2)(1/8πkc)[Im{E * j ∂ i E j } + Im{B * j ∂ i B j }], (i, j = x, y, z), With the electric and magnetic orbital momenta: respectively. < P O > contains the energy density w of the wave and, as such, points in the propagation direction OX of the wave, like both < g x > and the x-component, K > k, of its propagation wavevector. This, in agreement with [21], confers to the evanescent wave a superluminal group velocity; although of course the time-average energy flow < S x > propagates with speed less than c. Hence < P O x > pushes the aforementioned small particle, placed in the air on the interface, as radiation pressure along the x-propagation direction.
On the other hand, the electric and magnetic imaginary momenta of (25) are: (P S I e ) i = 1 8πkc Re{∂ j (E * i E j )} , Re{∂ j (B * i B j )} , Which is (35) when r → a .