Abstract
Considering time-harmonic optical fields, we put forward the complex helicity and its alternating flow, together with their conservation equation: the complex helicity theorem. Its imaginary part constitutes a novel law that rules the build-up of what we establish as the reactive helicity through its zero time-average flow. Its associated reactive flow and the imaginary Poynting momentum that accounts for the accretion of reactive power are illustrated in two paradigmatic systems: evanescent waves and fields scattered from magnetodielectric dipolar nanoparticles. As for the former, we show that its reactive helicity may be experimentally observed as we introduce a reactive spin momentum and a reactive orbital momentum in terms of which we express the imaginary field momentum, whose transversal component produces an optical force on a magnetoelectric particle that, as we illustrate, may surpass and can be discriminated from the known force due to the so-called extraordinary momentum. We also uncover a nonconservative force on such a magnetoelectric particle, acting in the decay direction of the evanescent wave, and that may also be discriminated from the standard gradient force; thus making the reactive power of the wavefield also observable. Concerning the light scattered by magnetoelectric nanoparticles, we establish two optical theorems that govern the accretion of reactive helicity and reactive power on extinction of incident wave helicity and energy. Like a nule total, i.e., internal plus external, reactive power is at the root of a resonant scattered power, we show that a zero total reactive helicity underlies a resonant scattered helicity. These reactive quantities are shown to yield a novel interpretation of the two Kerker conditions, which we demonstrate to be linked to an absence, or minimum, of the overall scattered reactive energy. Further, we demonstrate that the first Kerker condition, under which the particle becomes dual on illumination with circularly polarized light, amounts to a nule overall scattered reactive helicity. Therefore these two reactive quantities are shown to underly the directivity of the particle scattering and emission. In addition, we discover a discriminatory property of the reactive helicity of chiral light incident on a chiral nanoparticle by excitation of the external reactive power. This should be useful for optical near field enantiomeric separation, an effect that we call reactive dichroism.
Highlights
Reactive quantities of electromagnetic fields, such as reactive and stored energy or the imaginary Poynting vector (IPV), are associated to physical entities that do not propagate in the environment, like evanescent waves in their varied forms as surface waves [1,2,3] or standing waves [3], along with near fields of RF antennas [3,4,5,6,7,8,9,10,12,13]
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 polarized standing waves; the former being identical to the socalled magnetoelectric energy density, introduced in Ref. [24] following symmetry arguments but without providing its undelying physical law
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 zero timeaverage, flow
Summary
Reactive quantities of electromagnetic fields, such as reactive and stored energy or the imaginary Poynting vector (IPV), are associated to physical entities that do not propagate in the environment, like evanescent waves in their varied forms as surface waves [1,2,3] or standing waves [3], along with near fields of RF antennas [3,4,5,6,7,8,9,10,12,13]. Its real part is the well-known continuity equation for the conservation of optical helicity [14,15,16,17,19], while its imaginary part is a novel law that describes the build-up of reactive helicity, whose flow has zero time average and 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 socalled magnetoelectric energy density, introduced in Ref. It underlines the observability of such incident reactive helicity, and its discriminatory property for enantiomer separation by near field optical techniques using structured illumination
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