Non-zero helicity extinction in light scattered from achiral (or chiral) small particles located at points of null incident helicity density
NNon-zero helicity extinction in light scattered fromachiral (or chiral) small particles located at pointsof null incident helicity density . Manuel Nieto-Vesperinas [email protected]
December 2016
Abstract.
Based on a recent unified formulation on dichorism and extinction ofhelicity on scattering by a small particle, dipolar in the wide sense, magnetodielectricor not, chiral or achiral, we show that such extinction is enhanced not only atresonances of the polarizabilities, but also due to interference between left andright circularly polarized components of the incident wave, which contributes withappropriate parameters of the illuminating field, even if the particle is achiral and isplaced at points of the incident field at which the local incident helicity density is zero.This phenomenon goes beyond standard circular dichroism (CD), and we analyzeit in detail on account of the values of the several quantities, both of the incident lightand the particle, involved in the process. In addition, this interference produces a termin the helicity extinction that remarkably yields information on the real parts of theelectric and/or magnetic polarizabilities, which are not provided by CD, of which thathelicity extinction phenomenon may be considered a generalization.PACS numbers: 42.25.Ja, 33.55.+b, 78.20.Ek, 75.85.+t
1. Introduction
In recent times, the concept of circular dichroism (CD) [1, 2, 3] has been extendedto the extinction by scattering (or diffraction), transmission, and/or absorption bynanostructures that may or may not be chiral [4, 5, 6, 7, 8], and procedures to enhanceits weak signal from absorbing molecules has been proposed by either enhancing thehelicity of the illuminating field [9], interposing a resonant particle, either chiral orachiral, between the molecule near field and the detecting tip [10, 11, 12]; or reinforcingCD from nanostructures by creating near field hot spots between sets of plasmonic a r X i v : . [ phy s i c s . op ti c s ] M a r on-zero helicity extinction in light scattered from achiral (or chiral) small particles located at points of null incident helicity density V O inclusions [14], or byfostering the interplay between electric and magnetic dipoles of the excited molecule[15]. Also a helicity optical theorem (HOT) has recently been established [16] showingthat dichroism phenomena are particular effects resulting from a fundamental law ofelectrodynamics: the conservation of electromagnetic helicity, [17, 9, 18, 19, 20, 21, 22],also lending sufficient conditions to produce chiral fields by scattering [8], and providinganswers to long-standing questions on the interplay between the chirality of fields andthat of matter [23, 24].The helicity of quasi-monochromatic, i.e. time-harmonic, light waves, which arethose addressed in this paper, is equivalent to their chirality [25]. The latter being aterm employed in [9] and that, having subsequently became of widespread use in theliterature, we shall also consider here. As stated in [16], for these time-harmonic fieldsboth magnitudes just differ by a factor: the square of the wavenumber. However, forgeneral time-dependent fields both quantities have a different physical nature and henceare not equivalent. This distinction being important in matter. Also, as noticed in [18],the helicity has dimensions of angular momentum whereas the chirality does not.In this paper we exploit the equations for the extinction of helicity and energy thatwe established in a previous work, where a unified formulation of helicity extinctionand dichroism beyond the CD concept, was put forward. Hence we now show thatCD may be generalized to 3D polarized fields, for which we introduce a helicityextinction factor g , a particular case of which is the standard CD dissymmetry factor . Inaddition, we further analyze the extinction of helicity on scattering of 3D polarized fieldspossessing a longitudinal component, and whose projection in the plane transversal tothe propagation direction has elliptic polarization, (namely is the sum of a left circular(LCP) and a right circular (RCP) wave). This helicity extinction may be generatednot only, as in CD, by the cross electric-magnetic polarizability that characterizes theparticle chirality, or by the incident helicity density, but also by an interference factorthat mixes the LCP and RCP components; a phenomenon in which the above mentionedcross-polarizability plays no role, and whose existence was already shown in [8]. We shallstudy it in detail here.In this way, we discuss how g assesses the helicity extinction in comparisonwith that of energy. We analyze this under different values of the polarizabilitiesof a particle that we initially assume of rather general characteristics; namely, bi-isotropic, magnetodielectric and chiral, (we shall later relax this latter property) inthe resonant regions of its polarizabilities. In addition we analyze this extinction fordifferent local values of the incident helicity density; also assessing the contibution ofthe aforementioned interference to this helicity extinction, in comparison with that ofthe particle chirality and the incident helicity density, as well as of the polarizabilityresonances, that we have so chosen in this study in order to enhance these effects.Among the illuminating fields whose electric and magnetic vectors fulfill theconditions leading to this interference effect, addressed in Sections 2 and 3, we shall on-zero helicity extinction in light scattered from achiral (or chiral) small particles located at points of null incident helicity density
2. 3D polarized fields with LCP and RCP transversal polarization
We address the spatial parts E and B of the electric and magnetic vectors ofquasimonochromatic fields in their complex representation. Their scattering in amedium of refractive index n = √ (cid:15)µ by a particle, that we generally considermagnetodielectric, chiral and bi-isotropic, dipolar in the wide sense [28, 29], is thuscharacterized by its polarizabilities, that for e.g. a sphere are: α e = i k a , α m = i k b , α em = i k c , α me = i k d = − α em . k = nω/c = 2 πn/λ . Where a , b and c = − d stand for the electric, magnetic, and magnetoelectric first Mie coefficients, respectively[30]. The electric and magnetic dipole moments, p and m , induced in the particle bythis incident field are: p = α e E − α me B , m = α me E + α m B . (1)Based on the angular spectrum decomposition of optical wavefields into LCP (sign: +;the notation of [31] is followed) and RCP (sign: − ) plane wave components that weestablished in [8], (we must remark that we have recently found that this representationwas also reported in [19]), both the incident and the scattered fields may be decomposedinto the sum of an LCP and an RCP 3D wavefield. Then we address incident fields E and B , (which we shall subsequently consider to be optical beams), expressible as thesum of 3D polarized fields whose transversal polarization is LCP and RCP, respectively,thus holding: E ( r ) = E + ( r ) + E − ( r ); B ( r ) = B + ( r ) + B − ( r ) = − ni [ E + ( r ) − E − ( r )]; (2)by which we express the dipolar moments as p ( r ) = p + ( r ) + p − ( r ) , m ( r ) = m + ( r ) + m − ( r ) . (3)with p ± ( r ) = ( α e ± niα me ) E ± ( r ); m ± ( r ) = ( α me ∓ niα m ) E ± ( r ) . (4)The 3D polarized wavefields of Eq. (2) are not just plane waves or transversally polarizedbeams. In a XY Z -Cartesian framework, E and B [cf. Eq. (2)] have, in general, a z -component, while that in the XY -plane is elliptically polarized. As shown below,we illustrate these electromagnetic fields by the sum of two beams propagating along OZ : LCP and RCP, respectively; both circular polarizations holding in the XY -planetransversal to the beam z -axis. In addition, both beams have a Cartesian componentalong OZ . on-zero helicity extinction in light scattered from achiral (or chiral) small particles located at points of null incident helicity density B and E is essential for the effects that we next obtain.This is the reason why we choose, among other possiblities, an illuminating Bessel beamin Section 4.
3. The extinction of incident helicity and energy on scattering. Beyondcircular dichroism
Using a Gaussian system of units, the densities of helicity , H , -or k − times the chirality -,and energy , W , (understood as a time-averaged in this work), of this incident field are: H ( r ) = ( (cid:15)/ k )[ | E + ( r ) | − | E − ( r ) | ] , (5)(see e.g. [8, 19]), and W ( r ) = ( (cid:15)/ π )[ | E + ( r ) | + | E − ( r ) | ] , (6)respectively. In what follows (cid:60) and (cid:61) denote real and imaginary parts, respectively.The HOT that expresses the conservation of helicity is [16]2 πcµ (cid:60){− (cid:15) p · B ∗ + µ m · E ∗ } = 8 πck (cid:15) (cid:61) [ p · m ∗ ] + W a H . (7)The left side of (7) constitutes the extinction of helicity of the incident wave on scatteringwith the particle. This extinction is shown in the right side of (7) to be divided up intothe total helicity scattered or radiated by the object, (i.e. the first term in this right side)and the rate of helicity dissipation W a H (see Eqs. (8), (11) and (12) of [16]), or convertedhelicity , (see Sections 3.2, 3.3 and 4 of [21]), on interaction with the scattering body.As shown by the right side of (7), as the light interacts with the particle suchextinction may convey a selective dissipation of helicity W a H which adds to a resultingtotal helicity of the scattered field. This latter fact agrees with [20, 21].We should recall the analogy of the HOT with the well-known standard opticaltheorem (OT) for energies ω (cid:61) [ p · E ∗ + m · B ∗ ] = ck n [ (cid:15) − | p | + µ | m | ] + W a . (8)The left side of (8) is the energy extinguished from the illuminting field, or rate ofenergy excitation in the scattering object. The first term in the right side constitutesthe total energy scattered by the dipolar object, whereas W a stands for the rate ofenergy absorption by the object from the illuminating wave.On employing Eqs.(1)-(4), the extinction of incident helicity [cf. Eq.(7)]:(2 πc/µ ) (cid:60){− (cid:15) p · B ∗ + µ m · E ∗ } , which henceforth we denote as W ext H , is expressedas [8] W ext H ( r ) ≡ πcµ {(cid:61){ [ p + ( r ) + in m + ( r )] · E + ∗ ( r ) − [ p − ( r ) − in m − ( r )] · E − ∗ ( r ) } +2 (cid:60){ α e − n α m }(cid:61){ E − ( r ) · E + ∗ ( r ) }} on-zero helicity extinction in light scattered from achiral (or chiral) small particles located at points of null incident helicity density
5= 2 πcµ { k(cid:15) (cid:61){ α e + n α m }H ( r ) + 16 π (cid:114) µ(cid:15) (cid:60){ α me }W ( r )+2 (cid:60){ α e − n α m }(cid:61){ E − ( r ) · E + ∗ ( r ) }} ; (9)whereas Eqs.(1)-(4) yield for the extinction of incident energy [cf.Eq.(8)]: ( ω/ (cid:61) [ p · E ∗ + m · B ∗ ], which we write as W ext [8], (a slightly different notation is used for thisquantity in [8]): W ext ( r ) ≡ ω {(cid:61){ [ p + ( r ) + in m + ( r )] · E + ∗ ( r )+[ p − ( r ) − in m − ( r )] · E − ∗ ( r ) } +2 (cid:61) ( α e − n α m ) (cid:60){ E − ( r ) · E + ∗ ( r ) }} = ω { π(cid:15) (cid:61){ α e + n α m }W ( r ) + 4 k (cid:114) µ(cid:15) (cid:60){ α me }H ( r )+2 (cid:61){ α e − n α m }(cid:60){ E − ( r ) · E + ∗ ( r ) }} . (10)In (9) and (10) r is the position vector of the center of the particle immersed inthe illuminating wavefield. Eqs. (9) and (10) are fundamental as they establish theconnection of the extinction of helicity W ext H and energy W ext of the incident wavewith the densities of incident helicity H and energy W , and with the chirality of thedipolar particle, characterized by α me . They remarkably show how the incident H and W contribute to W ext H and W ext with their roles exchanged with respect to thepolarizability factors in the corresponding term where they appear.Notice from the right side of (7) that W ext H contains both the total scattered helicityand the incident helicity dissipation (or conversion). Similarly, from (8) one sees that W ext contains the total scattered energy as well as the dissipation of incident energy inthe particle. In particular, if both W a H and W a are zero, W ext H and W ext represent thetotal scattered helicity and energy, respectively.Since we are here interested in the rate of extinction of helicity, we observe in (9)that W ext H , apart from being due to the incident helicity density H coupled with thedissipative part of the electric and magnetic polarizabilities, is generated by a couplingof the incident energy density W with the particle chirality through (cid:60){ α me } . Moreover,of special importance is that, as shown by the third term (cid:60){ α e − n α m }(cid:61){ E − · E + ∗ } in Eq. (9), placing the small particle at a position r in the illuminating wave, anincident field with no helicity density at r may give rise to an extinction rate of helicityon interaction with the particle, not only -as well-known- due to the particle chiralitythrough the term with (cid:60){ α me }W , but also, and this is the new feature addressed in thiswork, because of the interference coupling factor E − · E + ∗ . I.e. a non-zero W ext H willbe generated at r even if the incident helicity H ( r ) = 0 and the particle is not chiral( α me = 0). Moreover, as (cid:60){ α e } and (cid:60){ α m } are usually larger than their imaginarycounterparts at non-resonant λ , this interference term acquires special importance formolecules [8, 16].In this respect, and in contrast with the above argument for W ext H if H = 0, there isno analogous reasoning for a non-zero energy extinction W ext , Eq. (10), if the incident on-zero helicity extinction in light scattered from achiral (or chiral) small particles located at points of null incident helicity density W is null, since this would convey that E − = E + = H = 0 andthus W ext = 0, as it should.However, in (9) (cid:60){ α e } and n (cid:60){ α m } appear in a substraction, and thus competewith each other in their contribution to the last term of (9), which becomes zero when α e = n α m , namely when the particle is dual [16].Notice that similar arguments exist for (cid:61){ α e } and (cid:61){ α m } in connection with thethird term of the energy extinction, Eq. (10). It is also worth remarking that when { E − · E + ∗ } = 0, Eqs. (9) and (10) reduce to those standard of the HOT and the OT,respectively, (see e.g. Section V of [16]).Eqs. (9) and (10) govern a generalized dichroism phenomenon and hence accountfor CD as a particular case. Namely, by defining the ratio g = W ext H / W ext , (11)which we shall name the helicity extinction factor , it is straightforward to see that eitherwhen the particle is dual, or when the interference terms of these equations vanishlike e.g. for elliptically polarized plane waves, (for which E − · E + ∗ = 0 since then nolongitudinal z -component exists), then choosing as usually done: | E + | = | E − | , one has g = (cid:112) (cid:15)/µ λ g CD , where g CD = 2( W ext + −W ext − ) / ( W ext + + W ext + ) is the the well-known dissymmetry factor of standard CD which from (10) results in the well-known expressionin terms of the particle polarizabilities [1, 2, 3, 9]: g CD = 4 n (cid:60){ α me } / ( (cid:61){ α e } + n (cid:61){ α m } ).Hence the CD phenomenon is one of the several consequences of the HOT and thusof the conservation of electromagnetic helicity. Namely, while standard CD is observedby illuminating the particle, or structure, with a LCP plane wave only, and separatelywith a RCP one; subsequently substracting the corresponding scattered energies as: (cid:112) (cid:15)/µ ( W ext + − W ext − ); CD may identically be observed on a unique illumination bya wave of the form (2) with no longitudinal component along the OZ -propagationdirection, (e.g. a plane wave), linearly polarized in the (transversal) XY - plane, (orgenerally with | E + | = | E − | ), therefore whose LCP and RCP components do notinterfere with each other; ie. E − · E + ∗ = 0. The extinction of helicity, normalized to thewavelength λ , is identical to the above mentioned difference of LCP and RCP energies.We should remark that the HOT also account for the illumination of anobject with those so-called superchiral fields produced by the superposition of twocounterpropagating CPL plane waves of amplitudes E and E of opposite helicityas put forward in [9], (which on the basis of recent studies [23, 24] we prefer to callfields enhancing the dissymmetry factor). However it is known [32] that this methodis limited to particles -or molecules- with α m (cid:39)
0, because in such configuration g CD = 4 (cid:15) (cid:60){ α me } / [ (cid:61){ α e } ( E − E ) / ( E + E ) + n (cid:61){ α m } ( E + E ) / ( E − E )]. Sothat when (cid:61){ α m } = 0 the usually extremely small dissymmetry factor of standardCD, (often as small as 10 − for molecules), may be enhanced, as seen from this latterexpression of g CD , just by choosing E (cid:39) E , as proposed in [9], (or by making E (cid:39) − E when (cid:61){ α e } = 0); but it is evident that these choices of E and E cannot enhance g CD if both (cid:61){ α e } and (cid:61){ α m } are non-zero. on-zero helicity extinction in light scattered from achiral (or chiral) small particles located at points of null incident helicity density (cid:60){ α e − n α m }(cid:61){ E − ( r ) · E + ∗ ( r ) } of (9) is the only one among those expressing the extinction of helicity, Eq. (9), that provides information onthe real parts of the polarizabilities . No other term of (9) contain such information. Infact, there is a well-known lack of this information in the aforementioned dissymmetryfactor of standard CD, addressed above. Therefore, this paper shows how creatingexperimental conditions for this interference factor to exist, provides a source ofinformation on (cid:60){ α e } and (cid:60){ α m } through the extinction of helicity, Eq. (9), andits associated extinction factor g .
4. Illustration with a Bessel beam
The contribution of 2 (cid:60){ α e − n α m }(cid:61){ E − · E + ∗ } in (9) to the helicity extinction, while (cid:60){ E − · E + ∗ } = 0 in (10), thus increasing the helicity extinction factor g , is nextillustrated with an incident beam propagating along OZ , elliptically polarized in the XY -plane, and with longitudinal component along the z -propagation direction [8]. Inthis case ∂ z (cid:39) ik z ; the wavevector being written in Cartesian components as k = ( K , k z ); K = (cid:112) k x + k y , k z = √ k − K ). The electric vector is expressed in terms of the vectorpotential A ± [26, 27] as: E ± = ik z A ± + ik z ∇ ( ∇ · A ± ) . (12) A ± ( r ) = 1 ik z (ˆ x ± i ˆ y ) u ( r ) e ik z z . (13) u ( r ) = u ± ( R, z ) e ilφ . R = (cid:112) x + y . (14)So that using ∇ ( ∇ · A ± ) (cid:39) ik z ( ∇ · A ± )ˆ z one has E ± = e ik z z [(ˆ x ± i ˆ y ) u + i ˆ z k z ( ∂ x u ± i∂ y u )] . (15) B ± = ∓ ni E ± . (16)Which fulfills both ∇ · E ± = 0 and Eq. (2). We shall address the Bessel functionof integer order: u ± ( R, z ) = e ± J l ( KR ), ( e ± being constant amplitudes) which, from(12) - (16) and after a calculation using the recurrence relation: J l − ( x ) + J l +1 ( x ) =(2 l/x ) J l ( x ), leads to the form (2) for a Bessel beam, whose components E ± are LCPand RCP, respectively, in the XY -plane transversal to its z -direction of propagation.Viz.: E ± ( r ) = e ± e i ( k z z + lφ ) [ J l ( KR )(ˆ x ± i ˆ y ) ∓ iKk z exp( ± iφ ) J l ± ( KR )ˆ z ] . (17)Eq. (17) coincides with those of [26, 27, 33] characterizing Bessel beams. We havenevertheless undertaken here the derivation of this kind of beams from a first basis inorder to guarantee that this field fulfils the important condition (2), [(see Eqs. (15) and(16)]. on-zero helicity extinction in light scattered from achiral (or chiral) small particles located at points of null incident helicity density | E + ( r ) | ± | E − ( r ) | = 2( | e +0 | ± | e − | ) J l ( KR )+ K k z [ J l +1 ( KR ) | e +0 | ± | e − | J l − ( KR )] . (18)On the other hand, the factor (cid:60) ( (cid:61) ) { E − · E + ∗ } reduces to the contribution of the field z -component: (cid:60) ( (cid:61) ) { E − · E + ∗ } = (cid:60) ( (cid:61) ) { E − z · E + ∗ z } = − K k z J l − ( KR ) J l +1 ( KR ) (cid:60) ( (cid:61) ) { e − e + ∗ e − iφ } . (19)Therefore, either of these quantities, (cid:60) [ · ] or (cid:61) [ · ], may be made arbitrarily small (orzero) depending on the choice of parameters e − and e +0 for the beam in the factor (cid:60) ( (cid:61) ) { e − e + ∗ e − iφ } . Since according to [33], (see Fig. 6 in this reference), this beamrotates a particle of diameter 1 µm to 6 µm placed in the inner ring of maximum intensityin about 16 s per revolution, we shall assume the signal detection time large enoughfor the azimuthal angle φ of the particle center position r not to contribute to thisfactor, so that we just consider the quantity (cid:60) ( (cid:61) ) { e − e + ∗ } . Hence choosing for example e − /e +0 = ± a exp( ibπ/ a and b being real, the value of (cid:60) ( (cid:61) ) { E − · E + ∗ } will oscillateabout zero as cos( bπ/
2) (sin( bπ/ r of the particle in the beam, andthus of the argument KR , one will have in Eqs. (9) and (10) the third terms, whose (cid:60) ( (cid:61) ) { E − · E + ∗ } factor is given by (19), comparable, or not, to the first and secondterms whose ( | E + ( r ) | ± | E − ( r ) | ) factor is (18). This is seen observing the factor( K /k z ) J l − ( KR ) J l +1 ( KR ) in (19) which may be made either much larger or smallerthan the term of J l ( KR ) which is the dominant contribution to (18).This latter important fact will be seen in Section 5 by choosing two differentpositions of the particle in the beam, i.e. two distinct values of KR .
5. Example: Enhancements in the extinction of helicity on scattering witha resonant particle, aither chiral or not
To better illustrate these effects we address them at resonant wavelengths, so that thereis field enhancement on interaction with the particle, which in principle we considergenerally magnetodielectric and chiral. We shall later relax the latter property. Wehave found in the recent work [34] a particle model with these characteristics, and thuswe consider it useful for our illustration. Its linear dimension is not larger than 204 nm.(See details of this particle, made of a composite metal (silver)-dielectric in vacuum, n = 1, in Fig. 3 of [34]). Both helicity dissipation, or conversion, W a H , and energyabsorption W a , are susceptible of taking place, as previously emphasized concerningthe right sides of (7) and (8). However, as stated before, in this paper we are interestedin the left sides of those two optical theorems, and hence on the extinctions W ext H and W ext , Eqs. (9) and (10), respectively. on-zero helicity extinction in light scattered from achiral (or chiral) small particles located at points of null incident helicity density λ = 1 . µ m, as shown in Fig. 1,where we have fitted them from their numerical values, obtained in [34], to functions of λ ; this enabling us to straightforwardly employ them in Eqs. (9) and (10) . We choose K = 0 . k , and set l = 1, e +0 = 1, e − = i , i.e. (cid:60){ e − e + ∗ } = 0, hence (cid:60){ E − · E + ∗ } = 0,and so is the third term of (10) for W ext ; this allows to enhancing the value of g even ifthe incident helicity density H ( r ) = ( (cid:15)/ k )[ | E + | − | E − | ] is very small and the particlewere achiral, as shown below. - - - - - Ralfa_ee x( )Ralfa_mm x( )Ialfa_me x( ) 1.571.47 x λ (μm) - - - - - Ialfa_ee x( )Ialfa_mm x( )Ralfa_me x( ) 1.571.47 x λ (μm)
7 5.25 3.5 1.75 0 -1.75 -3.5 -5.25 -7
Re α e Re α m Im α me X -2 (μm )
10 7.5 5 2.5 0 -2.5 -5 -7.5 -10
Im α e Im α m Re α me X -2 (μm ) Figure 1. (Color online). Real and imaginary parts of the polarizabilities α e , α m and α me of an example of chiral particle near the resonant wavelength λ = 1 . µ m. Thesequantities are functionally fitted from those of Fig. 4 of [34]. At the different wavelengths, the radial coordinate R of the particle center in thebeam transversal section is adjusted to two alternative values of KR . One is KR (cid:39) . J l ( KR ) = J ( KR ) , (and thus the contributionof this factor to Eqs. (9) and (10) through the first term in the right side of (18) isnegligible). The other alternative value is KR (cid:39) .
25, which is close to the first zero of J l − ( KR ) = J ( KR ), (and hence by virtue of (19) the contribution of the (cid:61){ E − · E + ∗ } factor in (9) is negligible). The latter is like the situation of standard CD.These values of KR also give a hint on the range R of approximate distancesbetween minima of the beam intensity across its section, versus the size of the particle. R (cid:39) . λ/ π = 895 nm for λ (cid:39) . µm , which is well above the linear size of theparticle, which as said above is no larger than 204 nm ; and thus allows enough spatialresolution of its position, since this size is well below the width R of the circles ofintensity minima and maxima in the beam section, (see also [33]).Hence, at KR = 2 .
25 one has that | E + | + | E − | = 8 π W dominates over all other on-zero helicity extinction in light scattered from achiral (or chiral) small particles located at points of null incident helicity density . a.u. ) while (cid:61){ E − · E + ∗ } = − . (cid:60){ E − · E + ∗ } = 0,and | E + | − | E − | = 0 .
09. On the other hand, for KR = 3 .
75 one sees that | E + | + | E − | no longer dominates since it is about 0 . a.u. ), while (cid:61){ E − · E + ∗ } = − . | E + | − | E − | = 0 .
008 and (cid:60){ E − · E + ∗ } = 0. Therefore these two choices of kR conveya very small incident helicity H .Fig. 2 exhibits the spectra of the rate of helicity extinction W ext H , of energyextinction W ext , (both scaled by 10 / πc ), and helicity extinction factor g = W ext H / W ext ,for KR = 3 .
75 (upper graph) and KR = 2 .
25 (lower graph) for a chiral particle withpolarizabilities seen in Fig. 1. We also show the same scaled quantities, now denotedas W extnχ H , W extnχ , and g nχ , for an almost achiral particle, ( α me (cid:39) α e and α m as the former chiral one, but whose cross electric-magneticpolarizability α me has been somewhat artificially scaled to 1 /
10 of the α me values ofthe chiral particle. (We choose the letter χ in the subindex from the Greek χ(cid:15)ιρ for”hand”).As seen from Eq. (9) and Fig. 2 (above), for KR = 3 .
75, even when the particleis achiral and the incident helicity density is locally zero, namely at points R fullfiling KR = 3 . we confirm that the interference factor (cid:61){ E − · E + ∗ } may be essential toyield an appreciable helicity extinction rate W extnχ H and a resonant helicity extinctionfactor | g nχ | >
1, ( g nχ = − .
15 at λ (cid:39) . µm in this illustration). This is one of themain results of this work, and is in contrast with standard circular dichroism in which E − · E + ∗ = 0, and objects with zero, or a purely imaginary α me , with no selectivehelicity dissipation, would produce no helicity extinction and therefore a zero value of g in absence of incident chirality density.In this respect we remark that the appreciable helicity extinction factors g and g nχ observed in Fig. 2 (above), may also be influenced by shifts, (which depend on theparticle morphology), between the resonant peaks of the helicity and energy extinctionrates.The results of Fig. 2 (above) should be compared with those when the factor (cid:61){ E − · E + ∗ } is negligible, and so is the third term of Eq. (9). These are plotted in Fig.2 (below) for KR = 2 .
25, showing that W extnχ H is extremely small compared to W extnχ ,and hence g nχ is almost zero, ( | g nχ | ≤ . W ext H and g shown in Fig. 2 (below) for this KR = 2 .
25 when the particle is chiral, i.e. (cid:60){ α me } (cid:54) = 0, and thus in Eq. (9) the second term contributes, leading to significantlylarger peaks of these quantities, ( g (cid:39) − . λ (cid:39) . µm ), as in standard dichroism.On the other hand, both Figs. 2 (above and below) show that at a chosen valueof KR , W ext and W extnχ coincide with each other; i.e. at a given position of theparticle within the beam, W ext is not affected by the value of α me . This is due tothe above shown almost negligible | E + | − | E − | , and hence small H , for these chosen KR . Nonetheless when KR = 2 . W ext , W extnχ and W ext H considerably increase throughthe factor W in (9) and (10). This is expected from the discussion in Section 4 and inthis Section 5 above, since when KR = 2 .
25 the factor | E + | + | E − | = 8 π W dominatesover all other of Eqs. (9) and (10), while (cid:60) ( (cid:61) ) { E − · E + ∗ } remains very small, again on-zero helicity extinction in light scattered from achiral (or chiral) small particles located at points of null incident helicity density gggggg λ (μm)
3 2.25 1.5 0.75 0 -0.75 -1.5 -2.25 -3 W H W s g Wn q H W nq s g nq gggggg λ (μm ) W H ext W ext g W n χ H ext W n χ ext g n χ Figure 2. (Color online). (10 / πc ) W ext H , (10 / πc ) W ext , and g for the chiral particleof Figs. 1; as well as (10 / πc ) W extnχ H , (10 / πc ) W extnχ , and g nχ when that particle ismade achiral, ( α me = 0). Above: KR = 3 .
75. Below: KR = 2 . on-zero helicity extinction in light scattered from achiral (or chiral) small particles located at points of null incident helicity density K /k z ) J l − ( KR ) J l +1 ( KR ) in (19) is much smaller than the firstterm proportional to J l ( KR ) in (18) when KR = 2 .
25. Namely, we stress that in (9)and (10) (cid:60) ( (cid:61) ) { E − · E + ∗ } , Eq. (19), can be either comparable or much smaller than W ,Eq. (18), according to the value of KR .Another consequence of this latter discussion is that the relative values: g/ W ext and g nχ / W extnχ are significantly larger in Fig. 2 (above) than in Fig. 2 (below). Thisonce again highlights the relevance of the interference term with factor (cid:60) ( (cid:61) ) { E − · E + ∗ } in (9) and (10) when the terms proportional to W do not dominate.Finally, it should be reminded that, as stated in Section 3, (cid:60){ α e } and n (cid:60){ α m } appear substractacted from each other in the last term of (9). Consequently, andalthough not shown here for brevity, we observe that the amplitude of the peaksof both W ext H and W extnχ H increses as either (cid:60){ α m } or (cid:60){ α e } diminishes. Somethinganalogous occurs with the extinction of energy (10) as regards the imaginary parts ofthe polarizabilities
6. Conclusions
The concept of circular dichroism has been extended by addressing the rate of extinctionof helicity W ext H , whose extinction factor g has been introduced and generalizes thestandard CD dissymmetry factor. The parameter g monitors the rate of helicityextinction versus that of energy under different values of the polarizabilities of a generallymagnetodielectric particle, either chiral or not, (i.e. for the cross electric-magnetic one α me ranging from large to almost zero); also considering the local value of the incidenthelicity H . Thus both W ext H and g assess the contibution of the remarkable interferencefactor (cid:61){ E − · E + ∗ } to such helicity extinction in comparison with that of α me , H , andthe resonances of the polarizabilities that we addressed in this study in order to enhancethese effects. Notice in passing that an analogous analysis may be made with the factor (cid:60){ E − · E + ∗ } versus W , α me , and H , as regards its contribution to the energy extinctionrate W ext .When the incident fields are optical beams with LCP and RCP transversalcomponents, the factor { E − · E + ∗ } reduces to that of interference of the longitudinalcomponents. We have illustrated this with a Bessel beam. Interestingly, due to thisinterference, helicity extinction does not necessarily involve neither particle chiralitynor a non-zero local value of the incident helicity density; i.e. for α me = 0 and givenparameters of the illuminating beam, one may find positions r of the particle in thebeam where this local helicity density is H ( r ) = 0 while the aforementioned interferenceterm gives rise to a non-zero extinction of helicity W ext H . Also, and importantly, thisinterference phenomenon is mediated by (cid:60){ α e } and (cid:60){ α m } thus yielding a source ofinformation on these latter quantities, which was not provided by standard CD .Finally, although we have studied these phenomena in general bi-isotropic dipolarparticles, namely those magnetodielectric and chiral, the contribution of the 2 (cid:60){ α e − n α m }(cid:61){ E − · E + ∗ } term to an extinction of incident helicity, Eqs. (7) and (9), [as on-zero helicity extinction in light scattered from achiral (or chiral) small particles located at points of null incident helicity density (cid:61){ α e − n α m }(cid:60){ E − · E + ∗ } term to an extinction of incidentenergy, Eqs. (8) and (10)], may also be observed in purely electric ( α m = 0) or magnetic( α e = 0) particles. In this context, of special importance will be further research andobservation of these effects in high index dielectric particles, that possess ramarkablyunique optically induced electric and magnetic dipole resonances [35, 36] and that somuch interest are generating as low-loss elements of an increasingly active new area ofmicro and nano-optics [37, 38]. Acknowledgments
Work supported by MINECO, grants FIS2012-36113-C03-03, FIS2014-55563-REDC andFIS2015-69295-C3-1-P. The author thanks an anonymous referee for many interestingcomments that contributed to improve this report.
References [1] J. A. Schellman, ” Circular Dichroism and Optical Rotation”, Chem. Rev. , 323-331 (1975).[2] D. P. Craig, and T. Thirunamachandran, Molecular Quantum electrodynamics: An Introductionto Radiation Molecule Interactions , Dover, New York, 1998.[3] L. D. Barron,
Molecular Light Scattering and Optical Activity , Cambridge University Press,Cambridge, 2004.[4] C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tnnermann, T. Pertsch, and F. Lederer,”Asymmetric Transmission of Linearly Polarized Light at Optical Metamaterials”, Phys. Rev.Lett. , 253902 (2010).[5] A. O. Govorov, Z. Fan, P. Hernandez, J. M. Slocik and R. R. Naik, ”Theory of Circular Dichroism ofNanomaterials Comprising Chiral Molecules and Nanocrystals: Plasmon Enhancement, DipoleInteractions, and Dielectric Effects”, Nano Lett. , 13741382 (2010).[6] X. Zambrana-Puyalto, X. Vidal and G. Molina-Terriza, ”Angular momentum-induced circulardichroism in non-chiral nanostructures”, Nature Comm. , 137-143 (2016).[8] M. Nieto-Vesperinas, ”Chiral optical fields: A unified formulation of helicity scatteredfrom particles and dichroism enhancement ”, Phil. Trans. Roy. Soc. Lond. A, (in press).arXiv:1609.07889v1 (2016).[9] Y. Tang and A. E. Cohen , ”Optical Chirality and Its Interaction with Matter”, Phys. Rev. Lett. , 163901 (2010).[10] D. V. Guzatov and V.V. Klimov, ”The influence of chiral spherical particles on the radiation ofoptically active molecules”, New J. Phys. , 123009 1-19 (2012).[11] A. Garcia-Etxarri and J. A. Dionne, ”Surface-enhanced circular dichroism spectroscopy mediatedby nonchiral nanoantennas”, Phys. Rev. B , 235409 (2013).[12] H. Alaeian, and J. A .Dionne, ”Controlling electric, magnetic, and chiral dipolar emission withPT-symmetric potentials”, Phys. Rev. B , 245108 1-8 (2015).[13] H. Wang, Z. Li, H. Zhang, P. Wang and S. Wen, ”Giant local circular dichroism within anasymmetric plasmonic nanoparticle trimer ”, Sci. Rep. , 8207 (2015).[14] T. T. Lv, Y. X. Li, H. F. Ma, Z. Zhu, Z. P. Li, C. Y. Guan, J. H. Shi, H. Zhang and T. J. Cui,”Hybrid metamaterial switching for manipulating chirality based on VO2 phase transition”, Sci.Rep. , 23086 (2016).[15] L. Hu, X. Tian, Y. Huang, X. Wang and Y. Fang, ”Quantitatively analyzing the mechanism of on-zero helicity extinction in light scattered from achiral (or chiral) small particles located at points of null incident helicity density giant circular dichroism in extrinsic plasmonic chiral nanostructures by the interplay of electricand magnetic dipoles”, Nanoscale , 3720-3728 (2016).[16] M. Nieto-Vesperinas, ”Optical theorem for the conservation of electromagnetic helicity:Significance for molecular energy transfer and enantiomeric discrimination by circulardichroism”, Phys. Rev. A , 023813 1-8 (2015).[17] D.M. Lipkin, ”Existence of a New Conservation Law in Electromagnetic Theory”, J. Math. Phys. , 696-700 (1964).[18] R. P Cameron, S.M Barnett, and A. M. Yao, ”Optical helicity, optical spin and related quantitiesin electromagnetic theory”, New. J. Phys. , 053050 1-16 (2012).[19] K. Y. Bliokh, and F. Nori, ”Characterizing optical chirality”, Phys. Rev. A. , 106-110 (2015).[25] S. M. Barnett, R. P. Cameron and A. M. Yao, ”Duplex symmetry and its relation to theconservation of optical helicity”, Phys. Rev. A , 013845 (2012).[26] S. M. Barnett and L. Allen, ”Orbital angular momentum and nonparaxial light beams”, OpticsComm. , 670-678 (1994).[27] L. Allen, M. J. Padgett and M. Babiker, ”The orbital angular momentum of light”. In Prog. Opt. , E. Wolf, ed., (Elsevier, Amsterdam, 1999), pp. 291-372.[28] I. Sersic, M. A. van de Haar, F. B. Arango, and A. F. Koenderink, ”Ubiquity of optical activityin planar metamaterial scatterers”, Phys. Rev. Lett. , 3021-3024(2015).[30] C. F. Bohren and D.R. Huffman, Absorption and Scattering of Light by Small Particles , J. Wiley,New York, 1983.[31] J. D. Jackson,
Classical Electrodynamics , 3rd edition, John Wiley, New York, 1998.[32] J. S. Choi and M. Cho, ”Limitations of a superchiral field”, Phys. Rev. A , 063834 1-22 (2012).[33] K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt and K. Dholakia, ”Orbital angularmomentum of a high-order Bessel light beam” J. Opt. B: Quantum Semiclass. Opt. S82- S89(2002).[34] D. E. Fernandes and M. G. Silveirinha, ”Single beam optical conveyor belt for chiral particles”,Phys. Rev. Applied , 014016 (2016).[35] A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Lukyanchuk, B. N. Chichkov, ”Optical responsefeatures of Si-nanoparticle arrays”. Phys. Rev. B 82 , 045404 (2010).[36] A. Garcia-Etxarri, R. Gomez-Medina, L.S. Froufe-Perez, C. Lopez, L. Chantada, F. Scheffold, J.Aizpurua, M. Nieto-Vesperinas and J.J. Saenz, ”Strong magnetic response of submicron siliconparticles in the infrared”. Opt. Express , 4815-4826 (2011).[37] M. Decker and I. Staude, ”Resonant dielectric nanostructures: a low-loss platform for functionalnanophotonics”, J. Opt. , 103001 (2016).[38] A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, B. Lukyanchuk,”Optically resonant dielectric nanostructures”, Science354