Circular dichroism induced by Fano resonances in planar chiral oligomers
Ben Hopkins, Alexander N. Poddubny, Andrey E. Miroshnichenko, Yuri S. Kivshar
CCircular dichroism induced by Fano resonances in planar chiral oligomers
Ben Hopkins, Alexander N. Poddubny,
2, 3
Andrey E. Miroshnichenko, and Yuri S. Kivshar
1, 3 Nonlinear Physics Centre, Australian National University, Canberra, ACT 2601, Australia ∗ Ioffe Physical-Technical Institute of the Russian Academy of Sciences, St. Petersburg 194021, Russia National Research University for Information Technology,Mechanics and Optics, St. Petersburg 197101, Russia
We present a general theory of circular dichroism in planar chiral nanostructures with rotationalsymmetry. It is demonstrated, analytically, that the handedness of the incident field’s polarizationcan control whether a nanostructure induces either absorption or scattering losses, even when thetotal optical loss (extinction) is polarization-independent. We show that this effect is a consequenceof modal interference so that strong circular dichroism in absorption and scattering can be engineeredby combining Fano resonances with planar chiral nanoparticle clusters.
I. INTRODUCTION
The difference in absorption of left- and right-handedcircularly-polarized (LCP and RCP) light by chiral struc-tures has long been utilized for applications in molec-ular chemistry [1, 2], pharmaceuticals [3, 4], and op-tics [5, 6]. More recently, advances in nanotechnologyhave resulted in new types of chiral nanoantennas, col-loidal nanoparticles, and metamaterials; nanostructuredsystems which provide unprecedented freedom to pro-duce chiral responses that substantially exceed thoseof conventional materials. [7–13] It is, however, planar nanostructures that have the least restrictions in design,particularly when considering fabrication, and, therefore,they are more suitable to capitalize on design freedomand produce the most effectual chiral responses. Yet,such two-dimensional nanostructures cannot be truly chi-ral because they have an inherent plane of mirror sym-metry. In combination with reciprocity, this is knownto prevent any difference in the total optical losses un-der LCP or RCP incident fields – a difference known as circular dichroism . Here, however, we consider separatecontributions from the radiative and dissipative compo-nents of a system’s total loss. As we show, these com-ponents of the total loss, the scattering and absorptioncross-sections, are not constrained by reciprocity and cansubsequently exhibit circular dichroism in planar chiralgeometries and metasurfaces (see Fig. 1). Here we fo-cus on circular dichroism in the absorption cross-section.This definition is warranted from a practical perspectiveas, for instance, the absorption cross-section will encom-pass nonlinearity [14], photocurrent generation [15], heat-ing [16], fluorescence [17] and photocatalysis [18]; quan-tities that are both measurable and foreseeably useful.The effect of chirality on scattering and absorption isgenerally understood by how it impacts either the far-field scattering matrices or the effective medium param-eters of a given structure [19, 20]. However, these two ap-proaches have been tailored (historically) towards char-acterizing the chiral properties of conventional materials ∗ [email protected] and do not provide information on what is physically oc-curring in the system. This means that it can be highlynontrivial to deduce how a general change in geometrywill effect the far-field scattering matrices or effective pa-rameters. In an attempt to address this, some recentworks have considered the properties of the near-field ofcertain chiral structures to guide design empirically [21]or attempted to describe the response of chiral arrange-ments of nanorod dimers analytically [22, 23]. Note here,the relation between the chiral response and the geome-try is ultimately rooted in the current distribution thatis induced by the given incident field. Specifically, thesecurrents will encapsulate the complete optical responseof any system in both the near- and far-field regions,and should thereby provide a deeper understanding ofthe observed optical responses. The approach we presenthere will be to simplify the analysis of structures usinggeometric symmetry rather than analytical approxima-tions. In this regard, there have been significant workson planar chiral metasurfaces and metamaterials whoseconstituent meta-atoms have a discrete rotational sym-metry [24, 25]. A number of key advantages of uti-lizing rotational symmetry in applications that requirepolarization-selective operation have further been pro-posed. In particular: invariance of scattering and absorp-tion loss to all linear polarizations[26] [27], zero trans-mitted circular cross polarization [28] and others [29–32].Therefore, while imposing symmetry conditions necessar-ily restricts the range of chiral nanostructures that can beconsidered for a given application, the remaining subsetof nanostructures (which are both chiral and rotationally-symmetric at the same time) are desirable and have beenjust as effective at exhibiting chiral optical responses. Wehave subsequently focused here on symmetries that arerotationally symmetric and lack any mirror plane paral-lel to the rotational symmetry axis. We do not considergeometries that have a mirror plane parallel to their rota-tional symmetry axis, such as in the C nv and D nh pointgroups. This is because such mirror symmetry opera-tions are able to transform an LCP plane wave into anRCP plane wave (and vice versa ), thereby leaving the ge-ometry unchanged when flipping the handedness of theexcitation. This would subsequently make the optical a r X i v : . [ phy s i c s . op ti c s ] O c t LCPRCP
Scattering lossesAbsorption losses (a)(b)
FIG. 1. A schematic of the circular dichroism in absorption observed for planar structures excited by a normally incident planewave. The total energy loss (extinction) by a planar chiral structure is conserved due to reciprocity. But, the particular lossmechanism can be swapped between (a) scattering and (b) absorption with the handedness of the incident field polarization. responses to LCP and RCP plane waves symmetrically-equivalent. We therefore consider the following symme-try groups: C n , D n , S n and C nh (see Fig. 2). Whilethe C n and D n point groups are chiral, it is importantto acknowledge that S n and C nh point groups are, infact, achiral ; a result of their symmetry under improperrotation and planar mirror reflection operations, respec-tively. We focus on the C nh point group in the maintext, corresponding to geometries that are chiral in onlytwo dimensions, a property which is often referred to as planar chiral . We also consider only the incident wavespropagating along the symmetry axis. In this paper, we C S D C FIG. 2. Examples of geometries that have discrete rotationalsymmetry, specifically (clockwise the top left): cyclic group C n , dihedral group D n , cyclic group with a horizontal mirrorsymmetry C nh and symmetric group S n . These examples arefor n = 4, however our approach is applicable to any geometrywhere n ≥ demonstrate how symmetry considerations can allow us to analytically extricate meaningful information from thecurrents induced in C nh structures without calculatingthe explicit current distributions. From this foundation,we are able to link the origin of non-reciprocal circulardichroism in the absorption cross-section to modal inter-ference and Fano resonances. II. RECIPROCITY AND CHIRALITY
To provide broadly-applicable arguments about theinduced currents in any geometry, we will work fromfrequency-domain Maxwell’s equations, in the absence ofmagnetization and magnetic sources for simplicity. Letour scattering structure be some localized distribution ofa material having a permittivity not equal to the back-ground permittivity ( (cid:15) ), in which a current distribution( J ) is induced by an externally-applied field ( E ). Wecan then express the total electric field within the mate-rial in terms of the externally-applied field and the fieldradiated by the induced current distribution [33] E ( x ) = E ( x ) − iω (cid:90) V ¯ G ( x , x (cid:48) ) J ( x (cid:48) ) d x (cid:48) , (1)the volume V is all space and ¯ G is a generalized dyadicGreen’s function, written in terms of the source dyadic( ¯L ) and the free space Green’s function ¯ G ( x , x (cid:48) ) = ω µ P . V . (cid:20) ¯I + 1 k ∇∇ (cid:21) e ik | x − x (cid:48) | π | x − x (cid:48) | − ¯L δ ( x − x (cid:48) ) (cid:15) . (2)where ω is the angular frequency of light and P.V. impliesa principle value exclusion of x = x (cid:48) when performing theintegration in Eq. 1. For ease of notation, we can also de-fine a tensor permittivity ( ¯ (cid:15) ) in terms of the conductivity( ¯ σ ) and electric susceptibility ( ¯ χ ) to relate the inducedcurrent to the total electric field ¯ (cid:15) ≡ ( ¯ χ + 1) (cid:15) − ¯ σ iω ⇒ J ( x ) = − iω [ ¯ (cid:15) ( x ) − (cid:15) ] E ( x ) . (3)As we are only considering x in the volume ( V s ) withnon-background permittivity, we can rewrite Eq. 1 as anintegral equation for the induced current iω E ( x ) = − [ ¯ (cid:15) ( x ) − (cid:15) ] − J ( x ) + (cid:90) V s ¯ G ( x , x (cid:48) ) J ( x (cid:48) ) d x (cid:48) . (4)This integral equation is applicable to the fields ofany arbitrary structure in the absence of magnetization.Now, for our purposes, Eq. 4 has an associated eigenmodeequation, which has solutions v i that satisfy iωλ i v i ( x ) = − [ ¯ (cid:15) ( x ) − (cid:15) ] − v i ( x ) + (cid:90) V s ¯ G ( x , x (cid:48) ) v i ( x (cid:48) ) d x (cid:48) . (5)These eigenmodes can be calculated numerically for sim-ple systems based on discrete dipole approximation [34],but also for arbitrary continuous structures [35]. Nowwe can apply symmetry analysis to the structure un-der consideration. Each irreducible representation of thestructure’s highest-order symmetry group will describethe transformation properties of a distinct set of eigen-modes. In the absence of accidental degeneracies, eachof these eigenmodes will have a unique eigenvalue with adegeneracy level equal to the dimension of the associatedirreducible representation [36]. For our purposes we areconsidering the C n , S n , D n and C nh symmetry groups,and, specifically, the two-dimensional E irreducible rep-resentations, which describe the transformation proper-ties of any normally-incident planewave ( E in Eq. 4)under the given group’s symmetry operations. Becauseof this, only the eigenmodes which transform accordingto these E irreducible representations can be excited. Itis now important to notice here that we are working ina complex space. It implies that the two-dimensional E irreducible representations of the C n , S n and C nh groupsare written as two, one-dimensional irreducible represen-tations that are complex conjugates of each other (forexample, see Table I). In regard to notation, we will referto these two, one-dimensional irreducible representationsas E + and E − (i.e. where E + ∗ = E − ).Taking into account that E + and E − are different ir-reducible representations, each will describe the transfor-mation properties of a distinct set of eigenmodes, whichwe refer here to as { v + } and { v - } , respectively. From thegeometrical symmetry perspective, the eigenmodes asso-ciated with different irreducible representations shouldnot be degenerate [36]. However, such eigenmodes canbecome degenerate in some systems where additional C E C C A E (cid:26) φφ φ φ φ = e πi/ TABLE I. The character table of the 3-fold rotational ( C )symmetry group. Note here, the two-dimensional E represen-tation is made up of two, one-dimensional irreducible repre-sentations that are complex conjugates of each other. symmetries are present. One of the key symmetries ofelectromagnetic theory that is neglected in a purely ge-ometric argument is Lorentz reciprocity[37]. In the fol-lowing we will prove that every eigenmode in E + willalways be degenerate with one eigenmode of the asso-ciated E − (and vice versa ), purely due to the inherentLorentz reciprocity of the Maxwell’s equations.We start by taking an eigenmode ( v + i ), which trans-forms according to a one-dimensional E + irreducible rep-resentation. We also impose the standard normalizationcondition (cid:90) V s v + i ∗ · v + i d V = 1 , (6)where the dot denotes a vector dot product. Impor-tantly, if v + i transforms under symmetry operations ac-cording to E + , we know that v + i ∗ must transform accord-ing to the complex conjugate irreducible representation, E − , because any operator describing a geometric sym-metry operation in a Euclidean space will be real. Wecan therefore write v + i ∗ as some linear combination ofthe eigenmodes, { v - } , which transform according to the E − irreducible representation v + i ∗ = (cid:88) j b ij v - j b ij ∈ C . (7)By substituting Eq. 7 into Eq. 6 we can rewrite thenormalization of v + i as (cid:88) j b ij (cid:18)(cid:90) V s v - j · v + i d V (cid:19) = 1 . (8)For Eq. 8 to hold we know that there must be at leastone j such that (cid:90) V s v - j · v + i d V (cid:54) = 0 . (9)We now consider the role of the Lorentz reciprocity,a consequence of which is that both the dyadic Greensfunction and the permittivity tensor must be symmetric,albeit complex and not necessarily Hermitian ¯ G ( x , x (cid:48) ) = ¯ G ( x (cid:48) , x ) , ¯ G = ¯ G T , ¯ (cid:15) = ¯ (cid:15) T . (10)Due to this symmetry, it is possible to write the overalloperator of the eigenvalue equation (Eq. 5) as a matrixin the normal form shown by Gantmacher [38]. A re-sult of this is that nondegenerate eigenmodes must beorthogonal under unconjugated projections [39] (cid:90) V s v α · v β d V = 0 when λ α (cid:54) = λ β . (11)Hence Eq. 9 can be true only for some v + i is if there ex-ists an eigenmode v - j that is degenerate with v + i . More-over, we must always be able to find at least one v - j thatsatisfies Eq. 9 for each v + i . However, suppose now thatmultiple v - k satisfied Eq. 9 for the one v + i ; this wouldmean that each of these v - k are degenerate with everyother v - k . Such unenforced degeneracies between eigen-modes are accidental degeneracies and can be excludedfrom the general consideration. We can therefore ex-pect each v - j to be degenerate with a single v + i (and viceversa ). For ease of notation, we will use the subscriptconvention that v + i is degenerate with v - i . We can nowcombine Eqs. 6 and 7 to write the following relation b ij (cid:90) V s v + i · v - j d V = δ ij , (12)where δ ij is a Kronecker delta function. This com-pletes the proof, we have shown that Lorentz reciprocityforces degeneracy between pairs of eigenmodes of com-plex conjugate irreducible representations in any sym-metry group. For our purposes, we will specifically iden-tify the E irreducible representations of C n , S n and C nh symmetry groups ( n ≥
3) as being subject to this ar-gument (the degeneracy already exists in D n from ge-ometry alone). This result is applicable to all associatedsymmetric scattering systems irrespective of their specificdimensions or constituent materials.To emphasize the significance of this degeneracy, wehave to relate the eigenmodes to the applied fields thatexcite them. The behavior of the E + and E − irreduciblerepresentations in C n , S n and C nh symmetry groups isthat symmetric rotations are described by uniform phaseshifts that are complex conjugates of each other. Atnormal incidence, it is the RCP and LCP plane wavesthat behave in this way and they can be, therefore, as-signed to E + and E − , respectively. As such, we can de-fine the eigenmodes { v + i } to correspond to those excitedby an RCP plane wave and { v - } to those excitable byan LCP plane wave. Therefore the degeneracy we havejust proven means that the modes that can be excitedby LCP and RCP plane waves are degenerate. As such,changing the polarization of the incident plane wave be-tween LCP and RCP cannot result in the excitation ofmodes that were previously forbidden by symmetry (darkmodes). Moreover, the degeneracy of left- and right-handed eigenmodes highlights an important conceptualpoint on why the handedness of the incident field is ableto affect optical response. While LCP and RCP planewaves are enantiomers, the degenerate eigenmodes they respectively excite cannot themselves be enantiomers be-cause the geometry is not conserved under reflections.This permits nontrivial differences to exist in the distri-bution of eigenmodes that are excited by LCP and RCPplane waves. In other words, circular dichroism and otherchiral scattering effects will be due to the varying magni-tudes and phases of the excitations of degenerate eigen-modes. To now derive physically meaningful informationfrom the current distributions, we need to calculate theextinction and absorption cross-sections to define circu-lar dichroism. These two cross-sections can be calculatedfrom current distributions similar to point-dipole systemsin Ref. [40] by defining an infinite number of point dipoles { p } , where each dipole moment is defined for some in-finitesimal volume d V over which J can be considered asconstant p x = − iω (cid:90) d V J ( x (cid:48) )d x (cid:48) = − iω J ( x )d V . (13)Using the definition for induced current in Eq. 3 we canfurther define the polarizability of these dipoles as ¯ α x = [ ¯ (cid:15) ( x ) − (cid:15) ] d V . (14)The extinction can then be determined from the pro-jection of the complex conjugated incident field onto theinduced current σ ext = k(cid:15) | E | Im (cid:40)(cid:88) x E ∗ · p x (cid:41) = 1 | E | (cid:114) µ (cid:15) Re (cid:26)(cid:90) V s E ∗ · J d V (cid:27) . (15)Similarly, we can express the absorption in terms of theintensity of the induced currents by keeping only the low-est order of d Vσ abs = − k(cid:15) | E | (cid:88) x p ∗ x · (cid:18) Im (cid:8) ¯ α − x (cid:9) + k π (cid:19) · p x = − ωc(cid:15) | E | (cid:90) V s J ∗ · Im (cid:110) ( ¯ (cid:15) ( x ) − (cid:15) ) − (cid:111) · J d V . (16)Hence, it is straightforward now to calculate the circu-lar dichroism in extinction and absorption cross-sectionsfrom the currents that are induced by circularly-polarizedplane waves with opposite handedness. Importantly, dueto the optical theorem, the extinction cross-section de-pends only on the forward scattering amplitude whereasthe scattering and absorption cross-sections depend onthe far-field scattering at all angles (or, equivalently, onthe full structure of the near-field). This already suggeststhat the scattering and absorption cross-sections will bemore sensitive to the polarization in the structures withreduced symmetry, a hypothesis which we will confirm inthe coming arguments.
III. PLANAR CHIRAL OLIGOMERS
Circular dichroism is traditionally defined as a differ-ence in optical loss between LCP and RCP plane wavespropagating in the same direction. However, given thefreedom of two polarizations and two propagation di-rections, there are four distinct excitations that can beapplied by circularly-polarized plane waves. These fourcircularly-polarized plane waves be can represented interms of their polarization depicted spatially along thepropagation direction, where each of them resembles ei-ther one of two oppositely-handed helices (seen in Fig. 3).The remaining distinction is then the direction such a‘polarization helix’ rotates in time. This depiction of anincident field corresponds to the spatial distribution ofthe applied electric field, which is what we use in our in-tegral equation approach for the induced currents (Eq. 4).In the expressions for the applied electric field, seen asinsets in Fig. 3(a)-(d), we use the notation of two super-script plus and/or minus signs to indicate the sign of theimaginary unit in the polarization vector and exponen-tial, respectively. This notation highlights which planewaves are complex conjugates (reciprocal), e.g. E ( ±∓ ) is the complex conjugate of E ( ∓± ) .The reason circular dichroism is defined for one prop-agation direction only ( i.e. two, not four, excitations) isreciprocity, which equates the extinction cross-section ofoppositely propagating plane waves and, thus, the differ-ence in extinction cross-section between LCP and RCPplane waves does not depend on the propagation direc-tion [41]. This invariance between reciprocal plane wavescan be observed in the electric field distributions ( E ( x ))that are complex conjugates of each other. So, if we thenstart from the earlier analysis of the induced currents instructures with C n , D n , S n or C nh symmetry, we areable to express the current ( J ) induced by a circularly-polarized plane wave ( E ) in terms of eigenmodes E = (cid:88) i a i λ i v + i ⇒ J = (cid:88) i a i v + i . (17)Using Eq. 7, we can also define the current ( J (cid:48) ) inducedby the complex conjugate field ( E ∗ ) E ∗ = (cid:88) i,j a ∗ i λ ∗ i b ij v - j ⇒ J (cid:48) = (cid:88) i,j a ∗ i b ij λ ∗ i λ j v - j . (18)This shows that the overall excitation of each degenerateeigenmode pair is not necessarily the same between E and E ∗ . However, despite the change in eigenmode exci-tation, we can use the result in Eq. 12 with Eqs. 17 and18 to equate the extinction (Eq. 15) of complex conjugateapplied fields (as expected) (cid:90) V s E ∗ · J d V = (cid:90) V s E · J (cid:48) d V . (19)This, therefore, demonstrates that the total excitationof each nondegenerate eigenmode can change between re-ciprocal plane-waves while the extinction is conserved. The conservation of extinction means a structure with aplanar reflection symmetry (such as in C nh symmetry)cannot exhibit circular dichroism in extinction. Rigor-ously, applying a symmetric reflection operator (ˆ σ h ) tothe global reference frame of the generic system in Eq. 17will not change the scatterer’s geometry by definition, butthe applied field and the induced current become ˆ σ h E and ˆ σ h J when expressed in terms of the new referenceframe. Given that it is a unitary operation, the ˆ σ h op-erator then cancels when evaluating the extinction forˆ σ h E and ˆ σ h J , leaving the extinction unchanged (cid:90) V s (ˆ σ h E ) ∗ · (ˆ σ h J ) d V = (cid:90) V s E ∗ · J d V . (20)As the reflection operator changes the propagation di-rection of the incident field while leaving the polarizationunchanged, ˆ σ h E corresponds to an oppositely-handedpolarization helix to that of E ( cf. Fig. 3(a) and (b),and Fig. 3(c) and (d)). Therefore Eq. 20, in conjunc-tion with Eq. 19, means that all four excitations from acircularly-polarized plane-wave will produce the same ex-tinction cross-section and, subsequently, this shows thatno circular dichroism can occur in extinction for struc-tures with C nh symmetry. Our argument also holds forstructures with S n symmetry if we substitute the ˆ σ h re-flection operation for the ˆ S n improper rotation operation.As such, we have actually shown that neither C nh or S n symmetries permit circular dichroism in extinction. Morespecifically, we demonstrated that this conserved extinc-tion does not require there to be a trivial difference be-tween the currents induced by reciprocal plane-waves.Up to this point, the direction that an applied fieldrotates in time has been discounted because of the reci-procity condition in Eq. 19. However it is worth acknowl-edging that this result is only applicable to extinctionand, subsequently, in the presence of material losses, reci-procity does not constrain the absorption and scatteringcross-sections independently. Therefore, by introducingmaterial losses to produce an absorption cross-section,neither the absorption or scattering cross-section are nec-essarily invariant under reciprocal plane-waves. This isof particular interest for planar chiral scattering geome-tries, such as those with C nh symmetry, which we haveshown cannot produce circular dichroism in the extinc-tion cross-section due to the combination of reciprocityand planar reflection symmetry. Indeed, the combina-tion of polarization independent extinction and varyingabsorption requires that a resonance in absorption for onecircular polarization must coincide with a suppression inthe scattering cross-section to balance the conserved ex-tinction.To consider the origin of such an effect, we will beginby proving that the total induced current intensity canonly change between reciprocal excitations if the struc-ture’s eigenmodes are nonorthogonal. Specifically, if weassume that the scattering structure’s eigenmodes are or-thogonal to each other, we can define the excitations ofeach eigenmode by reciprocal excitations explicitly using (a) (b)(c) (d) FIG. 3. Schematic illustration of the electric field excitations that can be exerted by a circularly-polarized plane wave givenfour different combinations of propagation direction and polarization sign. complex projections E = (cid:88) i (cid:18)(cid:90) V s v + i ∗ · E d V (cid:19) v + i ⇒ J = (cid:88) i (cid:18)(cid:90) V s v + i ∗ · E d V (cid:19) v + i λ i , (21) E ∗ = (cid:88) i (cid:18)(cid:90) V s v - ∗ i · E ∗ d V (cid:19) v - i ⇒ J (cid:48) = (cid:88) i (cid:18)(cid:90) V s v - ∗ i · E ∗ d V (cid:19) v - i λ i . (22)Similarly we can also define b ij in Eq. 7 as b ij = (cid:90) V s v - j ∗ · v + i ∗ d V = (cid:18)(cid:90) V s v - j · v + i d V (cid:19) ∗ . (23)Notably, from Eq. 11, this means b ij is zero if i (cid:54) = j .Then we can relate the excitations of each eigenmode,under complex conjugate incident fields, by substitutingEq. 23 into Eq. 7 (cid:18)(cid:90) V s v + i ∗ · E d V (cid:19) ∗ = (cid:18)(cid:90) V s v + i · v - i d V (cid:19) (cid:90) V s v - ∗ i · E ∗ d V . (24)At the same time, Eq. 23, in conjunction with Eq. 12,means that the real projection of one eigenmode onto itsdegenerate partner has unit magnitude (cid:18)(cid:90) V s v - i · v + i d V (cid:19) − = (cid:18)(cid:90) V s v - i · v + i d V (cid:19) ∗ . (25) To combine this all, we start from the definitions of J and J (cid:48) in Eqs. 21 and 22, then use the relations in Eqs. 24and 25, in addition to our original assumption that alleigenmodes are orthogonal, to get the result that (cid:90) V s J (cid:48)∗ · J (cid:48) d V = (cid:90) V s J ∗ · J d V . (26)For geometries made of isotropic materials, this meansthat circular dichroism in absorption (Eq. 16) ceases toexist between complex conjugate excitations. In otherwords, we have proven that absorption circular dichro-ism can only exist between reciprocal plane-waves if suchgeometries have nonorthogonal eigenmodes. This pecu-liar requirement means circular dichroism in absorptioncan be considered as an interference effect in the samesense that Fano resonances are. Specifically, in Ref. [34],it was shown that the nonzero projections of one eigen-mode onto another eigenmode are able to fully describeFano resonances. We can now establish a link betweencircular dichroism in absorption and this eigenmode over-lap by relating it to the b ij coefficient using Eqs. 7 and12 (cid:90) v + i ∗ · v + j d V = b ij (cid:90) v - j · v + j d V . (27)This shows us that b ij is proportional to the overlap be-tween the nondegenerate eigenmodes v + i and v + j . If wethen refer to Eqs. 17 and 18, we can see that the excita-tion of nondegenerate eigenmodes should vary more be-tween reciprocal excitations given the presence of a large b ij . In other words, we should observe significant circu-lar dichroism in absorption and scattering cross-sectionswhen there is a large eigenmode overlap. Moreover, alarge eigenmode overlap is known to exist, quite promi-nently, at a Fano resonance. So, to produce circulardichroism in absorption, we can take a structure knownto produce Fano resonances and alter it to be planar chi-ral. In Fig. 4 we begin with a gold heptamer which isknown to support Fano resonances [42] and then alter thenanoparticles in the outer ring to make it both planar chi-ral and having C nh symmetry. The choice of parameterswas chosen so that the central particle resonance is over-lapped with that of the outer ring [43]. As the outer ringin Fig 4 is a planar chiral structure, it will experience dif-ferent current distributions in response to LCP and RCPplane waves (see Eqs. 17 and 18), but in isolation it doesnot experience significant circular dichroism. Howeverthe collective structure exhibits a Fano resonance, whichleads to significant circular dichroism in absorption andscattering. The extent of the circular dichroism is, infact, sufficient to swap the dominance of scattering andabsorption cross-sections using polarization. This there-fore supports our derivation that nonreciprocal circulardichroism in absorption is an interference effect.In Fig. 4, all nanoparticles are made from 20 nmthick gold, the central disk has a diameter of 140nmand each triangular nanoparticle has major and minoraxes of 100 nm and 60 nm (respectively). The triangu-lar nanoparticles have been placed at a radius of 170 nmaway from the center of the disk and the major axis isoriented 65 ◦ off the radial vector. LCP and RCP are de-fined relative to a vector pointing out of the page. Allsimulations were performed using CST Microwave Studioand gold data was taken from Johnson and Christy [45].We also demonstrate that the effect is robust and is notdependent on precise parameters. In Fig. 5(a) and (b),we plot the difference between LCP and RCP absorp-tion cross-sections when varying both the diameter of theouter ring of triangular nanoparticles and their angularorientation. It can be seen that the significant splittingbetween absorption from LCP and RCP plane waves, ob-served first in Fig. 4, is evident for a wide range of par-ticle arrangements. Furthermore, in Fig. 5(c), we showthat the same effect will occur even when changing thenumber of constituent nanoparticles that make up theoligomer. This subsequently shows that circular dichro-ism in absorption is a robust and widely achievable fea-ture of geometries that produce modal interference.To investigate the associated physical dependencies ofcircular dichroism in the absorption cross-section, weneed to identify a physical characteristic that causesnonorthogonal eigenmodes. This is in fact quite straight-forward if we neglect the retardation of coupling withinthe structure, which can be done either resorting to thequasistatic approximation ( k →
0) or by working in thenear-field limit ( k | x − x (cid:48) | →
0) for small systems. Inthese cases, the free space Green’s function becomes en-tirely real. So we can take the complex conjugate of theeigenmode equation of our system (Eq. 5) to get the fol- lowing equation iωη i v + i ∗ ( x ) = − ( ¯ (cid:15) ( x ) − (cid:15) ) − v + i ∗ ( x ) + (cid:90) V s ¯ G ( x , x (cid:48) ) v + i ∗ ( x (cid:48) ) d x (cid:48) , (28)where η i = 1 iω (cid:104) ( ¯ (cid:15) ∗ ( x ) − (cid:15) ∗ ) − − ( ¯ (cid:15) ( x ) − (cid:15) ) − (cid:105) − λ ∗ i . (29)Notably, the complex conjugate of any eigenmode ( v + i ∗ )will be an eigenmode in its own right if there is only oneuniform and isotropic material in the structure ( i.e. if ¯ (cid:15) ( x ) → (cid:15) ). Then, given v + i ∗ is an eigenmode, we canutilize Eqs. 6 and 7 to get the result that b ij = δ ij and v - i = v + i ∗ . (30)From Eq. 11, this is sufficient to ensure that the eigen-modes are orthogonal. So, if a structure is made of asingle isotropic material, nonorthogonal eigenmodes canonly exist if there is retarded coupling within the struc-ture. This means, for instance, that dichroism betweenreciprocal, circularly-polarized plane-waves in absorptiondoes not occur for a single-material structure in the qua-sistatic approximation. However, if one wants to workin the quasistatic regime and still observe the circulardichroism in absorption, the nonorthogonality of eigen-modes can instead be the result of having anisotropic orinhomogeneous materials ( i.e. even if we neglect retar-dation). This can be seen from the fact that any per-mittivity distribution ¯ (cid:15) ( x ) represents the geometry andtherefore has to be invariant under symmetry operations.So, if we were to define an incident field as E (cid:48) ( x ) = − [ ¯ (cid:15) ( x ) − (cid:15) ] − v + i ( x ) . (31)then this incident field would transform under symmetryoperations according to v + i ( x ) and we could subsequentlyexpress it as some linear combination of the eigenmodes { v + } − [ ¯ (cid:15) ( x ) − (cid:15) ] − v + i ( x ) = (cid:88) j u ij v + j ( x ) . (32)Provided that v + i is not, by chance, an eigenmode of( ¯ (cid:15) ( x ) − (cid:15) ) − , there has to be at least one k (cid:54) = i such that u ik is nonzero. Referring back to our original eigenmodeequation (Eq. 5), this means that the Green’s functionintegral of v + i must produce equal and opposite compo-nents of each v + k to counterbalance the v + k componentscreated by the permittivity distribution. Physically thismeans that inhomogeneous or anisotropic, lossy mate-rials induce coupling between eigenmodes. To demon-strate that this makes at least one pair of eigenmodesnonorthogonal, we make the assumption that all eigen-modes are orthogonal. Then, substituting Eq. 32 into += Cross Sections (um ) Chirality Density
LCP (a)(b)(c) (d)
RCP
FIG. 4. Simulations demonstrating the role of interference for inducing circular dichroism in the absorption cross-section ofa planar chiral heptamer with C h symmetry. We observe the creation of significant circular dichroism in absorption in thevicinity of the Fano resonance, as is predicted given it corresponds to high modal overlap. Additionally, on the right hand side,we show that this circular dichroism can also be observed in the magnitude of the near-field chirality density[44] at the Fanoresonance. Here the chirality density induced by an RCP plane-wave is less than half of that for the LCP case. The calculationparameters are given in text. Eq. 5, requires that there is a nonzero projection between v + k and the scattered field of v + i (cid:90) V s v + k ∗ ( x ) · (cid:18)(cid:90) V s ¯ G ( x , x (cid:48) ) v + i ( x (cid:48) ) d x (cid:48) (cid:19) d x (cid:54) = 0 . (33)We can then consider each v + i as a linear combination ofthe eigenmodes { g } (with eigenvalues { γ } ) of the Green’sfunction integral v + i ( x ) = (cid:88) j f ij g j ( x ) ⇒ (cid:90) V s ¯ G ( x , x (cid:48) ) v + i ( x (cid:48) ) d x (cid:48) = (cid:88) j γ j f ij g j ( x ) . (34) When neglecting retardation, ¯ G becomes real symmet-ric and, hence, the eigenmodes { g } are orthogonal andeach coefficient f ij is the complex projection of g j onto v + i . The orthogonality of { g } in Eq. 33 then requiresthat there is at least one eigenmode g l that exists in thelinear combinations for both v + i and v + k . Thus, we canexplicitly write the complex inner product between v + i and v + k using their respective decompositions into theeigenmodes { g } (cid:90) V s v + i ∗ · v k d V = (cid:88) l f ∗ il f kl (where f il , f kl (cid:54) = 0) . (35)As such, the projection between eigenmodes will gener-ally be nonzero. For instance, nonorthogonality would beguaranteed if there was only a single shared eigenmode( g l ) in both v + i and v + k . So, our original assumption, thatall eigenmodes in { v + i } are orthogonal, is broken and wehave therefore shown that at least one pair of nondegen-erate eigenmodes will be nonorthogonal if the geometryconsists of inhomogeneous and/or anisotropic materials.This also builds on our earlier result (Eq. 26) that circulardichroism in geometries made of isotropic materials canonly occur when there are nonorthogonal eigenmodes, be-cause the exception to that condition was the presence ofanisotropic materials, which we have just shown leads tononorthogonal eigenmodes independently. Therefore weknow that nonorthogonal eigenmodes are always neces-sary for nonreciprocal circular dichroism in absorption.Furthermore, we have derived that such nonorthogonaleigenmodes are a result of either retardation of couplingbetween currents in the structure or from anisotropic orinhomogeneous materials.It is finally important to acknowledge that small dif-ferences in absorption and scattering have recently beenobserved numerically for a chiral oligomer with planar re-flection symmetry [46]. Additionally, recent experimentalobservations have also reported a difference in heat gen-eration from a planar chiral structure under excitationby LCP and RCP light [47]. These investigations sup-port our derivations on the circular dichroism presentedin this pape, but the observed difference in absorptionwas hypothesized to be a consequence of polarization-dependent near fields, which is not necessarily sufficentto produce a difference in the total absorption as shownin [27, 30]. It is nonetheless worth acknowledging thatan alternate option is to measure circular dichroism inscattering and absorption from the far-field. As demon-strated in Ref. [48], the scattering and absorption crosssections can be experimentally measured from far-fieldlight by using spatial modulation and interferometry toobserve scattering from the interference of scattered fieldwith the incident field. It has additionally been demon-strated in Ref. [49] that measuring the far-field extinction phase , in addition to amplitude, allows for measurementof the absorption cross section. Yet, we should also ac-knowlege that simpler measurements would be able toobserve signs of circular dichroism from the scatteringcross section. Measurements of transmission or reflectionusing a high numerical apeture lens have the capacity tocapture a large portion of the scattering cross sectionand will therefore depend on LCP or RCP light. Indeed,there are a number of options to experimentally observethe circular dichroism presented here.0 (a) (b) (c) Orientation2angle2( θ ) Ring2diameter2(d) Number2of2particles2(N) θ =-60º θ =-30º θ =0º θ =30º θ =60º θ =90º d=400nmd=360nmd=320nmd=280nmd=240nmd=200nm N=3N=4N=5N=6 FIG. 5. Simulations of the absorption cross-section produced by LCP and RCP plane waves incident on the same planar chiraloligomer seen in Fig. 4, when: (a) rotating the triangular nanoparticles in 30 ◦ increments, (b) varying the diameter of thering of triangular nanoparticle in 40nm increments and (c) reducing the number of nanoparticles. It can be seen that circulardichroism in absorption is most dependent on the orientation of the triangular nanoparticles, but the effect nonetheless existsfor a wide range of structures. All dimensions are the same as Fig. 4, except where mentioned otherwise. IV. CONCLUSIONS
We have presented a rigorous analytical study of circu-lar dichroism in nanostructures with rotational symme-try. It was shown that, because of reciprocity-enforceddegeneracies, chiral scattering behavior cannot involvethe excitation of modes that are inaccessible (dark) de-pending on polarization’s handedness. This observa-tion led to the distinction of two forms of circular dichro-ism: the traditional form that originates from spatially-distinct excitations and is regularly observed in extinc-tion, and the second form originating from spatially-identical excitations that rotate in opposite directionstemporally (such as from reciprocal plane waves) and canbe observed in absorption and scattering. To explain thepeculiarities of the second form, we have shown that itcan occur only if the scatterer has nonorthogonal eigen-modes. Necessary criteria for a scattering structure tohave nonorthogonal eigenmodes was then shown to be re-tardation of coupling between currents and/or the use ofmultiple materials. Notably, these will also be necessary criteria for Fano resonances and other such modal inter-ference effects that rely on nonorthogonal eigenmodes. Aconsequence of this analysis is that circular dichroism inabsorption would be amplified at locations of significantmodal interference, such as Fano resonances. To demon-strate a manifestation of this circular dichroism effect, wehave proposed a planar chiral nanoparticle heptamer thatexhibits a Fano resonance and observed that significantabsorption circular dichroism occurs in the vicinity of theFano resonance. Our conclusions subsequently suggestthat there is a key relationship between the modal inter-ference and circularly-dichroic scattering in both linearand nonlinear responses of planar chiral systems.
V. ACKNOWLEDGEMENTS
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