 # Microscopic origin of the chiroptical response of optical media

### The generalized coupled oscillator model

We model the microscopic CO response of optical media at the molecular unit cell level using two lossy coupled electron oscillators. The two oscillators are assumed to be arbitrarily located and oriented relative to each other, and interacting with an arbitrarily polarized light at oblique incidence with electric field

$E⇀0ei(k⇀∙r⇀−ωt)$

(Fig. 1A), where

$k⇀$

and ω are the wave vector and frequency of the incident light, respectively. These coupled oscillators constitute a single molecular unit cell described by a pair of fully vectoral second-order coupled differential equations

$∂t2u⇀1+γ1∂tu⇀1+ω12u⇀1+ζ2,1u2û1=−em*(E⇀0⋅û1)û1ei(k⇀⋅r⇀1−ωt)$

(1.1)

$∂t2u⇀2+γ2∂tu⇀2+ω22u⇀2+ζ1,2u1û2=−em*(E⇀0⋅û2)û2ei(k⇀⋅r⇀2−ωt)$

(1.2) Fig. 1 Generalized coupled oscillator model space. (A) Representation of an arbitrarily oriented incident plane-wave of wave vector k⇀=−k(âxsin θ0 cos ϕ0+âyk sin θ0 sin ϕ0+âzk cos θ0) originating from a source placed at infinity. (B) Molecular unit cell consisting of two oscillators u⇀1 and u⇀2 located at distances δr1 and δr2, respectively, from the molecular center of mass, O′, which is located at a distance r⇀0 from the origin O. Each oscillator is arbitrarily oriented with respect to the other. (C) Coordinate system with the origin (O′) corresponding to the molecular center of mass. The oscillator displacement from O′ is given by δr⇀i=δri(âxsin ξi cos ψi+âysin ξi sin ψi+âzcos ξi) for i = 1,2. (D) The origin here corresponds to oscillator center of mass (O″), which is positioned at a distance δr⇀i from the molecular center of mass (O′). The orientation of each oscillator is described by the unit vector ûi=âxsin θi cos ϕi+âysin θi sin ϕi+âzcos θi for i = 1,2.

Each oscillator

$u⇀i$

is characterized by an oscillation amplitude ui(ω, t), resonant frequency ωi, damping factor γi, and cross-coupling strength ζi, j(ω), representing the electromagnetic interaction between the oscillators, for i, j = 1,2. The oscillator locations are given by

$r⇀i=r⇀0+δr⇀i$

, with

$δr⇀i$

being the oscillator displacement from the molecular center of mass

$r⇀0$

(Fig. 1, B to D). Furthermore, the electron oscillators are described by a charge e and an effective mass m*.

Inserting the time harmonic expressions

$u⇀1(ω,t)=û1u1e−iωt$

and

$u⇀2(ω,t)=û2u2e−iωt$

into Eqs. 1.1 and 1.2 and using the substitution

$Ωk=ωk2−ω2−iγkω$

for k = 1,2 give closed-form solutions for the two oscillation amplitudes expressed as (section S1)

$u1(ω)=−em*[Ω22(E⇀0⋅û1)eik⇀⋅δr⇀1−ζ2,1(E⇀0⋅û2)eik⇀⋅δr⇀2Ω12Ω22−ζ1,2ζ2,1]eik⇀⋅r⇀0$

(2.1)

$u2(ω)=−em*[Ω12(E⇀0⋅û2)eik⇀⋅δr⇀2−ζ1,2(E⇀0⋅û1)eik⇀⋅δr⇀1Ω12Ω22−ζ1,2ζ2,1]eik⇀⋅r⇀0$

(2.2)

Using Eqs. 2.1 and 2.2, the medium’s current density response

$J⇀(ω,t)$

to the driving source field can be calculated as (section S2)

$J⇀(ω,t)=−iε0ωωp2Ω12Ω22−ζ1,2ζ2,1{[Ω22(E⇀0⋅û1)−ζ2,1(E⇀0⋅û2)e−ik⇀⋅(δr⇀1−δr⇀2)]û1+[Ω12(E⇀0⋅û2)−ζ1,2(E⇀0⋅û1)eik⇀⋅(δr⇀1−δr⇀2)]û2}ei(k⇀⋅r⇀−ωt)$

(3)where

$ωp=ne2/m*ε0$

is the plasma frequency, ε0 is the permittivity of free space, and n is the molecular unit cell density. By rearranging Eq. 3, the current density response can be simplified as

$J⇀(ω,t)=−iωε0χE⇀0ei(k⇀∙r⇀−ωt)$

, showing

$J⇀$

to be proportional to the product of the incident source field with a susceptibility tensor χ containing elements χi, j with i, j = x, y, z. The susceptibility tensor can be expressed in terms of a modified dielectric tensor ϵ(k, ω) and a nonlocality tensor Γ(k, ω) as χ(k, ω) = ϵ(k, ω) + ikΓ(k, ω), where the modified dielectric tensor is related to the dielectric tensor as ϵ(k, ω) = ε(k, ω) − Ι (29). The nonlocality tensor has previously been identified as related to the optical activity by the relations ORD = ωRe{Γ}/2c and CD = 2ωIm {Γ}/c, where c is the speed of light in free space (20). Full expressions for χ(k, ω) along with derivations of expressions for ϵ(k, ω) and Γ(k, ω) are given in section S3.

Because the relationship between the far- and near-field CO response is typically approximated as

$TRCP−TLCP∝∣J⇀RCP∣2−∣J⇀LCP∣2$

, we express the CO response calculated using the model as

$CO=∣J⇀RCP∣2−∣J⇀LCP∣2$

, where

$J⇀RCP$

and

$J⇀LCP$

indicate the current density response of the optical medium to RCP and LCP light, respectively. Expanding this term results in a concise expression for CO given as (section S4)

$CO/ε02ω2=(χ⇀n×χ⇀n*)⋅(E⇀0×E⇀0*)$

(4)

Equation 4 is expressed using the Einstein summation notation summed over n = x, y, z, where each susceptibility vector

$χ⇀n$

contains elements χn,k for k = x, y, z and is related to the dielectric and nonlocality vectors by

$χ⇀n=ϵ⇀n+ikΓ⇀n$

(29). Note that the expression for CO is nonzero only if both (i) the incident source field is elliptically or circularly polarized and (ii) the susceptibility terms are complex, which occurs in the presence of either damping in the optical medium, γ1 or γ2 ≠ 0, or spatial separation between the oscillators along the direction of source propagation,

$k⇀∙(δr⇀1−δr⇀2)≠0$

(section S3). Setting the two oscillators’ orientation parallel to the xy plane (θ1 = θ2 = π/2) and inserting this into Eq. 4 give

$CO=ε02ω2[(ϵ⇀n×ϵ⇀n*)+ik(Γ⇀n×ϵ⇀n*−ϵ⇀n×Γ⇀n*)]⋅(E⇀0×E⇀0*)$

. This expression can be rewritten as the sum of two components, CO = ΔA = ΔAϵ,ϵ + ΔAΓ,ϵ, where

$ΔAϵ,ϵ/ε02ω2=(ϵ⇀n×ϵ⇀n*)⋅(E⇀0×E⇀0*)$

(5.1)

$ΔAΓ,ϵ/ε02ω2=2ikRe{Γ⇀n×ϵ⇀n*}⋅(E⇀0×E⇀0*)$

(5.2)

Here, ΔAϵ,ϵ is determined by the source interaction with the dielectric tensor and ΔAΓ,ϵ is determined by the source interaction with both the nonlocality and dielectric tensors. In the limit where the spatial separation between the oscillators is much smaller than the wavelength,

$k⇀∙(δr⇀1−δr⇀2)≪1$

, eqs. S12.1 to S12.9 and S13.1 to S13.9 show that the dielectric tensor ϵ(k, ω) only depends on ω, whereas the nonlocality tensor Γ(k, ω) becomes directly proportional to

$k̂$

. This suggests an interesting dichotomy: The response ΔAϵ,ϵ is largely influenced by the source frequency corresponding to a temporal dispersion in the system, whereas ΔAΓ,ϵ is influenced by the direction of the incident field corresponding to a spatial dispersion in the system. Consistent with this, we show the dependence of ΔAϵ,ϵ on the angular separation between the oscillators in the direction of source electric field rotation and of ΔAΓ,ϵ on the separation between oscillators in the direction of the source propagation.

By further simplification, Eqs. 5.1 and 5.2 can be rewritten as (section S4)

$ΔAϵ,ϵ=2ε02ω2∣E0∣2cosθ0Im{ϵxx*ϵxy+ϵyx*ϵyy}$

(6.1)

$ΔAΓ,ϵ=2ε02ω2∣E0∣2cosθ0Re{k[(ϵxyΓxx*−ϵxxΓxy*)+(ϵyyΓyx*−ϵyxΓyy*)]}$

(6.2)

Note that, in the absence of damping,

$ϵi,j=ϵi,j*$

for i, j = x, y, Eq. 6.1 reduces to ΔAϵ,ϵ = 0. Furthermore, for an isotropic medium, the diagonal elements of the dielectric tensor are equal and the oscillator coupling is symmetric (ζ1,2(ω) = ζ2,1(ω)), resulting in ϵxx = ϵyy and ϵxy = ϵyx, respectively. Substituting these in Eq. 6.1 results in

$Im{ϵxx*ϵxy+ϵyx*ϵyy}=0$

, or equivalently ΔAϵ,ϵ = 0. Therefore, both damping and anisotropy in an optical medium are necessary to achieve a ΔAϵ,ϵ type CO response. This conclusion is consistent with previous observation that absorption plays a critical role in generating a CO response (22, 23). Moreover, a CO response of the ΔAϵ,ϵ type has also been observed in lossy two-dimensional anisotropic plasmonic media (21, 30). We associate ΔAϵ,ϵ to the absorption-based CO response described earlier, COabs, noting again that this type of response is not related to optical activity. For the second response type, ΔAΓ,ϵ, of Eq. 6.2 to be nonzero, a finite coupling between the oscillators is required, ζ1,2(ω) ≠ 0 and ζ2,1(ω) ≠ 0. Note that even for an isotropic medium with nonzero symmetric coupling (ζ1,2(ω) = ζ2,1(ω)), nonlocality constants become Γxx = Γyy = 0 and Γxy = −Γyx (section S3), resulting in a nonzero ΔAΓ,ϵ response. Hence, coupling between oscillators is a necessary condition to achieve a ΔAΓ,ϵ type CO response—a conclusion that is consistent with both the predictions of the Born-Kuhn model (20, 29) and the treatment of bi-isotropic chiral media presented in (31). We associate ΔAΓ,ϵ to the COOA type response described earlier, which is fundamentally related to optical activity.

Further insights into the ΔAϵ,ϵ and ΔAΓ,ϵ response types can be achieved by expressing them in terms of the fundamental oscillator parameters of Eqs. 1.1 and 1.2. By inserting expressions for the dielectric (eqs. S12.1 to S12.9) and nonlocality (eqs. S13.1 to S13.9) constants into Eqs. 6.1 and 6.2, and assuming ϕ1 = 90° for simplicity, ΔAϵ,ϵ and ΔAΓ,ϵ can be expressed as

(7.1)

(7.2)where the multiplication factor κ is defined as

By allowing the two oscillators to have the same damping coefficient, γ1 = γ2 = γ, and assuming the spatial separation between them to be much smaller than the wavelength,

$k⇀∙(δr⇀1−δr⇀2)≪1$

, Eqs. 7.1 and 7.2 reduce to

(8.1)

(8.2)

We illustrate the behavior of these two CO response types in Eqs. 8.1 and 8.2 by applying them to two Au nanocuboids, acting as oscillators, aligned parallel to the xy plane (with ϕ1 = 90° and ϕ2 = 45°) excited with a source field normally incident on the structure at angles, θ0 = 0° and θ0 = 180° (Fig. 2A). We assume the two Au nanocuboids, separated along the direction of source propagation (z) by a distance dz = d1,zd2,z = 200 nm and located at d1,y = d2,x = 100 nm, to exhibit resonance at wavelengths λ1 = 750 nm and λ2 = 735 nm with ζ1,21) = ζ2,12) = 1.6 × 1029 s−2. The following values for the plasma frequency, ωp = 1.37 × 1016 s−1, and damping coefficient, γ = γ1 = γ2 = 1.22 × 1014 s−1, for Au in the near-infrared region are used (32). ΔAϵ,ϵ and ΔAΓ,ϵ plotted versus incident wavelength λ0 (Fig. 2, B and C) for the two source angles θ0 illustrates that the presence of an inversion in the sign of ΔAϵ,ϵ as θ0 is rotated by 180°, which is consistent with Eq. 8.1, where ΔAϵ,ϵ0 + π) = −ΔAϵ,ϵ0). Previous observations of inversion in the sign of the far-field CO response due to θ0 rotation suggest an absence of optical activity in the underlying medium (21, 30), verifying our observations, whereas the lack of sign change in the ΔAΓ,ϵ due to θ0 rotation, where ΔAΓ,ϵ0 + π) = ΔAΓ,ϵ0), is indicative of optical activity (30). The total response, ΔA, plotted for θ0 = 0° and θ0 = 180°exhibits an asymmetric spectral line shape due to the competing contributions from the ΔAϵ,ϵ response, which exhibits a single-fold symmetric line shape, and the ΔAΓ,ϵ response, which exhibits a twofold symmetric line shape (Fig. 2D), indicating the presence of both COOA and COabs in the total CO response. Fig. 2 Dependence of the CO response of nanocuboid bi-oscillator system on source angles θ0 and ϕ0. (A) Relative orientation of the incident light of wave vector k⇀ with respect to the two nanocuboid oscillators. The two oscillators, represented by u⇀1 and u⇀2 , are oriented parallel to the x–y plane (θ1 = θ2 = π/2) with azimuth angles ϕ1 = 90° and ϕ2 = 45°, respectively. The nanocuboids are located at d1, z = d2, z = 100 nm with d1, y = d2, x = 100 nm, and for simplicity, d1, x = d2, y = 0 nm was assumed. The nanocuboid parameters were chosen such that they exhibit resonance at wavelengths of λ1 = 750 nm and λ2 = 735 nm, with coupling strengths ζ1,2(ω1) = ζ2,1(ω2) = 1.6 × 1029 s−1. (B) The calculated ΔAϵ,ϵ response at source angles θ0 = 0° and 180° (note that ϕ0 is undefined at these values of θ0) exhibits a onefold symmetric line shape and experiences an inversion in sign when the incident angle is changed from 0° to 180°. a.u., arbitrary units. (C) The corresponding ΔAΓ,ϵ response calculated under the same conditions exhibits a twofold symmetric line shape and does not experience an inversion in sign for a θ0 change from 0° to 180°. (D) The total CO response ΔA = ΔAϵ,ϵ + ΔAΓ,ϵ for the two source angles does not show any symmetry in the spectral line shape due to the presence of competing contributions from both ΔAϵ,ϵ and ΔAΓ,ϵ response types. (E to G) CO response for the oscillator configuration and orientations in (A) calculated at θ0 = 45° for two azimuth angles ϕ0 = 0° and 180°. (E) The calculated ΔAϵ,ϵ response does not change sign when the incident angle ϕ0 is changed from 0° to 180°. (F) The corresponding ΔAΓ,ϵ response, however, exhibits an inversion in sign for a 180° change in the source azimuth. At these source angles, ΔAϵ,ϵ exhibits a onefold symmetric line shape, whereas ΔAΓ,ϵ is asymmetric. (G) The total CO response ΔA = ΔAϵ,ϵ + ΔAΓ,ϵ also exhibits an asymmetric line shape due to the presence of both ΔAϵ,ϵ and ΔAΓ,ϵ contributions.

Analogous to the dependence of ΔAϵ,ϵ and ΔAΓ,ϵ responses on θ0, further insight can be achieved by analyzing the dependence of the CO response on the azimuth angle ϕ0 (for any θ0, except at θ0 = 0° and 180°, where ϕ0 is undefined). For an identical configuration of Fig. 2A, ΔAϵ,ϵ and ΔAΓ,ϵ plotted versus incident wavelength λ0 (Fig. 2, E to G) for two source azimuth angles ϕ0 = 0° and 180° (at θ0 = 45°) illustrates the presence of an inversion in the sign of ΔAΓ,ϵ instead, as ϕ0 is rotated by 180°. This follows from Eqs. 8.1 and 8.2, where ΔAϵ,ϵ0 + π) = ΔAϵ,ϵ0) and ΔAΓ,ϵ0 + π) = −ΔAΓ,ϵ0), respectively. This inversion in the ΔAΓ,ϵ response can be further described by assuming d1,z = d2,z = 0 nm to make a two-dimensional structure wherein the spatial dispersion dependence

$k⇀∙(δr⇀1−δr⇀2)$

of Eq. 8.2 simplifies to kd sin θ0( sin ϕ0 − cos ϕ0), for the two oscillators located equidistant from the origin (d = d1,y = d2,x), demonstrating the dependence of ΔAΓ,ϵ on ϕ0.

In addition to the dependence of the CO response on excitation direction, θ0 and ϕ0, we analyze its dependence on various oscillator parameters including the angular orientation between the two oscillators along the xy plane, by varying angle ϕ2 at ϕ1 = 90°, and the difference between coupling terms ζ2,1(ω) − ζ1,2(ω), oscillator frequencies Δω = ω1 − ω2, and damping coefficients Δγ = γ1 − γ2. For this analysis, we assume the light to be normally incident (θ0 = 0°) on the two Au nanocuboids, of lengths l1and l2, that are aligned parallel to the xy plane with d1,y = l1, d2,x = l2 and placed in a planar arrangement with d1,z = d2,z = 0 nm. In such a planar configuration at normal incidence,

$k⇀∙(δr⇀1−δr⇀2)=0$

, resulting in ΔAΓ,ϵ = 0 (Eq. 8.2). Last, by setting the two resonant wavelengths to be λ1 = 750 nm and λ2 = 735 nm (corresponding to Δω/γ = 0.42), and assuming ζ1,2(ω) = ζ2,1(ω), the dependence of ΔAϵ,ϵ on ϕ2 exhibits a peak response at ϕ2 = 45° (Fig. 3A). Note that this observation that a planar two-dimensional plasmonic structure can exhibit a COabs type CO response, not related to optical activity, is consistent with (30) and is also in agreement with the findings of Eftekhari and Davis (21). In their work, they also note, without explanation, an experimental finding of a peak CO response occurring at ϕ2 = 52° rather than the expected ϕ2 = 45°. A simple inclusion of a nonzero coupling difference, ζ2,1 − ζ1,2, between the two oscillators in the model accounts for this behavior wherein by plotting ϕ2 that maximizes the ΔAϵ,ϵ response as a function of ζ2,1 − ζ1,2 at ω = 2.43 × 1015 s−1 (Fig. 3B), we show that the presence of asymmetric oscillator coupling causes the maximum peak to occur at values other than ϕ2 = 45°. The ΔAϵ,ϵ response can also be maximized by optimizing the oscillator frequencies, wherein for ζ1,2 − ζ2,1 = − 5.2 × 1028 s−2 corresponding to ϕ2 = 52°, the model also predicts a peak ΔAϵ,ϵ for Δω/γ = 0.74 (Fig. 3C). This includes the underlying dependence of the multiplication factor κ(ω) on the difference between the normalized oscillator frequencies Δω/γ (fig. S2). Last, the model predicts a CO response for light normally incident on a geometrically achiral system if asymmetric absorption is present (γ1 ≠ γ2)—a scenario easily achieved by depositing two different metal types for each of the cuboids (Fig. 3D). Using dissimilar metals to achieve inhomogeneous damping on a geometrically achiral structure has been shown to exhibit a CO response (33). Fig. 3 Dependence of the CO response of nanocuboid bi-oscillator system on oscillator parameters. CO response of the two oscillators, under normal incidence excitation (θ0 = 0°), oriented parallel to the x–y plane (θ1 = θ2 = π/2) and arranged in a planar arrangement with d1, z = d2, z = 0 nm and d1, y = d2, x = 100 nm. In this planar configuration at normal incidence, ΔAΓ, ϵ = 0. (A) Dependence of ΔA = ΔAϵ, ϵ on the angular orientation between the two oscillators in the x–y plane calculated by varying ϕ2 at ϕ1 = 90°. The oscillators are designed to exhibit resonance at wavelengths of λ1 = 750 nm and λ2 = 735 nm, and assuming ζ1,2(ω) = ζ2,1(ω), the peak ΔAϵ, ϵ response is shown to occur at ϕ2 = 45°. (B) Orientation angle of the second oscillator ϕ2 (at ϕ1 = 90°) at which ΔAϵ, ϵ is maximized for a nonzero difference in coupling coefficients, ζ1,2 − ζ2,1, plotted here at ω = 2.43 × 1015 s−1. (C) ΔAϵ, ϵ dependence on the normalized difference in resonant frequencies (Δω)/γ at ζ1,2 − ζ2,1 = − 5.2 × 1028 s−2 corresponding to ϕ2 = 52°. A peak ΔAϵ, ϵ response is achieved at (Δω)/γ = 0.74. (D) ΔAϵ, ϵ dependence at normal incidence on a geometrically achiral system (l1 = l2) for oscillators of the same metal corresponding to γ1 = γ2 (red line) and of dissimilar metals corresponding to γ1 ≠ γ2 (blue line).

Last, we verify the validity of our generalized model by applying it to the structure and excitation conditions studied using the Born-Kuhn model in (20). We assume the two Au nanocuboids in Fig. 2A to be of equal lengths (l), aligned orthogonal to each other (ϕ1 = 90° and ϕ2 = 0°) with d1,y = d2,x = l/2 and separated by a distance dz along the z direction, resulting in ω1 = ω2 = ω and Ω1 = Ω2 = Ω (fig. S3A). Note that, for consistency, the cuboid lengths l were scaled to shift the resonance wavelengths to λ1 = λ2 = 1300 nm. Illumination of the structure at normal incidence, θ0 = 0°, under these conditions results in ΔAϵ,ϵ = 0 (from Eq. 8.1). Also, as expected, due to this lack of COabs contribution, ΔA = ΔAΓ,ϵ plotted versus incident wavelength λ0 (fig. S3B) exhibits a twofold symmetric line shape and is consistent with the results of (20). Moreover, by applying the geometrical and oscillator parameters to the configuration of fig. S2A, one could calculate the reduced dielectric and nonlocality tensor elements (section S6). Applying these to Eq. 6.2 and plotting the resulting ΔAΓ,ϵ versus λ0 result in the same response (fig. S3B), confirming the predictions of our generalized model as well as its consistency with the Born-Kuhn model (20).

### Experimental results

The model described above provides a comprehensive theoretical framework to study the origin and characteristics of various CO response types in both two- and three-dimensional optical media under arbitrary excitation conditions. A common performance metric associated with far-field CO measurements is circular diattenuation (CDA), a normalized form of the CO response expressed as CDA = (TRCPTLCP)/(TRCP + TLCP). CDA also corresponds to the normalized m14 element of the Mueller matrix, so it can be directly extracted from spectroscopic ellipsometry measurements (34). Note that Mueller matrix spectroscopy also presents an accurate method for distinguishing between the COOA and COabs contributions in a far-field CO measurement; however, this requires measurement of both m14 and m41 elements (17). As shown below, we verify through model calculations that both CDA and ΔA represent the same optical phenomenon; hence, for the simplicity of analysis, we present the following experimental measurements and comparisons with model predictions in the CDA format. Note that an alternate metric based on measuring optical chirality flux has recently been proposed as a quantitative far-field observable of the magnitude and handedness of the near-field chiral density in a nanostructured optical medium (35). Measured using a technique referred to as chirality flux spectroscopy, it corresponds to the third Stokes parameter, which is directly related to the degree of circular polarization of the scattered light in the far field (36) and carries information of the chiral near fields. For the purpose of discussion in this article, and its consistency with existing literature, we limit our analysis to measurements using the more prevalent metric of CO (or equivalently CDA) obtained from traditional CD spectroscopic measurements.

We experimentally characterize three planar cuboid configurations (Fig. 4A, left column) by measuring their far-field CDA response under various excitation conditions and compare them to predictions of the model. Respective expressions for ΔAϵ,ϵ and ΔAΓ,ϵ in the three configurations, assuming d1,z = d2,z = 0 nm and γ1 = γ2 = γ (Eqs. 8.1 and 8.2), are listed in Fig. 4A (right column). Note that the

$k⇀∙(δr⇀1−δr⇀2)$

term in these planar configurations simplifies to kd sin θ0(sin ϕ0 − cos ϕ0). The devices, consisting of an array of two Au nanocuboids (thickness t = 40 nm) of varying lengths (l1 and l2) and alignments (varying ϕ2 at ϕ1 = 90°), were fabricated on a fused-silica substrate using electron beam lithography and liftoff (see Materials and Methods and section S7). The pitch of the array (p= 375 nm) was chosen to minimize coupling between adjacent bi-oscillator unit cells. The devices were characterized using a spectroscopic ellipsometer between free-space wavelengths of λ0 = 500 and 1000 nm under illumination at θ0 = 45° for various azimuth angles ϕ0 (see Materials and Methods). The first device consisted of the two Au nanocuboids arranged orthogonal to each other (ϕ1 = 90° and ϕ2 = 0°) and were designed to be of different lengths (l1 = 120 nm and l2 = 100 nm placed at d1,y = d2,x = 100 nm, respectively). Because l1 and l2 determine both the resonant frequencies (ω1 and ω2) and the cross-coupling strengths (ζ1,2 and ζ2,1), setting l1l2 constitutes a general configuration where both ΔAϵ,ϵ and ΔAΓ,ϵ type contributions can be present in a single CDA measurement. The corresponding CDA spectra (Fig. 4B) measured at ϕ0 = 0°, 90°, and 135° (blue plots) and at 180° offset from these angles (red plots) show an inversion in the sign, indicating the response to primarily result from ΔAΓ,ϵ. However, note that the CDA measurements at these angles slightly lack the twofold symmetry in the spectral line shape, a result of a minor ΔAϵ,ϵ contribution. For ϕ0 = 45° and 225°, the spectra lack the sign inversion, indicating the response to primarily result from ΔAϵ,ϵ, which also follows from Fig. 4A, where ΔAΓ,ϵ = 0 at these two ϕ0 angles. This result is further validated by fabricating a device consisting of Au nanocuboids of equal lengths (l1 = l2 = 120 nm), wherein the CDA spectra at ϕ0 = 45° and 225° show no CO response, because both ΔAΓ,ϵ = ΔAϵ,ϵ = 0, confirming the predictions of the model (Fig. 4A). Moreover, by setting l1 = l2, the twofold symmetry in the CDA line shape at ϕ0 = 0° (180°), 90° (270°), and 135° (315°) is recovered, indicating the response to now only consist of ΔAΓ,ϵ contribution, a signature of optical activity (Fig. 4C). Hence, it is possible for a geometrically achiral structure to exhibit optical activity under certain illumination conditions. It follows then due to reciprocity that optical activity may be detectable at large scattering angles when a source field is normally incident on a planar achiral structure. This phenomenon was recently confirmed by Kuntman et al. (37) using a scattering matrix decomposition method. Note that the similarity between the calculated CDA and ΔA response (plotted under the conditions of Fig. 4B and fig. S5) verifies our assumption that they are equivalent measurements and can be used interchangeably. Fig. 4 Experimental characterization of the CO response of two-dimensional planar Au nanocuboids. (A) Simplified ΔAϵ,ϵ and ΔAΓ,ϵ relations, calculated from Eqs. 8.1 and 8.2, for three planar nanocuboid configurations. Top row: The two oscillators are aligned orthogonal to each other (ϕ1 = 90° and ϕ2 = 0°) and are assumed to be of different lengths (l1 ≠ l2), corresponding to ω1 ≠ ω2 and ζ1,2(ω) ≠ ζ2,1(ω). In such a system, it is expected that both ΔAϵ,ϵ and ΔAΓ,ϵ contributions are present. Middle row: Same as above except with l1 = l2 resulting in ω1 = ω2 = ω0, ζ1,2 = ζ2,1. In this configuration, ΔAϵ,ϵ contribution is expected to be absent for excitation at any arbitrary angle of incidence. Bottom row: Same as above (l1 = l2) except that the two oscillators are oriented parallel to each other (ϕ1 = 90° and ϕ2 = 90°). Ignoring any optical resonance along the width of the nanocuboid, the model predicts both ΔAϵ,ϵ and ΔAΓ,ϵ to be absent, for excitation at any arbitrary angle of incidence. (B to D) Corresponding experimental CDA measurements for an array of planar Au nanocuboid bi-oscillators, illuminated with free-space light between wavelengths of λ0 = 500 and 1000 nm, as a function of incidence angle (varying ϕ0 at a fixed θ0 = 45°) for the three configurations shown in (A). Top-down scanning electron microscopy (SEM) images of unit cells consisting of the two Au nanocuboid oscillators, overlaid with the coordinate system and orientation of the in-plane wave vector of the incident light ( k⇀∥xy ) along the x–y plane, are shown at the top of each column. Scale bar, 120 nm in the SEM images. (B) Experimentally measured (solid lines) and model-calculated (dashed lines) CDA spectra for a sample consisting of Au nanocuboids of unequal lengths (l1 = 120 nm and l2 = 100 nm) oriented orthogonal to each other (ϕ1 = 90° and ϕ2 = 0°) at various ϕ0. The spectra at ϕ0 = 0°, 90°, and 135° (blue plots) and at 180° offset from these angles (solid red plots) show an inversion in the sign, which is absent for excitation at ϕ0 = 45° (225°). The CDA model plots were calculated assuming ζ2,1(ω1) = 6.4 × 1029 s−2 and ζ1,2(ω2) = 8.1 × 1029 s−2 at λ1= 750 nm and λ2 = 720 nm, respectively. (C) Equivalent CDA measurements and model calculations for a device with Au nanocuboids of equal lengths (l1 = l2 = 120 nm). As expected, the CDA response is absent from this device for excitation at ϕ0 = 45° (225°). Moreover, the response at other ϕ0 angles exhibits a twofold symmetric spectral line shape [absent from measurements in (B)], indicating the CDA to only result from ΔAΓ,ϵ contribution. Model parameters used in the calculations are ζ2,1(ω0) = ζ1,2(ω0) = 8.1 × 1029 s−2 at λ1 = λ2 = 745 nm. (D) Same as (C) except that the two Au nanocuboids are oriented parallel to each other (ϕ1 = 90° and ϕ2 = 90°). The CDA spectra at ϕ0 = 0° (180°) and 90° (270°) show no response, whereas the spectra at ϕ0 = 45° (225°) and 135° (315°) show a pronounced signal of the ΔAϵ,ϵ type (no sign inversion for ϕ0 rotation by 180°). The CDA response at latter angles, though not expected from the model predictions in (A), can be attributed to the coupling to optical resonances along the cuboid widths (w1 = w2 = 60 nm), acting as additional orthogonally oriented oscillators ( u⇀1′ and u⇀2′ ) in the system.

For a device with Au nanocuboids of equal lengths l1 = l2 = 120 nm, aligned parallel to each other (ϕ1 = 90° and ϕ2 = 90°), Eqs. 8.1 and 8.2 predict both ΔAϵ,ϵ and ΔAΓ,ϵ to be zero under illumination at θ0 = 45° for any ϕ0. Consistent with these predictions, while the CDA spectra measured at ϕ0 = 0°(180°) and 90° (270°) show no response, the spectra at ϕ0 = 45°(225°) and 135° (315°) show a pronounced signal of the ΔAϵ,ϵ type (no sign inversion for ϕ0 rotation by 180°; Fig. 4D). We attribute this phenomenon to originate from coupling to the optical resonances (

$u⇀1′$

and

$u⇀2′$

) along the cuboid widths (w1 = w2 = 60 nm), acting as additional orthogonally oriented oscillators in the system, resulting in a two-dimensional anisotropic optical system supporting two orthogonal elliptical eigenmodes (30). A circularly polarized light at non-normal incidence (θ0 ≠ 0° and 180°) projects an elliptically polarized field along the plane of the device (Fig. 5, A to D, red ellipse), which, at certain azimuth angles ϕ0, can access these elliptical eigenmodes (Fig. 5, A to D, dashed yellow ellipses). At ϕ0 = 0° (180°) or ϕ0 = 90° (270°), both orthogonal eigenmodes are accessed equally, resulting in the total CO response to be zero, whereas, at ϕ0 = 45°(225°) and 135° (315°), only one of the two eigenmodes can be excited, resulting in a strong CDA response. This dependence of peak ∣ΔAϵ,ϵ∣ on the azimuth angle ϕ0 is shown schematically in Fig. 5E. These results are also consistent with Fig. 5F, which follows from Eqs. 8.1 and 8.2, wherein incorporation of contributions from these additional oscillators results in a zero ΔAΓ,ϵ response, whereas the ΔAϵ,ϵ response is shown to stay proportional to (ζ1′,2 − ζ2,1′). Note that for the CDA calculations in Fig. 4 (B and C), only coupling between the oscillators along their long axis (

$u⇀1$

and

$u⇀2$

) was assumed. The absence of contributions from coupling between the oscillators along their short axis,

$u⇀1′$

and

$u⇀2′$

, in the calculations could explain the minor discrepancy between the calculated and experimentally measured CDA spectra. Fig. 5 Origin of the CO response from parallel nanocuboid oscillators through coupling along orthogonal oscillator dimensions. (A to D) Top-down SEM images of the device consisting of an array of Au nanocuboid oscillators oriented parallel to each other. Overlaid are the constitutive elliptical eigenmodes (dashed yellow curves) and the projected in-plane source electric field ( E⇀∥xy ), indicated by a red vector arrow that traces the red elliptical path for a circularly polarized light at non-normal incidence. Scale bar, 125 nm in the SEM images. (A and B) Orientation of the two eigenmodes relative to the source electric field at ϕ0 = 0° (180°) and 90° (270°), illustrating that they can be accessed equally. (C and D) Same as above, except at source azimuths ϕ0 = 45° (225°) and 135° (315°), illustrating that only one of the two eigenmodes can be accessed. (E) Dependence of ∣ΔAϵ,ϵ∣ on ϕ0 for the parallel nanocuboid oscillator configuration studied here. The orientation of the long- and short-axis oscillators ( u⇀i and u⇀i′ , respectively) corresponding to the length (li) and width (wi) of the two nanocuboids relative to ϕ0 is shown for clarity. (F) Top: Schematic illustrations of the two coupled oscillator contributions that result in a far-field CO response from parallel nanocuboid oscillators of equal lengths (l1 = l2) and widths (w1 = w2) upon illumination at θ0 = 45° and ϕ0 = 45° (225°) or 135° (315°). Note that u⇀1=u⇀2 and u⇀1′=u⇀2′ in this configuration leads to ζ1,2 = ζ2,1 as well as ζ1,2′ = ζ2,1′ and ζ2′,1 = ζ1′,2, resulting in ΔAϵ,ϵ response to be doubled (from Eq. 8.1, bottom). However, because of the inversion of the spatial dispersion term k⇀⋅(δr⇀1−δr⇀2) of Eq. 8.2, the ΔAΓ,ϵ contributions between these two configurations become equal and opposite, canceling each other out.

In addition, it is instructive to study the CO response of a device where the two Au nanocuboids of equal lengths are aligned such that ϕ1 = 90° and ϕ2 = 45° in a planar arrangement. Upon illumination of this structure at θ0 = 45° for various ϕ0, the measured CDA response shows neither any clear inversion in sign with 180° rotation of ϕ0 nor any apparent symmetry in the spectral line shape (fig. S6). This is because the various sub-oscillators (

$u⇀1$

,

$u⇀2$

,

$u⇀1′$

, and

$u⇀2′$

) in this system are aligned with respect to each other such that they can all be intercoupled, resulting in substantial contributions from both ΔAΓ,ϵ and ΔAϵ,ϵ. This serves as a simple example for a system where the measured far-field CO response is ambiguous, and its underlying origin can be difficult to interpret.

Last, until now, we have applied the model predictions to, and validated them against, existing literature and experimental CDA measurements on planar metallic nanocuboid oscillators. However, as mentioned earlier, a strong far-field CO response of the COaxial type has been observed in an all-dielectric metamaterial acting as a uniaxial or a biaxial medium, wherein symmetry breaking of the unit cell along the direction of source propagation enables asymmetric transmission of the two CP components of incident linearly polarized light (19, 24, 25). An additional deployment of geometric phase further enables independent phase-front manipulation of these two components (24, 38). We demonstrate the generality of the model by applying it to an all-dielectric optical medium with a mirror-symmetry breaking chiral unit cell that enables asymmetric transmission of the two CP components, but without a geometric phase (section S10), and illustrate the conditions under which the Poynting vectors associated with the LCP and RCP components of a linearly polarized light normally incident on an all-dielectric biaxial medium can propagate in different directions within the medium. A simple spatial filtering of either the LCP or the RCP on the exit side can result in a strong CO response, as shown in (25). Note that such a far-field CO response is not related to optical activity.