### 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

${\stackrel{\rightharpoonup}{E}}_{0}{e}^{i(\stackrel{\rightharpoonup}{k}\bullet \stackrel{\rightharpoonup}{r}-\mathrm{\omega}t)}$(Fig. 1A), where

$\stackrel{\rightharpoonup}{k}$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

$${\mathrm{\partial}}_{t}^{2}{\stackrel{\rightharpoonup}{u}}_{1}+{\mathrm{\gamma}}_{1}{\mathrm{\partial}}_{t}{\stackrel{\rightharpoonup}{u}}_{1}+{\mathrm{\omega}}_{1}^{2}{\stackrel{\rightharpoonup}{u}}_{1}+{\mathrm{\zeta}}_{2,1}{u}_{2}{\widehat{u}}_{1}=-\frac{e}{{m}^{*}}({\stackrel{\rightharpoonup}{E}}_{0}\cdot {\widehat{u}}_{1}){\widehat{u}}_{1}{e}^{i(\stackrel{\rightharpoonup}{k}\cdot {\stackrel{\rightharpoonup}{r}}_{1}-\mathrm{\omega}t)}$$(1.1)

$${\mathrm{\partial}}_{t}^{2}{\stackrel{\rightharpoonup}{u}}_{2}+{\mathrm{\gamma}}_{2}{\mathrm{\partial}}_{t}{\stackrel{\rightharpoonup}{u}}_{2}+{\mathrm{\omega}}_{2}^{2}{\stackrel{\rightharpoonup}{u}}_{2}+{\mathrm{\zeta}}_{1,2}{u}_{1}{\widehat{u}}_{2}=-\frac{e}{{m}^{*}}({\stackrel{\rightharpoonup}{E}}_{0}\cdot {\widehat{u}}_{2}){\widehat{u}}_{2}{e}^{i(\stackrel{\rightharpoonup}{k}\cdot {\stackrel{\rightharpoonup}{r}}_{2}-\mathrm{\omega}t)}$$(1.2)

Each oscillator

${\stackrel{\rightharpoonup}{u}}_{i}$ is characterized by an oscillation amplitude *u _{i}*(ω,

*t*), resonant frequency ω

*, damping factor γ*

_{i}*, and cross-coupling strength ζ*

_{i}_{i, j}(ω), representing the electromagnetic interaction between the oscillators, for

*i*,

*j*= 1,2. The oscillator locations are given by

, with

$\mathrm{\delta}{\stackrel{\rightharpoonup}{r}}_{i}$being the oscillator displacement from the molecular center of mass

${\stackrel{\rightharpoonup}{r}}_{0}$ (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

${\stackrel{\rightharpoonup}{u}}_{1}(\mathrm{\omega},t)={\widehat{u}}_{1}{u}_{1}{e}^{-i\mathrm{\omega}t}$and

${\stackrel{\rightharpoonup}{u}}_{2}(\mathrm{\omega},t)={\widehat{u}}_{2}{u}_{2}{e}^{-i\mathrm{\omega}t}$into Eqs. 1.1 and 1.2 and using the substitution

${\mathrm{\Omega}}_{k}=\sqrt{{\mathrm{\omega}}_{k}^{2}-{\mathrm{\omega}}^{2}-i{\mathrm{\gamma}}_{k}\mathrm{\omega}}$ for *k* = 1,2 give closed-form solutions for the two oscillation amplitudes expressed as (section S1)

$${u}_{1}(\mathrm{\omega})=\frac{-e}{{m}^{*}}\left[\frac{{\mathrm{\Omega}}_{2}^{2}({\stackrel{\rightharpoonup}{E}}_{0}\cdot {\widehat{u}}_{1}){e}^{i\stackrel{\rightharpoonup}{k}\cdot \mathrm{\delta}{\stackrel{\rightharpoonup}{r}}_{1}}-{\mathrm{\zeta}}_{2,1}({\stackrel{\rightharpoonup}{E}}_{0}\cdot {\widehat{u}}_{2}){e}^{i\stackrel{\rightharpoonup}{k}\cdot \mathrm{\delta}{\stackrel{\rightharpoonup}{r}}_{2}}}{{\mathrm{\Omega}}_{1}^{2}{\mathrm{\Omega}}_{2}^{2}-{\mathrm{\zeta}}_{1,2}{\mathrm{\zeta}}_{2,1}}\right]{e}^{i\stackrel{\rightharpoonup}{k}\cdot {\stackrel{\rightharpoonup}{r}}_{0}}$$

(2.1)

$${u}_{2}(\mathrm{\omega})=\frac{-e}{{m}^{*}}\left[\frac{{\mathrm{\Omega}}_{1}^{2}({\stackrel{\rightharpoonup}{E}}_{0}\cdot {\widehat{u}}_{2}){e}^{i\stackrel{\rightharpoonup}{k}\cdot \mathrm{\delta}{\stackrel{\rightharpoonup}{r}}_{2}}-{\mathrm{\zeta}}_{1,2}({\stackrel{\rightharpoonup}{E}}_{0}\cdot {\widehat{u}}_{1}){e}^{i\stackrel{\rightharpoonup}{k}\cdot \mathrm{\delta}{\stackrel{\rightharpoonup}{r}}_{1}}}{{\mathrm{\Omega}}_{1}^{2}{\mathrm{\Omega}}_{2}^{2}-{\mathrm{\zeta}}_{1,2}{\mathrm{\zeta}}_{2,1}}\right]{e}^{i\stackrel{\rightharpoonup}{k}\cdot {\stackrel{\rightharpoonup}{r}}_{0}}$$(2.2)

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

$\stackrel{\rightharpoonup}{J}(\mathrm{\omega},t)$to the driving source field can be calculated as (section S2)

$$\stackrel{\rightharpoonup}{J}(\mathrm{\omega},t)=\frac{-i{\mathrm{\epsilon}}_{0}{\mathrm{\omega}\mathrm{\omega}}_{p}^{2}}{{\mathrm{\Omega}}_{1}^{2}{\mathrm{\Omega}}_{2}^{2}-{\mathrm{\zeta}}_{1,2}{\mathrm{\zeta}}_{2,1}}\left\{\right[{\mathrm{\Omega}}_{2}^{2}({\stackrel{\rightharpoonup}{E}}_{0}\cdot {\widehat{u}}_{1})-{\mathrm{\zeta}}_{2,1}({\stackrel{\rightharpoonup}{E}}_{0}\cdot {\widehat{u}}_{2}){e}^{-i\stackrel{\rightharpoonup}{k}\cdot (\mathrm{\delta}{\stackrel{\rightharpoonup}{r}}_{1}-\mathrm{\delta}{\stackrel{\rightharpoonup}{r}}_{2})}]{\widehat{u}}_{1}+[{\mathrm{\Omega}}_{1}^{2}({\stackrel{\rightharpoonup}{E}}_{0}\cdot {\widehat{u}}_{2})-{\mathrm{\zeta}}_{1,2}({\stackrel{\rightharpoonup}{E}}_{0}\cdot {\widehat{u}}_{1}){e}^{i\stackrel{\rightharpoonup}{k}\cdot (\mathrm{\delta}{\stackrel{\rightharpoonup}{r}}_{1}-\mathrm{\delta}{\stackrel{\rightharpoonup}{r}}_{2})}\left]{\widehat{u}}_{2}\right\}{e}^{i(\stackrel{\rightharpoonup}{k}\cdot \stackrel{\rightharpoonup}{r}-\mathrm{\omega}t)}$$(3)where

${\mathrm{\omega}}_{p}=\sqrt{n{e}^{2}/{m}^{*}{\mathrm{\epsilon}}_{0}}$ is the plasma frequency, ε_{0} 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

, showing

$\stackrel{\rightharpoonup}{J}$ to be proportional to the product of the incident source field with a susceptibility tensor **χ** containing elements χ_{i, 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) + *ik***Γ**(*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 = ω*Re*{Γ}/2*c* and CD = 2ω*Im* {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

${\mathit{T}}_{\text{RCP}}-{\mathit{T}}_{\text{LCP}}\propto {\mid {\stackrel{\rightharpoonup}{J}}^{\text{RCP}}\mid}^{2}-{\mid {\stackrel{\rightharpoonup}{J}}^{\text{LCP}}\mid}^{2}$, we express the CO response calculated using the model as

$\text{CO}={\mid {\stackrel{\rightharpoonup}{J}}^{\mathit{RCP}}\mid}^{2}-{\mid {\stackrel{\rightharpoonup}{J}}^{\text{LCP}}\mid}^{2}$, where

${\stackrel{\rightharpoonup}{J}}^{\text{RCP}}$and

${\stackrel{\rightharpoonup}{J}}^{\text{LCP}}$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)

$$\text{CO}/{{\mathrm{\epsilon}}_{0}}^{2}{\mathrm{\omega}}^{2}=({\stackrel{\rightharpoonup}{\mathrm{\chi}}}_{n}\times {\stackrel{\rightharpoonup}{\mathrm{\chi}}}_{n}^{*})\cdot ({\stackrel{\rightharpoonup}{E}}_{0}\times {\stackrel{\rightharpoonup}{E}}_{0}^{*})$$(4)

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

contains elements χ_{n,k} for *k* = *x*, *y*, *z* and is related to the dielectric and nonlocality vectors by

(*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, γ_{1} or γ_{2} ≠ 0, or spatial separation between the oscillators along the direction of source propagation,

(section S3). Setting the two oscillators’ orientation parallel to the *x*–*y* plane (θ_{1} = θ_{2} = π/2) and inserting this into Eq. 4 give

. This expression can be rewritten as the sum of two components, CO = Δ*A* = Δ*A*_{ϵ,ϵ} + Δ*A*_{Γ,ϵ}, where

(5.1)

$$\mathrm{\Delta}{A}_{\mathrm{\Gamma},\mathrm{\u03f5}}/{{\mathrm{\epsilon}}_{0}}^{2}{\mathrm{\omega}}^{2}=2\mathit{\text{ikRe}}\{{\stackrel{\rightharpoonup}{\mathrm{\Gamma}}}_{n}\times {\stackrel{\rightharpoonup}{\mathrm{\u03f5}}}_{n}^{*}\}\cdot ({\stackrel{\rightharpoonup}{E}}_{0}\times {\stackrel{\rightharpoonup}{E}}_{0}^{*})$$(5.2)

Here, Δ*A*_{ϵ,ϵ} is determined by the source interaction with the dielectric tensor and Δ*A*_{Γ,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,

, 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

. This suggests an interesting dichotomy: The response Δ*A*_{ϵ,ϵ} is largely influenced by the source frequency corresponding to a temporal dispersion in the system, whereas Δ*A*_{Γ,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 Δ*A*_{ϵ,ϵ} on the angular separation between the oscillators in the direction of source electric field rotation and of Δ*A*_{Γ,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)

$$\mathrm{\Delta}{A}_{\mathrm{\u03f5},\mathrm{\u03f5}}=2{{\mathrm{\epsilon}}_{0}}^{2}{\mathrm{\omega}}^{2}{\mid {E}_{0}\mid}^{2}\text{cos}{\mathrm{\theta}}_{0}\mathit{\text{Im}}\{{\mathrm{\u03f5}}_{\mathit{xx}}^{*}{\mathrm{\u03f5}}_{\mathit{xy}}+{\mathrm{\u03f5}}_{\mathit{yx}}^{*}{\mathrm{\u03f5}}_{\mathit{yy}}\}$$(6.1)

$$\mathrm{\Delta}{A}_{\mathrm{\Gamma},\mathrm{\u03f5}}=2{{\mathrm{\epsilon}}_{0}}^{2}{\mathrm{\omega}}^{2}{\mid {E}_{0}\mid}^{2}\text{cos}{\mathrm{\theta}}_{0}\mathit{\text{Re}}\left\{k\right[({\mathrm{\u03f5}}_{\mathit{xy}}{\mathrm{\Gamma}}_{\mathit{xx}}^{*}-{\mathrm{\u03f5}}_{\mathit{xx}}{\mathrm{\Gamma}}_{\mathit{xy}}^{*})+({\mathrm{\u03f5}}_{\mathit{yy}}{\mathrm{\Gamma}}_{\mathit{yx}}^{*}-{\mathrm{\u03f5}}_{\mathit{yx}}{\mathrm{\Gamma}}_{\mathit{yy}}^{*})\left]\right\}$$(6.2)

Note that, in the absence of damping,

${\mathrm{\u03f5}}_{i,j}={\mathrm{\u03f5}}_{i,j}^{*}$ for *i*, *j* = *x*, *y*, Eq. 6.1 reduces to Δ*A*_{ϵ,ϵ} = 0. Furthermore, for an isotropic medium, the diagonal elements of the dielectric tensor are equal and the oscillator coupling is symmetric (ζ_{1,2}(ω) = ζ_{2,1}(ω)), resulting in ϵ* _{xx}* = ϵ

*and ϵ*

_{yy}*= ϵ*

_{xy}*, respectively. Substituting these in Eq. 6.1 results in*

_{yx}, or equivalently Δ*A*_{ϵ,ϵ} = 0. Therefore, both damping and anisotropy in an optical medium are necessary to achieve a Δ*A*_{ϵ,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 Δ*A*_{ϵ,ϵ} type has also been observed in lossy two-dimensional anisotropic plasmonic media (*21*, *30*). We associate Δ*A*_{ϵ,ϵ} to the absorption-based CO response described earlier, CO_{abs}, noting again that this type of response is not related to optical activity. For the second response type, Δ*A*_{Γ,ϵ}, of Eq. 6.2 to be nonzero, a finite coupling between the oscillators is required, ζ_{1,2}(ω) ≠ 0 and ζ_{2,1}(ω) ≠ 0. Note that even for an isotropic medium with nonzero symmetric coupling (ζ_{1,2}(ω) = ζ_{2,1}(ω)), nonlocality constants become Γ* _{xx}* = Γ

*= 0 and Γ*

_{yy}*= −Γ*

_{xy}*(section S3), resulting in a nonzero Δ*

_{yx}*A*

_{Γ,ϵ}response. Hence, coupling between oscillators is a necessary condition to achieve a Δ

*A*

_{Γ,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 Δ

*A*

_{Γ,ϵ}to the CO

_{OA}type response described earlier, which is fundamentally related to optical activity.

Further insights into the Δ*A*_{ϵ,ϵ} and Δ*A*_{Γ,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 ϕ_{1} = 90° for simplicity, Δ*A*_{ϵ,ϵ} and Δ*A*_{Γ,ϵ} can be expressed as

(7.1)

$$\mathrm{\Delta}{A}_{\mathrm{\Gamma},\mathrm{\u03f5}}=\mathrm{\kappa}\{[{\mathrm{\zeta}}_{2,1}({\mathrm{\omega}}^{2}-{\mathrm{\omega}}_{1}^{2})+{\mathrm{\zeta}}_{1,2}({\mathrm{\omega}}^{2}-{\mathrm{\omega}}_{2}^{2})\left]\text{sin}\right[\stackrel{\rightharpoonup}{k}\cdot (\mathrm{\delta}{\stackrel{\rightharpoonup}{r}}_{1}-\mathrm{\delta}{\stackrel{\rightharpoonup}{r}}_{2})]+{\mathrm{\zeta}}_{1,2}{\mathrm{\zeta}}_{2,1}\text{sin}[2\stackrel{\rightharpoonup}{k}\cdot (\mathrm{\delta}{\stackrel{\rightharpoonup}{r}}_{1}-\mathrm{\delta}{\stackrel{\rightharpoonup}{r}}_{2})]\text{sin}{\mathrm{\varphi}}_{2}\}\text{cos}{\mathrm{\varphi}}_{2}$$(7.2)where the multiplication factor κ is defined as

$$\mathrm{\kappa}(\mathrm{\omega})=2{{\mathrm{\epsilon}}_{0}}^{2}{\mathrm{\omega}}^{2}{\mathrm{\omega}}_{p}^{4}{\mid {E}_{0}\mid}^{2}\text{cos}{\mathrm{\theta}}_{0}/{\mid \left[\right({\mathrm{\omega}}_{1}^{2}-{\mathrm{\omega}}^{2})-i{\mathrm{\gamma}}_{1}\mathrm{\omega}]\left[\right({\mathrm{\omega}}_{2}^{2}-{\mathrm{\omega}}^{2})-i{\mathrm{\gamma}}_{2}\mathrm{\omega}]-{\mathrm{\zeta}}_{1,2}{\mathrm{\zeta}}_{2,1}\mid}^{2}$$By allowing the two oscillators to have the same damping coefficient, γ_{1} = γ_{2} = γ, and assuming the spatial separation between them to be much smaller than the wavelength,

$$\mathrm{\Delta}{A}_{\mathrm{\u03f5},\mathrm{\u03f5}}=\mathrm{\kappa}\mathrm{\omega}\mathrm{\gamma}({\mathrm{\omega}}_{2}^{2}-{\mathrm{\omega}}_{1}^{2})\text{sin}{\mathrm{\varphi}}_{2}\text{cos}{\mathrm{\varphi}}_{2}+\mathrm{\omega}\mathrm{\gamma}({\mathrm{\zeta}}_{1,2}-{\mathrm{\zeta}}_{2,1})\text{cos}{\mathrm{\varphi}}_{2}$$

(8.1)

$$\mathrm{\Delta}{A}_{\mathrm{\Gamma},\mathrm{\u03f5}}=\mathrm{\kappa}\stackrel{\rightharpoonup}{k}\cdot (\mathrm{\delta}{\stackrel{\rightharpoonup}{r}}_{1}-\mathrm{\delta}{\stackrel{\rightharpoonup}{r}}_{2})\left[{\mathrm{\zeta}}_{2,1}\right({\mathrm{\omega}}^{2}-{\mathrm{\omega}}_{1}^{2})+{\mathrm{\zeta}}_{1,2}({\mathrm{\omega}}^{2}-{\mathrm{\omega}}_{2}^{2})+2{\mathrm{\zeta}}_{1,2}{\mathrm{\zeta}}_{2,1}\text{sin}{\mathrm{\varphi}}_{2}]\text{cos}{\mathrm{\varphi}}_{2}$$(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 *x*–*y* plane (with ϕ_{1} = 90° and ϕ_{2} = 45°) excited with a source field normally incident on the structure at angles, θ_{0} = 0° and θ_{0} = 180° (Fig. 2A). We assume the two Au nanocuboids, separated along the direction of source propagation (*z*) by a distance *d _{z}* =

*d*

_{1,z}−

*d*

_{2,z}= 200 nm and located at

*d*

_{1,y}=

*d*

_{2,x}= 100 nm, to exhibit resonance at wavelengths λ

_{1}= 750 nm and λ

_{2}= 735 nm with ζ

_{1,2}(ω

_{1}) = ζ

_{2,1}(ω

_{2}) = 1.6 × 10

^{29}s

^{−2}. The following values for the plasma frequency, ω

*= 1.37 × 10*

_{p}^{16}s

^{−1}, and damping coefficient, γ = γ

_{1}= γ

_{2}= 1.22 × 10

^{14}s

^{−1}, for Au in the near-infrared region are used (

*32*). Δ

*A*

_{ϵ,ϵ}and Δ

*A*

_{Γ,ϵ}plotted versus incident wavelength λ

_{0}(Fig. 2, B and C) for the two source angles θ

_{0}illustrates that the presence of an inversion in the sign of Δ

*A*

_{ϵ,ϵ}as θ

_{0}is rotated by 180°, which is consistent with Eq. 8.1, where Δ

*A*

_{ϵ,ϵ}(θ

_{0}+ π) = −Δ

*A*

_{ϵ,ϵ}(θ

_{0}). Previous observations of inversion in the sign of the far-field CO response due to θ

_{0}rotation suggest an absence of optical activity in the underlying medium (

*21*,

*30*), verifying our observations, whereas the lack of sign change in the Δ

*A*

_{Γ,ϵ}due to θ

_{0}rotation, where Δ

*A*

_{Γ,ϵ}(θ

_{0}+ π) = Δ

*A*

_{Γ,ϵ}(θ

_{0}), is indicative of optical activity (

*30*). The total response, Δ

*A*, plotted for θ

_{0}= 0° and θ

_{0}= 180°exhibits an asymmetric spectral line shape due to the competing contributions from the Δ

*A*

_{ϵ,ϵ}response, which exhibits a single-fold symmetric line shape, and the Δ

*A*

_{Γ,ϵ}response, which exhibits a twofold symmetric line shape (Fig. 2D), indicating the presence of both CO

_{OA}and CO

_{abs}in the total CO response.

Analogous to the dependence of Δ*A*_{ϵ,ϵ} and Δ*A*_{Γ,ϵ} responses on θ_{0}, further insight can be achieved by analyzing the dependence of the CO response on the azimuth angle ϕ_{0} (for any θ_{0}, except at θ_{0} = 0° and 180°, where ϕ_{0} is undefined). For an identical configuration of Fig. 2A, Δ*A*_{ϵ,ϵ} and Δ*A*_{Γ,ϵ} plotted versus incident wavelength λ_{0} (Fig. 2, E to G) for two source azimuth angles ϕ_{0} = 0° and 180° (at θ_{0} = 45°) illustrates the presence of an inversion in the sign of Δ*A*_{Γ,ϵ} instead, as ϕ_{0} is rotated by 180°. This follows from Eqs. 8.1 and 8.2, where Δ*A*_{ϵ,ϵ}(ϕ_{0} + π) = Δ*A*_{ϵ,ϵ}(ϕ_{0}) and Δ*A*_{Γ,ϵ}(ϕ_{0} + π) = −Δ*A*_{Γ,ϵ}(ϕ_{0}), respectively. This inversion in the Δ*A*_{Γ,ϵ} response can be further described by assuming *d*_{1,z} = *d*_{2,z} = 0 nm to make a two-dimensional structure wherein the spatial dispersion dependence

of Eq. 8.2 simplifies to *kd* sin θ_{0}( sin ϕ_{0} − cos ϕ_{0}), for the two oscillators located equidistant from the origin (*d* = *d*_{1,y} = *d*_{2,x}), demonstrating the dependence of Δ*A*_{Γ,ϵ} on ϕ_{0}.

In addition to the dependence of the CO response on excitation direction, θ_{0} and ϕ_{0}, we analyze its dependence on various oscillator parameters including the angular orientation between the two oscillators along the *x*–*y* plane, by varying angle ϕ_{2} at ϕ_{1} = 90°, and the difference between coupling terms ζ_{2,1}(ω) − ζ_{1,2}(ω), oscillator frequencies Δω = ω_{1} − ω_{2}, and damping coefficients Δγ = γ_{1} − γ_{2}. For this analysis, we assume the light to be normally incident (θ_{0} = 0°) on the two Au nanocuboids, of lengths *l*_{1}and *l*_{2}, that are aligned parallel to the *x*–*y* plane with *d*_{1,y} = *l*_{1}, *d*_{2,x} = *l*_{2} and placed in a planar arrangement with *d*_{1,z} = *d*_{2,z} = 0 nm. In such a planar configuration at normal incidence,

, resulting in Δ*A*_{Γ,ϵ} = 0 (Eq. 8.2). Last, by setting the two resonant wavelengths to be λ_{1} = 750 nm and λ_{2} = 735 nm (corresponding to Δω/γ = 0.42), and assuming ζ_{1,2}(ω) = ζ_{2,1}(ω), the dependence of Δ*A*_{ϵ,ϵ} on ϕ_{2} exhibits a peak response at ϕ_{2} = 45° (Fig. 3A). Note that this observation that a planar two-dimensional plasmonic structure can exhibit a CO_{abs} 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 ϕ_{2} = 52° rather than the expected ϕ_{2} = 45°. A simple inclusion of a nonzero coupling difference, ζ_{2,1} − ζ_{1,2}, between the two oscillators in the model accounts for this behavior wherein by plotting ϕ_{2} that maximizes the Δ*A*_{ϵ,ϵ} response as a function of ζ_{2,1} − ζ_{1,2} at ω = 2.43 × 10^{15} s^{−1} (Fig. 3B), we show that the presence of asymmetric oscillator coupling causes the maximum peak to occur at values other than ϕ_{2} = 45°. The Δ*A*_{ϵ,ϵ} response can also be maximized by optimizing the oscillator frequencies, wherein for ζ_{1,2} − ζ_{2,1} = − 5.2 × 10^{28} s^{−2} corresponding to ϕ_{2} = 52°, the model also predicts a peak Δ*A*_{ϵ,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 (γ_{1} ≠ γ_{2})—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*).

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 (ϕ_{1} = 90° and ϕ_{2} = 0°) with *d*_{1,y} = *d*_{2,x} = *l*/2 and separated by a distance *d _{z}* along the

*z*direction, resulting in ω

_{1}= ω

_{2}= ω and Ω

_{1}= Ω

_{2}= Ω (fig. S3A). Note that, for consistency, the cuboid lengths

*l*were scaled to shift the resonance wavelengths to λ

_{1}= λ

_{2}= 1300 nm. Illumination of the structure at normal incidence, θ

_{0}= 0°, under these conditions results in Δ

*A*

_{ϵ,ϵ}= 0 (from Eq. 8.1). Also, as expected, due to this lack of CO

_{abs}contribution, Δ

*A*= Δ

*A*

_{Γ,ϵ}plotted versus incident wavelength λ

_{0}(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 Δ

*A*

_{Γ,ϵ}versus λ

_{0}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 = (*T*_{RCP} − *T*_{LCP})/(*T*_{RCP} + *T*_{LCP}). CDA also corresponds to the normalized *m*_{14} 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 CO_{OA} and CO_{abs} contributions in a far-field CO measurement; however, this requires measurement of both *m*_{14} and *m*_{41} elements (*17*). As shown below, we verify through model calculations that both CDA and Δ*A* 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 Δ*A*_{ϵ,ϵ} and Δ*A*_{Γ,ϵ} in the three configurations, assuming *d*_{1,z} = *d*_{2,z} = 0 nm and γ_{1} = γ_{2} = γ (Eqs. 8.1 and 8.2), are listed in Fig. 4A (right column). Note that the

term in these planar configurations simplifies to *kd* sin θ_{0}(sin ϕ_{0} − cos ϕ_{0}). The devices, consisting of an array of two Au nanocuboids (thickness *t* = 40 nm) of varying lengths (*l*_{1} and *l*_{2}) and alignments (varying ϕ_{2} at ϕ_{1} = 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 λ_{0} = 500 and 1000 nm under illumination at θ_{0} = 45° for various azimuth angles ϕ_{0} (see Materials and Methods). The first device consisted of the two Au nanocuboids arranged orthogonal to each other (ϕ_{1} = 90° and ϕ_{2} = 0°) and were designed to be of different lengths (*l*_{1} = 120 nm and *l*_{2} = 100 nm placed at *d*_{1,y} = *d*_{2,x} = 100 nm, respectively). Because *l*_{1} and *l*_{2} determine both the resonant frequencies (ω_{1} and ω_{2}) and the cross-coupling strengths (ζ_{1,2} and ζ_{2,1}), setting *l*_{1} ≠ *l*_{2} constitutes a general configuration where both Δ*A*_{ϵ,ϵ} and Δ*A*_{Γ,ϵ} type contributions can be present in a single CDA measurement. The corresponding CDA spectra (Fig. 4B) measured at ϕ_{0} = 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 Δ*A*_{Γ,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 Δ*A*_{ϵ,ϵ} contribution. For ϕ_{0} = 45° and 225°, the spectra lack the sign inversion, indicating the response to primarily result from Δ*A*_{ϵ,ϵ}, which also follows from Fig. 4A, where Δ*A*_{Γ,ϵ} = 0 at these two ϕ_{0} angles. This result is further validated by fabricating a device consisting of Au nanocuboids of equal lengths (*l*_{1} = *l*_{2} = 120 nm), wherein the CDA spectra at ϕ_{0} = 45° and 225° show no CO response, because both Δ*A*_{Γ,ϵ} = Δ*A*_{ϵ,ϵ} = 0, confirming the predictions of the model (Fig. 4A). Moreover, by setting *l*_{1} = *l*_{2}, the twofold symmetry in the CDA line shape at ϕ_{0} = 0° (180°), 90° (270°), and 135° (315°) is recovered, indicating the response to now only consist of Δ*A*_{Γ,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 Δ*A* response (plotted under the conditions of Fig. 4B and fig. S5) verifies our assumption that they are equivalent measurements and can be used interchangeably.

For a device with Au nanocuboids of equal lengths *l*_{1} = *l*_{2} = 120 nm, aligned parallel to each other (ϕ_{1} = 90° and ϕ_{2} = 90°), Eqs. 8.1 and 8.2 predict both Δ*A*_{ϵ,ϵ} and Δ*A*_{Γ,ϵ} to be zero under illumination at θ_{0} = 45° for any ϕ_{0}. Consistent with these predictions, while the CDA spectra measured at ϕ_{0} = 0°(180°) and 90° (270°) show no response, the spectra at ϕ_{0} = 45°(225°) and 135° (315°) show a pronounced signal of the Δ*A*_{ϵ,ϵ} type (no sign inversion for ϕ_{0} rotation by 180°; Fig. 4D). We attribute this phenomenon to originate from coupling to the optical resonances (

and

${\stackrel{\rightharpoonup}{u}}_{2}^{\prime}$) along the cuboid widths (*w*_{1} = *w*_{2} = 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 (θ_{0} ≠ 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 ϕ_{0}, can access these elliptical eigenmodes (Fig. 5, A to D, dashed yellow ellipses). At ϕ_{0} = 0° (180°) or ϕ_{0} = 90° (270°), both orthogonal eigenmodes are accessed equally, resulting in the total CO response to be zero, whereas, at ϕ_{0} = 45°(225°) and 135° (315°), only one of the two eigenmodes can be excited, resulting in a strong CDA response. This dependence of peak ∣Δ*A*_{ϵ,ϵ}∣ on the azimuth angle ϕ_{0} 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 Δ*A*_{Γ,ϵ} response, whereas the Δ*A*_{ϵ,ϵ} response is shown to stay proportional to (ζ_{1′,2} − ζ_{2,1′}). Note that for the CDA calculations in Fig. 4 (B and C), only coupling between the oscillators along their long axis (

and

${\stackrel{\rightharpoonup}{u}}_{2}$) was assumed. The absence of contributions from coupling between the oscillators along their short axis,

${\stackrel{\rightharpoonup}{u}}_{1}^{\prime}$and

${\stackrel{\rightharpoonup}{u}}_{2}^{\prime}$, in the calculations could explain the minor discrepancy between the calculated and experimentally measured CDA spectra.

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 ϕ_{1} = 90° and ϕ_{2} = 45° in a planar arrangement. Upon illumination of this structure at θ_{0} = 45° for various ϕ_{0}, the measured CDA response shows neither any clear inversion in sign with 180° rotation of ϕ_{0} nor any apparent symmetry in the spectral line shape (fig. S6). This is because the various sub-oscillators (

,

${\stackrel{\rightharpoonup}{u}}_{2}$,

${\stackrel{\rightharpoonup}{u}}_{1}^{\prime}$, and

${\stackrel{\rightharpoonup}{u}}_{2}^{\prime}$) in this system are aligned with respect to each other such that they can all be intercoupled, resulting in substantial contributions from both Δ*A*_{Γ,ϵ} and Δ*A*_{ϵ,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 CO_{axial} 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.