## Abstract

The potential for enhancing the optical activity of natural chiral media using engineered nanophotonic components has been central in the quest toward developing next-generation circular-dichroism spectroscopic techniques. Through confinement and manipulation of optical fields at the nanoscale, ultrathin optical elements have enabled a path toward achieving order-of-magnitude enhancements in the chiroptical response. Here, we develop a model framework to describe the underlying physics governing the origin of the chiroptical response in optical media. The model identifies optical activity to originate from electromagnetic coupling to the hybridized eigenstates of a coupled electron-oscillator system, whereas differential absorption of opposite handedness light, though resulting in a far-field chiroptical response, is shown to have incorrectly been identified as optical activity. We validate the model predictions using experimental measurements and show them to also be consistent with observations in the literature. The work provides a generalized framework for the design and study of chiroptical systems.

## INTRODUCTION

Chirality is the geometric property of an object being nonsuperimposable on its mirror image along any symmetry axis and is ubiquitous in the natural world. For example, sugars, proteins, and deoxyribonucleic acids are chiral molecules essential to the functioning and continuation of biological processes. The two variants of a chiral molecule, known as enantiomers, are chemically identical but structured in either a left-handed or a right-handed arrangement. Biological systems on Earth have evolved to prefer left-handed enantiomers—a property referred to as homochirality (*1*). A comprehensive understanding of the evolutionary mechanisms responsible for homochirality remains elusive, but investigations are yielding insights into the origins of life on Earth (*2*) and even in the search for extraterrestrial life (*3*). Many biochemical processes, to function correctly, also require a particular handedness enantiomer. This is observed in the metabolism of pharmaceuticals such as thalidomide (*4*) and penicillamine (*5*), wherein one enantiomer produces medicinal effects and the other toxicity. Thus, enantiomer discrimination techniques such as circular dichroism (CD) spectroscopy are essential for minimizing the toxic effects of medications (*6*, *7*), developing effective treatments for diseases (*8*, *9*), and probing the nature of chiral systems (*10*). In addition to enantiomer discrimination, CD spectroscopy also provides information on protein secondary structures crucial to understanding protein folding (*11*, *12*). This understanding benefits the development of treatments for several deadly diseases such as Alzheimer’s, Parkinson’s, and some cancers (*13*). However, the inherently weak CD response from natural molecular systems, coupled with the limited sensitivity of conventional CD spectroscopic techniques, has placed an upper limit on the overall detection sensitivity. In recent years, engineered ultrathin nanoscale optical devices, composed of an array of metallic or dielectric nanostructures, have been used to enhance the CD response of natural chiral media by several orders in magnitude, suggesting the possibility of next-generation CD spectroscopic techniques with substantially improved measurement sensitivities (*14*, *15*). However, the underlying phenomena governing the microscopic origin of the chiroptical (CO) response from nano-optical devices are still not well understood. Here, we present, and experimentally validate, a generalized model that identifies the fundamental origin of optical activity in a chiral medium and unifies the distinct CO phenomenon observed in literature under a single theoretical framework.

CD is a measure of the optical activity in a CO medium and is characterized by the differential absorption between right and left circularly polarized light (RCP and LCP, respectively). Because chiral media exhibit circular birefringence, optical activity can also be characterized by the degree of rotation of a linearly polarized light as it propagates through it—a phenomenon commonly referred to as optical rotary dispersion (ORD). CD and ORD are both synonymous with optical activity because they originate from the same quantum mechanical phenomenon and are related to each other through the Kramers-Kronig transformation (*16*). We define a generalized far-field CO response of an optical medium as the differential transmission (or reflection) response to RCP and LCP source fields, quantitatively expressed for transmission measurements as CO(ω) = *T*_{RCP}(ω) − *T*_{LCP}(ω), where *T*_{RCP} (*T*_{LCP}) is the spectral intensity transmission for illumination with an RCP (LCP) light. As we demonstrate in this paper, a far-field CO response does not always correspond to CD and can originate from other microscopic phenomenon not related to optical activity. Hence, careful consideration must be given to the interpretation of CO measurements (*17*–*19*).

We identify three primary CO response types that are experimentally characterized and theoretically studied within the framework of an all-purpose, generalized coupled oscillator model described in the next section. We demonstrate optical activity to fundamentally originate from the accessibility of RCP and LCP light to the hybridized energy-shifted eigenstates of a coupled electron-oscillator system—a result that is consistent with the predictions of the Born-Kuhn model (*20*). Subtracting the two energy-shifted spectral responses from one another, upon illumination with RCP and LCP light, respectively, results in a far-field CO response associated with optical activity, which we hereafter refer to as CO_{OA}. Differential absorption to opposite handedness light, not related to optical activity but originating from near-field absorption modes in planar chiral media, has also been shown to produce a far-field CO response, which we refer to as CO_{abs} (*21*, *22*). In contrast to CO_{OA}, CO_{abs} results from a difference in amplitudes between the transmission (or reflection) spectra without any associated spectral shift when subjected to illumination with opposite handedness light (*23*). Last, by using birefringence in an all-dielectric metamaterial acting as a uniaxial or a biaxial medium, a strong far-field CO response has been observed through spatial filtering of either the RCP or the LCP light (*19*, *24*, *25*). This response type, referred to here as CO_{axial}, is also not associated with optical activity in the underlying optical medium. Because the three response types can be present in a single CO measurement, we express the total CO response of optical media as CO = CO_{OA} + CO_{abs} + CO_{axial}, where CO_{OA} ≠ CO_{abs} ≠ CO_{axial}. Note that these phenomena have been separately observed experimentally (*20*–*27*), and the former two are analytically described in previous works (*20*, *22*, *28*); however, independent models have been used to describe them without any clear relation between them. No analytical model has yet successfully described the various types of CO responses observed in literature under a single comprehensive theoretical framework. The model developed here provides an analytical foundation for a generalized CO response from an optical medium and suggests easy-to-implement methods for identifying the presence of, and distinguishing between, the distinct phenomena present in a CO measurement that may or may not originate from optical activity. The model predictions are experimentally validated using far-field CO measurements on engineered nanoscale plasmonic devices at optical frequencies and are shown to also be consistent with observations in the literature.

## RESULTS

### 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

Each oscillator *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

*e*and an effective mass

*m**.

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

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

Equation 4 is expressed using the Einstein summation notation summed over *n* = *x*, *y*, *z*, where each susceptibility vector _{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, *x*-*y* plane (θ_{1} = θ_{2} = π/2) and inserting this into Eq. 4 give *A* = Δ*A*_{ϵ,ϵ} + Δ*A*_{Γ,ϵ}, where

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, **ϵ**(*k*, ω) only depends on ω, whereas the nonlocality tensor **Γ**(*k*, ω) becomes directly proportional to *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)

Note that, in the absence of damping, *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}*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

*= Γ*

_{xx}*= 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

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,

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 *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, *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 *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 (*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 (

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 (*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.

## DISCUSSION

In conclusion, we have developed a comprehensive analytical model to study the microscopic origin of the CO response in optical media. Closed-form expressions for the various microscopic phenomena governing the far-field CO response are shown to provide intuitive insights when systematically studied for various sample geometries and optical excitation conditions. Optical activity, CO_{OA}, characterized in the far field by spectrally shifted transmission (or reflection) curves due to the accessibility of RCP and LCP light to hybridized eigenmodes, is shown to originate at the microscopic scale when coupled oscillators are spatially separated along the direction of source propagation. Differential absorption, CO_{abs}, another CO response type unrelated to optical activity, is characterized in the far field by amplitude-shifted transmission (or reflection) curves due to the presence of distinct near-field absorption modes for RCP and LCP light. CO_{abs} is shown to occur when the oscillators, in the presence of loss, are angularly separated along the direction of source electric field rotation. The third CO response type, CO_{axial}, is characterized in the far field by the spatial separation of RCP and LCP light. CO_{axial} is shown to occur when the Poynting vectors associated with the characteristic RCP and LCP waves of a biaxial medium are angularly offset. Both analytical and experimental methods provided here suggest a simple method for identifying the presence of, and distinguishing between, these various CO response types. As engineered chiral optical media become an essential component of advanced technologies such as enhanced CD spectroscopy, identification of the microscopic behavioral differences in the far-field optical response has become increasingly crucial. The generalized theoretical framework presented here is expected to aid in the application-specific design and study of engineered CO systems.

## MATERIALS AND METHODS

### Device fabrication

The Au nanocuboid structures were fabricated on 500-μm-thick fused-silica substrates. Polymethyl methacrylate (PMMA) resist (100 nm thick) was spun-coated on the substrates, followed by deposition of 20-nm Al film using thermal evaporation as an anti-charging layer. Electron beam lithography at 100 keV was then used to expose the nanocuboid patterns. After exposure, the Al layer was removed using a 60-s bath in a tetramethylammonium hydroxide–based developer followed by a 30-s rinse in deionized water. PMMA was developed for 90 s in methyl isobutyl ketone, followed by a 30-s rinse in isopropyl alcohol. Electron beam evaporation was used to deposit a 2-nm-thick Ti adhesion layer, followed by a 40-nm-thick Au film. A 12-hour soak in acetone was used for liftoff, revealing the completed cuboid structures on the substrate surface. The fabrication steps are schematically outlined in fig. S4.

### Optical characterization

For experimental characterization, the samples were illuminated from free space at wavelengths between λ_{0}= 500 and 1000 nm at a fixed angle θ_{0} = 45° for various source azimuth angles ϕ_{0}. The incident light was focused on the sample to a spot size (along the long axis) of ≈400 μm, and the incident polarization was controlled using an achromatic wave plate. The CDA spectra were directly measured, using a spectroscopic ellipsometer in reflection mode, by extracting the *m*_{14} element of the Mueller matrix.

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/10/eaav8262/DC1

Section S1. Generalized coupled oscillator model parameter definitions

Section S2. Current density response calculation

Section S3. Homogeneous material parameters

Section S4. CO response calculation

Section S5. Dependence of the κ on the difference between oscillator frequencies

Section S6. CO response for two identical, orthogonally oriented nanocuboids at finite *d _{z}*

Section S7. Device fabrication

Section S8. Comparison between calculated CDA and Δ*A* spectral response

Section S9. CO response of 45° oriented cuboids of equal lengths

Section S10. Asymmetric transmission CO response of all-dielectric media

Fig. S1. Arrangement of bi-oscillator molecular unit cells in a representative volume of optical media.

Fig. S2. Dependence of the multiplication factor κ on the difference between oscillator frequencies.

Fig. S3. CO response of orthogonally oriented identical nanocuboids in a three-dimensional arrangement.

Fig. S4. Nanofabrication process steps.

Fig. S5. Comparison between calculated CDA and Δ*A* spectral response.

Fig. S6. Experimental measurements of the CO response of 45° oriented cuboids of equal lengths.

Fig. S7. Isofrequency surfaces and Poynting vectors for the eigenmodes of a biaxial medium.

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**

**Funding:**M.S.D., W.Z., and A.A. acknowledge support under the Cooperative Research Agreement between the University of Maryland and the National Institute of Standards and Technology, Center for Nanoscale Science and Technology, award number 70NANB14H209, through the University of Maryland.

**Author contributions:**The experiments were designed and performed by M.S.D., W.Z., and A.A. Simulations were performed by M.S.D. with further analysis by W.Z., J.K.L., H.J.L., and A.A. Device fabrication and characterization were performed by M.S.D. and W.Z. All authors contributed to the interpretation of results and participated in manuscript preparation.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

- Copyright © 2019 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).