Plasmon-enhanced circular dichroism spectroscopy of chiral drug solutions

We investigate the potential of surface plasmon polaritons at noble metal interfaces for surface-enhanced chiroptical sensing of dilute chiral drug solutions. The high quality factor of surface plasmon resonances in both Otto and Kretschmann configurations enables the enhancement of circular dichroism differenatial absorption thanks to the large near-field intensity of such plasmonic excitations. Furthermore, the subwavelength confinement of surface plasmon polaritons is key to attain chiroptical sensitivity to small amounts of drug volumes placed around $\simeq 100$ nm by the metal surface. Our calculations focus on reparixin, a pharmaceutical molecule currently used in clinical studies for patients with community-acquired pneumonia, including COVID-19 and acute respiratory distress syndrome. Considering realistic dilute solutions of reparixin dissolved in water with concentration $\leq 5$ mg$/$ml, we find a circular-dichroism differential absorption enhancement factor of the order $\simeq 20$ and chirality-induced polarization distortion upon surface plasmon polariton excitation.


I. INTRODUCTION
Chiral sensing plays a key role in pharmaceutics because the specific enantiomeric form of drugs affects their functionality and toxicity 1 .Current state-of-theart techniques capable of measuring enantiomeric excess of chiral mixtures, e.g., nuclear magnetic resonance 2 , gas chromatography 3 and high performance liquid chromatography 4 , provide advanced tools for drug safety, but are designed to operate with macroscopic drug volumes and are not suitable for real-time analysis.Photonic devices can potentially overcome such limitations, but technological advancement in this direction is hampered by the inherently weak chiroptical interaction, which requires ml volumes in order to attain sensitivity to the enantiomeric imbalance of chiral drug solutions through, e.g., polarimetry 5 or electronic/vibrational circular dichroism (CD) 6,7 .Diverse approaches are currently investigated to enhance chiroptical interaction, as in particular ultrafast and nonlinear chiroptical spectroscopy techniques [8][9][10][11] or the exploitation of superchiral fields 12,13 that can be engineered via metasurfaces and nanophotonic structures [14][15][16] .However, the actual exploitation of superchiral fields for novel chiral sensing schemes remains elusive, mainly because most of the experimental sensing demonstrations in literature rely on chiral nanoplasmonic subtrates.The major issue is that when a chiral sample is placed in the near-field of such nanostructures, the detected CD signal is contaminated by the chiroptical response of the nanostructure itself, which acts as a strong background noise and masks the weak chiroptical signal of the actual sample 17 .It has to be noted that chiral drugs diluted in solutions are continuously deformed by the conformational fluctuations of the molecule in interaction with the solvent at finite temperature, modulating over time their chiroptical response.Concurring electric and magnetic resonances in nanostructures provide enhanced CD thanks to the dual enhancement of both electric and magnetic fields 18 .While such schemes are promising for single molecule chiral discrimination, they are difficult to implement for extended drug solutions because CD enhancement is attained only at specific hot spots where electric and magnetic field lines are strong, parallel and π/2 shifted 17,18 .Over a complementary direction, metalbased nanophotonic structures enable plasmon-enhanced CD spectroscopy 19 , which has been predicted for isolated molecules 20 , while it has been adopted mainly to characterise chiral nanoparticles [21][22][23] .
Here we predict that CD differential absorption (CDDA) by dilute drug solutions can get enhanced by a factor f CDDA ≃ 20 via the excitation of surface plasmon polaritons (SPPs) at a planar interface between a noble metal and the chiral sample.In particular, we focus on an isotropic assembly of reparixin (an inhibitor of the CXCR2 function attenuating inflammatory responses 24 that has been adopted in clinical trials for the treatment of hospitalized patients with COVID-19 pneumonia 25 ) dissolved in water with dilute number molecular density of the order n mol = n R mol + n S mol ≃ 10 −2 nm −3 (corresponding to a concentration of 5 mg/ml), where n  respectively.Because SPPs possess a momentum greater than the one of radiation in vacuum, their excitation requires a radiation momentum kick, which can be provided either by a grating 26,27 or by a prism through attenuated total reflection in Kretschmann 28 or Otto 29 configurations.However, the adoption of gratings to excite SPPs modifies significantly the SPP dispersion 30 , and in turn here we focus on silica prism coupling.We observe that CDDA ∆A = A R − A L upon right/left circular polarization excitation is proportional to the enantiomeric excess density ∆n mol = n S mol − n R mol , and in turn by measuring the CDDA it is possible to retrieve ∆n mol .Furthermore, we observe that the maximum plasmoninduced CDDA enhancement factor f CDDA ≃ 20 is attained with silver in the Otto configuration, while other noble metals like gold and copper provide a smaller f CDDA .The vacuum wavelength (λ) dependence of all macroscopic optical parameters [dielectric permittivity ϵ r (λ), magnetic permeability µ r (λ) and chiral parameter κ(λ)] of the isotropic chiral mixture are calculated from first principles.This is accomplished by perturbatively solving the density matrix equations accounting for the leading electronic and vibrational absorption peaks and by averaging the obtained polarizability tensors over the random molecular orientation through the Euler rotaton matrix approach 31,32 .We evaluate plasmon-enhanced CDDA ∆A by classical electrodynamics calculations accounting for (i) silica prism in the Otto and Kretschmann SPP coupling schemes, (ii) the macroscopic bi-anisotropic response of the chiral mixture, and (iii) the dielectric response of noble metals.Our predictions of efficient plasmon-enhanced CDDA by isotropic drug solutions suggest novel avenues for chiroptical sensing in nanoscale environments.
For the present study, it is necessary to make use of quantum molecular observables (i.e., electric and mag- netic moments, electronic and vibrational excitation energies) evaluated for the solvated drug, i.e., aqueous reparixin in our calculations.This is not a trivial task because such quantum molecular observables should be perceived as genuinely quantum expectation values averaged (in statistical-mechanical terms) over a large number of reparaxin/solvent configurations.
For this purpose we have adopted a combination of computational approaches, described to some extent in the Supporting Information and briefly illustrated in Fig. 1.The adopted methods combine semi-classical Molecular Dynamics (MD) simulations (necessary for the reparaxin-solvent conformational sampling), quantum-chemical (QM) calculations and Perturbed Matrix Method (PMM) [33][34][35] which, using the information obtained from MD simulations and QM calculations, provides the observables of interests expressed as ensemble averages.This approach is necessary for including in our model subtle but relevant effects produced by the semi-classical atomic-molecular motions (both reparaxin and water).

II. ELECTRONIC AND VIBRATIONAL STRUCTURE OF REPARIXIN IN WATER
In Figs.2a-c we illustrate in more detail the key-steps for evaluating the quantum-molecular observables of interest.As previously described in Fig. 1 and in the supporting information, the reparixin electric/magnetic transition dipole moments and the corresponding absorption UV/vibrational spectra are evaluated as ensemble averages of solvated reparaxin in water.These high probability conformation states are representative of the free-energy basins b k , where k = 1 − 4, located through MD simulations and ED analysis and depicted in Fig. 2 in the case of reparixin S enantiomer.We underline that, due to selection rules, we consider only two vibrational states (labelled by the index ν = 0, 1) producing resonant interaction with external radiation in the midinfrared (mid-IR) with vibrational energy hω v independent over the reparixin conformation state k.Indeed, we found that hω v is practically unaffected by the molecular configuration state, i.e., the nature of the vibrational mode (calculated as described in the supporting information) coincides in the four conformation states.For every enantiomer labelled by the index a = R, S, the radiation-unperturbed vibronic structure is approximated by independent electronic/vibrational Hamiltonians Ĥel,vib where hω i (k) and hω v (ν + 1/2) are the electronic/vibrational energy eigenvalues with eigenstates |i(k, a)⟩ and |ν(a)⟩.For flexible solvated molecules such as reparixin, the effect of the vibronic transitions in the UV spectral shape is negligible if compared to the one produced by the chromophore and solvent conformational transitions.In turn, the lack of fine structure in the electronic transitions is due to the effect of the semi-classical molecular vibrations and solvent perturbations producing a spectral broadening that prevents their observation.Furthermore, we account only for the electronic states i = 1 − 5 because we find that the corresponding energy levels are the only ones producing resonant electronic absorption peaks in the ultraviolet (UV).Fig. 2b illustrates the schematic vibronic structure of reparixin for a generic molecular conformation state.In the electric/magnetic dipole approximation, the molecule interacts with external radiation through the electronic/vibrational perturbing Hamiltonians given by the expressions where E(r, t), B(r, t) are the external radiation electric and magnetic induction fields and indicate the dyadic representation of electric/magnetic electronic/vibrational dipole operators in the set of radiation-unperturbed electronic/vibrational energy states.We emphasize that, for the particular case of reparixin, due to the co-presence of two electrons with opposite spins in the ground state, the static magnetic dipole moments are vanishing for every electron eigenstate (in the Born-Oppenheimer approximation), while static electric dipole moments are finite.In turn, the calculation of vibrational magnetic transition dipole moments m 01 requires going beyond the Born-Oppenheimer approximation, see supporting information for additional details.Note that transition electric dipole moments are

III. MACROSCOPIC BI-ANISOTROPIC RESPONSE OF REPARIXIN IN WATER
For every conformation state k and enantiomeric form a, radiation-induced electron/vibrational dynamics of chiral drugs (reparixin in our calculations) is in turn governed by the density matrix equations where ρk,a = ρel,vib k,a (r, t) is the time-dependent electronic/vibrational density matrix and L(ρ el,vib k,a ) is the Lindblad operator accounting for interaction with the thermal bath.The Lindblad relaxation rates are obtained by fitting the bandwidth of the absorption spectra, see supporting information for further details.For monochromatic external radiation fields V(r, t) = Re V 0 (r)e −iωt with carrier vacuum wavelength λ = 2πc/ω, where c is the speed of light in vacuum and V = E, B, respectively, Eq. ( 4) is solved perturbatively at first-order yielding ρel,vib k,a (r, t) 36 .The expectation value of the induced electric/magnetic dipole moments expressed as an ensemble average considering all the reparaxin conformation states for each enantiomer form a (see Fig. 1), is in turn given by o el,vib , where p(k) = 0.25, see Fig. 2a,  and o = d, m, respectively.Owing to the random molecular orientation, the induced electric/magnetic dipole moments averaged over arbitrary rotations through the Euler rotation matrix approach 31,32 are given by the expressions where ϵ 0 , µ 0 are the vacuum permittivity and permeability, respectively, and α e (ω), α b (ω), α    have been extracted by the frequency bandwidth of the UV absorption spectrum and are reported in Tab.S1 along with the electronic transition energies, the electric/magnetic transition dipole moment moduli and the chiral projections expressed as ensemble averages.While in principle such a procedure might provide inconsistent results, we find that averaging over time the full microscopic polarizabilities (we calculate ensemble averages as time averages in our ergodic assumptions) or adopting individually averaged electric/magnetic transition dipole moduli, chiral projections and transition energies provides consistent results, see Fig. S1 for comparison.
In the limit of dilute drug solution, reparixin dissolved in water can be treated as an effective-medium with bianisotropic macroscopic response by introducing the polarization P(r, t) = All plots are obtained for a silver substrate with dielectric constant ϵm(λ) (taken from Ref. 39 ), fixed solvate aqueous reparixin thickness d = 500 nm and fixed total molecular number density n mol = 10 −2 nm −3 .The dashed white curves in (b,c) indicate the SPP dispersion relation.

bi-anisotropic dispersion relations
where ϵ r (λ) = ϵ H 2 O (λ) + n mol α e (λ)/ϵ 0 is the relative dielectric permittivity, κ(λ) = i mol .Note that ϵ r (λ) = ϵ H 2 O (λ) + ∆ϵ r (λ) accounts for both the dielectric permittivity of the solvent (water) ϵ H 2 O (λ) 37 and the calculated correction produced by reparixin ∆ϵ r (λ).In Fig. 3a-f we illustrate the dependence of ∆ϵ r (λ), κ(λ) and µ r (λ) of a solution of pure S reparixin enantiomers dissolved in water (n (R) mol = 0) over the molecular number density n mol and the vacuum wavelength of impinging radiation λ.Note that the chiroptical response of reparixin is affected by the tails of electronic (resonant at λ ≃ 250 nm) and vibrational transitions (resonant at λ ≃ 6µm).The real part of ϵ r (λ) is the leading contribution to the refractive index dispersion n(λ) = Re ϵ r (λ) , which is dominated by the refractive index of water due to the dilute molecular number density 10 −3 nm −3 < n mol 10 −1 nm −3 , see Figs. 3a,d.The chirally extinction coefficient k ext (λ) = Im ϵ r (λ) is affected mainly by the imaginary part of ϵ r (λ) is also dominated by the absorption of water, see Figs. 3a,d.The relative magnetic permeability µ (λ) arises from the reparixin magnetic response but is chirally insensitive and provides only a tiny correction to the mixture absorption and dispersion, see Figs. 3b,e.The dependence of the complex dimensionless chiral parameter κ(λ) over vacuum wavelength λ and molecular number density n mol is depicted in Figs.3c,f.Note that the chiral parameter is responsible for the rotatory power (proportional to Re[κ(λ)]) and CD (proportional to Im[κ(λ)]), and is maximised at the electronic/vibrational resonances of reparixin.

IV. PLASMON-ENHANCED CD SPECTROSCOPY
Fig. 4a illustrates the considered Otto coupling scheme composed of a silica prism and a noble metal substrate separated by a distance d (100nm< d < 1µm) embedding the chiral drug solution to be analysed, reparixin dissolved in water in our calculations.We consider monochromatic radiation with vacuum wavelength λ, incidence angle θ and arbitrary polarization given by the superposition of right (s = +1) and left (s = −1) circular components, impinging from a BK7 silica prism with relative dielectric permittivity ϵ g (λ) 38 (z < 0, see Fig. 4a) on a layer of reparixin dissolved in water [with bi-anisotropic response given by Eqs.(7a,7b)] placed at 0 < z < d and a metal substrate with dielectric constant ϵ m (λ) in z > d.Because circular polarization waves are eigenfunctions of Maxwell's equations in the considered chiral medium, we decompose the impinging wave on circular polarization unit vectors ê(0) s (θ) = (cosθê x + isê y − sinθê z )/ √ 2 and investigate the scattering dynamics of circular polarization waves by analytically solving macroscopic Maxwell's equations accounting for the distinct polarization and magnetization fields in every medium of the considered geometry.We obtain where the subscript/superscript "o" indicates the considered Otto configuration, Θ(z) is the Heaviside step 0,±1 are the projections of the impinging vectorial amplitude over ê(0) is the polarization-dependent wave-vector z-component leading to optical activity and k g,m = k 2 0 ϵ g,m − k 2 x .In order to calculate the s-dependent forward (σ = +1) and backward (σ = −1) amplitudes within the chiral medium T,s ) field amplitudes, we apply the boundary conditions (BCs) for the continuity of (i) the normal components of the displacement vector and induction magnetic field, and (ii) the tangential components of the electric and magnetic fields at the interfaces z = 0, d.Such BCs provide an 8 × 8 inhomogeneous system of algebraic equations for the field amplitudes, see Eq. (S1) in the supporting information, which we invert numerically, obtaining E and κ = 0, electromagnetic excitations of the system, see Fig. 4a, can be split into independent transverse magnetic (TM) and transverse electric (TE) components.Conversely, owing to the chirality of reparixin enantiomers, non-racemic mixtures such that n TM over λ, θ considering silver as a metal substrate.Note that the absorbance is maximised at the SPP excitation curve which is indicated by the white lines in Figs.4b,c.As a consequence of SPP resonant absorption CDDA is enhanced at the SPP dispersion curve, see Fig. 4c where we plot ∆A (k) as a function of λ, θ.For right/left circular polarization excitation, also a Fabry-Perot resonance is excited at 600nm< λ < 800nm and 50 • < θ < 60 • , which enhances CDDA by a factor f CDDA ≃ 10.

V. DISCUSSION
Non-racemic mixtures such that n mol produce mixed polarization dynamics analogously to optical rotation in bulk chiral media.The reflected radiation polarization can be fully tracked through the Stokes parameters whose dependence over the incidence angle θ is depicted in Figs.5a-c at λ = 700 nm for a silver substrate.At the SPP excitation angle θ SPP ≃ 70 • polarization modulation is maximised for both circular and TM excitation polarization.Moreover, at θ ≃ 50 • polarization dynamics is also enhanced due to Fabry-Perot resonance of solvated reparixin.Such a phenomenon does not occur when the drug solution is probed in the Kretschmann coupling scheme, schematically depicted in Fig. 6a.Here, ana- lytical solutions of Maxwell's equations provide where Note that in the expressions above the roman subscript/superscript "k" indicates the considered Kretschmann configuration.Again, applying BCs one gets an 8 × 8 inhomogeneous system of algebraic equations for the field amplitudes, see Eq. (S10) in the supporting information, which we invert numerically, obtaining T,±1 for any impinging field components E − .Similarly to the Otto coupling scheme, the TM absorbance TM and (c) ∆A (k) over λ, θ considering a silver thin film with thickness d m = 80 nm.Note that, similarly to the Otto coupling scheme, the TM absorbance and the CDDA are maximised at the SPP excitation curve θ SPP (λ), indicated by the white lines in Figs.6b,c.Conversely to the Otto coupling scheme, Fabry-Perot resonances are not excited due to the intermediate metallic film.However, polarization dependent total interal reflection between glass and aqueous reparixin produces a CDDA absorption peak at θ ≃ 61 • .It is worth emphasizing that such a phenomenon occurs only thanks to the subwavelength metallic thickness d m << λ, enabling effective evanescent coupling between glass and aqueous reparixin.Also, note that the Kretschmann CDDA is about two orders of magnitude smaller than the Otto CDDA for identical total molecular number density n mol and enantiomeric number density imbalance ∆n mol .We illustrate the dependence of the Kretschmann Stokes parameters [S over the incidence angle θ in Figs.6d-f at λ = 700 nm for a silver thin film with thickness d m = 80 nm and a chi- ral medium with n mol = 10 −2 nm −3 and ∆n mol = n mol .Again, note that the Stokes parameters modulation is much weaker in the Kretschmann coupling scheme, similarly to CDDA.
In order to quantify the plasmon-induced CDDA enhancement, we define the factor f CDDA is the absorbance of a layer of reparixin dissolved in water in the absence of the nanophotonic structure and β Figs. 7a-c we depict the vacuum wavelength dependence of the CDDA enhancement for silver, gold and copper interfaces in the Otto coupling scheme.Note that the maximum plasmon-induced CDDA enhancement f CDDA ≃ 20 is attained in the Otto configuration at λ ≃ 400 nm adopting silver as metal substrate, see Fig. 7c.However, for λ > 500 nm, gold and copper provide CDDA enhancement factors f CDDA ≃ 5 comparable with silver.For the considered dilute molecular number density n mol ≃ 10 −2 nm −3 , we find that plasmon-enhanced CD spectroscopy is a viable chiroptical sensing platform able to discern ≲ 10 13 R, S enantiomers in the considered V int .We note that, in the considered Otto and Kretschmann coupling schemes, the transmitted intensity vanishes due to absorption because we assumed infinitely extended metal/chiral-sample for z > d, d m .However, in practical experimental realizations, the physical dimensions of the metal/chiral-sample is finite and the transmitted intensity is non-vanishing.In turn, experimentally, absorbance can be measured as A = 1 − I R /I 0 − I T /I 0 , where I R,T,0 indicate the measured reflected (I R ), transmitted (I T ), and impinging (I 0 ) intensities.CDDA can be in turn measured as the absorbance difference upon left/right impinging polarization.The dissymmetry factor can be measured as the CDDA divided by the averaged absorbance upon left and right impinging polarization.We emphasize that, while the CDDA is enhanced due to SPP excitation, the dissymmetry factor remains unaffected by the nanostructure.Indeed, the role played by SPPs lies in the enhancement of the local electric field enabling both higher averaged absorbance and higher CDDA, while maintaining their ratio (dissymmetry factor) unaffected.In turn, the proposed CD spectroscopic schemes are distinct from superchirality approaches [12][13][14][15][16] aiming at enhancing the dissymmetry factor .Nevertheless, importantly the enhanced CDDA enables reducing the volume of the drug solution to be analysed, as illustrated above.

VI. CONCLUSIONS
We have investigated the potential of SPPs at interfaces between noble metals and a water solution of reparixin for plasmon-enhanced CD spectroscopy.The considered drug solution is composed of a realistic molecule adopted in clinical studies for patients with communityacquired pneumonia, including COVID-19 and acute respiratory distress syndrome.Our calculations of reparixin microscopic observables are based on MD simulations, TD-DFT and PMM simulations enabling the evaluation of time-averaged permanent and transition electric/magnetic dipole moments.Macroscopic optical parameters of the isotropic chiral mixture are calculated by perturbatively solving the density matrix equations.Our results indicate that SPP excitation enables plasmoninduced CDDA enhancement of the order f CDDA ≃ 20.

4 FIG. 1 .
FIG. 1. Step 1. MD simulation of reparaxin in water.Step 2. (2a) Conformational analysis of the trajectory from Step 1 through Essential Dynamics (see supporting information).(2b) Extraction of the (four) reparaxin conformation states, representative of high-probability basins (see, Fig. 2), for TD-DFT calculations (in the gas-phase) aimed at obtaining the unperturbed basis-set, of size Ns, necessary for Step 3c.Step 3. (3a) For each reparaxin conformation state a further MD simulation is carried out with solute kept frozen and the solvent allowed to move.3b The electric field exerted by the solvent is evaluated onto the reparaxin center of mass at each frame of the trajectory of the previous section.3c The basis set obtained in the Step 2b and the electric field from the Step 3b Perturbed Matrix Method (see supporting information) is applied for evaluating, at each frame of the trajectory of Step 3a, the instantaneous expectation value of interest (electric dipole, electronic energy, vibrational energy) or the instantaneous values of electric or magnetic transition moment.The latter are used to obtain the UV and/or CD spectrum, either electronic or vibrational.Step 4. According to the probability of conformation state basins, all the above observables (including the spectra), the observables -assuming an ergodic behaviour of the produced trajectories -are calculated as time-averages and represent ensemble averages.

2 FIG. 2 .
FIG. 2. (a) Calculated relative free-energy at T = 298 K of the reparixin pure S enantiomer (dissolved in water) through MD simulations followed by Essential Dynamics analysis (see supporting information for details), indicating the co-existence of four distinct, statistically relevant conformation state basins b k , where k = 1 − 4. P 1 and P 2 represent the projections of the solvated reparaxin position vectors (sampled during the MD simulation) onto the eigenvectors of the all-atoms covariance matrix (see supporting information for additional details).(b) Sketch of reparixin vibronic dynamics produced by a driving optical laser field in the electric/magnetic dipole approximation.(c) Temporal evolution of electronic transition dipole moduli |d 1,2 |, |m 1,2 |, and chiral projection |d 1,2 • m 1,2 | (in atomic units) of a single reparixin molecule in the b 1 conformation state calculated at each temporal frame (expressed in nanoseconds) of the MD simulation through PMM calculations (see supporting information for additional details).We emphasize that the MD simulation box size is ≃ 3 nm lateral size and ≃ 27 nm 3 volume, and is unrelated to the 100 nm thickness of the chiral sample used.
real polar vectors u(k, a) ∈ ℜ 3 , while transition magnetic dipole moments are purely imaginary axial vectors w(k, a) ∈ ℑ 3 , where we indicate with u(k, a) either d 1i (k, a) or d 01 (a), and with w(k, a) either m 1i (k, a) or m 01 (a), depending over the electronic/vibrational dynamics considered.We define the reflection operator R n = I − 2nn by the plane perpendicular to the unit vector n where reflection symmetry is broken.Transition dipole moments of opposite enantiomers can be calculated by applying the reflection operator: u(k, S) = R nu(k, R), w(k, S) = −R nw(k, R).In turn, while the modulus of the transition dipole moments is unaffected by the enantiomeric type |u(k, S)|= |u(k, R)|, |w(k, S)| = |w(k, R)|, the scalar product u(k, S) • w(k, S) = −u(k, R) • w(k,R) flips sign for opposite enantiomeric forms.Such a quantity, which we call chiral projection, is in turn chirally sensitive as it involves the scalar product of a polar and an axial vector.In Fig.2cwe report the MDinduced temporal evolution of electronic contributions are the linear isotropic electric, magnetic and mixing polarizabilities of every molecular enantiomer, given by the expressions α (a) e,b,m (ω) = C (vib) e,b,m (a)F (vib) (ω) + (j,e,b,m (k, a)F (el) j,k (ω).

FIG. 4 .
FIG. 4. (a) Schematic representation of plasmon-enhanced CD in the Otto configuration.The considered chiral sample is a solution of S and R reparixin enantiomers dissolved in water with molecular density n mol .(b,c) Dependence of (b) TM absorbance A (o) TM (θ, λ) (for TM linear polarization excitation) and (c) CDDA ∆A (o) (θ, λ) = A R (θ, λ) − A L (θ, λ) (for circular polarization excitations) over the vacuum wavelength λ and the angle of incidence θ of the impinging radiation in the Otto coupling scheme illustrated in (a).The plots refer to a pure S reparixin enantiomer dissolved in water (with enantiomeric number density imbalance ∆n mol = n (S) mol − n (R) mol = n mol ).(d) Dependence of CDDA over the enantiomeric number density imbalance ∆n mol = n (S) mol − n (R) mol for several distinct incidence angles θ and fixed excitation vacuum wavelength λ = 800 nm.All plots are obtained for a silver substrate with dielectric constant ϵm(λ) (taken from Ref.39 ), fixed solvate aqueous reparixin thickness d = 500 nm and fixed total molecular number density n mol = 10 −2 nm −3 .The dashed white curves in (b,c) indicate the SPP dispersion relation.
)/ϵ 0 c is the chiral parameter and µ r (λ) = 1 + µ 0 n mol α b (λ) is the relative magnetic permeability of the chiral mixture [see supplementary information, where we report a MATLAB script for the calculation of ϵ r , µ r and κ].Because the mixing polarizability flips sign for opposite enantiomers, α (R) m = −α (S) m , the chiral parameter is proportional to the enantiomeric number density imbalance ∆n mol = n (S) mol − n (R) mol .Hence, optical activity vanishes for racemic mixtures where n for any impinging field components E (o) 0,±1 .Note that, for racemic mixtures such that n (S) mol = n (R) mol produce mixed polarization dynamics.The absorbance of the system upon right (s = +1, E (o) 0,− = 0) and left (s = −1, E (o) 0,+ = 0) circular polarization excitation is in turn given by A (o)

FIG. 5 .
FIG. 5. Reflected polarization dynamics considering a silver substrate with dielectric constant ϵm(λ) 39 , vacuum wavelength λ = 700 nm, solvated aqueous reparixin with thickness d = 500 nm and total molecular number density n mol = 10 −2 nm −3 .(a) Evolution of the Stokes parameters as a function of incidence angle θ in the Poincaré sphere for right-circular impinging polarization and pure S solvated aqueous reparixin with enantiomeric number density imbalance ∆n mol = n (S) mol − n (R) mol = n mol .(b,c) Stokes parameters variation between pure S and pure R solvated aqueous reparixin upon (b) right-circular and (c) TM impinging polarization.

FIG. 6 .
FIG. 6.(a) Schematic representation of plasmon-enhanced CD in the Kretschmann configuration.The considered chiral sample is a solution of S and R reparixin enantiomers dissolved in water with molecular density n mol .(b,c) Dependence of (b) TM absorbance A TM (θ, λ) and (c) CDDA ∆A(θ, λ) = A R (θ, λ) − A L (θ, λ) (for circular polarization excitation) over the vacuum wavelength λ and angle of incidence θ of the impinging radiation.Both plots are obtained for a silver thin film with thickness dm = 80 nm, dielectric constant ϵm(λ) (taken from Ref. 39 ) and a chiral medium with total molecular number density n mol = 10 −2 nm −3 and enantiomeric number density imbalance ∆n mol = n (S) mol − n (R) mol = n mol .(d) Evolution of the Stokes parameters as a function of incidence angle θ in the Poincaré sphere for right-circular impinging polarization, pure S solvated aqueous reparixin with enantiomeric number density imbalance ∆n mol = n (S) mol − n (R) mol = n mol and impinging vacuum wavelength λ = 700 nm.(e,f) Stokes parameters variation between pure S and pure R solvated aqueous reparixin upon (e) right-circular and (f) TM impinging polarization for impinging vacuum wavelength λ = 700 nm.

FIG. 7 .
FIG. 7. Plasmon-induced CDDA enhancement factor f CDDA (λ) = max θ |∆A (o,k) |/|∆A bulk | in the Otto coupling scheme for (a) copper, (b) gold and (c) silver substrates.The physical parameters used are identical to the ones used in Figs.4a,b, except the metal type included in ϵm, obtained from Ref.39 for all the considered noble metals.
|d 12 |, |m 12 | and the modulus of the chiral projection |d 12 • m 12 | of the S reparixin enantiomer in conformation state k = 1.Note that such quantities, constituting the key microscopic ingredients producing optical activity, see below, depend over time due to MD.For this reason, in our quantum-mechanical calculations of the macroscopic bi-anisotropic response we use ensemble-averaged < |d 12 | >, < |m 12 | > and < |d 12 • m 12 | > (calculated as time-averages over the ≃ ns MD timescale in our ergodic assumption, see below.