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# Plasmon-enhanced circular dichroism spectroscopy of chiral drug solutions

Matteo Venturi  Department of Physical and Chemical Sciences, University of
L’Aquila, Via Vetoio, 67100 L’Aquila, Italy  Raju Adhikary  Department of
Physical and Chemical Sciences, University of L’Aquila, Via Vetoio, 67100
L’Aquila, Italy  Ambaresh Sahoo  Department of Physical and Chemical Sciences,
University of L’Aquila, Via Vetoio, 67100 L’Aquila, Italy  Carino Ferrante
CNR-SPIN, c/o Dip.to di Scienze Fisiche e Chimiche, Via Vetoio, Coppito
(L’Aquila) 67100, Italy  Isabella Daidone  Department of Physical and Chemical
Sciences, University of L’Aquila, Via Vetoio, 67100 L’Aquila, Italy  Francesco
Di Stasio  Istituto Italiano di Tecnologia, Via Morego 30, Genova 16136, Italy
Andrea Toma  Istituto Italiano di Tecnologia, Via Morego 30, Genova 16136,
Italy  Francesco Tani  Max Planck Institute for the Science of Light,
Staudtstr. 2, 91058 Erlangen, Germany  Hatice Altug  Institute of
Bioengineering, Ecole Polytechnique Federale de Lausanne (EPFL), Lausanne
1015, Switzerland  Antonio Mecozzi  Department of Physical and Chemical
Sciences, University of L’Aquila, Via Vetoio, 67100 L’Aquila, Italy
Massimiliano Aschi  Department of Physical and Chemical Sciences, University
of L’Aquila, Via Vetoio, 67100 L’Aquila, Italy  Andrea Marini  Department of
Physical and Chemical Sciences, University of L’Aquila, Via Vetoio, 67100
L’Aquila, Italy  CNR-SPIN, c/o Dip.to di Scienze Fisiche e Chimiche, Via
Vetoio, Coppito (L’Aquila) 67100, Italy<EMAIL_ADDRESS>
###### Abstract

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.

††preprint: AIP/123-QED

##  I Introduction

Chiral sensing plays a key role in pharmaceutics because the specific
enantiomeric form of drugs affects their functionality and toxicity
ChiralDrugs . Current state-of-the-art techniques capable of measuring
enantiomeric excess of chiral mixtures, e.g., nuclear magnetic resonance NMR ,
gas chromatography Chromatography  and high performance liquid chromatography
HPLC , 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
Polarimetry  or electronic/vibrational circular dichroism (CD) VCD ; VCDD .
Diverse approaches are currently investigated to enhance chiroptical
interaction, as in particular ultrafast and nonlinear chiroptical spectroscopy
techniques BeaulieuNatPhys2018 ; CohenPRX2019 ; SmirnovaNatPhot2019 ;
Smirnova2022  or the exploitation of superchiral fields TangPRL2010 ;
TangScience2011  that can be engineered via metasurfaces and nanophotonic
structures MohammadiACSPhot2018 ; Pellegrini2018 ; Gilroy2019 . 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 Mohammadi2019 . 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 ACSPhotonics2021 . 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 $\pi/2$
shifted Mohammadi2019 ; ACSPhotonics2021 . Over a complementary direction,
metal-based nanophotonic structures enable plasmon-enhanced CD spectroscopy
Nesterov2016 , which has been predicted for isolated molecules Govorov2011 ,
while it has been adopted mainly to characterise chiral nanoparticles Fan2010
; Slocik2011 ; Hu2019 .

Figure 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 $N_{\rm s}$, 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. Figure 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 ${\rm b}_{k}$, 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 $|{\bf d}_{1,2}|$,
$|{\bf m}_{1,2}|$, and chiral projection $|{\bf d}_{1,2}\cdot{\bf m}_{1,2}|$
(in atomic units) of a single reparixin molecule in the ${\rm b}_{1}$
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 $\simeq 3$ nm lateral size and $\simeq 27$ nm3 volume, and is
unrelated to the $100$ nm thickness of the chiral sample used.

Here we predict that CD differential absorption (CDDA) by dilute drug
solutions can get enhanced by a factor $f_{\rm CDDA}\simeq 20$ 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 Bertini2007  that has been adopted in clinical trials
for the treatment of hospitalized patients with COVID-19 pneumonia Landoni2022
) dissolved in water with dilute number molecular density of the order $n_{\rm
mol}=n_{\rm mol}^{\rm R}+n_{\rm mol}^{\rm S}\simeq 10^{-2}$ nm-3
(corresponding to a concentration of $5$ mg$/$ml), where $n_{\rm mol}^{\rm
R,S}$ indicate the number densities of R and S enantiomers, 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 Park2003 ; Offerhaus2005  or by a prism through attenuated
total reflection in Kretschmann Kretschmann  or Otto Otto  configurations.
However, the adoption of gratings to excite SPPs modifies significantly the
SPP dispersion Hooper2004 , and in turn here we focus on silica prism
coupling. We observe that CDDA $\Delta A=A_{\rm R}-A_{\rm L}$ upon right/left
circular polarization excitation is proportional to the enantiomeric excess
density $\Delta n_{\rm mol}=n_{\rm mol}^{\rm S}-n_{\rm mol}^{\rm R}$, and in
turn by measuring the CDDA it is possible to retrieve $\Delta n_{\rm mol}$.
Furthermore, we observe that the maximum plasmon-induced CDDA enhancement
factor $f_{\rm CDDA}\simeq 20$ is attained with silver in the Otto
configuration, while other noble metals like gold and copper provide a smaller
$f_{\rm CDDA}$. The vacuum wavelength ($\lambda$) dependence of all
macroscopic optical parameters [dielectric permittivity $\epsilon_{\rm
r}(\lambda)$, magnetic permeability $\mu_{\rm r}(\lambda)$ and chiral
parameter $\kappa(\lambda)$] 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
Andrews2004 ; Valev2022 . We evaluate plasmon-enhanced CDDA $\Delta A$ 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 magnetic 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) Aschi2001 ; Amadei2009 ; Carrillo2017
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 $\nu=0,1$) producing resonant interaction with external radiation in
the mid-infrared (mid-IR) with vibrational energy $\hbar\omega_{\rm v}$
independent over the reparixin conformation state $k$. Indeed, we found that
$\hbar\omega_{\rm v}$ 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 $\hat{\cal H}_{0}^{\rm el,vib}(k,a)$, given by

$\displaystyle\hat{\cal H}_{0}^{\rm
el}(k,a)=\sum\limits_{i=1}^{5}\hbar\omega_{i}(k)|i(k,a)\rangle\langle
i(k,a)|,$ (1a) $\displaystyle\hat{\cal H}_{0}^{\rm
vib}(a)=\sum\limits_{\nu=0,1}(\nu+1/2)\hbar\omega_{\rm
v}|\nu(a)\rangle\langle\nu(a)|,$ (1b)

where $\hbar\omega_{i}(k)$ and $\hbar\omega_{\rm v}(\nu+1/2)$ are the
electronic/vibrational energy eigenvalues with eigenstates $|i(k,a)\rangle$
and $|\nu(a)\rangle$. 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

$\displaystyle\hat{\cal H}_{\rm I,el}^{(k,a)}({\bf r},t)=-{\bf E}({\bf
r},t)\cdot\hat{\bf d}_{\rm el}(k,a)-{\bf B}({\bf r},t)\cdot\hat{\bf m}_{\rm
el}(k,a),$ (2a) $\displaystyle\hat{\cal H}_{\rm I,vib}^{(a)}({\bf r},t)=-{\bf
E}({\bf r},t)\cdot\hat{\bf d}_{\rm vib}(a)-{\bf B}({\bf r},t)\cdot\hat{\bf
m}_{\rm vib}(a),$ (2b)

where ${\bf E}({\bf r},t)$, ${\bf B}({\bf r},t)$ are the external radiation
electric and magnetic induction fields and

$\displaystyle\hat{\bf d}_{\rm el}(k,a)=\sum\limits_{i,j=1}^{5}{\bf
d}_{i,j}(k,a)|i(k,a)\rangle\langle j(k,a)|,$ (3a) $\displaystyle\hat{\bf
m}_{\rm el}(k,a)=\sum\limits_{i,j=1}^{5}{\bf
m}_{i,j}(k,a)|i(k,a)\rangle\langle j(k,a)|,$ (3b) $\displaystyle\hat{\bf
d}_{\rm vib}(a)=\sum\limits_{\nu,\nu^{\prime}=0,1}{\bf
d}^{\nu,\nu^{\prime}}(a)|\nu(a)\rangle\langle\nu^{\prime}(a)|,$ (3c)
$\displaystyle\hat{\bf m}_{\rm vib}(a)=\sum\limits_{\nu,\nu^{\prime}=0,1}{\bf
m}^{\nu,\nu^{\prime}}(a)|\nu(a)\rangle\langle\nu^{\prime}(a)|$ (3d)

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 ${\bf m}^{01}$ requires going beyond the Born-Oppenheimer
approximation, see supporting information for additional details. Note that
transition electric dipole moments are real polar vectors ${\bf
u}(k,a)\in\Re^{3}$, while transition magnetic dipole moments are purely
imaginary axial vectors ${\bf w}(k,a)\in\Im^{3}$, where we indicate with ${\bf
u}(k,a)$ either ${\bf d}_{1i}(k,a)$ or ${\bf d}^{01}(a)$, and with ${\bf
w}(k,a)$ either ${\bf m}_{1i}(k,a)$ or ${\bf m}^{01}(a)$, depending over the
electronic/vibrational dynamics considered. We define the reflection operator
${\cal R}_{\hat{\bf n}}=\mathbb{I}-2\hat{\bf n}\hat{\bf n}$ by the plane
perpendicular to the unit vector $\hat{\bf n}$ where reflection symmetry is
broken. Transition dipole moments of opposite enantiomers can be calculated by
applying the reflection operator: ${\bf u}(k,{\rm S})={\cal R}_{\hat{\bf
n}}{\bf u}(k,{\rm R})$, ${\bf w}(k,{\rm S})=-{\cal R}_{\hat{\bf n}}{\bf
w}(k,{\rm R})$. In turn, while the modulus of the transition dipole moments is
unaffected by the enantiomeric type $|{\bf u}(k,{\rm S})|=|{\bf u}(k,{\rm
R})|$, $|{\bf w}(k,{\rm S})|=|{\bf w}(k,{\rm R})|$, the scalar product ${\bf
u}(k,{\rm S})\cdot{\bf w}(k,{\rm S})=-{\bf u}(k,{\rm R})\cdot{\bf w}(k,{\rm
R})$ 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. 2c we report the MD-
induced temporal evolution of electronic contributions $|{\bf d}_{12}|$,
$|{\bf m}_{12}|$ and the modulus of the chiral projection $|{\bf
d}_{12}\cdot{\bf m}_{12}|$ 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 $<|{\bf d}_{12}|>$, $<|{\bf
m}_{12}|>$ and $<|{\bf d}_{12}\cdot{\bf m}_{12}|>$ (calculated as time-
averages over the $\simeq$ ns MD timescale in our ergodic assumption, see
below.

Figure 3:  (a-c) Vacuum wavelength $\lambda$ and (d-f) number molecular
density $n_{\rm mol}$ dependencies of (a,d) relative dielectric permittivity
correction $\Delta\epsilon_{\rm r}$, (b,e) relative magnetic permeability
correction $\mu_{\rm r}-1$, and (c,f) chiral parameter $\kappa$ of a water
solution containing only the S enantiomer of reparixin for (a-c) fixed
molecular number density $n_{\rm mol}=10^{-2}$ nm-3 and (d-f) fixed vacuum
wavelength $\lambda=600$ nm.

##  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

$\frac{d\hat{\rho}_{k,a}}{dt}=\frac{1}{i\hbar}\left[\hat{\cal H}_{0}^{\rm
el,vib}(k,a)+\hat{\cal H}_{\rm
I,el,vib}^{(k,a)},\hat{\rho}_{k,a}\right]+\hat{\cal L}(\hat{\rho}_{k,a}),$ (4)

where $\hat{\rho}_{k,a}=\hat{\rho}_{k,a}^{\rm el,vib}({\bf r},t)$ is the time-
dependent electronic/vibrational density matrix and $\hat{\cal
L}(\hat{\rho}_{k,a}^{\rm el,vib})$ 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 ${\bf V}({\bf
r},t)={\rm Re}\left[{\bf V}_{0}({\bf r})e^{-i\omega t}\right]$ with carrier
vacuum wavelength $\lambda=2\pi c/\omega$, where $c$ is the speed of light in
vacuum and ${\bf V}={\bf E},{\bf B}$, respectively, Eq. (4) is solved
perturbatively at first-order yielding $\hat{\rho}_{k,a}^{\rm el,vib}({\bf
r},t)$ Govorov2012 . 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 ${\bf o}^{\rm el,vib}_{a}=\sum\limits_{k=1}^{4}p(k){\rm
Tr}\left[\hat{\rho}_{k,a}^{\rm el,vib}\hat{\bf o}(k,a)\right]$, where
$p(k)=0.25$, see Fig. 2a, and ${\bf o}={\bf d},{\bf m}$, respectively. Owing
to the random molecular orientation, the induced electric/magnetic dipole
moments averaged over arbitrary rotations through the Euler rotation matrix
approach Andrews2004 ; Valev2022  are given by the expressions

$\displaystyle\langle{\bf d}_{a}\rangle=\epsilon_{0}{\rm
Re}\left\\{\left[\alpha_{\rm e}{\bf E}_{0}+c\alpha_{\rm m}^{(a)}{\bf
B}_{0}\right]e^{-i\omega t}\right\\},$ (5a) $\displaystyle\langle{\bf
m}_{a}\rangle=\epsilon_{0}c{\rm Re}\left\\{\left[-\alpha_{\rm m}^{(a)}{\bf
E}_{0}+c\alpha_{\rm b}{\bf B}_{0}\right]e^{-i\omega t}\right\\},$ (5b)

where $\epsilon_{0}$, $\mu_{0}$ are the vacuum permittivity and permeability,
respectively, and $\alpha_{\rm e}(\omega)$, $\alpha_{\rm b}(\omega)$,
$\alpha_{\rm m}^{(a)}(\omega)$ are the linear isotropic electric, magnetic and
mixing polarizabilities of every molecular enantiomer, given by the
expressions

$\displaystyle\alpha_{\rm e,b,m}^{(a)}(\omega)$ $\displaystyle=$
$\displaystyle{\cal C}_{\rm e,b,m}^{\rm(vib)}(a){\cal F}^{\rm(vib)}(\omega)+$
$\displaystyle+$
$\displaystyle\sum\limits_{k=1}^{4}p(k)\sum\limits_{j=2}^{5}{\cal C}_{j,{\rm
e,b,m}}^{\rm(el)}(k,a){\cal F}_{j,k}^{\rm(el)}(\omega).$

In the expression above $p(k)=0.25$, ${\cal C}_{\rm
e}^{\rm(vib,el)}=2\langle|{\cal D}^{\rm(vib,el)}|^{2}\rangle/3\hbar$, ${\cal
C}_{\rm b}^{\rm(vib,el)}=2\langle|{\cal M}^{\rm(vib,el)}|^{2}\rangle/3\hbar$
and ${\cal C}_{\rm m}^{\rm(vib,el)}=2i\langle{\rm Im}[{\cal
D}^{\rm(vib,el)}\cdot{\cal M}^{\rm(vib,el)}]\rangle/3\hbar$, where
$\langle(...)\rangle$ indicates the ensemble-average (calculated as time-
average over the $\simeq$ ns MD time-scale in our ergodic assumption), ${\cal
D}^{\rm(vib)}={\bf d}^{01}$, ${\cal D}^{\rm(el)}_{j}={\bf d}_{1j}$, ${\cal
M}^{\rm(vib)}={\bf m}^{01}$ and ${\cal M}^{\rm(el)}_{j}={\bf m}_{1j}$. The
angular frequency dependence of the polarizabilities is accounted by ${\cal
F}^{\rm(vib)}(\omega)=\displaystyle\sum\limits_{\sigma=\pm
1}\sigma/(\omega+\sigma\langle\omega_{\rm v}\rangle+i\gamma_{\rm v}/2)$ and
${\cal F}_{j,k}^{\rm(el)}(\omega)=\displaystyle\sum\limits_{\sigma=\pm
1}\sigma/[\omega+\sigma\langle\omega_{j}(k)-\omega_{1}(k)\rangle+i\gamma_{j}^{\rm(e)}(k)/2]$,
where $\gamma_{\rm v}=0.012$ fs-1 is the relaxation rate of the only reparixin
vibrational transition $\nu=0\rightarrow 1$ with energy $\hbar\omega_{\rm
v}=0.218$ eV (corresponding to the transition wavelength $\lambda=5.69$
$\mu$m). The calculated electric/magnetic transition dipole moments ${\bf
d}^{01},{\bf m}^{01}$ provide $\sqrt{\langle|{\bf
d}^{01}|^{2}\rangle}/d_{0}=0.105$, $\sqrt{\langle|{\bf
m}^{01}|^{2}\rangle}/m_{0}=0.398$ and $\langle{\rm Im}[{\bf d}^{01}({\rm
S})\cdot{\bf d}^{01}({\rm S})]\rangle/d_{0}m_{0}=0.007$, where
$d_{0}=|e|a_{0}$, $a_{0}=\hbar/m_{\rm e}c\alpha$ is the Bohr radius,
$m_{0}=|e|\hbar/2m_{\rm e}$ is the Bohr magneton, $\alpha$ is the fine-
structure constant and $e,m_{\rm e}$ are the electron charge and mass. The
electronic relaxation rates $\gamma_{j}^{\rm(e)}$ 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.

Figure 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 $n_{\rm mol}$. (b,c)
Dependence of (b) TM absorbance $A_{\rm TM}^{\rm(o)}(\theta,\lambda)$ (for TM
linear polarization excitation) and (c) CDDA $\Delta
A^{\rm(o)}(\theta,\lambda)=A_{\rm R}(\theta,\lambda)-A_{\rm
L}(\theta,\lambda)$ (for circular polarization excitations) over the vacuum
wavelength $\lambda$ and the angle of incidence $\theta$ 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 $\Delta n_{\rm mol}=n_{\rm mol}^{\rm(S)}-n_{\rm
mol}^{\rm(R)}=n_{\rm mol}$). (d) Dependence of CDDA over the enantiomeric
number density imbalance $\Delta n_{\rm mol}=n_{\rm mol}^{\rm(S)}-n_{\rm
mol}^{\rm(R)}$ for several distinct incidence angles $\theta$ and fixed
excitation vacuum wavelength $\lambda=800$ nm. All plots are obtained for a
silver substrate with dielectric constant $\epsilon_{\rm m}(\lambda)$ (taken
from Ref. SilverRef ), fixed solvate aqueous reparixin thickness $d=500$ nm
and fixed total molecular number density $n_{\rm mol}=10^{-2}$ nm-3. The
dashed white curves in (b,c) indicate the SPP dispersion relation.

In the limit of dilute drug solution, reparixin dissolved in water can be
treated as an effective-medium with bi-anisotropic macroscopic response by
introducing the polarization ${\bf P}({\bf r},t)=\sum\limits_{a={\rm
R,S}}n_{\rm mol}^{(a)}\langle{\bf d}_{a}({\bf r},t)\rangle$ and magnetization
${\bf M}({\bf r},t)=\sum\limits_{a={\rm R,S}}n_{\rm mol}^{(a)}\langle{\bf
m}_{a}({\bf r},t)\rangle$ fields. Such vectorial fields depend over the
orientation-averaged Andrews2004 ; Valev2022  electric/magnetic induced dipole
moments $\langle{\bf d}_{a}\rangle$, $\langle{\bf m}_{a}\rangle$ and the
number molecular densities $n_{\rm mol}^{\rm(R,S)}$ of the two enantiomeric
forms $a={\rm R,S}$. In turn, the macroscopic displacement vector ${\bf
D}({\bf r},t)=\epsilon_{0}{\bf E}({\bf r},t)+{\bf P}({\bf r},t)$ and magnetic
field ${\bf H}({\bf r},t)={\bf B}({\bf r},t)/\mu_{0}-{\bf M}({\bf r},t)$ are
given by the bi-anisotropic dispersion relations

$\displaystyle{\bf D}=\epsilon_{0}{\rm Re}\left\\{\left[\epsilon_{\rm
r}(\lambda){\bf E}_{0}({\bf r})-i\mu_{0}c\kappa(\lambda){\bf H}_{0}({\bf
r})\right]e^{-i\omega t}\right\\},$ (7a) $\displaystyle{\bf B}=\mu_{0}{\rm
Re}\left\\{\left[\frac{i\kappa(\lambda)}{\mu_{0}c}{\bf E}_{0}({\bf
r})+\mu_{\rm r}(\lambda){\bf H}_{0}({\bf r})\right]e^{-i\omega t}\right\\},$
(7b)

where $\epsilon_{\rm r}(\lambda)=\epsilon_{\rm H_{2}O}(\lambda)+n_{\rm
mol}\alpha_{\rm e}(\lambda)/\epsilon_{0}$ is the relative dielectric
permittivity, $\kappa(\lambda)=i\sum\limits_{a={\rm R,S}}n_{\rm
mol}^{(a)}\alpha_{\rm m}^{(a)}(\lambda)/\epsilon_{0}c$ is the chiral parameter
and $\mu_{\rm r}(\lambda)=1+\mu_{0}n_{\rm mol}\alpha_{\rm b}(\lambda)$ is the
relative magnetic permeability of the chiral mixture [see supplementary
information, where we report a MATLAB script for the calculation of
$\epsilon_{\rm r},\mu_{\rm r}$ and $\kappa$].

Because the mixing polarizability flips sign for opposite enantiomers,
$\alpha_{\rm m}^{\rm(R)}=-\alpha_{\rm m}^{\rm(S)}$, the chiral parameter is
proportional to the enantiomeric number density imbalance $\Delta n_{\rm
mol}=n_{\rm mol}^{\rm(S)}-n_{\rm mol}^{\rm(R)}$. Hence, optical activity
vanishes for racemic mixtures where $n_{\rm mol}^{\rm(R)}=n_{\rm
mol}^{\rm(S)}$. Note that $\epsilon_{\rm r}(\lambda)=\epsilon_{\rm
H_{2}O}(\lambda)+\Delta\epsilon_{\rm r}(\lambda)$ accounts for both the
dielectric permittivity of the solvent (water) $\epsilon_{\rm
H_{2}O}(\lambda)$ WaterRef  and the calculated correction produced by
reparixin $\Delta\epsilon_{\rm r}(\lambda)$. In Fig. 3a-f we illustrate the
dependence of $\Delta\epsilon_{\rm r}(\lambda)$, $\kappa(\lambda)$ and
$\mu_{\rm r}(\lambda)$ of a solution of pure S reparixin enantiomers dissolved
in water ($n_{\rm mol}^{\rm(R)}=0$) over the molecular number density $n_{\rm
mol}$ and the vacuum wavelength of impinging radiation $\lambda$. Note that
the chiroptical response of reparixin is affected by the tails of electronic
(resonant at $\lambda\simeq 250$ nm) and vibrational transitions (resonant at
$\lambda\simeq 6\mu$m). The real part of $\epsilon_{\rm r}(\lambda)$ is the
leading contribution to the refractive index dispersion $n(\lambda)={\rm
Re}\left[\sqrt{\epsilon_{\rm r}(\lambda)}\right]$, which is dominated by the
refractive index of water due to the dilute molecular number density $10^{-3}$
nm${}^{-3}<n_{\rm mol}<10^{-1}$ nm-3, see Figs. 3a,d. The chirally insensitive
extinction coefficient $k_{\rm ext}(\lambda)={\rm Im}\left[\sqrt{\epsilon_{\rm
r}(\lambda)}\right]$ is affected mainly by the imaginary part of
$\epsilon_{\rm r}(\lambda)$ and is also dominated by the absorption of water,
see Figs. 3a,d. The relative magnetic permeability $\mu_{\rm r}(\lambda)$
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
$\kappa(\lambda)$ over vacuum wavelength $\lambda$ and molecular number
density $n_{\rm mol}$ is depicted in Figs. 3c,f. Note that the chiral
parameter is responsible for the rotatory power (proportional to ${\rm
Re}[\kappa(\lambda)]$) and CD (proportional to ${\rm Im}[\kappa(\lambda)]$),
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 $d$
($100$nm$<d<1\mu$m) embedding the chiral drug solution to be analysed,
reparixin dissolved in water in our calculations. We consider monochromatic
radiation with vacuum wavelength $\lambda$, incidence angle $\theta$ and
arbitrary polarization given by the superposition of right ($s=+1$) and left
($s=-1$) circular components, impinging from a BK$7$ silica prism with
relative dielectric permittivity $\epsilon_{\rm g}(\lambda)$ SilicaRef
($z<0$, 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 $\epsilon_{\rm m}(\lambda)$ 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 $\hat{\bf e}_{s}^{(0)}(\theta)=({\rm
cos}\theta\hat{\bf e}_{x}+is\hat{\bf e}_{y}-{\rm sin}\theta\hat{\bf
e}_{z})/\sqrt{2}$ 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

Figure 5: Reflected polarization dynamics considering a silver substrate with
dielectric constant $\epsilon_{\rm m}(\lambda)$SilverRef , vacuum wavelength
$\lambda=700$ nm, solvated aqueous reparixin with thickness $d=500$ nm and
total molecular number density $n_{\rm mol}=10^{-2}$ nm-3. (a) Evolution of
the Stokes parameters as a function of incidence angle $\theta$ in the
Poincaré sphere for right-circular impinging polarization and pure S solvated
aqueous reparixin with enantiomeric number density imbalance $\Delta n_{\rm
mol}=n_{\rm mol}^{\rm(S)}-n_{\rm mol}^{\rm(R)}=n_{\rm mol}$. (b,c) Stokes
parameters variation between pure S and pure R solvated aqueous reparixin upon
(b) right-circular and (c) TM impinging polarization.

$\displaystyle{\bf E}_{\rm o}({\bf r},t)=\sum\limits_{s=\pm 1}{\rm
Re}\left\\{\left[{\bf E}_{<}^{({\rm o},s)}({\bf r})\Theta(-z)+\right.\right.$
(8) $\displaystyle\left.\left.+{\bf E}_{\rm in}^{({\rm o},s)}({\bf
r})\Theta_{\rm in}(z)+{\bf E}_{>}^{({\rm o},s)}({\bf
r})\Theta(z-d)\right]e^{-i\omega t}\right\\},$ $\displaystyle{\bf H}_{\rm
o}({\bf r},t)=\sum\limits_{s=\pm 1}{\rm
Re}\left\\{\frac{s}{i\mu_{0}c}\left[\sqrt{\epsilon_{\rm g}}{\bf E}_{<}^{({\rm
o},s)}({\bf r})\Theta(-z)+\right.\right.$ (9)
$\displaystyle\left.\left.+\frac{\sqrt{\epsilon_{\rm r}}}{\sqrt{\mu_{\rm
r}}}{\bf E}_{\rm in}^{({\rm o},s)}({\bf r})\Theta_{\rm
in}(z)+\sqrt{\epsilon_{\rm m}}{\bf E}_{>}^{({\rm o},s)}({\bf
r})\Theta(z-d)\right]e^{-i\omega t}\right\\},$

where the subscript/superscript “${\rm o}$” indicates the considered Otto
configuration, $\Theta(z)$ is the Heaviside step function, $\Theta_{\rm
in}(z)=\Theta(z)-\Theta(z-d)$,

$\displaystyle{\bf E}_{<}^{({\rm o},s)}({\bf
r})=\left[E_{0,s}^{\rm(o)}\hat{\bf e}_{s}^{(0)}e^{ik_{\rm g}z}+E_{{\rm
R},s}^{\rm(o)}\hat{\bf e}_{s}^{\rm(R)}e^{-ik_{\rm g}z}\right]e^{ik_{x}x},$
(10a) $\displaystyle{\bf E}_{\rm in}^{({\rm o},s)}({\bf
r})=\sum\limits_{\sigma=\pm
1}E_{\sigma,s}^{\rm(o)}\left[\sigma\hat{e}_{x}-i\frac{k_{0}}{\beta_{s}}(\kappa-s\sqrt{\epsilon_{\rm
r}\mu_{\rm r}})\hat{e}_{y}+\right.$
$\displaystyle\left.-\frac{k_{x}}{\beta_{s}}\hat{e}_{z}\right]e^{ik_{x}x+i\sigma\beta_{s}z},$
(10b) $\displaystyle{\bf E}_{>}^{({\rm o},s)}({\bf
r})=\frac{1}{\sqrt{2}}E_{{\rm T},s}^{\rm(o)}\left[\sqrt{1-\frac{\epsilon_{\rm
g}}{\epsilon_{\rm m}}{\rm sin}^{2}\theta}\hat{\bf e}_{x}+is\hat{\bf
e}_{y}+\right.$ $\displaystyle\left.-\sqrt{\epsilon_{\rm g}/\epsilon_{\rm
m}}{\rm sin}\theta\hat{\bf e}_{z}\right]e^{ik_{x}x+ik_{\rm m}z},$ (10c)

Figure 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 $n_{\rm
mol}$. (b,c) Dependence of (b) TM absorbance $A_{\rm TM}(\theta,\lambda)$ and
(c) CDDA $\Delta A(\theta,\lambda)=A_{\rm R}(\theta,\lambda)-A_{\rm
L}(\theta,\lambda)$ (for circular polarization excitation) over the vacuum
wavelength $\lambda$ and angle of incidence $\theta$ of the impinging
radiation. Both plots are obtained for a silver thin film with thickness
$d_{\rm m}=80$ nm, dielectric constant $\epsilon_{\rm m}(\lambda)$ (taken from
Ref. SilverRef ) and a chiral medium with total molecular number density
$n_{\rm mol}=10^{-2}$ nm-3 and enantiomeric number density imbalance $\Delta
n_{\rm mol}=n_{\rm mol}^{\rm(S)}-n_{\rm mol}^{\rm(R)}=n_{\rm mol}$. (d)
Evolution of the Stokes parameters as a function of incidence angle $\theta$
in the Poincaré sphere for right-circular impinging polarization, pure S
solvated aqueous reparixin with enantiomeric number density imbalance $\Delta
n_{\rm mol}=n_{\rm mol}^{\rm(S)}-n_{\rm mol}^{\rm(R)}=n_{\rm mol}$ and
impinging vacuum wavelength $\lambda=700$ 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
$\lambda=700$ nm.

$\omega=2\pi c/\lambda$ is the angular frequency, $k_{0}=\omega/c$, $E_{0,\pm
1}^{\rm(o)}$ are the projections of the impinging vectorial amplitude over
$\hat{\bf e}_{\pm 1}^{(0)}$, $\hat{\bf e}_{s}^{\rm(R)}=(-{\rm
cos}\theta\hat{\bf e}_{x}+is\hat{\bf e}_{y}-{\rm sin}\theta\hat{\bf
e}_{z})/\sqrt{2}$, $k_{x}=k_{0}\sqrt{\epsilon_{\rm g}}{\rm sin}\theta$ is the
conserved impinging ${\bf k}$-vector $x$-component,
$\beta_{s}=\sqrt{k_{0}^{2}(\kappa-s\sqrt{\epsilon_{\rm r}\mu_{\rm
r}})^{2}-k_{x}^{2}}$ is the polarization-dependent wave-vector $z$-component
leading to optical activity and $k_{\rm g,m}=\sqrt{k_{0}^{2}\epsilon_{\rm
g,m}-k_{x}^{2}}$. In order to calculate the $s$-dependent forward
($\sigma=+1$) and backward ($\sigma=-1$) amplitudes within the chiral medium
$E_{\sigma,s}^{\rm(o)}$, the reflected ($E_{{\rm R},s}^{\rm(o)}$) and
transmitted ($E_{{\rm T},s}^{\rm(o)}$) 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 $8\times 8$ inhomogeneous system of algebraic equations for the
field amplitudes, see Eq. (S1) in the supporting information, which we invert
numerically, obtaining $E_{{\rm R},\pm 1}^{\rm(o)},E_{{\rm T},\pm 1}^{\rm(o)}$
for any impinging field components $E_{0,\pm 1}^{\rm(o)}$. Note that, for
racemic mixtures such that $n_{\rm mol}^{\rm(S)}=n_{\rm mol}^{\rm(R)}$ and
$\kappa=0$, 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 $n_{\rm mol}^{\rm(S)}\neq n_{\rm mol}^{\rm(R)}$
produce mixed polarization dynamics. The absorbance of the system upon right
($s=+1$, $E_{0,-}^{\rm(o)}=0$) and left ($s=-1$, $E_{0,+}^{\rm(o)}=0$)
circular polarization excitation is in turn given by
$A_{s}^{\rm(o)}(\theta,\lambda)=1-\sum\limits_{\sigma=\pm 1}|E_{{\rm
R},\sigma}^{\rm(o)}|^{2}/|E_{0,s}^{\rm(o)}|^{2}$, from which we calculate the
CDDA $\Delta A^{\rm(o)}(\theta,\lambda)=A_{+}^{\rm(o)}-A_{-}^{\rm(o)}$. The TM
absorbance $A_{\rm TM}^{\rm(o)}(\theta,\lambda)=1-\sum\limits_{\sigma=\pm
1}|E_{{\rm R},\sigma}^{\rm(o)}|^{2}/|E_{0}|^{2}$ is obtained by calculating
$E_{{\rm R},\sigma}^{\rm(o)}$ for TM impinging polarization, i.e.,
$E_{0,+}^{\rm(o)}=E_{0,-}^{\rm(o)}=E_{0}$. In Fig. 4b we depict the dependence
of $A_{\rm TM}^{\rm(o)}$ over $\lambda,\theta$ considering silver as a metal
substrate. Note that the absorbance is maximised at the SPP excitation curve

$\theta_{\rm SPP}(\lambda)\simeq{\rm arcsin}\left\\{{\rm
Re}\sqrt{\frac{\epsilon_{\rm r}(\lambda)\epsilon_{\rm
m}(\lambda)}{\epsilon_{\rm g}(\lambda)[\epsilon_{\rm g}(\lambda)+\epsilon_{\rm
m}(\lambda)]}}\right\\},$ (11)

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 $\Delta A^{\rm(k)}$ as a function of $\lambda,\theta$. For
right/left circular polarization excitation, also a Fabry-Perot resonance is
excited at $600$nm$<\lambda<800$nm and $50^{\circ}<\theta<60^{\circ}$, which
enhances CDDA by a factor $f_{\rm CDDA}\simeq 10$.

Figure 7: Plasmon-induced CDDA enhancement factor $f_{\rm CDDA}(\lambda)={\rm
max}_{\theta}|\Delta A^{\rm(o,k)}|/|\Delta A_{\rm bulk}|$ 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 $\epsilon_{\rm m}$, obtained from Ref. SilverRef  for all the
considered noble metals.

##  V Discussion

Non-racemic mixtures such that $n_{\rm mol}^{\rm(S)}\neq n_{\rm mol}^{\rm(R)}$
produce mixed polarization dynamics analogously to optical rotation in bulk
chiral media. The reflected radiation polarization can be fully tracked
through the Stokes parameters $S_{0}^{(\rm o)}=|E_{{\rm R},+}^{(\rm
o)}|^{2}+|E_{{\rm R},-}^{(\rm o)}|^{2}$, $S_{1}^{(\rm o)}=2{\rm Re}[E_{{\rm
R},+}^{(\rm o)}E_{{\rm R},-}^{(\rm o)*}]$, $S_{2}^{(\rm o)}=-2{\rm Im}[E_{{\rm
R},+}^{(\rm o)}E_{{\rm R},-}^{(\rm o)*}]$, $S_{3}^{(\rm o)}=|E_{{\rm
R},+}^{(\rm o)}|^{2}-|E_{{\rm R},-}^{(\rm o)}|^{2}$, whose dependence over the
incidence angle $\theta$ is depicted in Figs. 5a-c at $\lambda=700$ nm for a
silver substrate. At the SPP excitation angle $\theta_{\rm SPP}\simeq
70^{\circ}$ polarization modulation is maximised for both circular and TM
excitation polarization. Moreover, at $\theta\simeq 50^{\circ}$ 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,
analytical solutions of Maxwell’s equations provide

$\displaystyle{\bf E}_{\rm k}({\bf r},t)=\sum\limits_{s=\pm 1}{\rm
Re}\left\\{\left[{\bf E}_{<}^{({\rm k},s)}({\bf r})\Theta(-z)+\right.\right.$
(12) $\displaystyle\left.\left.+{\bf E}_{\rm in}^{({\rm k},s)}({\bf
r})\Theta_{\rm in}(z)+{\bf E}_{>}^{({\rm k},s)}({\bf
r})\Theta(z-d)\right]e^{-i\omega t}\right\\},$ $\displaystyle{\bf H}_{\rm
k}({\bf r},t)=\sum\limits_{s=\pm 1}{\rm
Re}\left\\{\frac{s}{i\mu_{0}c}\left[\sqrt{\epsilon_{\rm g}}{\bf E}_{<}^{({\rm
k},s)}({\bf r})\Theta(-z)+\right.\right.$ (13)
$\displaystyle\left.\left.+\sqrt{\epsilon_{\rm m}}{\bf E}_{\rm in}^{({\rm
k},s)}({\bf r})\Theta_{\rm in}(z)+\frac{\sqrt{\epsilon_{\rm
r}}}{\sqrt{\mu_{\rm r}}}{\bf E}_{>}^{({\rm k},s)}({\bf
r})\Theta(z-d)\right]e^{-i\omega t}\right\\},$

where

$\displaystyle{\bf E}_{<}^{({\rm k},s)}({\bf
r})=\left[E_{0,s}^{\rm(k)}\hat{\bf e}_{s}^{(0)}e^{ik_{\rm g}z}+E_{{\rm
R},s}^{\rm(k)}\hat{\bf e}_{s}^{\rm(R)}e^{-ik_{\rm g}z}\right]e^{ik_{x}x},$
(14a) $\displaystyle{\bf E}_{\rm in}^{({\rm k},s)}({\bf
r})=\sum\limits_{\sigma=\pm
1}\frac{1}{\sqrt{2}}E_{\sigma,s}^{\rm(k)}\left[\sigma\sqrt{1-\frac{\epsilon_{\rm
g}}{\epsilon_{\rm m}}{\rm sin}^{2}\theta}\hat{\bf e}_{x}+is\hat{\bf
e}_{y}+\right.$ $\displaystyle\left.-\sqrt{\epsilon_{\rm g}/\epsilon_{\rm
m}}{\rm sin}\theta\hat{\bf e}_{z}\right]e^{ik_{x}x+i\sigma k_{\rm m}z},$ (14b)
$\displaystyle{\bf E}_{>}^{({\rm k},s)}({\bf r})=E_{{\rm
T},s}^{\rm(k)}\left[\hat{e}_{x}-i\frac{k_{0}}{\beta_{s}}(\kappa-s\sqrt{\epsilon_{\rm
r}\mu_{\rm r}})\hat{e}_{y}+\right.$
$\displaystyle\left.-\frac{k_{x}}{\beta_{s}}\hat{e}_{z}\right]e^{ik_{x}x+i\beta_{s}z}.$
(14c)

Note that in the expressions above the roman subscript/superscript “${\rm k}$”
indicates the considered Kretschmann configuration. Again, applying BCs one
gets an $8\times 8$ inhomogeneous system of algebraic equations for the field
amplitudes, see Eq. (S10) in the supporting information, which we invert
numerically, obtaining $E_{{\rm R},\pm 1}^{\rm(k)},E_{{\rm T},\pm 1}^{\rm(k)}$
for any impinging field components $E_{0,\pm 1}^{\rm(k)}$. Hence, in the
Kretschmann coupling scheme, the absorbance upon right ($s=+1$,
$E_{0,-}^{\rm(k)}=0$) and left ($s=-1$, $E_{0,+}^{\rm(k)}=0$) circular
polarization excitation is given by
$A_{s}^{\rm(k)}(\theta,\lambda)=1-\sum\limits_{\sigma=\pm 1}|E_{{\rm
R},\sigma}^{\rm(k)}|^{2}/|E_{0,s}^{\rm(k)}|^{2}$, from which we calculate the
Kretschmann CDDA $\Delta
A^{\rm(k)}(\theta,\lambda)=A_{+}^{\rm(k)}-A_{-}^{\rm(k)}$. Similarly to the
Otto coupling scheme, the TM absorbance $A_{\rm
TM}^{\rm(k)}(\theta,\lambda)=1-\sum\limits_{\sigma=\pm 1}|E_{{\rm
R},\sigma}^{\rm(k)}|^{2}/|E_{0}|^{2}$ is obtained by calculating $E_{{\rm
R},\sigma}^{\rm(k)}$ for TM impinging polarization, i.e., by setting
$E_{0,+}^{\rm(k)}=E_{0,-}^{\rm(k)}=E_{0}$. In Figs. 6b,c we depict the
dependence of (b) $A_{\rm TM}^{\rm(k)}$ and (c) $\Delta A^{\rm(k)}$ over
$\lambda,\theta$ considering a silver thin film with thickness $d_{\rm m}=80$
nm. Note that, similarly to the Otto coupling scheme, the TM absorbance and
the CDDA are maximised at the SPP excitation curve $\theta_{\rm
SPP}(\lambda)$, 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 $\theta\simeq 61^{\circ}$. It is worth emphasizing that such a phenomenon
occurs only thanks to the sub-wavelength metallic thickness $d_{\rm
m}<<\lambda$, 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 $n_{\rm mol}$ and enantiomeric number density imbalance $\Delta n_{\rm
mol}$. We illustrate the dependence of the Kretschmann Stokes parameters
[$S_{0}^{(\rm k)}=|E_{{\rm R},+}^{(\rm k)}|^{2}+|E_{{\rm R},-}^{(\rm
k)}|^{2}$, $S_{1}^{(\rm k)}=2{\rm Re}[E_{{\rm R},+}^{(\rm k)}E_{{\rm
R},-}^{(\rm k)*}]$, $S_{2}^{(\rm k)}=-2{\rm Im}[E_{{\rm R},+}^{(\rm k)}E_{{\rm
R},-}^{(\rm o)*}]$, $S_{3}^{(\rm k)}=|E_{{\rm R},+}^{(\rm k)}|^{2}-|E_{{\rm
R},-}^{(\rm k)}|^{2}$ ] over the incidence angle $\theta$ in Figs. 6d-f at
$\lambda=700$ nm for a silver thin film with thickness $d_{\rm m}=80$ nm and a
chiral medium with $n_{\rm mol}=10^{-2}$ nm-3 and $\Delta n_{\rm mol}=n_{\rm
mol}$. 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 $f_{\rm CDDA}(\lambda)={\rm max}_{\theta}|\Delta
A^{\rm(o,k)}(\theta,\lambda)|/|\Delta A_{\rm bulk}(\lambda)|$, where $\Delta
A_{\rm bulk}(\lambda)=e^{-2{\rm Im}\beta_{-}^{(\rm b)}d}-e^{-2{\rm
Im}\beta_{+}^{(\rm b)}d}$ is the absorbance of a layer of reparixin dissolved
in water in the absence of the nanophotonic structure and $\beta_{\pm}^{(\rm
b)}=k_{0}(\sqrt{\epsilon_{\rm r}\mu_{\rm r}}\mp\kappa)$ . In 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 $f_{\rm CDDA}\simeq 20$ is attained in the
Otto configuration at $\lambda\simeq 400$ nm adopting silver as metal
substrate, see Fig. 7c. However, for $\lambda>500$ nm, gold and copper provide
CDDA enhancement factors $f_{\rm CDDA}\simeq 5$ comparable with silver. For
the considered dilute molecular number density $n_{\rm mol}\simeq 10^{-2}$
nm-3, we find that plasmon-enhanced CD spectroscopy is a viable chiroptical
sensing platform able to discern $\lesssim 10^{13}$ ${\rm R},{\rm S}$
enantiomers in the considered $V_{\rm int}$. 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
$z>d,d_{\rm m}$. 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
$A=1-I_{\rm R}/I_{0}-I_{\rm T}/I_{0}$, where $I_{\rm R,T,0}$ indicate the
measured reflected ($I_{\rm R}$), transmitted ($I_{\rm T}$), 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
TangPRL2010 ; TangScience2011 ; MohammadiACSPhot2018 ; Pellegrini2018 ;
Gilroy2019  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 community-acquired 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 plasmon-
induced CDDA enhancement of the order $f_{\rm CDDA}\simeq 20$.

## Supporting information

See the supporting information for the theoretical details on the reparixin
quantum observables calculations, the system of algebraic equations for field
amplitudes in the Otto and Kretschmann configurations, and a MATLAB script for
the calculation of the macroscopic bi-anisotropic response of reparixin
dissolved in water.

###### Acknowledgements.

This work has been partially funded by the European Union - NextGenerationEU
under the Italian Ministry of University and Research (MUR) National
Innovation Ecosystem grant ECS00000041 - VITALITY - CUP E13C22001060006, the
Progetti di ricerca di Rilevante Interesse Nazionale (PRIN) of the Italian
Ministry of Research PHOTO (PHOtonics Terahertz devices based on tOpological
materials) 2020RPEPNH, and the European Union under grant agreement No
101046424. Views and opinions expressed are however those of the author(s)
only and do not necessarily reflect those of the European Union or the
European Innovation Council. Neither the European Union nor the European
Innovation Council can be held responsible for them.
The authors acknowledge fruitful discussions with Jens Biegert, Patrice
Genevet, Michele Dipalo, Giovanni Melle, Sotirios Christodoulou, Anna Maria
Cimini and Domenico Bonanni.

## Author declarations

### Conflict of Interest

The authors have no conflicts to disclose.

## Author Contributions

Matteo Venturi: Data curation (equal); Formal analysis (equal); Methodology
(supporting); Investigation (equal); Software (supporting); Visualization
(supporting); Writing - original draft (equal); Writing - review and editing
(equal). Raju Adhikary: Data curation (equal); Formal analysis (equal);
Methodology (supporting); Investigation (equal); Software (supporting);
Visualization (supporting); Writing - original draft (supporting); Writing -
review and editing (equal). Ambaresh Sahoo: Data curation (supporting); Formal
analysis (equal); Methodology (supporting); Investigation (equal); Software
(supporting); Visualization; Writing - original draft (supporting); Writing -
review and editing (equal). Carino Ferrante: Methodology (supporting);
Investigation (supporting); Project administration (supporting); Supervision
(supporting); Visualization (supporting); Writing - original draft
(supporting); Writing - review and editing (equal). Isabella Daidone: Formal
analysis (supporting); Methodology (supporting); Investigation (supporting);
Software (supporting); Resources (supporting); Supervision (supporting);
Writing - original draft (supporting); Writing - review and editing (equal).
Francesco Di Stasio: Conceptualization (supporting); Funding acquisition
(equal); Project administration (supporting); Writing - review and editing
(equal). Andrea Toma: Conceptualization (supporting); Funding acquisition
(equal); Project administration (supporting); Writing - review and editing
(equal). Francesco Tani: Conceptualization (supporting); Funding acquisition
(equal); Project administration (supporting); Writing - review and editing
(equal). Hatice Altug: Conceptualization (supporting); Funding acquisition
(equal); Project administration (supporting); Writing - review and editing
(equal). Antonio Mecozzi: Formal analysis (supporting); Methodology
(supporting); Investigation (supporting); Writing - review and editing
(equal). Massimiliano Aschi: Data curation (supporting); Formal analysis
(supporting); Methodology (equal); Investigation (supporting); Software;
Resources; Supervision (equal); Writing - original draft (supporting); Writing
- review and editing (equal). Andrea Marini: Conceptualization; Formal
analysis (equal); Funding acquisition (equal); Investigation (equal);
Methodology (equal); Project administration; Software (supporting);
Supervision (equal); Visualization (supporting); Writing - original draft
(equal); Writing - review and editing (equal).

## Data Availability Statement

The data that support the findings of this study are available within the
article [and its supplementary material].

## References

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