Nanophotonic chiral sensing: from principles to practice

SeokJae Yoo; Q-Han Park

doi:10.3788/PI.2025.R08

1 Introduction

Nanophotonics research has long been dedicated to confining light into progressively smaller regions. These advancements have also enabled the detection of deep-subwavelength-scale objects, such as molecules and nanoparticles, at extremely low concentrations^[1–29]. Notably, nanophotonics has successfully identified molecular species and quantified their concentrations down to the single-molecule level^[30–46]. More recently, the field has set its sights on obtaining structural information about molecules. In particular, significant efforts have been made to distinguish the handedness of chiral molecules—structures that lack mirror symmetry (Fig. 1)—even at low concentrations^[47–64].

Figure 1.Schematic drawing for nanophotonic chiral sensing. Chiral molecules (top) and nanophotonic sensors (bottom; depicted by gold nanorods as an example) interact with each other, resulting in various phenomena listed in the figure. Nanophotonic sensors can be achieved by various optical systems such as nanoparticles, plasmonic structures, photonic crystals, and metasurfaces.

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Chirality is a ubiquitous property in nature, particularly in molecules and chemical compounds^[65–69]. Measuring molecular chirality with high sensitivity is crucial for understanding various molecular properties^[68,69]. Structural chirality, i.e., broken mirror symmetry arising from a chiral arrangement in the spatial positions of constituent atoms within a molecule, manifests in both electronic and vibrational quantum states^[69]: the former is referred to as natural electronic optical activity, typically observed in ultraviolet (UV) and visible frequencies, while the latter, natural vibrational optical activity, occurs in the infrared (IR) range. The term “natural” signifies that these optical activities do not require external perturbations, such as magnetic fields, to be observed^[69]. Natural electronic optical activity is valuable for obtaining structural information about biomolecules, including the secondary structures of proteins. Meanwhile, natural vibrational optical activity is essential for determining absolute configurations and reaction mechanisms^[70–73]. Both natural electronic and vibrational optical activity also play a key role in the development of enantiopure drugs, which consist of only one enantiomer from a pair of chiral molecules^[74–77]. In addition to natural optical activities, strong magnetic fields break time-reversal symmetry and can induce magnetic electronic optical activity, especially in metal-based catalysts and metalloproteins, even in the absence of structural chirality, providing insights into reaction coordinates^[78–81].

However, traditional optical chiral sensing techniques, such as circular dichroism (CD), optical rotatory dispersion (ORD), vibrational CD (VCD), Raman optical activity (ROA), and magnetic CD (MCD), require high concentrations and bulk molecular volumes due to the inherently weak nature of chiroptical signals; for instance, the CD signals of most molecules are only on the order of $10^{- 5}$ to $10^{- 3}$ relative to their conventional absorption^[69,82–84]. Given the significance of various optical activity phenomena, the goal of nanophotonic chiral sensing—achieving ultralow concentration and even single-molecule detection—is particularly compelling^[47–62].

Nanophotonic chiral sensing relies on spectroscopic features that arise from various interaction mechanisms between chiral molecules and nanophotonic structures. Figure 1 illustrates typical interaction phenomena. The “sensor-to-molecule” configuration (Fig. 1) refers to the engineering of near-field optical environments, e.g., strong intensity and helicity density, by the sensor, in order to directly influence the optical response of nearby molecules. Within this near-field environment, molecular CD can be enhanced^{[63,64,82–100]}, or molecular emission can be modified through weak (i.e., the Purcell effect) and strong coupling regimes^[101–106]. Additionally, nanostructures provide binding sites for molecules^[107–111]. On the other hand, the “molecule-to-sensor” configuration (Fig. 1) refers to the influence of chiral molecules on the resonance modes of nanophotonic sensors, enabling resonance-based sensing mechanisms^{[48,112–119]}. The scattering of near-fields by chiral molecules affects the sensor’s resonance, inducing CD in the nanophotonic structure^{[107,112,120,121]}. An ensemble of chiral molecules can further contribute to chiral refraction and diffraction^[122,123], resulting from the wavefield splitting^[124–126]. Moreover, chiral molecules with appropriate functional groups can modify the surface properties of nanostructures^[107–111].

In this review, we explore nanophotonic chiral sensing from fundamental principles to practical applications. While chiral sensing has attracted significant interest in the nanophotonics community, previous reviews have primarily enumerated known sensing mechanisms^{[47–49,55,57,59,127]}. In contrast, our review provides a complete picture of nanophotonic chiral sensing by systematically connecting fundamental principles to practical implementations, covering the most recent studies on chirality of light and matter, chiral sensing mechanisms, and experimental realizations. Here, we first introduce the electromagnetic description of chiral molecules using a quantum mechanical framework (Sec. 2). Next, we examine the chiral nature of light and discuss key electromagnetic field properties, such as optical chirality, optical helicity, spin angular momentum, and zilches (Sec. 3). Section 3 includes a thorough discussion on the mathematical structures of the chirality of light. Although recent studies on nanophotonic chiral sensing have focused on optical chirality, helicity, and spin, possibilities of novel nanophotonic sensing techniques may be discovered in the complicated mathematical properties of chiral light. We then review how nanophotonic structures interact with chiral molecules to provide handedness information as illustrated in Fig. 1, also highlighting important studies in the discipline (Sec. 4). Finally, we discuss how molecular chirality can be optically measured and how one can implement chiroptical spectroscopy instrumentation, offering guidance for researchers interested in implementing nanophotonic chiral sensing (Sec. 5).

2 Electromagnetics of Molecular Chirality

Nanophotonic chiral sensing uses the interaction between light and molecules. To understand the light-matter interaction, we first need to know the electromagnetic description of chiral molecules in both microscopic and macroscopic views (Fig. 2). In this section, we first introduce a macroscopic theory to describe an ensemble of chiral molecules as a homogeneous medium in Sec. 2.1. The macroscopic medium parameters, such as the permittivity $\tilde{ε}$ (or the refractive index $\tilde{n}$ ) and the chirality parameter $\tilde{κ}$ , enable the explain of the phenomena related to the light propagation, e.g., absorption, transmission, and refraction, as well as the complicated interaction between chiral molecule ensembles and nanophotonic sensors. Then, we describe multipole moment theory for a single chiral molecule and its interaction with light in Sec. 2.2. The microscopic theory is the basis of the macroscopic theory (Sec. 2.1). Microscopic multipole moments explain absorption, emission, and scattering of a single molecule.

$Schematic drawing for the electromagnetic description of molecular chirality. Microscopic view provides the multipole moments of each molecule (E1, electric dipole moment; E2, electric quadrupole moment; M1, magnetic dipole moment). Macroscopic view provides the medium parameters of the molecule ensemble. In the macroscopic medium level, left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) lights experience different refractive indices, n˜±κ˜, respectively.$

Figure 2.Schematic drawing for the electromagnetic description of molecular chirality. Microscopic view provides the multipole moments of each molecule (E1, electric dipole moment; E2, electric quadrupole moment; M1, magnetic dipole moment). Macroscopic view provides the medium parameters of the molecule ensemble. In the macroscopic medium level, left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) lights experience different refractive indices, $\tilde{n} \pm \tilde{κ}$ , respectively.

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2.1 Macroscopic View: Electromagnetic Theory

2.1.1 Electromagnetic constitutive relations

For isotropic chiral molecules, their electric dipole (E1) moment $p$ and magnetic dipole (M1) moment $m$ can be written in the vector forms^[83] $\tilde{p} = \tilde{α} \tilde{E} + i \tilde{G} \tilde{H},$ (1) $\tilde{m} = \tilde{χ} \tilde{H} - i \tilde{G} \tilde{E},$ (2)while the electric quadrupole (E2) moment [Eq. (45)] does not need to be considered because it does not involve the natural and vibrational optical activities of isotropic media. The E2 moment is meaningful for anisotropic molecules. Note that the prefactors are adjusted by pulling $i$ out from $\tilde{G}$ , and using $\tilde{H}$ instead of $\tilde{B}$ in Eqs. (1) and (2). The tilde notation indicates complex quantities.

At a macroscopic level, it is convenient to use the displacement field $D$ and the magnetic field $B$ instead of the microscopic dipole moments, Eqs. (1) and (2). The fields $D$ and $B$ are related to $E$ and $H$ by the electromagnetic constitutive relations for the isotropic chiral media^[124,125]: $\tilde{D} = \tilde{ε} \tilde{E} + i \tilde{κ} \tilde{H} / c,$ (3) $\tilde{B} = \tilde{μ} \tilde{H} - i \tilde{κ} \tilde{E} / c,$ (4)with the permittivity $\tilde{ε}$ , the permeability $\tilde{μ}$ , and the chirality parameter $\tilde{κ}$ of a given chiral medium.

The Lorenz-Lorentz formula [Eq. (8)]^[127] provides a connection between the microscopic dipole moments [Eqs. (1) and (2)] and the macroscopic constitutive relations [Eqs. (3) and (4)]^[128]. Here, we introduce the derivation of the Lorenz-Lorentz formula [Eq. (8)]^[127]. Suppose molecules are described by Eqs. (1) and (2), while they are suspended in the background medium (e.g., a solution or a solid matrix) of the permittivity $ε_{b}$ and the permeability $μ_{b}$ . We want to describe the resulting system as a homogeneous chiral medium described by $\tilde{ε}$ , $\tilde{μ}$ , and $\tilde{κ}$ . Upon the incident fields ${\tilde{E}}_{0}$ and ${\tilde{H}}_{0}$ , the dipolar scattering introduces local fields (or the so-called Lorentzian field): $(\begin{matrix} {\tilde{E}}_{L} \\ {\tilde{H}}_{L} \end{matrix}) = (\begin{matrix} {\tilde{E}}_{0} \\ {\tilde{H}}_{0} \end{matrix}) + \frac{1}{3} (\begin{matrix} 1 / ε_{b} & 0 \\ 0 & 1 / μ_{b} \end{matrix}) (\begin{matrix} {\tilde{P}}_{e} \\ {\tilde{P}}_{m} \end{matrix}),$ (5)where ${\tilde{P}}_{e}$ ( ${\tilde{P}}_{m}$ ) is the electric (magnetic) polarization induced by the local fields: $(\begin{matrix} {\tilde{P}}_{e} \\ {\tilde{P}}_{m} \end{matrix}) = n (\begin{matrix} \tilde{p} \\ \tilde{m} \end{matrix}) = n (\begin{matrix} \tilde{α} & - i \tilde{G} \\ + i \tilde{G} & \tilde{χ} \end{matrix}) (\begin{matrix} {\tilde{E}}_{L} \\ {\tilde{H}}_{L} \end{matrix}) .$ (6)where $n$ measures how many molecules are present in a given volume, i.e., the number density of molecules. The electric and magnetic polarizations ${\tilde{P}}_{e}$ and ${\tilde{P}}_{m}$ define the fields $\tilde{D}$ and $\tilde{B}$ in the chiral medium by the relation $(\begin{matrix} \tilde{D} \\ \tilde{B} \end{matrix}) = (\begin{matrix} ε_{b} & 0 \\ 0 & μ_{b} \end{matrix}) (\begin{matrix} {\tilde{E}}_{0} \\ {\tilde{H}}_{0} \end{matrix}) + (\begin{matrix} {\tilde{P}}_{e} \\ {\tilde{P}}_{m} \end{matrix}),$ (7)and it again should be consistent with the constitutive relations for the incident fields, $\tilde{D} = \tilde{ε} {\tilde{E}}_{0} + i \tilde{κ} {\tilde{H}}_{0} / c$ and $\tilde{B} = \tilde{μ} {\tilde{H}}_{0} - i \tilde{κ} {\tilde{E}}_{0} / c$ . Combining Eqs. (5 )–(7), we obtain the Lorenz-Lorentz formula relating the microscopic polarizabilities $\tilde{α}$ , $\tilde{χ}$ , and $\tilde{G}$ to the macroscopic medium parameters $\tilde{ε}$ , $\tilde{μ}$ , and $\tilde{κ}$ : $(\begin{matrix} \tilde{ε} & - i \tilde{κ} \\ i \tilde{κ} & \tilde{μ} \end{matrix}) = (\begin{matrix} ε_{0} & 0 \\ 0 & μ_{0} \end{matrix}) + n {[I - \frac{n}{3} (\begin{matrix} \tilde{α} & - i \tilde{G} \\ i \tilde{G} & \tilde{χ} \end{matrix}) (\begin{matrix} 1 / ε_{b} & 0 \\ 0 & 1 / μ_{b} \end{matrix})]}^{- 1} (\begin{matrix} \tilde{α} & - i \tilde{G} \\ + i \tilde{G} & \tilde{χ} \end{matrix}) .$ (8)

For small $n$ , i.e., a dilute solution, Eq. (8) is approximated to $(\begin{matrix} \tilde{ε} & - i \tilde{κ} \\ i \tilde{κ} & \tilde{μ} \end{matrix}) = (\begin{matrix} ε_{b} & 0 \\ 0 & μ_{b} \end{matrix}) + n (\begin{matrix} \tilde{α} & - i \tilde{G} \\ + i \tilde{G} & \tilde{χ} \end{matrix}) .$ (9)

2.1.2 Wavefield decomposition

To describe light-matter interactions in a chiral medium characterized by Eqs. (3) and (4), it is convenient to transform the electromagnetic fields into a circular polarization basis, i.e., a theoretical technique known as wavefield decomposition. In this framework, the fields in the chiral media, governed by the constitutive relations in Eqs. (3) and (4), can be decomposed into two distinct wavefields, i.e., $\tilde{E} = {\tilde{E}}_{+} + {\tilde{E}}_{-}$ and $\tilde{H} = {\tilde{H}}_{+} + {\tilde{H}}_{-}$ . Each wavefield satisfies Maxwell’s equations in an equivalent dielectric medium^[124]: $\nabla \times {\tilde{E}}_{\pm} - i ω {\tilde{μ}}_{\pm} {\tilde{H}}_{\pm} = 0,$ (10) $\nabla \times {\tilde{H}}_{\pm} + i ω {\tilde{ε}}_{\pm} {\tilde{E}}_{\pm} = 0,$ (11)with the corresponding dielectric constitutive relations, ${\tilde{D}}_{\pm} = {\tilde{ε}}_{\pm} {\tilde{E}}_{\pm}$ and ${\tilde{B}}_{\pm} = {\tilde{μ}}_{\pm} {\tilde{H}}_{\pm}$ . The equivalent parameters for each wavefield are given by ${\tilde{ε}}_{\pm} = \tilde{ε} (1 \pm {\tilde{κ}}_{r}),$ (12) ${\tilde{μ}}_{\pm} = \tilde{μ} (1 \pm {\tilde{κ}}_{r}),$ (13)where the relative chirality parameter is ${\tilde{κ}}_{r} = \tilde{κ} / \tilde{n}$ . The corresponding refractive index is $\tilde{n} = \sqrt{\tilde{μ} \tilde{ε} / μ_{0} ε_{0}}$ . The impedances and refractive indices for the wavefields are expressed as ${\tilde{η}}_{\pm} = \sqrt{\frac{{\tilde{μ}}_{\pm}}{{\tilde{ε}}_{\pm}}},$ (14) ${\tilde{n}}_{\pm} = \tilde{n} (1 \pm {\tilde{κ}}_{r}) = \tilde{n} \pm \tilde{κ},$ (15)respectively. The dispersion of the wavefield photons is given by ${\tilde{k}}_{\pm} = ω \sqrt{{\tilde{ε}}_{\pm} {\tilde{μ}}_{\pm}} = k (1 \pm {\tilde{κ}}_{r}) = k_{0} {\tilde{n}}_{\pm} .$ (16)

In a homogeneous chiral medium, these wavefields remain independent. For plane waves, the wavefields correspond to the left- and right-handed circularly polarized light, providing a natural basis for analyzing chiral optical interactions.

2.2 Microscopic View: Quantum Mechanical Multipole Theory

2.2.1 Multipole interaction Hamiltonian

To understand the multipolar nature of chiral molecules, we adopt a microscopic quantum mechanical approach. Here, we summarize the quantum mechanical multipole theory that describes the molecular response to optical fields, described by the scalar potential $Φ$ and the vector potential $A$ ^[69,129]. To derive the semiclassical expressions for the multipole moments, we begin with the Hamiltonian describing charged particles within a molecule. Consider a molecule consisting of $N$ charged particles (e.g., ions and electrons), each with position $r^{(α)}$ , mass $m^{(α)}$ , and charge $q^{(α)}$ , where $α = 1, 2, 3, \dots, N$ . The total Hamiltonian of the molecule is given by^[69,129] $H = \sum_{α = 1}^{N} {\frac{1}{2 m^{(α)}} {[Π^{(α)} - q^{(α)} A (r^{(α)}, t)]}^{2} + q^{(α)} Φ (r^{(α)}, t)} + V,$ (17)where $Π^{(α)}$ is the generalized momentum operator, and $V$ represents the potential of the molecule in the absence of external fields. Expanding Eq. (17) in terms of vector potential $A$ , the Hamiltonian can be written as $H = H^{(0)} + H^{(1)} + H^{(2)}$ , where each term is defined as $H^{(0)} \equiv \sum_{α = 1}^{N} \frac{{(Π^{(α)})}^{2}}{2 m^{(α)}} + V,$ (18) $H^{(1)} \equiv \sum_{α = 1}^{N} {- \frac{q^{(α)}}{2 m^{(α)}} [2 A (r^{(α)}, t) \cdot Π^{(α)} - i ℏ \nabla^{(α)} \cdot A (r^{(α)}, t)] + q^{(α)} ϕ (r^{(α)}, t)},$ (19) $H^{(2)} \equiv \sum_{α = 1}^{N} {- \frac{{(q^{(α)})}^{2}}{2 m^{(α)}} A^{2} (r^{(α)}, t)} .$ (20)

Here, the commutator $Π \cdot A - A \cdot Π = - i ℏ \nabla \cdot A$ is used in obtaining Eq. (19), ensuring the proper quantum mechanical treatment of the interaction between charged particles and the optical field. The influence of the optical fields on the molecule arises from the first- and second-order perturbation Hamiltonians, given by Eqs. (19) and (20). These terms describe the fundamental interactions that govern the molecular response to electromagnetic waves, capturing both dipole and higher-order multipolar contributions in chiral optical processes.

We expand $H^{(1)}$ [Eq. (19)] and $H^{(2)}$ [Eq. (20)] in terms of multipoles. To do so, we first express the electric and magnetic fields about an arbitrary origin as $E_{i} (r, t) = E_{i} (0, t) + {[\nabla_{j} E_{i} (r, t)]}_{0} r_{j} + \frac{1}{2} {[\nabla_{k} \nabla_{j} E_{i} (r, t)]}_{0} r_{j} r_{k} + \dots,$ (21) $B_{i} (r, t) = B_{i} (0, t) + {[\nabla_{j} B_{i} (r, t)]}_{0} r_{j} + \frac{1}{2} {[\nabla_{k} \nabla_{j} B_{i} (r, t)]}_{0} r_{j} r_{k} + \dots .$ (22)

Similarly, the corresponding scalar and vector potentials can also be written as $Φ (r, t) = Φ (0, t) - E_{i} (0, t) r_{i} - \frac{1}{2} {[\nabla_{j} E_{i} (r, t)]}_{0} r_{i} r_{j} + \dots,$ (23) $A_{i} (r, t) = ε_{i j k} {\frac{1}{2} B_{j} (0, t) r_{k} + \frac{1}{3} {[\nabla_{l} B_{j} (r, t)]}_{0} r_{k} r_{l} + \dots} .$ (24)

These expansions are formulated using the Barron-Gray gauge^[69,129], which differs from both the Coulomb gauge, $\nabla \cdot A = 0$ , and the Lorenz gauge, $\nabla \cdot A + c^{- 2} \partial Φ / \partial t = 0$ . The advantage of choosing the Barron-Gray gauge is that it ensures the electric and magnetic fields expansions [Eqs. (21) and (22)] share the same mathematical structure, simplifying their formulation and interpretation.

Utilizing the expanded forms of the scalar and vector potentials [Eqs. (23) and (24)], the first-order Hamiltonian $H^{(1)}$ [Eq. (19)] can be expressed in terms of multipole moments^[69,129], providing a systematic approach to describing chiral electromagnetic interactions at the subwavelength scale. Explicitly, $H^{(1)} = q Φ (t) - {\hat{p}}_{i} E_{i} (t) - \frac{1}{2} {\hat{q}}_{i j} E_{i j} (t) - {\hat{m}}_{i} B_{i} (t) + \dots,$ (25)with the total charge $q = \sum_{α = 1}^{N} q^{(α)}$ , the electric dipole (E1) moment operator ${\hat{p}}_{i} = \sum_{α = 1}^{N} q^{(α)} {\hat{r}}_{i}^{(α)}$ , the electric quadrupole (E2) moment operator ${\hat{q}}_{i j} = \sum_{α = 1}^{N} q^{(α)} {\hat{r}}_{i}^{(α)} {\hat{r}}_{j}^{(α)}$ , and the M1 moment operator ${\hat{m}}_{i} = \sum_{α = 1}^{N} (q^{(α)} / 2 m^{(α)}) {\hat{l}}_{i}^{(α)}$ , where the angular momentum operator $\hat{l} = \hat{r} \times \hat{Π}$ . In Eq. (25), terms are retained up to the E2-M1 order, employing the field gradient notation, $E_{i j} (t) \equiv {[\nabla_{j} E_{i} (r, t)]}_{0}$ . While the second-order Hamiltonian $H^{(2)}$ [Eq. (20)] leads to a series of magnetic susceptibilities, our focus remains on the first-order contributions associated with the multipole moments, as they play a central role in nanophotonic chiral interactions.

2.2.2 Multipole moments and polarizabilities

The multipole moments are given by the expectation values of their corresponding operators. For a time-harmonic field, $\tilde{E} = {\tilde{E}}_{0} e^{i (k \cdot r - ω t)}$ , the induced electric dipole (E1), electric quadrupole (E2), and magnetic dipole (M1) moments can be expressed as^[69,129] ${\tilde{p}}_{i} \equiv ⟨ n (t) | {\hat{p}}_{i} | n (t) ⟩ - p_{i}^{(0)} = {\tilde{α}}_{i j} {\tilde{E}}_{j} (t) + \frac{1}{2} {\tilde{a}}_{i j k} {\tilde{E}}_{j k} (t) + {\tilde{G}}_{i j} {\tilde{B}}_{j} (t) + \dots,$ (26) ${\tilde{q}}_{i j} \equiv ⟨ n (t) | {\hat{q}}_{i j} | n (t) ⟩ - q_{i j}^{(0)} = {\tilde{a}}_{i j k} {\tilde{E}}_{k} (t) + \dots,$ (27) ${\tilde{m}}_{i} \equiv ⟨ n (t) | {\hat{m}}_{i} | n (t) ⟩ - m_{i}^{(0)} = {\tilde{χ}}_{i j} {\tilde{B}}_{j} (t) + {\tilde{g}}_{i j} {\tilde{E}}_{j} (t) + \dots,$ (28)where the multipole polarizabilities, ${\tilde{α}}_{i j} = {\tilde{α}}_{i j}^{'} + i {\tilde{α}}_{i j}^{''}$ , ${\tilde{a}}_{i j k} = {\tilde{α}}_{i j k}^{'} + i {\tilde{α}}_{i j k}^{''}$ , ${\tilde{G}}_{i j} = {\tilde{G}}_{i j}^{'} + i {\tilde{G}}_{i j}^{''}$ , ${\tilde{a}}_{i j k} = {\tilde{a}}_{i j k}^{'} + i {\tilde{a}}_{i j k}^{''}$ , ${\tilde{χ}}_{i j} = {\tilde{χ}}_{i j}^{'} + i {\tilde{χ}}_{i j}^{''}$ , and ${\tilde{g}}_{i j} = {\tilde{g}}_{i j}^{'} + i {\tilde{g}}_{i j}^{''}$ , are defined as ${\tilde{α}}_{i j}^{'} = \frac{2}{ℏ} \sum_{s \neq n} ω_{s n} {\tilde{Z}}_{s n} Re (⟨ n | {\hat{p}}_{i} | s ⟩ ⟨ s | {\hat{p}}_{j} | n ⟩) = {\tilde{α}}_{j i}^{'},$ (29) ${\tilde{α}}_{i j}^{''} = \frac{2}{ℏ} \sum_{s \neq n} ω {\tilde{Z}}_{s n} Im (⟨ n | {\hat{p}}_{i} | s ⟩ ⟨ s | {\hat{p}}_{j} | n ⟩) = - {\tilde{α}}_{j i}^{''},$ (30) ${\tilde{a}}_{i j k}^{'} = \frac{2}{ℏ} \sum_{s \neq n} ω_{s n} {\tilde{Z}}_{s n} Re (⟨ n | {\hat{p}}_{i} | s ⟩ ⟨ s | {\hat{q}}_{j k} | n ⟩) = {\tilde{a}}_{j k i}^{'},$ (31) ${\tilde{a}}_{i j k}^{''} = \frac{2}{ℏ} \sum_{s \neq n} ω {\tilde{Z}}_{s n} Im (⟨ n | {\hat{p}}_{i} | s ⟩ ⟨ s | {\hat{q}}_{j k} | n ⟩) = - {\tilde{a}}_{j k i}^{''},$ (32) ${\tilde{G}}_{i j}^{'} = \frac{2}{ℏ} \sum_{s \neq n} ω_{s n} {\tilde{Z}}_{s n} Im (⟨ n | {\hat{p}}_{i} | s ⟩ ⟨ s | {\hat{m}}_{j} | n ⟩) = {\tilde{g}}_{j i}^{'},$ (33) ${\tilde{G}}_{i j}^{''} = \frac{2}{ℏ} \sum_{s \neq n} ω {\tilde{Z}}_{s n} Im (⟨ n | {\hat{μ}}_{i} | s ⟩ ⟨ s | {\hat{m}}_{j} | n ⟩) = - {\tilde{g}}_{j i}^{'},$ (34) ${\tilde{χ}}_{i j}^{'} = \frac{2}{ℏ} \sum_{s \neq n} ω_{s n} {\tilde{Z}}_{s n} ⟨ n | {\hat{m}}_{i} | s ⟩ ⟨ s | {\hat{m}}_{j} | n ⟩ = {\tilde{χ}}_{j i}^{'},$ (35) ${\tilde{χ}}_{i j}^{''} = \frac{2}{ℏ} \sum_{s \neq n} ω {\tilde{Z}}_{s n} ⟨ n | {\hat{m}}_{i} | s ⟩ ⟨ s | {\hat{m}}_{j} | n ⟩ = - {\tilde{χ}}_{j i}^{''},$ (36)where $ω_{s n}$ is the transition frequency between the states $s$ and $n$ . The tilde notation indicates complex quantities. The spectral function ${\tilde{Z}}_{s n} = f + i g$ , governing molecular absorption, is given by $f = \frac{ω_{s n}^{2} - ω^{2}}{{(ω_{s n}^{2} - ω^{2})}^{2} + ω^{2} Γ_{s n}^{2}},$ (37) $g = \frac{ω Γ_{s n}}{{(ω_{s n}^{2} - ω^{2})}^{2} + ω^{2} Γ_{s n}^{2}},$ (38)which account for the imaginary and real parts of the response, respectively. The symmetries of the polarizability tensors in their complex forms are given by ${\tilde{α}}_{i j} = {\tilde{α}}_{j i}^{*},$ (39) ${\tilde{a}}_{i j k} = {\tilde{a}}_{j k i}^{*},$ (40) ${\tilde{G}}_{i j} = {\tilde{g}}_{j i}^{*},$ (41) ${\tilde{χ}}_{i j} = {\tilde{χ}}_{j i}^{*},$ (42) ${\tilde{a}}_{i j k} = {\tilde{a}}_{i k j} .$ (43)

These symmetry relations arise due to the Hermiticity of the multipole moment operators, as shown in Eqs. (29 )–(36). Additionally, the symmetry of the E2 polarizability in Eq. (43) arises from the intrinsic permutation symmetry of the electric quadrupole (E2) moment operator.

2.2.3 Multipole absorption and circular dichroism

Cohen systematically analyzed how the structure of light affects the molecular absorption and CD^[85]. He derived the multipole contributions to the molecular absorption rate for a plane wave, $\tilde{E} = E^{(0)} \exp [i (k \cdot r - ω t)] \exp (i ϕ)$ , employing multipole moments: ${\tilde{p}}_{i} = {\tilde{α}}_{i j} {\tilde{E}}_{j} + \frac{1}{3} {\tilde{A}}_{i j k} {\tilde{E}}_{j k} + {\tilde{G}}_{i j} {\tilde{B}}_{j},$ (44) ${\tilde{q}}_{i j} = {\tilde{A}}_{i j k} {\tilde{E}}_{k},$ (45) ${\tilde{m}}_{i} = - {\tilde{G}}_{i j} E_{j},$ (46)

which differ slightly from Eqs. (26 )–(28) due to the choice of real molecular eigenfunctions in Eqs. (44 )–(46). The total absorption rate, considering contributions up to the E1-E2-M1 order, $Γ = ⟨ E_{i} {\dot{p}}_{i} + E_{i j} {\dot{q}}_{i j} / 3 + B_{i} {\dot{m}}_{i} ⟩$ , can be decomposed into three components: $Γ_{E 1 - E 1} = ω α_{i j}^{''} E_{i}^{(0)} E_{j}^{(0)},$ (47) $Γ_{E 1 - E 2} = \frac{2 ω}{3} A_{i j k}^{''} {⟨ E_{i} E_{j k} ⟩}_{t},$ (48) $Γ_{E 1 - M 1} = G_{i j}^{'} {⟨ E_{i} {\dot{B}}_{j} - {\dot{E}}_{i} B_{j} ⟩}_{t},$ (49)where ${⟨ \dots ⟩}_{t}$ represents a time average, and the prime and double-prime symbols denote the real and imaginary parts, respectively. In Eqs. (47 )–(49), the complex fields can be expressed as ${\tilde{E}}_{i} = E_{i} + (i / ω) {\dot{E}}_{i}$ and ${\tilde{E}}_{i j} = E_{i j} + (i / ω) {\dot{E}}_{i j}$ .

$Γ_{E 1 - E 1}$ [Eq. (47)] describes linear absorption, originating solely from the E1 moment. $Γ_{E 1 - E 2}$ [Eq. (48)] accounts for the optical activity of oriented molecules, originating from the E2 moment induced by the electric field ${\tilde{E}}_{k}$ and the E1 moment component induced by the electric field gradient ${\tilde{E}}_{j k}$ . $Γ_{E 1 - M 1}$ [Eq. (48)] governs the optical activity of both oriented and randomly oriented molecules, resulting from the E1 moment induced by the magnetic field and the M1 moment induced by the electric field. $Γ_{E 1 - E 2}$ and $Γ_{E 1 - M 1}$ [Eqs. (48) and (49)] are the origins of natural optical activity in UV and visible frequencies, as well as vibrational optical activity in IR frequencies.

Chiroptical spectroscopy often focuses on liquid samples, where molecules are randomly oriented due to free tumbling in solution. In such cases, Eqs. (47 )–(49) simplify to $Γ_{E 1 - E 1} = ω α_{i i}^{''} E_{i}^{(0)} E_{i}^{(0)},$ (50) $Γ_{E 1 - E 2} = 0,$ (51) $Γ_{E 1 - M 1} = G^{'} {⟨ E_{i} {\dot{B}}_{i} - {\dot{E}}_{i} B_{i} ⟩}_{t} = - \frac{2 G^{'}}{ε_{0}} {⟨ C ⟩}_{t},$ (52)Here, we used ${⟨ a_{i j}^{''} ⟩}_{Ω} = a_{i i}^{''}$ , ${⟨ {\tilde{A}}_{i j k} ⟩}_{Ω} = - (ε_{i j k} {\tilde{A}}_{i j k}) \overset{\leftrightarrow}{ε} / 6$ , ${\tilde{A}}_{i j k} = {\tilde{A}}_{i k j}$ , and $ε_{i j k} {\tilde{A}}_{i j k} = 0$ to derive Eq. (51), where ${⟨ \dots ⟩}_{Ω}$ denotes the orientational average. In Eq. (52), ${⟨ G_{i j}^{'} ⟩}_{Ω} = G^{'} I / 3$ is used, and the optical chirality density $C$ is defined as^[83,130] $C \equiv \frac{ε_{0}}{2} E \cdot \nabla \times E + \frac{1}{2 μ_{0}} B \cdot \nabla \times B .$ (53)

Beyond natural and vibrational optical activities described by Eqs. (47 )–(52), magnetic optical activity can also be accessed and tuned via an external electromagnetic field. When a magnetic field $B_{k}$ is applied along the $k$ -axis, the resulting MCD is given by $Γ_{E 1 - E 1} = α_{i j k}^{'} E_{i} {\dot{E}}_{j} B_{k} .$ (54)

For randomly oriented molecules, the field term in Eq. (54) simplifies to $ε_{i j k} E_{i} {\dot{E}}_{j} B_{k} = (E \times \dot{E}) \cdot B .$ (55)

3 Chirality of Light

The molecular absorption rate, as discussed in Sec. 1.1, arises from the interaction between the induced multipole moments of molecules and the surrounding electromagnetic fields. Depending on the nature of these multipole moments, the local electric or magnetic fields couple to the corresponding moment coefficients, either with or without spatial or temporal derivatives. Notably, after averaging over molecular orientations, the absorption rate, Eq. (52), becomes proportional to the so-called “optical chirality” $C$ , Eq. (53). The quantity $C$ is one of the components of the zilch tensor, a set of 10 conserved quantities of the source-free electromagnetic field discovered by Lipkin in 1964^[131], initially believed to have no physical significance. However, after being interpreted as the local electromagnetic density governing the differential absorption rate of chiral molecules by Tang and Cohen^[83,85], it has gained significant attention, particularly in the context of local CD enhancement using metamaterials or nanostructures^{[50,63,64,132]}. As a measure of the handedness of light, $C$ is often compared to the helicity of circularly polarized light, though the two quantities differ in physical dimensions. In this section, we clarify both the similarities and differences between them.

3.1 Properties of Maxwell’s Equations in Free Space

We begin by considering Maxwell’s equations in free space, where no source terms are present: $\nabla \cdot (ε_{0} E) = 0,$ (56) $\nabla \cdot B = 0,$ (57) $\nabla \times E = - \frac{\partial B}{\partial t},$ (58) $\nabla \times B = ε_{0} μ_{0} \frac{\partial E}{\partial t} .$ (59)

To achieve a more symmetric form, we introduce the scaled quantities $\bar{t}$ , $\bar{r}$ , $\bar{E}$ , and $\bar{B}$ defined as $t = \sqrt{ε_{0} μ_{0}} \bar{t},$ (60) $r = \bar{r},$ (61) $E = \frac{1}{\sqrt{ε_{0}}} \bar{E},$ (62) $B = \sqrt{μ_{0}} \bar{B},$ (63)which absorbs the factors $ε_{0}$ and $μ_{0}$ , leading to the reformulated Maxwell’s equations^[133] $\nabla \cdot \bar{E} = 0,$ (64) $\nabla \cdot \bar{B} = 0,$ (65) $\nabla \times \bar{E} = - \frac{\partial \bar{B}}{\partial \bar{t}},$ (66) $\nabla \times \bar{B} = \frac{\partial \bar{E}}{\partial \bar{t}} .$ (67)

We note the following key properties of Maxwell’s equations in free space that are particularly relevant to optical chirality.

3.1.1 Property 1: curly tower

We consider the successive applications of the curl operator to the electric and magnetic fields, defined as ${\bar{E}}^{(k)} = {(\nabla \times)}^{k} \bar{E},$ (68) ${\bar{B}}^{(k)} = {(\nabla \times)}^{k} \bar{B},$ (69)where $k$ represents the number of curl operations applied. It is straightforward to verify that these fields satisfy Maxwell’s equations^[133] $\nabla \cdot {\bar{E}}^{(k)} = 0,$ (70) $\nabla \cdot {\bar{B}}^{(k)} = 0,$ (71) $\nabla \times {\bar{E}}^{(k)} = - \frac{\partial {\bar{B}}^{(k)}}{\partial \bar{t}},$ (72) $\nabla \times {\bar{B}}^{(k)} = \frac{\partial {\bar{E}}^{(k)}}{\partial \bar{t}} .$ (73)

3.1.2 Property 2: dual symmetry

We introduce the complex vector $Ψ = \bar{E} + i \bar{B}$ , known as the Riemann-Silberstein vector^[134], where the real and imaginary parts correspond to the electric and magnetic fields, respectively. Using this representation, Maxwell’s equations in free space can be expressed in a more compact form as $i \frac{\partial Ψ}{\partial \bar{t}} = \nabla \times Ψ .$ (74)

Note that this equation remains invariant under a constant phase shift $Ψ^{'} = Ψ e^{i θ}$ , or equivalently, under the corresponding rotation: ${\bar{E}}^{'} \to \bar{E} \cos θ - \bar{B} \sin θ,$ (75) ${\bar{B}}^{'} \to \bar{E} \sin θ + \bar{B} \cos θ .$ (76)

This invariance corresponds to the celebrated continuous dual symmetry of free-space electric and magnetic fields^[135–137].

3.1.3 Property 3: vector potentials

An interesting case arises when $k = - 1$ , which leads to the introduction of vector potentials for both $E$ and $B$ ^[138]: ${\bar{E}}^{(- 1)} = - \bar{C},$ (77) $\bar{E} = - \nabla \times \bar{C},$ (78) ${\bar{B}}^{(- 1)} = \bar{A},$ (79) $\bar{B} = \nabla \times \tilde{A} .$ (80)

Although defining ${\bar{E}}^{(- 1)} = \bar{C}$ without the minus sign would provide a more symmetrical formulation, we retain it for consistency with established notations in the literature.

By imposing the Coulomb gauge condition for free fields in which the vector potential is divergenceless and scalar potentials vanish, the vector potentials also satisfy Maxwell’s equations $\nabla \cdot \bar{A} = 0,$ (81) $\nabla \cdot \bar{C} = 0,$ (82) $\nabla \times \bar{A} = - \frac{\partial \bar{C}}{\partial t},$ (83) $\nabla \times \bar{C} = \frac{\partial \bar{A}}{\partial t} .$ (84)

These vector potentials are related to the physical vector potentials and electric and magnetic fields through the following relation: $D = ε_{0} E = - \nabla \times C,$ (85) $B = \nabla \times A,$ (86) $C = \sqrt{ε_{0}} \bar{C},$ (87) $A = \sqrt{μ_{0}} \bar{A} .$ (88)

3.2 Helicity and Spin

As we will see later, the continuous dual symmetry in Eqs. (75) and (76) is generated by the optical helicity, defined as $h = \frac{1}{2} (\sqrt{\frac{ε_{0}}{μ_{0}}} A \cdot \nabla \times A + \sqrt{\frac{μ_{0}}{ε_{0}}} C \cdot \nabla \times C) .$ (89)

In general, helicity quantifies the degree to which field lines twist and wrap around one another. In fluid mechanics, helicity is defined as the scalar product of the fluid flow velocity $v$ and vorticity $\nabla \times v$ providing a measure of the knottedness of vortex lines^[139,140]. In plasma physics, magnetic helicity^[141,142] is defined by the first term in Eq. (89) and plays a crucial role in understanding plasma stability and dynamics^[143].

Interestingly, when integrated over all space and time-averaged, magnetic helicity coincides with optical helicity in Eq. (89). However, the expression of optical helicity including both terms, i.e., Eq. (89), is preferred because it ensures that $h$ remains invariant under continuous duality symmetry: $C \to C \cos θ + A \sin θ,$ (90) $A \to - C \sin θ + A \cos θ .$ (91)

Furthermore, this formulation satisfies the local conservation law for optical helicity $h$ . To verify the conservation, we take the time derivative of $h$ , yielding $\frac{\partial h}{\partial t} = \frac{1}{2} [\sqrt{\frac{ε_{0}}{μ_{0}}} (\dot{A} \cdot B + A \cdot \dot{B}) - \sqrt{\frac{μ_{0}}{ε_{0}}} (\dot{C} \cdot D + C \cdot \dot{D})] = \frac{1}{2} [\sqrt{\frac{ε_{0}}{μ_{0}}} (- E \cdot \nabla \times A - A \cdot \nabla \times E) - \sqrt{\frac{μ_{0}}{ε_{0}}} (H \cdot \nabla \times C + C \cdot \nabla \times H)] .$ (92)

Using the vector identity $\nabla \cdot (E \times A) = A \cdot (\nabla \times E) - E \cdot (\nabla \times A)$ , we find the conservation law $\frac{\partial h}{\partial t} + \nabla \cdot (c S) = 0,$ (93)where $S$ represents the optical spin vector: $S = \frac{1}{2} (ε_{0} E \times A + μ_{0} H \times C) .$ (94)

The optical spin vector remains invariant under the continuous dual transformation, Eqs. (90) and (91). The conservation law in Eq. (93) clarifies the nature of optical spin as a vector quantity transporting helicity, and likewise, electromagnetic momentum transports electromagnetic energy density.

3.3 Optical Chirality

By applying Property 1 from Sec. 3.1.1, it is evident that helicity conservation can be naturally extended to higher-order cases: $\frac{\partial}{\partial t} h^{(k)} + \nabla \cdot c S^{(k)} = 0 (k \geq 0),$ (95) $h^{(0)} = h,$ (96) $S^{(0)} = S,$ (97)where $h^{(k)} = \frac{1}{2} (\sqrt{\frac{ε_{0}}{μ_{0}}} B^{(k - 1)} \cdot \nabla \times B^{(k - 1)} + \sqrt{\frac{μ_{0}}{ε_{0}}} D^{(k - 1)} \cdot \nabla \times D^{(k - 1)}),$ (98) $S^{(k)} = \frac{1}{2} ε_{0} (E^{(k)} \times B^{(k - 1)} - B^{(k)} \times E^{(k - 1)}) .$ (99)

Particularly, for $k = 1$ , we have $h^{(1)} = \frac{1}{2 c} (\frac{1}{μ_{0}} B \cdot \nabla \times B + ε_{0} E \cdot \nabla \times E),$ (100) $S^{(1)} = \frac{1}{2} ε_{0} [(\nabla \times E) \times B - (\nabla \times B) \times E] .$ (101)

Note that $c h^{(1)}$ is precisely the “optical chirality density $C^{'}$ introduced by Tang and Cohen in Eq. (53). Thus, optical helicity and optical chirality represent the first two cases in an infinite series of conserved quantities associated with continuous dual symmetry^[133]. While they are physically distinct, both quantities characterize the handedness of electromagnetic fields. To explore their relationship, we consider a specific case of a monochromatic plane wave expressed in terms of complex fields, $E = Re (\tilde{E} e^{- i ω t}), B = Re (\tilde{B} e^{- i ω t})$ . Under these conditions, the optical chirality $C$ is given by $C = c h^{(1)} = \frac{1}{2} (\frac{1}{μ_{0}} B \cdot \nabla \times B + ε_{0} E \cdot \nabla \times E) = \frac{ε_{0}}{2} (B \cdot \frac{\partial}{\partial t} E - E \cdot \frac{\partial}{\partial t} B) = \frac{ω ε_{0}}{2} Im (\tilde{E} \cdot {\tilde{B}}^{*}) .$ (102)

Similarly, the helicity is expressed as $h = \frac{1}{2} (\sqrt{\frac{ε_{0}}{μ_{0}}} A \cdot B - \sqrt{\frac{μ_{0}}{ε_{0}}} C \cdot D) = \frac{1}{2 ω} \sqrt{\frac{ε_{0}}{μ_{0}}} Im (\tilde{E} \cdot {\tilde{B}}^{*}),$ (103)where we utilize the relations between the electromagnetic fields and vector potentials in the Coulomb gauge, $E = - \partial A / \partial t, B = - μ_{0} \partial C / \partial t$ as given in Eqs. (85) and (86). Consequently, for a monochromatic plane wave, optical chirality is found to be proportional to helicity by a factor of $ω^{2} / c^{2} = k_{0}^{2}$ . This result is expected, as in the case of a monochromatic plane wave with wave vector $k_{0}$ , the curl operator can be replaced by the wave vector ( $\nabla \times \to i k_{0} \times$ ), allowing for a straightforward demonstration that these quantities become proportional. This leads to the general relation $h^{(k)} = - k_{0}^{2} h^{(k - 1)} .$ (104)

However, this proportionality no longer holds in the presence of superposed plane waves^[144], or in subcycle chiroptical processes^[145].

3.4 Conservation Laws from Noether’s Theorem and Conserved Quantities

According to Noether’s theorem^[146], every continuous symmetry of the action corresponds to a conservation law. Therefore, the continuous dual symmetry of the free-space Maxwell’s equations suggests the existence of associated conservation laws. However, the conventional Lagrangian of the free-space Maxwell equations, $L = E^{2} - B^{2},$ (105)is not invariant under the continuous dual transformation, even though the resulting equation remains invariant. [Here, we use scaled field variables in Eqs. (60 )–(63), (87), and (88) to suppress $ε_{0}$ and $μ_{0}$ but without using the top bar notation]. This lack of invariance is not surprising, as the solution space is only a subset of the full field configuration constituting the action. One can modify the action to respect the symmetry, while still resulting in the same subspace of solutions. This has been achieved by treating the electric and magnetic vector potentials of the electric and magnetic fields as independent variables^[147], even though they are related through $\nabla \times C = - \nabla ϕ_{E} + \frac{\partial A}{\partial t} = E,$ (106) $\nabla \times A = - \nabla ϕ_{B} + \frac{\partial C}{\partial t} = B .$ (107)

The dependence of the vector potentials can be imposed at the action level by introducing a Lagrange multiplier. To better understand the symmetry of the action, we use the Lorentz-covariant formulation of electromagnetism in Minkowski spacetime with coordinates $x^{μ} = (t, x, y, z)$ , where Greek indices $μ, ν = 0, 1, 2, 3$ denote spacetime components, and the metric $η_{μ ν} = diag (1, - 1, - 1, - 1)$ . Using the notation $\partial_{μ} = (\partial / \partial t, \nabla)$ and four-vector potentials $A^{μ} = (ϕ_{M}, A)$ and $C^{μ} = (ϕ_{E}, C)$ , the field tensors are defined by $F^{μ ν} = \partial^{μ} A^{ν} - \partial^{ν} A^{μ},$ (108) $G^{μ ν} = \partial^{μ} C^{ν} - \partial^{ν} C^{μ},$ (109)which can be written in matrix form as $F^{μ ν} = [\begin{matrix} 0 & - E_{x} & - E_{y} & - E_{z} \\ E_{x} & 0 & - B_{z} & B_{y} \\ E_{y} & B_{z} & 0 & - B_{x} \\ E_{z} & - B_{y} & B_{x} & 0 \end{matrix}],$ (110) $G^{μ ν} = [\begin{matrix} 0 & - B_{x} & - B_{y} & - B_{z} \\ B_{x} & 0 & E_{z} & - E_{y} \\ B_{y} & - E_{z} & 0 & E_{x} \\ B_{z} & E_{y} & - E_{x} & 0 \end{matrix}] .$ (111)

In terms of covariant field tensors, the dependence of vector potentials in Eqs. (108) and (109) is expressed as^[147] $G^{μ ν} = {{}^{*}F}^{μ ν} \equiv \frac{1}{2} ε^{μ ν α β} F_{α β} .$ (112)

The continuous dual transformation with an infinitesimal angle is given as $δ A^{μ} = δ θ C^{μ},$ (113) $δ C^{μ} = - δ θ A^{μ},$ (114)which, in terms of field tensors, is written as $δ F^{μ ν} = δ θ G^{μ ν},$ (115) $δ G^{μ ν} = - δ θ F^{μ ν} .$ (116)

To incorporate this continuous symmetry into the action, we treat $A_{μ}$ and $C_{μ}$ as independent variables and introduce the vector potential constraint using a Lagrange multiplier. The modified Lagrangian for the free-space Maxwell’s equations is then given by $L = - \frac{1}{8} F^{μ ν} F_{μ ν} - \frac{1}{8} G^{μ ν} G_{μ ν} - \frac{1}{4} Λ_{μ ν} (G^{μ ν} - {{}^{*}F}^{μ ν}),$ (117)where $Λ_{μ ν} = - Λ_{ν μ}$ is the Lagrange multiplier. Notably, this Lagrangian is indeed invariant under the continuous dual transformation Eqs. (113)–(116) as well as an additional transformation: $δ Λ^{μ ν} = δ θ {{}^{*}Λ}^{μ ν} = \frac{δ θ}{2} ε^{μ ν α β} Λ_{α β} .$ (118)

One can readily verify the invariance of the resulting equations: i) $δ (\partial_{μ} F^{μ ν}) = δ θ \partial_{μ} G^{μ ν} = 0, δ (\partial_{μ} G^{μ ν}) = - δ θ \partial_{μ} F^{μ ν} = 0,$ (119)ii) $δ (G^{μ ν} - {{}^{*}F}^{μ ν}) = - δ θ (F^{μ ν} + {{}^{*}G}^{μ ν}) = - δ θ^{*} (- {{}^{*}F}^{μ ν} + G^{μ ν}) = 0,$ (120)where we have used the identity ${}^{* *}F = - F$ . Under the arbitrary variation of $A$ , $C$ , and $Λ$ , the Lagrangian changes by $δ L = \partial_{μ} {\frac{1}{2} [(F^{μ ν} - {{}^{*}Λ}^{μ ν}) δ A_{ν} + (G^{μ ν} - Λ^{μ ν}) δ C_{ν}]} - \frac{1}{2} [\partial_{μ} (F^{μ ν} - {{}^{*}Λ}^{μ ν})] δ A_{ν} - \frac{1}{2} [\partial_{μ} (G^{μ ν} - Λ^{μ ν})] δ C_{ν} - \frac{1}{4} (G^{μ ν} - {{}^{*}F}^{μ ν}) δ Λ_{μ ν} .$ (121)

The principle of least action leads to the constraint Eq. (112), which, together with the identity $\partial_{μ} {{}^{*}F}^{μ ν} = 0$ , results in the equation of motion: i) $\partial_{μ} F^{μ ν} = 0, \partial_{μ} G^{μ ν} = 0,$ (122)ii) $\partial_{μ} Λ^{μ ν} = 0, \partial_{μ} {{}^{*}Λ}^{μ ν} = 0 .$ (123)

Without loss of generality, we may set $Λ = 0$ .

The vanishing divergence in Eq. (121) gives rise to a local conservation law. The continuous dual transformation in Eqs. (113) and (114) then results in $δ L = δ θ \partial_{μ} [\frac{1}{2} (A_{α} G^{μ α} - C_{α} F^{μ α})] = 0,$ (124)where we assume that the vector potentials satisfy Eq. (122). Since $δ θ \neq 0$ , we obtain the local conservation law $\partial_{μ} h^{μ} = 0,$ (125) $h^{μ} = \frac{1}{2} (A_{α} G^{μ α} - C_{α} F^{μ α}) .$ (126)

Explicitly, the components $h^{0}$ and $h^{1}$ correspond to the helicity and spin vector in the Coulomb gauge, as described in Eqs. (92 )–(94). The total helicity $H$ and optical spin $S$ , defined as the integrals of the local helicity and spin vector over space, have been promoted to operators generating the symmetry transformations with quantized electric and magnetic fields $\hat{E}$ and $\hat{B}$ . Specifically^[148], $\exp (- \frac{i}{ℏ} θ H) \hat{E} \exp (\frac{i}{ℏ} θ H) \approx \hat{E} - θ \hat{B},$ (127) $\exp (- \frac{i}{ℏ} θ H) \hat{B} \exp (\frac{i}{ℏ} θ H) \approx \hat{B} + θ \hat{E},$ (128)and $\exp (- \frac{i}{ℏ} θ \cdot S) \hat{E} \exp (\frac{i}{ℏ} θ \cdot S) \approx \hat{E} - {(θ \times \hat{E})}^{⊥},$ (129) $\exp (- \frac{i}{ℏ} θ \cdot S) \hat{B} \exp (\frac{i}{ℏ} θ \cdot S) \approx \hat{B} - {(θ \times \hat{B})}^{⊥} .$ (130)

Here, the superscript $⊥$ denotes restriction to the transverse part of a vector defined by $V_{i}^{⊥} (r) = \int d^{3} r^{'} δ_{i j}^{⊥} (r - r^{'}) V_{i} (r^{'}),$ (131) $δ_{i j}^{⊥} (r) = \frac{2}{3} δ_{i j} δ (r) - \frac{1}{4 π r^{3}} (δ_{i j} - \frac{3 r_{i} r_{j}}{r^{2}}) .$ (132)

In the case of a plane wave, the symmetry transformation generated by the helicity operator in Eqs. (127) and (128) corresponds to an infinitesimal rotation of field vectors about the wavevector $k$ , while that of the spin operator in Eqs. (129) and (130) corresponds to an infinitesimal rotation about a fixed vector $θ$ . Remarkably, the Maxwell equations remain invariant under the transformation generated by the optical spin in Eqs. (129) and (130). Thus, not only is helicity a conserved quantity, but optical spin itself is also conserved.

This conservation arises due to the rank-3 tensor nature of helicity, which can be understood as a product of rank-1 and rank-2 tensors. In fact, helicity conservation is only part of a broader conservation law associated with rank-3 tensors: $\partial_{μ} N^{α β μ} = 0,$ (133)where $N^{α β μ}$ is defined as $N^{000} = h = \frac{1}{2} (A \cdot B - C \cdot E),$ (134) $N^{0 i 0} = S = \frac{1}{2} (E \times A + B \times C),$ (135) $N^{i j 0} = \frac{1}{2} [δ_{i j} (A \cdot B - C \cdot E) - A_{i} B_{j} - A_{j} B_{i} + C_{i} E_{j} + C_{j} E_{i}],$ (136) $N^{i j k} = δ_{i j} N^{00 k} + \frac{1}{2} (- A_{i} \partial_{k} C_{j} - A_{j} \partial_{k} C_{i} + C_{i} \partial_{k} A_{j} + C_{j} \partial_{k} A_{i}),$ (137) $N^{00 i} = N^{0 i 0},$ (138) $N^{0 i j} = N^{i j 0} .$ (139)

The conservation laws of helicity and optical spin correspond to $\partial_{μ} N^{00 μ} = 0$ and $\partial_{μ} N^{0 i μ} = 0$ , respectively. The remaining six conserved quantities $N^{i j 0}$ , corresponding to $\partial_{μ} N^{i j μ} = 0$ , are known as infra-zilch densities.

This situation can be compared to the energy conservation law, where the rank-2 energy-momentum tensor $T^{μ ν}$ satisfies $\partial_{μ} T^{μ ν} = 0$ . Local energy density $T^{00}$ , satisfying the conservation law $\partial_{μ} T^{μ 0} = 0$ , is transported by linear momentum $T^{0 i}$ following Poynting’s theorem. Similarly, linear momentum $T^{0 i}$ satisfies the conservation law $\partial_{μ} T^{μ i} = 0$ and is transported by the stress tensor $T^{i j}$ . By analogy, optical spin transports helicity, and infra-zilch transports optical spin.

Finally, we apply Property 1 to the conservation laws in Eq. (133). In particular, applying the curl operation once to each quantity, we obtain conservation laws $\partial_{μ} Z^{α β μ} = 0,$ (140)where $Z^{α β μ}$ is given by substituting $A \to B,$ (141) $C \to - E,$ (142) $E \to \nabla \times E,$ (143) $B \to \nabla \times B .$ (144)

This is the zilch tensor, a set of 10 conserved quantities for the source-free electromagnetic field, first discovered by Lipkin in 1964^[131]. In particular, $Z^{000}$ is the optical chirality, which appears in Eq. (53).

3.5 Physical Implications

Lipkin’s zilch, once considered to lack physical significance, gained renewed interest after Tang and Cohen linked it to optical chirality in the context of light-chiral matter interactions^[83]. This revival also spurred further exploration of helicity and optical spin and their relationship to optical chirality. As previously noted, for a monochromatic plane wave, these quantities become proportional when averaged over one time period. However, they remain fundamentally distinct in general. To clarify their physical meaning, we note that helicity and optical spin both have the dimension of angular momentum density, whereas optical chirality has that of force density. This implies that helicity and optical spin, which are interconnected through the conservation laws in Eqs. (93), (125), and (126), are fundamentally linked to the angular momentum of electromagnetic fields. It is well established that the total angular momentum of the field, obtained by integrating the angular momentum density, $r \times (E \times B)$ , over all space, can be expressed as $J = \int r \times (E \times B) d^{3} r = \int \frac{1}{2} (E \times A + B \times C) d^{3} r + \int \frac{1}{2} [E_{i} (r \times \nabla) A_{i} + B_{i} (r \times \nabla) C_{i}] d^{3} r = S + L .$ (145)

That is, the total angular momentum consists of the sum of optical spin $S$ and orbital angular momentum $L$ ^[148,149]. To better understand the relationship between spin and angular momentum and their respective conservation laws, we begin with the covariant generalization of the total angular momentum density from Eq. (145): $M^{α β γ} = x^{α} Θ^{β γ} - x^{β} Θ^{α γ} Θ^{α β} = F^{α γ} F_{γ}^{β} - \frac{1}{4} g^{α β} F^{γ δ} F_{γ δ} .$ (146)

Here, $Θ^{α β}$ is the gauge-invariant energy-momentum tensor. The local conservation of energy momentum ( $\partial_{β} Θ^{α β} = 0$ ) and the symmetry of the tensor ( $Θ^{α β} = Θ^{β α}$ ) lead to the local conservation law of the total angular momentum $\partial_{γ} M^{α β γ} = 0$ . This gauge-invariant tensor $Θ^{α β}$ is obtained via the Belinfante symmetrization of the canonical energy-momentum tensor $T^{α β}$ . The canonical tensor $T^{α β}$ , derived from Noether’s theorem for translational symmetry, is not gauge-invariant. The difference between these two tensors $Θ^{α β}$ and $T^{α β}$ defines the spin tensor $S^{α β γ} = F^{γ α} A^{β} - F^{γ β} A^{α},$ (147)such that $Θ^{α β} - T^{α β} = - \partial_{γ} (A^{α} F^{β γ}) = \frac{1}{2} \partial_{γ} (S^{β γ α} + S^{α γ β} - S^{α β γ}) .$ (148)

The canonical angular momentum tensor ${\tilde{M}}^{α β γ}$ , derived from Noether’s theorem for rotational symmetry, has the form ${\tilde{M}}^{α β γ} \equiv x^{α} T^{β γ} - x^{β} T^{α γ} + S^{α β γ} = L^{α β γ} + S^{α β γ},$ (149)being related to $M^{α β γ}$ by ${\tilde{M}}^{α β γ} = M^{α β γ} + \partial_{η} K^{α β γ η}, K^{α β γ η} = \frac{1}{2} [- x^{α} (S^{γ η β} + S^{β η γ} - S^{β γ η}) + x^{β} (S^{γ η α} + S^{α η γ} - S^{α γ η})] .$ (150) ${\tilde{M}}^{α β γ}$ also satisfies the local conservation law $\partial_{γ} {\tilde{M}}^{α β γ} = 0$ and presents a covariant version of Eq. (145), namely, ${\tilde{M}}^{α β γ} = L^{α β γ} + S^{α β γ}$ , as a sum of orbital angular momentum $L^{α β γ}$ and spin $S^{α β γ}$ . Both $M^{α β γ}$ and ${\tilde{M}}^{α β γ}$ result in the same integrated total angular momentum. However, the local conservation $\partial_{γ} {\tilde{M}}^{α β γ} = 0$ does not imply the local conservation of spin density. In fact, $\partial_{γ} S^{α β γ} = T^{α β} - T^{β α} \neq 0$ . This seems to contradict the conservation of the spin density described in Eqs. (133 )–(139). To resolve this paradox, we note that the conservation of spin density in Eq. (133) is given for the spin vector $S$ , which is related to the spin tensor $S^{α β γ}$ by $S_{k} = (1 / 2) ε_{k i j} S^{i j 0}$ . In the radiation gauge $A_{0} = 0, \nabla \cdot A = 0$ , one can readily check that $\frac{1}{2} ε_{k i j} (\partial_{γ} S^{i j γ}) = \partial_{0} S_{k} + \partial_{m} (\frac{1}{2} ε_{k i j} S^{i j m}) = - \partial_{m} (\frac{1}{2} ε_{k i j} A^{j} \partial^{i} A^{m}),$ (151)which confirms the conservation of the spin vector in free space where the radiation gauge condition holds. Just as the spin operator generates the rotation of fields in Eqs. (129) and (130), the angular momentum operator induces the rotation of coordinates, such that $\exp (- \frac{i}{ℏ} θ \cdot L) \hat{E} \exp (\frac{i}{ℏ} θ \cdot L) \approx \hat{E} + {(θ \cdot (r \times \nabla) \hat{E})}^{⊥},$ (152) $\exp (- \frac{i}{ℏ} θ \cdot L) \hat{B} \exp (\frac{i}{ℏ} θ \cdot L) \approx \hat{B} + {(θ \cdot (r \times \nabla) \hat{B})}^{⊥} .$ (153)

We apply the transversality restriction from Eqs. (131) and (132) to the vector potentials A and C. This ensures that the expressions for optical spin (S) and orbital angular momentum (L) in Eq. (145) are gauge invariant and separately conserved. Furthermore, the transformations in Eqs. (129), (130), (152), and (153) leave Maxwell’s equations invariant. Thus, while S and L do not satisfy the angular momentum commutation relations, they should be considered separately meaningful observables, a view that contrasts with common belief^[150,151].

Also, the local conservation laws of helicity and optical chirality have significant physical implications. This becomes more evident when extending these conservation laws to general cases involving macroscopic matter or external source terms. In the presence of macroscopic matter with permittivity $ε$ and permeability $μ$ , we define the helicity density and optical spin as $h = \frac{1}{2} (\sqrt{\frac{ε}{μ}} A \cdot \nabla \times A + \sqrt{\frac{μ}{ε}} C \cdot \nabla \times C),$ (154) $\tilde{S} = c S = \frac{1}{2} (\sqrt{\frac{ε}{μ}} E \times A + \sqrt{\frac{μ}{ε}} H \times C) .$ (155)

Then, an explicit calculation yields $\frac{\partial h}{\partial t} + \nabla \cdot \tilde{S} = \frac{1}{2} \nabla (\sqrt{\frac{ε}{μ}}) \cdot (E \times A) + \frac{1}{2} \nabla (\sqrt{\frac{μ}{ε}}) \cdot (H \times C) - \sqrt{\frac{ε}{μ}} E \cdot B + \sqrt{\frac{μ}{ε}} H \cdot D .$ (156)

Similarly, we extend the definition of optical chirality to a non-magnetic medium as $h^{(1)} = \frac{1}{2} (\sqrt{\frac{ε}{μ_{0}}} B \cdot \nabla \times B + \sqrt{\frac{μ_{0}}{ε}} E \cdot \nabla \times E),$ (157) ${\tilde{S}}^{(1)} = \frac{1}{2} \sqrt{\frac{ε}{μ_{0}}} [(\nabla \times E) \times B - (\nabla \times B) \times E] .$ (158)

After performing an explicit calculation, we obtain $\frac{\partial h^{(1)}}{\partial t} + \nabla \cdot {\tilde{S}}^{(1)} = \frac{1}{2} \sqrt{\frac{ε}{μ_{0}}} (\nabla \times B) \cdot (\frac{\nabla ε}{ε} \times E) + \frac{1}{2} \nabla (\sqrt{\frac{ε}{μ_{0}}}) [(\nabla \times E) \times B - (\nabla \times B) \times E] .$ (159)

Equations (156) and (159) demonstrate that the conservation law holds when the impedance or permittivity remains constant. Specifically, as evident from Eq. (156), it has been noted that conservation is preserved, and helicity can be properly defined even in heterogeneous media, provided they maintain uniform impedance^[152]. However, in general, the presence of matter introduces impedance mismatches or discontinuities in permittivity, causing the right-hand side to become nonzero, effectively acting as a source term^[100]. This source term clearly violates the local conservation law, accounting for the change of otherwise conserved total charge.

In quantum field theory, the total charge, obtained by integrating a local density over all space, is considered a physical observable. However, the local density itself, despite satisfying a conservation law, is generally not considered physically measurable. This is due to the inherent disturbance a measurement would cause, a consequence of the uncertainty principle.

In contrast, recent advancements in modern optics, particularly at the nanoscale, have challenged this view. It has been argued that local densities, such as the helicity or chirality density in a local conservation law, are indeed physically measurable using localized probes^[153]. For example, by employing nano-sized particles to map local polarization ellipticity or phase gradients, one can experimentally determine the local densities of spin and orbital angular momentum.

Remarkably, such measurements can distinguish between dual-asymmetric and dual-symmetric local densities. This capability suggests that local conservation laws and the concept of dual symmetry in free space are not merely theoretical constructs but are physically significant. Their role, especially in the context of light-matter interactions at the nanoscale, therefore warrants further investigation.

4 Nanophotonic Chiral Sensing

Nanophotonic structures are capable of manipulating the chirality of light (Sec. 3) at deep-subwavelength scales. Given that the multipole moments of chiral molecules interact with distinct quantities of the electromagnetic field (Sec. 2), such structures offer a powerful means to tailor molecular chiroptical responses. In this section, we outline the fundamental principles of nanophotonic chiral sensing based on molecular absorption (Sec. 4.1) and emission (Sec. 4.2), followed by a discussion of recent progress in their experimental implementations (Sec. 4.3).

4.1 Chiral Sensing via Differential Absorption

Transmission-type measurement (Fig. 3) is a typical way to measure CD (see Sec. 5.1 for more details). It measures transmittance $T = I_{T} / I_{0}$ , a ratio of the transmitted intensity $I_{T}$ to the incident intensity $I_{0}$ . Energy conservation requires $T = 1 - R - A$ , where $R = I_{R} / I_{0}$ and $A = I_{A} / I_{0}$ are reflectance and absorptance, respectively. $I_{R}$ and $I_{A}$ are the reflected and absorbed intensities, respectively. For a chiral sample, transmittance and absorptance become different, i.e., $T_{\pm}$ and $A_{\pm}$ , according to the two opposite-handed circularly polarized lights. On the other hand, reflectance $R_{+} = R_{-}$ is the same because the chirality does not break the time-reversal symmetry. Therefore, the difference in transmittance, $Δ T = T_{+} - T_{-}$ , is equivalent to that in absorptance, i.e., CD, $Δ A = A_{+} - A_{-}$ . Note that the same still holds for chiral nanostructures. Therefore, a transmission-type setup is a primary choice to measure the CD of nanostructure-molecule coupled systems, as in CD measurements for isolated molecules. In the following subsections, we discuss the principle of chiral sensing via differential absorption measurement.

Figure 3.Transmission-type CD measurement. LCP and RCP lights are transmitted separately through the chiral sample (i.e., chiral molecules and/or a nanophotonic sensor), and their difference in transmission is recorded. The difference in transmission is equivalent to that in absorption, CD.

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Before proceeding to the following subsection, it is important to note that interpreting CD spectra obtained via transmission-type measurements in the presence of nanophotonic structures requires careful consideration. In conventional CD measurements of isolated molecules, the spectral lineshapes and the sign of the CD signal provide direct stereochemical information about the analyte. In contrast, the CD spectra of chiral molecule-nanophotonic hybrids often reflect a complex interplay between the molecular responses and the optical resonances of the nanostructure. As a result, most previous studies have relied on identifying molecular chirality through modifications in the spectral lineshapes, while pure enhancement of CD signal intensity without spectral distortion has remained relatively rare.

4.1.1 Chiral absorption of a single molecule in the near-field: the “superchiral” field

Nanophotonic chiral sensing has been inspired by the introduction of optical chirality density, $C$ [Eqs. (53) and (102) for general electromagnetic fields and monochromatic harmonic fields, respectively], which enhances the magnitude of CD signals^[83]. The absorption rates of a chiral molecule for two oppositely circularly polarized light states ( $\pm$ ) are given by $A^{\pm} = \frac{ω}{2} α^{''} | \tilde{E} | \pm G^{''} ω Im ({\tilde{E}}^{*} \cdot \tilde{B}),$ (160)where $α$ and $G$ denote the electric polarizability and the electric-magnetic mixed polarizability, respectively. The double prime indicates the imaginary part. Here, we recall Eq. (102), which defines the optical chirality density $C$ for monochromatic fields: $C = c h^{(1)} = \frac{ω ε_{0}}{2} Im (\tilde{E} \cdot {\tilde{B}}^{*}) .$ (161)

It follows that the magnitude of CD, i.e., $Δ A = A^{+} - A^{-}$ , is determined by the optical chirality density $C$ [Eq. (161)] at the point where the molecule is located. On the other hand, the dissymmetry factor $g$ , which represents the ratio of CD to conventional absorption, is defined as $g \equiv \frac{A^{+} - A^{-}}{(A^{+} - A^{-}) / 2} = - (\frac{G^{''}}{α^{''}}) (\frac{2 C}{ω U_{e}}),$ (162)where $U_{e} = ε_{0} {| \tilde{E} |}^{2} / 4$ is the electric part of the energy density. The term “superchiral field” originally referred to a field configuration that enhances the dissymmetry factor $g$ by minimizing $U_{e}$ while maintaining a finite $C$ . For instance, Cohen achieved high- $g$ chiroptical spectroscopy using a mirror geometry that generated a superchiral field [Fig. 4(a)]. However, the term “superchiral field” has since come to describe near-fields exhibiting an optical chirality density $C$ greater than the optical chirality density of circularly polarized plane waves $C_{CPL}$ .

Figure 4.Enhancement of single-molecule CD by the superchiral fields. (a) The first experimental realization of the dissymmetry factor enhancement by the mirror geometry^[84]. (b) The limit of localized surface plasmon resonance to enhance the optical chirality density (left) with the uniform sign (c.f. the dipolar field enhancement)^[154]. (c) An example of the uniform optical chirality density in plasmonic structures^[155]. (d) The uniform optical chirality density enhancement in dielectric nanoparticles^[63].

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In nanophotonics, there is significant interest in enhancing $C$ beyond $C_{CPL}$ without sign flipping, as a sign reversal of $C$ leads to the cancellation of CD enhancement. In addition, the sign-uniform $C$ can provide an opportunity for enhanced CD spectroscopy for chiral molecules because the resulting chiroptical signals with the sign-uniform $C$ restore the handedness information of the molecular analytes. However, achieving sign-uniform $C$ using nanophotonic structures is challenging. For example, dipolar localized surface plasmon resonances (LSPRs) cannot produce sign-uniform optical chirality^[154], as illustrated in Fig. 4(b). To provide the sign uniform $C$ enhancement, sophisticated plasmonic structures^[155] [e.g., Fig. 4(c)] and dielectric nanoparticles with equal electric and magnetic polarizabilities^[63] [e.g., Fig. 4(d)] have been suggested.

Although the concept of optical chirality density $C$ has driven research in nanophotonic chiral sensing, it fails to fully capture the complex interactions between molecules and nanostructures. Consequently, experimental studies have reported CD sensitivities significantly higher than those predicted based on optical chirality density [Eq. (161)].

4.1.2 Chiral absorption of the medium: the induced and inherent CD

To overcome the limitation of the theory based on the optical chirality density [Eq. (161)], the chiral medium effect on the nanophotonic sensors is taken into account. By Poynting’s theorem for the harmonic field^[156], the energy absorption rate by the medium of a volume $V$ is given by $P_{abs} = - (ω / 2) \int_{V} Im (\tilde{E} \cdot {\tilde{D}}^{*} - \tilde{B} \cdot {\tilde{H}}^{*}) d V$ . Combining it with the constitutive relations for the chiral media, $\tilde{D} = \tilde{ε} \tilde{E} + i \tilde{κ} \tilde{H} / c$ and $\tilde{B} = \tilde{μ} \tilde{H} - i \tilde{κ} \tilde{E} / c$ , we can write $P_{abs} (\tilde{E}, \tilde{H}) = P_{abs, E} (\tilde{E}) + P_{abs, M} (\tilde{H}) + P_{abs, C} (\tilde{E}, \tilde{H})$ , where each component is given by $P_{abs, E} (\tilde{E}) = \frac{ω}{2} \int_{V} Im (\tilde{ε}) {| \tilde{E} |}^{2} d V,$ (163) $P_{abs, M} (\tilde{H}) = \frac{ω}{2} \int_{V} Im (\tilde{μ}) {| \tilde{H} |}^{2} d V,$ (164) $P_{abs, C} (\tilde{E}, \tilde{H}) = \frac{ω}{c} \int_{V} Im (\tilde{κ}) Im (\tilde{E} \cdot {\tilde{H}}^{*}) d V .$ (165)

$P_{abs, E}$ , $P_{abs, M}$ , and $P_{abs, C}$ describe the absorption of the electric energy, the magnetic energy, and the chiral energy, respectively. In natural materials, $P_{abs, M} = 0$ because of the inherently negligible optical magnetism. $P_{abs, C}$ is proportional to the optical chirality density $C$ (and the optical helicity density $h$ ) for the harmonic field.

Suppose a nanosensor of a volume $V_{NS}$ provides the near-field to chiral molecule analytes, and they are electromagnetically coupled with each other. Chiral molecule analytes occupy a volume $V_{A}$ in the vicinity of the nanosensor. Upon excitation by circularly polarized light, the optical responses of the chiral system [i.e., a nanosensor structure and/or chiral molecules as illustrated in Fig. 5(a)] can be different. Therefore, the electromagnetic fields ${\tilde{E}, \tilde{H}}$ can be replaced with ${{\tilde{E}}_{\pm}, {\tilde{H}}_{\pm}}$ , where the subscript $\pm$ denotes the circular polarizations. According to the absorption components [Eqs. (163) and (165)] and the integration volume, we can decompose the total CD $Δ P_{abs} ({\tilde{E}}_{\pm}, {\tilde{H}}_{\pm}) = P_{abs} ({\tilde{E}}_{+}, {\tilde{H}}_{+}) - P_{abs} ({\tilde{E}}_{-}, {\tilde{H}}_{-})$ into^[112] $Δ P_{abs, E; NS} ({\tilde{E}}_{\pm}) \equiv P_{abs, E; NS} ({\tilde{E}}_{+}) - P_{abs, E; NS} ({\tilde{E}}_{-}) = \frac{ω}{2} \int_{V_{NS}} Im (\tilde{ε}) ({| {\tilde{E}}_{+} |}^{2} - {| {\tilde{E}}_{-} |}^{2}) d V,$ (166) $Δ P_{abs, E; A} ({\tilde{E}}_{\pm}) \equiv P_{abs, E; A} ({\tilde{E}}_{+}) - P_{abs, E; A} ({\tilde{E}}_{-}) = \frac{ω}{2} \int_{V_{A}} Im (\tilde{ε}) ({| {\tilde{E}}_{+} |}^{2} - {| {\tilde{E}}_{-} |}^{2}) d V,$ (167) $Δ P_{abs, C; A} ({\tilde{E}}_{\pm}, {\tilde{H}}_{\pm}) \equiv P_{abs, C; A} ({\tilde{E}}_{+}, {\tilde{H}}_{+}) - P_{abs, C; A} ({\tilde{E}}_{-}, {\tilde{H}}_{-}) = \frac{ω}{c} \int_{V_{A}} Im (\tilde{κ}) [Im ({\tilde{E}}_{+} \cdot {\tilde{H}}_{+}^{*}) - Im ({\tilde{E}}_{-} \cdot {\tilde{H}}_{-}^{*})] d V .$ (168)

Figure 5.Decomposition of CD in a molecule-nanosensor system^[112]. (a) Schematic drawing for the system consisting of a molecule film-coated (green) gold nanodisk array (gold). (b) Inherent CD of molecules is enhanced by the strong near-field of the nanostructure. (c) CD is induced by the presence of chiral molecules in the vicinity of the nanostructure. Decomposed CD of the system coupled to (d) ORD-only molecules ( $κ = 0.0 01$ ) and (e) CD-only molecules ( $κ = 0.0 01 i$ ).

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$Δ P_{abs, E; NS} ({\tilde{E}}_{\pm})$ [Eq. (166)] is the induced CD of the nanosensor [Fig. 5(c)]. This term does not vanish, and it also contributes to the total CD although the nanosensor is achiral, while the molecule analytes are chiral, resulting from the chirality transfer. $Δ P_{abs, E; A} ({\tilde{E}}_{\pm})$ [Eq. (167)] is the induced CD of the molecule’s analytes. $Δ P_{abs, C; A} ({\tilde{E}}_{\pm}, {\tilde{H}}_{\pm})$ [Eq. (168)] is the inherent CD of the molecule analytes that can be enhanced by the optical chirality density of the near-field [Fig. 5(b)]. This term [Eq. (168)] is the extension of Cohen’s original suggestion, Eq. (52)^[83,85].

The following are conclusions of the chiral absorption described by Eqs. (166 )–(168): (i) the total CD has three contributions, Eqs. (166 )–(168), and thus we conclude that one needs to consider the induced and inherent CD together to explain the total CD for the nanosensor/chiral molecule system [Figs. 5(d) and 5(e)]; (ii) Eqs. (166) and (167) are not explicitly dependent on the optical chirality density $C \propto Im ({\tilde{E}}_{\pm} \cdot {\tilde{H}}_{\pm}^{*})$ . The electric field splitting, ${\tilde{E}}_{+} \neq {\tilde{E}}_{-}$ , resulting in strong induced CD $Δ P_{abs, E; NS} ({\tilde{E}}_{\pm})$ , can be more important, especially when the inherent molecular chirality, $Im (\tilde{κ})$ , is small, as shown in Fig. 5(d); (iii) If the field splitting is large, the difference in the optical chirality density $C \propto [Im ({\tilde{E}}_{+} \cdot {\tilde{H}}_{+}^{*}) - Im ({\tilde{E}}_{-} \cdot {\tilde{H}}_{-}^{*})]$ determines the inherent CD, rather than $C$ itself.

4.1.3 Spectral shift by dielectric and chiral effects

Nanophotonic sensors exploit spectral shifts of the resonance to detect small impurities nearby, such as molecules and nanoparticles^[1,11,13]. Chiral sensing also uses spectral shifts, but their shifting behavior is more complicated than dielectric sensing. Figures 6 and 7 illustrate diagrams of spectral shifts by dielectric and chiral analytes. The resonance frequency of the nanosensor is depicted by two horizontal bars. We first consider an achiral nanosensor coupled with analytes in Fig. 6. Figure 6(a) shows the energy diagram of the nanosensor before and after coupling to the analyte. Suppose the bare nanosensor is resonant at the frequency $ω_{0}$ . Upon coupling to the molecular analyte, the energy of the nanosensor is taken by the molecules, resulting in the resonance frequency shift. Molecules nearby change the refractive index $\tilde{n} = \sqrt{\tilde{ε}}$ as well as the real part of the chirality parameters $κ$ . The change in the refractive index red-shifts the resonance frequency to $ω_{0} - δ ω_{0}^{(n)}$ . This dielectric shift obeys the well-known formula^{[1,107,157–161]} $\frac{δ ω_{0}^{(n)}}{ω_{0}} = - {\frac{Δ ε \int_{A} {| \tilde{E} (r) |}^{2} d^{3} r}{\int_{NS} (\partial ε / \partial ω) {| \tilde{E} (r) |}^{2} d^{3} r} |}_{ω = ω_{0}},$ (169)

$(a) Energy diagram of the achiral nanosensor resonance ω0 before coupling to molecule analytes (black). The optical responses of the achiral nanosensor to LCP and RCP incidences are identical. After coupling, two effects are involved; the dielectric effect due to the refractive index n decreases the nanosensor resonance frequency ω0−δω0(n). The chiral effect due to the chirality parameter κ increases or decreases the resonance frequency ω0−δω0(n)±δω0(κ) according to the nanosensor characteristic, while it makes the nanosensor resonance circular dichroic (red and blue). (b) Spectral lineshapes of the absorption before (black) and after (red and blue) coupling. The resulting CD spectrum shows an asymmetric lineshape.$

Figure 6.(a) Energy diagram of the achiral nanosensor resonance $ω_{0}$ before coupling to molecule analytes (black). The optical responses of the achiral nanosensor to LCP and RCP incidences are identical. After coupling, two effects are involved; the dielectric effect due to the refractive index $n$ decreases the nanosensor resonance frequency $ω_{0} - δ ω_{0}^{(n)}$ . The chiral effect due to the chirality parameter $κ$ increases or decreases the resonance frequency $ω_{0} - δ ω_{0}^{(n)} \pm δ ω_{0}^{(κ)}$ according to the nanosensor characteristic, while it makes the nanosensor resonance circular dichroic (red and blue). (b) Spectral lineshapes of the absorption before (black) and after (red and blue) coupling. The resulting CD spectrum shows an asymmetric lineshape.

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Figure 7.Energy diagram of the chiral nanosensor resonance $ω_{0}$ before coupling to molecule analytes.

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where $Δ ε = ε_{A} - 1$ is the increase in the real part of the relative permittivity of the molecule analyte $ε_{A}$ . For chiral molecular analytes, the resonance shifts are bi-directional for two opposite-handed circularly polarized lights. These chiral shifts are described by^[107] $\frac{δ ω_{0}^{(κ)}}{ω_{0}} = - {\frac{2 Δ κ \int_{A} Im [\tilde{E} (r) \cdot \tilde{H} (r)] d^{3} r}{c \int_{NS} (\partial ε / \partial ω) {| \tilde{E} (r) |}^{2} d^{3} r} |}_{ω = ω_{0}},$ (170)where $Δ κ$ is the increase in the real part of the chirality parameter of the molecule analyte. As shown in Eq. (170), the direction of the chiral shifts, i.e., sign of $δ ω_{0}^{(κ)}$ , is determined by both the handedness of molecules and the sign of $Im (\tilde{E} \cdot \tilde{H})$ provided by the near-field of the nanosensor. Similar expressions corresponding to Eqs. (169) and (170) have been derived in another literature^[113]. Figure 6(b) shows a typical spectral lineshape of the nanosensor/molecule-coupled system. Suppose the nanosensor has the Lorentzian absorption lineshape, $g (ω)$ [Eq. (38)]. Resulting from the chiral shifts $\pm δ ω_{0}^{(κ)}$ , the absorption spectra split into two, while their strengths become asymmetric [Fig. 6(b)]. The CD spectrum shows the asymmetric dip-to-peak lineshape that is somewhat similar to the real part of the Lorentzian $f (ω)$ [Eq. (37)], but it is fundamentally irrelevant to the Lorentzian.

In addition to the chiral shifts, Eq. (170), the mode strength can be perturbed by the chiral molecules nearby. The perturbed mode strengths by the dielectric and chiral effects are expressed by^[107] $δ a^{(n)} = 2 i Q \frac{δ ω_{0}^{(n)}}{ω_{0}},$ (171) $δ a^{(κ)} = 2 i Q \frac{δ ω_{0}^{(κ)}}{ω_{0}},$ (172)respectively. Q is the quality factor of the mode.

Chiral nanosensors share the same principles in Eqs. (169) and (170) for the spectral shifts by the molecule analytes. As shown in Fig. 7, the chiral nanosensor has two different resonance frequencies $ω_{0 \pm}$ because of its structural chirality. Then, the dielectric shifts by molecules, Eq. (169), in the chiral nanosensor are also asymmetric because the local fields ${\tilde{E}}_{+}$ are ${\tilde{E}}_{-}$ are different. The chiral shifts, Eq. (170), are also different and bidirectional. In the chiral nanosensors, it is likely to obtain spatially uniform $Im (\tilde{E} \cdot \tilde{H})$ in their vicinity. The uniformity of $Im (\tilde{E} \cdot \tilde{H})$ in the chiral nanosensors is a clear advantage because one does not need to consider the exact location of molecules. Another important advantage of the chiral nanosensors is their ability to distinguish the molecular handedness.

4.1.4 Microscopic theory for nanophotonic chiral sensing

As shown in Secs. 4.1.2 and 4.1.3, the optical response of nanophotonic sensors can be perturbed by the presence of chiral molecules nearby. In this section, we introduce microscopic theories to explain the chiral molecular perturbations to the nanophotonic sensor response. In 2010, Govorov, Naik, and their coworkers suggested a quantum mechanical theory of the molecule-plasmon Coulomb interaction [Figs. 8(a)–8(d)], explaining an emergence of new CD peaks at plasmonic resonance when chiral molecules are coupled to achiral plasmonic structures^[162–164]. CD of chiral molecules is proportional to the E1-M1 interference, ${CD}_{molecule} \propto Im (p \cdot m)$ , as discussed in Sec. 2. According to their theory, CD of a chiral molecule-nanoparticle (NP) complex can be described by^[162,163] ${CD}_{molecule-NP} = a Im [(\hat{P} \cdot p) \cdot m] + b F (p, m),$ (173)

Figure 8.(a)–(d) Plasmon-induced CD by the Coulomb interaction between chiral molecules and plasmonic nanoparticles^[162]. (a) Normalized extinction spectra of a chiral molecule (black: the E1 moment $\vec{μ}$ and the M1 moment $\vec{m}$ ), a silver nanoparticle (blue), and a gold nanoparticle (red). (b) Field enhancement near each particle. (c) Normalized CD and ORD spectra of the chiral molecule. (d) Normalized CD spectra of chiral-molecule-coupled silver (blue) and gold (red) nanoparticles. (e) Nanophotonics-induced CD by the electromagnetic interaction between chiral media and nanostructures. Differential scattering cross sections of a gold nanoparticle embedded in the ORD-alone chiral medium^[120].

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where the coefficients $a$ and $b$ describe the geometry, material properties (e.g., the material permittivity), and the frequency of light $ω$ . The first term explains the intramolecular dissipation affected by the electric field enhancement matrix $\hat{P}$ . The second term originates from the chiral electric field in an NP that is induced by the electric dipole of the chiral molecule. The function $F (p, m)$ characterizes the geometry of the complex. As shown in Fig. 8(d), numerical calculations based on Eq. (173) demonstrate that new CD peaks arise for the complex at the plasmonic resonance frequencies.

The electromagnetic theory has also been suggested to explain the emergence of CD responses at the nanophotonic resonance frequencies. For example, the chiral Mie theory was presented by Bohren^[165,166] and our group^[120]. In the chiral Mie theory [Fig. 8(e)]^[120], an achiral nanoparticle of the permittivity $ε_{1}$ is immersed in a chiral medium of the permittivity $ε$ and the chirality parameter $κ$ . The chirality parameter $κ$ is assumed to be purely real, i.e., the medium exhibits ORD alone. Figure 8(e) demonstrates the differential scattering cross sections, $Δ C_{sca}$ , of a gold nanoparticle in the chiral medium for two opposite-handed circularly polarized excitations. Our analytic theory suggests that the achiral nanoparticle can exhibit circular dichroic scattering if it interacts electromagnetically with the chiral medium in its vicinity. To explain the electromagnetic interaction between the chiral medium and the arbitrary nanostructures, we also implement the numerical simulation of the chiral medium described by Eqs. (3) and (4) using the finite-element method (FEM)^[112,167].

4.2 Chiral Sensing via Differential Emission

The nanophotonic structures can modify the fluorescence characteristics of the photon emitter such as molecules and quantum dots. This cavity-modification effect on the photon emission is known as the Purcell effect. When chiral fluorophores couple to the resonant nanophotonic structure, our group suggested that the Purcell effect becomes circular dichroic^[101]. In the absence of the resonant nanostructure, i.e., free space, the polarization-dependent spontaneous decay rate of a chiral fluorophore has the form $Γ_{0, \pm} = Γ_{0, p} + Γ_{0, m} \pm Δ Γ_{0} / 2$ , where the first two terms correspond to E1 and M1 transitions, while the last term corresponds to circularly polarized emission, resulting from the natural electronic activity described by E1-M1 interference. The circular difference in the free-space decay rate is given by^[168] $Δ Γ_{0} = - \frac{4 ω^{3} n}{3 π ε_{0} ℏ c^{4}} Im [p \cdot m] .$ (174)

On the other hand, the (quasi)normal mode formalism provides a formula for the cavity-modified differential decay rate for a molecule at the position $r = r_{0}$ ^[101]: $\frac{Δ Γ (r_{0})}{Δ Γ_{0}} = F_{C} \frac{ω_{0}^{2}}{ω^{2}} \frac{ω_{0}^{2}}{ω_{0}^{2} + 4 Q^{2} {(ω - ω_{0})}^{2}} \frac{C (r_{0})}{C_{\max}} η_{C} .$ (175)

Here, we define the chiral Purcell factor: $F_{C} = \frac{1}{4 π^{2}} {(\frac{λ_{0}}{n})}^{3} (\frac{Q}{V_{C}}) .$ (176)

In Eq. (175), we also define the extended optical chirality: $C (r_{0}; u, v) = - \frac{3 n^{2} ε_{0} ω}{2} Im {[u \cdot \tilde{E} (r_{0})] [v \cdot \tilde{B} (r_{0})]},$ (177)reflecting the molecular orientation, $p = p u$ and $m = m v$ . Rotational average reduces Eq. (177) to the conventional optical chirality $C$ . The orientation factor is defined by $η_{C} = C (r_{0}; u, v) / C (r_{0})$ . $C_{\max}$ in Eq. (175) denotes the optical chirality density at its maximum position. In Eq. (176), we also define the chiral mode volume: $V_{C} = - \frac{ω U}{c C_{\max}} = \frac{U}{u_{C, \max}} .$ (178)

The conventional mode volume represents the spatial confinement of the light energy density near the cavity because it is defined by a ratio of the total energy to the energy density. On the other hand, the definition of the chiral mode volume $V_{C}$ implies the spatial confinement of the chiral part of the energy density $u_{C} = c C_{\max} / ω$ near the cavity. One can use the chiral Purcell factor $F_{C}$ , Eq. (176), to characterize the cavity performance to enhance a chiral emitter’s circularly polarized fluorescence or photoluminescence.

Figure 9 shows the enhancement of fluorescent CD by the hypothetical helicity-preserving Fabry-Perot cavity^[101]. Ordinary mirrors change the helicity of light upon reflection, and thus, the Fabry-Perot cavity formed by the mirrors cannot support the chiral Purcell enhancement. On the other hand, in the cavity formed by the hypothetical helicity-preserving mirrors, the chiral Purcell factors can exceed unity. The realistic cavities supporting the chiral Purcell effect have also been suggested (e.g., a fishnet structure^[101]) and realized (e.g., a twisted photonic crystal^[102] and a chiral quantum metamaterial^[103]).

Figure 9.Chiral Purcell-enhancement of fluorescent CD in a hypothetical helicity-preserving Fabry-Perot cavity^[101]. Cavity resonances provide the CD enhancement, i.e., $Δ Γ / Δ Γ_{0} > 1$ .

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4.3 Literature Review for Experimental Realizations

Explosive interest in nanophotonic chiral sensing in the past decade has brought numerous experimental realizations. In this subsection, we introduce remarkable experimental works that have realized highly sensitive nanophotonic chiral sensors. Figure 10(a) shows chiral sensing by gold nanoislands^[114]. Gold nanoislands were simply obtained by gold evaporation without an adhesion layer on glass. Riboflavin molecules are embedded in poly(methyl methacrylate) (PMMA), and they are spin-coated on the gold nanoislands. The CD spectrum shows the emergence of a plasmon-induced CD peak around 600 nm, while bare riboflavin molecule films exhibit CD peaks at $\sim 350$ and $\sim 450 nm$ . The plasmon-induced CD peak at $\sim 600 nm$ is observable for bilayer riboflavin, which is not possible to detect in a bare riboflavin film. This work also demonstrated that poly-L-lysine molecules whose CD features appear only in the UV range can be detected by gold nanoislands.

Figure 10.Absorption-based nanophotonic chiral sensing. (a) Plasmon-induced CD of riboflavin bilayer-coated gold nanoislands^[114]. (b) Optical chirality enhancement of gold gammadion arrays and their chiral sensing^[82]. (c) Zeptomole-level chiral sensing by twisted optical metamaterials^[168].

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Figure 10(b) shows a pioneering work on nanophotonic chiral sensing by Hendry and Kadodwala in 2010^[82]. They fabricated gold gammadion arrays using e-beam lithography. They successfully detected various chiral molecule species, such as tryptophan, concanavalin A, β-lactoglobulin, BSA, myoglobin, and hemoglobin, at high sensitivity. They claimed that their CD signals are $10^{6}$ times greater than those in conventional CD spectroscopy. Their original intention to prepare gammadion structures was to obtain the optical chirality density $C$ with the uniform sign, resulting in molecular CD enhancement as proposed by Tang and Cohen^[83]. However, experimentally observed dissymmetry was much larger than the prediction based on the enhanced optical chirality density $C$ . It is likely that the CD enhancement by the gammadion array was mediated by Coulomb and electromagnetic interaction discussed in Sec. 3.1.3.

Figure 10(c) presents microfluidic chiroptical sensing by twisted optical metamaterials. Gold nanorods are twisted by $\pm 60 °$ to form chiral metamaterials. E-beam lithography and an etch-back planarization method are used to fabricate the twisted optical metamaterials. Propanediol, irinotecan hydrochloride (the anticancer drug), and monolayer protein (Concanavalin A) were detected at high sensitivity. They claimed that they sense chiral molecules down to zeptomole levels.

In addition to the absorption-based chiral sensing, scattering-based chiral sensing can provide few-molecule level detection (please see Sec. 5.4.2 for details on circular differential scatterometry and its instrumentation). For example, a nanoparticle-on-mirror (NPoM) structure can form an ultracompact cavity in its nanogap between the particle and the metallic substrate. A single nanoparticle can host four chiral molecules, i.e., helical self-assembled monolayer (SAM), within the nanogap. The nanogap of NPoM provides not only strong enhancement of the electric field and the optical chirality density, but also additional chiroptical enhancements by tunnel electrons^[169].

Our group and our collaborators also demonstrated highly sensitive chiral sensing using a chiral nanostructure^[107]. Synthesis of chiral gold nanoparticles, namely, helicoids, was reported by Lee et al. in 2018^[170]. We first fabricated nanopatterned PDMS with $10^{8}$ nanowells. Then, the synthesized helicoids are loaded on the nanowells in the PDMS to form a periodic array of helicoids with a 400 nm pitch [Figs. 11(a) and 11(b)]. Each helicoid shows chiral LSPRs around the wavelength region, 500–700 nm, but the helicoid array has another chiral resonance at the longer wavelength of $\sim 900 nm$ , originating from the photonic crystal nature [Fig. 11(b)]^[171–175]. The unique property of this photonic crystal resonance of the helicoid array is the uniform optical helicity $h$ at the array plane [Fig. 11(c)], and thus we named it collective CD for this resonance. We demonstrated chiral sensing for proline, glucose, sVAMP2, and miR-21 using this sensor platform [Fig. 11(d)]. The $10^{- 13} M$ (0.1 pM)-level sensitivity was achieved.

Figure 11.Nanophotonic chiral sensing using collective circular dichroism of a gold helicoid array^[107]. (a) Schematic drawing for the sensing mechanism. (b) Electron (left) and optical (right) microscope images of the helicoid array. (c) Near-field profiles of the uniform optical helicity density $h$ within the array plane upon LCP and RCP incidences. (d) Sensitivity of the helicoid array sensor for various chiral molecule species.

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Luminescence-based chiral sensing has also been demonstrated by a quantum-dot-coupled chiral metamaterial [Fig. 12(a)]^[103]. Using an injection molding technique, a gold thin film hosts periodic shuriken-shaped holes over a large area of $500 μ m \times 500 μ m$ . Each shuriken structure can host six molecules corresponding to $\sim 900 zmol$ over the whole metamaterial area. By the presence of chiral molecules, luminescence properties become significantly changed. The numerical simulations [the right panel of Fig. 12(a)] suggest that the change in luminescence spectra by chiral molecules can be attributed to the chiral Purcell effect we discussed in Sec. 3.2. In addition to luminescence-based chiral sensing, it has been reported that incorporation of chiral fluorescent molecules with nanophotonic structures leads to chiral light-emitting devices, such as chiral OLEDs^[176] and chiral lasers^[177] [Figs. 12(b) and 12(c)]. Although these chiral light-emitting devices did not aim to chiral sensing, they provide another intriguing direction in active chiral sensing devices.

Figure 12.Light emission by chiral-molecule-nanostructure complexes. (a) Luminescence-based chiral sensing by chiral quantum metamaterials^[103]. By the chiral Purcell effect, quantum dots composed of metamaterials show different luminescence by the presence of chiral molecules, allowing the sub-zeptomole level sensitivity. (b) Circularly polarized organic light-emitting-diodes (CP-OLEDs)^[176] and their spectra of the CP photoluminescence (CPPL) and the PL dissymmetry factor $g_{PL}$ . Circularly polarized light (CPL) is emitted by chiral Frenkel excitons and/or chiral plasmons. (c) Chiral laser by the Fabry-Perot cavity with gain media, fluorescein (FITC) binding with l-tryptophan and green fluorescent proteins (GFPs)^[177].

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5 Measurements and Implementations

We have discussed the principle of nanophotonic sensing and the properties of chiral light so far. On the other hand, the chiroptical measurements are sometimes challenging in practice, even though the nanophotonic sensor amplifies (or modulates) inherently weak chiral signals of molecule analytes. In this section, we introduce important concepts and instrumentations in chiroptical spectroscopy.

5.1 Circular Dichroism and Optical Rotatory Dispersion

Conventional CD and ORD spectroscopy are based on the absorption of light by a sample. Before discussing CD and ORD, we first discuss absorption spectroscopy, and the important measures frequently used.

Commercial ultraviolet-visible (UV-VIS) spectrophotometers usually measure the transmission of light through a sample much thicker than the wavelength of light, i.e., an optically thick sample. Absorption within the sample is dominant, while reflection at the air-sample interfaces is relatively negligible (Fig. 13). Therefore, the measurement of transmission is equivalent to that of absorption.

Figure 13.CD measurement for the bulk sample using conventional CD spectrophotometers. Intensities of two circularly polarized lights ( $I_{\pm}$ ) are plotted. Lights experience different attenuation coefficients $μ_{\pm}$ within the chiral molecule sample according to their handedness.

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Here, we first review important measures used in UV-VIS spectrophotometry. Light propagating through the dielectric medium with the complex refractive index, $\tilde{n} = n + i k$ , is described by the electric field $\tilde{E} = E_{0} \exp (i \tilde{n} k_{0} z)$ , where $E_{0}$ , $k_{0}$ , and $z$ are the amplitude of the incident light, the vacuum wavenumber $k_{0} = 2 π / λ$ at the wavelength $λ$ , and the position, respectively. If the light enters the medium, and then propagates along the distance $l$ , the intensity is written as $I \equiv {| \tilde{E} |}^{2} = I_{0} \exp (- 2 k k_{0} l)$ , where $I_{0} \equiv E_{0}^{2}$ is the intensity of the incident light. Then, transmittance $T$ is defined by the intensity ratio of the transmitted to the incident light, i.e., $T \equiv \frac{I}{I_{0}} = \exp (- μ l),$ (179)where the optical density or the attenuation coefficient is defined by $μ \equiv 2 k k_{0} = 4 π k / λ$ . In chemistry and biology, the common logarithm of transmittance is called the absorbance $A$ , and the Beer-Lambert law, $A \equiv - \lg T = ε c l,$ (180)relates the absorbance to the molar absorption coefficient $ε$ and the molar concentration $c$ of the sample. Note that the molar absorption coefficient $ε$ is an intrinsic property of a given molecule. From Eq. (179), we can write $A = - \lg T = - \ln T / \ln 10 = μ l / \ln 10$ . Combining it with Eq. (180), we obtain the relation for the imaginary part of the refractive index $k$ : $Im (\tilde{n}) = k = \frac{2.303 λ c ε}{4 π},$ (181)where $\ln 10 \approx 2.303$ . Putting Eq. (181) back into the Beer-Lambert law, Eq. (180), we are also able to write $A = \frac{4 π}{\ln 10} \frac{Im (\tilde{n}) l}{λ} = \frac{4 π}{2.303} \frac{k l}{λ} .$ (182)

We are now ready to discuss CD spectroscopy for chiral samples. CD spectroscopy measures differences in absorbance ( $Δ A \equiv A_{+} - A_{-}$ ) or molar absorption coefficient ( $Δ ε \equiv ε_{+} - ε_{-}$ ) for two opposite circularly polarized lights, where the plus and minus signs denote left and right circular polarization, respectively. On the other hand, isotropic chiral samples have the circular birefringent refractive index, ${\tilde{n}}_{\pm} = \tilde{n} \pm \tilde{κ}$ . Note that the circular birefringence is a direct result of the chiral constitutive relations, $\tilde{D} = \tilde{ε} \tilde{E} + i \tilde{κ} \tilde{H} / c$ and $\tilde{B} = \tilde{μ} \tilde{H} - i \tilde{κ} \tilde{E} / c$ [Eqs. (3) and (4) in Sec. 2]. By circular birefringence of isotropic chiral samples and Eq. (182), we can write explicitly CD, i.e., $Δ A$ , as $Δ A = \frac{4 π}{\ln 10} \frac{Δ (Im \tilde{n}) l}{λ} = \frac{8 π}{2.303} \frac{Im (\tilde{κ}) l}{λ} .$ (183)

By Eq. (183), we can conclude that the CD measurement provides information of the imaginary part of the chirality parameter $Im (κ)$ of the sample.

When the linearly polarized light passes through a chiral sample, it becomes the elliptically polarized light with the ellipticity $θ$ of the polarization ellipse. Interestingly, $Δ A$ (or $Δ ε$ ) measured in CD spectroscopy is related to the ellipticity $θ$ , and thus CD is often written in terms of $θ$ in deg. The ellipticity is defined by $\tan θ = (| {\tilde{E}}_{-} | - | {\tilde{E}}_{+} |) / (| {\tilde{E}}_{-} | + | {\tilde{E}}_{+} |)$ . Then, it becomes $\tan θ = \frac{\exp {[Im ({\tilde{n}}_{+}) - Im ({\tilde{n}}_{-})] k_{0} l} - 1}{\exp {[Im ({\tilde{n}}_{+}) - Im ({\tilde{n}}_{-})] k_{0} l} + 1} = \frac{\exp [2 Im (\tilde{κ}) k_{0} l] - 1}{\exp [2 Im (\tilde{κ}) k_{0} l] + 1} .$ (184)

Since $κ ≪ 1$ for chiral molecules, also resulting in $\tan ψ \approx ψ$ , Eq. (184) can be approximated to $θ \approx Im (\tilde{κ}) k_{0} l = 2 π \frac{Im (\tilde{κ}) l}{λ} .$ (185)

Combining Eq. (185) with Eq. (183), we can relate the ellipticity in deg to CD ( $Δ A$ ) as follows: $θ \approx (\frac{180 °}{π rad}) (\frac{\ln 10}{4} Δ A) = 32.98 Δ A .$ (186)

Also, the molar ellipticity in $\deg \cdot {cm}^{2} / dmol$ , $[θ] = \frac{100 θ}{c l} = 100 (\frac{180 °}{π rad}) (\frac{\ln 10}{4} Δ ε) = 3298 Δ ε,$ (187)is also often used in chemistry. The dissymmetry factor, a ratio of CD to the conventional absorption, $g \equiv \frac{Δ ε}{(ε_{+} + ε_{-}) / 2} = \frac{Δ A}{(A_{+} + A_{-}) / 2} = \frac{2 Im (\tilde{κ})}{k},$ (188)is another important measure in CD spectroscopy. Note that the dissymmetry factor $g$ cannot exceed two by definition. In natural molecules, CD is much smaller than absorption, and $g$ is useful when determining whether or not CD is measurable in the instrumental sensitivity.

On the other hand, the ellipticity $θ$ , Eq. (186), is a function of $Im (κ)$ , which is related to $Re (κ)$ by the Kramers-Kronig relation. The quantity corresponding to Eq. (186), $α = 2 π \frac{Re (\tilde{κ}) l}{λ},$ (189)provides ORD. It is straightforward to find that Eq. (189) is equivalent to ORD by its definition, $α = (ϕ_{+} - ϕ_{-}) / 2 = [Re ({\tilde{n}}_{+}) - Re ({\tilde{n}}_{-})] k_{0} l / 2$ , where the phase $ϕ_{\pm} = Re ({\tilde{n}}_{\pm}) k_{0} l$ .

We also emphasize that the experimental measures in UV-VIS spectrophotometry and CD spectroscopy we introduced here are valid only when a liquid sample with a thick optical pathlength $l ≫ λ$ . Also, one should be careful to use these experimental measures in UV-VIS spectrophotometry when measuring samples with large reflections such as plasmonic nanostructures and metasurfaces.

5.2 Signal-to-Noise Ratio in CD Measurements

CD is much smaller than the absorption in natural molecules. We should maximize the signal-to-noise ratio, SNR. In conventional CD measurements for a liquid chiral sample, one can control the absorbance $A$ of the sample by varying concentration $c$ (recall the Beer-Lambert law $A = ε c l$ ). Here, we find an optimal $A$ to maximize SNR in CD measurements. To find the optimal condition, we need to express SNR in terms of $A$ . We first need to write CD in terms of $Δ I \equiv I_{+} - I_{-}$ and $I = (I_{+} + I_{-}) / 2$ as follows: $Δ A \equiv A_{+} - A_{-} = \lg (\frac{I_{+}}{I_{-}}) = \lg (\frac{Δ I / I}{2 - I_{+} / I} + 1) \approx \frac{1}{2 \ln 10} \frac{Δ I}{I} .$ (190)

Although CD is recorded by the ellipticity $θ$ (in deg) in convention, what we actually measure is the transmitted intensities $I_{+}$ , $I_{-}$ , and $I$ according to the polarization of light. Equation (190) then gives $Δ I \approx 2 \ln 10 I Δ A$ . SNR is proportional to^[178] $σ (A) \equiv \frac{Δ I}{\sqrt{I}} = 2 \ln 10 \sqrt{I_{0}} (\frac{Δ ε}{ε}) (\sqrt{10^{- A}} A) .$ (191)

Here, we used the relations $Δ A = (Δ ε) (c d) = (Δ ε) (A / ε)$ and $I / I_{0} = 10^{- A}$ in deriving Eq. (191). Then, $d [σ (A)] / d A = 0$ gives an optimal absorbance, $A_{opt} = 0.8686$ , for maximizing SNR in CD spectroscopy. It corresponds to $T = 1 3.5 3 %$ . For a strongly absorbing sample, Eq. (191) becomes linear when $T > \sim 30 %$ . Then, one can find the asymptotic expression for Eq. (191): $\frac{σ (T)}{σ (A_{opt})} = 1.359 (1 - T) .$ (192)

At $T = 63 %$ (i.e., $A = 0.2$ ), $σ (T) / σ (A_{opt})$ decreases to $\sim 50 %$ .

5.3 Instrumentation of CD Spectroscopy

5.3.1 Direct subtraction method

The most straightforward and cheapest realization of the CD measurement is to obtain CD by direct subtraction between two absorptions of left- and right-circularly polarized lights. Figure 14(a) shows an optical arrangement for the subtraction setup.

Figure 14.CD instrumentations. (a) Direct subtraction, (b) self-interference, and (c) polarization modulation methods (S, source; LP, linear polarizer; QWP, quarter-wave plate; D, detector; M, monochromator; R, retarder; LI, lock-in amplifier). Polarization states of light are depicted by arrows in the optical path. In (b), the second linear polarizer is tilted by a small angle $θ = δ$ with respect to the optic axis of the first linear polarizer. In (c), the polarization state of the light after the retarder continuously varying at a frequency $ω$ .

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One can determine an appropriate light source according to the spectral range of interest; mercury, Xe, and halogen lamps are typical choices in the UV-VIS spectral range. Note that commercial CD spectrophotometers also use them. For the near-infrared (NIR) spectral range, a halogen lamp and a globar can be used. Also, a supercontinuum laser can be an option because it provides a relatively flat spectral lineshape over a broad range from VIS to NIR.

A set of an LP and a QWP makes the beam circularly polarized. A QWP angle of $\pm 45 °$ relative to the LP axis gives two opposite circular polarizations. There are some practical considerations in the set of the polarizer and the wave plate: (i) The optical components usually have anti-reflection (AR) coatings that work only for a limited spectral range; (ii) Linear polarizers are not perfect in the real world. They have a finite extinction ratio, a ratio between the intensities of two orthogonal linear polarizations. Film polarizers are cheap, compact, and easy to use, but their extinction ratio is relatively low. On the other hand, crystal polarizers have a better extinction ratio $> 100, 000 : 1$ and work in a much broader spectral range; (iii) One has to use an achromatic quarter wave plate, while ordinary quarter wave plates are monochromatic. However, the retardance of the achromatic quarter plate is also not exactly 0.25 in the working spectral range. One should always keep this imperfect retardance in mind. Then, a spectrograph and an array camera detector measure absorptions of two opposite circularly polarized lights by a chiral sample. For the detector, one should choose appropriate models supporting the spectral range of interest.

A critical disadvantage of the subtraction method is the purity of circular polarization. Since CD is $10^{- 5} - 10^{- 6}$ times smaller than conventional absorption, even a small deviation of light’s polarization from circular polarizations can lead to a huge inaccuracy in the CD result.

5.3.2 Self-interference method

The use of linearly polarized light rather than circularly polarized light in chiroptical spectroscopy can be an alternative to the subtraction method. Linear polarization is not only much easier to obtain but also robust to errors in polarization and dispersion. Linear polarization is a half-and-half mixture of two opposite circular polarizations with the same amplitude.

The simplest realization of the ORD measurement setup is to use two linear polarizers as shown in Fig. 14(b). Polarizers before and after the sample are called a polarizer and an analyzer, respectively. The polarizer makes the incident light polarized linearly along the $x$ -axis, i.e., ${\tilde{E}}_{0} = E_{0} \hat{x}$ . Then, the $x$ -polarized light passes through the sample of the path length $l$ . The transmitted light ${\tilde{E}}_{T}$ is given by ${\tilde{E}}_{T} = E_{0} [e^{i (\tilde{n} + \tilde{κ}) k_{0} l} (\hat{x} + i \hat{y}) + e^{i (\tilde{n} - \tilde{κ}) k_{0} l} (\hat{x} - i \hat{y})] = 2 E_{0} e^{i \tilde{n} k_{0} l} [\cos (\tilde{κ} k_{0} l) \hat{x} - \sin (\tilde{κ} k_{0} l) \hat{y}],$ (193)where $E_{0}$ is the amplitude of the incident light. Since the chirality parameter $\tilde{κ}$ of the sample is small in general, the expression for the transmitted light, Eq. (193), can be approximated to ${\tilde{E}}_{T} \approx 2 E_{0} e^{i \tilde{n} k_{0} l} (\hat{x} - \tilde{κ} k_{0} l \hat{y}) .$ (194)

If we measure the $y$ -component using the orthogonally arranged analyzer, we obtain $I \propto {| \tilde{κ} |}^{2}$ . However, it not only mixes the real and imaginary parts of $\tilde{κ}$ , but also is too small to detect because of the smallness of $\tilde{κ}$ . Instead, we can take advantage of the vectorial property of the electric field. We put an analyzer to detect chiral signals at an angle $θ$ with respect to the $x$ -axis, resulting in the detected field: ${\tilde{E}}_{D} \approx 2 E_{0} e^{i \tilde{n} k_{0} l} (\cos θ \hat{x} - \tilde{κ} k_{0} l \sin θ \hat{y}) .$ (195)

Then, we set the angle $θ = δ$ deviated slightly from the $x$ -axis, providing the intensity: ${\begin{cases} I (δ) = 4 e^{- 2 k k_{0} l} {| 1 - \tilde{κ} k_{0} l δ |}^{2} \\ I_{0} \approx 4 e^{- 2 k k_{0} l} [1 - 2 Re (\tilde{κ}) k_{0} l δ] I_{0} \end{cases},$ (196)where we ignore the second-order term $\propto {| \tilde{κ} |}^{2} δ^{2}$ because it is a product of two small values. $I_{0} \equiv E_{0}^{2}$ is the intensity of the incident light. Finally, we can determine the real part of the chirality parameter, i.e., ORD, $Re (\tilde{κ}) = \frac{1}{2 k_{0} l δ} \frac{Δ I (δ)}{I (0)},$ (197)in terms of experimentally obtained $I (0) = 4 e^{- 2 k k_{0} l} I_{0}$ and $Δ I \equiv I (0) - I (δ)$ . The advantage of this method is achromaticity over a broad spectral range if crystal polarizers are used. Another advantage is to suppress shot noise since the measurables $I (0)$ and $I (δ)$ are proportional to the incident intensity $I_{0}$ . Therefore, this technique falls into the category of a homodyne detection. Note that the self-interference method introduced in this section is similar to the measurement of the magneto-optic Kerr effect (MOKE)^[179].

There are two ways to obtain the imaginary part of the chirality parameter, $Im (\tilde{κ})$ , i.e., CD. The first way is a numerical method based on the causality of $\tilde{κ}$ . Since the chirality parameter $\tilde{κ}$ is a causal quantity, it obeys the Kramers-Kronig relations^[124] $Re [\tilde{κ} (ω)] = \frac{1}{π} P \int_{- \infty}^{\infty} \frac{Im [\tilde{κ} (ω^{'})]}{ω^{'} - ω} d ω^{'},$ (198) $Im [\tilde{κ} (ω)] = - \frac{1}{π} P \int_{- \infty}^{\infty} \frac{Re [\tilde{κ} (ω^{'})]}{ω^{'} - ω} d ω^{'},$ (199)where $P$ denotes the Cauchy principal value. The Kramers-Kronig transformation (equivalent to the Hilbert transformation) of Eq. (197) yields $Im (\tilde{κ})$ . Experiments are always band-limited, and thus we need to truncate the integrals, Eqs. (198) and (199). One should notice that the truncated Kramers-Kronig transformation introduces numerical artifacts, especially near the ends of the measured ORD spectrum.

The second way is to put a QWP in front of the analyzer. The QWP provides a phase shift of $π / 2$ to the $y$ -component. It results in the detected field ${\tilde{E}}_{D} \approx 2 E_{0} e^{i \tilde{n} k_{0} l} (\cos θ \hat{x} - i \tilde{κ} k_{0} l \sin θ \hat{y}) .$ (200)

The detected intensity is then given by $I (δ) = 4 e^{- 2 k k_{0} l} {| 1 - i \tilde{κ} k_{0} l δ |}^{2} \approx 4 e^{- 2 k k_{0} l} [1 + 2 Im (\tilde{κ}) k_{0} l δ],$ (201)resulting in ORD: $Im (\tilde{κ}) = - \frac{1}{2 k_{0} l δ} \frac{Δ I (δ)}{I (0)} .$ (202)

In this way, one should keep a nonperfect $π / 2$ -phase shift of achromatic QWP in mind as in the direct subtraction method.

5.3.3 Polarization modulation method

As we mentioned in Sec. 5.3.1, the direct subtraction method, achieving pure circular polarization, is challenging. Instead, the retarder can be rotated while detecting transmitted optical signals continuously. The polarization of the probe beam transitions from left circularly polarization (LCP) to linear polarization (LP), then to right circularly polarization (RCP), and back to LCP as the retarder is rotated. In the direct subtraction method, we cannot specify the retarder angle providing the exact circular polarization because of the mechanical errors. On the other hand, in the polarization modulation method, the polarization state of light must pass through LCP and RCP at some time by the retarder of the peak retardance $δ_{0}$ that rotates at a frequency $ω$ . The current recorded by the detector has the form $i_{\det} (t) = i_{DC} + i_{AC} (t)$ , where direct current (DC) and alternating current (AC) components are respectively given by^[180] $i_{DC} = Q I_{0} (T_{L} + T_{R}) = Q I_{0} (10^{- A_{L}} + 10^{- A_{R}}),$ (203) $i_{DC} (t) = Q I_{0} (T_{L} - T_{R}) \sin (δ_{0} \sin ω t) = Q I_{0} (10^{- A_{L}} - 10^{- A_{R}}) \sin (δ_{0} \sin ω t),$ (204)while varying the retardance over time. DC and AC currents correspond to the conventional absorption and CD, respectively. $Q$ is a coefficient determined by the detector quantum efficiency and gain. The difference between the maximum and the minimum gives CD. Since the AC signal is much weaker than the DC signal, a lock-in technique is useful to obtain inherently weak CD with high accuracy. Figure 14(c) shows an optical arrangement of the polarization modulation method with the lock-in amplifier.

5.4 Special Techniques

5.4.1 Fluorescence-detected circular dichroism and circularly polarized luminescence spectroscopy

Light-emitting molecules are elevated to the excited state by absorbing light and then return to the lower state while emitting light. This process is called fluorescence. As we discussed, chiral molecules absorb circularly polarized lights differently. Therefore, a difference in absorption, i.e., CD, leads to a difference in fluorescence. Fluorescence-detected circular dichroism (FDCD) spectroscopy measures the difference in fluorescence to obtain CD. Figure 15(a) shows the instrumentation of FDCD; two opposite circularly polarized lights excite the chiral sample as in conventional CD spectroscopy, and then the emission spectra $F_{L}$ and $F_{R}$ for two excitations are recorded. Note that the polarization of the fluorescence is not resolved in FDCD. Then, the measured FDCD signals $S_{F} = K [(F_{L} - F_{R}) / (F_{L} + F_{R})]$ can be described by^[182] $S_{F} = K \frac{(Δ ε_{F} / ε_{F}) - 2 R}{2 - (Δ ε_{F} / ε_{F}) R},$ (205) $R = \frac{A_{L} (1 - 10^{- A_{R}}) - A_{R} (1 - 10^{- A_{L}})}{A_{L} (1 - 10^{- A_{R}}) + A_{R} (1 - 10^{- A_{L}})},$ (206)

Figure 15.(a) FDCD spectroscopy and (b) CPL spectroscopy. S, source; LP, linear polarizer; QWP, quarter wave plate; D, detector. Polarization states of light are depicted by arrows in the optical path. A set of the linear polarizer and the quarter wave plate is used in this figure for the sake of simplicity, but the polarization modulation with a retarder and a detector coupled to a lock-in amplifier can be used^[181].

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where $K$ is the instrument constant. $ε_{F}$ is the molar extinction coefficient of the fluorophore, while $Δ ε_{F} = ε_{L} - ε_{R}$ is the corresponding molar circular dichroism. The ellipticity of the fluorescence signal can be expressed as $θ_{F}^{0} = - 28.65 (S / K)$ .

For a single fluorescent species, FDCD signals give the same information in CD, i.e., $θ_{F}^{0} = - 14.32 (Δ A / A) = - 14.32 (Δ ε_{F} / ε_{F})$ . On the other hand, FDCD is especially useful when multiple species are present in the sample or when energy transfer between species occurs.

CPL spectroscopy is another technique measuring chiral molecule fluorescence. In CPL spectroscopy, however, unpolarized or arbitrarily polarized light excites the chiral sample, and then left- and right-circular polarization components of the emission are separately recorded [Fig. 15(b)]. Using this instrumentation, CPL can measure the chirality of the excited states, while CD and FDCD measure that of the ground states^[183–185]. Both excitation and detection have polarization resolution, while FDCD and CPL spectroscopy has polarization resolution only in excitation or detection. This technique has been used in solid-state physics, especially to characterize valley polarization of two-dimensional gapped Dirac materials such as transition metal dichalcogenides (TMDs).

5.4.2 Circular differential scatterometry

Scatterometry measures light intensity scattered by a single nanoparticle [Fig. 16(a)]. Plasmonic nanoparticles have scattering cross sections much larger than their physical cross sections because of the localized surface plasmon resonance (LSPR). Therefore, it provides the scattering spectrum with a low signal-to-noise ratio. The dark-field technique is commonly used in scatterometry; the incident light impinges onto the substrate at an oblique angle large enough, while the objective lens collects light within its numerical aperture. For this purpose, the dark-field condenser in commercial microscopes is useful because it makes the beam ring-shaped, blocking the normal incident components of the beam. As a result, the image background becomes dark, while the light scattered by the nanoparticles is bright, contrasting with the scattering from the background. If a slit or iris is used to isolate the single-particle image, the single-particle scattering spectrum can be obtained using a spectrograph.

Figure 16.(a) Circular differential scatterometry^[186] and (b) its application to single-particle sensing of plasmonic nanoparticle-protein complexes^[109].

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To achieve the chiral resolution for the scatterer analyte, the scattering spectra can be obtained upon two opposite-handed circularly polarized excitations. This technique is called circular differential scatterometry [Fig. 16(a)]. Circular differential scatterometry was first suggested to measure circular dichroic scattering of chiral plasmonic nanoparticles, e.g., chiral gold nanorod dimers^[187], plasmonic oligomers^[188], and helicoids^[186]. It is also applied to measure CD properties of plasmonic nanoparticle aggregates formed by protein [Fig. 16(b)]^[109]. A recent work has used circular differential scatterometry for NPoM structures, and it has reported four-molecule-level chiral sensing^[169].

The advantage of this technique is to reduce the molecular number compared to the conventional CD spectroscopy requiring centimeter- or millimeter-long cuvettes. Chiral scattering of nanoparticles is affected only by molecules nearby, and thus, the number of molecules can be limited to the surface area of the nanoparticle.

5.4.3 Chiral optical force

We have discussed chiroptical measurement techniques based on chiral absorption, fluorescence, and scattering. Other properties such as the optical force and refraction can be used for chiral sensing. First of all, upon the excitation by chiral near-fields, chiral molecules with E1 and M1 moments $\tilde{p} = \tilde{α} \tilde{E} + i \tilde{G} \tilde{H}$ and $\tilde{m} = \tilde{χ} \tilde{H} - i \tilde{G} \tilde{E}$ [Eqs. (1) and (2)] experience the enantiomeric force described by^[189] $F_{C} = 4 π [- ω Re (\tilde{G}) \nabla h - c Im (\tilde{G}) (\nabla \times p - 2 k^{2} s) - \frac{4 ω k^{4}}{3 n} [ε Re (\tilde{G} {\tilde{χ}}^{*}) s_{m} + μ Re (\tilde{G} {\tilde{α}}^{*}) s_{e}]],$ (207)where $h$ , $p$ , $s$ , $s_{m}$ , and $s_{e}$ correspond to the optical helicity density, the optical moment density, the spin angular momentum density, the magnetic part of $s$ , and the electric part of $s$ , respectively. As already shown in Eq. (207), the direction of the enantiomeric force $F_{C}$ is determined by a combination of the handedness of molecules ( $\tilde{G}$ ) and the near-field ( $\nabla h$ , $\nabla \times p$ , $s$ , $s_{m}$ , and $s_{e}$ ). Therefore, a mixture of the chiral molecules can be mechanically sorted by the engineered near-field according to their handedness.

5.4.4 Chiral refraction

Chiral refraction of light in chiral media can also be used for molecular sensing. For example, the Snell’s law for the dielectric/chiral interface becomes [Fig. 17(b)]^[122,190] $n_{0} \sin θ_{0} = n_{\pm} \sin θ_{\pm},$ (208)where $n_{0}$ and $θ_{0}$ denote the refractive index of the incident medium and the incident angle, respectively. $n_{\pm} = Re (\tilde{n} \pm \tilde{κ})$ is the refractive indices for two opposite-handed circularly polarized lights. The chiral Snell’s law, Eq. (208), shows that the refracted light is split into two angles $θ_{\pm}$ .

$Chiral (a) optical force on a chiral object[189] and (b) refraction of light in a chiral medium[122].$

Figure 17.Chiral (a) optical force on a chiral object^[189] and (b) refraction of light in a chiral medium^[122].

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The photonic spin Hall effect, split refraction of two opposite-handed circularly polarized lights by the phase-gradient metasurface, also enables chiral sensing^{[123,191,192]}. For the phase-gradient metasurface, the generalized Snell’s law is given by $n_{0} \sin θ_{0} = n_{1} \sin θ_{1} - (λ / 2 π) (d Φ / d x)$ , where $d Φ / d x$ is the phase gradient along the interface direction $x$ . With careful design of a metasurface, e.g., in Ref. [120], the generalized Snell’s law becomes chiral, i.e., $\sin θ_{\pm} = \pm λ / P,$ (209)where $P$ is the period of the metasurface unit cell. Chiral refraction phenomena, described by Eqs. (208) and (209), are interfacial effects, and thus they can miniaturize both the sensor size and the analyte volume.

6 Conclusion and Outlook

Nanophotonic chiral sensing has attracted explosive interest in recent years. As discussed in this review, as well as in other articles providing comprehensive summaries, nanophotonic chiral sensors have pushed the detection limit to extremely low concentrations, ultimately aiming for single-molecule-level sensing. However, several challenges remain in nanophotonic chiral sensing. (i)Many studies have focused on chiral species exhibiting electronic optical activity, which predominantly appears in the UV frequency range^[47]. However, nanophotonic sensors typically operate in the visible and NIR regions. In the UV range, dielectric materials suffer from significant Ohmic losses, whereas noble metal surface plasmons occur primarily in the visible range^{[21,193–199]}. As a result, many nanophotonic sensors rely on the weak, extended tail of ORD in the visible spectrum, which limits their performance. Achieving more sensitive detection would require the development and application of UV plasmons and/or dielectric metasurfaces^{[197,198,200]}. On the other hand, both electronic and vibrational optical activities share the same fundamental principle of chiroptical response, as described by Eq. (49), despite their different response frequencies^[69]. Thus, the development of infrared chiral sensors presents an intriguing research direction.(ii)It is well established theoretically that molecular orientation plays a crucial role in absorption, fluorescence, and coupling to the resonance modes of nanophotonic sensors^[85,101]. However, experimental demonstrations of these effects have been lacking due to significant experimental challenges to control molecular orientations in a deterministic manner.(iii)Nanophotonic chiral sensing has achieved remarkable success in detecting enantiomers at extremely low concentrations. However, extracting detailed information on molecular and electronic structures from the measured spectral lineshapes remains a significant challenge^[201,202]. Especially, the deconvolution of CD spectra of protein in conventional chiroptical spectroscopy is of high interest. The CD spectrum of a protein is a composite signal arising from various structural elements, such as $α$ -helices, $β$ -sheets, and random coils. To extract meaningful structural information, deconvolution techniques are employed to separate the contributions of different secondary structures. Deconvolution of CD spectra typically involves comparing the experimental spectrum with a reference dataset of known protein structures. Computational algorithms, such as singular value decomposition (SVD) and principal component analysis (PCA), are used to estimate the proportion of each secondary structure^[68]. Several established methods, including CONTINLL^[203,204], SELCON^[205,206], and CDSSTR^[205–209], are commonly applied for this purpose. A website for CD deconvolution is also available^[210–213]. On the other hand, to the best of our knowledge, these CD deconvolutions have never been performed in nanophotonic chiral sensing. CD deconvolution in nanophotonic chiral sensing should address two challenges: i) chiroptical spectra obtained by nanophotonic chiral sensors include both the molecule and the sensor information, while it is not straightforward to decompose them; ii) no algorithms for nanophotonic-sensor-coupled chiral molecules have been presented. Once achieving CD deconvolution at extremely low concentrations using nanophotonics, it would bring breakthroughs to biological research.

Before concluding, it is worth mentioning that chirality has garnered considerable attention not only in nanophotonic chiral sensing^[127] but also across a wide range of disciplines, including negative-index metamaterials^[214], holography^[215–217], encryption^[218], plasmonic molecules^[48,219,220], and bound states in the continuum^[221]. Ongoing advances in these areas are expected to further elucidate the fundamental roles of chirality in light-matter interactions and facilitate the emergence of novel photonic functionalities.

Acknowledgments

Acknowledgment. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (Nos. RS-2023-00254920 and RS-2025-00514894), the Korea Institute for Advancement of Technology (KIAT) grant funded by the Korea Government (MOTIE) (No. RS00411221, HRD Program for Industrial Innovation), and a grant of the Korea-US Collaborative Research Fund (KUCRF), funded by the Ministry of Science and ICT and Ministry of Health & Welfare, Republic of Korea (No. RS-2024-00468463).

Category: Review Articles

Received: Mar. 14, 2025

Accepted: Jul. 21, 2025

Published Online: Aug. 26, 2025

The Author Email: SeokJae Yoo (seokjaeyoo@inha.ac.kr), Q-Han Park (qpark@korea.ac.kr)

DOI:10.3788/PI.2025.R08

CSTR:32396.14.PI.2025.R08