Chiral-Linear Optical Interference

• Chiral-linear optical interference is the phenomenon where electromagnetic fields interact with chiral media to yield handedness-dependent optical responses critical for selective molecular sensing.
• Advanced methodologies like transfer-matrix, scattering-matrix, and Green-tensor analyses are employed to model polarization-specific transmission and enhance circular dichroism in structured cavities.
• This field drives innovations in chiral polaritonics and enantio-selective photonics, enabling optical devices with near-unity helicity dissymmetry and improved stereochemical control.

Chiral-linear optical interference encompasses the phenomena and mechanisms by which interference effects between electromagnetic fields and chiral matter yield polarization- and handedness-dependent optical responses. This includes circular dichroism, optical rotation, chiral optical forces and torques, and the emergence of highly selective chiral modes in structured photonic environments. Chiral-linear optical interference is central to emerging fields such as chiral polaritonics, nonreciprocal photonics, and enantio-selective molecular sensing. Advanced implementations employ high-finesse cavities, planar and three-dimensional chiral metamaterials, and sophisticated cavity/polarimeter architectures that exploit and maximize the dissymmetry between electromagnetic helicities, pushing the boundaries of chiral light–matter interaction and its technological impact.

1. Theoretical Foundations and Formulation

The classical electromagnetic description of chiral-linear optical interference originates from Maxwell’s equations with generalized constitutive relations incorporating chiral (Pasteur) terms. The constitutive equations for a Pasteur medium (isotropic, lossless, linear chiral material) can be written as: $$\mathbf{D} = \epsilon \mathbf{E} + i\frac{\kappa}{c}\mathbf{H}, \quad \mathbf{B} = \mu \mathbf{H} - i\frac{\kappa}{c}\mathbf{E}$$ where $\epsilon$, $\mu$ are the permittivity and permeability, and $\kappa$ is the chirality (Pasteur) parameter. This leads to eigenmodes given by superpositions of left- and right-circularly polarized (LCP/RCP) waves, each with distinct refractive indices,

$n_{\pm} = n \pm \kappa$

and polarization-dependent transmission, reflection, and absorption across interfaces and through multilayer structures.

For multilayered optical systems such as Fabry–Pérot (FP) cavities filled with chiral media, three complementary approaches are commonly employed for predictive modeling (Mauro et al., 2023):

• Transfer-matrix formalism: Used to recursively compute the fields and their polarization content across layered interfaces for arbitrary stacks.
• Scattering-matrix method: Relates incoming and outgoing amplitudes at all ports, explicitly capturing helicity-conversion and reciprocity constraints.
• Green-tensor analysis: Provides a local field response formalism and connects directly to transmission and input–output relations, with formal analogies to mesoscopic quantum transport.

This mathematical infrastructure supports the quantitative prediction of chiral-linear optical interference phenomena such as cavity-enhanced circular dichroism (CD), polarization-dependent cavity-polariton splittings, and the engineering of optical dissymmetry via structured mirrors and resonators.

2. Symmetry Constraints and Polarization-Selective Interference

Symmetry plays a central role in dictating the allowed and forbidden chiral-linear optical responses:

• Lorentz reciprocity: Ensures that transmission and reflection matrices obey specific transposition relations between LCP and RCP channels. In matrix notation, this manifests as

$$T_{\leftarrow}^T = \sigma_z\, T_{\rightarrow}\, \sigma_z$$

• Time-reversal symmetry (TRS): Imposes further constraints on the scattering matrix, leading to

$S = \sigma_z (S^*)^{-1} \sigma_z$

in circular polarization basis.

In conventional FP cavities with metallic mirrors, each reflection reverses the helicity of the optical field, preventing the buildup of handedness-selective intensity beyond that determined by direct transmission. Conversely, the introduction of helicity-preserving (or selectively-preserving) photonic crystal dielectric mirrors enables the construction of cavities that maintain or even enhance a net helicity imbalance (Mauro et al., 2023). In such systems, the differential response becomes tunable by the design of the mirror S-matrix and the propagation phase, and can be optimized for strong chiroptical effects such as the formation of chiral cavity-polaritons with pronounced splitting and dissymmetry.

3. Polarization-Dependent Transmission and Enhanced Circular Dichroism

The chiral nature of the medium causes left and right circular polarization states to accumulate distinct phases and attenuation during propagation. For a chiral slab of length $L$, the transmission coefficients follow a generalized Beer–Lambert law: $$T_{\perp,\pm}(\omega) = T_\perp(\omega) \exp\left[-\alpha_\pm(\omega) L\right]$$ where $\alpha_\pm = 2(\omega/c) \,\mathrm{Im}[n_\pm]$ are polarization-specific attenuation coefficients.

The key figure of merit is the differential circular transmission (DCT) or normalized chiroptical signal,

$$\mathcal{DCT} = 2\, \tanh\left[\frac{L}{2}(\alpha_{-} - \alpha_{+})\right]$$

which is maximized by increasing both the chiral absorption contrast (through large $\kappa$ or molecular concentration) and the effective path length (via cavity enhancement or multipass). In the context of FP cavities with helicity-preserving mirrors, it is possible to reach chiroptical dissymmetry (imbalance $\gamma$) exceeding $95\%$ for certain mode classes, a result directly relevant to enantio-selective photonics (Rebholz et al., 14 Jul 2025).

Table 1: Chiroptical Signal Enhancement: Metallic vs. Helicity-Preserving FP Cavities

Mirror Type	Helicity Reversal	Max. Dissymmetry Achievable	CD Enhancement	Suitable for Chiral-Sensing?
Metallic (Ag, Au)	Yes	Limited (mirror cancels)	No	Ineffective
Dielectric-HP (PC)	No (HP design)	$\gtrsim95\%$	Yes	Highly effective

The enhanced differential response forms the physical basis for advanced chiral-sensing strategies in microcavities and for the manipulation of optical activity and stereochemistry under strong light–matter coupling.

4. Cavity-Polaritons and Mode Engineering in Chiral Cavities

When the coupling between a chiral medium and an optical cavity mode exceeds cavity dissipation, hybridization leads to the emergence of chiral cavity-polariton branches (upper and lower polaritons) (Mauro et al., 2023). In metallic-mirror FP cavities, the effect of helicity reversal prevents any significant polarization imbalance. In contrast, with properly designed HP photonic crystal mirrors, cavity resonances can be engineered such that:

• The S-matrix for the round-trip operator strongly favors one helicity, suppressing the other.
• The eigenmode decomposition of the round-trip matrix $S_{rt}$ yields vectors with high helicity content (nearly pure LCP or RCP), quantified by $\Lambda(v_i)$.
• For modes with zero in-plane momentum (parallel to the optical axis), the field inside the cavity can be nearly entirely of one helicity. For instance, at resonance and optimal cavity length, a dissymmetry as high as $0.953$ in $\gamma$ is obtained at $44.21$ THz (Rebholz et al., 14 Jul 2025).

The interplay between chiral scatterer design (such as planar lattices of silver helices with multipole-locked responses) and the angular/diffraction selection by the mirror structure ensures selective buildup and resonance of a single helicity, enabling unprecedented control over the local optical environment for chiral matter.

5. Technological Implications and Research Frontiers

These recent advances in chiral-linear optical interference and chiral cavity engineering have broad implications:

• Ultra-sensitive enantio-selective photonics: The ability to generate modes with nearly complete helicity dissymmetry enables optical separation, detection, and discrimination of chiral molecules at sensitivities beyond conventional CD approaches.
• Cavity-modified stereochemistry: The photonic environment can be engineered to bias reaction outcomes and ground-state populations for chiral species, opening new avenues in photo-induced asymmetric synthesis.
• Quantum and topological chiroptics: Parity and time-reversal symmetry constraints, as well as engineered nonreciprocity, intersect with the emergence of nontrivial chiral polaritonic states and device functionalities for quantum technologies.
• Mode-selective chiral force enhancement: Exploiting chirality-selective field profiles within engineered cavities and near-field platforms supports high-fidelity optical manipulation (forces, torques) on chiral nanostructures and molecules, with applications in microfluidics and optomechanics.

The main challenges and future directions include the scalable nanofabrication of high-chirality scatterer lattices, loss mitigation for increased Q-factor and sensitivity, and the integration of chiral cavity systems into photonic circuits and lab-on-chip platforms for real-world chiral analysis and control.

6. Summary Table: Key Elements in Chiral-Linear Optical Interference

Element/Concept	Description	Reference
Pasteur Medium Constitutive Laws	Maxwell equations with chiral (off-diagonal) terms	(Mauro et al., 2023)
Polarization-Selective Cavity Side	Metallic (helicity-reversing), HP dielectric PC (helicity-preserving)	(Mauro et al., 2023, Rebholz et al., 14 Jul 2025)
Transmission/Absorption Formulae	Chiral Beer-Lambert law, DCT expressions	(Mauro et al., 2023)
Symmetry Constraints	Lorentz reciprocity and TRS relations for S-matrix	(Mauro et al., 2023)
Helicity Dissymmetry Enhancement	Near-unity in optimized chiral cavities	(Rebholz et al., 14 Jul 2025)
Chiral Polaritons	HP mirrors + strong coupling → chiral UP/LP branches	(Mauro et al., 2023)

These insights provide the foundation for understanding, exploiting, and engineering chiral-linear optical interference across classical and quantum regimes, laying the groundwork for next-generation chiroptical devices and applications.