Chiral plasmonics represents a specialized subfield of nanophotonics that investigates the interaction between circularly polarized light and nanostructures exhibiting geometric or optical chirality. The theoretical foundations of this field are rooted in classical electrodynamics, quantum mechanics, and symmetry considerations, which collectively explain the unique optical responses of chiral plasmonic systems. Key phenomena such as circular dichroism (CD) in nanostructures like helixes and gammadions arise from the broken mirror symmetry of these systems, leading to differential absorption of left- and right-handed circularly polarized light. Theoretical frameworks, including multipole expansions and the Born-Kuhn model, provide rigorous mathematical descriptions of these interactions, enabling the design of nanostructures with tailored chiroptical properties.
The origin of chiral plasmonic responses lies in the coupling between incident light and the collective oscillations of conduction electrons in metallic nanostructures. For a nanostructure to exhibit chirality, it must lack mirror symmetry, meaning it cannot be superimposed onto its mirror image. Helical and gammadion-shaped nanostructures are prime examples, as their three-dimensional geometry or planar arrangement breaks this symmetry. When circularly polarized light interacts with such structures, the induced plasmonic modes generate distinct charge distributions and near-field enhancements for left- and right-handed polarizations. This differential response manifests as circular dichroism, quantified by the difference in absorption between the two polarizations.
Multipole expansions serve as a powerful tool for decomposing the electromagnetic response of chiral nanostructures into contributions from various electric and magnetic moments. In conventional plasmonics, the dipole approximation often suffices to describe the optical properties of small nanoparticles. However, chiral systems require higher-order multipoles, including electric quadrupoles and magnetic dipoles, to fully capture their complex near- and far-field interactions. The multipole expansion of the induced current density in a nanostructure can be written as a series of terms, each corresponding to a specific moment. For chiral systems, cross-coupling between electric and magnetic dipoles plays a critical role in generating CD signals. This coupling is described by the mixed electric-magnetic dipole term, which vanishes in achiral systems due to symmetry constraints.
The Born-Kuhn model offers a simplified yet insightful approach to understanding chiral light-matter interactions. Originally developed to explain optical activity in molecular systems, this model has been adapted to describe plasmonic nanostructures. It consists of two coupled oscillators representing plasmonic modes with orthogonal polarization states. The coupling between these oscillators breaks the symmetry of the system, leading to a splitting of resonance frequencies for left- and right-handed circularly polarized light. The resulting CD spectrum exhibits a characteristic bisignate line shape, with positive and negative peaks corresponding to the split resonances. The magnitude of the CD signal depends on the strength of the coupling between the oscillators, which in turn is determined by the geometric parameters of the nanostructure, such as the pitch of a helix or the arm length of a gammadion.
For helical nanostructures, theoretical analysis often employs a combination of analytical models and numerical simulations. The helical geometry supports propagating surface plasmon polaritons that follow the winding path of the structure, leading to a phase delay between different points along the helix. This phase delay introduces a handedness-dependent propagation constant, resulting in different effective refractive indices for left- and right-handed circularly polarized light. The CD spectrum of a helix typically shows a strong dependence on the pitch-to-diameter ratio, with optimal chirality occurring at specific geometric parameters. Numerical methods such as finite-difference time-domain (FDTD) simulations are frequently used to solve Maxwell's equations for these complex geometries, providing detailed insights into the near-field distributions and far-field scattering patterns.
Gammadion-shaped nanostructures, though planar, exhibit chirality due to their lack of mirror symmetry in the plane of the structure. The four-armed geometry supports localized plasmon resonances that interact coherently, leading to a net chiral response. Theoretical studies have shown that the CD signal in gammadions arises from the interference between plasmon modes excited in different arms of the structure. The relative orientation and length of the arms determine the spectral position and magnitude of the CD peaks. Unlike helixes, gammadions often exhibit multiple CD peaks corresponding to different plasmonic modes, making their theoretical analysis more involved.
The role of retardation effects in chiral plasmonics cannot be overlooked, particularly for nanostructures with dimensions comparable to the wavelength of light. Retardation leads to phase differences in the excitation of different parts of the nanostructure, which can enhance or suppress chiral responses depending on the geometry. Theoretical treatments of retardation effects typically involve full-wave electromagnetic simulations, as analytical models become intractable for complex geometries. The interplay between retardation and near-field coupling gives rise to phenomena such as Fano resonances in chiral systems, where narrow spectral features emerge due to interference between broad and narrow plasmonic modes.
Quantum mechanical effects also contribute to the chiroptical response of plasmonic nanostructures, especially at small sizes where nonlocal effects become significant. Classical electrodynamics fails to account for the spatial dispersion of the dielectric function at nanometer scales, necessitating quantum corrections. Theoretical approaches such as the hydrodynamic model or time-dependent density functional theory (TDDFT) have been employed to describe these effects. Quantum corrections are particularly important for accurately predicting the CD spectra of ultrasmall chiral nanoparticles, where electron spill-out and surface effects dominate the optical response.
Theoretical studies have also explored the influence of material properties on chiral plasmonic effects. Noble metals like gold and silver are commonly used due to their strong plasmonic responses in the visible and near-infrared regimes. However, alternative materials such as aluminum or doped semiconductors offer tunability across different spectral ranges. The dielectric function of the material, which describes its response to electromagnetic fields, directly affects the plasmon resonance frequencies and quality factors. Theoretical models incorporating material dispersion relations provide valuable guidance for selecting materials with desired chiroptical properties.
Recent advances in computational methods have enabled the theoretical investigation of more complex chiral plasmonic systems, including oligomers, twisted metamaterials, and three-dimensional assemblies. These systems often exhibit collective chiral responses that emerge from the interaction between individual building blocks. Coupled dipole approximations and generalized Mie theory have been extended to handle these cases, allowing researchers to predict and optimize their optical properties. Machine learning techniques are also being explored for inverse design of chiral nanostructures, where desired CD spectra are used as inputs to generate optimal geometries.
The theoretical understanding of chiral plasmonics has significant implications for applications such as polarization-sensitive photodetectors, enantioselective sensing, and optical metamaterials. By providing a rigorous foundation for predicting and interpreting chiroptical phenomena, these theoretical frameworks guide the development of next-generation nanophotonic devices with tailored optical functionalities. Future theoretical challenges include extending these models to account for nonlinear chiral effects, nonreciprocal responses, and quantum plasmonic phenomena in hybrid light-matter systems.