Plasmonic lattice resonances emerge in periodic arrays of metallic nanostructures when localized surface plasmon resonances (LSPRs) interact with diffractive orders of the array. These collective modes arise from the coherent coupling between individual nanoparticle resonances and the far-field radiation mediated by the lattice periodicity. The resulting hybridized modes exhibit narrow spectral linewidths and enhanced near-field intensities, making them attractive for applications in sensing, nonlinear optics, and light-matter interactions.

Theoretical investigations of plasmonic lattice resonances often begin with the analysis of single-particle LSPRs, which depend on the material, size, and shape of the nanostructures. For noble metals like gold and silver, the dielectric function exhibits strong frequency dispersion in the visible and near-infrared ranges, leading to resonant absorption and scattering. When these nanoparticles are arranged in a two-dimensional lattice with subwavelength spacing, the scattered fields from each particle interfere constructively or destructively depending on the wavevector matching conditions. This interference gives rise to diffractive coupling, where the lattice period determines the onset of Rayleigh anomalies, marking the emergence of collective resonances.

Band structure calculations provide a framework for understanding the dispersion relations of plasmonic lattice modes. Using numerical methods such as finite-difference time-domain (FDTD) simulations or coupled dipole approximations, the photonic band structure can be computed for infinite periodic arrays. The band diagram reveals avoided crossings between the LSPR and the diffractive orders, indicating strong coupling and the formation of new hybrid modes. The width and position of these bands depend on the lattice symmetry, particle geometry, and interparticle spacing. For example, square lattices exhibit distinct polarization-dependent modes due to their fourfold symmetry, while hexagonal lattices support degenerate modes at high-symmetry points in the Brillouin zone.

Quasi-bound states in the continuum (quasi-BICs) represent a special class of plasmonic lattice resonances with theoretically infinite quality factors. These states occur when a resonant mode becomes decoupled from the radiative continuum due to symmetry protection or destructive interference. In practice, imperfections or intentional symmetry breaking transform ideal BICs into quasi-BICs with finite but high quality factors. Theoretical studies employ multipole expansions or scattering matrix formalisms to identify the conditions for quasi-BIC formation. For instance, in arrays of asymmetric nanoparticles, the interference between electric and magnetic dipole moments can lead to a suppression of radiative losses, resulting in narrowband resonances.

The coupling between localized and extended states in plasmonic lattices can be analyzed using temporal coupled-mode theory. This approach models the energy exchange between the LSPR and the diffractive orders through phenomenological coupling coefficients. The resonance linewidth is determined by the balance between radiative and nonradiative damping, with the former being strongly influenced by the lattice geometry. By tuning the array parameters, such as the period or particle arrangement, the coupling strength can be optimized to achieve desired spectral features.

Numerical simulations reveal that the near-field enhancement in plasmonic lattices is highly sensitive to the interplay between individual particle resonances and collective lattice effects. Near the quasi-BIC condition, the electromagnetic field localization can exceed that of isolated nanoparticles by several orders of magnitude. This enhancement is accompanied by a steep dispersion in the band structure, leading to slow-light effects and enhanced light-matter interactions. Theoretical models based on multiple scattering theory or Green's function methods provide insights into the spatial distribution of these enhanced fields and their dependence on the lattice parameters.

The role of material losses in plasmonic lattices is another critical aspect of theoretical investigations. Ohmic losses in the metal limit the achievable quality factors and near-field intensities. Advanced computational approaches, such as electron energy-loss spectroscopy simulations, quantify the nonradiative damping channels and their impact on the resonance linewidth. By comparing different materials or core-shell geometries, theoretical studies guide the design of low-loss plasmonic lattices with optimized performance.

Higher-order lattice resonances, involving coupling to multiple diffractive orders, exhibit complex spectral and angular dependencies. These modes often appear at shorter wavelengths or larger wavevectors, where the lattice period becomes comparable to the wavelength. Band structure calculations for these regimes require careful consideration of the reciprocal lattice vectors and their coupling to the plasmonic modes. The resulting hybrid states can exhibit Fano-like line shapes due to interference between different scattering pathways.

Theoretical studies also explore the nonlinear optical response of plasmonic lattices, where the collective modes enhance harmonic generation or multiphoton processes. Nonlinear coupled-mode theories or hydrodynamic models for electron dynamics predict the efficiency of these processes under resonant excitation. The interplay between the lattice symmetry and nonlinear susceptibility tensors leads to polarization-dependent effects, which can be harnessed for frequency conversion or optical switching.

In summary, theoretical investigations of plasmonic lattice resonances provide a comprehensive understanding of the underlying physics governing their optical properties. Through band structure calculations, quasi-BIC analysis, and advanced numerical modeling, researchers can predict and optimize the behavior of these systems for various applications. The insights gained from these studies continue to drive the development of new plasmonic materials and architectures with tailored optical responses.