Perovskite materials have emerged as a promising class of semiconductors for nonlinear optical (NLO) applications due to their exceptional optoelectronic properties, tunable compositions, and strong light-matter interactions. Their unique crystal structures and electronic configurations enable efficient nonlinear processes such as two-photon absorption (TPA), harmonic generation, and Kerr effects, making them attractive for applications in photonics, imaging, and ultrafast optics. This article explores the fundamental NLO properties of perovskites, focusing on the underlying mechanisms, structure-property relationships, and insights from ultrafast spectroscopy.

The nonlinear optical response of perovskites arises from their highly polarizable electronic structures and strong excitonic effects. The perovskite crystal structure, typically represented as ABX3, consists of a cationic framework (A and B sites) and an anionic sublattice (X site). The B-site cations, often lead or tin, contribute to the strong spin-orbit coupling and high carrier mobility, while the X-site halides or chalcogenides influence the bandgap and exciton binding energy. The flexibility in chemical composition allows precise tuning of the NLO properties by adjusting the A, B, or X components. For instance, hybrid organic-inorganic perovskites (HOIPs) like CH3NH3PbI3 exhibit large nonlinear susceptibilities due to the interplay between the organic cations and inorganic framework.

Two-photon absorption is a key NLO phenomenon in perovskites, where simultaneous absorption of two photons promotes an electron to a higher energy state. The TPA coefficient in perovskites is significantly enhanced compared to conventional semiconductors, with reported values ranging from 5 to 50 cm/GW depending on the material composition and excitation wavelength. The strong TPA response is attributed to the high density of states near the band edges and the large transition dipole moments. Lead-based perovskites, such as CsPbBr3, demonstrate particularly high TPA cross-sections due to the heavy atom effect, which enhances spin-orbit coupling and intermediate-state lifetimes. Mixed-halide perovskites further allow spectral tuning of TPA peaks, enabling broadband nonlinear absorption.

Harmonic generation, including second-harmonic generation (SHG) and third-harmonic generation (THG), is another critical NLO property of perovskites. SHG is highly sensitive to crystal symmetry, with non-centrosymmetric perovskites exhibiting strong second-order nonlinearities. For example, formamidinium lead iodide (FAPbI3) in its polar phase shows a second-order susceptibility (χ(2)) comparable to that of LiNbO3, a benchmark NLO material. THG, on the other hand, is less dependent on symmetry and is observed even in centrosymmetric structures due to third-order nonlinearities. The THG efficiency in perovskites is influenced by the exciton resonance and the electronic band structure, with CsPbCl3 demonstrating strong THG signals near the bandgap energy. The harmonic generation efficiency can be further enhanced by engineering the perovskite dimensionality, as reduced-dimensional perovskites exhibit quantum confinement effects that amplify nonlinear responses.

The Kerr effect, a third-order nonlinear process, describes the intensity-dependent refractive index change in perovskites. This effect is crucial for all-optical switching and modulation applications. The nonlinear refractive index (n2) of perovskites is typically in the range of 10^-15 to 10^-13 cm²/W, with lead halide perovskites showing higher values due to their large exciton binding energies and strong Coulomb interactions. The Kerr effect in perovskites is also influenced by the interplay between electronic and thermal contributions, with ultrafast laser studies revealing sub-picosecond response times for the electronic component. Mixed-cation perovskites, such as CsMAFA-based compositions, exhibit tunable Kerr nonlinearities by varying the cation ratios, enabling optimization for specific wavelength regimes.

Structure-property relationships play a pivotal role in understanding and optimizing the NLO performance of perovskites. The dimensionality of the perovskite structure, from 3D bulk to 2D layered and quasi-2D forms, significantly impacts the nonlinear response. 2D perovskites, with their naturally formed quantum wells, exhibit enhanced excitonic effects and larger binding energies, leading to stronger TPA and harmonic generation compared to their 3D counterparts. The organic spacer layers in 2D perovskites also introduce additional dielectric confinement, further amplifying the nonlinearities. In contrast, 3D perovskites benefit from higher carrier mobility and broader absorption spectra, making them suitable for broadband NLO applications. The role of defects and grain boundaries in polycrystalline perovskites is another critical factor, as these can either enhance or suppress nonlinear effects depending on their nature and density.

Ultrafast spectroscopy techniques have provided deep insights into the dynamics of NLO processes in perovskites. Transient absorption spectroscopy reveals that the TPA in perovskites is dominated by excitonic states, with exciton-exciton interactions contributing to the nonlinear absorption at high excitation densities. Pump-probe measurements have shown that the Kerr effect in perovskites has both instantaneous electronic and slower thermal components, with the electronic response occurring on femtosecond timescales. Time-resolved photoluminescence studies further highlight the role of hot carriers and multi-exciton generation in enhancing NLO properties, particularly under high-intensity excitation. Coherent anti-Stokes Raman scattering (CARS) has been employed to probe the vibrational modes contributing to the nonlinear polarizability, revealing strong coupling between electronic and lattice degrees of freedom.

The composition and stoichiometry of perovskites also play a crucial role in their NLO behavior. Halide mixing, for instance, allows continuous tuning of the bandgap and nonlinear coefficients. Iodide-rich perovskites exhibit stronger TPA at longer wavelengths, while bromide-rich compositions show higher harmonic generation efficiencies in the visible range. Cation engineering, such as partial substitution of formamidinium with cesium, can stabilize the perovskite phase while maintaining high nonlinearities. Double perovskites, where two different B-site cations are incorporated, offer additional opportunities for tailoring NLO properties through band structure engineering. These materials often exhibit reduced toxicity and improved stability while retaining competitive nonlinear performance.

Environmental factors such as temperature and pressure also influence the NLO properties of perovskites. Low-temperature studies have revealed sharpening of excitonic features and enhanced nonlinearities due to reduced phonon scattering. High-pressure experiments demonstrate that the perovskite structure can undergo phase transitions, leading to abrupt changes in NLO coefficients. For example, the SHG signal in CH3NH3PbI3 disappears above a critical pressure as the material transitions to a centrosymmetric phase. Such pressure-dependent studies provide valuable insights into the structure-property relationships and the potential for strain engineering to modulate NLO responses.

In summary, perovskites exhibit rich and tunable nonlinear optical properties driven by their unique crystal structures, excitonic effects, and compositional flexibility. Two-photon absorption, harmonic generation, and Kerr effects are strongly influenced by the perovskite dimensionality, stoichiometry, and defect landscape. Ultrafast spectroscopy has been instrumental in unraveling the dynamic processes underlying these nonlinear phenomena. The ability to tailor perovskites at the molecular level offers unprecedented opportunities for designing materials with optimized NLO performance for next-generation photonic technologies. Future research directions may explore novel perovskite compositions, heterostructures, and nanostructured forms to further push the boundaries of nonlinear optics.