Third-order Nonlinear Optical Properties Research Articles

Experimental Investigation of the Optical Nonlinearity of Laser-Ablated Titanium Dioxide Nanoparticles Using Femtosecond Laser Light Pulses

In this report, the nonlinear optical (NLO) properties of titanium dioxide nanoparticles (TiO2 NPs) have been explored experimentally using femtosecond laser light along with the Z-scan approach. The synthesis of TiO2 NPs was carried out in distilled water through nanosecond second harmonic Nd:YAG laser ablation. Characterization of the TiO2 NPs colloids was conducted using UV-visible absorption spectroscopy, transmission electron microscopy (TEM), inductively coupled plasma (ICP), and energy-dispersive X-ray spectroscopy (EDX). The TEM analysis indicated that the size distribution and average particle size of the TiO2 NPs varied from 8.3 nm to 19.1 nm, depending on the laser ablation duration. The third-order NLO properties of the synthesized TiO2 NPs were examined at different excitation laser wavelengths and incident powers through both open- and closed-aperture Z-scan techniques, utilizing a laser pulse duration of 100 fs and a high repetition rate of 80 MHz. The nonlinear absorption (NLA) coefficient and nonlinear refractive (NLR) index of the TiO2 NPs colloidal solutions were found to be influenced by the incident power, excitation wavelength, average size, and concentration of TiO2 NPs. Maximum values of 4.93 × 10⁻⁹ cm/W for the NLA coefficient and 15.39 × 10⁻15 cm2/W for the NLR index were observed at an excitation wavelength of 800 nm, an incident power of 0.6 W, and an ablation time of 15 min. The optical limiting (OL) effects of the TiO2 NPs solution at different ablation times were investigated and revealed to be concentration and average size dependent. An increase in concentration results in a more limiting effect.

Investigation of the Impact of Clitoria ternatea Dye on the Growth and Characterization of Lithium Paranitrophenolate Trihydrate Single Crystals for Nonlinear Optical Applications

Incorporating dyes into crystalline structures has emerged as a promising strategy for enhancing material properties in solid-state devices. In this study, a single crystal of lithium p-nitrophenolate trihydrate (LiPNPT) doped with Clitoria ternatea L. (CT) flower dye was successfully synthesized and its properties were compared with those of undoped LiPNPT crystals. Energy-dispersive X-ray spectroscopy (EDAX) confirmed the elemental composition while single crystal X-ray diffraction (XRD) elucidated the structural properties. The optical characteristics were evaluated using UV-Visible and Photoluminescence analyses. Fourier-transform infrared (FTIR) and FT-Raman spectroscopy provided insights into the vibrational modes and molecular interactions. The CT-LiPNPT crystal exhibited enhanced thermal stability, mechanical strength and a higher decomposition temperature compared to the undoped crystal. Dielectric studies were conducted to assess the electrical properties. Notably, the second-harmonic generation (SHG) efficiency of the CT-LiPNPT crystal was 2.39 times higher than that of the undoped crystal and showed a higher laser damage threshold. Z-scan analysis revealed significant third-order nonlinear optical properties and optical limiting behavior in the doped crystal. These comprehensive investigations demonstrate that the CT dye-doped LiPNPT possesses improved characteristics, rendering it a more reliable candidate for optoelectronic and nonlinear optical device applications.

An extensive investigation of structural, linear, and ultrafast third-order nonlinear optical properties of a novel trimethoxy anthracene chalcone: Experimental and DFT study

An extensive investigation of structural, linear, and ultrafast third-order nonlinear optical properties of a novel trimethoxy anthracene chalcone: Experimental and DFT study

Synthesis and characterization of piperazine nitrate monochloride semi-organic single crystal for NLO applications

A semiorganic single crystal of piperazine nitrate chloride (PNC) was grown utilising the gradual evaporation solution technique at ambient temperature. The crystal lattice properties and molecular structure of the formed crystal of PNC were identified via X-ray diffraction examination of a single crystal and are consistent with the monoclinic crystal system with space group P21/n. Using Hirshfeld surface analysis, intra and intermolecular interactions were displayed. The FTIR spectrum analysis presented here examines the vibratory patterns of several functional sections found within the PNC crystal. The crystal exhibits a low absorbance throughout the entire visible spectrum according to studies of UV–vis-NIR absorbance and calculated band gap as well as optical constants. TG/DTA analysis was used to identify the stability and breakdown properties. This refractive index was computed using the prism coupling method. At different temperatures studies of the dielectric properties were conducted and the Vickers hardness technique was employed to investigate the mechanical properties. The grown crystal's laser-damaged threshold value was found. The Z-scan method investigates a PNC crystal’s third-order NLO behaviours. This method makes it possible to calculate the material's linear and nonlinear refractive indices. The PNC crystal exhibits strong third-order nonlinear optical properties, crucial for applications in optical switching and NLO devices.

Engineering the third-order nonlinear optical absorption properties of two-dimensional layered materials

Engineering the third-order nonlinear optical absorption properties of two-dimensional layered materials

Third-order nonlinear optical properties based on triphenylamine derivative-based covalent organic frameworks

Nonlinear optics (NLO) is an extremely critical research area. In recent years, covalent organic frameworks (COFs) have attracted much attention as a potential NLO material. In this study, a series of COF materials with different D-A effects were successfully synthesized based on the excellent electron-donating properties of the triphenylamine group and were prepared into D-A COFs\\KBr sheet materials that can be used for third-order NLO by the KBr−pressing method. The results of the open-aperture Z-scan technique showed that all COFs\\KBr sheets exhibited saturable absorption (SA) behavior at 3 μJ, with nonlinear absorption coefficients (β) of −1.64, −1.23, and −2.02 × 10−9 mW−1 for PT-PA, PT-BD, and PT-TZ, respectively. Theoretical calculations show that the electron transport distance of PT-PA is shorter than that of PT-BD, while the introduction of S atoms makes the D-A effect of PT-TZ stronger than that of PT-BD, and thus the SA properties of PT-PA and PT-TZ are better than those of PT-BD. These works indicate that the COFs\\KBr sheets have a potential application as high-performance NLO materials and also provide an option for COFs powder application in the field of third-order NLO.

Investigation on Growth, Structural, Spectral, DFT, MEP Computational Studies of Novel Methyl Triphenylphosphonium Bromide HydrateSingle Crystal for Third-Order Nonlinear Optical Limiting Applications

A novel organic nonlinear optical (NLO) single crystal, Methyl Triphenylphosphonium Bromide Hydrate (MTPB), grown for the first time using the slow evaporation solution technique at room temperature. Comprehensive structural analysis by single crystal X-ray diffraction reveals that MTPB crystallizes in the monoclinic system with space group P2₁/n, and its lattice parameters have been accurately determined. Structural refinement was performed using the full matrix least squares technique with the SHELXL-97 program. To characterize the molecular structure and interactions, FT-IR and FT-Raman spectra provided functional group identification, and 2D fingerprint and Hirshfeld surface analyses offered insights into hydrogen bonding interactions within the crystal. The novelty extends to computational analysis, where the molecule's geometry was optimized using DFT-B3LYP with a 6-31++G(d,p) basis set, allowing comparison between theoretical vibrational frequencies and experimental data. Additionally, the third-order NLO properties were assessed via the Z-scan technique, marking the first use of MTPB in optical limiting applications. This work's innovation lies in the successful synthesis, comprehensive characterization, and demonstration of MTPB's significant potential as an optical limiter, paving the way for new NLO applications.

Synthesis, spectroscopical and DFT studies on third order NLO crystal: Sodium bis(malonato)borate monohydrate (SBMBM)

The single crystal of sodium bis(malonato)borate monohydrate (SBMBM) was grown using the Slow Evaporation Solution Technique(SEST). To analyze the crystal lattice parameters of the grown SBMBM crystal, Single Crystal X-Ray Diffraction analysis were used. The results shows that the crystal belongs to Triclinic crystal system with space group P21/n with lattice parameters A = 7.9058 (4) Å, B = 8.2979 (5) Å, and C = 14.6473 (9) Å, where α = 90°, β = 101.565 (2)°, and γ = 90° The theoretical calculation utilizing HF/LANL2DZ and the experimental results agree rather well. The molecular optimized geometry, FT-IR, and the energy gap between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) were calculated using the HF/LANL2DZ basis set. The several functional groups in the SBMBM crystal are investigated by FT-IR spectrum. Studies of UV–visible NIR transmittance indicate that the crystal has a high transmittance throughout the whole visible spectrum. Using the Z-scan technique, third-order nonlinear optical (NLO) property of SBMBM crystal is analysed, and the linear and nonlinear refractive index are computed. χ3 = 4.49 × 10−6esu is the third-order nonlinear optical susceptibility.

Third-order nonlinear optical properties of metal oxide nanoparticles and nanocomposites: A review

This paper reviews the vibrant field of nonlinear optical (NLO) properties of metal oxide nanoparticles and nanocomposites. It includes a historic introduction which discusses the birth and successive developments of NLO. An attempt is made to give a summary in a nutshell about the basic concepts of nonlinear optics and different NLO phenomena with special attention to third-order phenomenon and [Formula: see text]-scan technique. The review also touches on the topic of science and technology of metal oxide nanoparticles and nanocomposites, various synthesis processes that include co-precipitation, micro-emulsion, sol–gel route, etc. have been discussed. Metal oxide nanoparticles and nanocomposites are introduced with a deep focus on the origin of various physical mechanisms and optical properties. The origin of NLO properties from these important structures has also been discussed. Some recently studied metal oxide nanocomposites along with their NLO coefficients are also given.

Bismorpholin-4-ium tetrabromo-cadmium metal-organic single crystal delivering Crystal structure, physicochemical properties and Characterizations for Optoelectronic Applications

For the development of nonlinear optical (NLO) materials, compositions of cadmium chloride (CC) and Morpholinium bromide (MB) molecules were utilized to form metal-organic materials. Morpholinium bromide was incorporated with cadmium metal to have electron charge transfer by constructing acentric structures that take advantage of strong third-order nonlinear optical properties. Bis Morpholinium cadmium bromide (BMCB) crystals were synthesized by a slow evaporation method (SEST). The BMCB crystal belongs to monoclinic P21/c space group with four molecules in the unit cell (a = 6.7607(4) Å, b = 16.5811(2) Å, c = 14.9744(2) Å, β=94.136(3) ° and Z = 4). The grown BMCB crystal phase purity and crystallinity were further investigated using powder X-ray diffraction (PXRD) analysis. Using Fourier transform infrared (FTIR) spectroscopy, the functional groups and their vibrational states were evaluated. From the results obtained from UV–vis-NIR spectral analysis, it is clear that the title compound is transparent in the visible range with the lower cut-off wavelength of 254 nm. The TG-DTA measurement has been used to evaluate thermal stability. Microhardness measurements were carried out to evaluate the mechanical stability of the grown crystal. The third-order nonlinear optical (NLO) characteristics were studied using Z-scan technique. The grown crystal exhibits outstanding third-order NLO response and wide laser damage threshold (LDT), which are crucial characteristics for developing practical NLO devices. The findings consequently open a novel path for the logical design of enormous, thermally stable and NLO-active metal-organic complexes for NLO applications.

Crystal growth, linear optical, thermal, mechanical and third-order nonlinear optical properties of 1,3-diphenyl prop‑2-en-1-one single crystals

This work brings out the viability of the chalcone derivative (E)-1,3-diphenyl-2-propen-1-one (DPP) for industrial applications in nonlinear optical technology and device fabrication, through investigations on its crystal structure and nonlinear optical properties. The unit cell features of the DPP single crystals grown by slow evaporation solution growth method have been determined and hkl values indexed. The optical transparency of DPP in the near infrared and visible range is found to be admirable and an optical bandgap of 3.42 eV is revealed in the UV–Vis-NIR analysis. The work hardening coefficient (n) of DPP, determined as 2 from Vickers microhardness measurement, shows the moderate mechanical strength of the crystal. The Laser Damage Threshold value of 3.9 GW/cm2 sheds light on the high tolerance of DPP to optical damage and suggests its suitability in device fabrication. The potential of DPP crystal in nonlinear optical applications is suggested from its superior value of 1.16×10−7 esu for third order susceptibility and appealing values of other nonlinear optical parameters.

An analysis of the synthesis, crystal structure, optical, spectroscopic, mechanical, self-focusing, and third-order nonlinear optical properties of Kojic acid single crystals

A single, excellent crystal of Kojic acid (KA) was developed utilising the traditional approach of gradual evaporation in a solution-growing procedure. A variety of characterisation techniques were utilised to evaluate the characteristics of the crystals generated. Single crystal x-ray diffraction (SXRD) was utilised to identify the crystal structure. The crystalline arrangement of the KA crystal was confirmed using a powder x-ray diffraction (PXRD) analysis. Experiments with UV–vis spectrometers revealed that the KA crystal has a lower wavelength at which it blocks UV light but has outstanding light transmission throughout the visible spectrum. The KA single crystal has an energy band gap (Eg) of 2.5 eV. The Z-scan method was used to examine the third order susceptibility, nonlinear refraction, and nonlinear absorption coefficient of the synthesised KA crystal. The dielectric characteristics and AC conductivity were examined, with a particular emphasis on their connection to resistance at room temperature. The presence of appreciable levels of kojic acid was confirmed by energy-dispersive spectrometry. The crystal’s thermal behaviour was investigated using differential thermal analysis (DTA) and thermogravimetric (TG) methods. The nucleation parameters and controlled nucleation rate were calculated using normal theoretical research methods.

Growth, characterization and cognition of 2-cyanopyridinium perchlorate single crystals for nonlinear optical applications.

The fascinating electronic applications attracted researchers to explore the field of nonlinear optical (NLO) materials. The slow evaporation of solvent technique (SEST) was employed to grow the 2-cyanopyridinium perchlorate (2-CPPC) NLO single crystals. The cell parameters of the grown 2-CPPC crystal are confirmed by the single crystal X-ray diffraction (SCXRD) study. The powder X-ray diffraction studies confirm the crystallinity of 2-CPPC crystals, and the peaks were indexed. The computation for the geometry optimization, HOMO-LUMO energy gap, global reactivity parameters, natural bond orbital (NBO) analysis, polarizability, and hyperpolarizability of the 2-CPPC molecule was done using B3LYP (6-311G basis set) functional of DFT method. The experimental FTIR and UV-Vis results of the 2-CPPC compound were compared with the simulated results. The second harmonic generation (SHG) study for the 2-CPPC crystal was employed using Kurtz-Perry powder technique. Single beam Z-scan technique using He-Ne laser is used to study the third-order NLO properties.

Rational analysis of hydrogen bonding interaction in phenazine, 2-hydroxynaphthalene (1:1) cocrystal: from molecular modeling to photophysical properties.

Organic cocrystals have a wide range of applications in the field of optics due to their photo responsive property. We present here a newly synthesized phenazine 2-hydroxynaphthalene (1:1) cocrystal, its structural and theoretical calculations which tend to the nonlinear optical property. In the crystal structure of the title cocrystal, the phenazine and 2-hydroxynaphthalene molecules from one- and two-dimensional supramolecular frameworks via O‒H…N hydrogen bonds and C‒H…N, C‒H…π interaction, respectively. The phenazine molecules from an infinite off-set stacking through π…π interactionin the three-dimensional molecular packing of the title cocrystal. The contribution of intermolecular interaction in the three-dimensional molecular packing and the interaction energy calculation is studied by the Hirshfeld surface analysis. The molecular geometry retrieved from the experimental X-ray diffraction analysis is in good agreement with the theoretically calculated parameters. Further, the molecular electrostatic potential (MEP) and frontier molecular orbital (FMO) analysis have been carried out to study the charge distribution and molecular reactive mechanism. Third-order nonlinear optical property of the cocrystals has been analyzed by Z-scan measurements. The determined nonlinear optical absorption coefficient value 6.442 × 10-05 (m/W) and the nonlinear refractive index value - 5.535 × 10-2 (m/W) suggest that the crystalline solid can be a good choice of potential nonlinear optical material. The crystal structures of phenazine 2-hydroxynaphthalene cocrystal was solved by direct methods procedure using SHELXS program and refined by full-matrix least square procedure on F2 using SHELXL-2018 program on Olex2 software. The computational calculation has been carried out using DFT/B3LYP quantum chemical function with triple zeta 6-311 + + basis set in the ground state molecular stability using Gaussian 09W program suite. The Hirshfeld surface analysis mapping, associated 2D fingerprint plot, and intermolecular molecular interaction energy calculations were carried out using CrystalExplorer (version 21.5) software.

Theoretical Study on the Correlation between Open-Shell Electronic Structures and Third-Order Nonlinear Optical Properties in One-Dimensional Chains of π-Radicals.

This paper theoretically investigated the correlation between the open-shell electronic structure and third-order nonlinear optical (NLO) properties of one-dimensional (1D) stacked chains of π-radicals. By employing the finite N-mer models consisting of methyl or phenalenyl radicals with different stacking distances, we evaluated the average and standard deviation of diradical characters yi for N-mer models of π-radicals (yav and ySD). Then, we estimated these diradical characters at the limit of N → ∞. These y-based indices were helpful in discussing the correlation between the open-shell electronic structures and the second hyperpolarizability per dimer at the limit N → ∞, γ∞ for the 1D chains with stacking distance alternation (SDA). The calculated γ∞ values and the polymer/dimer ratio γ∞/γ(N = 2) were enhanced significantly when both the stacking distance and SDA are small. We also found that the spin-unrestricted long-range-corrected (LC-)UBLYP method with the range-separating parameter μ = 0.47 bohr-1 well reproduced the trend of γ∞ of this type of 1D chain estimated at the spin-unrestricted coupled-cluster levels. The present study is expected to contribute to establishing the design guidelines for future high-performance open-shell molecular NLO materials.

Synthesis, structure, third-order nonlinear optical properties and Hirshfeld surface analysis of 18-crown-6 with dimethylpyridin-1-ium iodide isomers

Supramolecular complexes of 18-crown-6 with isomers of dimethylpyridin-1-ium iodide have been successfully synthesized and characterized using various analytical techniques, including single-crystal X-ray diffraction (SCXRD). SCXRD study revealed the crystal structures of bis(1,2-dimethylpyridin-1-ium iodide)-18-crown-6 (I) with a monoclinic structure in the centric space group P21/n, bis(1,3-dimethylpyridin-1-ium iodide)-18-crown-6 (II), and bis(1,4-dimethylpyridin-1-ium iodide)-18-crown-6 (III), both exhibiting triclinic structures in the centric space group Pī. The band gap energies are determined using diffuse reflectance data through the Kubelka-Munk algorithm. The thermal stability of these cocrystals was assessed through differential thermal and thermogravimetric studies, while their surface morphology was analyzed using scanning electron microscopy. The third-order nonlinear optical susceptibilities of cocrystals (I), (II), and (III) were measured at 3.54 × 10−6 esu, 4.05 × 10−6 esu, and 4.08 × 10−6 esu, respectively, indicating their promising utility in nonlinear optical applications.

Synthesis, Structural, Electrochemical, and DFT Studies of Highly Substituted Nonplanar Ni(II) Porphyrins and Their Intensity-Dependent Third-Order Nonlinear Optical Properties.

We designed and successfully synthesized highly substituted electron-deficient nonplanar Ni(II) porphyrins and their derivatives (1-7) in moderate to good yields. These derivatives were comprehensively characterized by various spectroscopic techniques and single-crystal X-ray diffraction (SCXRD) analysis. SCXRD analysis confirmed the structures of compounds 2, 4, and 7, adopting saddle-shape geometry. These nonplanar porphyrins demonstrated significant bathochromic shifts in their absorption spectra compared to parent NiTPP, attributed to the influence of bulky β-substituents and/or peripheral fusion. π-Extended porphyrins 6 and 7 displayed panchromatic absorption spectra extending into the NIR region. Porphyrins 6 and 7 demonstrated a profound anodic shift (∼400 mV) in their first reduction peak potentials compared to precursor NiTPP(NO2)Br6. The experimental absorption spectral pattern matches the simulated absorption spectra obtained from TD-DFT studies. The femtosecond laser intensity-dependent third-order nonlinear optical studies revealed that NiDFP(VCN)2Br6 (6) and NiDFP(VCN)2(PE)6 (7) displayed giant optical nonlinearities compared to the other porphyrins. Among all, NiDFP(VCN)2Br6 (6) possessed the highest two-photon absorption coefficient (β) and cross-section (σTPA) values in the range of 22-33 × 10-10 m/W and 3.77-6.95 × 106 GM, respectively. These findings suggest that the investigated nonplanar π-extended porphyrins are promising candidates for future optoelectronic applications.

Structural and ultrafast third-order nonlinearities of methyl and methoxy substituted anthracene chalcones: Z-scan, four-wave mixing, and DFT approaches

We report on the structural, thermal, linear, and ultrafast third-order nonlinear optical (NLO) properties of two novel anthracene chalcones: (2E)-1-(anthracen-9-yl)-3-(5-methylthiophen-2-yl)prop-2-en-1-one (5ML2SANC) and (2E)-1-(9-anthryl)-3-(2,4,5-trimethoxyphenyl)prop-2-en-1-one (245TMANC). The chalcones were synthesized by Claisen-Schmidt condensation reaction, and the single crystals were grown by the solvent evaporation method. The molecular structure was confirmed by FTIR and NMR spectroscopy, while the crystal structure was determined using the single crystal XRD. Both crystals belong to centrosymmetric monoclinic crystal system with space group P21/n. The Hirshfeld surface was analyzed to understand intermolecular interactions, and the band structures - including HOMO-LUMO levels, excited state energies, GCRDs and MEPs-were studied using DFT. The ultrafast third-order NLO properties were investigated by Z-scan and degenerate four-wave mixing (DFWM) techniques using Ti: Sapphire amplifier laser delivering ∼50 fs pulses at 800 nm (1 kHz, ∼4 mJ, 2 W). Two-photon absorption, positive nonlinear refraction, optical limiting and optical switching behaviors were observed by Z-scan measurements. The time-resolved DFWM show that the decay time of 5ML2SANC is ∼127 fs, while for 245TMANC it is ∼142 fs. The second hyperpolarizability (γ) measured by Z-scan, DFWM and the estimations from the DFT theory are found to be in good agreement (∼10−34 esu). The ultrafast optical response, significant NLO properties and thermal stability of the synthesized chalcones demonstrate their potential suitability in optical limiting and switching applications.

Integration of Open Metal Sites in an Amino-Functionalized Sm(III)-Organic Framework toward Enhanced Third-Order Nonlinear Optical Property.

A variety of new inorganic and organic materials have emerged to advance laser technologies and optical engineering. A rational design approach can contribute significantly to fabricating nonlinear optically active metal-organic frameworks (MOFs) by considering the underlying structure-property linkage. Here, it has been embarked on a study of novel samarium(III) MOF, ([Sm2(ata)3(DMF)4]·DMF (ata2-: 2-aminoterephthalate), abbreviated as NH2-Sm-MUM-4) with enhanced nonlinear optical (NLO) properties. The crystal structure of this MOF represents a 6-connected framework with a pcu topology and distinctive characteristics, including open metal sites, free amine groups, and great stability, making it suitable for third-order NLO activity. The nonlinear index of refraction (n2) revealed the self-focusing impacts of NH2-Sm-MUM-4 at different incident intensities. The highest value of n2 and β related to 10 mw power of incident intensity are 5.15 cm2/W and 2.65 cm/W, respectively. As far as the authors know, this is the first study examining the potential systematic structural-property associations in Sm-MOFs considering improved third-order NLO properties.

Efficient High-Order Harmonic Generation from the van der Waals Layered Crystal Copper Indium Thiophosphate.

Layered metal thio- and selenophosphates (MTPs) are a family of van der Waals gapped materials that exhibit a multitude of functionalities in terms of magnetic, ferroelectric, and optical properties. Despite the recent progress in terms of understanding the material properties of these compounds, the potential of MTPs as a material class yet needs further scrutiny, especially in terms of nonlinear optical properties. Recent reports of efficient low-order harmonic generation and extremely high third-order nonlinear optical properties in MTPs suggest the potential application of these materials in integrated nanophotonics. In this article, we investigate the high-order nonlinear response of bulk and exfoliated thin-film crystals of copper indium thiophosphate (CIPS) to intense mid-infrared fields through experimental and computational studies of high-order harmonic generation (HHG). From a driving laser source with a 3.2 μm wavelength, we generate odd and even harmonics up to the 10th order, exceeding the bandgap of the material. We note conversion efficiencies as high as 10-7 measured for the fifth and seventh harmonics and observe that the harmonic intensities follow a power law scaling with the driving laser intensity, suggesting a perturbative nonlinear optical origin of the observed harmonics for both bulk and thin flakes. Furthermore, first-principles calculations suggest that the generation of the highest harmonic orders results from electron-electron interactions, suggesting a correlation-mediated enhancement of the high-order optical nonlinearity.