Techniques for nonlinear optical characterization of materials: a review

Nonlinear Optical Properties of 2D Materials and their Applications

2D materials are a subject of intense research in recent years owing to their exclusive photoelectric properties. With giant nonlinear susceptibility and perfect phase matching, 2D materials have marvelous nonlinear light-matter interactions. The nonlinear optical properties of 2D materials are of great significance to the design and analysis of applied materials and functional devices. Here, the fundamental of nonlinear optics (NLO) for 2D materials is introduced, and the methods for characterizing and measuring second-order and third-order nonlinear susceptibility of 2D materials are reviewed. Furthermore, the theoretical and experimental values of second-order susceptibility χ(2) and third-order susceptibility χ(3) are tabulated. Several applications and possible future research directions of second-harmonic generation (SHG) and third-harmonic generation (THG) for 2D materials are presented.

Thermal lens Z-scan measurements: theoretical and experimental uncertainties for low and high fluorescence quantum yields

The single-beam Z-scan thermal lens technique is conducted to evaluate the fluorescence quantum yield of various solutions in the case of high-moderate absorption, considering both scenarios: solutions with substantial fluorescence and solutions with high thermal efficiency but low fluorescence. An analytical calculation is performed to determine the uncertainties associated with the random errors introduced by optical detectors. The results reveal that solutions with low fluorescence lead to a significant error, whereas higher fluorescence can help in decreasing the uncertainty. Additionally, the issue of random errors arising when multiple measurements are needed to accurately estimate the fluorescence of a solution will be discussed in different situations.

Nonlinear Refraction, Excited State Absorption, and Multiphoton Absorption of 1,3-Thiazolium-5-thiolate Mesoionic Compound

We synthesized the mesoionic compound 2-(4-chlorophenyl)-3-methyl-4-(4-methylphenyl)-1,3-thiazole-5-thiolate and measured its refractive and absorptive nonlinear optical response in different temporal and spectral regimes. The experiments were performed by using the Z-scan technique with two pulsed light sources: the second harmonic (at 532 nm) of a mode-locked and Q-switched Nd-YAG laser (100 ps, 10 Hz) and a Ti: Sapphire laser system (100 fs, 1 kHz) operating at 800 nm. The observation and characterization of nonlinear refraction, two- and three-photon absorption, and excited state absorption of the mesoionic compound dissolved in dimethyl sulfoxide, in different concentrations, are presented and discussed with basis on the population redistribution in a three-energy-level model that allows the determination of the parameters which characterize the nonlinear response.

Intensity-Dependent Optical Response of 2D LTMDs Suspensions: From Thermal to Electronic Nonlinearities

The nonlinear optical (NLO) response of photonic materials plays an important role in the understanding of light-matter interaction as well as pointing out a diversity of photonic and optoelectronic applications. Among the recently studied materials, 2D-LTMDs (bi-dimensional layered transition metal dichalcogenides) have appeared as a beyond-graphene nanomaterial with semiconducting and metallic optical properties. In this article, we review most of our work in studies of the NLO response of a series of 2D-LTMDs nanomaterials in suspension, using six different NLO techniques, namely hyper Rayleigh scattering, Z-scan, photoacoustic Z-scan, optical Kerr gate, and spatial self-phase modulation, besides the Fourier transform nonlinear optics technique, to infer the nonlinear optical response of semiconducting MoS2, MoSe2, MoTe2, WS2, semimetallic WTe2, ZrTe2, and metallic NbS2 and NbSe2. The nonlinear optical response from a thermal to non-thermal origin was studied, and the nonlinear refraction index and nonlinear absorption coefficient, where present, were measured. Theoretical support was given to explain the origin of the nonlinear responses, which is very dependent on the spectro-temporal regime of the optical source employed in the studies.

Intensity correlation scan (IC-scan) technique to characterize the optical nonlinearities of scattering media

Light scattering, whether caused by desired or spurious elements, is considered one of the main phenomena that present great challenges for the nonlinear (NL) optical characterization of turbid media. The most relevant disturbing factor is the random deformation suffered by the spatial intensity distribution of the laser beam due to multiple scattering. In this work, we report the intensity correlation scan (IC-scan) technique as a new tool to characterize the NL optical response of scattering media, by taking advantage of light scattering to generate speckle patterns sensitive to wavefront changes induced by the self-focusing and self-defocusing effects. Peak-to-valley transmittance curves, with a higher signal-to-noise ratio, are obtained by analyzing the spatial intensity correlation functions of the different speckle patterns, even in very turbid media where conventional NL spectroscopy techniques fail. To demonstrate the potential of the IC-scan technique, the NL characterization of colloids that contain a high concentration of silica nanospheres as scatterers, as well as gold nanorods, which act as NL particles and light scatterers, was performed. The results show that the IC-scan technique is more accurate, precise and robust to measure NL refractive indices in turbid media, overcoming limitations imposed by well-established Z-scan and D4σ techniques.

Near-infrared ultrafast third-order nonlinear optical response of 2D NbS2, NbSe2, ZrTe2, and MoS2

By employing the optical Kerr gate technique at 800 nm with 180 fs pulses at 76 MHz, we evaluated the third-order nonlinear optical response of two-dimensional (2D) semiconducting MoS₂, semimetallic ZrTe₂, and metallic NbS₂ and NbSe₂. The modulus of the nonlinear refractive index was measured to range from 8.6 × 10^-19 m²/W to 5.3 × 10^-18 m²/W, with all materials' response time limited by the pulse duration. The physical mechanism to explain the ultrafast response time's origin considers the nature of the 2D material, as will be discussed.

Nonlinear Optical Properties of Zinc Oxide Nanoparticle Colloids Prepared by Pulsed Laser Ablation in Distilled Water

The nonlinear optical properties of zinc oxide nanoparticles (ZnONPs) in distilled water were measured using a femtosecond laser and the Z-scan technique. The ZnONPs colloids were created by the ablation of zinc bulk in distilled water with a 532 nm Nd: YAG laser. Transmission electron microscopy, an ultraviolet-visible spectrophotometer, and atomic absorption spectrophotometry were used to determine the size, shape, absorption spectra, and concentration of the ZnONPs colloids. The nonlinear absorption coefficient and nonlinear refractive index were measured at different excitation wavelengths and intensities. The nonlinear absorption coefficient of the ZnONPs colloids was found to be positive, caused by reverse saturable absorption, whereas the nonlinear refractive index was found to be negative due to self-defocusing in the ZnONPs. Both laser parameters, such as excitation wavelength and input intensity, and nanoparticle features, such as concentration and size, were found to influence the nonlinear optical properties of the ZnONPs.

Faris H, Ahmed H, Mamdouh S, Thambiratnam K, Mohamed T. Using Femtosecond Laser Pulses to Explore the Nonlinear Optical Properties of Au NP Colloids That Were Synthesized by Laser Ablation

In this study, we experimentally investigated the nonlinear optical properties of Au nanoparticles (Au NPs) that were prepared in pure distilled water using the laser ablation method. The Au NPs were prepared using a nanosecond Nd:YAG laser with an ablation time of 5 or 10 min at a constant laser energy of 100 mJ. The structure and the linear optical properties of the Au NPs were investigated using a transmission electron microscope (TEM) and UV-visible spectrophotometer analysis, respectively. The TEM measurements showed that the average size of the Au NPs varied from 20.3 to 14.1 nm, depending on the laser ablation time. The z-scan technique was used to investigate the nonlinear refractive index (n2) and nonlinear absorption coefficient (γ) of the Au NPs, which were irradiated at different excitation wavelengths that ranged from 740 to 820 nm and at different average powers that ranged from 0.8 to 1.6 W. The Au NP samples exhibited a reverse saturable absorption (RSA) behavior that increased when the excitation wavelength and/or incident laser power increased. In addition, the Au NPs acted as a self-defocusing material whenever the excitation wavelength or incident power were modified.

Interferometric measurements of nonlinear refractive index in the infrared spectral range

This study presents the development and application of interferometric technique for the measurement of nonlinear refractive index of optical materials, while directly accounting for experimentally determined laser pulse shape and beam profile. The method was employed in a systematic study of nonlinear refractive index on a series of common optical materials used in near and mid-IR spectral range, where experimental data on nonlinear material properties is still scarce. The values of nonlinear refractive index were determined at 1.03 µm, 2.2 µm, and 3.2 µm. The measurement results are compared to the values determined by previous studies (where available), and the influence of cascaded second-order nonlinearities is discussed.

Molecular modeling and nonlinear optical properties of new isostructural halogenated dihydroquinolinones

Two new isostructural halogenated dihydroquinolinones were synthesized and characterized by single crystal X-ray diffraction.

Numerical Simulation of the Whole Thermal Lensing Process with Z-Scan-Based Methods Using Gaussian Beams

A general study of the diffracted far field due to thermal lens heating using Gaussian beams is presented. The numerical simulation considers the whole system, including both the optical and the thermal parameters. It is shown that the existing simplified relations found in the literature and used up to now only give the order of magnitude of the thermo-optical coefficients. More accurate, simplified formulas are derived to measure the induced thermal phase shift when working with Z-scan-based methods. Temperature estimation in absorbing media turn out to be more reliable whether using time-resolved or steady-state techniques. The extension of the calculation to the image formation in a 4f system is also addressed.

Efficient direct mapping of the nonlinear optical response via modulated Airy beams

We report a scheme to achieve efficient direct mapping of the nonlinear optical response into a spatial beam profile. Compared with previous methods where a standard two-dimensional Airy beam was used as a probe, a modulated beam configuration allows for an improved mapping efficiency, stemming from the induced nonlinearity caused by the applied modulation. We find that the mapping efficiency along different orientations is highly related to the beam patterns and the type of nonlinearity. The improvement of the mapping quality and new, to the best of our knowledge, features found in simulations are further verified in experiments by testing a photorefractive nonlinearity. Our results represent a further step towards an effective tool for the direct measurement of the nonlinear optical response with low power consumption.

Fifth-order optical nonlinear response of semiconducting 2D LTMD MoS2

The effective fifth-order susceptibility, ${\chi}_{\rm eff}^{(5)}$, of two-dimensional (2D) semiconducting layered transition metal dichalcogenide (LTMD) molybdenum disulfide (${\rm MoS}_2$) is reported here for the first time, to the best of our knowledge. Using the $ Z $-scan technique with a laser operating at 800 nm, 1 kHz, 100 fs, we investigated the nonlinear behavior of ${\rm MoS}_2$ suspended in acetonitrile (concentration, 70 µg/ml). The effective nonlinear refractive index ${{n}_{4,{eff}}} = - ({7.6 \pm 0.5}) \times {10^{- 26}}\; {{\rm cm}^4}/{{\rm W}^2}$, proportional to ${\rm Re}{\chi}_{\rm eff}^{(5)}$, was measured for monolayer ${\rm MoS}_2$ nanoflakes, prepared by a modified redox exfoliation method. We also determined the value of the nonlinear refractive index ${{n}_2} = + ({4.8 \pm 0.5}) \times {10^{- 16}}\;{{\rm cm}^2}/{\rm W}$, which is related to the material's effective third-order optical susceptibility real part, ${Re\chi}_{\rm eff}^{(3)}$. For comparison, we also investigated the nonlinear response of tungsten disulfide (${\rm WS}_2$) monolayers, prepared by the same method and suspended in acetonitrile (concentration, 40 µg/ml), which only exhibited the third-order nonlinear effect in the same intensity range, up to ${120}\;{{{\rm GW}/{\rm cm}}^2}$. Nonlinear absorption was not observed in either ${\rm MoS}_2$ or ${\rm WS}_2$.

Second-harmonic generation enhancement in monolayer transition-metal dichalcogenides by using an epsilon-near-zero substrate

Monolayer transition-metal dichalcogenides (TMDCs) present high second-order optical nonlinearity, which is extremely desirable for, e.g., frequency conversion in nonlinear photonic devices. On the other hand, the atomic thickness of 2D materials naturally leads to low frequency converted intensities, highlighting the importance of designing structures that enhance the nonlinear response for practical applications. A number of methods to increase the pump electric field at 2D materials have been reported, relying on complex plasmonic and/or metasurface structures. Here, we take advantage of the fact that unstructured substrates with a low refractive index naturally maximize the pump field at a dielectric interface, offering a simple means to promote enhanced nonlinear optical effects. In particular, we measured second harmonic generation (SHG) in MoS₂ and WS₂ on fluorine tin oxide (FTO), which presents an epsilon-near zero point near our 1550 nm pump wavelength. Polarized SHG measurements reveal an SHG intensity that is one order of magnitude higher on FTO than on a glass substrate.

Nonlinear photoacoustic response of suspensions of laser-synthesized plasmonic titanium nitride nanoparticles

A nonlinear photoacoustic (PA) response from solutions of 40 nm plasmonic titanium nitride nanoparticles (NPs) synthesized by laser ablation in a liquid environment (acetone) is reported. Using a photoacoustic Z-scan with 5 ns pumping pulses, values of effective nonlinear absorption (NLA) coefficients β_PA,eff were measured and found to be 3.27±0.17 × 10^-8, 6.41±0.32 × 10^-8, and 3.22±0.16 × 10^-8 for 600, 700, and 800 nm pumping wavelengths, respectively. To take into account the influence of nonlinear scattering, absorption-dependent PA measurements were carried out together with the optical Z-scan, and the obtained data were compared. The origin of the effective absorptive nonlinearity is discussed based on combined NLA in NPs, nonlinear scattering, and bubble generation triggered by NP-mediated light absorption. Potential applications include biomedical diagnostics and therapy.

Nonlinear optical properties of arsenic telluride and its use in ultrafast fiber lasers

We report the first investigation results of the nonlinear optical properties of As2Te3. More specifically, the nonlinear optical absorption properties of the prepared α-As2Te3 were investigated at wavelengths of 1.56 and 1.9 μm using the open-aperture (OA) Z-scan technique. Using the OA Z-scan technique, the nonlinear absorption coefficients (β) of α-As2Te3 were estimated in a range from (- 54.8 ± 3.4) × 104 cm/GW to (- 4.9 ± 0.4) × 104 cm/GW depending on the irradiance of the input beam at 1.56 μm, whereas the values did from (- 19.8 ± 0.8) × 104 cm/GW to (- 3.2 ± 0.1) × 104 cm/GW at 1.9 μm. In particular, the β value at 1.56 μm is an order of magnitude larger than the previously reported values of other group-15 sesquichalcogenides such as Bi2Se3, Bi2Te3, and Bi2TeSe2. Furthermore, this is the first time report on β value of a group-15 sesquichalcogenide at a 1.9-μm wavelength. The density functional theory (DFT) calculations of the electronic band structures of α-As2Te3 were also conducted to obtain a better understanding of their energy band structure. The DFT calculations indicated that α-As2Te3 possess sufficient optical absorption in a wide wavelength region, including 1.5 μm, 1.9 μm, and beyond (up to 3.7 μm). Using both the measured nonlinear absorption coefficients and the theoretically obtained refractive indices from the DFT calculations, the imaginary parts of the third-order optical susceptibilities (Im χ(3)) of As2Te3 were estimated and they were found to vary from (- 39 ± 2.4) × 10-19 m2/V2 to (- 3.5 ± 0.3) × 10-19 m2/V2 at 1.56 μm and (- 16.5 ± 0.7) × 10-19 m2/V2 to (- 2.7 ± 0.1) × 10-19 m2/V2 at 1.9 μm, respectively, depending on the irradiance of the input beam. Finally, the feasibility of using α-As2Te3 for SAs was investigated, and the prepared SAs were thus tested by incorporating them into an erbium (Er)-doped fiber cavity and a thulium-holmium (Tm-Ho) co-doped fiber cavity for both 1.5 and 1.9 μm operation.

Synthesis, Optical, and Morphological Studies of ZnO Powders and Thin Films Fabricated by Wet Chemical Methods

Zinc oxide nanoparticles were prepared from Zn₅(CO₃)₂(OH)₆ precursor, capped with poly(vinylpyrrolidone) (PVP), and annealed at 600 °C. The obtained powders were characterized by a powder X-ray diffraction (PXD), transmission electron microscopy (TEM), scanning electron microscopy (SEM), UV–visible spectroscopy (UV–vis), Raman spectroscopy, infrared spectroscopy (IR), thermal analysis (TGA/DTA), and third-order nonlinear (NL) optical measurement. Morphological evaluation by TEM and SEM measurements indicated that the precursor micro-particles are ball-shaped structures composed of plates with a thickness of approximately 10 nm. ZnO thin films, as well as ZnO/polymer multilayer layouts, were obtained by wet chemical methods (spin- and dip-coating). Surface topography and morphology of the obtained films were studied by SEM and AFM microscopy. Films with uniformly distributed ZnO plates, due to the erosion of primary micro-particles were formed. The fabricated specimens were also analyzed using a spectroscopic ellipsometry in order to calculate dielectric function and film thickness.

Dynamic compressibility and third-order optical nonlinearities in carbon/metal-based nanofluids

The influence of the superposition of two high-irradiance optical beams on the mechanical properties exhibited by carbon nanotubes decorated with platinum nanoparticles was analyzed. The change in density, compressibility modulus and acoustic velocity in the samples suspended in acetone and ethanol was estimated by measuring the nonlinear refractive index tested by a two-wave mixing experiment. The nanotubes were prepared by a spray pyrolysis processing route and the metal decoration was carried by chemical vapor deposition. High-Resolution Transmission Electron Microscopy studies confirmed the multiwall nature of the carbon nanotubes; while energy-dispersive X-ray spectroscopy reveals the separated presence of platinum nanoparticles incorporated to the hybrid nanostructures. An Nd-YAG laser system emitting at 532 nm wavelength with 4 ns pulse duration was used for conducting the third-order nonlinear optical evaluations by a standard optical Kerr gate technique. Comparative experiments showed that the composition of the liquid solution plays an important role in the manipulation of the density exhibited by the nanofluids. Remarkably, the incorporation of Pt in the tubes originates stronger changes of the mechanical characteristics induced by optical nonlinearities in the nanofluids irradiated by nanosecond pulses. Within this work, it is highlighted that potential applications for developing multivalent logic operations by fuzzy mechano-optic effects exhibited by nanofluids can be contemplated.

2D materials towards ultrafast photonic applications

Two-dimensional materials are now excelling in yet another arena of ultrafast photonics, including optical modulation through optical limiting/mode-locking, photodetectors, optical communications, integrated miniaturized all-optical devices, etc.

Sr4Pb1.5Sb5O5Se8: a new mid-infrared nonlinear optical material with a moderate SHG response

The crystal structure and second-harmonic generation response of the new nonlinear optical material, Sr₄Pb_1.5Sb₅O₅Se₈, were examined.

Exploring adaptive optics on focus-scan for nonlinear materials characterization

Z-scan is a well-established technique used to measure the nonlinear refractive index and absorption coefficient of thin and transparent materials. The method requires the displacement of the sample along the focus of a laser beam. Therefore, the Z-scan is not suitable for experiments where the sample cannot be axially translated. Here, we explore a deformable mirror to create controllable defocus aberrations, translating the focus of the beam through the sample, alternatively to the sample displacement. The technique is based on time behavior analysis of the light beam transmitted by the nonlinear sample, at different defocus configurations. The method is validated by measuring reference samples (CS₂ and SiO₂) and comparing them with the conventional Z-scan technique.

Chalcone as Potential Nonlinear Optical Material: A Combined Theoretical, Structural and Spectroscopic Study

In this work, we propose the synthesis of a novel bromine chalcone (E)-3-(2-bromophenyl)-1-(2phenylsulfonylamine)-phenyl)prop-2-en-1-one (BRC) that has been crystallized by slow evaporation technique. The second order molecular optical scattering and two-photon absorption(2PA) spectrumof the BRC molecule dissolved in Dimethyl Sulfoxide (DMSO) were evaluated by using Hyper Rayleigh Scattering and femtosecond tunable Z-Scan techniques. The first order hyperpolarizability of BRC dissolved in DMSO was estimated by using a simplified two-level model, in which one- and two-photon absorption parameters were used as input information to the model. The BRC crystal was characterized from single crystal X-ray diffraction (DRX) and spectroscopy analyzes. Also, the thermogravimetric analyses and the fluorescence spectra were obtained. In addition, an ab-initio calculation method, which includes the Møller-Plesset Perturbation Theory (MP2) and the Density Functional Theory (DFT) at CAM-B3LYP level,was used to estimate the crystal linear refractive index and the third-order electric susceptibility. Also, the average first hyperpolarizability of BRC molecules dissolved in DMSO was calculated and compared with the experimental results. The obtained values are good and qualify BRC crystal as a potential candidate for application in nonlinear optical devices.

Time-resolved inline digital holography for the study of noncollinear degenerate phase modulation

Recent works demonstrated that digital time-resolved holography is the prospective approach to study nonlinear light-matter interaction processes. In this Letter, we present a straightforward inline holographic approach for studying degenerate phase modulation induced by an inclined collimated pump beam in the isotropic sample. The method is based on a minimization of the difference between experimentally acquired data and simulated inline holograms obtained from a numerical model of pump-probe interaction in optical nonlinear media. A sophisticated experimental data processing algorithm is implemented to provide high sensitivity and a signal-to-noise ratio eligible for soft interaction with a collimated pump beam. The integral phase shift determined by our method can be used to estimate the nonlinear refractive index and the relaxation time for material with a low damage threshold. We validated our approach for the case of soda-lime and BK7 glasses.

Rapid and high precision measurement of opto-thermal relaxation with pump-probe method

Opto-thermal relaxation is one of the most important properties of nonlinear optical materials. Rapid and high precision measurement of this parameter is vital in both fundamental research and applications. Current measurement uses either complicated structure with poor precision or high power heating source with low efficiency. Here, we propose a pump-probe method (PPM) to optically measure the thermal relaxation using whispering gallery mode (WGM) microcavities. When the pump laser shines on a microcavity, the materials absorb the input power resonantly and heat up. Then the heat dissipates from the cavities to the surroundings. The opto-thermal effect induces a refractive index change reflected in the signal light transmission spectra. By analyzing the curve character of the transmission spectra of the signal response in the spontaneous relaxation process, the thermal relaxation time can be rapidly measured with high precision. Additionally, we systematically verify the PPM using microtoroids under various pump powers and at various locking points of the signal laser mode. The small rate of refractive index changes (∼10^-8) can be discerned with an input pump power as low as 11.816 μW. Hence, the PPM can be used to detect refractive index perturbation, like gas or liquid sensing, temperature fluctuations with ultra-high sensitivity and be applied to optical materials analysis efficiently.

Analytic methods to find beating transitions of asymmetric Gaussian beams in GNLS equations

In a simple model of propagation of asymmetric Gaussian beams in nonlinear waveguides, described by a reduction to ordinary differential equations of generalized nonlinear Schrödinger equations with cubic-quintic (CQ) and saturable (SAT) nonlinearities and a graded-index profile, the beam widths exhibit two different types of beating behavior, with transitions between them. We present an analytic model to explain these phenomena, which originate in a 1:1 resonance in a 2 degree-of-freedom Hamiltonian system. We show how small oscillations near a fixed point close to 1:1 resonance in such a system can be approximated using an integrable Hamiltonian and, ultimately, a single first order differential equation. In particular, the beating transitions can be located from coincidences of roots of a pair of quadratic equations, with coefficients determined (in a highly complex manner) by the internal parameters and initial conditions of the original system. The results of the analytic model agree with the numerics of the original system over large parameter ranges, and allow new predictions that can be verified directly. In the CQ case, we identify a band of beam energies for which there is only a single beating transition (as opposed to 0 or 2) as the eccentricity is increased. In the SAT case, we explain the sudden (dis)appearance of beating transitions for certain values of the other parameters as the grade-index is changed.

Third- and high-order nonlinear optical properties of an intramolecular charge-transfer compound

An oligo(phenylenevinylene) bridged intramolecular charge-transfer (ICT) compound, (TCNQ)2OPV3, has been synthesized and its third- and fifth-order nonlinear optical properties have been determined by measurement with the 4f system with a phase-object.

Linear and nonlinear optical characterization of self-assembled, large-area gold nanosphere metasurfaces with sub-nanometer gaps

We created centimeter-scale area metasurfaces consisting of a quasi-hexagonally close packed monolayer of gold nanospheres capped with alkanethiol ligands on glass substrates using a directed self-assembly approach. We experimentally characterized the morphology and the linear and nonlinear optical properties of metasurfaces. We show these metasurfaces, with interparticle gaps of 0.6 nm, are modeled well using a classical (without charge transfer) description. We find a large dispersion of linear refractive index, ranging from values less than vacuum, 0.87 at 600 nm, to Germanium-like values of 4.1 at 880 nm, determined using spectroscopic ellipsometry. Nonlinear optical characterization was carried out using femtosecond Z-scan and we observe saturation behavior of the nonlinear absorption (NLA) and nonlinear refraction (NLR). We find a negative NLR from these metasurfaces two orders of magnitude larger (n_2,sat = -7.94x10^-9 cm²/W at I_sat,n2 = 0.43 GW/cm²) than previous reports on gold nanostructures at similar femtosecond time scales. We also find the magnitude of the NLA comparable to the largest values reported (β_2,sat = -0.90x10⁵ cm/GW at I_sat,β2 = 0.34 GW/cm²). Precise knowledge of the index of refraction is of crucial importance for emerging dispersion engineering technologies. Furthermore, utilizing this directed self-assembly approach enables the nanometer scale resolution required to develop the unique optical response and simultaneously provides high-throughput for potential device realization.

D4σ curves described analytically through propagation analysis of transverse irradiance moments

The far-field changes in the beam width of an intense light beam propagating inside optical materials are investigated aiming the measurement of the nonlinear properties of the samples. The beam width is characterized in this Letter in terms of the transverse irradiance moments (TIMs) that allow an accurate description of the beam interaction and propagation. TIMs are rigorously related to the paraxial beam propagation parameters, and it is also possible to consider elliptical and astigmatic beams. Experimental data with very good signal-to-noise ratio were obtained for the reference materials carbon disulfide and fused quartz.