## NL refractive optical characterization of transparent media

Figure 4 shows the experimental results obtained by the CA Z-scan, D4σ and IC-scan techniques performed to characterize the refractive NL response of pure ethanol. A good signal-to-noise ratio is observed in the curves because scattering particles (or defects) are not present in the NL medium. In Fig. 4(a) and Fig. 4(b), by fitting the experimental results using the nonlocal model (with \(m=0.1\)) described in the Numerical simulation section, the NL refractive indices \({n}_{2}^{ethanol}=-\left(2.8\pm 0.4\right)\times 1{0}^{-8} {\text{cm}}^{\text{2}}/\text{W}\) (for Z-scan) and \(-\left(2.2\pm 0.3\right)\times 1{0}^{-8} {\text{cm}}^{\text{2}}/\text{W}\) (for D4σ), were obtained. The difference between the values measured by CA Z-scan and D4σ is probably due to the asymmetry or imperfection in the Gaussian beam profile that causes alterations when analyzing the transmittance through a small aperture or the transverse irradiance (second-order) moments, respectively. Nevertheless, reports in the literature using the local NL model \(\left(m=2.0\right)\) to fit the experimental curves showed \({n}_{2}\) values lower than those obtained by us (see for example [56]). However, because the thermal contribution is dominant in continuous or quasi-continuous excitation, the nonlocal NL model presents better results when compared to the experimental curves, as shown in Fig. 4. Comparisons between the local and nonlocal models to fit the experimental curves are described in the Supplementary Material.

The IC-scan curves, shown in Fig. 4(c), were obtained by calculating the maximum value of \({g}_{self}^{\left(2\right)}\) as a function of the sample position, when a light diffuser is used as WDS in the far field (5 cm before the CCD). Profiles similar to those of Z-scan, with a peak-valley structure, are observed in the IC-scan curves, starting with \({g}_{self, max}^{\left(2\right)}=2\) in the linear regime and increasing (decreasing) with beam self-focusing (self-defocusing) inside the NL medium [57]. To validate the experimental IC-scan results, numerical simulations were performed using the \({n}_{2}^{ethanol}=-2.2\times 1{0}^{-8} {\text{cm}}^{\text{2}}/\text{W}\) and \(m=0.1\), which coincides with the values obtained by D4σ (also being close to those of Z-scan). The solid lines in Fig. 4(c) reveal the good agreement between the experimental results and the numerical simulation of well-characterized transparent NL media.

In a simpler way, the NL refractive index in IC-scan can also be obtained by using an *external reference method*, provided that the NL parameters (\(m\) and \({n}_{2}\)) of a reference material are known. For instance, Fig. 4(d) shows the evolution of \(\varDelta {g}_{self, max}^{\left(2\right)}\), from peak to valley, as a function of the incident intensity for pure ethanol. A linear behavior with slope \({S}_{ethanol}=\left(2.55\pm 0.05\right)\times {10}^{-2}{\text{cm}}^{\text{2}}/\text{kW}\) is obtained for intensities up to 42 kW/cm2. Assuming that \({n}_{2}\) is known for pure ethanol (reference material), similar to the Z-scan and D4σ techniques, where \(\varDelta {T}^{p-v}\propto \varDelta {\varphi }^{NL}\) and \(\varDelta {m}_{2}^{p-v}\propto \varDelta {\varphi }^{NL}\), respectively, in IC-scan we propose that \(\varDelta {g}_{self, max}^{\left(2\right)}\propto \varDelta {\varphi }^{NL}=k{n}_{2}{L}_{eff}I\). Thus, the NL refractive index for a different material can be obtained by using the relationship: \({n}_{2}^{j}=\left({S}_{j}/{S}_{ref}\right){n}_{2}^{ref}\), where the subscripts *ref* and *j* represent the reference and the new NL media, respectively. This proposal was applied to the study of the NL refraction of pure methanol, which also exhibits a linear dependence of \(\varDelta {g}_{self, max}^{\left(2\right)}\) on the incident intensity (see Fig. 2S). By using \({S}_{methanol}=\left(3.60\pm 0.04\right)\times {10}^{-2}{\text{cm}}^{\text{2}}/\text{kW}\), calculated from the linear fit of the experimental results, and the \({n}_{2}^{ethanol}\) measured by IC-scan, it is possible to find \({n}_{2}^{methanol}=-\left(3.1\pm 0.3\right)\times 1{0}^{-8} {\text{cm}}^{\text{2}}/\text{W}\), which is very close to the \({n}_{2}\) values obtained by Z-scan \(\left(-\left(3.3\pm 0.2\right)\times 1{0}^{-8} {\text{cm}}^{\text{2}}/\text{W}\right)\) and D4σ \(\left(-\left(2.7\pm 0.4\right)\times 1{0}^{-8} {\text{cm}}^{\text{2}}/\text{W}\right)\), demonstrating the reliability of the IC-scan technique. The \({n}_{2}^{methanol}\) calculated by the *external reference method* coincides with the value obtained from the numerical fit \(-\left(3.0\pm 0.2\right)\times 1{0}^{-8} {\text{cm}}^{\text{2}}/\text{W}\) described in the Numerical simulation section (see Fig. 2S of the supplementary material).

It is important to mention that the expression to calculate \({n}_{2}^{j}\), using the *external reference method*, is modified by a multiplicative factor when the reference and new NL material do not present the same nonlocality factor, *m*. In this work, the dependence of \(\varDelta {g}_{self, max}^{\left(2\right)}\) on *m* can be calculated numerically, as was done in previous works for Z-scan [51, 52]. However, new studies are being developed to analytically describe IC-scan curves.

## NL refractive optical characterization of turbid media

Although IC-scan can be used to characterize the NL refractive response of transparent media, the advantages of IC-scan over other techniques become relevant when measuring the NL response of scattering media. First, colloids containing SiO2 NPs suspended in ethanol were prepared as described in the NL media section. Since Rayleigh scattering is more predominant in the blue spectral region, large NP’s volume fractions were needed to induce from weak to moderate scattering at 788 nm. For instance, Fig. 5(a) illustrates a good signal-to-noise ratio in the CA Z-scan and D4σ curves for *f* = 8.2 × 10− 3 and *I* = 22.2 kW/cm2. The experimental curves were fitted using the nonlocal nonlinearity model, obtaining as result a NL refractive index that coincides with that of pure ethanol \(\left({n}_{2}^{ethanol}\right)\). The analysis indicates that, under the excitation conditions used here, SiO2 NPs play the role of light scatterers with negligible nonlinearity, i.e. \({n}_{2}^{Si{O}_{2}-colloid}=\left(1-f\right){n}_{2}^{ethanol}=99\%\left({n}_{2}^{ethanol}\right)\).

In addition, the experimental curves show that the higher the NPs concentration (*f* = 4.1 × 10− 2), the lower the signal-to-noise ratio and, consequently, the greater the deviation against the theoretical (without dispersion) model, as shown in Fig. 5(b). A compilation of the effective NL phase shift, \(\varDelta {\varphi }_{eff}^{NL}\), measured by CA Z-scan and D4σ for pure ethanol and colloids containing SiO2 NPs can be seen in Fig. 5(c) and Fig. 5(d), respectively. Notice, in both techniques, that \(\varDelta {\varphi }_{eff}^{NL}\) for pure ethanol and the colloid with *f* = 8.2 × 10− 3 are very close to each other, in agreement with Fig. 5(a), for intensities up to 40 kW/cm2. Nevertheless, for *f* = 4.1 × 10− 2, the slope of the \(\varDelta {\varphi }_{eff}^{NL}\) curve as a function of \(I\) for CA Z-scan (D4σ) is 2.6 (2.2) times less than that of pure ethanol, contradicting what is expected through \({n}_{2}^{Si{O}_{2}-colloid}=\left(1-f\right){n}_{2}^{ethanol}=96\%\left({n}_{2}^{ethanol}\right)\). In the latter case, it is evident that the scattering is responsible for causing distortions in the profile of the CA Z-scan and D4σ curves, leading to inadequate measurements of the NL refractive index in turbid media. Table 1 shows the \({n}_{2}^{Si{O}_{2}-colloid}\) obtained for both techniques using the nonlocal nonlinearity model.

On the other hand, the IC-scan curves show greater robustness against the scattering caused by SiO2 NPs, even for larger concentrations, as shown in Fig. 5(e) and 5(f). For visualization purposes, all IC-scan curves were set to start at \({g}_{max}^{\left(2\right)}\left(z=-20 \text{mm}\right)=2.0\), although the curves show a decrease in the maxima of the correlation functions due to contrast decrease caused by particle-induced light scattering. The vertical shift does not modify the results obtained since the measurement of \({n}_{2}^{Si{O}_{2}-colloid}\) by using the *external reference method* is based on the analysis of the intensity dependence of \(\varDelta {g}_{self, max}^{\left(2\right)}\). Figure 5(g) shows that the slope of the \(\varDelta {g}_{self, max}^{\left(2\right)}\) curves for the different concentrations of SiO2 NPs behaves in a similar way to that observed in \(\varDelta {\varphi }_{eff}^{NL}\) for Z-scan and D4σ. As expected, similar values of \({n}_{2}\) were found for pure ethanol and the colloid with *f* = 8.2 × 10− 3. Meanwhile, for the colloid with *f* = 4.1 × 10− 2, the \({n}_{2}^{Si{O}_{2}-colloid}\) was 1.7 times lower than that of pure ethanol, as reported in Table 1. Although the \({n}_{2}^{Si{O}_{2}-colloid}\) obtained by IC-scan for the largest concentrations differs from the expected theoretical value \(\left(96\%{n}_{2}^{ethanol}\right)\), its accuracy is higher than that of the Z-scan and D4σ techniques.

An adaptation in the analysis methodology of the speckle patterns captured by the CCD, in the IC-scan technique, allows to achieve more exact measurements of the NL refractive index in strong-scattering media. For this purpose, instead of the self-correlation function, it is proposed to analyze the 2D spatial intensity cross-correlation function \(\left({g}_{cross}^{\left(2\right)}\left(\varDelta r\right)=\frac{⟨\int {d}^{2}r{I}_{1}\left(r\right){I}_{2}\left(r+\varDelta r\right)⟩}{\int {d}^{2}r⟨{I}_{1}\left(r\right)⟩⟨{I}_{2}\left(r+\varDelta r\right)⟩}\right)\) between the far-field intensity transverse profiles (*I*1 and *I*2) induced in two different regimes. The first regime is excited at low incident intensities (*I* = 0.1 kW/cm2), where linear scattering effects exist but refractive nonlinearities are negligible. Meanwhile, in the second regime, the incident intensities (*I* > 1.0 kW/cm2) are high enough to excite both linear and NL effects. Therefore, the cross-correlation function allows to analyze the statistical properties of the speckle patterns that were modified only by NL refraction effects.

The black curves in Fig. 5(e) and 5(f) show the new IC-scan profiles for SiO2 colloids obtained by analyzing the maximum values of the \({g}_{cross}^{\left(2\right)}\) as a function of the sample position. It is important to mention that the 2D spatial intensity cross-correlation function was also calculated from 50 consecutive images captured for *I*1 (linear regime) and *I*2 (NL regime). Notice from Fig. 5(h) that the \(\varDelta {g}_{cross, max}^{\left(2\right)}\) between the peak and the valley for pure ethanol and the SiO2 colloid with *f* = 8.2 × 10− 3 evolve in a similar way with the increase of the incident intensity, in accordance with the other techniques. Even more interesting is that for intensities up to 15 kW/cm2, the IC-scan technique using the cross-correlation function is the only methodology that, as expected, shows that the slope of the \(\varDelta {g}_{cross, max}^{\left(2\right)}\) curve for the colloid with *f* = 4.1 × 10− 2 is close to that of pure ethanol. As a result, \({n}_{2}^{Si{O}_{2}-colloid}=-\left(2.1\pm 0.1\right)\times 1{0}^{-8} {\text{cm}}^{\text{2}}/\text{W}\) is obtained for the most concentrated SiO2-colloid, corresponding to ~ 96% of the value obtained for pure ethanol, as indicated in Table 1. The results reveal the potential of the IC-scan technique to remove the contribution of linear scattering in the analysis of intensity cross-correlations, allowing a correct measurement of the NL refractive index in turbid media.

For *I* > 15 kW/cm2, it is observed that for the colloid with *f* = 4.1 × 10− 2, \(\varDelta {g}_{cross, max}^{\left(2\right)}\) also deviates significantly from the values found for pure ethanol, indicating the contribution of some new NL phenomenon that influences the characterization of the NL refractive behavior. To understand the origin of the change in the slope of the \(\varDelta {g}_{cross, max}^{\left(2\right)}\) versus *I* curve, experiments to characterize the behavior of the scattered light intensity with the increase of the laser intensity were performed. In these experiments, a cell with 1.0 mm thickness, containing SiO2 colloids, was located in the focus of a 10 cm lens, identical to that used in the Z-scan, D4σ and IC-scan experiments. The scattered light was collected in a direction nearly perpendicular to the propagation direction of the incident laser beam by using a microscope objective, a plano-convex lens and a photodetector, as schematized in Fig. 5(i).

Figure 5(j) shows the dependence of the scattered light intensity (at 788 nm) with the incident laser intensity for the SiO2 colloids. Notice that for *f* = 8.2 × 10− 3, the scattered light intensity (*I*scat) presents a linear behavior (red line) versus the incident intensity that extends up to ~ 40 kW/cm2. However, similar to Fig. 5(h) for *f* = 4.1 × 10− 2, *I*scat exhibits a significant deviation from the linear behavior for *I* > 15 kW/cm2. This NL scattering contributions can be understood from the Rayleigh-Gans model [58], by expressing the scattering coefficient as: \({\alpha }_{scat}={g}_{s}{\left(\varDelta n\right)}^{2}\), where \(\varDelta n\) represents the difference between the effective refractive indices of the NP and the host medium, and \({g}_{s}\) is an intensity-independent parameter, but depends on the size, shape and concentration of the NPs and the optical wavelength. By considering the NL refractive behavior of the colloids \(\left(\varDelta n=\varDelta {n}^{L}+\varDelta {n}_{2}^{eff}I\right)\), it is possible to find expressions for the linear \(\left({\alpha }_{scat}^{L}={g}_{s}{\left[{\varDelta n}_{L}\right]}^{2}\right)\) and NL \(\left({\alpha }_{scat}^{NL}=2{g}_{s}{\varDelta n}_{L}{\varDelta n}_{2}\right)\) scattering coefficients, with \({\alpha }_{scat}={\alpha }_{scat}^{L}+{\alpha }_{scat}^{NL}I\). Since the NL contribution of the SiO2 NPs was considered small compared to the solvent, \({\varDelta n}_{2}\) corresponds mainly to the NL refractive index of ethanol, which became significant for higher intensities. Thus, as shown in Table 1, \({\alpha }_{scat}<0\), decreasing the linear scattering coefficient for high intensities and corroborating the results of Figs. 5(h) and 5(j). Therefore, in addition to the IC-scan technique allowing scattering-free NL refraction measurements, it also has the ability to distinguish linear and NL scattering contributions.

A similar study was performed with Au-NRs colloids, where the nonlinearity is dominated by the thermal response of the nanoparticles. However, due to the NR’s dimensions, a relevant contribution from linear scattering is present in the *l*-LSP band [59], which makes the Au-NRs behave as both scatterers and NL particles. This dual behavior of the Au-NRs makes its NL characterization, using the Z-scan technique, suffer from scattering-induced wavefront distortions that cause erroneous NL refractive index measurements. In fact, Figs. 6(a)-6(c) exhibit CA Z-scan curves whose signal-to-noise ratio decreases drastically with increasing volume fraction. An even more critical result is that \({\varDelta T}_{p-v}^{Z-scan}\) for the Au-NRs colloid with *f* = 7.5 × 10− 5 is less than for the more dilute colloids. By fitting the experimental curves using the nonlocal nonlinearity model (green solid lines), NL refractive indices that do not obey monotonic growth as a function of Au-NR’s concentration are obtained, as shown in Table 2. Thus, the NL characterization of these scattering media, by using the Z-scan technique, contradicts the expected optical behavior in effective media theories whose nonlinearity is dominated by the NP’s response. A clear example is the Maxwell-Garnett theory [49, 60], where the effective third-order susceptibility is the result of the contributions of the host medium and the NPs susceptibilities, weighted through the volume fraction. Figures 6(d)-6(f) show the D4σ curves for Au-NR colloids where, despite exhibiting a better signal-to-noise ratio, \(\varDelta {m}_{2}^{p-v}\) also does not grow proportionally to the increase in Au-NR concentration.

In contrast, the IC-scan curves, obtained through the analysis of the self- [Figs. 6(g)-(i)] and cross-correlation [Figs. 6(j)-(l)] functions, show excellent signal-to-noise ratios for all concentrations explored in this work. Furthermore, Fig. 7 exhibits a monotonic linear increase of \(\varDelta {g}_{max}^{\left(2\right)}\propto \varDelta {\varphi }^{NL}=k{n}_{2}{L}_{eff}I\) with incident intensity, as expected for both IC-scan configurations. Regarding the concentration dependence of the NL refractive index, measurements using the CA Z-scan [Figs. 6(a)-(c)] or self-correlation IC-scan [Fig. 7(a)] show that for larger volume fractions, \(\varDelta {\varphi }^{NL}\) decreases or saturates, respectively, due to strong scattering. Nevertheless, when the IC-scan technique is applied using the cross-correlation functions [Fig. 7(b)], \(\varDelta {g}_{cross,max}^{\left(2\right)}\) presents a linear behavior with the NP’s volumetric fraction, preserving the validity of models such as the Maxwell-Garnett one to study the NL response of composite media. Therefore, the studies with Au-NR’s colloids reinforce the potential of the cross-correlation IC-scan technique to measure the NL refractive indices of turbid media.