## Abstract

We report spatial self- and cross-phase modulation effects in metal-dielectric nanocomposites (MDNCs) whose nonlinear (NL) response is dominated by the quintic or septic refractive nonlinearity. The MDNCs consist of silver nanoparticles (NPs) suspended either in acetone or carbon disulfide and their effective NL susceptibilities are controlled by adjusting the volume fraction occupied by the NPs and the incident laser intensity. A theoretical treatment based on the Maxwell-Garnett model was developed to include contributions up to the seventh-order susceptibility showing a very good agreement with the experimental results.

© 2014 Optical Society of America

## 1. Introduction

Metal-dielectric nanocomposites (MDNCs) is attracting large interest due to their high nonlinear (NL) optical susceptibility, fast response and the possibility of controlling the optical behavior changing the volume fraction, *f*, of the metallic nanoparticles (NPs). Glasses, polymers and liquids containing metallic NPs are illustrative examples of MDNCs of interest for photonics [1–10]. The NL response of a MDNC is described by effective susceptibilities containing information on the host and the NPs. Therefore, the induced polarization of a MDNC is described by a power series in the optical field having the effective susceptibilities, ${\chi}_{eff}^{(2N+1)}$, $N=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}\dots $, as coefficients of the expansion. Up to now the majority of studies were focused on ${\chi}_{eff}^{(3)}$ in centro-symmetric MDNCs [7–14].

Nowadays, due to the development of short pulse lasers it is possible to use large optical fields without destruction of samples and the contributions of high-order nonlinearities (HON) become detectable. Due to the mismatch between the dielectric function of the NPs and the host, there is an enhancement of the electromagnetic field that depends on the laser frequency and its detuning from the NPs localized surface plasmon resonance (LSPR). Therefore, high values of the effective susceptibilities are observed that may favor detection of effects related to HON either associated to direct NL processes or due to cascade processes [15–20].

Control of the MDNCs’ effective susceptibilities changing the volume fraction, *f*, was reported for colloids with metallic NPs [14–16] and it was demonstrated a way to cancel the third-order refractive index, ${n}_{2}\propto \mathrm{Re}{\chi}_{eff}^{(3)}$, to obtain a system with non-nulled ${n}_{4}\propto \mathrm{Re}{\chi}_{eff}^{(5)}$ [21].

In this paper we apply the procedure of [21] and report the investigation of spatial self-phase modulation (SPM) and spatial cross-phase modulation (XPM) in MDNCs presenting quintic and septic refractive nonlinearities, in the absence of contributions due to the lower order NL refractive indices. The experimental results were corroborated with calculations based on a generalization of the NL Maxwell-Garnett (MG) model [22].

Section 2 describes the samples preparation and the experimental setups used. Section 3 presents a generalization of the MG model to include high order contributions and experiments of SPM and XPM based on HON are reported. By changing the value of *f* and the laser intensity it was possible to demonstrate nonlinearity management of the MDNCs studied. Section 4 presents a summary of the results.

## 2. Experimental details

Colloids containing silver NPs suspended in acetone and carbon disulfide (CS_{2}) were prepared by chemical reduction methods, following the procedures of [23] and [14], respectively. Sample A (Ag NPs in acetone) was obtained by diluting 90 mg of AgNO_{3} in 500 ml of water at 100°C; 10 ml of 1% sodium citrate solution was added for reduction the Ag^{+} ions, and later was boiled and strongly stirred for 1 h. The preparation of sample B (Ag NPs in CS_{2}) consisted of adding dropwise 3.75 ml of a 0.03 mol/l AgNO_{3} aqueous solution to a 0.05 mol/l N(C_{8}H_{17})_{4}Br solution in toluene at 100 drops/min. After 10 min, 50 μl of dodecanethiol (dodecanethiol-to-silver molar ratio of 2:1) were introduced in the mixture. Then 3.1 ml of a freshly prepared NaBH_{4} aqueous solution (0.4 mol/l) was rapidly added to the mixture. The reacting medium was stirred for 3 h, and the organic layer was extracted. The resulting dodecanethiol-stabilized Ag NPs were precipitated by adding ethanol and cooling to −18 °C for 4 h. Finally, the NPs were centrifuged, washed several times with ethanol and redispersed in toluene. Homogeneous distributions of spherical NPs with average diameter of (9.0 ± 2.2) nm and (6.0 ± 3.0) nm for sample A and sample B, respectively, were obtained after laser ablation of the pristine colloids for 1 h using a Nd: YAG laser (532 nm, 8 ns, 85 mJ/pulse, 10 Hz) following the method of [24].

Colloids with *f* varying from $0.5\times {10}^{-5}$ to $2.5\times {10}^{-4}$ for sample A, and from $0.2\times {10}^{-5}$ to $4.5\times {10}^{-5}$ for sample B, were obtained by adding 20 µl to 300 µl of the Ag-water suspension in 1 ml of acetone, and adding 2 µl to 40 µl of the Ag-toluene suspension in 1 ml of CS_{2}, respectively.

The linear absorption spectra of the samples were measured from 200 nm to 800 nm using a commercial spectrophotometer. The light source for the NL measurements was the second harmonic of a Q-switched and mode-locked Nd: YAG laser (80 ps, 532 nm). Single pulses at 10 Hz were selected using a pulse-picker.

Effective NL refractive indices and NL absorption coefficients were measured using the Z-scan technique [25]. The laser beam was focused by a 10 cm focal distance lens (confocal length: 4.7 mm) on a sample with thickness of 1 mm, contained in a quartz cell. Slow photodetectors placed in the far-field region with adjustable apertures in front of them were used to measure the beam intensity transmitted by the sample. The aperture radius, *r _{a}*, is related to its transmittance by $S=1-\mathrm{exp}\left(-2{r}_{a}^{2}/{w}_{a}\right)$, with

*w*being the beam radius at the aperture positions.

_{a}*Closed-aperture*(

*CA*), $S<1$, and

*open-aperture*(

*OA*), $S=1$, schemes were used to determine the NL refractive indices and the NL absorptive coefficients. A reference channel was used to improve the signal-to-noise ratio as in [26].

SPM experiments were performed by focusing the laser beam with a 2 cm focal length lens, producing a beam waist of 7 µm at the focus. The far-field diffraction patterns were captured by a CCD camera located $\sim 20\text{\hspace{0.17em}}cm$ from the exit plane of the sample (cell thickness: 1 mm). For the XPM experiments, counter-propagating pump and probe beams were used with a 5 cm long cell. A CCD camera, located $\sim 5\text{\hspace{0.17em}}cm$ from the exit plane of the sample, monitored the transverse beam profile.

## 3. Results and discussion

#### 3.1 Generalized Maxwell-Garnett model

To understand the experimental results we developed an extension of the Maxwell-Garnett model [22] including the contributions of ${\chi}_{eff}^{(3)}$, ${\chi}_{eff}^{(5)}$ and ${\chi}_{eff}^{(7)}$. The colloid is supposed to be homogeneous, $f\ll 1$, the NPs are spherical and their diameter, *a*, is smaller than their relative distance, *b*. The light wavelength, $\lambda $, satisfies $\lambda >b>a$. The optical field, ${E}_{0}$, is considered uniform on each particle. Under these conditions the optical polarization, in the quasi-static approximation, can be written as $P={P}_{h}+\frac{1}{V}{\displaystyle \sum _{i=1}^{{N}_{p}}{p}_{i}}$, where ${P}_{h}$ is associated to the host, ${N}_{p}$ is the number of NPs inside the volume *V* and ${p}_{i}={\epsilon}_{h}{\sigma}_{i}{E}_{0}$ is the induced dipole moment of each NP; ${\sigma}_{i}=3{v}_{i}\beta $ is the NP polarizability, with $\beta =\left(\frac{{\epsilon}_{np}-{\epsilon}_{h}}{{\epsilon}_{np}+2{\epsilon}_{h}}\right)$, where ${\epsilon}_{np}$ $\left({\epsilon}_{h}\right)$ is the dielectric function of the NPs (host) and ${v}_{i}$ is the NP volume; $\beta $ depends on $\left|{E}_{0}\right|$ through ${\epsilon}_{np}$ and ${\epsilon}_{h}$. Hence, the optical polarization is given by:

*i*-th susceptibility of the NPs and ${\chi}_{h}^{(3)}$ belongs to the host, $\u3008{\left|{E}_{0}\right|}^{2}\u3009$ correspond to the mean square modulus of the applied electric field. The contributions of ${\chi}_{h}^{(2N+1)}$, $N\ge 2$, were neglected considering the experimental results herein reported. The numerical coefficients of Eqs. (2) and (3) correspond to the degeneracy factors for the

*i*-th order process in the convention of [27]. Due to the small value of

*f*the intensity dependent susceptibility is written as $\chi \left({\left|{E}_{0}\right|}^{2}\right)={\chi}_{h}+3\beta f$ where the NL terms in

*f*were neglected because: ${\left(\beta f\right)}^{n}\ll 1+\beta f$, $n\ge 2$. The mean squared modulus of the electric field inside the NP is represented by $\u3008{\left|{E}_{np}\right|}^{2}\u3009={\left|\eta \right|}^{2}{\left|{E}_{0}\right|}^{2}$ and $\eta =3{\epsilon}_{h}/\left({\epsilon}_{np}+2{\epsilon}_{h}\right)$ is the local field factor.

The effective NL susceptibilities of different orders are determined by performing two series expansions in terms of ${\left|{E}_{0}\right|}^{2}$ and ${\left|{E}_{np}\right|}^{2}$. The first expansion consists in expressing $\chi \left(y\right)$ as a Taylor series up to third order in $y={\left|{E}_{0}\right|}^{2}$ that assumes the form

In order to obtain an expression of the electric field inside the NPs in terms of ${E}_{0}$, a second expansion was performed for ${\left|\eta \right|}^{2}$ as function of $z=\sqrt{\u3008{\left|{E}_{np}\right|}^{2}\u3009}$ written as

The expansion of $\chi $ as a Taylor series up to third order in ${z}^{2}$ is given by

Expressing the effective dielectric function of the MDNC as ${\epsilon}_{eff}={\epsilon}_{h}+{\displaystyle \sum _{n-odd}\frac{n!\text{\hspace{0.17em}}\text{\hspace{0.17em}}{2}^{1-n}}{\left[\left(n-1\right)/2\right]!\left[\left(n+1\right)/2\right]!}{\chi}_{eff}^{(n)}{\left|{E}_{0}\right|}^{n-1}}$, we obtain the effective susceptibility as a function of ${\left|{E}_{0}\right|}^{2}$ that can be written, according to [27], as

Finally, by comparison of Eqs. (11) and (12), using the coefficients given by Eqs. (5)-(7) , we obtain expressions for the effective third-, fifth- and seventh-order susceptibilities as

It can be seen from Eq. (13) that ${\chi}_{eff}^{(3)}$ may be cancelled adjusting *f* and the result is independent of the laser intensity; however, $\mathrm{Re}{\chi}_{eff}^{(3)}$ and $\mathrm{Im}{\chi}_{eff}^{(3)}$ are canceled for different *f* values. Equation (14) and (15) indicate that the NPs contribution to ${\chi}_{eff}^{(5)}$ and ${\chi}_{eff}^{(7)}$ are due to ${\chi}_{np}^{(3)}$, ${\chi}_{np}^{(5)}$, and ${\chi}_{np}^{(7)}$, with different powers. Observe that the contributions of the NPs susceptibilities are enhanced due to the high powers of $L$ and $\left|L\right|$. It is important to notice that in the case of fifth- and seventh-order effective susceptibilities, we have to vary *f* and the incident intensity, to determine the conditions for destructive interference between the susceptibilities of different orders. We remark that $\chi \left({\left|{E}_{0}\right|}^{2}\right)$ is linearly proportional to *f* because terms proportional to ${f}^{n}$, $n\ge 2$, were neglected. This result is corroborated by the experimental measurements reported below corresponding to ${10}^{-6}<f<{10}^{-4}$.

The NL response of the NPs is mainly due to the *s-*electrons because the *d*-band is $\sim 4\text{\hspace{0.17em}}eV$ above the Fermi level and the energy of the incident photons is $2.34\text{\hspace{0.17em}}eV$.

#### 3.2 Z-scan experiments and theoretical analysis

Figure 1(a) and 1(b) show the linear absorbance spectra of samples A and B for $f=4.0\times {10}^{-5}$. The features at $\sim 400\text{\hspace{0.17em}}nm$ are due to the LSPR in the silver NPs. Also shown in Fig. 1 are the spectra of pure acetone and CS_{2} that present large transparency window from the near ultraviolet/visible to the near infrared. The LSPR is broader in the CS_{2} based colloid due to the large chemical interaction of sulfur and the Ag surface; this is a typical result for Ag-CS_{2} colloid with homogeneous NPs size distribution as illustrated by the transmission electron microscope image presented in [14].

Figure 2 shows the Z-scan traces for samples A and B. Figure 2(a) illustrates the results obtained for sample A using the *CA* scheme corresponding to $f=0.8\times {10}^{-5}$, $1.3\times {10}^{-5}$, $2.0\times {10}^{-5}$ and $3.0\times {10}^{-5}$. *OA* Z-scan profiles did not show NL absorption for these volume fractions. However, for $3.3\times {10}^{-5}<f\le 5.3\times {10}^{-5}$ the colloid presented saturated absorption as shown in Fig. 2(b) due to the small detuning between the laser frequency and the LSPR. Z-scan measurements were also performed for sample B with *f* varying from $0.4\times {10}^{-5}$ to $4.5\times {10}^{-5}$ as displayed in Figs. 2(c) and 2(d), respectively.

Sign-reversal of the NL refractive index as a function of *f* can be seen for both samples in the interval $1.3\times {10}^{-5}<f<3.3\times {10}^{-5}$.

Experiments were also performed using different laser intensities. The samples were investigated in order to determine ${n}_{2}$, ${n}_{4}$, and ${n}_{6}$ as well as to the illustrate their behavior for the *f* value that corresponds to ${n}_{2}=0$. Figure 3(a) shows the NL refractive response of sample A for $f=1.6\times {10}^{-5}$ (${n}_{2}=0$ and ${n}_{4}=+3.2\times {10}^{-25}\text{\hspace{0.17em}}c{m}^{4}/{W}^{2}$). For laser intensities up to $4\text{\hspace{0.17em}}GW/c{m}^{2}$ no feature is observed in the *CA* Z-scan trace while for $9\text{\hspace{0.17em}}GW/c{m}^{2}$ a positive effective NL refractive index is observed due to the HON. On the other hand, when $f\ne 1.6\times {10}^{-5}$ and for laser intensity of $\sim 2\text{\hspace{0.17em}}GW/c{m}^{2}$ larger *CA* Z-scan signal, dominated by the negative third-order nonlinearity, is observed [21]. This behavior is illustrated in Fig. 3(b) which shows the Z-scan trace obtained for $4\text{\hspace{0.17em}}GW/c{m}^{2}$. For larger intensities clear indication of HON contribution is observed in the Z-scan traces as illustrated in Fig. 3(b) corresponding to ${n}_{2}=-5.5\times {10}^{-15}$, ${n}_{4}=+2.1\times {10}^{-24}\text{\hspace{0.17em}}c{m}^{4}/{W}^{2}$ and ${n}_{6}=-2.6\times {10}^{-34}\text{\hspace{0.17em}}c{m}^{6}/{W}^{3}$ for $f=5.0\times {10}^{-5}$. The values of ${n}_{2}$, ${n}_{4}$, and ${n}_{6}$ mentioned above were obtained as described below. The results for sample B is analogous to the ones shown in Fig. 3.

The solid lines in Fig. 3 correspond to the normalized laser transmittance, measured in the far field as function of the sample's position, described by [15]

*CA*Z-scan traces, present a polynomial behavior versus

*I*. From these measurements the NL parameters of third and higher order were obtained, as in [15, 16]. A combination of the Levenberg-Marquardt and the least-squares minimum method were used to minimize errors.

For sample B a NL refractive medium with ${n}_{2}=0$, ${n}_{4}=+1.3\times {10}^{-22}\text{\hspace{0.17em}}c{m}^{4}/{W}^{2}$ and ${n}_{6}=-5.7\times {10}^{-31}\text{\hspace{0.17em}}c{m}^{6}/{W}^{3}$ was obtained for $f=1.8\times {10}^{-5}$. The *OA* profiles, shown in Figs. 2(b) and 2(d), correspond to ${\alpha}_{2}=-4.9\times {10}^{-10}\text{\hspace{0.17em}}cm/W$, ${\alpha}_{4}=+1.4\times {10}^{-19}\text{\hspace{0.17em}}{c{m}^{3}/W}^{2}$ and ${\alpha}_{6}=-1.7\times {10}^{-29}\text{\hspace{0.17em}}{c{m}^{5}/W}^{3}$ for sample A with $f=5.0\times {10}^{-5}$; and ${\alpha}_{2}=-1.0\times {10}^{-9}\text{\hspace{0.17em}}cm/W$, ${\alpha}_{4}=+3.9\times {10}^{-18}\text{\hspace{0.17em}}{c{m}^{3}/W}^{2}$ and ${\alpha}_{6}=-1.3\times {10}^{-26}\text{\hspace{0.17em}}{c{m}^{5}/W}^{3}$ for sample B with $f=2.0\times {10}^{-5}$. The results for sample A are the same reported in [21].

Further results for sample B are summarized in Fig. 4. Figure 4(a) illustrates the destructive interference between the ${n}_{2}\propto \mathrm{Re}{\chi}_{eff}^{(3)}$ and ${n}_{4}\propto \mathrm{Re}{\chi}_{eff}^{(5)}$ contributions versus *f*. For $f=3.3\times {10}^{-5}$ we obtain a NL refractive MDNC with ${n}_{6}=-1.1\times {10}^{-30}\text{\hspace{0.17em}}c{m}^{6}/{W}^{3}$ (septic refractive nonlinearity). Analogously, a NL refractive MDNC with ${n}_{4}=+1.1\times {10}^{-22}\text{\hspace{0.17em}}c{m}^{4}/{W}^{2}$ (quintic refractive nonlinearity), is shown in Fig. 4(b) for $f=1.5\times {10}^{-5}$ due to destructive interference between the third- and seventh-order contributions. Figure 4(c) exhibits a NL absorptive behavior with ${\alpha}_{6}=-7.5\times {10}^{-27}\text{\hspace{0.17em}}{c{m}^{5}/W}^{3}$, for $f=1.2\times {10}^{-5}$ and $I=2.5\times {10}^{8}\text{\hspace{0.17em}}W/c{m}^{2}$. Using Eqs. (13)-(15) we obtained the third-order susceptibility of the host, ${\chi}_{h}^{(3)}=2.9\times {10}^{-20}+i\text{\hspace{0.17em}}3.41\times {10}^{-22}\left({m}^{2}/{V}^{2}\right)$in agreement with [25, 29], and the NL susceptibilities for the NPs given by ${\chi}_{np}^{(3)}=-6.4\times {10}^{-16}+i\text{\hspace{0.17em}}2.0\times {10}^{-16}\left({m}^{2}/{V}^{2}\right)$, ${\chi}_{np}^{(5)}=-1.37\times {10}^{-33}-i\text{\hspace{0.17em}}2.2\times {10}^{-33}\left({m}^{4}/{V}^{4}\right)$ and ${\chi}_{np}^{(7)}=-4.1\times {10}^{-51}+i\text{\hspace{0.17em}}5.1\times {10}^{-51}\left({m}^{6}/{V}^{6}\right)$ for the sample B. Notice that the value of ${\chi}_{np}^{(3)}$ is in agreement with [14] that was determined in the absence of HON. We recall that the silver NPs susceptibility depends on the stabilizing agents attached to the NPs surface as well as the host solvent [30, 31]. It is important to observe that ${n}_{2}$, ${n}_{4}$ and ${n}_{6}$ depend linearly with *f* as obtained in Eqs. (13)-(15). Terms proportional to ${f}^{2}$ due to cascade contributions [18] were not observed in the presents experiments.

#### 3.3 Spatial self-phase modulation (SPM) in a colloid with quintic refractive nonlinearity

SPM is a phenomenon due to interaction of a laser beam with the NL medium inducing an intensity dependent refractive index that is spatially modulated. Focusing/defocusing and conical diffraction due to SPM were observed for various materials (see for instance [32–40]). For high intensities SPM is influenced by HON and the simultaneous presence of third- and fifth-order nonlinearities allow the propagation of spatial solitons in liquids [41].

Here, to investigate the SPM effects in a quintic composite (${n}_{2}={n}_{6}=0$;${n}_{4}\ne 0$), we examined the transverse beam profile in the far-field region after propagation through the NL samples. Figures 5(a)-5(c) show diffraction patterns observed in the beam transmitted by sample A (with $f=1.6\times {10}^{-5}$) due to the SPM effect. The *f* value used corresponds to a medium having NL response dominated by $\mathrm{Re}{\chi}_{eff}^{(5)}$ with $\mathrm{Re}{\chi}_{eff}^{(3)}=0$. The diffraction patterns observed in the far-field consist of concentric rings that vary in number, thickness and positions depending on the fifth-order NL phase-shift, $\Delta {\Phi}_{0}^{\left(5\right)}$, and laser intensity.

To describe the conical diffraction we used the Fraunhofer approximation of the Fresnel-Kirchhoff diffraction integral, given by

*R*is the wavefront radius of curvature,

*w*is the beam radius at the entrance plane of the NL medium, and

_{p}*D*is the distance from the exit plane of the cell and the far-field detection plane.

Figures 5(d)-5(f) show the intensity distribution versus the radial coordinate (black line) obtained from CCD images processed as intensity matrices. The red lines correspond to the numerical results obtained from Eq. (17), considering the NL phase-shift due to ${n}_{4}$ that was obtained from the Z-scan experiments. A low-intensity background is observed in the experimental profile due to linear light scattering. Notice that the number of rings, their thickness and the spacing are in good agreement with the numerical results.

Figures 5(g) and 5(h) illustrate the SPM effect in sample A for $f=1.6\times {10}^{-5}$. Notice that different diffraction patterns are observed when the sample is located of 0.3 mm before $\left(R<0\right)$ and 0.3 mm after $\left(R>0\right)$ the focal plane, respectively. The curves in Figs. 5(i)-5(j) show that the analysis of the experimental images (black line) are consistent with the theoretical result (red line), for $R\approx \pm \text{\hspace{0.17em}}0.4\text{\hspace{0.17em}}mm$. The present experimental results are in agreement with the numerical results of [32] that predicted a far-field diffraction pattern formation that depends on the sign of the product between the wavefront curvature radius and the NL phase-shift induced in the sample. It is important to notice that diffraction patterns, due to pure ${\chi}_{eff}^{(5)}$, are reported here for the first time, presenting good agreement with theoretical predictions.

#### 3.4 Spatial cross-phase modulation (XPM) experiment in a colloid with septic nonlinearity

To investigate the XPM due to the septic refractive index (${n}_{2}={n}_{4}=0$;${n}_{6}\ne 0$) the laser beam was split into pump and probe beams with beam's waist of ${w}_{0}=100\text{\hspace{0.17em}}\mu m$ (Rayleigh length $\sim 6\text{\hspace{0.17em}}cm$) in the entrance face of the sample. The two beams propagate in opposite directions along sample B (cell length: 5 cm). The pump beam intensity ${I}_{pump}={10}^{8}\text{\hspace{0.17em}}W/c{m}^{2}$ and $f=3.3\times {10}^{-5}$ were chosen; the probe-to-pump intensities ratio was 1:10 to ensure that the NL phase shift is due to the pump beam only.

Figure 6 shows the pump beam profile after propagation through sample B with $f=3.3\times {10}^{-5}$. Figure 6(a) exhibits the beam profile at the far-field region for ${I}_{pump}={10}^{6}\text{\hspace{0.17em}}W/c{m}^{2}$, while Fig. 6(b) shows the spatial broadening by a factor of $\sim 2$, for ${I}_{pump}={10}^{8}\text{\hspace{0.17em}}W/c{m}^{2}$, due to ${n}_{6}=-1.1\times {10}^{-30}\text{\hspace{0.17em}}c{m}^{6}/{W}^{3}$. Figure 6(c) shows the intensity distribution of the pump beam transverse profile (black lines), obtained from Figs. 6(a)-6(b). The red and blue lines correspond to the calculated profile for pump intensity of ${10}^{6}$ and ${10}^{8}\text{\hspace{0.17em}}W/c{m}^{2}$, respectively, using the Eq. (18) considering ${E}_{2}=0$.

The XPM effects due to ${\chi}_{eff}^{(7)}$ were also studied by changing the relative distance $\left(x/{w}_{0}\right)$ between the centers of the incident pump and probe beams. Figures 7(a)-7(c) shows the experimental probe beam profiles after propagation through sample B, with $f=3.3\times {10}^{-5}$, in presence of pump beam, ${I}_{pump}={10}^{8}\text{\hspace{0.17em}}W/c{m}^{2}$, for different values of $\left(x/{w}_{0}\right)$. The pink and white lines represent the positions of incident probe and pump beams, respectively. Figures 7(d)-7(f) show the intensity profile of the probe beam without the pump beam (red dashed lines) and in the presence of the pump beam (black lines), obtained from the images shown in Figs. 7(a)-7(c). Notice in particular a partial focusing of the probe beam induced by the pump beam although the sample presents negative value of ${n}_{6}$. This effect is analogous to the third-order induced focusing effect reported in [39] where the sample presented a self-defocusing behavior at the laser frequency used.

The mathematical description of the XPM experiment is provided by the coupled NL Schrödinger equations, Eqs. (18) and (19), considering two counter-propagating beams. Using the slowly varying approximation, the equations are

*f*, is obtained from Eqs. (13)-(15).

Figures 7(g)-7(i) show the theoretical probe beam transverse profile, obtained from Eqs. (18) and (19), using the NL parameters measured in the Z-scan experiments (${n}_{2}={n}_{4}=0$;${n}_{6}=-1.1\times {10}^{-30}\text{\hspace{0.17em}}c{m}^{6}/{W}^{3}$) when $f=3.3\times {10}^{-5}$, ${I}_{probe}={10}^{7}\text{\hspace{0.17em}}W/c{m}^{2}$ and ${I}_{pump}={10}^{8}\text{\hspace{0.17em}}W/c{m}^{2}$. A good agreement is observed between experimental and theoretical results that support the given interpretation.

## 4. Conclusion

In summary, we reported conditions for obtaining a MDNC presenting either fifth- or seventh-order refractive index, adjusting the volume fraction occupied by silver NPs with respect to the host volume and the incident laser intensity. Spatial self- and cross-phase modulation were demonstrated due to the effective fifth- and seventh-order nonlinearities and their characterization were reported. The experimental procedure to control the dominant nonlinearity was supported by the Maxwell-Garnett model that was generalized to include effects of fifth- and seventh-order.

## Acknowledgments

We acknowledge financial support from the Brazilian agencies Conselho Nacional de Desenvolvimento Cientifico e Tecnológico (CNPq) and the Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco (FACEPE). The work was performed in the framework of the National Institute of Photonics (INCT de Fotônica) project and PRONEX/CNPq/FACEPE.

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