## Abstract

High-energy metallic ions were implanted in silica matrices, obtaining spherical-like metallic nanoparticles (NPs) after a proper thermal treatment. These NPs were then deformed by irradiation with Si ions, obtaining an anisotropic metallic nanocomposite. An average large birefringence of 0.06 was measured for these materials in the 300-800 nm region. Besides, their third order nonlinear optical response was measured using self-diffraction and P-scan techniques at 532 nm with 26 ps pulses. By adjusting the incident light’s polarization and the angular position of the nanocomposite, the measurements could be directly related to, at least, two of the three linear independent components of its third order susceptibility tensor, finding a large, but anisotropic, response of around 10^{−7} esu with respect to other isotropic metallic systems. For the nonlinear optical absorption, we were able to shift from saturable to reverse saturable absorption depending on probing the Au NP’s major or minor axes, respectively. This fact could be related to local field calculations and NP’s electronic properties. For the nonlinear optical refraction, we passed from self-focusing to self-defocusing, when changing from Ag to Au.

©2009 Optical Society of America

## 1. Introduction

The flourishing of plasmonics nowadays has attracted growing interest on metallic nanoparticles (NPs) and their optical properties, which are determined by their surface plasmon (SP) resonances and depend on the NP´s geometry and environment. Following this tendency, great efforts have been devoted to tune the optical response of this kind of systems by controlling their size, shape, surroundings, etc [1–11]. For instance, one can cite the quest to achieve what has been called ‘lab-on-a-chip’ platforms [12], consisting on nanoscaled optical components all assembled into a chip; or the search to obtain perfect lenses [13] and hyperlenses [14] to image objects at subdiffraction-limit scales, as happens with the optical nanoantenna [15]. Within the same trend, surfaces displaying omnidirectional absorption have been already obtained [16], leading the way to devising very efficient photovoltaic cells in which the totally absorbed light will allow a high rate of electron-hole pair production. Some other examples are the use of metal nanoparticles for coherent feedback in random lasers [17], the recently proposed nanoscale source of optical fields based on surface plasmons, the so-called spaser [18], or the design of optical tweezers in a subwavelength size, again based on surface plasmons [19].

Very recently, we have reported on the control of the deformation of metallic NPs by Si irradiation, obtaining metallic nanocomposites composed of ellipsoidal NPs aligned along the Si irradiation’s direction and with an aspect ratio dependent on the Si fluence [6]. We also showed that these anisotropic nanosystems show a large form or optical birefringence of around 0.1 for wavelengths close to their SP resonances [20]; now, we are reporting on the birefringence for the 300-800 nm region. However, one of the most promising endowments of these anisotropic systems comes from their nonlinear optical properties, since they have exhibited so far the largest cubic nonlinearities under resonance conditions, which is due to the strengthening of the local field at SP resonance frequencies [21–23], and since they can be implicated in optoelectronics [24] and quantum-information devices [25–32], for example. Nevertheless, a complete understanding of the physical reasons behind the nonlinear optical response of metallic NPs for all the possible scenarios (NPs size and shape, wavelength used, pulse duration, etc.) is still lacking. For instance, although some works have started to clear the path to [33–35], there is not yet an unambiguous trend indicating whether the nanocomposite will show saturable or reverse saturable absorption near resonance [36–47]. In such a direction, studying the physical contribution of the separate components of the third-order susceptibility tensor would add more information about the physics behind the optical response of metallic nanocomposites. In particular, as it will be shown below for nanoellipsoids, this tensor has only three linear independent components when probing with a fully-degenerate wave mixing and two of them can be associated to the major and minor axes of the ellipsoid, respectively [47,48]. In this way, for the wavelength chosen, we will be near or close to the position of the surface plasmon resonance of the metallic NPs, helping to get more elements in order to identify whether there is a hot electron contribution or not.

Therefore, in this work, apart from the birefringence measurements, we obtained, by self-diffraction [49] and P-scan [50] methods at 532 nm and 26 ps, the real and the imaginary parts of the linear independent components of *χ*
^{(3)} for anisotropic Ag- and Au-nanocomposites, which were obtained by controlled deformation with Si irradiation [6]. For the nonlinear optical refraction, we passed from self-focusing to self-defocusing, when changing from Ag to Au. In the case of Au, we could shift from saturable to reverse saturable absorption by probing the Au NP’s major or minor axes, respectively. By following local field calculations presented earlier by Lamarre et al. [34], we could also verify the ratio between the measured real and imaginary parts of the components associated with the major and the minor axes. Finally, the electronic contribution from hot electrons to this nonlinear optical response will be discussed accordingly to the pulse duration, the intensity and the position of the wavelength used in our setup, with respect to the SP resonances of the anisotropic nanocomposites.

## 2. Theoretical analysis

#### 2.1 Birefringence analysis

As it has been shown previously [20], we have used an ellipsometric technique to measure the light transmission through our anisotropic samples when placed and rotated between crossed and parallel polarizers. The mathematical details about the analysis of these ellipsometric measurements are given throughout in [9]. However, we recall the main expressions for the measured intensities when using the experimental setup shown in Fig. 1 , which has been used to obtain the results shown below in Table 1 .

For the transmitted light, when the axes of the polarizer-analyzer system are aligned, it can be shown [9] that the detected intensity at a given in-plane polarization *α* is given by

*A*and

_{s}*A*are the measured amplitude transmission factors for each eigenpolarization,

_{p}*L*is the interaction length, i.e. the thickness of the NPs layer, and

*λ*is the free-space incident wavelength. We will measure these two intensities, obtaining${A}_{p}^{2}$, ${A}_{s}^{2}$ and the birefringence of the nanocomposite as explained into the results section.

#### 2.2 Third order nonlinear polarization for anisotropic metallic nanocomposites

Since we are dealing with anisotropic systems, it is also necessary rewrite the third order nonlinear polarization in terms of the appropriate macroscopic susceptibility tensor, i.e. a tensor that includes such anisotropy. Within the next lines, we will obtain the third order nonlinear polarization for metallic nanocomposites showing uniaxial symmetry, when measuring by means of a totally degenerate wave mixing setup. Such an expression will put this polarization in terms of the nonzero, independent components of the macroscopic susceptibility tensor and, most importantly, in terms of the angular position of the composite.

According to Ref. 47, the third order nonlinear polarization is written in general as

On the other hand, for an uniaxial system, aligned but not oriented (*D _{∞}* symmetry) [48], the susceptibility tensor has only 11 nonzero elements, 10 of which are independent, for a non-degenerate wave mixing. In the case of a single degeneracy, only 8 nonzero elements remain, 7 of which are independent. But, most interestingly, in fully degenerate wave mixing, where ${\omega}_{1}={\omega}_{2}=\omega $ and ${\omega}_{3}=-\omega $, it remains only 3 nonzero, independent components, given by ${\chi}_{1111}^{\left(3\right)}$, ${\chi}_{1133}^{\left(3\right)}$ and ${\chi}_{3333}^{\left(3\right)}$ [48]. In consequence, the nonlinear polarization of a general uniaxial system, for the fully degenerate case, may be written as

Now, in order to determine the components of the susceptibility tensor for an uniaxial material by using the last expression and fully degenerate wave mixing, it is necessary to choose the laboratory coordinate system such that the *z*-axis is parallel to the optical axis of the nanocomposite. However, such a coincidence it is not obvious for the anisotropic metallic nanocomposites studied in this work, since the metallic nanoparticles are embedded into a SiO_{2} matrix and then deformed in the direction of a Si ions beam, becoming an unixial system [6,11,20]. In consequence, the electric field of the incident light beam will always make an angle *θ* to this optical axis. Therefore, the most convenient way of performing the calculations is giving preference to the deformed NP coordinate system, expressing the incident electric field in this frame and then coming back to the laboratory frame. To do this in the simplest manner, one can choose the laboratory and the NP systems such that their *x*-axes coincide, the *y*-axis of the laboratory frame be parallel to the wavevector of the incident light, the electric field be parallel to the *z*-axis of the laboratory, and that this last make an angle *θ* to the NP *z*-axis. This choice is shown in detail in Fig. 2
.

As it was established above, Eq. (5) gives the nonlinear polarization of the all system in its main axes; consequently, for the anisotropic metallic nanocomposite, this polarization may be expressed in the NP frame by rewriting the electric field in that system, coming back later to the laboratory frame. Thus, Eq. (5) may be written in the *xyz*-frame of the NP, as

*y’z’*, the nonlinear polarization is expressed as

Nevertheless, for the type of materials in consideration, it is necessary to take into account the refraction of the light. Then, the simplest is performing one measurement of wave mixing at normal incidence and other one such that the electric field is parallel to the major axis of the NP, if possible due to the refraction; or, such that the angle between them is the smallest possible. In conclusion, one has to perform at least three measurements in order to determine the three nonzero, independent components of the third order susceptibility tensor of an anisotropic metallic nanocomposite: one measurement should be performed at normal incidence with the electric field in the same plane of the major axis of the NP, making an angle between them given by the angle of deformation of the NPs by the Si ion beam (Fig. 3(a) ); a second one, at normal incidence too, but the electric field parallel to the minor axis of the NPs, which is achieved by rotating the sample by 90° with respect to the previous case (Fig. 3(b)); finally, a last measurement such that the electric field and the NP’s major axis are parallel, or the angle between them is the minimum allowed by refraction (Fig. 3(c)). This last measurement is achieved by rotating the sample from case (a) with respect to an axis perpendicular to the plane of Fig. 3.

From what has been said above, one can rewrite Eq. (8) as

*θ*between the NP and the incident electric field, after considering light refraction. This last expression allows us finally to write

## 3. Experimental

#### 3.1 Synthesis and deformation of metallic nanoparticles

As reported before [6,11,20], high-purity silica glass plates were implanted at room temperature with 2 MeV Ag^{2+} (or Au^{2+}) ions at a fluence of (4.7 ± 0.4) × 10^{16} ions/cm^{2} for Ag, and (2.6 ± 0.2) × 10^{16} and (5.0 ± 0.4) × 10^{16} ions/cm^{2} for Au. The Ag ions were implanted at 45.0° ± 0.5° off normal for the nonlinear measurements and at 0° for the birefringence measurements, while the Au ions were implanted at 0° for both cases. The corresponding metal concentrations for these fluences and these angles of implantation are 0.0379 and 0.0268 for Ag, respectively; while for Au are 0.0312 and 0.06, respectively. The depth of the Ag NPs layer was (0.61 ± 0.03) μm with a FWHM of (0.33 ± 0.03) μm for implantation at 45° and (0.90 ± 0.03) μm with a FWHM of (0.45 ± 0.03) μm for implantation at 0°, while for Au the depth was (0.70 ± 0.02) μm and the FWHM was (0.10 ± 0.02) μm. After implantation, the samples were thermally annealed for 1 hr in a reducing atmosphere 50%H_{2} + 50%N_{2} at a temperature of 600°C for Ag. In the case of Au, an oxidizing atmosphere (air) was used for 1 hr at 1100°C. The metal implanted distributions and fluences were determined by Rutherford Backscattering Spectrometry (RBS) measurements using a 3 MeV ^{4}He^{+} beam for Ag and 2 MeV ^{4}He^{+} beam for Au. Afterwards, the silica plate was cut into several pieces and each piece was irradiated at room temperature with 8 MeV Si ions for Ag and 10 MeV Si ions for Au. The Si irradiation was performed under an angle off normal of *θ* = (45.0° ± 0.5°) or of (80.0° ± 0.5°) for both, Ag and Au. Each sample was irradiated at different Si fluences in the range of 10^{15} Si/cm^{2} for Ag and of 10^{16} Si/cm^{2} for Au, in order to induce a shape deformation of the NPs [3]. Ion implantation, RBS analysis and Si irradiation were performed using the IFUNAM’s 3 MV Tandem accelerator NEC 9SDH-2 Pelletron facility.

#### 3.2 Optical measurements

Optical absorption and ellipsometric measurements were performed with an Ocean Optics Dual Channel S2000 UV-visible spectrophotometer. For the ellipsometric technique, we used the setup shown in Fig. 1 and measured the birefringence in the range of 300-800 nm.

The third order nonlinear optical response for a thin nonlinear optical media with strong absorptive response can be obtained by identifying the vectorial self-diffraction intensities generated by two incident waves [49]. In this work, we firstly measured the nonlinear optical absorption using a P-scan technique [50], and later we used these results in order to fit the experimental data obtained by scalar self-diffraction experiments. The measurements were performed at IFUNAM’s Nonlinear Optics laboratory using a Nd-YAG PL2143A EKSPLA system at λ = 532 nm with a pulse duration of 26 ps (FWHM) and linear polarization. The maximum pulse energy in the experiments was 0.1 mJ, while the intensity rate between the two beams of the self-diffraction setup, *I*
_{1}:*I*
_{2}, was 1:1. The radius of the beam waist at the focus in the sample was measured to be 0.1 mm. The obtained nonlinear results are the average of enough time-distanced single-pulse measurements, well below the ablation threshold, in order to avoid thermal effects from accumulated pulses and assure reversible and reproducible nonlinear optical effects. Both setups are schematized in Fig. 4
, where RPD represents a photodetector used for monitoring the laser stability; PD1 and PD2 are photodetectors for measuring the optical transmittance of the sample, while PD3 detects the self-diffraction signal. The mirrors were placed in order to obtain the same optical path for the two incident beams. We calibrated the self-diffraction measurements using a CS_{2} sample, which is a well known nonlinear optical media with $\left|{\chi}_{eff}^{\left(3\right)}\right|$
*= 1.9 × 10 ^{−12}esu [*
26

*].*For the single beam transmittance measurement in the P-scan experiments, we blocked one of the beams in the same experimental setup,

*as*indicate into the Fig.

In order to perform both self-diffraction and P-scan measurements, according to what has been said at the end of Section 2.2, we started by positioning the sample as indicated in Fig. 4, which would correspond to Fig. 3(a) (*normal incidence, vertical* in Tables 2
to 5
) after measuring, we rotated the sample by 90° with respect to the normal of the sample, as indicated by the inferior arrow, which would correspond to Fig. 3(b) (*normal incidence, horizontal* in Tables 4
and 5); and, finally, coming back to the initial case, we rotated the sample the necessary to have the electric field of the incident light parallel to the major axis of the NPs, if allowed by the matrix’s refraction, with respect to an axis perpendicular to the plane of Fig. 4, as indicated by the superior arrow. This last case would correspond to Fig. 3(c) (*15° incidence, vertical* in Tables 4 and 5). We performed this last procedure for the Au samples; however, in the case of the Ag samples, we proceeded slightly different since it was impossible, due to the refraction of the light, to obtain ${\chi}_{3333}^{\left(3\right)}$ according to Fig. 3(c). Then, the measurements were performed for normal incidence and incidence at 45*°. In the first case, it corresponds to*
*Fig. 3(a)*
*(nor*mal incidence, vertical in Tables 2 and 3
). In the second case (45°), the signal was measured for two positions, one such that the substrate containing the Ag nanocomposites was rotated 45° as indicated by the superior arrow, but the incident optical polarization was kept parallel to the NP’s major axis (*45° incidence, vertical* in Tables 2 and 3, corresponding to Fig. 3(c)), addressing mainly ${\chi}_{3333}^{\left(3\right)}$
*; *and the other in a similar way, but the incident optical polarization was kept parallel to the NP’s minor axis (45°, horizontal in Tables 2 and 3, substituting what it had been described for Fig. 3(b)), mainly related to ${\chi}_{1111}^{\left(3\right)}$. This has been achieved by rotating the sample 45° with respect to the axis normal to the setup, taking the first measurement and then, rotating the sample around its normal by 180°, measuring again.

## 4. Results and discussion

#### 4.1 RBS, optical absorption and electron microscopy characterization

Figure 5 shows the RBS spectra of the Ag and Au nanocomposites, after the corresponding annealing treatments and before the irradiation with Si ions. We have proceeded routinely in this way, since we have verified before that the metal distributions are not affected by the posterior Si irradiation. From these spectra, we can clearly see that the metallic NPs distribute inside the silica in a Gaussian way, where the position of its maximum gives the depth of the layer of NPs, while its FWHM gives the thickness of it. Typical optical absorption spectra of the Ag and Au anisotropic nanocomposites are also included. They show the SPs corresponding to the minor and the major axis of the Si-deformed NPs.

Figure 6
shows the deformation of the Ag NPs after Si ion irradiation. As it has been mentioned earlier, the metal concentrations for the fluences and the angles of implantation described before are 0.0379 and 0.0268 for Ag, respectively; while for Au are 0.0312 and 0.06, respectively. From appropriate HRTEM measurements, performed at the *Laboratorio Central de Microscopía* from IFUNAM, with a 200 KV JEOL-2010FEG in contrast-Z mode, we sorted out a Gaussian size distribution in the 1-6 nm range, with an average diameter of 3.6 nm and FWHM of 2.25 nm for spherical-like Au NPs; while for spherical-like Ag NPs, the Gaussian size distribution was in the range of 1-12 nm, with an average diameter of 4.4 nm and FWHM of 7.0 nm. On the other hand, for the Si-deformed NPs, we observed that not all the NPs are elongated, that were the smallest NPs that remained spherical-like and it was a large number of them, affecting then the aspect ratio distribution; and that the elongated ones show a prolate spheroid shape, with the major axis aligned along the ion irradiation direction (Fig. 6(a)). Besides, the equivalent diameter distribution is very close to that of the spherical-like NPs, indicating that the NP volume is, in general, conserved. For Au, this equivalent diameter is distributed between 2 and 6 nm, with an average equivalent diameter of 3.8 nm, while the aspect ratio distribution goes from 1 to 2 centered at 1.37. For Ag, the equivalent diameter distributes between 1 and 12 nm, with an average of 5.4 nm; while the aspect ratio is between 1 and 3 centered at 2 [51]. Although the actual deformation mechanism is still under discussion, what we have argued before [6] and nowadays is that, following the D’Orleans scheme [52], the embedded NPs under study in this work fall in a size range such that, when irradiating with Si ions, they melt and flow into the ion track, therefore being deformed as a result of the Si irradiation, according to a thermal spike model. Besides, we have corroborated that, in general, spherical and deformed NPs have a fcc symmetry (Fig. 6(b)) [53], which will be an important fact regarding the discussion below for the nonlinear optical results.

#### 4.2 Birefringence results

We have performed the birefringence measurements in the range of 300-800 nm, i.e. in the visible region, mainly. Figures 7
and 8
show typical measurements in such a range for Ag and Au, respectively. For *α* = 0 and *α* = *π*/2, we obtain from Eq. (1), for each of the wavelengths, ${A}_{p}^{2}$ and ${A}_{s}^{2}$, respectively. While, on the other hand, from Eq. (2), with *α* = *π*/4, we get the maximum measured birefringence for each wavelength as

There are two things that can be sorted out from these last Figs. From Fig. 7(a), one can observe what we have remarked in Ref. 20, that is, how the intensity as a function of the angle reverses its behavior when the wavelength passes from one SP resonance to the other. On the contrary, due to the small separation between the two plasmons of the deformed Au NPs, it is not possible to observe this reversal in Fig. 8(a). As well, this small separation does not allow observing two peaks for the intensity as a function of the wavelength in Fig. 8(b), while they are clearly differentiated in Fig. 7(b), each one related to each of the two plasmons of the deformed Ag NPs; although the respective wavelengths do not match exactly, but this mismatching will be discussed just below.

In Fig. 9
we show typical birefringences computed with Eq. (15) as a function of wavelength for the nanocomposites studied in this work, that is, Ag NPs deformed at 45 and 80°, and Au NPs deformed at 45°, as well as their absorption spectra. As it can be seen, in general, these anisotropic metallic nanocomposites exhibit a large birefringence from 0.04 to 0.07 in the optical region, being bigger for Ag than for Au nanocomposites. However, when comparing the peak positions to the SP positions for each case, it is evident that there is not a match between them. This fact is resumed in Table 1, where we show the corresponding nanocomposite’s angle of deformation, including the SP resonance positions for each case, and the birefringence peak positions. In Ref. 20, we had argued that the physical reason behind this birefringence was a dichroism effect. In this work, we still argue the same since, from our measurements, the birefringence for pure SiO_{2} matrix, not deformed NPs and outside the 300-800 nm region for deformed NPs is null. But we recognize that something not clearly understood is happening according to what is presented in Fig. 9 and Table 1. According to local field calculations performed with the extended Maxwell-Garnett theory, for the nanocomposite parameters presented in Section 4.1 (metal concentrations of 0.0268 for Ag and 0.0312 for Au, aspect ratios of 2 for Ag and 1.37 for Au, matrix dielectric permittivity from standard data for fused silica, Ag and Au dielectric permittivities from [54]), one can obtain curves for *Δn* and *Δk*, for both Ag and Au, showing the same mismatching between their maxima as that observed in Fig. 9, i.e. in general, the birefringence maxima are red-shifted with respect to the absorption maxima. Theoretical calculations concerning near field considerations around the deformed NPs might give some insight on this mismatching. Besides, from these same calculations, but for the dispersion relations, we found that, at 532 nm, for Ag, the slope is negative for both *k _{x}* and

*k*, but also for both

_{z}*n*and

_{x}*n*. On the other hand, for Au the slope is negative for

_{z}*k*but positive for

_{x}*k*, while it is positive for both

_{z}*n*and

_{x}*n*, indicating anomalous dispersion. In particular, these facts for

_{z}*k*can be observed in Fig. 9 for Ag NPs deformed at 45° and for Au, or even in Figs. 5(c) and 5(d).

_{i}#### 4.3 Nonlinear optical results

According to what has been shown above and to our previous results [6,11,20], the nanocomposites thus fabricated show uniaxial symmetry. However, although the metallic anisotropic NPs are all oriented in the same direction, as shown in Fig. 6, they do not exhibit polar order since they do not possess intrinsic dipolar moment. In consequence, as it has been shown into the second part of the theoretical section, for nonlinear optical measurements using a fully degenerate setup, i.e. *χ*
^{(3)}(-*ω*;*ω*,*ω*,*−ω*), the tensor has only three independent nonzero components: ${\chi}_{1111}^{\left(3\right)}$, ${\chi}_{1133}^{\left(3\right)}$ and ${\chi}_{3333}^{\left(3\right)}$. By following Fig. 3 and its discussion, we can perform the necessary measurements to obtain these three elements, show the anisotropy of the nonlinear optical properties of the nanocomposites, and therefore their usefulness for straightforward nonlinear applications, as we have already shown for linear optical properties [6,20]. Then, we measured the P-scan transmitted and the scalar self-diffracted intensities while varying the angular position of the nanocomposites according to Fig. 3, for a fixed polarization of the incident beams. Firstly, we measured the single beam optical transmittance in order to obtain the nonlinear optical absorption coefficient for each position of the sample. The fitting was made considering that the optical absorption is described by the expression *α* = *α*
_{0} + *βI*; where *α*
_{0} and *β* represent the linear and nonlinear absorption coefficients, respectively; and *I* is the incident intensity. Then, the self-diffraction experiments were performed and the transmitted and self-diffracted intensities were measured for each case. For the evaluation of the $\left|{\chi}_{eff}^{\left(3\right)}\right|$corresponding value, we considered that nonlinear absorption takes place during the self-diffraction experiments. Thus, the numerical fitting was made by following the analysis of degenerate two-wave mixing in the stationary regime for a thin nonlinear medium [55], but taking into account for the numerical simulations, the results obtained for the nonlinearity of absorption for each case.

$\left|{\chi}_{eff}^{\left(3\right)}\right|$, $\mathrm{Re}\left({\chi}_{eff}^{\left(3\right)}\right)$ and $\mathrm{Im}\left({\chi}_{eff}^{\left(3\right)}\right)$ values obtained for each sample, for each angular position, according to what has been described in Section 3.2, are shown in Tables 2 and 3 for Ag and in Tables 4 and 5 for Au. According to Eq. (13), since the refraction was not a problem for Au case, the two last columns of Table 4 correspond actually to $\left|{\chi}_{1111}^{\left(3\right)}\right|$ and $\left|{\chi}_{3333}^{\left(3\right)}\right|$, respectively. Similarly, the two last columns of Table 5 correspond to the imaginary and the real parts of these same components, respectively It is worth remarking, that the self-diffraction and the P-scan signals were measured also for isotropic metallic nanocomposites, i.e. spherical-like Ag and Au NPs not deformed with Si, for each angular position as mentioned before, finding practically the same value for the nonlinear response, respectively.

From Tables 2 and 4, the difference in the measured values for the anisotropic nanocomposites, for each position, is quite clear. Therefore, from the given association of them to the different components of the nanocomposite’s third-order susceptibility tensor, *χ*
^{(3)}, i.e. for Ag and Au, the vertical measurements are associated to the major axis component, ${\chi}_{3333}^{\left(3\right)}$, while the horizontal ones to the minor axis component, ${\chi}_{1111}^{\left(3\right)}$; for similar geometric parameters of nanocomposites, we can establish the inequality ${\left|{\chi}_{eff}^{\left(3\right)}\right|}_{\text{minor axis}}<{\left|{\chi}_{eff}^{\left(3\right)}\right|}_{\text{isotropic}}<{\left|{\chi}_{eff}^{\left(3\right)}\right|}_{\text{major axis}}$. These results are valid considering that the magnitude of the nonlinear optical response seems to be strongly dependent on the size and shape of the NPs [56], and our deformation technique for obtaining nanoellipsoids is a direct way of modifying both. One obvious implication that we can sort out from this result, is that the lesser the size of the NP that sees the electric field of the incident light, the smaller the third order nonlinear optical response. This could be related to the near-field enhancement associated with the shape of the NP, indicating that the largest enhancement is obtained for the major axis. This result has been already indicated in the literature for Au nanorods [57].

On the other hand, from Tables 3 and 5, the anisotropy of the nonlinear absorption and refraction is quite clear for these anisotropic nanocomposites. Just by a simple 90° rotation of the sample, one can obtain a more intense nonlinear refraction or one can pass from reverse saturable to saturable absorption. Going deep into the details, the real part of *χ*
^{(3)}, $\mathrm{Re}\left({\chi}_{eff}^{\left(3\right)}\right)$, is positive for Ag but negative for Au, showing an opposite behavior for the nonlinear refraction, namely, a change from self-focusing to self-defocusing. In order to explain the resulting differences for the focusing properties of these materials, we are still performing a deeper research about the mechanisms involved in the optical Kerr effect for this kind of nanostructures. With respect to the imaginary part, it is negative for Ag, for both axes; but it goes from positive to negative for Au, when switching from the minor to the major axis. The imaginary part results, although at first sight may seem contradictory to other recent results, where saturable absorption has been obtained for both axes of Au nanorods [37], or a flipping from saturable to reverse saturable absorption in Ag nanodots that has been observed when increasing the input irradiance [58]; indicate instead that a more elaborated analysis has to be done. One first step is to perform some local field calculations in order to get some insight about the dependence of the nonlinear optical response on the metal concentration and the NPs deformation [34], while a second step is to widely discuss the electron dynamics for metallic NPs [58] and the hot electron contribution to the nonlinear optical response [33,35,45,59]. Before attacking these two possibilities, it is worth recalling the discussion at the end of Section 4.2, that is, at 532 nm, the Ag linear absorption spectra have negative slope for both minor and major axes, and saturable absorption (negative) was obtained for the nonlinear case for both axes. On the other hand, for Au, the slope is negative for the absorption related to the minor axis, while is positive for the major one. In this case, the nonlinear absorption changes from reverse saturable to saturable (from positive to negative), respectively. Again, from Kramers-Kroning relations, this change in the slope of the linear absorption band would imply a change from normal to anomalous dispersion, and then a change of sign for the nonlinear absorption should be expected. We are about to perform the nonlinear optical measurements as a function of the wavelength, in order to look for an inflexion point for the nonlinear absorption, which could help us to enlighten this issue in a better way.

Following with the discussion, let us undertake first the local field calculations. According to Ref. 34, the real and the imaginary parts of the third order nonlinear susceptibility, for an anisotropic metallic nanocomposite, depend on the metal volume concentration as

*p*is the metal volume concentration, reaching a maximum value of 20 vol. % to respect the limit of validity of the extended Maxwell-Garnett theory,

*ε*is the matrix dielectric permittivity, ${\chi}_{m}^{\left(3\right)}$ is the metal third order susceptibility, ${A}_{0}={\epsilon}_{d}+{L}_{j}\left(\mathrm{Re}{\epsilon}_{m}-{\epsilon}_{d}\right)$, ${B}_{0}={L}_{j}\mathrm{Im}{\epsilon}_{m}$, ${A}_{p}={\epsilon}_{d}+{L}_{j}\left(1-p\right)\left(\mathrm{Re}{\epsilon}_{m}-{\epsilon}_{d}\right)$, ${B}_{p}={L}_{j}\left(1-p\right)\mathrm{Im}{\epsilon}_{m}$,

_{d}*ε*is the metal dielectric permittivity, ${L}_{z}=\genfrac{}{}{0.1ex}{}{1-{e}^{2}}{{e}^{2}}\left(\genfrac{}{}{0.1ex}{}{1}{2e}\mathrm{ln}\left(\genfrac{}{}{0.1ex}{}{1+e}{1-e}\right)-1\right)$ and ${L}_{x}={L}_{y}=\genfrac{}{}{0.1ex}{}{1-{L}_{z}}{2}$ are the corresponding depolarization factors for the major and the minor axes, respectively; and, finally, ${e}^{2}=1-\left({b}^{2}/{a}^{2}\right)$ is the ellipticity of the deformed NPs, with

_{m}*b*and

*a*being the length of the minor and the major axes, respectively. These Eqs. may be thought as similar to Eq. (13), just in the sense that they distinguish the anisotropy of the nanocomposite, the first one by considering its symmetry arguments, and the last two by considering the anisotropic enhancement of the nonlinear susceptibility due to local field effects. Given this context, as Lamarre et al. have done [34], the best way of comparing this theoretical formalism to the experimental results, and be then able to quantify the anisotropy of the nonlinear susceptibility, is calculating the ratio between the real (imaginary) part of the third order susceptibility along both, the major and the minor axes, as well as the ratio between the real (imaginary) part along the major axis and those measured for the spherical-like NPs. We consider that Au anisotropic nanocomposites are the most interesting case of this work because of the angle of deformation, 80° in this case, which allows us to measure directly the third order susceptibilities corresponding to the major and the minor axes of the deformed NPs; and because of the change of sign for the nonlinear absorption when scanning one or the other axis. In consequence, we used the following parameters for the theoretical formalism given by Eqs. (16) and (17).

*ε*= 2.1316, $\mathrm{Re}{\epsilon}_{m}=-6.508$, $\mathrm{Im}{\epsilon}_{m}=1.71$ [34]. We justify this choice since the Au parameters were obtained for samples prepared in a similar way as ours; the shapes are the same, spherical-like and prolate spheroids; and because, despite the difference in sizes, the absorption spectra are quite similar [37]. For the metal third order susceptibility, we also take the considerations made at [33,60], where a positive imaginary part is deduced [33] and measured [60], and a 5 times smaller, negative real part is calculated [60]; therefore, we take the normalized values, $\mathrm{Im}{\chi}_{m}^{\left(3\right)}=5$ and $\mathrm{Re}{\chi}_{m}^{\left(3\right)}=-1$, to perform our calculations. However, we will come back to this later, when discussing the electron dynamics, since this fact would have also implications about the role played by the hot electrons of metallic NPs. Taking the metal concentrations given above for Au, i.e. 0.0312 and 0.06 for the two fluences used, respectively. Figure 10 shows the real (Fig. 10(a)) and the imaginary (Fig. 10(b)) parts of the nonlinear optical susceptibility for these parameters, for these filling factors and for the minor and the major axes, according with Eqs. (16) and (17), with respect to the aspect ratio,

_{d}*a*/

*b*.

From this Fig. we can remark the anisotropy of the optical response of the nanocomposite when switching from the minor to the major axis, reaching an increment even of two orders of magnitude. Furthermore, there is clearly a change of sign depending on the aspect ratio and, as in [34], a range of aspect ratios giving the largest anisotropy can be also identified. These theoretical results coincide with the inequality established above, ${\left|{\chi}_{eff}^{\left(3\right)}\right|}_{\text{minor axis}}<{\left|{\chi}_{eff}^{\left(3\right)}\right|}_{\text{isotropic}}<{\left|{\chi}_{eff}^{\left(3\right)}\right|}_{\text{major axis}}$, since it can be observed that the absolute value of the nonlinearities for the minor axis, (*j* = *x*), decreases when increasing the aspect ratio, with respect to the absolute value of the nonlinearities for the spherical-like NP (aspect ratio of 1). This behavior is reversed for the major axis. It is very important to notice here that the larger anisotropy for the nonlinear optical response comes, for both the filling factors presented in this work, for just a slight deformation of the metallic NPs (the calculations were also performed for far larger aspect ratios going to 100). This is a remarkable fact since all the recent plasmonics developments, in particular all the results concerning nanoantennas, have been obtained for rather large aspect ratios. One would think that, given our results, the attention should come back to almost spherical-like NPs. There is also an increment in the nonlinear optical response when increasing the volume concentration; however, this would have always the limit of validity of Maxwell-Garnett theory. By performing the comparison between the experimental and the theoretical nonlinear susceptibilities ratios, the best aspect ratio matching the experimental results is, for *p* = 0.0312, 1.58 for the imaginary part and 1.33 for the real part; for *p* = 0.06, the best fittings are 1.54 for the imaginary part, and 1.28 for the real part. All these values agree well with the aspect ratio distributions obtained experimentally [51] and mentioned in Section 4.1 (average aspect ratio of 1.37). It is worth remarking that a matching value of 1.02 was also found for most of the cases, which should not be very surprising since there were a large number of small spherical-like NPs, as indicated also above. These values can also be taken as an evidence of the inhomogeneity of the anisotropic nanocomposites found from the electron microscopy characterization. Summarizing this part, we can say that the experimental results agree well with the local field calculations, allowing a more systematic search of the optimum deformation of the metallic NPs, in order to obtain, by this technique, the best anisotropic nonlinear nanocomposite.

Concerning the electron dynamics and the hot electron contribution, according to Guillet et al. [35], for a fixed total energy absorbed by a Au NP showing a fcc structure, the longer the pulse, the lesser the hot electron contribution. However, what we have found from a previous work [59] and from this one, specifically from the negative sign of the nonlinear absorption, is, apparently, just the contrary, i.e., the longer the pulse, the larger the hot electron contribution. Nevertheless, it is necessary to say that our experiments were performed with a maximum intensity ranging into the GW/cm^{2}, i.e., slightly beyond the regime where the hot electron contribution to the nonlinear optical response of the nanocomposite would be a pure third order nonlinear effect. On the other hand, there is a sort of coincidence regarding the spectral dependence of the nonlinear optical response since, as these authors argue; there would be a change of sign in the nonlinear optical absorption for long pulses, besides of the dependence on the linear absorption. This applies to our case in the sense that we observe such a change of sign for a clear variation of the linear optical absorption, when switching from the minor to the major axis of the deformed Au NPs. Finally, since we are very close to the interband transition threshold and also to the SPR corresponding to both axes of the NP, we would expect that the hot electron contribution dominates over the intraband and interband ones, as concluded in [33] and [35]. Nevertheless, as it was mentioned before when showing the presence of the anomalous dispersion, and from what Guillet et al. [35] asserts about not taking for granted the conclusions obtained in the already classical work of Hache, et al. [33], it is still necessary undertaking a systematic experimental study of the spectral and intensity dependences of the third order nonlinear optical response from metallic NPs embedded in a dielectric matrix, as suggested by Guillet et al. [35], and this should be done for several pulse durations. In particular, we are now performing such a study for 26 ps pulses in a wide spectral range, and for spherical-like and deformed Au and Ag NPs.

## 5. Conclusions

The ensemble of results obtained throughout this work show the control we have achieved in designing anisotropic metallic nanocomposites, which present large and anisotropic linear and nonlinear optical responses, adding to them supplementary value for potential technological applications. For the linear part, we have measured a large birefringence in the region going from 300 to 800 nm. Concerning the nonlinear optical response, we have first contributed to the tensor analysis of the third order nonlinear optical response for an anisotropic uniaxial nanocomposite. Subsequently, we showed how the third-order nonlinear optical response varies, and even shifts from saturable to reverse saturable absorption in the case of Au, when modifying the angular position of the sample with respect to the incident beams and their polarization. Furthermore, self-focusing was detected in the case of Ag, while the opposite, self-defocusing, was observed for Au. We showed also how the nonlinear optical anisotropy is maximized for just a small deformation of the metallic NP. Finally, these measurements could be associated with the different nonzero components of the third-order susceptibility tensor of the metallic nanocomposite, which are only three in the degenerate case, allowing besides to establish the inequality ${\left|{\chi}_{eff}^{\left(3\right)}\right|}_{\text{minor axis}}<{\left|{\chi}_{eff}^{\left(3\right)}\right|}_{\text{isotropic}}<{\left|{\chi}_{eff}^{\left(3\right)}\right|}_{\text{major axis}}$, which could be associated with near-field enhancement considerations.

## Acknowledgments

We acknowledge the partial financial supports from DGAPA-UNAM, through grants No. IN108807-3, No. IN119706-3 and No. IN108407; and CONACyT-Mexico, through grants No. 80019, No. 82708, No. 79152 and No. 50504. V. Rodríguez-Iglesias and H.-G. Silva-Pereyra acknowledge specially the support from CONACyT and DGEP-UNAM for their Ph. D. scholarships. We are very grateful to the reviewers’ comments since they helped to a great improvement of the discussion of our results.

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