1. Introduction

Silver nanoparticles have received increased attention in the past few years due to its variety of applications exploring the excitation of localized plasmons. Such applications include plasmon-controlled fluorescence [1, 2], multiphoton plasmon-resonance microscopy [3], and femtosecond filamentation and supercontinuum generation [4]. Besides photonics applications, fundamental studies on nonlinear optical properties of colloids have been performed [5, 6, 7, 8, 9, 10, 11]. Several of the aforementioned works concentrate on water solutions of silver nanoparticles, which by itself has great importance since biological materials are prone to have a great amount of water. Therefore, it is important to understand the role played by metal nanoparticles in water solutions. On the investigation of colloids nonlinear properties different techniques and spectro-temporal excitation regimes have been reported, ranging from 400nm to 1064nm, and from nanosecond to femtosecond pulse duration. In the literature most of the work in the femtosecond regime explores low intensity (GW/cm²) and low energy (nJ) light sources. An exception is found in reference [4], which uses intensities close to the TW/cm ² regime and energies of few (µJ).

In this work we report on the investigation of the nonlinear optical properties of water based colloids prepared with silver particles with maximum excitation intensity of 0.18TW/cm ² and energy of 0.1µJ. The pump wavelength, 800nm, provides a one photon nonresonant excitation mechanism, but a 2+1 mechanism, with the 400nm plasmon resonance in silver playing a role, can also make a contribution. We therefore measure an effective third order nonlinearity. A conventional Z-scan technique was employed to characterize the third-, fifth- and seventh-order nonlinear optical susceptibility and its dependence with volume fraction occupied by the silver nanoparticles. Experimental findings were supported by a theoretical model.

2. Experimental details

In order to analyze the nonlinear interaction of light with an aqueous suspension of silver nanoparticles, the samples in colloidal form were prepared in house. The colloid was synthesized according to the method described in [12]. In short, 90mg of AgNO ₃ were diluted in 500mL of water at 100C; 10mL of solution of 1% sodium citrate was added for reducing Ag, and later boiled and strongly stirred for 1h. Afterwards, this procedure was followed by laser ablation using the second harmonic of a Q-switched Nd:YAG laser (8ns, 10Hz), for one hour (approximate time for 10mL). This allowed us to obtain silver nanoparticles of 9nm average size. Filling factors varying from 4×10^-6 to 1.5×10^-4 were obtained. The filling factor is defined as the ratio of the total volume occupied by the nanoparticles divided by the total volume of the solution. The characterizations of the prepared colloids were performed by measuring their UV-visible absorption spectra and by analyzing the nanoparticle size distribution through electron microscopy imaging. Low-intensity absorption spectra over the 200-800nm were obtained using a using a CCD spectrophotometer (DV-Z500 Beckman), with the colloid sample contained inside a quartz cell 2mm thick.

To characterize the optical nonlinearities, the well known Z-scan technique was used [13], exploiting the wavefront distortion (self phase changes) of the beam that propagates inside a nonlinear medium. The laser beam was focused by a 15cm focal distance lens, and the studied material was placed inside a 2mm thick quartz cell mounted in a translation stage that moved along the beam propagation direction z. The z ordinate, determined by the beam propagation direction, where z<0 corresponds to locations of the sample between the focusing lens and its focal plane. By measuring the variation of the transmitted beam intensity through a small circular aperture placed in front of a detector, in the far-field region, one can determine the sign and magnitude of the nonlinear refractive index and the nonlinear absorption coefficient of the analyzed medium. The aperture size r_a is related to S, the linear aperture transmittance, by S=[1-exp(2r²_a/w²_a)], with w_a denoting the beam radius at the aperture. A small aperture Z-scan experiment, corresponding to S≪1, is employed on the measurement of the real components of the nonlinear optical susceptibilities. A wide or absent aperture S=1 is necessary for the determination of the nonlinear absorption coefficients. A digital scope was used in connection with a computer to record the signal.

The experiments were performed using an amplified femtosecond laser system (Coherent Libra) delivering 80 f s optical pulses at 800nm, excellent beam quality (M²<1.5) and low pulse to pulse intensity fluctuations (energy stability less than 1% RMS for an 8 hour period). In the Z-scan setup the intensity at the beam focus was inferred to be 0.18TW/cm ², for a beam waist of 20µm. The pulse energy was 0.1µJ.

3. Experimental results and discussion

Before carrying out the characterization of the optical nonlinearities, the linear absorption spectra of the colloid was obtained. Figure 1 presents the colloid absorption spectrum, clearly showing the Ag plasmon resonance at 400nm, and the inset illustrates the size particle distribution, obtained by electron microscopy imaging.

From those results, the dielectric constant of the 9nm silver nanoparticles and the colloid filling factor can be inferred. Using the values for thick Ag films [14] and the classical Mie expression for the absorption cross section to model the experimental colloid’s absorption spectrum [15, 16], we determined that the calculated silver dielectric constant was ε_NP(9nm)=-28.4+7.49i.

3.1. Third-order susceptibility of nano silver colloids.

The colloid third order nonlinear susceptibility can be described through the generalized Maxwell Garnett model [17]. Here, the silver nanoparticles are assumed to be spheres of radius a embedded in a host having dielectric constant ε_h. The characteristic distance between the inclusions is b, an it is assumed that a<b<λ. The medium is considered to be macroscopically isotropic, and the linear effective dielectric constant (without intensity-dependence) can be written as

with the local field factor β given by

where ε_NP and ε_h denote the linear dielectric constant of the silver nanoparticles and the host material (water), respectively. The water dielectric constant is given by ε_h=n²₀ (where n ₀=1.333).

 

Fig. 1. Absorption spectrum of the colloidal silver nanoparticle (Sample thickness:2mm). The inset in each figure illustrates the particles size distribution.

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Considering the nonlinearities of the host (water) and the silver nanoparticles, and also considering that the contributions will arise from both one photon or a (2+1) mechanism, due to the plasmon resonance at 400nm, is possible to show that the effective third-order susceptibility for small values of f is given by [17]:

where χ⁽³⁾_NP and χ⁽³⁾_h are the diagonal nonlinear susceptibilities of the nanoparticles and the host, respectively, and P=(ε_NP+2ε_h)/3ε_h. Moreover, introducing numerical values ε_h and ε_NP for 9nm diameter silver particles in the Eq. (3), the effective susceptibility can be written as

with C ₁=1.94 and C ₂=0.9. One can observe observe that χ⁽³⁾_eff increase linearly with f. Therefore the nonlinear refraction index, n₂, and the nonlinear absorption coefficient, α₂, of the colloid can be expressed by

Experimentally, through the Z-scan technique, the values of n ₂ and α₂ can be determined by measuring the difference between the normalized peak and valley transmittance (ΔT). For small phase distortion and small aperture (S≪1) [13],

The on-axis phase shift at the focus, Δϕ₀, is defined as

where λ is the laser wavelength, I being the on-axis irradiance at focus, L is the sample length, L_eff=[1-exp(-α₀L)]/α₀, and α₀ is the linear absorption coefficient of the colloid.

The transmission change for the open aperture configuration (S=1) is given by [13]

The Z-scan results obtained for different colloid are presented in Fig. 2 and Fig. 3. Figure 2(a) shows Z-scan traces for the closed aperture scheme (S≪1), which provides the n ₂ values, whereas the Fig. 2(b) shows results corresponding to the open aperture scheme (S=1) allowing the measurement of α ₂ for different filling factors. The measured values of n ₂ e α ₂, are plotted in Figs. 3(a) and 3(b) as a function of f.

 

Fig. 2. Closed-aperture (a) and open-aperture (b) z-scan traces obtained at 800nm for different filling factors f (laser peak intensity 0.18TW/cm²). The curves correspond to f=1.5×10^-6, f=2.4×10^-6, f=4.3×10^-6, and f=5.8×10^-6.

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The heavy line in Figs. 3(a) and 3(b) are the theoretical fitting obtained using Eqs. (5) and (6). Furthermore, exploring the theoretical line of Fig. 3, it was possible to estimate the value of the 9nm silver nanoparticle third order nonlinearity, χ⁽³⁾_NP≈(2.2+i0.4)×10^-16 m ²/V ².

The possible origin for the nonlinearity is mainly due to hot electrons, since in this regime (150 f s, kHz repetition rate) thermal or cumulative effects are definitely ruled out. Also, because both nonlinear refraction and nonlinear absportion are positive, this result is in good agreement with the work reported by Hamanaka and co-workers [18], which worked in the same regime (150 f s) using silver nanoparticles in a glass host. The work of ref. [11], in the picosecond and almost resonant (for one photon) regime, reported negative nonlinear refraction and nonlinear absorption.

3.2. High-order nonlinear optical properties of nano silver water colloids.

Third-order nonlinear optical effects give rise to index changes proportional to the beam intensity (Δn∝I). However, for so-called higher order effects where (Δn∝I^m), with m>1, contributions of χ⁽⁵⁾ and χ⁽⁷⁾ or higher order terms in the nonlinear susceptibility expansion are present. If only the third-order contributions are presents, the ratio |ΔT|/I should be a constant. On the other hand, if there are fifth- and seven-order contributions, the ratio |ΔT|/I have a linear and quadratic dependence with I, respectively [19].

 

Fig. 3. Experimental values for nonlinear refractive index n₂ (a) and nonlinear absorption α₂ (b), as a function of filling fraction f for the 9nm colloidal silver nanoparticle.

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For highly concentrated silver colloids (f>3×10^-5) we observed experimentally the contributions of fifth and seven-order nonlinear susceptibilities. The intensity dependence of |ΔT|/I for high concentrated silver colloids (f=3.0×10^-5, f=5.0×10^-5, f=8.5×10^-5 and f=14.5×10^-5) are shown in Fig. 4.

 

Fig. 4. Intensity dependence of the ratio ΔT/I, for the nonlinear refraction (a-d) and nonlinear absorption (a1-d1). The solid lines corresponding to the theoretical fitting using the Kothari model [20]. The filling fractions are f=3.0×10^-5 (a), f=5.0×10^-5 (b), f=8.5×10^-5 (c) and f=14.5×10^-5 (d).

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In order to infer the magnitude and the nanoparticle’s concentration dependence of the high-order nonlinear parameters we followed the procedure given in [11], where a theoretical treatment was developed to describe the behavior of ΔT (Z-scan results) as a function of laser intensity. In what follows, we summarize, for completeness, the derivation leading to the main equations of [11], and apply to our experimental results.

The mean peak-valley separation obtained from the z-scan data for small aperture (S≪1) is Δz_p,v ≅ 1.4z₀, where z₀ is the confocal parameter. Therefore the transmittance change can be written as ΔT_p,ν≅0.396ΔΦ⁽³⁾ ₀+0.198ΔΦ⁽⁵⁾ ₀+0.102ΔΦ⁽⁷⁾ ₀ [11]. The on-axis phase shift at the focus are given by

The total transmittance change for open aperture (S=1) can be written as [11]:

where the nonlinear refractive index (n ₂, n ₄, n₆) and absorption coefficients (α₂, α₄, α₆) are related to the colloid nonlinear susceptibilities.

As measured the silver nanoparticles exhibit a large nonlinearity. Therefore, as described in reference [20], it is reasonable to expect that the nanoparticle’s dielectric constant is intensity dependent. Reported theoretical approaches have assumed that the particle’s dielectric constant undergo an intensity dependent perturbation. As the colloid NP concentration increases, the dielectric constant intensity dependent perturbation becomes relevant to the medium nonlinearities. Therefore, the effective dielectric constant of the colloid can be expressed as ε_eff=ε^Linear_eff+ε^NL_eff(I). According with the Kothari’s model [20], the effective dielectric function of a Maxwell-Garnett colloidal geometry can be expressed as a function of the local field, E_Local, as

where a₀=4π(χ³_NP/(ε_NP+2ε_h)), P has been defined before and the effective third-order susceptibility is given by Eq.(3).

It is common to describe the optical properties of a medium by a mean macroscopic field, instead of a local field. The local field E_Local, is related to the mean macroscopic field, E, by E=PE_Local(1+a₀|E_Local|²). Moreover, the effective dielectric function of a Maxwell-Garnett colloidal geometry, Eq. (12), can be written in terms of a mean macroscopic field, expanded into a power series of |E|². The resulting expression is

where [11]

The (*) indicates the complex conjugate. As shown by Eqs. (13), (14a), (14b) the nonlinear behavior of colloid effective dielectric function is determined by the colloid third-order nonlinear optical susceptibility. Furthermore, the Kothari’s model [20] can be modified, introducing a χ⁽⁵⁾_NP contribution through χ⁽⁵⁾_eff. Therefore, the total dielectric function electrical can be written as

where [11]

In Eq. (15), we do not consider the χ⁽⁵⁾_NP contribution to higher order terms of ε_total. Therefore the following association can be made:

Using a combination of Levenberg-Marquardt and least squares minimum algorithms [21], it was possible to fit the ratio |ΔT|/I shown in Fig. (4) (solid curves). Exploring Eqs. (15), (10b) and (10c) and the Z-scan experimental data, it is possible obtain the values of n ₂, n ₄, n ₆, α₂, α₄, α₆ and the corresponding nonlinear susceptibilities, for different NP concentration. These results are summarized in Table 1. An error of 25% was estimated for the fitting parameters.

Table 1. Values of the real and imaginary part of the colloids for different filling factors, f, obtained from the experimental values shown in Fig.(4)

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4. Conclusion

We have studied effective third-order and higher order nonlinearities of silver colloids in a high intensity one-photon nonresonant femtosecond excitation regime. The colloid nonlinear refractive index was experimentally determined and the dependence with the filling factor was theoretically described. For low concentration of NP (f<3×10^-5), the third order nonlinearity of the colloid is a linear function of the nanoparticle’s volume fraction, described by the generalized Maxwell-Garnett model for the 9nm size particles. As the colloid NP concentration increases the dielectric constant intensity dependence changes and therefore changing the medium nonlinearities. The fifth- and seventh- order susceptibilities were obtained for the most concentrated silver nanoparticle colloid, with 3×10^-5<f<1.4×10^-4, and the data was supported by a theoretical model. Positive values of nonlinear index refraction and nonlinear absorption and the intensity dependence were reported, leading to the conclusion that the main contribution for the nonlinearity arises from hot electrons, as reported in the literature [18]. The effective χ⁽³⁾ has contributions from one photon nonresonant excitation and a 2+1 mechanism, with the 400nm plasmon resonance in silver playing a role. Although it is difficult to separate these two mechanism in this kind of experiment, we have independent evidence of the contribution of this 2+1 process, as it causes fluorescence enhancement in colloids of biomaterial with silver nanoparticles [2]. The findings described here should be useful in supporting further understanding of continuum generation in silver-water colloids, which exploits excitation in the femtosecond regime, as well as could provide useful insights for multiphoton imaging using plasmon enhancement in metallic nanoparticles.