Materials

Chloroauric acid (HAuCl₄, 99.99 %), silver nitrate (AgNO₃, 99.8 %), sodium borohydride (NaBH₄, 99.99 %), tetraoctylammonium bromide (N(C₈H₁₇)₄Br, 98%) and dodecanethiol (CH₃(CH₂)₁₀CH₂SH ≥ 98%) were purchased from Sigma-Aldrich. Hexadecyltrimethylammonium bromide (C₁₆TAB), L-ascorbic acid (C₆H₈O₆) were purchased from Dinâmica. Deionized water (resistivity: 18 MΩ cm) was used in all syntheses. The magnetic stirrer and all glassware were washed with aqua-regia.

Synthesis of spherical Ag nanoparticles for suspension in carbon disulfide

Colloidal suspensions containing Ag-NS in carbon disulfide (CS₂) were prepared by a chemical synthesis method following a procedure previously reported (^{Reyna and de Araújo, 2014}). Initially, 3.75 mL of a 0.03 mol/L AgNO₃ aqueous solution were added to a 0.05 mol/L N(C₈H₁₇)₄Br solution in toluene, at a speed of 100 drops/min. After 10 min, 50 mL of dodecanethiol (dodecanethiol-to-silver molar ratio of 2:1) was introduced in the mixture, followed by rapid addition of 3.1 mL of a freshly prepared NaBH₄ aqueous solution (0.4 mol/L). 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 CS₂. In order to obtain a colloidal suspension containing homogeneous distribution of Ag-NS, the product of the chemical synthesis was subjected to laser photofragmentation, under slow stirring, using the second harmonic beam at 532 nm from a Nd: YAG laser (8 ns, 85 mJ/pulse, 10 Hz) for 1 h. The photofragmentation of the NPs occurs due to melting and vaporization because of the large absorption of the laser energy by the NPs and low heat-transfer to the hosting medium (^{Takami et al., 1999}; ^{Brito-Silva et al., 2010}).

Synthesis of gold nanorods in aqueous solutions

Three colloidal suspensions containing Au-NR, with different aspect ratios, were prepared by a two-step chemical synthesis: (i) the seed preparation and (ii) the nanorods growth. This seed-mediated growth method provides highyielding Au-NR with good control of size and morphology by addition of silver ions (^{Nikoobakht and El-Sayed, 2003}).

The first step of the synthesis consists in the preparation of gold seeds (with diameters between 2 nm and 4 nm) from the mixture of 5 mL of C₁₆TAB (200 mM) with 5 mL of HAuCl₄ solution (0.5 mM), under moderate agitation. Then, 0.600 mL of ice-cold NaBH₄ (10 mM) solution was added under vigorous stirring for 2 minutes. During this time, the solution changed from colorless to brownish-yellow.

The growth solution was obtained by mixing 50, 100 and 150 µL of AgNO₃ solution (4 mM) in 5 mL of C₁₆TAB (200 mM), to obtain Au-NR with aspect ratios of 1.8, 2.8 and 3.4, respectively. Immediately 5 mL of HAuCl₄ solution (1 mM) followed by 70 µL of L-ascorbic acid (78.8 mM) were added to the mixture and gently stirred after each addition. With addition of HAuCl₄ the solution displays an intense yellow color which changes rapidly to colorless when the L-ascorbic acid is added and after moderate shaking. Finally, 12 µl of the seed solution prepared in the first step were added to the growth solution of the Au-NR, maintaining the temperature between 27 °C and 30 °C overnight.

All colloids were subjected to centrifugation to remove by-products and to separate the Au-NR from spheres, cubes, prisms, and other undesirable morphologies. Three centrifugations were performed at 10,000 rpm during 30 min, also contributing to remove the excess of C₁₆TAB. Afterwards, the supernatant was discarded and the precipitate, containing the Au-NR, was redispersed in deionized water.

Morphological, quantitative and UV-VIS characterization

The morphology and average size of the Ag-NS and Au-NR were determined by using a 120 kV transmission electron microscope (TEM) - model: Tecnai Spirit Biotwin G2. Quantitative characterization of gold atoms contained in the Au-NR colloids was performed by using a Spectro Arcos inductively coupled plasma optical emission spectrometer (ICP-OES), with radial vision (SOP), from nanorod samples digested in aqua-regia.

The optical extinction spectra of Ag-NS and Au-NR were acquired using a ultraviolet-visible spectrophotometer (model: Varian Cary 5E) from 200 to 1500 nm.

Nonlinear optical experiments

The experiments were performed by using the second harmonic of a Q-switched and mode-locked Nd: YAG laser (80 ps, 532 nm, maximum pulse energy of 10 mJ, repetition rate: 4 Hz). The closed- and open-aperture schemes of the Z-scan technique were used to determine the nonlinear refractive indices and the nonlinear absorption coefficients, respectively (^{Sheik-Bahae et al., 1990}; ^{de Araújo et al., 2016}). To perform these measurements, the laser beam was focused inside a 1 mm thick quartz cell containing the colloidal samples, by using a 5 cm focal length lens. 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 samples. A reference channel was used to improve the signal-to-noise ratio of the measurements (^{Ma et al., 1991}; ^{de Araújo et al., 2016}).

Guiding and confinement of light-by-light (^{Reyna and de Araújo, 2016})

Colloids containing Ag-NS suspended in acetone behave as self-defocusing media for light at 532 nm (^{Reyna and de Araújo, 2017}). When the Ag-NS filling fraction is 3 × 10^-5 the colloid presents effective third-order susceptibility, χeff(3)= -8.3 + i2.7× 10-23m2V-2, and fifth-order effective susceptibility, χeff(5)= +(2.8 + i0.2) ×10-35m4V-4. The origin of the nonlinearity, measured using single picosecond pulses at low repetition rate, is electronic without contamination by thermal effects.

The stable propagation of vortex solitons (with unitary topological charge and 532 nm wavelength) was observed for 3 GW/cm². The shape and radius of the vortex beam were constant for propagation over 10 mm, which corresponds to ~ 6 Rayleigh lengths. The formation and propagation of the vortex soliton was corroborate by numerical simulations based on the modified nonlinear Schrödinger equation with the form 2ik ((dE_v)/(dz)) + ∆E_v = ((ω²)/(c²)) [3χeff(3)E_v |E_v |² + 10χeff(5)E_v |E_v |⁴], where E_v is the field amplitude of the vortex beam, ∆ is the Laplacian operator, z is the beam propagation direction, k = 2πn₀/λ, n₀ is the linear refractive index, c is the speed of light, λ and ω are the laser wavelength and angular frequency, respectively. The initial condition for the vortex field was E_v (r,θ,z = 0) ∝ exp(- (r²/w02) + iθ)tanh [r/(2w_v)], where r and θ are the polar coordinates, and w₀ (w_v) is the waist of the Gaussian background (vortex core).

Although the defocusing third-order contribution is dominant, the positive Reχeff(5) is crucial to have stable soliton propagation for long distance.

Figure 1 shows a simplified scheme of the experimental setup.

Note: P, polarizer; T, telescope; VPP, vortex phase plate; M, mirror; SF, spatial filter; BS, beam splitter and CCD (charge-coupled device) is a camera to record the transverse beam profile. The cell’s length is 10 mm. The HeNe laser has very small intensity and does not induce nonlinear effects.

Source: Creation of the authors.

Figure 1 Simplified experimental scheme. 

Due to the negative real part of the effective nonlinear optical susceptibility, the region illuminated by the vortex soliton has refractive index smaller than the dark core. Consequently, a weak probe beam from a HeNe laser was guided by the soliton as illustrated by Figure 2 (a-d), which shows the probe beam profile for propagation along the colloid cell containing Ag-NS suspended in acetone. The radial intensity distribution of the weak probe beam, shown in Figure 2 (e-h), illustrates the guiding and confinement of light induced by the optical vortex soliton.

Note: (e-h) Normalized intensity profile of the Gaussian beam in absence of the vortex beam (red circles) and obtained from (a-d) (blue circles). The solid lines were obtained by numerical solutions of Eq. (1).

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Figure 2 (a-d) Evolution of the transverse Gaussian (HeNe) beam profile as a function of the propagation distance z, when guided by the optical vortex soliton. 

The solid lines in Figure 2 (e-h) show the intensity profiles obtained by solving two coupled nonlinear Schrödinger equations given by

where j = v,G correspond to the vortex soliton and the Gaussian HeNe beam, respectively. The nonlinear contribution of the Gaussian beam is negligible because |E_G |² ≪ |E_v |². The parameters used in the theoretical fit were obtained from the Z-scan experiments for I_v = 3 GW/cm² and I_G = 10 kW/cm².

In summary, the propagation of the vortex soliton through the Ag-NS colloid produced a nonlinear change in the refractive index according to the vortex beam shape and an effective waveguide was formed. The imperfect coupling of the HeNe probe beam into the vortex core and the escape of light from the waveguide, due to the small refractive index difference induced be tween the vortex core and the illuminated area, provided a light intensity output of ≈22 % of the initial probe beam after 10 mm length propagation. By theoretical analysis of the experiments we concluded that the fifth-order refractive nonlinearity contribution is very important to obtain stable guiding and confinement of light-by-light for long distances.

The results herein reported are highly encouraging to try additional experiments in order to investigate further the light-by-light control for ultrafast signal processing using short pulse lasers propagating inside MDC.

Nonlinearity management of colloidal gold nanorods (^{Oliveira et al., 2019})

Several works highlight the importance of nonlinear materials based on plasmonic NPs for the development of new light-by-light control procedures due to their high optical susceptibility and ultrafast nonlinear response (^{Schuller et al., 2010}; ^{Lu et al., 2011}; ^{Reyna and de Araújo, 2015} and ²⁰¹⁶). However, plasmonic NPs suffer limitation in applications that require high efficiency due to intensity losses induced by linear and nonlinear optical absorption in the NPs. In this sense, anisotropic plasmonic NPs offer significant advantages, compared with the spherical NPs, due to the ability to tune their multiple LSP resonances, enabling the study of on- and off-resonance optical phenomena. For instance, Au-NR display two tunable resonance bands associated with the electron oscillations along the transverse and longitudinal directions of the nanorods. Theoretically, by using an extension of the linear Maxwell-Garnett model, adapted for elongated metal NPs suspended in a dielectric medium, it is possible to analyze the longitudinal LSP (L- LSP) that redshift with the increase of the aspect ratio (AR) of the nanorods, as shown in Figure 3 (a). The intense peak in the extinction spectrum clearly reveals that the linear optical behavior of Au-NR colloids, around the L-LSP, is dominated by absorption and/or scattering phenomena. Therefore, the AR of gold nanorods is a parameter that can be adjusted to tune the L-LSP and consequently it becomes possible to manage the intensity losses in plasmonic nanocomposites.

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Figure 3 Theoretical (a) linear absorption coefficient versus the wavelength and (b) enhancement factor for colloids containing gold nanorods as a function of its aspect ratio, for excitation at 532 nm, near to the T-LSP resonance. 

The LSP is also responsible for the local field enhancement, which has profound implications on the nonlinear properties of the composite medium, as can be analyzed by using the generalized Maxwell-Garnett model (^{Reyna and de Araújo, 2017}). For example, the effective third-order susceptibility of a plasmonic colloid is given by χeff3ω=χh3ω+f ξ(ω)χNP3ω, where χh3ω and χNP3ω represent the frequency dependent third-order susceptibility of the host and the nanoparticles, respectively, and f is the volume fraction occupied by metal NPs. The enhancement factor, ξ(ω) = ∑j,k=13 =  [( βj2/3)(|βk|2/3)], depends on the local field factor, β_j (ω) = (ε_h)/(ε_h + (ε_NP -ε_h )P_j), which is significantly increased in the spectral region near to the LSP resonances. For metal nanorods the depolarization factors are P₁ = ((1- e²)/e²) [(1/2e) ln ((1+ e)/(1- e)) - 1] and P₂ = P₃ = ((1 - P₁)/2) with e =1 - (AR)-2 , and the nonlinear behavior is enhanced in two spectral regions: near to the transverse LSP (T-LSP) and the L-LSP, while spherical NPs, with P₁ = P₂ = P₃= 1/3, exhibit only one LSP resonance. Further, by writing the enhancement factor as ξ= ∑j=13=( βj2/3)(|βj2|/3)+∑j≠k3( βj2/3)(|βk|2/3), we notice that the second term with the products β_j |β_k | reveals that by varying the AR one can control the nonlinear response around both resonance regions, due to the cooperative influence of the L-LSP and T-LSP bands. Based on this fact, Figure 3 (b) shows the enhancement factor dependence with the aspect ratio of Au-NR, near to the T-LSP resonance that was kept constant. Although larger values of ξ are obtained when the NPs shape approaches that of a sphere (AR = 1), it is possible to conclude that the AR of Au-NR is a parameter that allows control of the nonlinearity strength, by excitation of the MDC using a laser frequency close to the T-LSP resonance.

Therefore, by exploring the region near the T-LSP band would allow to dribble the large intensity losses, due to the L-LSP band, but enjoying the effects of the nonlinearity enhancement due to the contribution of both LSP bands (^{Oliveira et al., 2019}).

We measured the nonlinear susceptibility of aqueous colloids containing Au-NR, with three different aspect ratios, by applying the Z-scan technique, using a picosecond laser with frequency near the T-LSP resonance frequency.

Figure 4 shows the extinction spectra and TEM images of Au-NR colloids with average cross-sectional diameter of 15 ± 0.6 nm, corresponding to a fixed T-LSP resonance wavelength of ≈512 nm, and the L-LSP resonance tuned to 610 nm, 698 nm and 774 nm, by varying the aspect ratio to 1.8, 2.8 and 3.4, respectively.

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Figure 4 (a) Linear extinction spectra measured in quartz cells with thickness of 1 mm; (b)-(d) TEM images of colloidal dispersions of gold nanorods with aspect ratios of (b) 1.8, (c) 2.8 and (d) 3.4. 

In order to compare the nonlinear response as a function of the Au-NR aspect ratio, the three colloids were diluted to exhibit the same volume fraction (f = 10^-4). Such procedure was achieved by measuring the concentration of gold atoms (12 mM), using the ICP-OES technique, and subsequently calculating the average concentration of nanorods per mL (1.2 × 10¹³ for the AR of 3.4), using the Au-NR dimensions determined from the TEM images.

Figure 5 illustrates the evolution of the transmittance profiles obtained using the open-aperture (OA) Z-scan scheme as a function of input intensity, for Au-NR colloids. Notice that the curves show clear differences in the nonlinear behavior for different aspect ratios. For AR = 3.4 the colloid exhibits, at low intensities, a valley in the transmittance profile, related to a twophoton absorption (2PA) process. With the increase of intensity, the curves show a flat bottom valley followed by a peak formed at the bottom of the valley that indicates saturation of the 2PA process. For higher intensities, the increase in the transmittance peak indicates a saturable absorption (SA)like process related to absorption by excited-states. Colloids with Au-NR having AR equal to 2.8 and 1.8 show similar evolution up to intensities of 5.5 GW/cm², but with different nonlinear coefficients (see Table 1). The results corroborate the theoretical prediction that the nonlinear absorption, for excitation near the T-LSP band, can be controlled by changing the longitudinal dimension of the Au-NR but keeping the rod diameter fixed.

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Figure 5 OA Z-scan profiles as a function of the input intensity, for colloids with Au-NR of different aspect ratios. The numbers above each curve represent the peak intensity in GW/cm². The curves were shifted vertically to prevent overlap between them. 

For Au-NR with AR = 2.8, the Z-scan curve in Figure 5 shows a dip on the transmittance curve, at z=0, for intensities larger than 5.5 GW/cm², indicating the contribution of a new nonlinear process with origin associated to the nonlinear elastic light scattering as illustrated in Figure 6. Measurements of the scattered light as a function of the input intensity were performed by placing a microscope objective and a photodetector in a direction perpendicular to the propagation of the laser beam and fixing the sample at the focus of the 5 cm lens, which was used for the Z-scan experiments.

Note: The dashed, dotted and solid lines correspond to numerical fits using linear, cubic and quintic nonlinear scattering models. The lines in the yellow plane are the projections of the scattered intensities for each sample.

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Figure 6 Nonlinear scattered light intensity for the gold nanorods colloids, with different aspect ratios, versus the input intensity. 

Figure 6 shows that the variation in the AR also influences the behavior of nonlinear scattering, in the spectral region near the T-LSP resonance although the diameter of the Au-NR is not changed. The experimental results (circles), for each colloid, were fitted by linear (dashed lines), cubic (dotted lines) and quintic (solid lines) nonlinear scattering models and the best agreement was obtaining for the fifth-order process as the best fit.

From Figures 5 and 6, it was possible to adjust a mathematical expression that encompasses the nonlinear extinction exhibited by the Au-NR colloids, given by α(I) = α₀ + α^NL(I) + γ_sI², with γ_s being the fifth-order scattering coefficient. The linear and nonlinear absorption processes are described by α₀ and α^NL(I) = ((α₂/(1 + I/I_α) + (η / (1 + I/I_η)), where the first and second terms in the right hand side describe the 2PA saturation and SA-like process associated to excited-states (^{Cesca et al., 2012}). This model was used to fit the experimental data of Figure 5 using the parameters given in Table 1. The nonlinear parameters experimentally determined are very close to the theoretically estimated values for contributions of electronic transitions between the 5d and 6sp bands (^{Hache et al., 1988}) and hot-electron contributions, which describes the dynamics of the electronic thermalization process from the two-temperature model (^{Marine et al., 2013}), as discussed in (^{Oliveira et al., 2019}). Note that by controlling the AR and laser intensity (see Figure 3) it is possible to manage the nonlinear response of the Au-NR colloids by prevailing the SA effects (nanorods with AR = 3.4), the nonlinear scattering effect (nanorods with AR = 2.8, at high intensities), or to mask the nonlinear scattering effects by the strong SA process (nanorods with AR = 1.8). Based on these results, we foresee that the SA process in Au-NR opens new routes to manage intensity losses effects commonly reported as one of the main limitations of metal nanoparticles in applications of light-by-light control that require high efficiency.

Figure 7 shows that colloids with Au-NR exhibit significant refractive cubic-quintic nonlinear behavior, which is revealed by the double peak-valley profile of the CA Z-scan curves as recognized also in Ag-NS experiments (^{Reyna and de Araújo, 2017}). Hence, following the method of Gaussian decomposition and considering the simultaneous contribution of third- and fifth-order refraction, as described in ^{Oliveira et al. (2019)}, it is possible to obtain the best fits for the experimental data, represented by the solid lines in Figure 7. For instance, the third-order refractive indices (5.0 ± 0.3) × 10^-14, (7.0 ± 0.2) × 10^-14 and (8.0 ± 0.2) × 10^-14 cm²/W were obtained for the Au-NR with aspect ratio of 1.8, 2.8 and 3.4, respectively, while the fifth-order refractive index, (-4.6 ± 0.4) × 10^-23 cm⁴/W², remained nearly constant for all samples. The behavior of the CA Z-scan profiles reveals the strength by which high-order processes can modify the nonlinear optical behavior of Au-NR colloids.

Note: From bottom to top the intensities are 0.4, 0.8, 1.6, 2.4, 3.2 and 4 GW/cm². The curves were shifted vertically for clarity.

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Figure 7 CA Z-scan curves as a function of incident laser intensities for colloids with Au-NR having different aspect ratios.