We present second-harmonic generation (SHG) measurements and simulations from a silica matrix containing randomly distributed but aligned elongated silver nanoparticles (NPs). The composites were produced by a double ion-implantation process of silver nanoparticles followed by an irradiation with Si ions. It is demonstrated that one can model the experimental results by considering the sub-micrometric composite layer as a nonlinear media containing rod NPs for which the hyperpolarizability tensor is cylindrically symmetric along the NP long axis. The second-order macroscopic susceptibility of the composite originates from the coherent summation of the hyperpolarizabilities associated to each NP. We obtain analytical expressions for the p- and s-polarized effective susceptibility tensor as a function of experimental variables, such as the fundamental beam input polarization and sample orientation, and fitting parameters relating the cylindrically shaped hyperpolarizability. In addition, coherent SHG measurements on spherical nanoparticles resulting from the first ion-implantation process are also presented showing an isotropic polar behavior for the total SHG intensity where the p-polarized SHG intensity resulted to be the main contribution.
© 2011 OSA
In recent years, nanostructured materials composed of metal nano-particles have attracted much attention due to the possibility of using their nonlinear optical properties for photonic nanodevices [1, 2] and plasmonic circuitry [3, 4]. Their linear and nonlinear optical properties are dominated by collective electron-plasma oscillations, the so-called localized surface plasmon’s (SP’s), and a vast literature can be found elsewhere studying such properties [1–11]. In particular, we have studied the optical third-order nonlinearity of randomly oriented, but elongated and aligned in a preferential direction, silver nanoparticles (Ag-NPs) embedded in silica [9, 10] and now we extend our study to investigate the optical second-order nonlinearity in the same samples by means of second harmonic generation (SHG) experiments.
Metallic nano-particles such silver and gold are centrosymmetrical materials, their crystalline lattice structures are cubic face-centered and, in principle, no SHG from the bulk NP takes place in the electric dipole approximation. The SHG origin of such materials is attributed then to higher order interactions like electric quadrupole and magnetic dipole responses from the NPs bulk and/or electric dipole responses allowed from the NPs surfaces [5,6,11–15], where the inversion symmetry of the bulk material is broken. The latter response dominates in the specific case when the NP size is much smaller than the wavelength of the exciting (fundamental) beam  so that field retardation effects (no spatial dependence of the electromagnetic fields) are neglected. Therefore, the problem at the macroscopic level turns to be very similar to that of nonlinear media containing particles of non centrosymmetrical material apart from the interfacial second-order origin of the response. In a sense, the arrangement of the NPs in the array resembles that of the atoms in a crystal cell, where phase-matched SHG signal radiates in specific directions. This principle has been utilized for example on planar structures containing metallic 2D arrays of nanoparticles lacking inversion symmetry, providing coherent addition of the SH field where the efficiency of the process increased rapidly with decreasing nanoparticle size [7, 8].
In this context, in this paper we demonstrate that, similar to the analysis to derive molecular orientation information of smaller noncentrosymmetic units at interfacial monolayer’s or macromolecules systems using SHG/SFG experiments [16–22], we can treat the actual sub-micrometric layer containing randomly arranged but highly aligned anisotropically shaped Ag-NPs (elongated or nearly spherical) as a nonlinear media; where the origin of its macromolecular second-order susceptibility, χ(2), is the coherent contribution of the SH signal induced on every single nanoparticle.
2. Theoretical Analysis
In general, the second-order nonlinear polarization, Pi, induced by an incident electric field, E, for the bulk second-order susceptibility, , is given by . For materials composed of nonlinear optical scatterers much smaller than the wavelength of the fundamental beam, such as in the case of thin films composed of syntectic macromolecules or fibrillar proteins for instance [18–20,23], the origins of the bulk second-order susceptibility comes from the coherent summation of the molecular hyperpolarizability βi′j′k′ of the smaller molecules. In a similar way, due to the fact that the elongated NPs of our samples are at least a hundred times smaller than the wavelength of the fundamental light and in the limit of weak coupling between each nonlinear NP, we can express the macroscopic susceptibility of the thick layer containing NPs asFigure 1 shows a schematic representation of the Euler angles, where the z-axis is normal to the sample surface and the ζ-axis is along the NP long axis. Assuming the interface between the glass substrate and the NPs layer to be azimuthally isotropic (invariance under ψ- rotation), i. e. this interface does not contribute to the second harmonic signal, there are only three nonvanishing independent components of χ(2), χxxz = χyyz = χxzx = χyzy, χzxx = χzyy and χzzz.
It is common practice to calculate the average orientation of the sub-molecular units that give rise to the macroscopic susceptibility of nonlinear materials using SHG/SFG experiments. In the process, analytical expressions of the independent tensor components are typically written in terms of the pitch angle, θ, defined by the z-axis and the noncentrosymmetric subunit ζ-axis, and the resulting nonzero elements of the microscopic hyperpolarizability, βi′j′k′. Such expressions and their derivation are found elsewhere [16–19,24], and will not be rewritten here. Instead, for the purpose of this work, we followed the formalism used in Refs. [17, 18, 24] to derive expressions for the effective susceptibility tensor, , in terms of the experimental variables α and Φ. With α as the angle of the linearly polarized fundamental beam with respect to the plane of incidence, and, Φ, as the angle made by the projection of the NP long axis, ζ, over the xy plane and the fixed plane of incidence contained in xz. From Fig. 1 we can deduce that Φ =ψ + 90°. The total SH intensity is proportional to the sum of the effective susceptibility p and s as followsEq. 2 denote the output (fixed) and input (variable) polarization directions, respectively, and each component can be computed from 17]. However, we already know the orientation angle of the NPs and for simplicity, we will approximate this values to unity.
In the first approximation, we chose to model the elongated NPs as rod particles for which the hyperpolarizability tensor is cylindrically symmetric along ζ (invariance under ϕ-rotation). We also assume that the optical frequencies of the fundamental and SH beams are not in resonance with electronic transitions, so that there are only two nonvanishing independent components (βξξζ = βηηζ, and βζζζ). Under these conditions using Eqs. (1) and (3) we obtain
In Eqs. (4) and (5) r = βξξζ/βζζζ and a = Nβζζζ are our fitting parameters, while b = 1 – r. Note that , in the specific case when Φ = 0, this result is expected since under this configuration the input electric field finds isotropically shaped NPs. For all other configurations (Φ = 90, Φ = 180, and Φ = 270) and polar traces with different lobes of maximum SH are found for p- and s-SH, respectively, as will be shown in our discussions.
3. Experimental Section
3.1. Sample Preparation
The procedure to prepare samples with metallic nanoparticles is described elsewhere [10, 25] and here we will resume it briefly. Spherical silver nanoparticles (Sp Ag-NPs) were produced by a single ion implantation process using a 3 MV Tandem accelerator (NEC 9SDH-2 Pel-letron). 2 MeV Ag2+ ions were implanted into host matrices consisting of high-purity silica glass plates from NSG ED-C (Nippon silica glass), at room temperature. The samples were thermally annealed afterwards at 600 °C in a 50%N2 + 50% H2 reducing atmosphere. The measured Ag-ion fluence and projected range were 2.4 × 1017 Ag/cm2 and 0.9 μm, respectively. The result of this process was a film containing spherical-like silver nanoparticles, of around 6 nm in diameter. Figure 2 a shows a schematic of the resulting Sp Ag-NPs after the first ion implantation process.
In order to produce elongated and tilted silver nanopaticles (El Ag-NPs) some of the Sp Ag-NPs samples were subjected to a second ion implantation process. The samples were irradiated, at room temperature, at an angle of 45 ° with respect to the sample surface with 8 MeV Si ions of around 1 × 1016 Si/cm2 of fluence. With this energy the electronic stopping power for Sp Ag-NPs SiO2 is 200 times larger than the nuclear  and the ion projected range is 4.3 μm in SiO2, i.e. far beyond the location of the Ag-NPs. The resulting samples had a 0.5 μm thick layer containing randomly placed elongated nanoparticles, at a 1 μm depth inside the silica matrix, aligned in the direction of the second ion implantation . In other words, the particles long axes are tilted 45 ° with respect to the substrate normal and lay on the xz planes of the laboratory xyz shown in Fig. 2 b. When viewed from the front, the projection the El Ag-NPs long axes point in the direction we label as x. The El Ag-NPs were actually shaped as prolate spheroids, with an average minor axis diameter of 5 nm, and an aspect ratio of 1.7. The size distribution obtained from the statistics over 290 measurements shows a diameter distribution centered at 5.9 nm with a standard deviation of 1.1 nm. The TEM photograph shown in Fig. 2 c demonstrates both the different sizes of the NPs and their random distribution (but aligned in a preferred direction) obtained during the second ion implantation process. The inset shows the morphology of a single El Ag-NP. The average distance between the elongated NPs are estimated to be of around 25 nm and around 13 nm in the remaining spherical NPs, that is, one order of magnitude larger than the particles sizes, so no inter-particle coupling effects are expected in our experiments.
3.2. Sample Characterization
In order to characterize the NPs orientation in our samples, prior to our SHG experiments, we collected optical absorption spectra using linearly polarized light at two mutually orthogonal polarizations, one parallel (labeled as p) and the other perpendicular (labeled as s) to the plane of incidence. This experiment was performed at three different angles of incidence (0 ° and ± 45 °) and an UV-visible spectrophotometer was used to perform the measurements. Figure 2 shows the schematic of these experiments (a and b) and the respective absorption spectra (d and e).
The absorption spectra taken under different angle of incidence have no significant changes for the case of Sp Ag-NPs samples. This is shown in Fig. 2d, where a single SP resonance is found at approximately 400 nm. In contrast, from Fig. 2e we can see that for El Ag-NPs the absorption spectra depends on both the light polarization and the angle of incidence. At normal incidence the SP resonance of the particles is shifted to lower wavelengths for s-polarization (red line), while the absorption spectra splits into two spectrally separated SP resonances for p-polarization (black line). The shifted resonance at 365 nm obtained with s-polarized light is associated with the short-axis SP and can be explained by the decrease of the NPs size  during the second ion implantation process. The resonance at 570 nm obtained with p-polarized light is associated with the long-axis SP and its broadness can be explained by the different NPs sizes formed in the matrix, as can be seen in Fig. 2c. Note that the 365 nm resonance is also present and presumably invariant in the spectra taken at the three angles of incidence with p-polarized beams, i. e. at −45 ° (blue line), 0 ° (black line) and 45 ° (green line). While in the case of p-polarized light at 45 ° this result is obvious, since it resemble the case of s-polarized light at 0 °, the presence of this band in the other two cases, 0 ° and −45 °, can be attributed to a residual misalignment with respect to the direction of elongation of the particles or to an actual fraction of smaller spherical NPs remaining in the matrix after the second ion-implantation process (indicated by white arrows in Fig. 2c). Finally, a strong dependence of the 570 nm SP is obtained with p-polarized light for different angles of incidence being higher when the light propagation is orthogonal to the NPs axes (blue curve). This last result was the criteria used to characterize the NPs orientation used in the SHG experiments.
3.3. SHG experiments
SHG experiments in the reflection mode with a fixed angle of incidence were conducted using a Ti:Sapphire oscillator as the fundamental beam. The schematic representation of these experiments is shown in Fig. 3a. The laser delivered linearly polarized femto-second pulses with a wavelength centered at 825 nm (pulse width, 88 fs; repetition rate, 94 MHz). The angle of polarization, α, of the fundamental beam was rotated using a λ/2 wave-plate in order to trace the polar SH dependence of our samples. Using a 50 mm focal length lens, the beam was focused onto the Ag-NP sample at an angle of incidence θinc = 45 ° with respect the sample surface normal. The reflected SH signal was collected at 90 ° with respect to the incoming light using a second lens of 30 mm focal length. A color filter and a grating monochromator (not shown) were used to spectrally separate the SH signal from the fundamental light. Finally, the signal was detected via a photomultiplier tube connected to a current/voltage pre-amplifier circuit and a digital oscilloscope. The p-polarized SH (p-SH) and s-polarized SH (s-SH) intensities were also measured using a polarizer cube before signal collection. The sample was mounted on a rotation stage in order to vary the NPs long axis orientation by rotating the angle Φ. The four different angles used in our experiments are represented in Fig. 3b.
4. Results and Discussion
Figure 4 shows both the measured and simulated polar dependence of the total (black), p-polarized (red) and s-polarized (blue), SHG intensities obtained for the different 90°-shifted Φ configurations described in Sec. 3.1. Note that in the experiment, the four configurations produced detectable p- and/or s-SHG signal. We stress out that, in principle, even when the centrosymmetry of the ellipsoidal and spherical NPs is locally disrupted by its surface, the homogeneous polarizing field induces SHG of mutually canceling polarizations at opposite sides of the circular surface, neglecting then an overall dipolar SHG contribution [14, 15]. However, we have to bear in mind that no perfect ellipsoids (or spheres) are present in our samples and that the NP size is almost two orders of magnitude smaller that the excitation beam to consider the SH signal as a quadripolar contribution from the NPs bulk. In addition, according to rigorous calculations made by Valencia et. al. [30, 31], SHG radiation from centrosymmetric infinite cylinders is not symmetric in the back and front surfaces. They find a multi-lobe SHG pattern originated at the cylinder surface, where the angle made from the first SHG scattered lobes in the first surface is more pronounced as the cylinder width is decreased. Seemingly, Bachelier et. al.  modeled both the near-field of the harmonic amplitude and the far-field SH intensity distribution in spherical gold NPs, the two cases show anisotropic radiation patterns arising from the NP surface. Therefore, we attribute this signal to a nonlinear dipolar contribution arising from the NPs surface.
Figure 4a shows, for example, the case when the incident beam is polarized perpendicular to the NP long axis (see Fig. 3 a; Φ=0°). Here the cross section of the El Ag-NPs could be considered circularly shaped (from the incident fundamental beam point of view) and therefore no SHG signal is expected according to Eqs. 4–5. In this case, in addition to our argument that no perfect circularly shaped NPs cross section are present in the sample, we attribute this signal to a systematic misalignment in the experiment while rotating the samples as will be shown latter. In contrast, lobes of maximum SH intensity are found at α-angles near the NPs long axes. This is seen in Figs. 4, b and d (Φ=90° and Φ=270°), respectively, where the s-SH intensity is the main contribution of the total SH signal. Seemingly, the main contribution in Fig. 4c (Φ=180°) is the p-SH, instead. Note that Figs. 4 b and d are basically mirror images of each other(with y,or α =90°, as the symmetry axis) with total SHG maxima at ∼ 115° and ∼ 290°, for b, and ∼ 65° and ∼ 255° for d, respectively; the counterpart p-SH intensity has practically no contribution. Otherwise, from Fig. 4 c we can see that both the p- and s-SH intensities contribute to the total SH when the fundamental beam is perpendicular to the NPs long axes (Fig. 3 b; Φ =180°). The simulated polar traces are in a good agreement with the experimental results, this can be seen also in Fig.4. Table 1 shows specific values of the parameters r and a, used in Eqs. 4–5, that best fitted with the experimental data, where, Φsim, stands for the simulated value of Φ. The simulated data revealed two extra peaks in between the s-SH maxima (∼ 25 times lower), in Figs. 4 b and d, respectively, and their values are found also in Tab. 1. Seemingly, two extra peaks are also found in Fig. 4 c, but at much smaller values (∼ 300 times less). Note also that the values of r were fitted in the range of 1.4 < 1/r < 3.5 for cases 2 – 4 (see table), confirming that there is a stronger hyperpolarizability response for fields oscillating along the NPs long axis (i.e. βζζζ > βξξζ). These values are very similar to the values obtained in synthetic films consisting of helical (PBLG) macromolecules  and native fibrillar collagen , their hyperpolarizability are reported to be within an order of magnitude of that of crystalline quartz .
Note that simulating the different cases shown in Figs. 4b–c with the same parameters (a and r) would result in obtaining higher SHG maxima in Fig. 4c than in Figs. 4b and d. However, in the experiment we obtained less SHG signal in Fig. 4d and we attribute this result to the presence of less NPs within the point spread function of the fundamental beam and/or a minor hyperpolarizability value. The inhomogeneous NPs distribution in the composite film makes extremely challenging maintaining the same irradiated area in the experiment while rotating the sample. As a consequence, the value of the parameters a and r used to fit the experimental data were different. As can be seen from Tab. 1, in this experiment the parameters used in Fig. 4c resulted to be smaller than the respective parameter values used to fit the experimental data of Figs. 4b and d. Otherwise, it is interesting to note that the experimental case at Φ = 0° (Fig. 4a) is reproduced for angles Φsim close to 180° and r = 1.41, this simply indicates that we can simulate this result by assuming stronger hyperpoplarizability responses for fields oscillating perpendicular to the NPs long axis (i.e. βξξζ > βζζζ). The asymmetric polar dependence was obtained using Φsim confirming that our experimental results are most probable due to misalignment. Note also that both the p- and s-SHG intensities are comparable in magnitude, while for the case shown in Fig. 4 c the p-SH intensity is ∼ 5 times larger than s-SH.
In order to be sure that the results correlate indeed with the known (simulated) structure of the El Ag-NPs, SHG experiments were also made in Sp Ag-NPs. Figure 5 shows SH signal from samples with embedded Sp Ag-NPs. The total SHG (black) presents nearly isotropic polar trace where p-SH (red) is the maximum signal contribution. The s-SH intensity (blue) also contributes but the signal is ∼ 10 times lower than the p-SH counterpart. It presents a characteristic shape with maxima at 45°, 135°, 225° and 315°. We found the same dependence for different Φ values 0° and 90° (Fig. 5 a and b), respectively, indicating that the obtained polar traces are a characteristic of the Sp-Ag NPs. The fundamental and SH beams spectra for elongated (black) and spherical (red) NPs, with the respective absorption spectra (dotted curves) are also shown in Fig. 6a. The absorption traces indicate that for El Ag-NPs the SHG may be enhanced since the fundamental is close to resonance with the SP broad band (black dotted line) at 570 nm (see also blue curve on Fig. 2 e). Note, however, that the SHG suffers also absorption of about the same optical density reducing the signal. In contrast, the SP resonance of Sp Ag-NPs (red dotted line in Fig. 6 a; which is the red solid line in Fig. 2 d) is far away of the fundamental wavelength and therefore no enhancement effect is expected. In addition since the NPs sizes are small compared to the fundamental wavelength, this suggest that the SHG in Sp Ag-NPs samples arises mainly from the electric dipole surface contribution, owing to the actual non perfect spherical shape of the particles , and must be large enough to be detectable even after being absorbed with an optical density of ∼ 3. Otherwise, the quadratic dependence with respect to the input power obtained in both types of samples, Fig. 6 b (for El Ag-NPs) and c (for Sp Ag-NPs), indicates the typical coherent nature of nonlinear scatterers. In particular, the result obtained in Sp NPs indicates that the SH observed is not due to grating effects such the hyper Rayleigh scattering (HRS), where incoherent SHG is produced for which a typical linear dependence with respect to the input power is observed.
Our results are in accordance with earlier SHG experiments performed by Podlipensky et. al.  on elongated Ag-NPs. The main differences with respect to our experiments are that this group obtained equivalent intensities for both p- and s-SH signal and no measurable signal for spherical NPs was detected. Their SH experiments were performed with the fundamental beam at an angle of incidence close to the surface normal, 15° (SH collected in transmission), and the NPs long axes aligned along the surface normal. We consider that such experimental arrangement is close to the case shown in Fig. 4a, since the direction of the fundamental beam is also close to the direction of the NPs long axis and comparable intensities are obtained for p-and s-SH. As discussed above we believe that we are able to detect SH from Sp Ag-NPs due to the fact that we have smaller NPs sizes (at least 10 times smaller) with respect to Podlipensky samples. Note that Eqs. (4–5) do not explain this dependence, since they were obtained considering a hyperpolarizability tensor with cylindrical symmetry, however, it is interesting to note that our experimental results can be explained using analytical expressions obtained by Dadap et. al.  to describe SH Rayleigh Scattering from spheres of centrosymmetric material, where the intensities for vertical and horizontal SH are given by Ipα ∝ |a1|2 and Isα ∝ |a2|2 sin2α, respectively. In these expressions, p and s, stands for the horizontal and vertical polarization of the harmonic generated signals, respectively, α, is the fundamental input polarization, and, a1 and a2, are complex numbers related to the pure effective dipole contribution and quadruple contribution, respectively. For p-SHG the intensity is constant, independent of the input polarization angle α, while for s-SHG intensity the signal is maximum at α = (2n – 1)45° and vanishes at α = (n – 1)90°, with n = integer. Fig. 5 b has been intentionally altered in order to see such s-SHG polar behavior. In addition, being the p-SH intensity higher with respect to the s-SH counterpart in our experiments (then a1 >> a2), strongly supports our assumption that the SHG is dominated by dipolar contributions arising from the surface of each non-perfectly spherical Ag-NP and having a quadratic response with respect to the input power (observed in Fig. 6c) confirms their coherent summation.
Second-harmonic generation from composites containing randomly distributed but aligned elongated silver nanoparticles has been presented and modeled as a coherent summation of the microscopcic hyperpolarizability associated to each NP. Our experimental data suggest that the origin of the hyperpolarizability, in both elongated and spherical NPs, can be attributed mainly to a surface nonlinear contribution of each non-perfect ellipsoidal or spherical Ag-NP.
The authors wish to acknowledge the technical assistance of K. López, F. J. Jaimes, J. G. Morales and E. J. Robles-Raygoza. We would like to thank Dr. Eugenio Méndez for useful discussions. Finally, we also acknowledge the financial support from PAPIIT-UNAM IN103609; from CONACyT through grant 102937, from ICyT-DF through grant PICCT08-80, and (I. Rocha-Mendoza) from UC-MEXUS/CONACyT under collaborative research programs.
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