1. Introduction

An electromagnetic wave interacting with a metallic nanoparticle (NP) experiences a considerable electric field enhancement when its frequency is close to the surface plasmon resonance (SPR) of the free electrons of the NP. Thus the linear as well as nonlinear optical properties of materials composed of metallic NPs in dielectric matrix are often governed by the SPR. In particular, the nonlinear properties, which are important for applications like all-optical communication or computing, can be considerably boosted near to the SPR because the local field factor f has to be regarded for each electric field involved in the process; for instance, χ⁽³⁾-effects experience an increase proportional to f ⁴ at the SPR (in electric dipole approximation). As a consequence, dielectrics containing metal nanoparticles are a very interesting class of optical nonlinear materials, mainly for two reasons: (i) its third-order nonlinear susceptibility χ⁽³⁾ has a sub-picosecond response time [1,2], which has the potential for ultrafast optical switching processes, and (ii) the strong χ⁽³⁾ enhancement due to the SPR can be potentially tailored within a wide spectral range by controlling the material parameters like size, shape and filling factor of the NPs [3,4].

In this work we focus on the shape anisotropy of NPs, in particular uniformly oriented prolate spheroids. For these, the SPR splits into a longitudinal one (LSPR) for light polarization along the symmetry axis, and a transversal one (TSPR) for light polarization perpendicular to the symmetry axis. While the LSPR is very sensitive to NP shape, moving strongly towards lower frequencies with increasing aspect ratio c/a of long and short half axis (e.g., for Ag NPs in glass from ~400 nm to infrared wavelengths), the TSPR is only very slightly shifted to higher frequency with growing c/a [5]. While previous studies on non-spherical gold NPs [6–8] provided rather comprehensive insight into their nonlinear optical properties, the work reported on non-spherical silver NPs is limited to selected points of the problem of anisotropy of optical nonlinearity [9, 10]. Here, we report the results of femtosecond Z-scan experiments aiming at the geometry-dependent nonlinear optical properties of nano-composite materials containing uniformly oriented prolate Ag spheroids of different aspect ratio in glass matrix. By studying samples with different aspect ratio at different laser wavelengths and polarization, we will be able to demonstrate that the third order optical nonlinearity of glass-metal nanocomposites is strongly enhanced by the SPR of the nanoparticles and shows dispersion behavior as function of relative spectral distance between laser wavelength and resonance.

2. Experimental

The samples studied were flat glass sheets (thickness 200 µm) containing silver nanoparticles in an approximately 1-2 µm thick surface layer, as observed in scanning electron microscopic (SEM) images on sample cross sections; an exact definition of layer thickness is not possible because, due to preparation method, the NP have a concentration gradient [11]. Choosing an effective layer thickness of 1 µm, the peak extinction allows us to estimate the NP volume fraction p to be in the range of p ≈5∙10⁻³ (which is compatible with SEM results). The NPs are prolate spheroids, being uniformly oriented along the direction of mechanical stress applied during the thermo-mechanical stretching method used for preparation [12]. In brief, spherical nanoparticles in soda-lime silicate glass were prepared by Ag/Na ion-exchange followed by subsequent reduction in H₂ atmosphere. Stretching of the sample was done at 650°C by pulling at constant stress. Depending on the processing parameters, samples with different NP aspect ratios c/a can be prepared. For the present study we have chosen three different samples with increasing anisotropy, indicated by the peak position of the longitudinal plasmon bands (LSPR) observed at ≈450 nm, 550 nm and 1200 nm. This would correspond to NP aspect ratios of c/a ≈1.4, 2.4 and 6.5, respectively, if all the nanoparticles would have the same size and shape [12]. It is well-known, however, that owing to the NP formation process there is a considerable inhomogeneity of particle sizes; therefore the c/a values can only be considered as an estimation of the average NP aspect ratio. In the following, we will identify the different samples as LSPR_450, LSPR_550 and LSPR_1200 according to the approximate peak position of the LSPR band.

Femtosecond Z-scan technique was used for the nonlinear optical characterization of the metal-glass nanocomposites in a setup allowing the simultaneous determination of amplitude (open aperture-OA) and phase (closed aperture-CA) effects [13], from which real and imaginary part of χ⁽³⁾ can be determined. Two different laser sources operating at 1030 nm (pulse duration 250 fs, repetition rate 1 kHz) and 800 nm (100 fs, 1 kHz) were used as excitation wavelengths. The Rayleigh ranges z₀ of the laser beams were measured to be ~2 mm and ~3 mm, respectively for 800 nm and 1030 nm pump wavelength, which is in both cases one order of magnitude larger than the thickness of the sample. Hence a thin sample approximation is valid irrespective of the excitation parameters. The spatial intensity profile of the laser beams was circular Gaussian in good approximation. Both laser systems provide linearly polarized light, enabling to realize either a parallel (p_||) or perpendicular (p_⊥) orientation of long NP axis and laser polarization vector. In p_|| configuration the laser polarization direction is set parallel to the stretched direction of the sample which enables the selective excitation of the longitudinal plasmon band (LSPR), whereas in p_⊥ configuration the transverse plasmon band (TSPR) is addressed (see Fig. 1(a) ).

 

Fig. 1 (a) Illustration of definition of linear laser polarization: parallel means electric field vector of laser light parallel to the long axes of the uniformly oriented elongated Ag nanoparticles; (b), (c): schematic representation of surface plasmon resonances of the different samples and photon energies of the used lasers for comparison, shown for laser polarization parallel and perpendicular to the long axis of the nanoparticles.

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The experimental situation is shown in Fig. 1, for clarity in a schematic representation: Figs. 1(b) and 1(c) compare the energetic positions of the different LSPR and TSPR with the photon energies of the two laser systems. For single photon interaction (solid arrows), the experiments refer to excitation frequency clearly below resonance in most cases, with the exception of sample LSPR_1200 in p_|| configuration. The dotted arrows indicate possible two-photon resonance, which is almost precisely matched in the case of λ = 800 nm laser and p_⊥ configuration. The various combinations of sample, laser wavelength and polarization thus represent several relative energy distances of excitation and resonance, which can be used to study, at least qualitatively, the dispersion behavior of χ⁽³⁾. For this purpose, we will below use the abbreviations Δ_|| = ħ∙(ω_LSPR - ω_Laser) and Δ_⊥ = ħ∙(ω_TSPR - ω_Laser) to characterize the relative spectral position of the excitation laser; here, ω_LSPR and ω_TSPR denote the longitudinal and transverse plasmon resonance frequency of the respective sample.

3. Results and discussion

In order to investigate the effect of SPR on the third-order optical nonlinearities of our anisotropic nanocomposites, several series of polarization sensitive Z-scan experiments have been performed. A selection of interesting results obtained at 800 nm pump wavelength for the samples with relatively small aspect ratio, LSPR_450 and LSPR_550, are presented in Fig. 2 . The closed aperture (CA) scans (symmetrized if necessary, see below) are shown in Fig. 2(a), open aperture (OA) scans in Fig. 2(b); the main panels in both cases refer to the sample LSPR_550, the insets give the corresponding data for sample LSPR_450. The CA scans mostly indicate a positive nonlinear refractive index: self-focusing upon approaching the focus leads to increasing beam diameter at the aperture, which decreases transmittance at negative z/z₀ values; the only exception is the p_|| configuration on sample LSPR_550, which clearly indicates a negative nonlinear refractive index. Furthermore, the signal in the latter situation was obtained at a considerably lower peak pump intensity of 8 GW∙cm⁻², while the other signals were measured using peak intensities of 200 GW∙cm⁻² or 330 GW∙cm⁻², respectively.

 

Fig. 2 Closed (a) and open (b) aperture Z-scan signals using 800nm pump wavelength measured on the samples LSPR_550 (main panels) and LSPR_450 (insets). The red circles and blue squares represent the data points obtained in p_|| and p_⊥ configuration, respectively. The solid lines refer to numerical simulations to derive the nonlinear material parameters (see text below); (c) conventional extinction spectra of the LSPR of the two samples.

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A first qualitative explanation for these effects is found looking at the longitudinal plasmon extinction bands shown in Fig. 2(c): the considerable extinction of the LSPR-550 band at 800 nm apparently enables a (near) resonant interaction of the laser photons, while they are already far out of resonance with the NPs in sample LSPR_450. In both cases the TSPR bands are centered at ~390 nm, so that the non-resonant character of interaction between light and NPs is even more obvious.

For the OA scans (Fig. 2(b)) a similar behavior can be seen: only for the p_|| configuration on sample LSPR_550, a transmission increase (bleaching) is seen, all other cases lead to a dip in the measured signals around z = 0 due to nonlinear absorption. Again the near resonant case yields the strong bleaching signal at much lower peak intensity (~7 GW∙cm⁻²) than is required for comparably strong induced absorption in the other cases (250 – 350 GW∙cm⁻²). And, also like for the CA scans, the linear extinction spectra provide a reasonable qualitative explanation: apparently bleaching can only occur when the sample has a substantial initial extinction, while the observed transmission decrease most probably comes from two-photon absorption at the respective SPR (see Fig. 1).

This first series of experimental results can be summarized as follows: for non-resonant excitation, the glass-metal nanocomposites show positive nonlinear refraction and two-photon absorption; when the excitation comes sufficiently near to resonance on the low frequency side of the SPR, strongly enhanced negative nonlinear refraction and nonlinear bleaching are observed. For samples with a considerable degree of anisotropy (sufficient spectral separation of LSPR and TSPR), the sign of both real and imaginary part of χ⁽³⁾ can thus be switched by a simple rotation of laser polarization by 90°.

Prior to a detailed evaluation of the nonlinear parameters of the materials, we want to look into the question what happens for near resonant excitation on the high frequency side of the SPR. This case can be realized for this work studying the sample LSPR_1200 in p_|| configuration. In particular at the pump wavelength λ = 1030 nm, the energy distance to the SPR is only Δ_|| = −0.17 eV. As is clearly seen in Fig. 3(a) , this situation yields a positive nonlinear refraction, here obtained with a peak intensity of 25 GW∙cm⁻². Switching to non-resonant excitation by polarization change in this case does not change the sign, but only decreases the magnitude of the nonlinear refractive index considerably. OA scans at λ = 1030 nm on the sample LSPR_1200 (Fig. 3(b)) yielded a very strong bleaching in p_|| configuration, but no measurable transmission change with perpendicular polarization p_⊥. Both effects are also apparent, because the laser frequency is very close to a strongly absorbing resonance in the first case, while in the latter neither single- nor two-photon resonances are at hand (cf. Figure 1).

 

Fig. 3 Closed and open aperture Z-scan at pump wavelength λ = 1030 nm, in p_|| and p_⊥ configuration, for sample LSPR_1200. Red circles and blue squares are data points, solid lines are fit curves (see text for details).

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For a physical discussion of the observed phenomena, we first have to extract the nonlinear optical parameters of the materials in all experimental situations acquired. To do this, we first analyzed the open aperture scans, describing the nonlinear absorption change Δα(I) in first-order approximation by Δα_||,⊥(I) = β_||,⊥∙I, with a nonlinear absorption parameter β_||,⊥ which can take positive (e.g., in case of two-photon absorption) or negative values (for saturable absorption). Numerical solutions of the propagation equation dI/dz’ = I∙[α₀ + Δα(I)] were used to calculate best fit curves for the OA scans (shown also as curves in the Figs. above). The values of the nonlinear absorption parameters β_||,⊥ obtained from these best fits are given in Table 1 . No values could be determined at the excitation wavelength 1030 nm for β_|| in case of sample LSPR_450, and for β_⊥ for all samples; apparently due to the SPR being too far off two-photon resonance, no noticeable transmission changes could be obtained in OA scans up to the highest intensities (I₀ ≈350 GW∙cm⁻²) applied in this study.

Table 1. Spectral gap values and nonlinear absorption coefficients calculated for all samples at 800nm and 1030nm excitation.

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To evaluate the closed aperture data, we applied the symmetrization technique introduced by Yin et al. [14] and cross-checked the validity of this approach comparing the values of the nonlinear absorption coefficient obtained with the two different methods. The symmetrized pure refractive part of the Z-scan curves have then been fitted with the usual expression for Z-scan experiments in thin film approximation, where the sample transmittance T is given by [13]:

The values stated in Table 1 reveal several important findings: first, the nonlinear refractive index in p_⊥ configuration is always positive and has a nearly constant value of n_2⊥ ≈ 10⁻¹⁷ m²∙W⁻¹, irrespective of sample and laser wavelength. This is clearly the non-resonant case, which nonetheless shows an n₂ almost 3 orders of magnitude larger than in pure glass (which is typically of the order of n₂ ≈ 10⁻²⁰ m²∙W⁻¹). In p_|| configuration, nonlinear refraction is considerably increased (by a factor of 4.5 to 26) for the samples LSPR_550 and LSPR_1200; positive or negative values of n_2|| are found depending on whether the laser is exciting on the high (Δ_|| < 0) or low (Δ_|| > 0) frequency side of the SPR. For LSPR_450, n_2|| is even smaller than in the non-resonant case for both laser wavelengths, indicating that here already a small (negative) contribution from the SPR enhancement is involved. The positive nonlinear absorption parameters derived for 800 nm laser wavelength (β_|| and β_⊥ for LSPR_450; β_⊥ for the other two samples) are apparently due to two-photon absorption, while the much (up to three orders of magnitude) larger, negative values of β_|| for the samples LSPR_550 and LSPR_1200 are assigned to saturation (bleaching) of the surface plasmon oscillation.

The relative distances between SPR and laser frequency, Δ_||,⊥, can be used to look at the dispersion features of the χ⁽³⁾ of uniformly oriented, spheroidal Ag nanoparticles in glass matrix. To do this, we have converted the obtained nonlinear parameters into real and imaginary part of χ⁽³⁾ by taking into account the linear absorption of the sample [15]. The results for the p_|| case are plotted in Fig. 4 , as a function of Δ_||, which is comparable to a dispersion curve of χ⁽³⁾; it has to be regarded, though, that the different samples can actually not be compared quantitatively to each other because of the inhomogeneity of the real SPR bands, as caused by the polydispersity of NP sizes and shapes. The effective NP concentration which the laser is interacting with may thus be considerably different from sample to sample. Nonetheless the data show very clearly dispersion behavior as it is expected for a susceptibility around a resonance: the (negative) imaginary part increases towards the resonance (Δ_|| = 0) and is fairly symmetric, while the real part has the typical change of sign at resonance.

 

Fig. 4 Imaginary (a) and real (b) part of χ⁽³⁾ in esu units, calculated from the values of n and b (as given in Table 1) and the linear absorption coefficients at the respective laser wavelengths; the two smallest values in (b) are additionally shown after multiplication with a factor of 100 to make clear that they are positive. Dashed curves are arbitrarily chosen Lorentzian curves, as a guide to the eye to illustrate the dispersion behaviour around SPR (Δ_|| = 0, vertical lines).

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Though quantitative modeling appears not appropriate due to the individual sample differences, we want to compare our findings to simple theoretical approaches. Generally, it has been found in previous work that, for fs pulses, the main contribution to χ⁽³⁾ of silver-glass nanocomposites is the so-called hot electron contribution, whereas intraband transitions are generally much weaker, and interband transitions are in the UV for silver and thus not relevant in the spectral range studied here [16, 17]. The spectral dependence of this contribution can in simplest approximation be described in a two-level model as saturation of a homogeneously broadened transition, using saturation Intensity I_S. In such an approach, the intensity dependent change of total susceptibility is modeled by a Lorentzian-shaped χ⁽¹⁾, which is modified by a saturation denominator: <χ(I)> = χ⁽¹⁾·[1 + (I/I_S)]⁻¹; the averaged change due to the applied field is then calculated as Δχ = <χ(I)> − χ⁽¹⁾, which can finally be used to calculate χ⁽³⁾ via the relation Δχ = 3π χ⁽³⁾ |E_l|². Here E_l is the local field, which is obtained by considering the local field correction (E_l = f∙E₀). For moderate changes, this model yields for χ⁽³⁾ mainly the Lorentz shape of the linear susceptibility, but with a negative sign; this is qualitatively well compatible with our results, as demonstrated by the dashed curves in Fig. 4, which represent real and imaginary part of appropriately scaled Lorentzians.

The more realistic approach of using an effective medium theory, as successfully applied by Hamanaka and associates for spherical silver NPs [18], could in principle be transferred also to spheroidal NPs. However, this extension would mean to introduce an anisotropy (depolarisation) factor into the formula expressing Δχ (or, what is the same, Δε), individual for each sample; so in our case we could only compare two values (referring to the two laser wavelengths) with each model function which appears not really useful. Nonetheless, the results shown in [18] are in practice not so dramatically different from the simple two-level model (although the physical approach is completely different); the main difference is that Im χ⁽³⁾ also takes small positive values in a certain range on the low frequency side of the SPR. In fact, we do also have a small positive value of Im χ⁽³⁾ at Δ_|| = 1.21 eV. However, the physical situation is different: in our study, due to the used near IR pulses two-photon absorption at the SPR itself is addressed, whereas in the previous work of Hamanaka et al. [18] two laser photons can excite an interband transition.

For the real part of χ⁽³⁾, in contrast, we may assume the same physical origin for off-resonant as well as for (near) resonant interaction, namely the hot electron contribution of the Ag conduction band electrons. The latter has repeatedly been described as fairly independent of excitation frequency for bulk metals, so that the third-order nonlinearity of an effective medium with metal volume fraction p can be described as local field-corrected intrinsic nonlinearity of the metal (χ_m⁽³⁾) as given by [16, 17, 19]:

In that case, the dispersion behavior of χ⁽³⁾ of the nanocomposite is completely governed by the fourth power of the complex local field factor f. Thus, if we take our experimental value of Re χ⁽³⁾ (Δ_|| = 1.55 eV) as an approximation for f ≈1 (i.e., volume fraction p times intrinsic metal nonlinearity χ_m⁽³⁾), we can get an estimate for the local field factor. For instance, Re χ⁽³⁾ (Δ_|| = −0.17 eV) ≈150 ∙ Re χ⁽³⁾ (Δ_|| = 1.55 eV); therefore, a maximum local field enhancement of at least | f | ≈3.5 can be inferred, in good accordance with values obtained in previous work for silver NPs in glass. Thus, an effective medium approach appears well suitable to describe the third-order optical nonlinearity also in the case of uniformly oriented, spheroidal silver nanoparticles in glass.

4. Conclusions

In conclusion, our studies on the nonlinear optical properties of mechanically stretched nanoparticles in glass proved the strong impact of geometrical NP anisotropy. In particular, it could be shown that χ⁽³⁾ of these nanocomposite materials for interaction with femtosecond pulses is apparently based on the hot-electron contribution of the NPs, and shows a strong nonlinearity enhancement of more than two orders of magnitude with dispersion behavior around the surface plasmon resonance. This enhancement and dispersion can be identified with the local field factor f used in established effective medium theories to describe the local effective field interacting with the nanoscopic inclusions. As this complex factor (or its fourth power, respectively) can take also negative values, nonlinear refraction can switch sign going from low to high frequency side of the SPR, or from a near resonant to a non-resonant situation. The results of this work demonstrate clearly that the latter can be realized for near IR laser wavelengths simply by 90° polarization rotation. The same effect has been observed for nonlinear absorption where, however, the switch of sign can clearly be associated with a switch in the physical process from saturation of the one-photon transition at the LSPR to two-photon absorption at the TSPR.

In general, this polarization-dependent switch of sign of both nonlinear refraction and absorption observed with fs pulses at near IR wavelengths offers a great potential for application of glass-metal nanocomposites with intrinsic geometrical anisotropy in optoelectronics and photonics.