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The spectral map of the nonlinear absorption coefficient of glass-copper nanocomposite in the pump-probe scheme constructed with the use of a simple anharmonic oscillator model reproduced well the spectral map obtained in the experiment. It is shown that spectral features in nonlinear response of glass-metal nanocomposites (GMN) can be engineered by varying the size of nanoparticles. The pronounced dependence of the magnitude of the third-order nonlinearity on the particles size explains the diversity of experimental data related to nonlinear optical response of GMNs in different experiments. Performed modeling proves that silver GMN demonstrate much sharper spectral dependence than copper ones due to strong frequency dependence of local field enhancement factor for silver nanoparticles.
©2012 Optical Society of America
1. Introduction
Recent advances in nanotechnology gave birth to plasmonics [1–5], a new branch of the optical and material science that studies optical phenomena in metal nanostructures. Optical properties of such systems dominate surface plasmon modes capable of concentration and enhancement of the electromagnetic field in the proximity of the metal-dielectric interface. In glass-metal nanocomposites (GMN) comprising of nanosized metal inclusions in glassy matrix, the spectral position and strength of the surface plasmon resonance (SPR) is determined by properties of both metal particles and host matrix. This enables tailoring of the GMN optical properties by varying size, shape and packing density of the particles and by changing the matrix [6–8]. Such a tunability of optical properties along with strong optical nonlinearity of metals and compatibility with all-solid-state opto-electronic circuits makes GMN a key material for various plasmonic devices [9–11].
A combination of the strong but featureless optical nonlinearity of metal and SPR-enriched optical response of the composite has also made GMN a unique playground to study ensembles of highly localized hot electrons. The dynamics of hot electron ensembles can be visualized by using ultrafast nonlinear spectroscopy techniques [12]. In the time domain, the dynamics of light-excited electrons is usually visualized by pump-probe measurements, i. e. by studying temporal evolution of light-induced transmission change of the GMN at the excitation with ultrashort light pulses tuned to the SPR. In the frequency domain, the study of the nonlinear optical properties of GMN is often restricted to the measurements of the nonlinear refraction and absorption coefficients (i.e. to the measurements of the real and imaginary parts of the frequency degenerate third-order susceptibility, respectively) by Z-scan technique [13,14]. However study of the pronounced spectral features in the ultrafast nonlinear optical response [15,16] of the GMN in the vicinity of the SPR requires combining both approaches, i.e. the time-resolved measurements of the essentially non-degenerate nonlinear optical susceptibility at the excitation tunable over a wide frequency range.
If the volume fraction of metal in GMN does not exceed 10-15%, the optical properties of GMN are well described within the framework of the Maxwell Garnett (MG) effective medium approximation [17]. It is worth noting however that at higher volume fraction of metal another approaches should be used [18–20]. In particular MG approximation allows one to reproduce the observed in the experiment SPR-dominated linear absorption spectra of GMN at the metal volume fraction less than ~15% provided that dielectric functions of metal and glass matrix are known. Thus one may expect that MG approximation can also be employed for description of the nonlinear optical effects in such GMN. These effects originate from the anharmonic oscillations of conduction electrons in nanoparticles, while the spectral properties of nonlinear optical response of GMN being strongly influenced by the surface plasmon modes.
In this paper, we calculate the third-order nonlinear optical susceptibility χ(3) of GMN as a function of the pump and probe wavelengths using MG approximation. The obtained results allow us to interpret recently obtained spectral map of the imaginary part of χ(3) for copper-based GMN [21] and copper film [22]. We show in particular that conventional model of the anharmonic oscillator [23] satisfactory describes the wavelength dependence of χ(3) of the copper based GMN, while the spectral features of nonlinear absorption are strongly influenced by the volume concentration and size of the metal nanoparticles. This implies that comparison of the results obtained in different experiments requires detailed information on constituencies and composition of the GMN involved.
2. χ(3) of bulk copper and copper-based GMN
At relatively low (up to 15% [24]) volume concentration of metal, the dielectric coefficient of the GMN εGMN can be obtained in the framework of the Maxwell-Garnett relation [17] in the following form:
Here f is volume fraction of the metal particles, εG and εM are permittivity of glass matrix and metal, respectively. Linear optical absorption coefficient of the GMN [25] at frequency ω is then given by:When the pump wave at frequency ω1 propagates through the nonlinear media, the optical absorption coefficient of the GMN at frequency ω2 can be described as [25]
where I is intensity of the pump wave and α2(ω1,ω2) is the so-called nonlinear absorption coefficient. The nonlinear absorption coefficient of GMN is described by the imaginary part of the relevant third-order susceptibility. If the pump wave at frequency ω1 and probe wave at frequency ω2 are co-linearly polarized, α2(ω1,ω2) can be written as [25]while the third-order susceptibility of GMN, , is given by the following equation [26]:Here is nonlinear susceptibility of metal, and L(ω1,2) are local field factors that describe enhancement of the light waves at the frequencies ω1,2 in the vicinity of a spherical metal nanoparticle [27]:One can observe from Eq. (5) that the dependence of the light-induced absorption in GMN on the frequencies of the pump and probe is governed by the third-order susceptibility of the metal and the local field factors. The local field factors can be readily obtained from Eqs. (1), (6). In this paper, we will calculate nonlinear susceptibility of metal using anharmonic oscillator model [23].
The frequency dispersion of the dielectric function in noble metals can be described by combining the contributions to the permittivity from both free conduction electrons and interband transitions due to bound d-electrons [28]:
where ε∞ is the background high-frequency dielectric constant, ωpf (ωpb) and Γf (Γb) are plasma frequency and damping rate for free (bound) electrons, respectively. One may expect that the interplay of the free and bound electrons in copper may result in interesting spectral features in the dielectric function of the copper-based GMN.Figure 1 shows the real and imaginary parts of the copper permittivity calculated using Eq. (7) (red solid line) and obtained from the standard handbook data [29]. One can observe that at ω0 = 3.46·1015s−1, ωpf = 1.39·1016s−1, ωpb = 3.10·1015s−1, Γf = 1.61·1014s−1, and Γb = 4.68·1014s−1, GMN permittivity obtained from Eq. (7) well corresponds to the experimental data for light wavelength longer than 570nm. This indicates that in this spectral range, Eq. (7) can be employed for the modeling the optical properties of copper-based GMN.
Fig. 1 Imaginary (a) and real (b) components of the permittivity of copper calculated from Eq. (5) at ω0 = 3.46·1015s−1, ωpf = 1.39·1016s−1, ωpb = 3.10·1015s−1, Γf = 1.61·1014s−1, and Γb = 4.68·1014 s−1 (solid lines) and plotted according to the handbook [29] (dash lines).
In order to describe frequency dependence of the third-order susceptibility of bulk copper we assume that optical response of the bound electrons can be described in terms of the conventional anharmonic oscillator model [23]. Generally, the consideration is valid for longer, NIR and IR, wavelengths, however, the nonlinearity of noble metal nanoparticles at corresponding frequencies is low, for their resonance falls in visible range.
Since the bulk copper possesses the inversion centre we present the potential energy of the bound electron with mass m in the following form:
where x is the electron displacement from equilibrium position and ξ is the anharmonicity parameter. The motion of the bound electrons is described by the following equation of motion:where is the electric field in the medium due to presence of the pump and probe waves at the frequencies of ω1 and ω2, respectively. The perturbative solution on Eq. (9) results in the following equation for the third-order nonlinear optical susceptibility of copper:Thus the imaginary part of the third-order susceptibility of GMN,(frequency arguments in the and are omitted) can be also presented in terms of theanharmonicity parameter ξ. By using experimental data [21,22] on and , the anharmonicity parameter, associated with bound electrons in copper, is ξ = (4.3 ± 0.2)·1032nm−2s−2. Figure 2 shows the calculated (dash lines) and experimentally measured (solid lines) imaginary part of the third-order optical susceptibility of copper for pump wavelengths λ1 = 580nm and λ1 = 620nm. One may observe that at ξ = 4.32·1032nm−2s−2, ω0 = 3.46·1015s−1, ωpf = 1.39·1016s−1, ωpb = 3.10·1015s−1, Γf = 1.61·1014s−1, and Γb = 4.68·1014s−1 (see Fig. 1) the calculated from Eq. (10) well corresponds to that measured in experiment [22] at λ2>570nm.Fig. 2 Probe wavelength dependence of the nonlinear optical susceptibility of copper measured [22] (solid lines) and calculated from Eq. (4) for pump wavelengths λ1 = 580nm (upper curves) and λ1 = 620nm (lower curves). Anharmonicity parameter ξ = 4.32·1032nm−2s−2.
3. Dependence of the optical nonlinearity of GMN on the size of nanoparticles
In order to visualize the effect of the nanoparticles size on the GMN nonlinearity we calculated the real and imaginary part of the third-order susceptibility of the copper-based GMN with different parameters using Johnson and Christy data [29] for copper permittivity. The confinement of the conduction electrons in nanoparticle results in the increase of the momentum relaxation rate Γf as [30]
where Γf∞ is the damping rate for bulk metal, νF and R are Fermi velocity in the metal and the nanoparticle radius. This results in the dependence of the linear and nonlinear opticalproperties of GMN on the nanoparticle size. Results of the calculations of for copper-based GMN are presented in Fig. 4 . One can observe that the spectral map of is qualitatively different for GMNs composed of nanoparticles with radius 1nm and 5nm. Specifically the probe wavelength at which the third-order susceptibility of the GMN changes sign depends on the size of copper nanoparticles. This dependence is pronounced in silver-based GMN because for silver nanoparticles the local field factor L(ω) shows much sharper frequency dependence (see Fig. 5 ).Fig. 3 Calculated (a) and experimental (b) spectral map [21] for glass copper nanocomposite, anharmonicity parameter ξ = 4.32·1032nm−2s−2.
Fig. 4 Imaginary part of glass-copper nanocomposite third order susceptibility, , metal volume fraction f = 10−5, particles size is marked in the figures. The spectral position where changes its sign is shown with dashed line.
Fig. 5 Calculated spectral maps of real and imaginary part of local field enhancement factor for embedded in glass silver and copper nanoparticles of the same 15 nm radius.
The obtained results explain the diversity of experimental data on the nonlinearity of GMN [15, 26, 31–36], because even relatively small variations in the particle size may produce a considerable change in the spectral properties of the nonlinear response. The broadening of particle size distribution will result in smoothing resonant features and respective decrease in the magnitude of nonlinearity. In contrary the variation of the metal volume fraction, which is often considered as the most important characteristic ofnanoparticles, can only scale the magnitude of the GMN nonlinearity leaving the spectral map unchanged. Thus comparative analysis of data obtained in different experiments is possible only if the comprehensive information on GMN parameters is available. It should be noted that such information is especially important for silver-based GMN because the local field enhancement factor for silver nanoparticles is about four orders of magnitude higher than that of copper nanoparticles and has very sharp frequency dependence.
It is worth to mention, that used MG approximation is valid for GMN containing below 15 vol.% of metal. For higher metal content the model developed can be broaden using the Sheng theory [19], which contrary to wide-spread Bruggeman theory [20] allows consideration of GMN in the vicinity of resonance frequency.
4. Conclusion
We developed simple anharmonic oscillator model to describe dependence of the third-order nonlinearity for the GMN on the frequencies of the light waves involved in the nonlinear interaction. The model is valid for GMN which can be described in the frames of Maxwell Garnett effective media approximation, that is for up to ~15 vol.% metal content in the GMN. The calculated spectral map of the nonlinear absorption coefficient in the pump-probe scheme reproduced well that obtained in the experiment for the cooper-based GMN. The dependence of the GMN nonlinearity on the frequencies of the light waves involved implies that spectral features in nonlinear response of GMN is governed by both spectral dependence of the metal dielectric function and local field enhancement factor, and hence it can be engineered by varying the size of nanoparticles. The pronounced dependence of the magnitude of the third-order nonlinearity on the particles size explains the diversity of sometimes contradictory experimental data obtained in the investigations of the nonlinear optical response of GMN made using different manufacturing methods and experimental techniques. It is essential that silver GMN demonstrate much sharper spectral dependence than copper ones, and this could be explained by strong frequency dependence of local field enhancement factor for silver nanoparticles.
Acknowledgments
This study was supported by Russian foundation for Basic Research (project#10-02-91755), Joensuu University Foundation, Academy of Finland (projects #135815 and 137859), and EU (FP7 project "Nanocom").
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