1 Introduction

Composite materials containing metallic NPs have received an extraordinary attention due to their high NL optical susceptibilities [1, 2]. In composite materials with a Maxwell–Garnett topology, it is observed that the optical response depends of many variables like shape and NPs size, dielectric constant of the constituents, filling factor, NL susceptibilities of the inclusion and the host, and the wavelength and pulse duration of the excitation laser. The influence of each parameter on the linear and NL optical properties was studied from the theoretical and experimental point of view. For example, the dependence of the NL response on the NPs size was recently studied for silver and gold NPs using the degenerate electron gas model [3–5]. The influence of the dielectric constant of the host on the NL third-order susceptibility was studied and analyzed using the Maxwell–Garnett model [6, 7]. In addition, differences in the NL response were observed at different wavelength and pulse duration of the excitation laser in composites containing gold NPs [8]. The influence of the filling factor on the limit of low concentrations was studied, and the cancellations of the NL refraction and NL absorption with the increasing of the filling fraction were observed [9, 10]. In both cases, the contribution of the NL optical properties of the host played an important role to describe the reversal sign. In the case of composites containing high concentration of NPs, the NL optical properties were studied experimentally in the femtosecond, picosecond, and nanosecond regime [8, 11, 12].

Here we analyze theoretically the influence of high values of the filling factor in the linear and NL optical properties of gold NPs using the degenerate electron gas model and the Maxwell–Garnett model.

2 Linear optical properties

Consider a composite material containing metallic spherical NPs in a host with negligible optical response. The linear optical properties can be studied using the Maxwell–Garnett model, which establishes the relationship between the dielectric constant of the composite material, ε, with the dielectric constants of the NPs ε NP, and the host, ε h, as:

$$ \varepsilon = \varepsilon_{\text{h}} \left( {1 + \frac{3\beta p}{1 - \beta p}} \right) $$
(1)

where \( \beta = (\varepsilon_{\text{NP}} - \varepsilon_{\text{h}} )/(\varepsilon_{\text{NP}} + 2\varepsilon_{\text{h}} ) \) and p is the filling factor. The linear absorption coefficient, α, and linear refraction index, n, can be expressed as:

$$ n = \sqrt {\frac{{\varepsilon^{'} + \sqrt {\varepsilon^{'2} + \varepsilon^{''2} } }}{2}} $$
(2)
$$ \alpha = \frac{2\pi }{\lambda }\frac{{\varepsilon^{''} }}{n} $$
(3)

where λ is the wavelength, and ε′ and ε″ are the real and imaginary parts of ε, respectively. The dependence of the NPs dielectric constant on frequency and size was calculated previously for gold NPs using the Critical Point model including corrections to the damping parameter [3]. Therefore, the NPs dielectric constant can be written as:

$$ \varepsilon_{\text{NP}} (\omega , R) = \varepsilon_{{{\text{C}} . {\text{P}}}} (\omega ) + i\frac{{A\omega_{\text{p}}^{2} v_{\text{F}} }}{{\omega^{3} R}} $$
(4)

where \( \varepsilon_{{{\text{C}} . {\text{P}}}} (\omega ) \) is the dielectric constant of gold as a function of frequency calculated using the Critical Point model, A is the scattering parameter, ω p is the plasma frequency, v F is the Fermi velocity, \( \omega = 2\pi /\lambda \) is the frequency, and R is the NPs radius.

In the limit of low values of the filling factor, an analytical expression for the linear absorption coefficient can be calculated from (3) using (1) and (2), as follows:

$$ \alpha (\lambda ,R) = \frac{18\pi p}{\lambda }\varepsilon_{\text{h}}^{3/2} \frac{{\varepsilon_{\text{NP}}^{''} (\lambda ,R)}}{{\left( {\varepsilon_{\text{NP}}^{'} (\lambda ,R) + 2\varepsilon_{\text{h}} } \right)^{2} + \varepsilon_{\text{NP}}^{''2} (\lambda ,R)}} . $$
(5)

Equation (5) shows a linear dependence of the linear absorption coefficient on p, and a maximum value, known as plasmon resonance, when \( (\varepsilon_{\text{NP}}^{'} (\lambda ,R) + 2\varepsilon_{\text{h}} )^{2} + \varepsilon_{\text{NP}}^{''2} (\lambda ,R) \approx 0. \) In addition, the linear refraction index calculated from Eq. (2) can be approximated to \( n \approx \sqrt {\varepsilon^{'} } \) because \( \varepsilon^{''} \approx 0 \) in the visible spectrum. As the real part of the dielectric constant is practically constant, the linear refraction index is constant too.

However, at high values of p it is important to take into account the contribution of the polarizations of neighboring NPs to the optical response. Taking into account this consideration, the linear optical properties of a Maxwell–Garnett composite with different gold NPs size and filling factor in a vitreous host (ε h = 2.25) were calculated using Eqs. (1)–(3), and the results are shown in Figs. 1, 2, 3 and 4.

Fig. 1
figure 1

Spectral dependence of the absorption coefficient for different values of filling factor

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Fig. 2
figure 2

Dependence of the plasmon resonance peak on the filling factor for different values of the NPs size

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Fig. 3
figure 3

Spectral dependence of the linear refraction index for different values of the filling factor

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Fig. 4
figure 4

Spectral dependence of the linear refraction index for different values of the NPs size

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The linear absorption coefficient spectra are shown in Fig. 1 for different values of p. From these spectra, a red shift of the plasmon resonance peak (PRP) is observed when the filling factor is increased. The dependence of the PRP on the NPs size and p is shown in Fig. 2. The PRP is red shifted when the NPs size is increased and has a strong NL variation for p > 0.01. For p < 0.01, it is observed that the PRP is practically constant for a composite with constant NPs size. On the other hand, the dependence of the linear refraction index on the filling factor and the NPs size is shown in Figs. 3 and 4. For low values of p, the linear refraction index is practically constant in the visible spectrum as shown in Fig. 3. For high values of the filling factor, low and high values of the linear refraction index were predicted around the plasmon resonance and near IR region, respectively. In Fig. 4, the dispersion of the linear refraction index was calculated considering a constant value of p (p = 0.3) and increasing the NPs size. From the figure it is observed that the linear refraction index has values lower than 1 around the PRP, letting the absorption coefficient to have high values accordingly with Eq. (3). Besides, for wavelengths around of 600 nm, the linear refraction index is increased when the NPs size is increased, indicating high nonlinear response accordingly with the Miller’s rule [13]. Is important to note from Fig. 3 that, if the filling factor is more increased, negative values of n can be obtained as reported for similar nanostructures [14–16]. From the experimental point of view, measurements of the linear optical properties of composites containing gold NPs are reported in the literature, and the Maxwell–Garnett model presented here well describes the published results [17–19].

3 Nonlinear optical properties

For a composite with a Maxwell–Garnett topology and negligible optical response of the host, the NL optical properties can be calculated using the Maxwell–Garnett model and the degenerate electron gas model. For low values of the filling factor, and considering that the composite is in the presence of an external electric field E 0, the electric field inside of the NP is \( E_{\text{in}} = fE_{\text{o}} \), where the term \( f = 3\varepsilon_{\text{h}} /(\varepsilon_{\text{NP}} + 2\varepsilon_{\text{h}} ) \) is the local field factor. Under this condition, the third-order NL susceptibility of the composite material \( \chi_{\text{eff}}^{(3)} \), can be written as \( \chi_{\text{eff}}^{(3)} = pf^{2} \left| f \right|^{2} \chi_{\text{NP}}^{(3)} \), where \( \chi_{\text{NP}}^{(3)} \) is the third-order NL susceptibility of the NPs. Using this approach, the NL optical properties were calculated for composites containing silver and gold NPs [3, 4]. However, at high values of p, the electric field surrounding the NP is affected by the contributions of the other NPs. Then, the local field factor is corrected and the NL susceptibility can be written as \( \chi_{\text{eff}}^{(3)} = p\eta^{2} \left| \eta \right|^{2} \chi_{\text{NP}}^{(3)} , \) where \( \eta = (\varepsilon + 2\varepsilon_{\text{h}} )/(\varepsilon_{\text{NP}} + 2\varepsilon_{\text{h}} ) \) [20]. From this expression, the NL dependence of \( \chi_{\text{eff}}^{(3)} \) with p is given by the corrected local field factor, η.

The NPs third-order NL susceptibility can be calculated using the degenerate electron gas model, which describes the \( \chi_{\text{NP}}^{(3)} \) considering that an electron is in an infinitely deep spherical potential well of radius R. In the limit of 1 ≫ (R/λ) ≫ (v F/2πc), \( \chi_{\text{NP}}^{(3)} \) can be written as [5]:

$$ \chi_{\text{NP}}^{(3)} = \frac{2}{15}\left( {\frac{{e^{2} n}}{{m\omega^{2} }}} \right)\left( {\frac{eR}{\hbar \omega }} \right)^{2} \frac{{\Upgamma_{2} }}{{\Upgamma_{1} }}\left\{ {F_{3} - i\left[ {\frac{{2\Upgamma_{2} }}{\omega }F_{3} + \left( {\frac{\omega }{{2\Upgamma_{2} }}} \right)^{2} \left( {\frac{{v_{\text{F}} }}{R\omega }} \right)^{5} g_{3} } \right]} \right\} $$
(6)

where m is the electron mass, e is the electron charge, n is the electron density, ω = 2πc/λ is the frequency, c is the speed of light, ħ is the reduced Planck constant, v F is the Fermi velocity, F 3 ranges from 0.30 to 0.33 for particles varying between 2 and 15 nm, g 3 = 0.64, and Γ1 and Γ2 represent the relaxation rates for the population and coherence, respectively. The values of Γ1 and Γ2 were determined previously for composites containing gold NPs [3].

The calculations of the NL optical properties of a composite containing gold NPs with radii of 3 nm embedded in a host with a linear refraction index of 1.5 are given in Figs. 5 and 6. Results show changes in the sign of the real and imaginary part of the NL susceptibility. For example, at 532 nm (560 nm), the real (imaginary) part changes from negative (positive) to positive (negative), whereas the imaginary (real) part has a NL behavior without reversal of the sign. Changes in almost three orders of magnitude for the imaginary and real parts of the NL susceptibility at 650 and 625 nm from 1.8 × 10−10 and 3.0 × 10−10 esu for p = 0.1 to 1.31 × 10−7 and −2.1 × 10−7 esu for p = 0.5 were predicted, respectively.

Fig. 5
figure 5

Spectral dependence of the real part of the third-order NL susceptibility for different values of the filling factor

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Fig. 6
figure 6

Spectral dependence of the imaginary part of the third-order NL susceptibility for different values of the filling factor

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Influence of the NPs size on the NL response was calculated for p = 0.3, and the results are shown in Figs. 7 and 8. From figures, it is observed that the position of the maximum response is not altered, and that the NL response is nonlinearly increased when the NPs size is increased. The dependence of \( \chi_{\text{eff}}^{(3)} \) with R is given through the Eqs. 4 and 5 for \( \varepsilon_{\text{NP}} (\omega , R) \) and \( \chi_{\text{NP}}^{(3)} \), respectively.

Fig. 7
figure 7

Spectral dependence of the real part of the third-order NL susceptibility for different values of the NPs size

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Fig. 8
figure 8

Spectral dependence of the imaginary part of the third-order NL susceptibility for different values of the NPs size

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To understand the influence of the filling factor in the NL response, is important to observe that the corrected local field can be written as \( \eta = f/(1 - \beta p) \), where the term \( 1/(1 - \beta p) \) is the correction factor of f. Therefore, differences between the NL susceptibility at high and low values of the filling factor are given by the ratio \( \chi_{\text{eff, high}}^{(3)} /\chi_{\text{eff, low}}^{(3)} = (1 - \beta p)^{2} \left| {(1 - \beta p)} \right|^{2} \). At low values of p, the term \( 1 - \beta p \) is practically real and the sign of the NL susceptibility of the composite is given by the NL susceptibility of the NPs. However, at high values of p, the imaginary part of \( 1 - \beta p \) is not negligible and will be responsible by the dependence of the real and imaginary parts of \( \chi_{\text{eff}}^{(3)} \) on p.

From the experimental point of view, the Fig. 9 shows the measured values of the \( |\chi_{\text{eff}}^{(3)} | \) obtained in the femtosecond regime at 532 nm for composites with high values of the filling factor of gold NPs in SiO2 [8]. Accordingly, with the experiment, samples with high values of p were obtained by co-sputtering of Au and SiO2 in a multitarget magnetron sputtering system. The Au:SiO2 films were deposited on a fuse quartz substrate at 150 °C in an Ar atmosphere. Right after the deposition, the samples with different Au concentrations were annealed at 850 °C for 1 min in a rapid annealing furnace in the presence of Ar gas at 1 atm, and then cooled down in a flowing Ar atmosphere. The average size of the gold NPs was determined using X-ray diffraction data and the Scherrer equation. Small NPs with size less than 3 nm were expected for p < 0.25, and for filling factors greater than 0.25, the gold NPs size was in the range between 3 and 80 nm. Increasing the filling factor, the PRP shifted to the red due to the size increase of the gold NPs [21]. TEM images of these samples were not published; however, TEM images of films of Au:Al2O3 fabricated using the same technique reveals that the NPs size is similar for the Au:SiO2 films in the same range of concentrations [22]. For comparison, the Fig. 9 also shows the calculated values of \( |\chi_{\text{eff}}^{(3)} | \) for different NPs size. From the figure, it is observed that the measured values of \( |\chi_{\text{eff}}^{(3)} | \) are in the range of the \( |\chi_{\text{eff}}^{(3)} | \) calculated.

Fig. 9
figure 9

Dependence of \( \left| {\chi_{\text{eff}}^{(3)} } \right| \) on the filling factor for different NPs size. Experimental values from [7]

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As the model can describe the nonlinear third-order susceptibility of highly concentrated gold NPs, photonic applications can be explored through the figures of merit. For optical switching devices, \( T = 2\alpha_{2} \lambda /n_{2} < 1 \) must be satisfied, where \( \alpha_{2} \) and \( n_{2} \) are the NL absorption and NL refraction coefficients, respectively. The relations between the NL coefficients and the real and imaginary parts of the NL susceptibility are given by \( n_{2} = \frac{3}{{4\varepsilon_{0} n^{2} c^{2} }}\text{Re} [\chi_{\text{eff}}^{(3)} ] \) and \( \alpha_{2} = \frac{3\omega }{{2\varepsilon_{0} n^{2} c}}\text{Im} [\chi_{\text{eff}}^{(3)} ] \), where \( \varepsilon_{0} \) is the vacuum permittivity. Therefore, \( T = 8\pi \frac{{\text{Im} [\chi_{\text{eff}}^{(3)} ]}}{{\text{Re} [\chi_{\text{eff}}^{(3)} ]}} < 1 \) [23]. To satisfy this relation, the value of \( \text{Im} [\chi_{\text{eff}}^{(3)} ] \) must be closer to zero, which can be observed from Fig. 6 at 532 nm when the reversal sign of the \( \text{Im} [\chi_{\text{eff}}^{(3)} ] \) is obtained for filling factors between 0.2 and 0.3. For others, wavelengths is possible to tune the dispersion curve changing the value of p to obtain \( \text{Im} [\chi_{\text{eff}}^{(3)} ] \approx 0 \).

In addition, for a photonic device, it is expected to have a material with high value of \( \left| {\chi_{\text{eff}}^{(3)} } \right| \) and low value of α. Then, the figure of merit \( \left| {\chi_{\text{eff}}^{(3)} } \right|/\alpha \) was calculated at 532 nm as a function of the filling factor for different values of the NPs size, and results are shown in Fig. 10. Similar results were obtained for Au:SiO2 composites excited with a picosecond laser [21]. From the figure, it is observed that the figure of merit can be calculated up to a limit value of the filling factor depending of the NPs size. This limit is established by the linear absorption coefficient and the linear refraction index which were calculated using the Maxwell–Garnett model.

Fig. 10
figure 10

Dependence of \( \left| {\chi_{\text{eff}}^{(3)} } \right|/\alpha \) on the filling factor for different NPs size

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4 Conclusion

The calculations of the linear optical properties using the Maxwell–Garnett model reveal the limit of applicability of the model depending of the NPs size. For small NPs, the linear optical properties can be calculated for filling factors greater than for large NPs.

The NL third-order susceptibility calculated using the Maxwell–Garnett model and the degenerate electron gas model is in agreement with the experimental values reported in the literature, and predict the reversal sign of the NL response depending of the filling factor. For composites with high concentration of gold NPs, the requirements given by the figures of merit for optical devices can be fulfilled depending of the filling factor and the NPs size.