## Abstract

We study saturable absorption and the nonlinear contribution to the refractive index of metal-nanoparticle composites by using a modified self-consistent Maxwell-Garnett formalism for spherical nanoparticles and a generalization of the discrete-dipole formalism for particles of arbitrary shape and size. The results for fused silica doped with silver nanoparticles show that the saturation of loss of the composites is strongest near the surface plasmon resonance and the saturation intensity is in the range of 10 MW/cm^{2}. The nonlinear refraction index decrease with increasing intensity and its sign depends on frequency and filling factor. The predictions show that metal-nanoparticle composites can be used for mode locking of lasers in a broad spectral range down to 400 nm, where attractive saturable absorbers are still missing.

© 2010 Optical Society of America

## 1. Introduction

Over the past three decades remarkable progress has been achieved in the generation of extremely short pulses [1]. Most femtosecond lasers involve a saturable absorber for passive mode-locking or the Kerr-lens mechanism. Commonly applied saturable absorbers for passive mode-locking are semiconductors, especially multi-quantum wells (see e.g. [2]). Recently carbon nanotube mode-locking saturable absorbers (see e.g. [3, 4]) or graphene-based absorbers (see e.g. [5]) have gained much attention. However, these types of saturable absorbers can be used mainly for mode-locking of lasers in the spectral region above 700 nm. At least to our knowledge, in the short-wavelength range below 700 nm attractive saturable absorbers are still missing. In this paper, we theoretically study composites doped with metal nanoparticles (NPs) as an alternative class of saturable absorbers. The nonlinear properties of metal NPs have been extensively studied for applications in optics, medicine and biology because of their plasmon enhancement of the local electric field [6]. Several studies reported that in such composites the absorption becomes saturated in the region of the plasmon resonance (see e. g. [7–14]). The plasmon resonance of NP composites exhibit a sensitive dependence on their shapes and sizes and can be shifted to a wavelength range from the UV up to the IR, therefore a very broad frequency range for effective saturable absorbers can be realized. In previous theoretical papers (see e.g. [17, 18]), the nonlinear properties of NP composites has been studied both by analytical and numerical approaches for relatively low intensities below 1 MW/cm^{2} where the peculiarities of saturation effects at higher intensities were not taken into account. However, in this approach the total loss becomes negative for intensities above the range of MW/cm^{2} which is physically inconsistent. In particular, in [17] by using such approach valid only for low intensities we calculated the effective nonlinear optical susceptibility of dielectric composites containing NPs with different sizes and shapes. In the present paper we extend the range of validity of the theory to higher incident intensities, where the intensity-dependent change of the dielectric constant of the metal nanoparticles contained in the composites has to be taken into account. This approach allows to study the intensity-dependent nonlinear refraction and absorption in NP composites for different sizes and shapes of metal NPs in a realistic way for much higher intensities in the range of GW/cm^{2}. Our calculations are based on a self-consistent approach in which saturation effect are included both using the Maxwell-Garnet model for spherical NP particles with very small diameter and a modified discrete dipole approximation (DDA) [16] for NPs with different shapes and sizes. In particular, we study saturable absorption in fused silica doped with spherical silver NPs in the range of 400 nm, and in fused silica doped with silver nanorods in the range of 600 nm. Besides, we show that the nonlinear coefficient and the field enhancement factor are also saturated for the same intensities and frequencies. The physical origin of the saturation effect is related to the intensity-dependent intrinsic dielectric function of the NPs leading to a shift of the plasmonic resonance and therefore to a reduction of the effective nonlinear coefficients. In previous papers the saturation effect has been interpreted by the ground state plasmon bleaching related with the intrinsic electron dynamics in the metal NPs (see e. g. [12, 13, 15]). Note that at very high intensities above 7 GW/cm^{2} a regime of the reverse effect with increasing loss for increasing intensity has been observed [11] which was interpreted by two-photon interband electronic transitions in the metal. In the present paper we consider a lower intensity range where this effect can be neglected.

## 2. Self-consistent formalism of the intensity-dependent dielectric function of the composite

In the numerical simulation we use the Maxwell-Garnett theory for spherical NPs with very small diameter and the discrete-dipole approximation for nanorods and spherical NPs of arbitrary diameter. The used analytical and numerical methods are similar to those in our previous paper [17] but with an extension which takes into account the intensity-depending shift of the plasmon resonance leading to the saturation effects. The intensity-dependent dielectric function of the metal NPs is given by
${\varepsilon}_{m}\hspace{0.17em}=\hspace{0.17em}{\varepsilon}_{m0}\hspace{0.17em}+\hspace{0.17em}{\chi}_{m}^{(3)}{\left|{E}_{L}\right|}^{2}$, where *ɛ _{m}*

_{0}and ${\chi}_{m}^{(3)}$ are the (generally complex-valued) linear dielectric function and the third-order nonlinear susceptibility of the metal NPs and

*E*is the field within the particle.

_{L}For spherical particles the latter is given by *E _{L}* = 3

*ɛ*

_{h}E_{0}/(

*ɛ*+ 2

_{m}*ɛ*), where

_{h}*E*

_{0}is the incident field and

*ɛ*the linear dielectric function of the host medium. Combining the above equations, we obtain the corrected field enhancement factor

_{h}*x*=

*E*/

_{L}*E*

_{0}

By solving the above equation, the dielectric function of metal nanoparticles is self-consistently obtained. The resultant intensity-dependent dielectric function of the composite *ɛ*_{eff} can be calculated by the Maxwell-Garnett model equation

*f*is the filling factor. For relatively low intensities, using simple but lengthy algebra we obtain the Taylor expansion ${\varepsilon}_{\text{eff}}={\varepsilon}_{\text{eff}}^{(0)}+{\varepsilon}_{\text{eff}}^{(2)}I+{\varepsilon}_{\text{eff}}^{(4)}{I}^{2}\hspace{0.17em}+\hspace{0.17em}\cdots $ of the effective dielectric function of the composite, which predicts the emergence of fifth-order nonlinearity and coincides with the result of the generalized Maxwell-Garnett theory [18] and the T-matrix method [19]. However, this Taylor expansion can not be applied for intensities larger than 10 MW/cm

^{2}in the spectral range of the surface plasmon resonance (SPR), because the expansion diverges leading to a nonphysical transformation of loss into gain. Therefore, here we numerically solve the Eqs. (1) and (2). The total absorption

*α*= 2

*k*Im(

*n*

_{eff}), the nonlinear refractive index Δ

*n*= Re(Δ

*n*

_{eff}) and the absorption coefficient Δ

*α*= 2

*k*Im(Δ

*n*

_{eff}) can be obtained from the above equations, where

*k*is the wavenumber in free space and $\mathrm{\Delta}{n}_{\text{eff}}\hspace{0.17em}=\hspace{0.17em}\sqrt{{\varepsilon}_{\text{eff}}(I)}\hspace{0.17em}-\hspace{0.17em}\sqrt{{\varepsilon}_{\text{eff}}(0)}$ is the nonlinear change of effective refractive index of the composite. In Fig. 1, we show the total loss (a), (b), the change of effective refractive index (c) and absorption (d), saturation intensity and enhancement factor (e) in dependence on wavelength, light intensity and filling factor. The dielectric functions of silver and silica have been taken from [20] and the intrinsic third-order nonlinear susceptibility of silver NPs from [21]. The total loss presented in Fig. 1(a) shows a peak at the plasmon resonance and decreases with increasing intensity in the wavelength range from 412 to 520 nm. In Fig. 1(b) the total loss in dependence on the intensity for different filling factors at 430 nm is presented. It can be seen that with increasing intensity the loss decreases in a similar manner for all the filling factors, although the initial values of the loss differ by orders of magnitudes. The saturation intensity (defined as the intensity at which the linear loss is reduced by a factor of 2) is marked by the points in Fig. 1(b). At 430 nm, its value is about 100 MW/cm

^{2}for all filling factors. In Fig. 1(f) the saturation intensity (blue curve) and the field enhancement (red curve) are shown as a function of the wavelength. The saturation intensity of about 10 MW/cm

^{2}exhibits a minimum in the vicinity of the resonance, as expected intuitively and found in the previous experimental studies. At the same wavelength the field enhancement (red curve) shows a maximum. The nonlinear refractive index change Δ

*n*

_{eff}and the nonlinear coefficient

*n*

_{2}are presented in Fig. 1(c) and (e) and show a sign change from negative to positive around the SPR wavelength because of a phase difference between the field inside the NPs and the the external field due to the complex-valued character of the field enhancement factor [22]. For larger intensities the nonlinear contribution to the total loss saturates, which can be seen from its reduced values around SPR in Fig. 1(d). So far, nonlinear absorption of metal nanocomposite materials has been interpreted by the ground state plasmon bleaching and free-carrier absorption [15]. However, the above results show that it can be explained in a simple self-consistent approach within the framework of the Maxwell-Garnett model due to the intensity dependence of the intrinsic dielectric function of silver NPs.

Note that for large intensities in the range of GW/cm^{2}, the validity of the above described model is limited due to the empirically found observation that the dielectric function of metals cannot change by much more than unity.

## 3. NPs of arbitrary sizes and shapes: A modified discrete dipole approximation

Next we study the case of spherical NPs with a size comparable with the wavelength and non-spherical metal NPs with different sizes and shapes in a self-consistent formalism. Here, we apply a modification of the discrete-dipole approximation (DDA) [16].

In difference to the previously used approach we substitute the polarizability of small dipole by the self-consistent one, considering the change of dielectric function of the metal governed by Eqs. (1–2). The whole volume is divided into many small dipoles, and the local fields are given by [16]

**E**

*and*

_{j}**E**

_{0j}are the local and incident fields at the position of the

*j*-th small dipole, respectively,

*β*is the propagation constant in the surrounding medium,

*r*=|

_{jk}*r*–

_{j}*r*| is the distance between the

_{k}*j*-th and

*k*-th dipoles and

**P**

*=*

_{k}*α*

_{k}**E**

*. In the above equation the dipole polarizabilities*

_{k}*α*of

_{k}*k*-th dipoles are substituted by the the self-consistently corrected ones

*α*= (1 –

_{k}*x*)(3

_{k}*v*/4

_{k}*π*) depending on the intensity where

*v*is its volume and

_{k}*x*is given by

_{k}*x*depends on the local field

_{k}**E**

*and not on the incident field*

_{k}**E**

_{0k}. As we can see, the equation is now a nonlinear matrix equation and can be solved by the nonlinear optimization, such as the standard nonlinear conjugate gradient method [23]. Given the self-consistently obtained local field, we can calculate the effective complex permittivity by using the formula

*ɛ*

_{eff}= 〈

**D**〉/〈

*x*

**E**〉. In Fig. 2 the total loss (a), the nonlinear refraction index (b) and the nonlinear change of the refraction (c) and loss (d) are shown in dependence on the wavelength for a NP diameter of 40 nm. The behavior of these characteristics is very similar to the case of the Maxwell-Garnett approach (valid for very small NP diameter) as illustrated in Fig. 1. However, as can be seen in Fig. 2(e) the maximum of the total loss (e. g. the plasmon resonance) is shifted to larger frequencies with increasing NP diameter and its dependence on intensity also differs for different particle sizes [Fig. 2(f) and (g)]. The saturation intensity, shown in Fig. 2(d), exhibits a sensitive dependence on the particle diameter and differs for a diameter of 50 nm by more than one order of magnitude from the value in the Maxwell Garnet approach. The smallest saturation intensity for a diameter of 10 nm is about 10 MW/cm

^{2}. In Fig. 3 we consider as an example for nonspherical metal NPs a composite containing silver nanorods with a diameter of 30 nm and a length of 48 nm. In this nanostructure three plasmon modes are excited for our conditions. As can be seen in Fig. 3(a), the main peak for low intensities arising from the longitudinal dipole resonance is located at 602 nm while the other two peaks at shorter wavelengths are related with a quadrupole and transverse dipole mode. For low intensities, the absorption coefficient exhibits relatively sharp peaks, while it is smoothed and lowered with increasing intensity of the incident light. Because the quadrupole and transverse dipole SPR peaks are much weaker than that of the longitudinal dipole SPR, the saturation effect for those wavelengths is small, and at larger intensities absorption saturation is dominated by the longitudinal plasmonic dipole resonance. As we can see in Fig. 3(b), the minimum saturation intensity at 615 nm is about 12 MW/cm

^{2}which is in the same range as for spherical NP in Fig. 1 or Fig. 2 for small NP diameter. Fig. 3(c) and (d) show the spatial field enhancement distribution for very low intensities (c) and for an intensity of 100 MW/cm

^{2}(d). As seen the enhancement factors decrease with increasing intensity.

## 4. Conclusion

We studied the saturation of loss and the light-induced change of the refractive index in fused silica doped with silver nanoparticles. We used a self-consistent formalism in which saturation is included by the intensity-dependent intrinsic dielectric function of the metal NPs both within the frame of the generalized Maxwell-Garnet approach and a generalized discrete-dipole formalism for NPs with arbitrary shape. The numerical results show that the total absorption coefficient exhibit strong saturation behavior near the plasmon resonance. For spherical silver NPs with a diameter smaller than 20 nm the composite acts as a saturable absorber at 400 nm with a saturation intensity of about 10 MW/cm^{2}. Increasing the NP diameter leads to a small shift of the plasmon resonance and a larger increase of the saturation intensity. The wavelength range of saturated absorption can be significantly shifted in the whole ultraviolet, visible and near-infrared spectral region by using metal NPs with nonspherical shapes. As an example we studied a composite containing silver nanorods and predicted saturable absorption at 600 nm with a saturation intensity of 12 MW/cm^{2}. We also studied the light-induced changes of the refraction index and the field enhancement factor in the above given examples and predicted that both the nonlinear refraction index and the field enhancement factor decreases with increasing intensity.

Besides other possible applications the obtained results could be interesting for mode locking of lasers. In particular in the short-wavelength range below 700 nm advantageous saturable absorbers are still missing. As an example saturable absorbers for mode locking of high-power diode lasers in the spectral range of 400 nm are of high interest. Composites doped with metal NPs could enable the fabrication of new types of such broadband elements with ultrafast response time. In order to realize small linear loss, a thin layer of metal nanoparticles with small filling factors can be deposited on the surface of an appropriate substrate material both in reflection or transmission geometries yielding saturable absorbers with small modulation depth with a saturation intensity in the range of or larger than 10 MW/cm^{2}.

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