Linear and nonlinear optical characteristics of composites containing metal nanoparticles with different sizes and shapes

Kwang-Hyon Kim; Anton Husakou; Joachim Herrmann

doi:10.1364/OE.18.007488

1. Introduction

Metal nanoparticles (NP) play a central role in the emerging field of nanooptics and plasmonics. The interaction between light and metal NPs is dominated by charge-density oscillations on the surface of the particles, or localized plasmon resonances. These elementary electronic excitations have been the subject of extensive research, both fundamental and with a view to applications ranging from sensing and biomedicine to imaging and information technology (for a review see e.g [1–3].). To remark a few examples of exciting applications, recently plasmon enhancement has been used for Raman Scattering [4], high-order harmonic generation [5], intense photoelectron emission [6], nanolasers [7] and supercontinuum generation [8,9]. The plasmonic properties of nanoparticles exhibit a highly sensitive dependence on the shape, and the plasmonic resonance is shifted in nonspherical particles, especially rods, spheroids or triangular prisms to longer wavelengths [10,11]. A second important property of metal NPs is its intrinsically high nonlinear coefficient, which is about seven orders of magnitude higher than in fused silica. Therefore composites containing metal NPs are of special interest as nonlinear material for a multitude of applications.

In previous studies linear optical properties of composites containing metal nanoparticles, such as extinction, absorption and scattering matrix elements has been studied both by analytical [12] and numerical approaches [13]. While the analytical methods can only be applied for few types of particle shapes, the numerical methods can be used for arbitrary shapes. Especially, the discrete dipole approximation (DDA) [14–20] is a widely applied method with high calculation efficiency which allows the study of a wide range of structures with regularly and irregularly arranged metal NPs. In Refs. [21,22], the nonlinear optical properties in composites doped with metal nanospheres and spheroids in periodic arrangement has been studied.

The main purpose of this work is to study linear and nonlinear optical properties of composites containing in random arrangement noble-metal nanoparticles (NP) in dependence on the shape and size of the NPs which are of crucial importance for applications in nonlinear optical processes. Based on the discrete dipole approximation, we show that the nonlinear susceptibility is mainly determined by the NPs even for very small filling factors, it has a frequency-dependent sign and is only weakly influenced by the host material. For non-spherical NPs, such as rods and triangles, the plasmon resonance is red-shifted and different dipole and quadrupole resonances are excited, with optical parameters exhibiting several peaks and depending on the polarization of the incident light.

2. Effective medium approximation combined with the discrete dipole approximation

The enhancement of the nonlinear optical susceptibility of composites containing metal NPs can be evaluated by using the Maxwell-Garnet approach with local field corrections [23] and field enhancement factors [24–31]. However, this approach is valid only when the metal NPs are much smaller than the wavelength and for nanostructures with simple shapes (in particular for spheres, ellipsoids or coated ellipsoids), in which analytical expressions for the effective parameters can be found. In this work, we apply the discrete dipole approximation (or coupled dipole approximation) for the numerical simulation of the electromagnetic field inside and outside the NPs that is necessary for the calculation of the effective linear and nonlinear optical properties of the composites.

In the discrete dipole approximation (DDA), particles of arbitrary shapes are approximated in terms of N discrete dipoles associated with small spatial regions, where Np dipoles are situated inside of the nanoparticles and p is the nanoparticle volume fraction (filling factor). The dipole moment of each spatial region $d_{j}$ is given by $d_{j} = α_{j} E_{j}$ , where $α_{j}$ is the polarizability of the dipole, $E_{j}$ is the local field in this dipole. In the host material we can assume $α_{j} = 0$ , while the small dipoles inside the nanoparticle are characterized by the polarizabilities of small spherical nanoparticles in the host medium, as described by

α_{j} = \frac{3 v_{j}}{4 π} \frac{ε_{j} - ε_{h}}{ε_{j} + 2 ε_{h}},

where

ε_{j}

and

ε_{h}

are the dielectric function of j-th dipole and the host medium and

v_{j}

is the volume of j-th dipole, respectively. The electric field

E_{j}

at the location of the j-th dipole is the sum of the incident field

E_{j}

and the contributions from other dipoles, as given by [14,15]

E_{j} = E_{j}^{i n c} - \sum_{k = 1, j \neq k}^{N} {\hat{A}}_{j k} α_{k} E_{k},

where

E_{j}^{i n c}

is the incident field at the position of the dipole j, the summation goes over all dipoles with j≠k , respectively. The matrix coefficients

{\hat{A}}_{j k}

describe the influence of the dipole at the position k on the dipole at the position j and are given by

{\hat{A}}_{j k} = \frac{\exp (i β r_{j k})}{r_{j k}^{3}} [β^{2} (r_{j k} . r_{j k} - \hat{I}) + \frac{i β r_{j k} - 1}{r_{j k}^{2}} (3 r_{j k} . r_{j k} - \hat{I})],

where

r_{j k} = | r_{j} - r_{k} |

,

a . b

is a dyadic product formed by vectors a and b, and

β = ω / c

.

The linear system of Eqs. (2) with (3) is solved to obtain the field $E_{j}$ at the positions of the dipoles. After that, mixing rules $ε_{e f f} = 〈 D 〉 / 〈 f E 〉$ [25] and $χ_{e f f}^{(3)} = 〈 P^{N L} f E 〉 / {| 〈 f E 〉 |}^{2} {〈 f E 〉}^{2}$ [30] are applied to calculate the effective properties of the composite, where $〈 D 〉 = 〈 f ε E 〉$ , $P^{N L} = 〈 χ^{(3)} {| f |}^{2} f^{2} {| E |}^{2} E 〉$ , $f = 3 ε_{h} / (ε + 2 ε_{h})$ is the factor which relates the field inside the small nanoparticles and the local field, and $χ^{(3)}$ is the space-dependent third-order susceptibility. These mixing rules are applicable in the range of applicability of the Maxwell-Garnett theory, which extends up to filling factors of 0.2. In the host medium, the field enhancement factor is equal to 1. When the filling factor is very low and the particles are small spheres, the above equation gives the effective nonlinear polarization $〈 P^{N L} 〉 = p χ_{m}^{(3)} {| f |}^{2} f^{2} {| E_{}^{i n c} |}^{2} E_{}^{i n c}$ . This formula has been used to estimate the inherent nonlinear coefficient of noble metal particles from measurements [26].

Additionally, we calculate the extinction and absorption efficiencies, defined as $Q_{ext} = C_{ext} / (π a^{2})$ and $Q_{abs} = C_{abs} / (π a^{2})$ , where $a = {(3 V / (4 π))}^{1 / 3}$ is the geometrical effective radius of the particle, V is volume of the particle, and the extinction and absorption cross-sections are given by [14,15]

C_{ext} = \frac{4 π k}{{| E_{0} |}^{2}} \sum_{j = 1}^{N} Im (E_{j}^{i n c} * \cdot d_{j}),

C_{abs} = \frac{4 π k}{{| E_{0} |}^{2}} \sum_{j = 1}^{N} {Im [d_{j} \cdot (α_{j}^{- 1}) * d_{j} *] - \frac{2}{3} k^{3} {| d_{j} |}^{2}},

where the asterisk denotes complex conjugate.

3. Numerical simulations and discussion

For the simulation, we have developed a numerical code by using the DDA method [14,15] based on the fast Fourier transformation with the conjugate gradient method [16]. We have checked the applicability of our numerical code by comparison of the results with the experimental data on the losses of colloids containing silver nanoprisms [11]. The predicted absorption agrees well with the experimental findings, for example, the in-plane dipole resonance is positioned at 2.42 eV in the simulation and at 2.50 eV in the experiment, and the in-plane quadrupole resonance is at 3.06 eV in the simulation and 3.15 eV in the experiment.

First we study the permittivity ε, the absorption and extinction coefficient $Q_{ext}$ and $Q_{abs}$ , and the third-order susceptibility $χ^{(3)}$ of an aqueous solution containing spherical silver NPs in dependence on the diameter of the NPs.

Figure 1 shows extinction and absorption efficiency and effective dielectric function for diameters of 10 nm, 40 nm and 70 nm, respectively. Besides, results of the generalized Maxwell-Garnett (GMG) model which corresponds to an infinitely small size of the NPs are presented for comparison. The dielectric functions of silver and water are taken from [32,29]. The figure shows that the smaller the particle, the smaller the difference between the simulation and the GMG model. All curves show a resonant behavior in the vicinity of the wavelength of the surface plasmon resonance (SRP), where the imaginary part of the permittivity is significantly enhanced. Around the SPR, the effective permittivity exhibits anomalous dispersion that is characteristic for an absorption resonance. The width is nearly the same as the width of the SPR [Fig. 1(c)]. In Fig. 2 , we show the degenerate nonlinear susceptibility as a function of wavelength for different diameters of silver nanospheres. The nonlinear optical susceptibility of silver and water is taken from [8,9,28] and [33], respectively.

Fig. 1 Dependence of the effective linear optical parameters of aqueous colloid containing silver nanospheres on the particle size. In (a) the extinction efficiency $Q_{ext}$ , in (b) the absorption efficiency $Q_{abs}$ , in (c) the real and in (d) the imaginary part of effective permittivity ε are presented. The results of the generalized Maxwell-Garnett model (GMG) are presented by the red curve and numerical results are presented for NP diameters 10 nm (blue), 40 nm (green) and 70 nm (black). The filling factor is 3×10^-4.

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Fig. 2 Dependence of the real (a) and imaginary (b) part of the nonlinear optical susceptibility of aqueous colloid containing silver nanospheres on the particle size. The results of the generalized Maxwell-Garnett model (GMG) are presented by the red curve and numerical results are presented for NP diameters 10 nm (blue), 40 nm (green) and 70 nm (black). The other parameters are the same as in Fig. 1.

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As Fig. 2(a) and 2(b) show, even for low filling factors the nonlinearity of the metal NPs dominates over that of the host medium in a broad spectral region even rather far from the plasmon resonance. Only in the spectral range above 950 nm, the nonlinearity of the host medium dominates the nonlinear response, resulting in positive total nonlinearity. This only weakly depends on the host material, because the nonlinear susceptibilities of dielectric media are typically in the range of 10^-20 ~ 10^-22 m²/V² which is several orders smaller than the inherent nonlinear susceptibility of metals. We find that the maximum effective nonlinear susceptibility $χ^{(3)}$ is larger in those cases for which the ratio of peak absorption to the peak extinction is larger (at the parameters Q_abs/Q_ext = 0.989, 0.769 and 0.428 for particle diameters of 10, 40 and 70 nm, respectively). We relate this finding to the fact that both high absorption and high nonlinearity are associated with significant field enhancement in the metal nanoparticles.

Figures 1 and 2 show that the peaks of linear and nonlinear parameters become wider and lower and are red-shifted with the increase of the particle size due to the red-shift of the surface plasmon resonance wavelength.

Because the real part of nonlinear susceptibility of metal is negative while that of dielectric material is positive, the sign of the real part of nonlinear susceptibility in Fig. 2(a) is changed at a certain wavelength which is red-shifted with decreasing filling factor and increasing particle size.

To study the influence of the host media we presented in Fig. 3 the results for the nonlinear coefficient $Re [χ^{(3)}]$ of a composite of carbon disulfide (CS₂) with a higher nonlinear coefficient and silver nanospheres for different diameters in dependence on the wavelengths. The permittivity and third-order nonlinear susceptibility of carbon disulfide is taken from [28] in Fig 3.

Fig. 3 Effective nonlinear optical susceptibility of CS₂ colloid containing silver nanospheres with various diameters with a filling factor of 3×10^-5.

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In the figure, the results of the generalized Maxwell-Garnett (GMG) model describe the case of an infinitely small size of the particles. At 532 nm, the generalized Maxwell-Garnett model yields a real part of the third-order susceptibility of -1.42×10^-20 m²/V² corresponding to a nonlinear refractive index of -2.01×10^-14 cm²/W. This value agrees well with the experimental data (see Fig. 5(a) of reference [28]). One can see that the effective nonlinear coefficient changes the sign at a wavelength which increases with increasing particle size: at 590 nm, 693 nm and 824 nm for particle diameters of 10 nm, 40 nm and 70 nm respectively, while the GMG model predicts a sign change at 539 nm. For longer wavelengths the nonlinear coefficient approaches to that of the host material. Because the nonlinear susceptibility of carbon disulfide is much larger than that of water, the sign of the effective nonlinear susceptibility is changed at shorter wavelengths compared to such wavelength in the case of aqueous colloids.

Fig. 5 Linear and nonlinear optical parameters of fused silica doped with silver nanoparticles with different shapes, as nanospheres (a)-(c), nanorods (d)-(f) and nanotriangles (g)-(i). The diameters and side lengths are all the same and as much as 30 nm, the length of the nanorods is 40 nm, the thickness of the nanotriangles is 15 nm and the filling factor is 10^-4. All the quantities are averaged for all the possible polarization directions versus the orientations of nanoparticles in space.

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These results can be used to design a composite system with desirable wavelength-dependent nonlinear properties by the choice of appropriate host materials and diameters of the nanosheres. Let us now study the optical properties of composites containing non-sherical NPs with different shapes. The shape of the particles significantly influences the field intensity inside and out of the particles. In particular for non-spherical NPs the field intensity at the sharp edges is strongly increased. This can be seen in Fig. 4 which shows the field distribution inside and outside of silver nanotriangles in silica for different polarizations and wavelengths of incident light.

Fig. 4 Field distribution near silver nanotriangles with side length and thickness of 45 nm and 15 nm, respectively, in a silica composite The polarization of the incident field as shown in the inset is parallel to one side of the nanotriangle (a),(b) and in the direction of its bisector (c),(d).

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As shown in Fig. 5, the number of peaks of the nonlinear susceptibility increases for the nanoparticles with many sharp edges, which offer the possibility of control of the nonlinear properties. The amplitude of the resonance peaks of the susceptibility decreases for particles with several sharp edges, but the nonresonant susceptibility in the long-wavelength tail is enhancedfor such particles. The field redistribution and enhancement in this case results from excitation of both dipole resonances and quadrupole resonances. The complicated nature of the excitation results in sensitive dependence of the field distribution on the wavelength and polarization, as well as in the field variation inside of the nanoparticles.

In the case of nanotriangles and nanorods, their height is different from their side length and diameter, respectively. Therefore, in- and out-plane SPR wavelengths are different from each other contrary to the case of nanospheres. Correspondingly, the effective optical characteristics exhibit more than two peaks induced from the different dipole and quadrupole resonances arising from the different arrangement of nanoparticles versus polarization direction of the incident light. To clarify this phenomenon, in Fig. 6 the linear and nonlinear optical susceptibilities of silver nanorods are calculated for different polarization of the incident light. The geometry and material parameters are the same as in Fig. 5. In the case of a polarization perpendicular to the axis of the nanorods (in-plane polarization) the SPR wavelengths are at 475 nm (dipole resonance) and 399 nm (quadrupole resonance) while, in case of polarization parallel to the axis (out-of-plane polarization) they are at 562 nm (dipole-resonance) and at 357 nm (quadrupole-resonance). Near the corresponding wavelengths, the linear and nonlinear optical parameters are greatly increased. As a result, arbitrarily oriented nanoparticles in the composites exhibit an SPR-induced enhancement with lower and smoothed peaks.

Fig. 6 Dielectric function and nonlinear optical susceptibilities of fused silica doped with silver nanorods for different polarization of the incident light. The black curves represent the result for in-plane polarization (polarization is perpendicular to the axis of nanorod), the green curves refer to out-of-plane polarization (polarization is parallel to the axis), and the red curves are direction averaged quantities. The parameters are the same as in Fig. 5(d)–5(f).

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4. Conclusion

We calculated the effective linear and nonlinear optical parameters of materials doped with noble silver nanoparticles, using the effective medium theory combined with the discrete dipole approximation. We numerically evaluated the absorption and extinction cross-sections, the permittivity and the third-order nonlinear susceptibility of composites containing silver NPs in dependence on its size and shape and compared it with results calculated by using the generalized Maxwell-Garnett model. The effective dielectric function and the nonlinear susceptibility is significantly enhanced by the plasmon resonance which is shifted to longer wavelengths with increasing diameter of spherical NPs. The nonlinear susceptibility is mainly determined by the NPs even for very small filling factors, it has a frequency-dependent sign and is only weakly influenced by the host material. The linear and nonlinear optical parameters are also calculated for non-spherical NPs, such as rods and triangles. In this case the plasmon resonance is red-shifted and different dipole and quadrupole resonances are excited, with optical parameters exhibiting several peaks and depending on the polarization of the incident light. The results reported here demonstrate the possibility for a significant enhancement of the nonlinearity in a desired frequency range with a promising potential of applications in nonlinear optics.

Acknowledgements

We gratefully acknowledge financial support from Daimler-Benz-Foundation and German Research Foundation (DFG).

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Linear and nonlinear optical characteristics of composites containing metal nanoparticles with different sizes and shapes

Abstract

1. Introduction

2. Effective medium approximation combined with the discrete dipole approximation

3. Numerical simulations and discussion

4. Conclusion

Acknowledgements

References and links

Cited By

Figures (6)

Equations (5)

Optics Express