1. Introduction

Surface science relies on probing and characterization techniques. Second Harmonic Generation (SHG) has proven to be a surface sensitive measurement tool, hence it has been used for in situ investigations for several decades [1]. For centrosymmetric materials SH is generated at the surface of nano-particles, because symmetry is broken at that interface as Wang has demonstrated experimentally for the first time [2]. SHG has been applied to study the surface and buried micro-structure of colloidal nano-particles [3–13] or as probing technique in bio-physical experiments [14–18]. We have used SHG to determine the nonlinear susceptibility ${\chi }^{\left(2\right)}$ of a thin layer of Malachite Green [19], to study the layer of oriented water near polystyrene particles [20] and the organization of polyelectrolyte chains, depending on the salt concentration of the solvent [21].

SHG has also been used to investigate the surface of metals, where surface plasmons can be excited. First experimental and theoretical studies were performed on plane surfaces [22–27]. Recently studies on SHG or Hyper Rayleigh [28] scattering were performed on colloidal metallic nanoparticles [29–36] as well as on particles embedded in a matrix [37,38] and more complex plasmonic structures [39–42].

There exist several theoretical models for SHG from spherical nano-particles. The response of small particles can be modeled by Second-Harmonic Rayleigh Scattering [43–46]. For small particles of low refractive index contrast, which do not significantly perturb the incident field, also the Rayleigh-Gans-Debye approximation can be used [47–51]. Larger spheres or such with a high refractive index contrast require the application of a full Mie theory, based on which several numerical models have been developed for solid particles [18,19,52–54]. Only recently, a nonlinear Mie model for core-shell particles has been applied to describe the wavelength-dependence of SHG on nanoparticles with up to 100 nm outer diameter and with fixed gold and silver shells of 10 nm thickness [55,56]. Simulations were used to investigate the dependence of the SH signal on the refractive indices of the core and the surrounding.

Here, we apply a refined nonlinear Mie model [19] to investigate in detail the nature of the various plasmonic resonances, which influence the SHG process from dielectric-metallic core-shell particles with variable shell thickness. We analyze the resonant behavior of the Second Harmonic intensity when varying the geometry and the material parameters of the nanoparticle. We find that even the nonlinear response is solely dominated by plasmonic resonances at fundamental harmonic (FH) and the second harmonic (SH) frequencies.

As the aim of our paper is to enable an analysis of experimentally obtained scattering data a typical experimental set-up is introduced at the beginning. This allows introducing experimentally relevant parameters, which will later be discussed in more detail. After that the general behavior of SHG and its dependence on particle parameters is discussed. Because plasmonic resonances play a decisive role in SHG we discuss the excitation and propagation of plasmons.

2. Nonlinear Mie model

In most other nonlinear Mie models second harmonic generation is assumed to take place in a nonlinear core or at an interface, which all have spherical symmetry. Solutions for the scattering coefficients are than obtained by decomposing nonlinear products of vector spherical harmonics into the basis functions of the Mie model and by evaluating respective boundary conditions [52,53]. Here we use a different approach and start from a molecular Mie model that we have introduced earlier [19]: The nonlinear polarization

Mie theory, which is the basis of our model, relies on the expansion of all fields into a set of vector spherical harmonics ${\overrightarrow{M}}_{m,n}^{(j)}$ and ${\overrightarrow{N}}_{m,n}^{(j)}$ $j=1$$j=3$with expansion coefficients $a_{m,n}$, $b_{m,n}$, $c_{m,n}$, and $d_{m,n}$:

Standard Mie theory only evaluates one boundary condition for the incident electric field and scattered fields inside and outside the sphere

For metallic particles it is essential, that some modifications are added to Mie theory for stability [59,60]. The refractive index, which has a dominant imaginary part in case of metals, enters the equations through the argument of the Bessel functions. Bessel functions of an imaginary argument diverge exponentially and therefore make Mie scattering unstable for highly absorbing particles. In the original equations for the boundary conditions in Mie-Theory, the Bessel functions and their derivatives are therefore replaced by ratios of Bessel functions and logarithmic derivatives for which stable recursion formulas exist [58].

The polarization ${\overrightarrow{P}}^{\text{(2)}}\left({\overrightarrow{r}}_d\right)$ is resembled by dipoles at discrete positions ${\overrightarrow{r}}_d$. The electric fields of a dipole at position ${\overrightarrow{r}}_d$ are calculated from a dipole at the origin ${\overrightarrow{E}}_{}^{\text{Dipole}}\left(\overrightarrow{r}\right)={\overrightarrow{N}}_{0,1}^{\left(j=3\right)}\left(\overrightarrow{r}\right)$ by a translation theorem [61]. This results in different expansions, depending on $\overrightarrow{r}$:

3. Results and discussion

In typical experiments [20] the SH fields, generated at the surface of nanoparticles, are measured as a function of the scattering angle $\theta$ and for the polarizations of the incident and the detected light being either parallel (p) or perpendicular (s) to the scattering plane (Fig. 1). As high power pulses are required we assume the FH to be generated by a Ti:Sa-Laser at 800 nm wavelength. The intensity and the polarization of the scattered SH light at 400 nm is detected. The respective intensities are given for example by $I_{\text{sp}}\left(\theta \right)$, if the incident fundamental harmonic is s-polarized and the p-polarized second harmonic is detected under an angle θ with respect to the exciting laser pulse. $I_{\text{pp}}\left(\theta \right)$ is measured if both beams are p-polarized. Due to symmetry reasons $I_{\text{ps}}\left(\theta \right)$ and $I_{\text{ss}}\left(\theta \right)$ are identical to zero for spherical particles.

 

Fig. 1 a) typical experimental setup for angle-resolved SHG from colloidal core-shell particles. b) Angular scattering pattern $I_{\text{sp}}\left(\theta \right)$ (FH s-polarized, SH p-polarized) and $I_{\text{pp}}\left(\theta \right)$ (FH and SH p-polarized) for a silica particle (radius 100 nm) with a silver shell of 6.3 nm thickness (compare Fig. 2).

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Here we simulated the SH-intensities $I_{\text{pp}}\left(\theta \right)$ and $I_{\text{sp}}\left(\theta \right)$ for a nano-particle consisting of a silica core (radius 100 nm) and a silver shell, which is suspended in water. Dielectric constants are taken from Johnson and Christy [62]. In addition we first assume a dominating ${\chi }_{zzz}^{\left(2\right)}$ [55,63–65] for the nonlinear surface susceptibility of the silica-silver and the silver-water interfaces. We will later investigate the effect of the nonlinear coefficient on the SH-signal.

A typical angular dependent emission spectrum is displayed in Fig. 1. For symmetry reasons the SH emission in forward direction is always zero (but a nonzero intensity might be measured in the experiment due to a finite aperture and detector size), but characteristic peaks emerge for higher angles. Varying the thickness of the shell between 0 nm and 30 nm causes changes in the angular emission pattern, but mainly influences the total efficiency of SHG (Fig. 2). Resonances occur e.g. at 4 nm, 7 nm or 25 nm thickness and are even more pronounced, if an idealized material without losses is assumed for the metallic shell. When plasmon losses are included (${ϵ}_{\text{shell}}^{\text{400} \text{nm}}=-4.4+0.2i$, ${ϵ}_{\text{shell}}^{\text{800} \text{nm}}=-30.8+0.4i$), the resonant enhancement is reduced and the resonance at 25 nm shell thickness even disappears (Fig. 2).

 

Fig. 2 Angle-resolved SH-intensity $I_{\text{pp}}\left(\theta \right)$ and $I_{\text{sp}}\left(\theta \right)$ (a,b) and mean intensity, averaged over the detection angle (c,d) for a growing silver shell on a silica particle. For the metallic shell either idealized silver without plasmon losses ${ϵ}_{\text{shell}}^{\text{400} \text{nm}}=-4.4$, ${ϵ}_{\text{shell}}^{\text{800} \text{nm}}=-30.8$ (a,b,c) or realistic values, including plasmon losses ${ϵ}_{\text{shell}}^{\text{400} \text{nm}}=-4.4+0.2i$, ${ϵ}_{\text{shell}}^{\text{800} \text{nm}}=-30.8+0.4i$, are used (d).

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To understand these resonances, we have to keep in mind that SHG takes place in two steps. The first one includes the formation of the nonlinear polarization at the inner and outer interface of the core-shell particle. The nonlinear polarization is directly proportional to the square of the scattered FH wave that excites the particle. Hence, plasmonic resonances at FH frequency lead to a higher SH-intensity due to the increased field strength at the two interfaces.

The second step is calculating the electromagnetic field at SH frequency that is produced by the nonlinear polarization. To this end we decompose the nonlinear polarization into an ensemble of dipolar point sources each emitting an electromagnetic field at SH frequency. This emission process is again heavily determined by resonances of the particle, but now at SH frequency.

Field enhancement at such Mie resonances happens for both the radial ${\left|E_r\right|}^2$ and azimuthal ${\left|E_{\vartheta }\right|}^2$, ${\left|E_{\varphi }\right|}^2$ components of the electric field simultaneously. Also the inner (core-shell) and the outer (shell-surrounding) interfaces of the shell (Fig. 3) experience qualitatively the same field strength. Strength and width of resonances are heavily influenced by the shell thickness. FH resonances occur for a shell thickness below 10 nm. Their spectral position is equally reproduced by peaks of the SH-intensity. In contrast the linear extinction spectrum, which is usually measured does not show such a significant dependence on the shell thickness because the farfield is much less sensitive to resonant effects compared with the the fundamental harmonic nearfield at the surface of the sphere which finally causes the SH emission. For metals with higher plasmon losses, like gold, this difference between linear extinction spectra and field enhancement at the surface of the sphere becomes even more pronounced.

 

Fig. 3 Mean electric field ${〈E_r\left(\vartheta \right)〉}_{\vartheta }$ induced by a scattered plane wave (fundamental harmonic wavelength) at the inner ($r_{in}$) and outer ($r_{out}$) interface of a sphere with silica core of radius 100 nm and a silver shell of variable thickness with (${ϵ}_{\text{shell}}^{\text{800} \text{nm}}=-30.8+0.4i$) and without (${ϵ}_{\text{shell}}^{\text{800} \text{nm}}=-30.8$) damping. Additionally, the extinction at $\lambda =800 \text{nm}$ is given as a function of the shell thickness.

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A second possible reason for enhancement of SHG is the excitation of a resonance at SH frequency. The emission of a dipole is influenced by the sphere (Fig. 4). This influence can be very dramatic in the case of a plasmonic resonance: The emitting dipole is hardly visible and all radiation seems to emerge from the particle in a star-like pattern. For dipoles with radial orientation the generated tangential field distribution shows a minimum at the position of the dipole and on the opposite site of the sphere. When the dipole is parallel to the surface, maxima and minima switch places, so that a maximum is observed at the position of the dipole and on the opposite site of the sphere. Obviously the near-field of the dipole is imaged onto the opposite side of the sphere, showing that a spherical surface has perfect imaging properties even on a sub wavelength scale.

 

Fig. 4 Field distribution ${\log }_{10}{\left|\overrightarrow{E}\right|}^2$ emitted by a dipole radiating at 400 nm wavelength (a) and respective field distribution ${\log }_{10}{\left|E_{\vartheta }\right|}^2$ modified by the presence of a sphere (b,c). The sphere has a silica core of radius 100 nm, and silver shell of thickness $D=25 \text{nm}$. The shell thickness is tuned for the resonance $K=5$ with 10 lobes. For the metallic shell either idealized silver without plasmon losses ${ϵ}_{\text{shell}}^{\text{400} \text{nm}}=-4.4$ (b) or realistic ${ϵ}_{\text{shell}}^{\text{400} \text{nm}}=-4.4+0.2i$ is used (c).

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The situation of a single dipole close to a sphere is similar to single molecule spectroscopy performed near a metallic surface in order to profit from the enhanced local fields as they are employed in surface enhanced Raman scattering (SERS). Here, the environment influences the molecular emission properties [66]. Kuhn et al. [67] even used a spherical gold nanoparticle as an optical nanoantenna to enhance the fluorescence signal.

Although FH resonances are excited by a plane wave, but SH resonances by localized dipoles the respective field pattern are quite similar. In both cases each resonance exhibits a star-like pattern with a certain number of lobes. For decreasing shell thickness two more lobes arise at every new resonance (Fig. 5). This suggests that resonances are caused by surface plasmon polaritons running around the sphere. A resonance with $2K$ lobes occurs, whenever a multiple of the effective wavelength of the plasmon fits to the circumference of the sphere: $2\pi R=K{\lambda }_{\text{SPP}}$. Hence, resonances depend on both the diameter of the sphere and the effective wavelength of the plasmon. The latter one is mainly determined by the shell thickness. In our case the resonance corresponding to the longest plasmon wavelength at SH frequency occurs for $D=25 \text{nm}$ and explains the respective enhancement of the total SHG. For smaller shell thicknesses the plasmon wavelength decreases and further resonances are observed, at least for the lossless case (Fig. 5). But, plasmons are guided waves and do only interact with the farfield due to the curvature of the surface. The thinner the shell, the shorter is the plasmon wavelength and the stronger is the guidance. Therefore resonances at smaller shell thickness can be excited by nonlinear dipoles, but they are only visible close to the surface of the sphere (Fig. 5), but do not significantly contribute to an emission to the farfield.

 

Fig. 5 Tangential component of the intensity of a dipole (second harmonic wavelength) placed at the surface of a spherical particle with silica core of radius 100 nm and a silver shell (with and without damping) of variable thickness (compare Fig. 4 for the geometry). a) Intensity ${\left|E_{\vartheta }\left(\vartheta \right)\right|}^2$ of a dipole at the outer surface (without plasmonic losses, ${ϵ}_{\text{shell}}^{\text{400} \text{nm}}=-4.4$). The intensity is detected 20 nm above the surface of the sphere and shown as a function of the scattering angle $\vartheta$. b, c: Intensity ${〈{\left|E_{\vartheta }\left(\vartheta \right)\right|}^2〉}_{\vartheta }$ of a dipole, which is placed at the inner and outer interface of a sphere, without (b) and with (c) plasmonic losses (${ϵ}_{\text{shell}}^{\text{400} \text{nm}}=-4.4+0.2i$), detected in 2 cm distance.

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To test our assumption that resonances are caused by running plasmons we shortly discuss the case of a solid metallic particle because the wavelength of plasmons propagating at a pure metal interface is known analytically as

 

Fig. 6 Resonances with $2K$ lobes occurring at certain values of the permittivity ${\epsilon }_N^{\text{Sphere}}$ of a solid metallic sphere in comparison to the model based on the planar plasmon for $R=100 \text{nm}$ and ${\lambda }_{\text{Dipole}}=400 \text{nm}$.

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Hence, any enhancement of SHG on core-shell nanoparticles is solely based on resonances at FH or SH wavelength, which both are caused by plasmons running along the metallic film. As the plasmon wavelength critically depends on the shell thickness SHG provides an extremely sensitive tool to determine the thickness of nanoshells by in situ scattering experiments and by evaluating the spectral position of respective peaks (Fig. 7). This sensitivity is based on the strong response of SHG on the local near fields at the surface of the sphere. As discussed earlier conventional linear extinction spectra cannot provide this information with a comparable contrast in particular in case of metals with higher losses.

 

Fig. 7 SH-intensity (a) and extinction spectrum (b) as a function of the fundamental harmonic wavelength. The dashed lines show the SH-scattering cross section $C_{\text{ext}}\propto {\displaystyle \sum _{m,n}^{}\frac{(2n+1)}{k^2}(|a_{m,n}^{\text{SH}}{|}^2+|b_{m,n}^{\text{SH}}{|}^2})$. Three systems of silica particles with silver shell (including plasmon losses) of 5 nm, 7.5 nm, and 10 nm thickness are shown.

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Up to now we have assumed the nonlinear coefficient to have only a nonvanishing ${\chi }_{zzz}^{\left(2\right)}$ component, also because the various tensor elements characterizing the quadratically nonlinear response of metals are not well known. As spectral resonance positions are solely determined by the linear optical response the above obtained results do not depend on this choice. We now include further nonvanishing tensor components into the calculations. In all simulation ${\chi }^{\left(2\right)}$ of the inner interface is identical but inverted to ${\chi }^{\left(2\right)}$ of the outer interface. The different surface areas of the inner and outer interface are taken into account. All calculations show that if ${\chi }_{zzz}^{\left(2\right)}$ is present it dominates the nonlinear response (see Fig. 8). A pure ${\chi }_{xxz}^{\left(2\right)}$ or ${\chi }_{zxx}^{\left(2\right)}$ yield only a negligible SH-intensity. Varying the various components of the surface susceptibility ${\chi }^{\left(2\right)}$ affects the angular response. The position of the first maximum in pp-polarization (${\theta }_{\text{pp}}$) is only slightly affected by the actual choice of the nonlinear coefficients, ${\theta }_{\text{sp}}$ and the ratio of the intensities of the first maxima $I_{\text{pp}}/I_{\text{sp}}$ show a larger dependence, which potentially allows to extract more information about nonlinear coefficients from scattering data.

 

Fig. 8 Effect of the variation of ${\chi }_{}^{\left(2\right)}$ on the position of the first maximum in pp- and sp-polarization (${\theta }_{\text{pp}}$, ${\theta }_{\text{sp}}$), the ratio of the intensities of the first maximum ($I_{\text{pp}}/I_{\text{sp}}$) and the sum of the intensities of the first maxima in sp- and pp-polarization ($I_{\text{pp}}+I_{\text{sp}}$) for a silica particle ($R=100 \text{nm}$) with a silver shell (including plasmon losses) of $R=15 \text{nm}$ thickness. The triangular parameter space displays the relative amount of the three independent components of ${\chi }_{}^{\left(2\right)}$: ${\chi }_{zzz}^{\left(2\right)}$, ${\chi }_{zxx}^{\left(2\right)}$, and ${\chi }_{xxz}^{\left(2\right)}$. In the corners of the triangle, only the component written next to the corner is nonzero. Along the edges the two components are nonzero, their relative magnitude becomes larger the closer the point lies to the respective corner [19,20].

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4. Conclusion

We have successfully refined the nonlinear Mie model to account for metallo-dielectric core-shell particles. We found that for certain shell thicknesses SHG is extremely enhanced. We attribute this to the occurrence of plasmonic resonances at either the FH or SH frequency. Resonances at a low shell thickness arise from Mie-resonances of the nanoshells-particle excited the scattered FH plain wave. Enhancement of SHG at larger shell thicknesses is caused by resonances excited by the nonlinear dipoles oscillating at SH frequency. All resonances are caused by surface plasmons, which run on the surface of the spherical particle and are in resonance with the circumference of the sphere. Because the plasmon wavelength critically depends on the properties of the metallic layer SHG resonances of core-shell nanoparticles can be easily tuned by varying the thickness of the shell. In turn SHG might therefore become a convenient tool to study the growth and the evolution of the thickness of metallic shells on particles with much higher sensitivity than it can be provided by linear scattering experiments. Our simulations have also shown that the actual composition of the nonlinear tensor mainly influences the strength of the interaction with ${\chi }_{zzz}^{\left(2\right)}$ playing the dominant role, but does not influence the resonance positions. In contrast the angular distribution of the scattered SH field is affected, which in principal allows to deduce also individual tensor elements from angularly resolved SHG scattering data.