Linear and nonlinear light scattering and absorption in free-electron nanoclusters with diffuse surface: General considerations and linear response

Both linear and nonlinear scattering and absorption of a laser pulse by spherical nanoclusters with free electrons and with a diffuse surface are considered in the collisionless hydrodynamics approximation. The developed model of forced collective motion of electrons confined to a cluster permits one consistently to introduce into the theory all the sources of nonlinearity, as well as the inhomogeneity of the cluster near its boundary. Two different perturbation theories corresponding to different laser intensity ranges are developed in this context, and both cold metal clusters and hot laser-heated or -ionized clusters are considered within the same approach. In the present article, after developing the full nonlinear model, the linear response to the laser field of the free-electron cluster with diffuse surface is investigated in detail, especially the properties of the linear Mie resonance (width and position). Under certain conditions, depending on the various cluster parameters secondary resonances are found. The properties of resonance-enhanced third-order harmonic generation and nonlinear laser absorption and their dependence on the shape of the diffuse surface will be presented separately.

Figure 1

The model profiles of the diffuse surface $g_i(x)$ with different smoothness ($i=1-5$, see text) for spherical clusters, with $x=(r-R)/\sigma$.

Figure 2

The electron density $n_0(\rho )$ near the cluster surface as a result of solution of Eqs. (6) for cold neutral metal clusters in vacuum [with a halo] and in a surrounding dielectric matrix [without halo] (a) and hot laser heated/ionized clusters in vacuum (b). For (a)$A={10}^{-5}$, $\sigma /R=3\times {10}^{-2}$. For (b) $A={10}^{-3}$, $\sigma /R={10}^{-1}$, $\eta =0$, $0.1$, and $0.2$. For the ion density of the cluster surface region, the completely smooth profile $g_5(x)$ is used (see Fig. 1).

Figure 3

The reduced cross sections of linear scattering and absorption, respectively, ${\mathrm{\omega ̃}}^4\left|{\alpha }_1\right|{}^2$ [(a) and (c)] and $\mathrm{\omega ̃}\mathrm{Im}{\alpha }_1$ [(b) and (d)], as a result of numerical solution of Eqs. (38) for a neutral cluster ($\eta =0$) within the CCA ($A=0$), for ${\epsilon }_1=1$, for $\gamma ={10}^{-3}$ [(a) and (b)] and $\gamma ={10}^{-2}$ [(c) and (d)], and for different cluster surface thicknesses. In each panel, the four curves correspond to $\sigma /R={10}^{-3}$ (the solid curves), $\sigma /R={10}^{-2}$ (the dashed curves), $\sigma /R=3\times {10}^{-2}$ (the dotted curves), and $\sigma /R={10}^{-1}$ (the dash-dotted curves). For the ion density of the cluster surface region, the completely smooth profile $g_5(x)$ is used (see Fig. 1).

Figure 4

The reduced cross sections of linear scattering and absorption, respectively, (a) ${\mathrm{\omega ̃}}^4\left|{\alpha }_1\right|{}^2$ and (b) $\mathrm{\omega ̃}\mathrm{Im}{\alpha }_1$, as a result of numerical solution of Eqs. (38) for a neutral cluster ($\eta =0$) within the CCA ($A=0$) for $\sigma /R=0.01$ and with $\gamma ={10}^{-2}$, and for different dielectric permittivities ${\epsilon }_1$ of the surrounding. In each panel, the three curves correspond to ${\epsilon }_1=1$ (the solid curves), ${\epsilon }_1=2$ (the dashed curves), and ${\epsilon }_1=3$ (the dotted curves). For the ion density of the cluster surface region, the completely smooth profile $g_5(x)$ is used (see Fig. 1).

Figure 5

The relative shift of the calculated resonance Mie frequency ${\omega }_M$ of the metal cluster embedded into a dielectric surrounding with permittivity ${\epsilon }_1$ with respect to the classical Mie frequency ${\omega }_M^{(\mathit{cl})}={\omega }_p/\sqrt{1+2{\epsilon }_1}$ as a function of ${\epsilon }_1$. As in Fig. 4, $\sigma /R=0.01$ and $\gamma ={10}^{-2}$.

Figure 6

The reduced cross sections of linear scattering and absorption, respectively, (a) ${\mathrm{\omega ̃}}^4\left|{\alpha }_1\right|{}^2$ and (b) $\mathrm{\omega ̃}\mathrm{Im}{\alpha }_1$, as a result of the numerical solution of Eqs. (38) for a charged cluster in vacuum (${\epsilon }_1=1$) within the CCA ($A=0$) for $\sigma /R=0.03$ and with $\gamma ={10}^{-2}$ and for various outer ionization degrees $\eta$. In each panel, the four curves correspond to the following values of $\eta$: $\eta =0$ (solid curves), $\eta =0.1$ (dash curves), $\eta =0.3$ (dot curves), and $\eta =0.5$ (dash-dot curves). For the ion density of the cluster surface region, the completely smooth profile $g_5(x)$ is used (see Fig. 1).

Figure 7

The reduced cross sections of linear scattering and absorption, respectively, ${\mathrm{\omega ̃}}^4\left|{\alpha }_1\right|{}^2$ [(a) and (c)] and $\mathrm{\omega ̃}\mathrm{Im}{\alpha }_1$ [(b) and (d)], as a result of the numerical solution of Eqs. (47) and (48) for cold neutral metal clusters with an electron halo in vacuum (${\epsilon }_1=1$) beyond the CCA [$A={10}^{-7}$ for (a) and (b) and $A={10}^{-5}$ for (c) and (d)], with $\gamma ={10}^{-3}$ and for various cluster-surface diffusenesses $\sigma /R$. In each panel the four calculated curves correspond to the following values of $\sigma /R$: $\sigma /R=0.001$ (solid curves), $\sigma /R=0.003$ (dash curves), $\sigma /R=0.01$ (dot curves), and $\sigma /R=0.03$ (dash-dot curves). For the ion density of the cluster surface region, the completely smooth profile $g_5(x)$ is used (see Fig. 1).

Figure 8

The same as Fig. 7 but for $\gamma ={10}^{-2}$.

Figure 9

The reduced cross sections of linear scattering and absorption, respectively, ${\mathrm{\omega ̃}}^4\left|{\alpha }_1\right|{}^2$ and $\mathrm{\omega ̃}\mathrm{Im}{\alpha }_1$, as a result of numerical solution of Eqs. (47) and (48) for cold neutral metal clusters without electron halo beyond the CCA ($A={10}^{-6}$) in vacuum [${\epsilon }_1=1$, (a), (b), and (c)] and in a surrounding dielectric [${\epsilon }_1=2$, (d), (e), and (f)], with $\gamma ={10}^{-2}$ and for various cluster surface diffusenesses $\sigma /R=0.003$ [(a) and (d)], $\sigma /R=0.01$ [(b) and (e)], and $\sigma /R=0.03$ [(c) and (f)]. For the ion density of the cluster surface region, the completely smooth profile $g_5(x)$ is used (see Fig. 1).

Figure 10

The reduced cross sections of linear scattering and absorption, respectively, ${\mathrm{\omega ̃}}^4\left|{\alpha }_1\right|{}^2$ [(a) and (c)] and $\mathrm{\omega ̃}\mathrm{Im}{\alpha }_1$ [(b) and (d)], as a result of the numerical solution of Eqs. (47) and (48) for a hot laser-heated/ionized cluster with an electron halo in vacuum (${\epsilon }_1=1$) beyond the CCA ($A={10}^{-4}$) for $\sigma /R=0.1$ and with $\gamma =0.02$, and for various outer ionization degrees $\eta$. In (a) and (b), the six curves with increasing maxima correspond to the following increasing values of $\eta$: $0$, $0.01$, $0.015$, $0.02$, $0.025$, $0.03$, while in (c) and (d) the six curves with decreasing maxima correspond to the following increasing values of $\eta$: $0.04$, $0.05$, $0.1$, $0.2$, $0.3$, $0.4$. For the ion density of the cluster surface region, the completely smooth profile $g_5(x)$ is used (see Fig. 1). Increasing values of eta are bottom to top [(a) and (b)] and top to bottom [(c) and (d)].

Figure 11

The reduced cross sections of linear scattering and absorption, respectively, ${\mathrm{\omega ̃}}^4\left|{\alpha }_1\right|{}^2$ [(a) and (c)] and $\mathrm{\omega ̃}\mathrm{Im}{\alpha }_1$ [(b) and (d)], as a result of the numerical solution of Eqs. (47) and (48) for a hot laser-heated/ionized cluster with an electron halo in vacuum (${\epsilon }_1=1$) beyond the CCA ($A=3\times {10}^{-3}$) for $\sigma /R=0.03$ and with $\gamma =0.02$ and for various outer ionization degrees $\eta$. In (a) and (b), the five curves with decreasing low-frequency maxima in absorption [(b)] correspond, respectively, to the following increasing values of $\eta$: $0$, $0.01$, $0.02$, $0.03$, $0.05$, while in (c) and (d) the five curves correspond to the following values of $\eta$: $0.1$ (solid curves), $0.2$ (dashed curves), $0.3$ (dotted curves), $0.4$ (dashed-dotted curves), and $0.5$ (dash-dot-dot curves). For the ion density of the cluster surface region, the completely smooth profile $g_5(x)$ is used (see Fig. 1). In (a) and (b), increasing values of eta are top to bottom.