Quantization of surface charge density on hyperboloidal and paraboloidal domains with application to plasmon decay rate on nanoprobes

Field quantization in high curvature geometries help understanding the elastic and inelastic scattering of photons and electrons in nanostructures and probelike metallic domains. The results find important applications in high-resolution photonic and electronic modalities of scanning probe microscopy, nano-optics, plasmonics, and quantum sensing. We present a calculation of relevant photon interactions in both hyperboloidal and paraboloidal material domains. The two morphologies are compared for their plasmon dispersion properties, field distributions, and radiative decay rates, which are shown to be consistent with the corresponding quantities for the finite prolate spheroidal domains. The results are relevant to other material domains that model a nanostructure such as a probe tip, quantum dot, or nanoantenna.

Figure 1

Modeling systems and their potential distributions. (a) One sheet of a two-sheeted hyperboloid of revolution modeling a nanotip or a nanostructure with local curvature. Surface modes of momentum $\kappa$, e.g., excited by incoming photons $h\omega$, decay radiatively into a solid angle $d\mathrm{\Omega }$. The curvature of the tip apex is set by the ${\mu }_0$ defining the hyperboloidal surface. Here, ${\mu }_0=cos{\theta }_0$, where ${\theta }_0$ is the angle between the $z$ axis and an asymptote to the hyperboloidal surface such that small ${\theta }_0$ yields a sharp probe while ${\theta }_0\to \pi /2$ corresponds to $xy$ plane. The apex point $z_{\text{min}}=z_0{\mu }_0,$ near the focal point of the hyperboloid, is set by the scale factor $z_0$, as in Eq. (B1). (b), (c), and (d) show the spatial distribution of the lowest lying eigenmodes of the quasistatic electric potential for the three modeling domains investigated. For the same mode index $m$, optimizing the apex curvature overlap within the same spatial $zx$ domains, and analyzing the potential distribution, leads to the determination of the corresponding continuous eigenvalues $\lambda$ of the paraboloid (b) and $q$ of the hyperboloid (d), respectively, as well as the discrete eigenvalue $l$ of the prolate spheroid (c). The geometric parameters ${\eta }_0$, ${\mu }_0$, and ${\zeta }_0$ determines the form of the considered domains.

Figure 2

Paraboloidal and hyperboloidal nonretarded surface plasmon dispersion relations. The resonance values of the dielectric function $\varepsilon$ are shown for low lying modes as a function of the continuous eigenvalue $\lambda$ for a paraboloid (black) and $q$ for a hyperboloid (red). The surfaces of the paraboloid and hyperboloid are set by the parameter ${\eta }_0$ and ${\mu }_0$, respectively. The discrete modes are denoted by $m$ for the azimuthal oscillations.

Figure 3

The radiative decay rate of plasmons engendered on paraboloidal, hyperboloidal, and prolate spheroidal surfaces. The decay rate of paraboloidal plasmons (blue) is computed from Eq. (28) for the single eigenmode $m=0$ (right), $m=1$ (left) and $\lambda =1$ when the shape parameter is ${\eta }_0=60\mathrm{nm}$. Similarly, the decay rate of hyperboloidal plasmons (green) computed from Eq. (38) corresponds to the eigenmode $m=0$ and $q=0.2$ for a shape parameter ${\mu }_0=86\mathrm{nm}$. For comparison, the radiative decay rate (red) for plasmons excited on a prolate spheroidal surface is computed using Eq. (42) for ${\zeta }_0=65\mathrm{nm}$ and $m=0$, $l=1$, corresponding to the dipolar mode. To facilitate visual comparison within the same plot window, note that the spheroidal case for $m=0$ (right) has been multiplied by 0.006 and for the case $m=1$ (left) by 0.01. The composition of the specific parameters for comparison was guided by the potential distribution in each case.

Figure 4

Comparison of the higher-order modes' radiative decay rates for the three different cases described in Fig. 3. In the case of the prolate spheroidal plasmons, the emitted radiation pattern corresponds to the quadropolar charge density oscillations.

Figure 5

Curvature induced shift in the radiation pattern associated with the decay of plasmons excited on the hyperboloidal surfaces for the modes $m=0$ (left) and $m=1$ (right) and $q=0.2$. To facilitate comparison within the same plot window, the case for ${\mu }_1$ has been multiplied by 20 (left) and by 50 (right).

Figure 6

The spatial distribution of low and high lying eigenmodes of the quasistatic electric potential for the three modeling domains investigated. For the same mode index $m$, optimizing the apex curvature overlap within the same spatial $zx$ domains, and analyzing the potential distribution, leads to the determination of the corresponding continuous eigenvalues $\lambda$ of the paraboloid (top) and $q$ of the hyperboloid (middle), respectively, as well as the discrete eigenvalue $l$ of the prolate spheroid (bottom). The geometric parameters ${\eta }_0$, ${\mu }_0$, and $\zeta$ determines the form of the considered domains. For proper geometric and modal adjustments, the similarities in the potential distributions are clearly evident from the contour plots. The discontinuity in the contour lines near the symmetry axis of the hyperboloids is due to the singularity in the conical functions there.

Figure 7

Paraboloidal, hyperboloidal, and prolate spheroidal nonretarded surface plasmon dispersion relations. The resonance values of the dielectric function $\varepsilon$ are shown for low lying modes as a function of the continuous eigenvalue $\lambda$ for a paraboloid (top) and $q$ for a hyperboloid (middle), and as a function of the shape parameter $\zeta$ for a prolate spheroid. The surfaces of the paraboloid and hyperboloids are set by the parameter ${\eta }_0$ and ${\mu }_0$, respectively, while $\zeta$ defines the form of the spheroidal surface (bottom). The discrete modes are denoted by $m$ for the azimuthal oscillations and by $l$ in the spheroidal case.