Fano resonances and fluorescence enhancement of a dipole emitter near a plasmonic nanoshell

We analytically study the spontaneous emission of a single optical dipole emitter in the vicinity of a plasmonic nanoshell, based on the Lorenz-Mie theory. We show that the fluorescence enhancement due to the coupling between optical emitter and sphere can be tuned by the thickness ratio of the core-shell nanosphere and by the distance between the quantum emitter and its surface. In particular, we demonstrate that both the enhancement and quenching of the fluorescence intensity are associated with plasmonic Fano resonances induced by near- and far-field interactions. These Fano resonances have asymmetry parameters whose signs depend on the orientation of the dipole with respect to the spherical nanoshell. We also show that if the atomic dipole is oriented tangentially to the nanoshell, the interaction exhibits saddle points in the near-field energy flow. This results in a Lorentzian fluorescence enhancement response in the near field and a Fano line shape in the far field. The signatures of this interaction may have interesting applications for sensing the presence and the orientation of optical emitters in close proximity to plasmonic nanoshells.

Figure 1

An electric dipole emitter in the vicinity of an illuminated coated sphere with inner radius $a$ and outer radius $b$. There are two basic orientations for the electric dipole moment at ${\mathbf{r}}_0$: orthogonal (${\mathbf{d}}_0^{\perp })$ and parallel (${\mathbf{d}}_0^{\left|\right|}$) to the spherical surface. The sphere has optical properties $({\varepsilon }_1,{\mu }_1)$ for the core $(0<r\le a)$ and $({\varepsilon }_2,{\mu }_2)$ for the shell $(a<r\le b)$, where $\varepsilon$ ($\mu$) is the permittivity (permeability). The surrounding medium is (${\varepsilon }_0,{\mu }_0)$. The sphere and the dipole emitter are exposed to an incoming electromagnetic wave with electric field ${\mathbf{E}}_{\mathrm{in}}=E_0{e}^{ıkz}\widehat{\mathbf{x}}$ ($k=\omega \sqrt{{\varepsilon }_0{\mu }_0}$). The dipole response is represented by the electric field ${\mathbf{E}}_{\mathrm{dip}}$.

Figure 2

Optical cross sections associated with a silver (Ag) nanoshell illuminated by plane waves as a function of the excitation wavelength. The lossless dielectric core has refractive index $n_1=3.5$ and radius $a=50$ nm, whereas the Ag shell has radius $b=70$ nm and dispersive permittivity provided by Ref. [47]. The resonant peak in the cross sections around 770 nm is due to plasmon hybridization between the core-shell ($r=a$) and shell-external medium ($r=b$) interface plasmon modes, and has a Lorentzian line shape for the absorption cross section ${\sigma }_{\mathrm{abs}}$ and a Fano line shape for the scattering cross section ${\sigma }_{\mathrm{sca}}$. The inset shows that these peaks vanish for a homogenous Ag nanoparticle with radius $b=70$ nm. The quadrupolar resonance peak around 635 nm is also due to plasmon hybridization.

Figure 3

Spontaneous-decay rates for an optical emitter with transition wavelength ${\lambda }_0=780$ nm in the vicinity of a silver nanoshell as a function of the distance to the spherical surface $\mathrm{\Delta }r=r_0-b$. The nanoparticle consists of a dielectric core with $n_1=3.5$ and radius $a=50$ nm and a silver shell with radius $b=70$ nm. (a) The radiative (${\mathrm{\Gamma }}^{\mathrm{rad}}$) and nonradiative $\left({\mathrm{\Gamma }}^{\mathrm{nrad}}\right)$ decay rates for a dipole oriented tangentially $\left(\right|\left|\right)$ or normally $(\perp )$ to the spherical surface normalized by the decay rate in vacuum $\left({\mathrm{\Gamma }}_0\right)$. The inset shows that ${\mathrm{\Gamma }}_{\perp \left(\right|\left|\right)}^{\mathrm{rad}}\to {\mathrm{\Gamma }}_0$ and ${\mathrm{\Gamma }}_{\perp \left(\right|\left|\right)}^{\mathrm{nrad}}\to 0$ as $\mathrm{\Delta }r\gg b$. (b) The corresponding quantum efficiency $Q={\mathrm{\Gamma }}^{\mathrm{rad}}/\mathrm{\Gamma }$ for each orientation. The inset shows that $Q_{\perp \left(\right|\left|\right)}\to 1$ as $\mathrm{\Delta }r\gg b$.

Figure 4

The intensity and fluorescence enhancement factors for a quantum emitter with transition wavelength ${\lambda }_0=780$ nm as a function of the distance $\mathrm{\Delta }r$ to a silver nanoshell and excitations wavelengths. The core-shell nanoparticle consists of a dielectric core ($n_1=3.5$) with radius $a=50$ nm and a silver shell with radius $b=70$ nm. The excitation wavelengths $\lambda$ are chosen from Fig. 2, with ${\mathbf{E}}_{\mathrm{in}}\left(\lambda \right)$ being parallel to ${\mathbf{d}}_0^{\perp \left(\right|\left|\right)}$: 780 nm (emission wavelength), 770 nm (scattering resonance), 740 nm (cloaking or Fano dip in the scattering response), 635 nm (quadrupole resonance). The plots show the intensity enhancement factors (a) ${\mathcal{G}}_{\left|\right|}$ and ${\mathcal{G}}_{\perp }$ (inset), and the corresponding fluorescence enhancement factors (b) $F_{\left|\right|}$ and (c) $F_{\perp }$. The maximum fluorescence enhancement occurs for $\mathrm{\Delta }r\approx 5$ nm.

Figure 5

Fluorescence enhancement response associated with a quantum emitter (transition wavelength ${\lambda }_0=780$ nm) in the vicinity of a silver nanoshell. The inner sphere has refractive index $n_1=3.5$ and radius $a=50$ nm, whereas the silver nanoshell has radius $b=70$ nm. The plots show the fluorescence enhancement factor as a function of the excitation wavelength and the distance between the emitter and the spherical surface for a (a) tangentially and (b) normally oriented dipole, with ${\mathbf{d}}_0^{\perp \left(\right|\left|\right)}$ being parallel to ${\mathbf{E}}_{\mathrm{in}}$. From left to right, the peaks correspond to distances $\mathrm{\Delta }r$ ranging from 0 nm (the thicker line) to 50 nm (the thinner line). (c) The plot shows the Lorentzian line shape of $F_{\left|\right|}$ for $\mathrm{\Delta }r=5$ nm (maximum enhancement). The inset shows a Fano line shape for $\mathrm{\Delta }r=25$ nm with negative asymmetry parameter $q_{\left|\right|}=-2.9$. (d) The plot shows the Fano line shape of $F_{\perp }$ for $\mathrm{\Delta }r=5$ nm (maximum enhancement). The inset shows a Fano line shape for $\mathrm{\Delta }r=25$ nm with positive asymmetry parameter $q_{\perp }=2.2$.

Figure 6

Energy flow vector field (normalized Poynting vector) in the vicinity of a plasmonic Ag nanoshell interacting with an electromagnetic plane wave ($\lambda =770$ nm). The dielectric core has refractive index $n_1=3.5$ and radius $a=50$ nm, whereas the Ag shell has radius $b=70$ nm. (a) The $xz$ plane shows the presence of a saddle point in the energy flow in the $z$ axis around $z\approx 110$ nm ($\mathrm{\Delta }r=40$ nm). (b) The $yz$ plane shows the singular point along the $y$ direction, $z\approx 110$ nm.

Figure 7

The Fano asymmetry parameter of the fluorescence enhancement as a function of the distance between dipole and sphere. The system and parameters are the same as in Fig. 5. The two curves provide the values of $q_{\perp }^{\lambda }=-q_{\perp }^{\omega }$ [Eq. (10)] and $q_{\left|\right|}^{\lambda }=-q_{\left|\right|}^{\omega }$ [Eq. (11)] that fit the plots in Figs. 5 and 5, respectively. The parameters used in Eq. (12) are ${\sigma }_{\mathrm{sca}}^{\left(\max \right)}\approx 5.4\pi b^2, {\lambda }_{\mathrm{res}}\approx 770$ nm, and $q=7.53$, leading to ${\epsilon }^{\prime \prime }\approx 0.83$. To fit the Fano curves in Fig. 5, we consider $\epsilon \left(\lambda \right)\approx (\lambda -770)/10$ ($\lambda$ in nanometers) and the calculated asymmetry parameters. The inset shows $q_{\perp \left(\right|\left|\right)}$ in the range $10<\mathrm{\Delta }r<50$ nm.