1. Introduction

Controlling and enhancing molecular fluorescence plays a significant role in sensing, imaging, biochemistry, medicine, and in the design of antennas for nanophotonics [1–6]. Among the various techniques for controlling the emission of molecules, nanophotonics has remained at the forefront for decades, long before plasmonics became the active research field that it is today, through the concept of metal- or plasmon-enhanced fluorescence [7–10]. Metallic nanoparticles (NPs), with their localised surface plasmons (LSPs) producing strongly enhanced electromagnetic (EM) fields confined to subwavelength volumes [11], are now considered among the most efficient templates for manipulating light–matter interactions [12–15], especially since the controversy of both fluorescence enhancement [16] and quenching [17] being observed in plasmonic environments was resolved [18] by showing that the combined effect of modifying both the spontaneous emission rates (due to changes in the emitter environment, the well-known Purcell effect [19]) and the excitation rate of the molecule is what leads to the observed fluorescence rates.

Several theoretical and experimental works have addressed understanding and optimising molecular emission in the vicinity of single [20–29] or aggregated metallic NPs [30–34]. Part of the appeal of such arrangements is owed to the fact that placing fluorophores in the proximity of metallic NPs can be done with unprecedented control over the emitter orientation and distance, e.g. through DNA origami patterning [35–38] or with appropriate hosting molecules [39,40]. Nevertheless, despite the tremendous enhancement of the excitation rate due to the intense near fields around plasmonic NPs, Ohmic losses that lead to poor radiative properties can still be a hurdle, especially when quantum effects in the material response are important [41–43].

Alternative approaches are being sought in the attempt to deal with the problem of plasmonic losses [44,45]. In this direction, appropriately engineered plasmonic materials, and hybrid plasmonic–dielectric nanostructures that mix the light confinement of plasmonics with the high quality factors of microscale dielectric cavities, attract particular interest [46–48]. Another rapidly growing trend is to altogether shift focus towards all-dielectric nanostructures, and more specifically to high-index dielectrics [49], which combine relatively low absorptive losses with a plethora of different optical modes of electric, magnetic, toroidal or anapole character [50–54]. Mie resonances are explored in areas ranging from optical metamaterials [55,56] and highly directional antennas [57,58] to strong coupling with quantum emitters [59–63]. Regarding the weak emitter–NP coupling that concerns us here, the influence of Mie resonances on the properties of nearby electric or magnetic dipoles in terms of Purcell effect, fluorescence enhancement or quenching, and importance for spectroscopies, has already been studied in the past [64–69]. Most of the theoretical studies have been based on numerical simulations; even when analytic solutions were derived, they were restricted to very specific orientations and positions of the emitters around the dielectric NP [64], and mostly to observables related only to the emission properties of the system [69]. This paper is devoted to extending such analytic formulas to describe both excitation and emission of emitters at arbitrary positions and orientations around the NP, by providing closed-form expressions for a few characteristic cases and more lengthy equations for any other situation. These expressions are used to explore fluorescence enhancement near a spherical silicon NP, potentially layered in a core–shell arrangement. It is shown that the obtained enhancement rates are moderate, as compared to the case of plasmonics, but they survive for a much wider range of emitter positions and orientations, thus relaxing the requirement for precise positioning of the molecule around the nanophotonic antenna. Inclusion of a plasmonic component can contribute to further enhancing these fluorescence rates [70], while other improvement strategies might include tailoring NP shapes or sizes [71] or introducing NP interactions [65,72]. The present formalism only applies to spherical particles; nevertheless, it should be feasible to further extend it to collections of such spheres, e.g., through a multiple-scattering [73] or generalised Mie-theory [74] approach, and to nonspherical particles through the null-field (or extended boundary condition) method [75,76].

2. Mie theory-based analytic solutions

All calculations in this paper, both for far-field spectra and for near-field contour plots, are based on appropriate application or extension of the analytic Mie theory [77] for scattering by spherical particles. This powerful tool allows not only to study, accurately and efficiently in terms of computation times, single or layered spherical NPs excited by dipoles [78] or electron beams [79], but also explore their combination with two-dimensional materials [80], the role of quantum effects [81], and their response to circularly polarised light [82] and external magnetic fields [83]. In this section, the response of isotropic nanospheres to plane waves and electric dipoles will be considered. The main steps for each derivation will be outlined, with more details provided where necessary, aiming to offer an as self-sufficient description as possible. The section starts with plane-wave excitation, expansion of the incoming wave into spherical waves, and calculation of exact analytic expressions for the experimental observables, i.e., scattering and absorption cross sections, through the scattering $\mathbb {T}$ matrix of spherical, possibly stratified (onion-like) NPs. Subsequently, an electric point dipole of dipole moment $\mathbf {p}_{\mathrm {d}}$, placed at distance $\mathbf {r}_{\mathrm {d}}$ from the centre of the sphere is considered. Its EM field is again expanded into spherical waves, and solutions for its excitation rate, the power scattered or absorbed by the dipole–NP system, and the modification of the fluorescence rate due to the presence of the NP are obtained for some characteristic dipole positions and orientations.

2.1 Plane-wave excitation

Let us consider a spherical scatterer, of radius $R$, located at the origin of the local coordinate system. The EM properties of the sphere are described by a relative permittivity $\varepsilon _{1}$ and a relative permeability $\mu _{1}$ and, therefore, by a wavenumber $k_{1} = \frac {\omega }{c} \sqrt {\varepsilon _{1} \mu _{1}}$, where $\omega$ is the angular frequency and $c$ the velocity of light in vacuum. The particle is embedded in a medium of $\varepsilon$, $\mu$, and $k$, respectively, as shown in Fig. 1(a). The EM field inside and outside the sphere can be expanded into a sum of transverse spherical waves [84]

Here, ${j_{\ell }}$ and ${h_{\ell }^{+}}$ are the spherical Bessel and Hankel (of the first kind) functions, respectively [85], and $\mathbf {X}_{\ell m} (\widehat {\mathbf {r}})$ are the vector spherical harmonics [84], with $\widehat {\mathbf {r}}$ denoting collectively the dependence on the polar and azimuthal angles, $\theta$ and $\phi$. The subscripts $\ell$ and $m$ refer to the usual angular momentum indices, while the polarisation index $P = E, H$ refers to transverse electric and magnetic polarisation, respectively. The expansion of a transverse EM wave with electric field $\mathbf {E}$ and magnetic field $\mathbf {H}$ in spherical waves is written as [84]

 

Fig. 1. (a) A plane wave with electric field $\mathbf {E}$ and wavevector $\mathbf {k}$ propagating in a medium with permittivity $\varepsilon$ and permeability $\mu$, expanded into spherical waves with expansion coefficients $a_{P\ell m}^{0}$, is scattered by a sphere of radius $R$, described by a permittivity $\varepsilon _{1}$ and permeability $\mu _{1}$. The scattered and internal fields are expanded into spherical waves with expansion coefficients $a_{P\ell m}^{+}$ and $a_{P\ell m}^{\mathrm {in}}$, respectively. (b) An emitter (modelled as a point dipole of dipole moment $\mathbf {p}_{\mathrm {d}}$) at distance $\mathbf {r}_{\mathrm {d}}$ from the centre of the sphere of (a), and the appropriate spherical coordinate system. The electric field of the dipole is expanded into spherical waves with expansion coefficients $d_{P\ell m}^{0}$ and $d_{P\ell m}^{+}$. The inset shows the typical Jablonski diagram of the emitters considered here, modelled as three-level systems, with levels $S_{0}$, $S_{1}$, and $S_{2}$, and transition lifetime $\tau$. (c) The different $\mathbf {r}_{\mathrm {d}}$ and $\mathbf {p}_{\mathrm {d}}$ explored here, with respect to the NP and the incident field.

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To expand the incoming plane wave into spherical waves, we need the expansion of $\exp (\mathrm {i} \mathbf {k} \cdot \mathbf {r})$ ($\mathbf {k}$ being the wavevector) into the basis of spherical harmonics [85], $\exp \left (\mathrm {i} \mathbf {k} \cdot \mathbf {r} \right ) = 4\pi \sum _{\ell = 0}^{+\infty } \sum _{m = -\ell }^{\ell } \mathrm {i}^{\ell } {j_{\ell }} (kr) {Y_{\ell m}} (\widehat {\mathbf {r}}) {Y_{\ell m}^{\ast }} (\widehat {\mathbf {k}})$, where ${Y_{\ell m}}$ are the scalar spherical harmonics, and $\widehat {\mathbf {k}}$ denotes again the solid angle corresponding to the wavevector. Equating this expression with the expansion of an incoming spherical wave, and with use of the orthogonality relations of vector spherical harmonics [84], the expansion coefficients become [86]

At the interface between the two different media, the usual pill-box boundary conditions [84] need be satisfied. This means that the tangential components of the fields (two tangential directions for each field) must be continuous at the surface of the sphere, leading to a set of four boundary conditions. Since Eqs. (3) provide the expansion coefficients for the incident wave, all that is required is to calculate the $\mathbb {T}$ matrix that connects the amplitudes of the scattered field to those of the incident field, defined (in the case of spherical symmetry) through $a_{P \ell m}^{+} = T_{P\ell } a_{P\ell m}^{0}$. For arbitrary particle shapes, this matrix is in general non-diagonal [87], but the spherical symmetry of the problem simplifies things significantly, and the resulting matrix is independent of $m$ and diagonal with respect to $\ell$.

Applying the boundary conditions, and using the orthogonality relations for vector spherical harmonics [84], one can immediately derive the elements of the $\mathbb {T}$ matrix, i.e., the well-known Mie coefficients [77]

The scattering matrix is a particularly powerful tool, because all observables can be evaluated through it, and because one can investigate each of its elements separately to conclude about the origin of different resonances and features in the calculated optical spectra. Light scattering and absorption are usually described in terms of the corresponding cross sections, which express the fraction of the incident energy that is scattered or absorbed by the particle. The incident energy flow is given by the component of the Poynting vector along the direction of propagation. For a monochromatic wave, the time-averaged Poynting vector over a period is $\langle \mathbf {S} (\mathbf {r}) \rangle = \frac {1}{2} \mathrm {Re} [\mathbf {E} (\mathbf {r}) \times \mathbf {H}^{\ast } (\mathbf {r})]$, where $\mathrm {Re}$ denotes the real part of the expression in square brackets and $\ast$ complex conjugation, while the average power flowing through a surface $S$ is $P = \int _{S} \textrm {d}^{2} r \; \widehat {\mathbf {n}} \cdot \langle \mathbf {S} (\mathbf {r}) \rangle$, with $\widehat {\mathbf {n}}$ being the unit vector normal to the surface. For an incident plane wave of amplitude $E_{0}$, the energy flow normally to the direction of propagation is $\langle \mathbf {S}_{0} \rangle \cdot \widehat {\mathbf {k}} = \frac {1}{2} \tilde {Z} \left | {E_{0}} \right |^{2}$ [84]. The total scattered power can be obtained by integrating the Poynting vector of the scattered field only, over the surface $S$ of a sphere surrounding the particle. It is straightforward to obtain $P_{\mathrm {sc}} = \frac {1}{2 k^{2}} \tilde {Z} \sum _{P \ell m} \left | {a_{P \ell m}^{+}} \right |^{2}$. As a consequence of energy conservation, the power absorbed by the sphere will be equal to the opposite of the total energy flow through $S$, $P_{\mathrm {out}} = P_{\mathrm {sc}} - P_{\mathrm {ext}}$, where $P_{\mathrm {ext}} = - \frac {1}{2 k^{2}} \tilde {Z} \sum _{P \ell m} \mathrm {Re} \left (a_{P \ell m}^{+} a_{P \ell m}^{0\ast } \right )$ is the energy related to the interaction of the incident and scattered fields, called extinction. Again, from energy conservation, we have $P_{\mathrm {abs}} = P_{\mathrm {ext}} - P_{\mathrm {sc}}$, which means that the extincted energy (and the corresponding cross section) expresses the total energy provided to the system by the incoming wave, which can be either absorbed or scattered. Using the appropriate relations for the field amplitudes, the scattering and extinction cross sections of spherical NPs (normalised to the geometric cross section, $\pi R^{2}$) can be expressed in terms of the scattering matrix as [77]

2.2 Dipole excitation: excitation rate

Let us now turn to the interaction of light with an emitter, modelled as a three-level system, as shown in the inset of Fig. 1(b). The meaning of these three levels will be discussed later on. In the long-wavelength limit, the interaction of an emitter with light is accurately described by the Hamiltonian $\mathcal {H}_{\mathrm {int}} = - \hat {\mathbf {p}} \cdot \hat {\mathbf {E}}$ (here —and only here— hats represent operators), i.e., a term of coupling of the electric dipole moment of the emitter with the external electric field. According to Fermi’s golden rule, the excitation rate of the emitter is proportional to the square of the norm of the Hamiltonian element for the transition between the initial and final state [89]. In free space, the electric field responsible for the excitation is just the incident field, e.g., that due to a laser. When the environment of the emitter has been modified by the presence of an NP, what is relevant is the total field at the position of the emitter, outside the sphere, $\mathbf {E}_{\mathrm {out}}$. Assuming that the scatterer only affects the local environment but not the transitions of the emitter, we can model the emitter as an electric dipole of dipole moment $\mathbf {p}_{\mathrm {d}}$. Since we are usually interested in how the emitter properties have been modified compared to those in free space, we only need to calculate the excitation rate enhancement $\gamma _{\mathrm {exc}}/\gamma _{\mathrm {exc}}^{0}$ which, below saturation, is simply [18]

We already have at hand analytic expressions for the incoming and scattered field in spherical-wave expansions. Therefore, for the normalised excitation rate, we do not need the magnitude of the dipole moment of the emitter, but only its direction, as the denominator will always be proportional to $({p_{\mathrm {d}}}E_{0})^{2}$. For the dipole positions and orientations shown in Fig. 1(c) (whose choice was inspired from corresponding numerical calculations near plasmonic NPs in [90]), using the expansion coefficients of Eqs. (3) one can exactly work out the corresponding excitation rates. For configuration $A_{1}$, with $\mathbf {r}_{\mathrm {d}} \parallel \widehat {\mathbf {z}}$, $\mathbf {E}_{0} \parallel \widehat {\mathbf {z}}$, $\mathbf {p}_{\mathrm {d}} \parallel \widehat {\mathbf {z}}$, and $\mathbf {k} \parallel \widehat {\mathbf {x}}$ (so that the polar angle of $\mathbf {r}_{\mathrm {d}}$ in spherical coordinates $\theta _{\mathrm {d}} = 0$), we obtain

When the dipole is at the same position but oscillating along the propagation direction (configuration $A_{2}$), with $\mathbf {r}_{\mathrm {d}} \parallel \widehat {\mathbf {z}}$, $\mathbf {E}_{0} \parallel \widehat {\mathbf {z}}$, $\mathbf {p}_{\mathrm {d}} \parallel \widehat {\mathbf {x}}$, and $\mathbf {k} \parallel \widehat {\mathbf {x}}$ ($\theta _{\mathrm {d}} = \pi /2$) the corresponding expression becomes

In the two expressions above, $\mathbf {r}_{\mathrm {d}}$ was along $\widehat {\mathbf {z}}$, leading to electric field amplitudes involving spherical harmonics along this direction, i.e., with $\theta _{\mathrm {d}} = 0$, which allowed to eliminate the sum over $m$, taking advantage of the property of spherical harmonics ${Y_{\ell m}} (\theta = 0, \phi ) = \sqrt {(2\ell + 1)/ (4\pi )}~\delta _{m0}$, thus facilitating the analysis of the results in terms of multipole contributions. In the next couple of configurations this will no longer be the case. To continue using this property, we will thus rotate the coordinate system appropriately. For configuration B, with $\mathbf {r}_{\mathrm {d}} \parallel \widehat {\mathbf {y}}$, $\mathbf {E}_{0} \parallel \widehat {\mathbf {z}}$, $\mathbf {p}_{\mathrm {d}} \parallel \widehat {\mathbf {z}}$, and $\mathbf {k} \parallel \widehat {\mathbf {x}}$, a rotation by $\pi /2$ around the $x$ axis is required. Doing so, after some quite lengthy but straightforward algebra, we obtain

Similarly, at position C the dipole oscillates tangentially, with $\mathbf {r}_{\mathrm {d}} \parallel \widehat {\mathbf {x}}$, $\mathbf {E}_{0} \parallel \widehat {\mathbf {z}}$, $\mathbf {p}_{\mathrm {d}} \parallel \widehat {\mathbf {z}}$, and $\mathbf {k} \parallel \widehat {\mathbf {x}}$. To eliminate the sum over $m$ we will proceed with a rotation of the coordinate system by $\pi /2$ around the $y$ axis, to obtain

Finally, configuration D is the antipodal point of C. The excitation rate is the result of Eq. (10), multiplied by $(-1)^{\ell }$. Of course, apart from these special cases, any other random configuration can be calculated either by similar rotations, yielding expressions with spherical harmonics of $\theta _{\mathrm {d}}$ different from $0$ or $\pi /2$, or directly through the full Eqs. (3), in which case a summation over both $\ell$ and $m$ is required. To the best of our knowledge, apart from Eq. (7), the rest of the expressions in this subsection have not appeared in literature before.

2.3 Dipole excitation: emission rate

To describe the process of spontaneous emission, we will once more assimilate the emitter to a point dipole, which now acts as a source of EM radiation. For a dipole oscillating in time $t$ with angular frequency $\omega$ (in general different from the excitation frequency) in a homogeneous environment of permittivity $\varepsilon$ and permeability $\mu$, the induced polarisation is $\mathbf {P} (\mathbf {r}, t) = \mathrm {Re} [\mathbf {p}_{\mathrm {d}} \delta (\mathbf {r} - \mathbf {r}_{\mathrm {d}}) \exp (- \mathrm {i} \omega t)]$. The resulting EM field follows the wave equation in the presence of a polarisation,

To calculate the decay rate of the emitter, we can again use the time-averaged Poynting vector, similarly to how scattering and absorption cross sections were calculated. The total radiated power can be calculated by integrating the Poynting flux over the surface of a sphere large enough to contain both the dipole and the spherical scatterer. In this region, the total field has the form of outgoing spherical waves, with expansion coefficients $a_{P\ell m} = a_{P\ell m}^{+} + d_{P\ell m}^{+}$, to account for both the field scattered by the sphere and that of the dipole. The calculations are lengthy and depend on the orientation and position of the dipole. But any random orientation can be decomposed into two special cases, of a dipole oscillating radially or tangentially [along or normally to $\mathbf {r}_{\mathrm {d}}$, corresponding to excitation $A_{1}$ or $A_{2}$ in Fig. 1(c)]. In the former (radial) case [64],

Similarly, for the absorbed power, we need integrate the Poynting flux over the surface of the scatterer (assuming that this is where all energy losses take place). The corresponding expressions for the radial and tangential case are [64]

The observable that is probably of the greatest value when characterising emitters is their decay rate $\gamma$, which is different for any process during which photons are emitted; if the photons are radiated to the environment, we talk about the radiative decay rate $\gamma _{\mathrm {r}}$, while if they are lost (through absorption by the environment) we talk about the nonradiative decay rate $\gamma _{\mathrm {nr}}$. In any case, the decay rate can be obtained by the corresponding power $P$, if we assume that the emitter —though described as a classical point dipole— emits photons of energy $\hbar \omega$. Then, based on the assumption that power is radiated or absorbed in photons, the ratio $P/\hbar \omega$ is exactly the decay rate $\gamma$ [11]. The total decay rate of the emitter, in the absence of a scatterer, is the sum of decay rates due to radiation and due to other causes, $\gamma _{0\mathrm {r}}$ and $\gamma _{0\mathrm {nr}}$, respectively, with $\gamma _{0\mathrm {nr}}$ referring to intrinsic loss mechanisms of the emitter, such as many-photon processes, electron-transfer processes etc. The internal quantum yield of an emitter is defined as $q_{0} = \gamma _{0\mathrm {r}}/(\gamma _{0\mathrm {r}} + \gamma _{0\mathrm {nr}})$. The presence of the scatterer introduces an additional decay mechanism, through energy absorption. Then the quantum yield need be modified to contain the absorption rate [70],

3. Results and discussion

To demonstrate how the results derived above can be used, we turn to the system shown in Fig. 2(a), with an emitter placed in the vicinity of a silicon nanosphere. The positions and dipole orientations studied are the same as in Fig. 1(c) (plus one dipole at position A with its dipole moment forming an angle $0 < \theta _{\mathrm {d}} < \pi /2$ in the $xz$ plane). The emitter is modelled as a three-level system with ground state $S_{0}$ and two excited states, $S_{1}$ and $S_{2}$ [see Fig. 1(b)]. The incident field excites it to level $S_{2}$, which is a metastable state, and the emitter rapidly decays to $S_{1}$. After time $\tau$, the system decays back to $S_{0}$ by emitting a photon of energy equal to the difference between the two levels. In what follows, it will be assumed that $S_{2}$ and $S_{1}$ have practically the same energy, to ease the calculation of fluorescence enhancement spectra. This approximation can affect the exact calculated values, but not the underlying physics [41].

 

Fig. 2. (a) Schematic of a silicon nanosphere and an electric dipole at different positions and dipole orientations around it. (b) Normalised extinction, scattering and absorption cross section (black, blue and red line, respectively) for a silicon sphere of radius $R = 85$ nm in air. (c) Fluorescence enhancement for an electric dipole placed at distance ${r_{\mathrm {d}}} = 5$ nm from the surface of the sphere of (b), at positions corresponding to the colours of the dipoles in (a). (d) Normalised excitation rate (left-hand axis, grey line), quantum yield (right-hand axis, red line) and fluorescence (left-hand axis, black line) as a function of emitter–NP distance, for the silicon sphere of (b) at the magnetic-dipole resonance ($\hbar \omega = 1.843$ eV), for the three different orientations at position A denoted by the coloured vectors in the inset ($\theta _{\mathrm {d}} = \pi /4$ in the case of the brown arrow). (e) Emission radiation patterns in the $x-z$ plane, for the same dipole orientations as in (d), at the magnetic dipolar resonance. The scattered field magnitudes are expressed in arbitrary —but same in all three cases— units.

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Figure 2(b) shows the extinction spectrum of a silicon nanosphere with radius $R = 85$ nm, described by the experimental permittivity of [92], in air ($\varepsilon = \mu = 1$). Extinction is decomposed into its scattering (blue line) and absorption (red line) contributions, showing that, even when absorptive losses are fully taken into account, mesoscopic silicon NPs are excellent antennas for light. The modes of such spheres have been characterised in the past in terms of their multipolar contributions [50,60], with the lowest-energy one (at about $1.8$ eV) being of magnetic dipolar character, followed by a broad electric dipole and a sharp magnetic quadrupole. Unlike plasmonics, where the resonances are due to collective oscillations of free electrons, in this case the resonances emerge due to oscillating polarisation charges inside the NP, and the resulting displacement current loops due to retardation [93], leading to significant enhancement of the magnetic field [60].

According to the far-field spectra of Fig. 2(b), the magnetic dipolar resonance is a good candidate for enhancing fluorescence: it is a relatively narrow resonance with a huge difference between the scattering and absorption cross sections, implying that it will act as a good antenna for the radiated power. This can indeed be seen in the fluorescence enhancement spectra of Fig. 2(c), while the antenna effect is demonstrated by the high quantum yield of Fig. 2(d) (nearly unity, except for the last $3-4$ nm closest to the surface, where coupling with the non-radiative higher-order multipoles becomes important [94,95]). The overall fluorescence enhancement factor does not exceed a value of $\sim 15$ for any wavelength, which is a moderate enhancement as compared to plasmonics, where factors of the order of $200$ are easily achieved with single NPs [41]. The reason is that the electric-field enhancement around the NP is quite lower than in the case of plasmonics, typically around $5$, while factors of $20$ are common in plasmonics [60]. Interestingly, the largest enhancement at the energy of the magnetic dipole does not occur for an emitter with its dipole moment aligned to the incident field, but when the dipole moment forms an angle with it. On the contrary, such an agreement between maximum enhancement and external field happens at the energy of the electric dipolar resonance. To understand the differences between the two resonances, one has to refer to Eq. (6), and consider the orientation of the scattered field in each case. In the case of the electric dipolar resonance, the entire NP behaves like a dipole aligned to the external field, and the scattered field is dominated by its $z$ component. But in the case of the magnetic dipolar resonance, which is due to the circulation of a displacement current, the field has comparable $x$ and $z$ components, as can be seen in Fig. 3 for different components at different planes, leading to an efficient excitation of the emitter for any orientation. In fact, the most efficient excitation is obtained for the emitter at an angle $\theta _{\mathrm {d}}$ between $\pi /6$ and $\pi /3$ [see brown line (with $\theta _{\mathrm {d}} = \pi /4$) in Fig. 2(c), and the excitation rates in Fig. 2(d)], for which both components contribute equally.

 

Fig. 3. Near-field enhancement around the silicon NP of Fig. 2 under plane-wave excitation at $\hbar \omega =1.843$ eV. The top left contour shows the amplitude of the total field in the $x-z$ plane, at $y = 0$, while the bottom left contour shows the corresponding field in the $x-y$ plane, at $z = 90$ nm (emitter position), as shown in the schematics. The top- and bottom-right contours show the corresponding amplitude of the $x$ and $z$ component of the field only, respectively. All contours share the common colour scale in the middle.

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An appreciable fluorescence enhancement is also obtained for the orientation presented by the green arrow, where the dipole is aligned to the incident field, but the NP–emitter axis is along the direction of propagation. To understand this strikingly different response, one should again consider the form of the scattered field in this case. The magnetic dipolar resonance originates from a displacement current loop, circulating inside the NP. The electric field enhancement outside the sphere, which is responsible for the efficient excitation of the emitter, is the part of this displacement loop that has leaked outside the NP, and thus maintains the circulation. The field is thus predominantly along the $x$ direction on top of the NP, and its largest component is along the $z$ direction on the side of the NP.

Another interesting possibility that emerges from the results of Fig. 2(c) is to use dipole emitters to experimentally perform a multipole decomposition of the spectra of such NPs. Indeed, the black, brown and green lines in the figure show a striking resemblance with the individual multipolar contributions obtained when one analyses the extinction spectrum [50]. The black line (configuration $A_{1}$) exactly captures only the electric dipolar contribution, the brown line (intermediate angle) produces the strongest resonance for the case of the magnetic dipolar mode, while the green line (configuration $C$) is the only one that captures the magnetic quadrupolar term.

So far, we have mentioned many of the similarities and differences between dielectric and plasmonic NPs without, however, providing any particular example. To visualise this discussion, we repeat in Fig. 4 the same calculations as in Fig. 2, this time for a plasmonic NP. Silver, described here by the experimental bulk permittivity of Ref. [96], is a good choice for our purposes, because of its relatively low losses in the visible. The NP of Fig. 4 has a radius of $R = 50$ nm, for which the maximum normalised extinction is similar to that of the silicon sphere of Fig. 2, while the NP is also large enough to allow efficient excitation of an electric quadrupolar mode. It should also be an equally good antenna, as one can deduce from the relative contributions of scattering and absorption to the extinction spectrum, Fig. 4(b). But Fig. 4(c) clearly displays that, while for $A_{1}$ configuration the total enhancement is quite larger than in the case of silicon —for an emitter at the same distance from the NP surface— as soon as the emitter is moved, or even tilted, the enhancement factors become much worse than in Fig. 2. Most noteably, for the case of the electric quadrupolar resonance of the silver NP, we could not identify any position producing a decent enhancement —this is of course also related to the higher absorptive losses of this mode, as seen in Fig. 2(b).

 

Fig. 4. Same as Fig. 2, for a silver nanosphere with radius $R = 50$ nm, and an electric dipole placed at distance ${r_{\mathrm {d}}} = 5$ nm from its surface. The electric dipolar resonance of the sphere is at $3.154$ eV.

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The differences between dielectric and plasmonic NPs can be understood in terms of their different near-field properties, considering both the excitation and the emission process. The near-field responsible for the excitation of a dipole close to the silicon NP was discussed in Fig. 3, while similar profiles for the near-field of plasmonic NPs can be found in any textbook on nanophotonics [11]. But even without exploring the full field profile, one can tell from Fig. 4(d) (and the practically zero excitation rate for configuration $A_{2}$) that it is mostly the poor excitation of the emitter that leads to overall low enhancement factors. On the other hand, the dipole emission patterns of Figs. 2(e) and Figs. 4(e), which were obtained by calculating, in the far-field zone, the scattered field resulting from an excitation with the dipole field of Eq. (14), show that, as the emitter is tilted at the same position, the dielectric NP (Fig. 2) becomes a more efficient, and possibly more directional antenna, while for the plasmonic NP (Fig. 4) the radiation efficiency keeps decreasing. Consequently, for configurations other than $A_{1}$, both excitation and emission processes benefit more by the presence of a dielectric NP.

Finally, as discussed above, an interesting feature in Fig. 2(c) is the enhancement factor obtained at the magnetic-quadrupolar resonance, in the case of position C (green arrow). To better explore this response, it is preferable to resort to an NP where higher-order magnetic multipoles are clearly separated, such as a silicon nanoshell [60] like the one shown in Fig. 5(a). The complex NP comprises a silica core ($\varepsilon _{1} = 2.13$ describes now the core, and $\varepsilon _{2}$ is the permittivity of the silicon shell) of radius $70$ nm, covered by a silicon shell with external radius $R = 100$ nm, so that the thickness of the shell is $C = R - R_{1}$. In such an NP, the magnetic quadrupolar resonance appears as a sharp resonance around $2.3$ eV, dominated again by its scattering contribution, as can be seen in Fig. 5(b). The resulting fluorescence enhancement for an emitter lying along the $x$ axis approaches a factor of $5$, even surpassing the corresponding factor for the magnetic-dipole case, and it is again due to the direction of the scattered field at this resonance, and its effect on the excitation rate of the emitter. It is thus clear that dielectric NPs offer much flexibility to control the excitation and decay rates of quantum emitters, as the requirement for precise positioning and orientation of the emitter with respect to the photonic antenna can be relaxed.

 

Fig. 5. (a) Schematic of a silicon nanoshell with a core of silica with radius $R_{1}$, and total NP radius $R$ (and thus shell thickness $C = R - R_{1}$), and an electric dipole at different positions and dipole orientations around it. (b) Normalised extinction, scattering and absorption cross section (black, blue and red line, respectively) for a silica–silicon nanoshell with radii $R_{1} = 70$ nm and $R_{1} = 100$ nm, in air. The insets show the electric-field enhancement in the $x-z$ plane ($y = 0$) at the frequency of the magnetic dipolar resonance, and in the $y-z$ plane ($x = 107$ nm) at the frequency of the magnetic quadrupolar resonance, as sketched in the insets of (c). (c) Fluorescence enhancement for an electric dipole placed at distance ${r_{\mathrm {d}}} = 7$ nm from the surface of the sphere of (b), at positions corresponding to the colours of the dipoles in (a).

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4. Conclusion

In summary, analytic expressions for the excitation and emission of arbitrarily oriented electric dipoles near spherical NPs were derived, within a consistent Mie-theory framework. These expressions were used to study fluorescence enhancement of organic molecules, modelled as electric dipoles, near high-index dielectric nanospheres. It was shown that, while silicon NPs cannot compete with plasmonic ones in terms of absolute enhancement factors, they provide more flexibility in terms of emitter positioning and orientation. They combine the characteristics of excellent antennas for the emitted light, with decent excitation rates which, benefiting from the complex form of the scattered field, can survive for a wide range of dipole orientations. The existence of exact analytic solutions to obtain these results for any configuration should further facilitate the optimisation of the design of such dielectric nanoantennas.