1. Introduction

Fluorescence emission from molecules situated in the vicinity of metal surfaces has been studied intensively during the last decades. The phenomenon of enhancement and quenching of fluorescence as a result of modified dipole decay rates was investigated already more than 30 years ago. Both theoretical and experimental investigations first considered the influence of metallic 2D planar structures on fluorescent aggregates [1]. Later, metal island films instead of closed metal layers were used as substrates for dye fluorescence experiments [2]. Studies of these metal island - molecule systems revealed collective effects for larger separations between the metal surface and the molecules, as well as confined island - molecule interactions for shorter distances [3, 4]. These findings, in turn, led to the investigation of the coupling between isolated metal particles and fluorophores, as these systems show a much more complicated dependence than 2D film structures.

Light incident onto a metal particle can excite localized surface plasmon (LSP) resonances and thus enables the particle to

These processes were early comprehended theoretically [5] and investigated experimentally for ensembles containing a large number of particles [6].

In order to fully understand the interaction processes between metal particles and fluo-rophores, single molecule - single particle experiments have been performed by physically or chemically attaching molecules to metal spheres [7]. However, in such experiments it is difficult to examine the dependence of the fluorescence signal on the distance between molecule and particle. It was only after Kalkbrenner had demonstrated how single metal particles can be used as mobile scattering centers in near-field optical microscopy [8] that such experiments could be performed in a controlled way. The work of Anger [9] and Kühn [10] finally clarified the influence of the separation between a single molecule and the particle surface on the molecular fluorescence signal.

However, although the fundamental physics of the processes involved is well known, the impact of several parameters has remained unaddressed in the literature so far. Here we present a theoretical investigation of how the dipole orientation with respect to both the particle and the exciting field influences the fluorescence yield. Furthermore, we examine the response of the fluorescence intensity when the particle diameter, the excitation and emission wavelengths of the molecular dipole, and, most importantly, the dielectric constant of the embedding medium are varied.

The presence of a surrounding medium with a dielectric constant ε_m > 1 is known to have a drastic influence on the plasmonic response of a metal nanoparticle to illumination [11]. Such media, particularly water, are typically present when biological samples are investigated, and therefore this influence can not be neglected. Altogether, our results are relevant for a wide range of particle-based application fields, from single-particle-enhanced microscopy techniques to methods involving ensembles of nanoparticles in solutions or matrices, e.g. for bioanalytics.

This paper is structured as follows: In section 2 a description of our model and calculation method can be found. The discussion of the results starts with the variation of the orientation and position of the dipole with respect to the sphere surface (section 3). Section 4 deals with the influence of the particle diameter on the fluorescence enhancement. Then the impact of the dielectric constant of the particle - dipole environment as well as different excitation and emission wavelengths are discussed in section 5. A summary of our findings is presented in section 6.

2. Method

We used the multiple-multipole (MMP) [12] method to calculate the emission of a single point dipole (referred to as fluorescence in what follows) at a variable distance from an unsupported gold nanoparticle (GNP). The method allows the computation of the electromagnetic field distribution in the coupled particle - dipole system. It can be shown that this classical electrodynamic approach leads to the same results as a quantum electrodynamic treatment for all quantities involved in the fluorescence emission [13]. The fluorescence emission rate γ_em of the coupled particle - dipole system can be expressed as

with q the emission quantum efficiency of the coupled particle - dipole system, γ_exc the molecular excitation rate, E _exc the electric field vector at the position of the dipole, and p the dipole moment of the molecular transition giving rise to the fluorescence emission. Here we assume an intrinsic quantum efficiency of unity for the molecule, i.e., energy is only dissipated in the particle [14]. Furthermore, the molecule is weakly excited so as to be far from saturation.

The calculation was done in a two-step process. In the first step, we computed the electric field E _exc(λ,r) at the position of the dipole using Mie theory [15] and the experimental data of Johnson and Christy [16] for the dielectric function of gold. The electric field arises from the superposition of a linearly polarized plane wave incident on the particle and the induced particle polarization field.

In the second step, a classical Hertzian dipole emitter was used to excite the system from the position of the molecule. This approach allows us to explore the particle - molecule interaction. The emission of the molecule was taken to occur with the same polarization as the excitation field, but at a different wavelength λ_em. The interaction is treated by a 3D Poynting vector integration over the spherical surfaces depicted in Fig. 1. The time-averaged radiative energy flux P_rad out of sphere S ₁ containing the entire particle - dipole system is given by

 

Fig. 1. Geometry used for the calculation of the emission quantum efficiency q according to Eq. 4. The power P_rad emitted from the coupled particle - dipole system as well as the power P_nr dissipated in the gold particle are calculated by 3D Poynting vector integration over spherical surfaces S ₁ and S ₂, respectively.

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with E _exc and H ^* _exc being the electromagnetic field components at the surface of integration, and s _r a unit vector pointing radially away from the sphere surface. Furthermore, the nonradia-tive power P_nr will be dissipated into Ohmic losses within the GNP enclosed by sphere S ₂, and can be calculated in the same way:

Here, ŝ _r denotes a unit vector pointing radially into sphere S ₂. The emission quantum efficiency of the entire system is then given by the relation [5]

Thus, the necessary quantities q and γ_exc for the calculation of the emission rate γ_em according to Eq. 1 are at hand. Note that all curves shown in the following are normalized to the values obtained for an infinite distance between dipole and particle, i.e., the situation of a free molecule.

3. Fluorescence emission for different particle - dipole coupling configurations

Our first calculations concern the dependence of the fluorescence signal on the position and orientation of the dipole with respect to the particle surface. We choose a GNP of 80 nm diameter, plane wave excitation at λ_exc = 532 nm, and emission at λ_em = 560 nm. These values are fairly well approximating a single Rhodamine 6G molecule, for example. The surrounding medium of the system is considered to be air (dielectric constant ε_m = 1).

The molecular dipole is placed at different positions with respect to both the particle surface and the excitation field. The positions A to E and the respective dipole orientations are indicated by the dark arrows in Fig. 2 (a). Illumination is realized by a z-polarized plane wave incident from the left along the x axis. Cases A to D constitute dipoles positioned directly on one of the main cartesian axes x,y,z of the coordinate system. Case E represents a dipole positioned at the specific site in the vicinity of the particle where the electric field reaches its maximum. This spot is determined individually for each set of parameters throughout this report.

 

Fig. 2. (a) Calculations were carried out for five different positions A to E of the molecular dipole with respect to the particle and the excitation field. While positions A to D are located on one of the main cartesian axes, case E represents a dipole positioned at that specific site where the excitation field reaches its maximum. The dark arrows indicate the dipole oscillation direction considered. Note that case A was devided into A1 and A2, describing a radially and a tangentially oscillating dipole, respectively. Excitation occurs by a linearly polarized plane wave incident from the left along the x axis. (b) Plane-wave excitation of plasmon modes on a 80-nm GNP at λ_exc = 532 nm and (c) λ_exc = 400 nm. Note the tilt of the field maxima in the direction of propagation for the shorter wavelength (c).

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Due to the symmetry of the system, not all of the excitation field components E_x, E_y, or E_z have to be taken into account. For the polarization of the incident field considered here, no E_y excitation component occurs at any position. However, once the particle diameter is no longer small compared to λ_exc, plasmonic modes of higher order than the dipolar mode become important and contribute to the electric field distribution. In particular, we observe a finite E_x component. These higher-order modes and the interference of the scattered field with the incoming plane wave are responsible for a tilt of the position of the electric field maximum (case E) away from the poles of the sphere.

The tilt angle θ in the direction of propagation is shown in Fig. 2 (c). Due to the lack of any E_y component, the field maximum is always located in the x-z plane.

Figure 2 (a) also shows that case A has been split into A1 and A2. For A1, we assumed that the dipole is excited only by the radial component E_z, while only the tangential component E_x contributes to A2. For case E, the dipole was positioned into the maximum of the radial field component (as obtained from E_x and E_z) with the transition dipole being oriented in this direction. Furthermore, our calculations revealed that E_x vanishes completely in the equatorial region, independently of the excitation wavelength. Table 1 gives an overview of the occurring field components for the respective position. Vanishing components are labelled “0”, nonzero components that were not considered for excitation are marked with an “x”, and, finally, nonzero components assumed to excite the dipole are labelled by “✓”.

For all these excitation components, we calculated the normalized excitation rate λ_exc/λ ⁰ _exc, the quantum efficiency q, and the fluorescence rate λ_exc/λ ⁰ _exc of the coupled particle - dipole system. Figure 3 shows the dependence of these quantities on the distance d between the dipole and the metal surface. Since the condition r_s << λ_exc (r_s … sphere radius) is still fulfilled for the parameters considered so far, no higher-order plasmon modes occur on the particle. Consequently, both the field component E_x and the tilt angle θ are negligible. This will become important later (cf. section 5). Here, however, cases A1 and E look identical, while in case A2 the excitation field is extremely low.

Table 1. Due to the symmetry of the coupled particle - dipole system (see Fig. 2 (a)), not all excitation field components E_x,E_y,E_z have to be considered for the respective positions A1 to E. While no E_y excitation component exists, a finite E_x field component becomes observable besides E_z when the particle diameter is in the order of the wavelength of the incident wave. Dipole excitation due to E_z and E_x was treated separately for cases A1 and A2. For case E, E_x and E_z were used to calculate the maximum of the radial field. Vanishing field components are marked by “0”, nonzero components not considered for excitation by “x”, and nonzero components considered to excite the dipole by “✓”.

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All curves show that the emitted signal strength results from a competition between the exciting field and the quantum efficiency. Only in cases A1 and E is the field enhancement in the vicinity of the particle large enough to compensate for the decrease of q for distances in the range of 10 nm < d < 30 nm. This behaviour is due to the highest fields occurring radially at the poles of the GNP.

Similar results obtained with different calculation methods have been reported by other groups [5, 9, 10]. However, they considered predominantly case A1. This certainly is the most interesting configuration for experiments in which single particles are used as scanning probes (SPs). However, when a large number of molecules are adsorbed on a single GNP [17, 18], it is necessary to investigate all possible configurations and electric field distributions in order to quantify the detected fluorescence signal. Especially the quenching in cases B to D is not negligible in such a configuration. The curves allow for a general estimation of the 3D distance dependence of the fluorescence signal for an arbitrary particle - fluorophore experiment.

 

Fig. 3. Normalized excitation rate γ_exc/γ ⁰ _exc (dashed line), quantum efficiency q (dotted line), and normalized fluorescence rate γ_em/γ ⁰ _em (solid line) versus distance [nm], calculated for an 80-nm GNP and a dipole located at positions A1 to E (see Fig. 2). Excitation wavelength is λ_exc = 532 nm, dipole emission at λ_em = 560 nm.

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4. Particle diameter

In the next step, we reduced the GNP diameter from 80 nm to 30 nm while all other parameters were kept the same. Such a size reduction appears promising for SP applications, as the lateral resolution of SP microscopes first of all depends on the tip dimensions. However, our calculations reveal that, concerning the fluorescence enhancement properties, such small GNPs are not at all preferable. Figure 4 displays the results calculated for the configurations depicted in Fig. 2. Even in case A1, no fluorescence enhancement occurs, as the electric field in the vicinity of the particle is nowhere high enough to compensate for the drop of the quantum efficiency. Cases A2 and E have been omitted. From the results in section 3, no fluorescence enhancement can be expected for A2, and case E again is identical to case A1 due to the lack of higher-order plasmon modes.

 

Fig. 4. Normalized excitation rate γ_exc/γ ⁰ _exc (dashed line), quantum efficiency q (dotted line), and normalized fluorescence rate γ_em/γ ⁰ _em (solid line) versus distance [nm], calculated for a 30-nm GNP and a dipole located at positions A1, B, C, D (see Fig. 2). Excitation wavelength is λ_exc = 532 nm, dipole emission at λ_em = 560 nm. A1 (α) and A1 (β): Comparison of q and γ_exc/γ ⁰ _exc, respectively, for case A1 of the 30-nm GNP (solid line) and the 80-nm GNP (dotted line). As both quantities are lower for the 30-nm particle at short distances, no fluorescence enhancement occurs in case A1. Cases A2 and E have been omitted (see text).

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The curves in Fig. 4 can be understood if we take into account that the polarizability α of the GNP depends on the third power of its radius r_s [19]:

with ε ₀ the vacuum permittivity, ε(ω) the dielectric function of the metal, and ε_m the dielectric constant of the surrounding medium. Consequently, the electric field strength becomes size dependent, which in spherical coordinates (r, θ, φ) can be approximated in the quasi-static regime to [13]

with E ₀ e^iωt the incident plane wave, and n _r, n _θ the respective unit vectors.

According to Eq. 1 – 4, both the excitation rate and the emission quantum efficiency also exhibit this size dependence. They turn out to be much smaller for the 30-nm GNP at short distances (see lower graphs A1(α) and A1(β) in Fig. 4). This finally leads to quenching of the fluorescence and suggests 30-nm (and smaller) GNPs to be less favourable compared to their 80-nm pendants for particle - enhanced fluorescence microscopy.

A computation of the spectral dependence of the field enhancement factor by means of Mie theory proves that in the case of ε_m = 1.00 this holds for the entire wavelength range from 350 nm to 750 nm. Figure 5 shows the results of the calculation for (a) a 30-nm and (b) an 80-nm GNP, considering position E and different embedding media. The graphs reveal that the field enhancement at the smaller GNP nowhere exceeds that of the larger one for ε_m = 1.00. Moreover, the excitation wavelength of λ_em = 532 nm considered here produces maximal enhancement possible at a GNP embedded in air. However, for higher ε_m the situation changes as will be shown in the next section.

 

Fig. 5. Normalized radial field E_r/E ₀ at position E for (a) a 30-nm GNP, and (b) an 80-nm GNP, calculated for d = 0 (fluorophore at GNP surface). Position and height of the enhancement peak are shifted for different embedding media (air: ε_m = 1.00, water: ε_m = 1.69, immersion oil: ε_m = 2.25).

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5. Variation of embedding medium and excitation and emission wavelengths

The next parameter varied in our calculations is the dielectric constant of the surrounding medium. We considered an 80-nm GNP embedded in air (ε_m = 1.00), water (ε_m = 1.69), or immersion oil (ε_m = 2.25), again for λ_exc = 532 nm and λ_em = 560 nm [20]. The effect of higher-order plasmon modes had necessarily to be taken into account in these calculations. For increasing ε_m, the wavelength in the medium drops according to λ = 2πc/ω√ε_m, with c the vacuum speed of light, and ω the angular frequency of the incident light. As mentioned above, once the condition r_s << λ_exc is not fulfilled anymore, the region of the highest field is tilted away from the poles by a polar angle θ within the x-z plane. The field plots in Fig. 6 illustrate this situation.

Therefore, we first determined the site of the strongest electric field for each distance d from the surface (case E) in order to then exploit the maximum fluorescence. The distance dependence is shown for cases A1 and E in Fig. 6. Although the emission quantum efficiency drops for higher ε_m at the emission wavelength of 560 nm, the fluorescence signal reaches its maximum in water. This maximum is owed to the strong enhancement of the excitation field which outweighs the efficiency loss (see Fig. 5). The change of both q and γ_exc/γ ⁰ _exc is a consequence of the polarizability of the GNP depending on the embedding medium, as indicated by Eq. 5 and 6. As expected, no fluorescence enhancement occurs for cases B - D in any surrounding medium (data not shown).

 

Fig. 6. Normalized excitation rate γ_exc/γ ⁰ _exc (dashed line), quantum efficiency q (dotted line), and normalized fluorescence rate γ_em/γ ⁰ _em (solid line) versus distance [nm], calculated for an 80-nm GNP and a dipole located at positions A1 (right colum) and E (middle). The system is embedded in (a) air (ε_m = 1.00), (b) water (ε_m = 1.69), and (c) immersion oil (ε_m = 2.25). Excitation wavelength is λ_exc = 532 nm, while dipole emission occurs at λ_em = 560 nm [20]. Although q drops for higher ε_m, the fluorescence yield reaches its maximum in water due to the highly enhanced excitation field.

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For ε_m = 2.25 the wavelength of maximum field enhancement in Fig. 5 is much larger than the emission wavelength of 560 nm considered so far, namely 590 nm. Therefore, we extended the calculations to emission wavelengths of 560, 590, and 650 nm, as well as excitation wavelengths covering the range from 450 nm to the respective emission wavelength in steps of 10 nm. The results are plotted in Fig. 7. For a fixed emission wavelength, the fluorescence signal follows the excitation field in its spectral intensity distribution (cf. Fig. 5). Furthermore, the fluorescence intensity increases towards larger emission wavelengths, which is due to the increase of the emission quantum efficiency (data not shown). This gain is caused by decreased damping of the particle plasmon modes towards longer wavelengths, i.e., less energy is dissipated within the GNP. This finding is in accordance with previous reports concerning the spectral position of the radiative decay peak [21] – [24].

The fluorescence yield also increases with higher ε_m, although q diminishes for higher dielectric constants at the emission wavelengths considered here. As stated above, the increase in field enhancement due to the high ε_m outweighs this effect. Finally, the distance d for maximum fluorescence enhancement for a fixed ε_m also decreases slightly in the vicinity of the resonance due to the enhanced excitation field in this spectral region.

 

Fig. 7. Normalized fluorescence rate γ_em/γ ⁰ _em at 560 nm, 590 nm, and 650 nm for air (ε_m = 1.00), water (ε_m = 1.69), and immersion oil (ε_m = 2.25) [20], plotted as a function of distance d and excitation wavelength λ_exc. The fluorescence enhancement follows the excitation field in its spectral distribution.

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The above results indicate that the fluorescence output for fixed excitation and emission wavelengths can be maximized by optimizing the refractive index of the surrounding medium. Vice versa, for a given environmental refractive index, one can either tune the excitation wavelength or the dipole emission wavelength (by choice of fluorophore) to obtain a maximum fluorescence signal. The results presented in Fig. 7 allows one to make the optimum choice of excitation and emission wavelengths in order to achieve maximum fluorescence signal intensity for particle - enhanced fluorescence microscopy or scattering-type SNOM.

6. Conclusions

In conclusion, we have investigated the dependence of a single molecular fluorescent emitter on the distance to a small GNP. For experimentally preferred 80-nm gold spheres, we have shown that fluorescence enhancement is optimal for particle - molecule configurations, in which the dipole of the molecule oscillates perpendicularly to the GNP surface and parallel to the dominant localized plasmon mode on the particle. Additionally, we have investigated all other expedient particle - molecule configurations which all show fluorescence quenching. In many experiments, these quenching contributions can not be neglected. For such cases, the results allow a general assessment of the fluorescence signal.

Furthermore, the fluorescence yield drops drastically when the polarizability and consequently the near-field enhancement of the GNP as well as the emission quantum efficiency are reduced. For too small a particle volume, the nonradiative channel will dominate the molecular dipole decay, which makes 30-nm (and smaller) GNPs unsuitable for particle - enhanced fluorescence applications. It appears much more promising to exploit of the enormously enhanced fields of either ellipsoidal particles [25] or hot spots between two GNPs [26].

Finally, we have shown how the shift of the LSP resonance of the particle caused by a change of the environmental conditions (dielectric constant) leads to a spectral and intensity shift of the maximum fluorescence enhancement factor. In both respects, fluorescence intensity follows the field enhancement. Consequently, the fluorescence yield can be optimized for a given particle -molecule ensemble in a medium by varying either the excitation or emission wavelength or the refractive index of the ensemble environment.