1. Introduction

The fluorescence from a single molecule can be enhanced by a local probe acting as an optical antenna [1, 2, 3, 4], a device that efficiently converts the energy associated with free-propagating radiation to localized energy, and vice versa. The study of optical antennas has been undertaken only recently [5, 6, 7, 8, 9] and draws its inspiration from near-field optics [10] and nanoplasmonics [11]. In future technology, optical antennas are likely to be employed for improving the efficiency of photovoltaics, light emitting devices and biosensors. An excited molecule placed near an optical antenna can be viewed as a transmitter and, similarly, a molecule in its ground state and getting excited by the localized field near the optical antenna acts as a receiver. Hence, an antenna used to enhance the fluorescence from a single molecule acts as a two-way antenna [1]. In this article we investigate both experimentally and theoretically the spectral dependence of single molecule fluorescence enhancement near an optical antenna made of a simple gold or silver particle attached to a pointed glass tip. Although a spherical particle is not the most efficient antenna it has a simple geometry that makes a quantitative comparison between theory and experiment possible. The findings of our study can be readily extended to other antenna geometries.

The challenge associated with the design of optical antennas is the fact that a simple down-scaling of well-established radiowave and microwave antennas into the optical frequency regime is not possible. Traditional radiowave or microwave antennas operate at frequencies much smaller than the plasma frequency ω_p of the electrons in the metal. Consequently, the conduction electrons respond ‘instantaneously’ to an external electric field. On the other hand, optical frequencies are near ω_p and hence the response becomes strongly material and frequency dependent. As a consequence, while the crucial parameters of a microstrip antenna (e.g. antenna gain or radiation efficiency) operating at 1 GHz depend only on its geometry and not on the particular properties of its metal components, the properties of an optical antenna are strongly influenced by the material-specific plasmon resonances. Not suprisingly, therefore, antennas of the same geometry but made from different metals are expected to exhibit potentially very different behavior at optical frequencies. As optical antennas increasingly find use as probes for nanoscale optical spectroscopy, it becomes necessary to characterize and optimize them with respect to specific applications. The use of fluorescence to study optical antennas allows a quantitative understanding of the loss mechanisms that are present in real materials at optical wavelengths. In this article, we investigate the effect of a metal nanoparticle antenna on the fluorescence rate of a nearby single fluorophore, and demonstrate that material properties and the excitation wavelength are key parameters in determining the maximal enhancement of fluorescence caused by the antenna.

As shown in Fig. 1, we use a simple antenna geometry in the form of a spherical metal (gold or silver) nanoparticle attached to a glass tip. This choice makes the theoretical modeling of the system less challenging and the experimental fabrication of the antennas reproducible. In the next section, we theoretically analyze the fluorescence rate of a molecule placed near a metal nanoparticle. In particular, we present approximate analytical solutions that elucidate the role of the particle’s surface plasmon resonances on the radiative and non-radiative decay rates of the molecule. In section 3 we present our experimental results and compare the results with theory.

2. Theory

The fluorescence rate γ_em of a single molecule can be expressed as the product of excitation rate γ_exc and quantum yield q, defined as the fraction of radiative transitions from excited to ground state to the total decay rate. Treating the excitation and emission processes independently is legitimate because there is no coherence between the two processes. The fluorescence enhancement can then be expressed as

where the superscript ‘o’ indicates the corresponding free-space quantity. Below saturation, the excitation rate γ_exc is proportional to |E ∙ p|², with E being the local electric field and p the transition dipole moment. On the other hand, within the validity of Fermi’s Golden Rule, the decay rate of the excited molecule is expressed by the weighted sum of possible decay channels. The latter corresponds to the electromagnetic density of states defined by the Green’s function of the system that is embedding the molecule. Because the Green’s function is the response of a point source (dipole) it is possible to represent the emitting molecule by a classical dipole [12, 10]. Let us denote the decay rate of the excited molecule in absence of the optical antenna as γ^o = γ^o_r + γ^o_nr, where γ^o_r and γ^o_nr are the radiative and nonradiative decay rates of the excited molecule, respectively. Notice that γ^o_nr accounts for nonradiative losses inside the molecule only, i.e. it is an intrinsic property independent of antenna-molecule distance. The intrinsic quantum yield of the isolated molecule is defined as q^o = γ^o_r/(γ^o_r + γ^o_nr). The presence of the particle antenna, however, introduces an additional nonradiative rate γ_abs, thereby modifying the quantum yield to q = γ_r/(γ_r + γ_nr + γ_abs). Using the definition of q^o, this can be recast as

 

Fig. 1. (a) Schematic of the experiment. An optical antenna in the form of a gold or silver nanoparticle attached to the end of a pointed glass tip is interacting with individual fluorescent molecules excited at frequency ω ₁ and emitting at frequency ω ₂. The inset shows an SEM image of a 80nm gold particle attached to an HF etched glass tip. (b) Definition of coordinates used in the theoretical analysis.

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Here, γ_r is the radiative rate in presence of the optical antenna. We assumed that the antenna does not influence the intrinsic nonradiative rate, i.e. γ_nr = γ^o_nr. For large separations between molecule and particle we have γ_abs → 0 and γ_r → γ^o_rγ, and hence q = q^o. On the other hand, for a molecule with high intrinsic quantum yield (q^o = 1) we obtain q = γ_r/(γ_r + γ_abs) and hence γ_abs is the only nonradiative decay channel. Using the multiple multipole (MMP) method [10] we have calculated the different decay rates and the quantum yield for different particle-molecule configurations and for different particle materials following a theory outlined previously [1]. The results will be presented and discussed in the following. However, in order to establish an intuitive understanding we first introduce an approximate analysis which agrees surprisingly well with the rigorous calculations.

Let us first analyze the energy transfer rate to the particle (γ_abs). This rate can be deduced from the power P_abs that is radiated by the molecule and absorbed in the particle. Once P_abs is known, the normalized energy transfer rate is calculated as γ_abs/γ^o_r = P_abs/P_o, with P_o = |p|² ω ₂ k ³ ₂/(12πε_o) being the power radiated by a classical dipole with frequency ω ₂ and dipole moment p, and the associated wavevector being defined as k ₂ = ω ₂/c. For very small distances between molecule and particle the curvature of the particle’s surface can be neglected and the environment ‘as seen’ by the molecule is a plane interface. Using electrostatic image theory [10] the power emitted by the molecule and absorbed inside a half-space with frequency-dependent dielectric constant ε(ω ₂) can be calculated and the energy transfer rate becomes

where z is the molecule-surface separation and p = [p_x,P_y,P_z]. Equation 3 was derived by integrating the dissipated power density (ω ₂ ε_o/2)Im{ε}E∙E ^* E being the electrostatic dipole field, over the volume of the half-space. Higher-order correction terms to this expression can be derived as outlined in the literature [13, 14, 15].

The derivation of the radiative decay rate γ_r makes use of the fact that the scattered power P_r from a particle much smaller than the wavelength of light originates mostly from the particle’s induced dipole. Higher-order multipole terms give rise to dissipation. Using γ_r/γ^o_r = P_r/γ_o, the normalized power radiated by the system of molecule and particle is calculated as

where α⃡ is the particle’s polarizability, G⃡ is the free-space Green’s dyadic, n _p is the unit vector in direction of the molecule’s dipole moment, and r ₀ and r _p are the position vectors of molecule and particle, respectively. Equations 2–4 define the quantum yield q of the molecule placed next to a particle characterized by dielectric constant ε and polarizability α⃡. Our theory predicts that γ_r depends on the shape and material of the particle (ε and α⃡) whereas γ_abs depends only on the material properties (ε).

For a spherical particle, the quasi-static polarizability is calculated as [10]

where a is the particle radius and I⃡ the unit dyad. The polarizability in Eq. 5 can be extended to include radiation reaction [16] but the contribution of this additional term is marginal. Evaluation of Eq. 4 in the non-retarded (near-field) limit leads to

where, for simplicity, we assumed a molecule orientation such that p_x = p_y = 0. The enhancement of the radiative rate is at a maximum at z = 0. It amounts to |3ε/(ε + 2) |² for a vertically aligned molecule (p_x = p_y = 0) and to |3/(ε + 2)|² for a horizontally aligned molecule (p_z = 0). On the other hand, according to Eq. 3 the energy transfer rate at z = 0 goes to infinity and hence the quantum yield drops to zero. For small distances z, the radiative rate enhancement in the denominator of Eq. 2 can be ignored and the quantum yield becomes a simple function of distance and material. It is important to notice that γ_abs has a maximum at the frequency determined by ε(ω ₂)+ 1 = 0, whereas the maximum of γ_r occurs at ε(ω ₂)+2 = 0. For a spherical metal particle characterized by a free-electron gas (Drude model) the shape-dependent plasmon resonance given by ε(ω ₂) + 2 = 0 is red-shifted from the dissipative resonance of η(ω₂) + 1 = 0. It is not surprising, therefore, that maximum fluorescence enhancement occurs when the molecule emits even to the red of the plasmon resonance of the particle, rather than at it [17].

Let us now turn to the excitation rate γ_exc which is determined by both the incident laser field E _o and the secondary field E _s, i.e. the enhanced field by the particle. For weak excitation (below saturation) we have γ_exc ∝ |p ∙ [E _o(r _o,ω ₁) + E _s(r _o,(ω ₁)]|², where ω ₁ is the excitation frequency. We assume that the particle is small compared with the wavelength and that it can be represented by a polarizability α⃡(ω ₁). In this case, the secondary field can be expressed in terms of the Green’s function as E _s(r _o) = (k ² ₁/ε _o) G⃡(r _o,r _p;ω ₁) α⃡(ω ₁)E _o(r _p) and the enhancement of the excitation rate becomes

where n _Eo is the unit vector in direction of E _o. We used E _o (r _p) = E _o (r _o) because |r _p - r ₀| ≪ λ. For a spherical particle with polarizability α defined in Eq. 5 we obtain

As before, we retained only the near-field terms in G⃡ and assumed p_x = p_y = 0.It turns out that the expressions for γ_exc/γ^o_exc [Eq. 8] and γ_r/γ^o_r [Eq. 6] are identical, a property that is ascribed to reciprocity [3, 18, 19, 20, 21]. However, Eqs. 8 and 6 appear identical because the dipole orientation was restricted to point along a Cartesian axis. For arbitrary dipole orientations the two expressions are not identical because different dipole components interfere in Eq. 8 whereas in Eq. 6 they do not!

In order to estimate the validity of the approximations derived for the different rates we have compared the approximations with rigorous calculations based on the MMP method. Figure 2 shows the results for a molecule with q^o = 1 placed next to a 60 nm gold particle irradiated with light of wavelength at λ = 650nm. For simplicity, we assumed ω ₁ ≈ ω ₂. The dielectric constant is inferred from interpolated experimental data [22]. The dashed curves derived with the approximate formulas overlap nearly perfectly with the solid curves derived with the MMP method. This remarkable agreement validates the finding that fluorescence quenching (γ_abs) is largely independent of the particle geometry. Fluorescence quenching becomes important at small distances between molecule and particle surface and for such small distances the particle surface is well approximated by a plane boundary. Thus, the only degrees of freedom for minimizing γ_abs are the choice of material and the excitation frequency. On the other hand, besides a different spectral dependence, the radiative decay rate and the excitation rate are strongly dependent on the geometry of the particle and hence can be optimized by suitable optical antenna design. For larger particles we have found that the approximation becomes less accurate, mainly because of the insufficiency of retaining only the near-field terms in the Green’s function. Furthermore, the scattering efficiency of large particles becomes dependent on higher order multipoles and hence the representation by a simple polarizability is no longer sufficient.

Using the approximate theory outlined above, we have carried out a spectral analysis of the fluorescence enhancement for both silver and gold nanoparticles. As before, we assume p_x = p_y = 0 and q_o ≈ 1. Figure 3(a) shows that the enhancement is strongly wavelength dependent. At λ = 650 nm the enhancements for silver and gold particles differ only slightly whereas for bluer wavelengths, e.g. at λ = 488 nm, the enhancement for silver particles is predicted to be much stronger than the enhancement for gold particles. Figure 3(a) demonstrates that the maximum fluorescence enhancement is obtained for wavelengths red-detuned from the surface plasmon resonance. Figure 3(b) shows the calculated distance dependence of the fluorescence enhancement for an excitation and emission wavelength of λ = 488nm. Because this wavelength is blue-shifted from the plasmon resonance of a gold particle we expect no significant enhancement and this is indeed reflected by the curve for the gold particle.

 

Fig. 2. Fluorescence enhancement near a 60nm gold nanoparticle excited with wavelength λ = 650nm. (a) Quantum yield (blue) and excitation rate enhancement (red). (b) Emission enhancement. The emission wavelength is taken to be same as the excitation wavelength. The solid curves are exact results based on the MMP method and the dashed curves are approximations (Eqs. 3, 6, 8).

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3. Experiments

We have experimentally tested the predictions shown in Fig. 3 using two different excitation wavelengths (488nm and 637nm) and two different molecules (Alexa488 and Nile Blue). Single molecule samples were prepared by spincoating the molecules on a glass coverslip and then again spincoating a thin layer (<5nm) of polymer (PMMA) on top. The antenna, in this case an 80nm gold or silver particle, is raster-scanned over the polymer surface at variable heights. The theoretical curves shown in Fig. 3 do not account for the presence of a dielectric interface. As shown previously [1], the dielectric interface increases the fluorescence enhancement by a factor 2 – 3 which has to be taken into account when comparing the here derived theory with the experimental results presented in the following.

The molecules are excited by a standard confocal arrangement with a radially polarized excitation laser beam and a single-photon counting detector (c.f. Fig. 1(a)). The particle (antenna) can be positioned into the focus using a shear-force feedback mechanism. We can either generate a fluorescence rate image by raster-scanning the particle at a controllable but constant distance (typically ~5nm) over the sample surface, or we can position the particle over a selected molecule and record the fluorescence rate as a function of the particle-sample distance. The results of our experiments at using λ = 488nm excitation are summarized in Fig. 4 which shows fluorescence rate images recorded with both a gold particle (left) and a silver particle (right). In order to obtain roughly equal fluorescence intensities (~ 10kHz), the excitation power in Fig. 4(a) had to be increased by roughly ten times over the power used in Fig. 4(b). For the gold particle, the near-field enhancement is barely discernible whereas for the silver particle the near-field enhancement is so strong that the far-field fluorescence background is at the level of the detector’s dark counts. A comparison of the two images in Fig. 4 qualitatively validates the theoretical predictions displayed in Fig. 3. However, to be more quantitative we recorded the distance dependence of the fluorescence enhancement for both gold and silver particles.

 

Fig. 3. Calculated fluorescence enhancement near gold and silver nanoparticles of 80nm diameter. (a) Spectral dependence at a distance of z = 10nm. The arrow marks the plasmon resonance of a gold nanoparticle in air. The corresponding resonance for silver is at around 360nm.(b) Distance dependence at a wavelength of λ = 488nm. Experimental data from Ref. [22] was used for the dielectric constants of gold and silver.

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After identifying the orientations of various single molecules embedded in the polymer matrix from either confocal or near-field emission patterns [23, 24] we selected a vertically oriented molecule (p_x ≈ p_y ≈ 0) and positioned it into the laser focus and underneath the gold or silver particle. The fluorescence rate was then recorded as a function of the vertical position of the particle. In order to observe quenching at short distances the thickness of the PMMA layer was chosen to be sufficiently thin. Figure 5 shows recorded approach curves for two different combinations of particles and laser wavelengths. In Fig. 5(a) we used a silver particle to enhance the fluorescence of an Alexa488 molecule excited with a laser wavelength of λ =488 nm, and in Fig. 5(b) a gold particle was used in combination with a Nile Blue molecule and a wavelength of 637 nm. The silver particle gives an enhancement of a factor of 13 – 15 whereas the gold particle enhances the fluorescence about 8 – 9 fold. The ratio between these enhancements is consistent with the theoretical predictions of Fig. 3(a). Both approach curves shown in Fig. 5 clearly show that the fluorescence enhancement starts to drop for very short distances, indicative for the onset of quenching. Theory predicts that fluorescence should drop to zero when the molecule is in direct contact with the particle but this cannot be verified experimentally because of the finite thickness of the PMMA layer that is used to immobilize and orient the dye molecules.

In conclusion, we have presented a simple analytical model for calculating the fluorescence rate of a single molecule close to a nanoparticle antenna. The model highlights the role of antenna plasmon resonances in determining the relative strength of various rates involved in the fluorescence process, and its predictions are in good agreement with more rigorous MMP calculations. Our single molecule experiments validate the theory and demonstrate that nanoplas-monic enhancement of single molecule fluorescence is strongly frequency dependent. For molecules emitting to the blue of the surface plasmon resonance the fluorescence can be quenched whereas for molecules emitting to the red of the surface plasmon resonance the fluorescence can be strongly enhanced. We believe these findings are general and not only specific to spherical particles, and that they provide intuitive guidelines for efficient optical antenna design. Future work will concentrate on the development of optical antennas providing increased enhancement, stronger field localization, and tunable spectral properties.

 

Fig. 4. Near-field fluorescence rate images of Alexa488 molecules recorded using (a) an 80nm gold particle and (b) an 80nm silver particle excited at λ = 488nm. At this wavelength the enhancement for the silver particle is more than 10 times stronger than for the gold particle, consistent with the predictions of Fig. 3.. The image in (a) reflects the confocal excitation spot whereas the image in (b) represents the localized fields at the silver particle.

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Fig. 5. Fluorescence enhancement as a function of molecule-particle separation for two different particles (gold and silver) and two different excitation wavelengths. (a) Alexa488 molecule excited by a 80nm silver particle irradiated with λ = 488nm. (b) Nile Blue molecule excited by a 80nm gold particle irradiated with λ = 637nm. The unity baseline for fluorescence enhancement is determined using the measured fluorescence when the antenna is retracted far away from the molecule (∼ 500nm). Both curves show a decrease at short distances due to fluorescence quenching.

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