1. Introduction

Plasmonics, that is the properties and applications of surface plasmons in metal structures, has become one of the most rapidly expanding fields in nanoscience and a cornerstone of nanoop-tics. This development has become possible due to considerable progress in nanofabrication techniques and computational electromagnetics. However, the most important driver has been the potential for applications in areas such as the life sciences, photonics and solar energy research. Metal nanostructures exhibit localized surface plasmon resonances (LSPRs) determined primarily by their shape, size and material composition. But LSPRs are also utterly sensitive to the presence of nearby dielectric objects, and the optical response of such objects is in turn strongly affected by the localized electromagnetic field induced by the plasmon charge oscillation. These effects are the basis for one of the most exciting application opportunities in plasmonics, i.e. to use the localized plasmons to probe the presence of molecules that reside in the vicinity of the metal surface. This can be accomplished through amplification of the molecular vibrational fingerprint, like in surface-enhanced Raman scattering (SERS) and surface-enhanced infrared absorption (SEIRA), or by affecting molecular fluorescence properties. However, the most general approach is to monitor how the localized plasmon resonance itself changes in the presence of a molecule by measuring the color of the nanostructure [1–7]. This approach, which is essentially analogous to classical thin film surface plasmon resonance (SPR) refractive index sensing [8], is the focus of the present contribution.

The relation between optical response and the location of a molecular target relative to a plasmonic nanostructure is well established in the case of surface-enhanced spectroscopies. The amplitude of a SERS signal, for example, scales roughly as the fourth power of the local field-enhancement factor, which implies that “hot spots,” such as sharp tips or crevices between particles, may dominate completely [9]. Indeed, enhancement factors as high as ten orders-of-magnitude has been realized in single molecule SERS experiments [10–13]. But the position of an analyte molecule is also expected to affect the response of a LSPR refractive index sensor [14, 15]. This effect was recently analyzed theoretically by Davis et al., who studied the coupling between a plasmonic nanoparticle and a small dielectric sphere in the quasistatic limit [16, 17]. Here we present an alternative approach that results in a description of the LSPR shift in terms of changes in local field intensities due to the presence of the analyte. We begin by describing the analyzed system in terms of coupled point dipoles to provide a general background. Next, we describe a model based on the quasistatic approximation that is based on an analysis of local fields. It allows for estimation of the magnitude of the LSPR shift as a function of the location of the dielectric analyte and takes into account its material and geometrical size. As a test case, we assume that the metal resonator and analyte are both spherical in shape. Finally, we compare and discuss the analytical model in relation to Finite Difference Time Domain (FDTD) simulations of a gold nanorod and a nanodisk coupled to a small dielectric object.

2. Coupled dipole approximation

The interaction of two objects with dimensions smaller than the wavelength of the illuminating light may be analyzed within a coupled dipole approximation (CDA), which is a self-consistent solution to a problem of scattering electromagnetic fields by the system. The particles are assumed to be characterized only by an electric dipole polarization P_i = α_i(E_i0 – Σ_j≠iA_ijP_j) dependent on the local electric field coming from the source E_i0 and other dipoles A_ijP_j and its polarizability α_i. To make our point we drop all vector and matrix indices and look at the following set of equations for a dipole P₁ coupled to a dipole P₂ [18]:

Setting the denominator to zero: 1 – α₁α₂A₁₂A₂₁ = 0 allows us to calculate the mode frequencies of the interacting system. We assume the polarizability to be ${\alpha }_i=4\pi {\varepsilon }_0{r}_i^3\frac{{\varepsilon }_i-1}{{\varepsilon }_i+2}$, where i = 1,2, r_i is the radius of sphere i, the permittivities ɛ_i are in units of ɛ₀ which is the permittivity of free space. We get an expression for the permittivity of the resonator

From this simple analysis we see, that the expected resonance shift is proportional to the relative volumes of the analyte and resonator and depends on the polarizability of the analyte. If we take into account the fact that the dipole coupling terms determine the electric field acting onto the other dipole (Q ∼ A²), we see that the magnitude of the shift is proportional to the local electric field enhancement. Moreover, in the full case A is a tensor, so the dipole orientation with respect to the external field also influences the observed shift. Comparing this schematic result to the one obtained by Davis et al. [16] we note the qualitative agreement when neglecting the field orientation: both models reproduce the volume scaling and the saturation for large ɛ₁. Next, we will describe a way of calculating the resonance shift based on the local field intensities in the constituents of the resonant system, which is not limited to spherical geometries.

3. Quasistatic model

Let us consider an arbitrarily shaped metallic object in a dielectric medium illuminated by a plane wave of frequency ω. The extinction spectrum and resonance frequency ω₀ depend on its size, shape, and material properties as well as those of the surrounding medium. Binding of an additional object of permittivity different from the surrounding medium to its surface changes the resonance conditions, what can be observed by a slight shift of the spectrum. In the quasistatic limit, letting E⃗ denote the induced field due to an external field E⃗₀ impinging on the system, we let E⃗ = −∇⃗Ψ. We divide a large volume V into subvolumes V_i along material boundaries. Then, for each electric field E_i in V_i we define I_i ≡ ∫_Vi |E_i|² dV and in the quasistatic limit for the local fields we can write [21]

 

Fig. 1 Schematic view of a dielectric analyte (ɛ_d, V_d) attached to a metal sphere (ɛ_m(ω), V_m) surrounded by a dielectric medium (ɛ_s′, V_s′).

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We rewrite Eq. (10) into a form that allows for an explicit estimation of the resonance frequency shift from the electric fields, which may be obtained using different methods. The dielectric particle decreases the outside volume by V_d giving a new surrounding medium volume V_s′ = V_s – V_d and we can split the second term of II into two parts. Additionally, we assume, that the analyte is small compared to the resonator (V_d ≪ V_m), so that the perturbed field in the resonator is approximated by the field of the unperturbed one I′_m ≈ I_m, however, this approximation is used only in the second term of Eq. (10). These substitutions and a subtraction from both sides of Eq. (10) of all terms save the second of Eq. (10) give

When comparing Eq. (12) for the resonance shift in the quasistatic approximation with the result of the CDA (Eqs (5) and (6)) we find qualitative similarities. The easiest to notice is the permittivity dependence, which in both cases is determined by the assumed polarizability model of the dielectric analyte and is independent of the preferred method. The volume scaling, both resonator and analyte dependent, is explicit in the CDA, but it is not so obvious in the quasistatic approximation. On one hand, the resonator scaling in the denominator is explicit, as I_m is derived from an integral over the resonator. On the other hand, the analyte volume scaling is hidden in the in the spatial integrations which run over all subvolumes. The integral over the dielectric is clearly a function of the analyte volume and the remaining ones, due to the fact that the perturbation decays away from its source and is confined to its neighborhood, are also. Moreover, in the following section we examine the analyte volume scaling. The remaining important parameter is the electric field at the location of analyte adhesion. In both approaches this dependence is easy to quantify. When employing Eq. (12) to estimate the resonance shift we need to calculate the induced perturbation, which originates at the additional dielectric material. Its response, assumed to be dipolar in the simplest case, is proportional to the electric field at its location and using Eq. (12) we obtain a square dependence. Moreover, the interaction of a polarized material with its substrate-generated image is a function of the dipole orientation. An analysis of the A parameter in Eq. (6), which is hidden in Q, exposes a similar dipole orientation and power dependence.

4. Results and discussion

We will now apply this formula to a simple test case of a gold sphere (radius r_m = 30 nm) with a spherical analyte (r_d = 1 nm) limiting the number to three components m, s, and d (see Fig. 1). The considered gold sphere has maximum extinction at 585 THz and at this frequency permittivity ɛ_m = −3.21+2.91i and its derivative ${{\omega }_0\frac{\partial {\varepsilon }_m}{\partial \omega }|}_{{\omega }_0}=30.18+22.5\text{li}$. These values are calculated from experimental data of Johnson and Christy [22].

In Fig. 2 we plot cross-sections of spatial distributions for the numerator of Eq. (12) (i.e. the difference between the perturbed and unperturbed intensity distributions ɛ′_i|E′_i|² – ɛ_i|E_i|²) showing the magnitude and sign of the contributions for a parallel (β = 0) and perpendicular (β = π/2) field alignment in Fig. 2a and b, respectively. When β = 0 the analyte is located at the incident side of the metal sphere on the z-axis, while β = π/2 means that the analyte is on the side where the resonant field is the strongest. The energy redistribution due to analyte adhesion in the considered case is the following: the difference of the product of the permittivity and the integral between the perturbed and unperturbed fields in the metal gives a negative contribution to the total shift, while in the surrounding medium it has a positive sign. In the dielectric particle the sign depends on the permittivity of the analyte – in the case presented for ɛ_d = 4 the contribution is positive.

 

Fig. 2 Local values of ɛ′_i|E′_i|² – ɛ_i|E_i|², which integrated in individual subvolumes i give the numerator of Eq. (12), calculated for a dielectric analyte of ɛ_d = 4, r_d = 1 nm attached to a gold resonator (ɛ_m = −3.21+2.91i, r_m = 30 nm) in vacuum (ɛ_s = 1) for the local electric field in metal E_m (a) parallel and (b) perpendicular to the surface. The contribution from the metal is negative and causes a blueshift. The contributions from the surrounding medium and the analyte for ɛ_d = 4 are positive and cause a redshift. The vertical and horizontal black lines, in (a) and (b) respectively, separate the metal from the surrounding medium. The insets in (a) and (b) show the system scheme for β = 0 and β = π/2, respectively, where the dielectric particle (red circle) is attached to the metal resonator placed in an incident field E₀.

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To obtain these cross-sections as well as resonance shift plots in Fig. 3 we take into account that the analyte is much smaller than the resonator and that the local radius of curvature of the resonator is larger than the dielectric particle size. For a sphere the image factor is $\left({\varepsilon }_m-{\varepsilon }_s\right)/\left({\varepsilon }_m+\frac{l+1}{l}{\varepsilon }_s\right)$ and its response at close distances is determined by large l, which then approaches the planar image factor (ɛ_m – ɛ_s)/(ɛ_m + ɛ_s). This allows us to locally model the gold surface as flat. The analyte is placed in the near-field of the metal resonator and its dipolar response is determined by its polarizability α_d ∝ (ɛ_d – ɛ_s)/(ɛ_d + 2ɛ_s). The induced dipole, decomposed into parallel and perpendicular components relative to the local metal surface, changes the local electric field which we calculate in the quasistatic case using the method of images. The change to the field outside the resonator is generated by the original dipole and a mirrored one inside the metal, the change inside the metal by the original and an image outside, and inside the dielectric particle by the image in the metal and the presence of the particle itself.

 

Fig. 3 (a) Resonance shift $\Re \left\{\overline{\Delta \omega }\right\}$ as function of ɛ_d and β for an analyte of r_d = 1 nm attached to a gold resonator r_m = 30 nm in vacuum (ɛ_s = 1): black points – calculated with model, surface – Eq. (13) fitted to data, color scale for surface – residuals. (b)–(c) Total resonance shift $\Re \left\{\overline{\Delta \omega }={\sum }_{i=m,d,s^{\prime }}{\overline{\Delta \omega }}_i\right\}$, contributions from subvolumes $\Re \left\{{\overline{\Delta \omega }}_{i=m,d,s^{\prime }}\right\}$, and shift obtained with FDTD ${\overline{\Delta \omega }}_{\text{FDTD}}$ for (b) a parallel (β = 0) and (c) perpendicular (β = π/2) field. Notice the different scales for β = 0 and β = π/2

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Figure 3a shows the calculated resonance shift $\Re \left\{\overline{\Delta \omega }\right\}$ as a function of the analyte permittivity ɛ_d and the angle β with black points. The shift is a monotonic function of permittivity, because for an increasing ɛ_d the dipole moment increases asymptotically due to saturation of the dielectric sphere polarizability (full saturation is not shown here). Figure 3b and c show the total shift predicted by the model $\overline{\Delta \omega }$ (solid blue line) and calculated with FDTD ${\overline{\Delta \omega }}_{\text{FDTD}}$ (dashed black line with circles), and individual contributions from the three subvolumes ( $\overline{\Delta \omega }={\overline{\Delta \omega }}_{s^{\prime }}+{\overline{\Delta \omega }}_m+{\overline{\Delta \omega }}_d$) for β = 0 and π/2, respectively. We notice, that the model agrees quite well with the full FDTD calculation, although the latter is a fully-retarded method and takes into account the radiative components. For small ɛ_d the main redshift comes from the surrounding medium ${\overline{\Delta \omega }}_{s^{\prime }}$ and analyte ${\overline{\Delta \omega }}_d$ and is reduced by the remaining volume. For large ɛ_d the analyte volume also decreases the redshift. When the field is perpendicular to the surface instead of parallel, the electric field amplitude, determined by β, increases giving a larger shift.

The angular dependence is, in a first approximation, of the form ${\text{cos}}^2\beta +b{\left|\frac{{\varepsilon }_m}{{\varepsilon }_s}\right|}^2{\text{sin}}^2\beta$, where b is a parameter determining the relative sensitivity to the presence of an analyte placed in a parallel or perpendicular electric field with respect to the surface. If the shift is only proportional to the intensity of the local field |E|² determined by the position of the analyte, and by the permittivity of gold at the resonance frequency ω₀, then b = 1. To estimate b we fit the following function

Within the assumed model the frequency shift is proportional to the normalized volume V_d/V_m of the dielectric analyte and is confirmed in Fig. 4, where the largest considered normalized analyte volume corresponds to a radius of 2 nm. The field decays away from the gold sphere, so the resonance shift increases more slowly than a linear function and will approach asymptotically a maximum value in a similar manner as observed for gold nanoislands [23]. However, this last observation is beyond this quasistatic approximation as the analyte size would have to be comparable in size to the metal sphere.

 

Fig. 4 Resonance shift $\Re \left\{\overline{\Delta \omega }\right\}$ as function of normalized particle volume V_d/V_m and angle β calculated for a dielectric particle with ɛ_d = 2.

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To see how the method performs for commonly encountered sensors, we check how it compares with FDTD simulations for predicting resonance shifts for a disk-shaped resonator (Fig. 5a) with the following dimensions: radius 25 nm, thickness 25 nm, edge curvature 5 nm, which is illuminated by a linearly polarized plane wave. The analyte is modeled as a dielectric sphere with permittivity ɛ_d = 4 and radius of r_d = 1 nm and is arranged sequentially in a lattice on the surface of the disk. We calculate the extinction spectra using 3D-FDTD with nonuniform meshing equal to 0.5 nm for the gold disk and 0.2 nm for the dielectric analyte. Resonance shifts, shown in Fig. 5b, are obtained by fitting the data points and extracting the local maximum of the fitted function. Resonance shift values between the particle placements are obtained by interpolating with cubic splines. For the disk, the accuracy of the fit does not depend on the location of the analyte and is ±0.2 × 10⁻³ nm. The electric field distribution for the model is calculated at resonance (534.7 nm) of a lone disk and is used, in a similar scheme as discussed for the sphere, to estimate the peak shifts (Fig. 5c). We see, that the agreement is very good when the analyte is placed far away from the edges of the disk, however, decreases when moving away from the center. The largest mismatch is observed when the analyte is placed in the vicinity of the curved edge – the FDTD simulation yields a 1.5 × 10⁻² nm shift, while using the model we estimate a 1.0 × 10⁻². The reason behind this is the calculation scheme for the dipole moments of the analyte and its image – we assume a much larger local curvature of the surface of the metal resonator than the analyte. This condition is not fulfilled when the dielectric particle is near the rounded edge. However, a method of calculating electromagnetic fields created by an electric dipole in the vicinity of a sphere in the quasistatic approximation, presented by Zurita-Sánchez [24], may be modified to correct the model prediction.

 

Fig. 5 (a) Gold disk of radius 25 nm, thickness 25 nm, edge curvature 5 nm illuminated by a plane wave. Blue spheres indicate positions of analyte placement for FDTD calculations. (b) Resonance shift calculated using FDTD at positions from (a) and interpolated using cubic splines for easier comparison with model calculated shift. (c) Resonance shift calculated with the model. The color scale is the same for (b) and (c).

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A second comparison of model predicted shifts to FDTD obtained ones is carried out for a gold rod (radius of caps and rod 20 nm, length 100 nm) with dielectric particles (r_d = 1 nm, ɛ_d = 4) arranged at 13 positions along a line as indicated in Fig. 6. The longitudinal dimension of the rod relative to the resonance wavelength is about four times larger so the quasistatic condition is not fulfilled as strongly as in the previous case. Thus, there is a larger discrepancy of the quantitative results obtained via FDTD (simulations and extinction fitting) and model calculated data. The shifts are schematically indicated by lines following the surface of the rod where the analyte molecules are placed: the blue solid line represents fits to FDTD obtained shifts (maximum resonance shift is 3.7 × 10⁻³ nm, $\overline{\Delta \omega }=7.4\times {10}^{-6}$) with a 95% confidence interval shown using the green dashed line; the solid red line indicates model calculated shifts. Quantitative matching is not as good as in the test case of a sphere, but due to a larger radius of curvature of the resonator is better than for analytes near the disk edge.

 

Fig. 6 Gold rod 20 nm in radius with hemispherical caps of 100 nm long with schematically indicated positions of analyte placement illuminated with a plane wave polarized perpendicularly to the rod axis. The analytes are placed along a line parallel to the incident electric field as indicated by blue spheres. FDTD calculated/fitted resonance shifts are indicated by the blue line with 0.95 confidence interval (green dashed). The shift calculated using the model is shown by the red line.

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Our approach of calculating the plasmon resonance shift of a metallic resonator due to dielectric particle adhesion can be viewed in light of dipole – image-dipole interaction. An electric field parallel to the surface induces a parallel dipole moment in the analyte, while its image is antiparallel. For the electric field perpendicular to the surface, both the dipole and its image have the same sense. Both arrangements form an attractive system leading to a lower resonance frequency, but, in the case of the perpendicular field the interaction is weaker and for an outside field of the same amplitude as for a parallel field produces a smaller shift. However, the larger field enhancement counteracts this.

5. Conclusions

We have demonstrated a method of estimating the resonance shift of a simple metal object due to adhesion of a small dielectric analyte within the quasistatic approximation. It requires only a single, prior calculation of electromagnetic fields surrounding the resonator without the analyte using a numerical method, while its accuracy depends on the estimation quality of the analyte polarizability. Using FDTD simulations and the quasistatic model we have demonstrated the dependence of the plasmon resonance of metallic objects on the location of the added dielectric analyte. The magnitude of the shift is a function of geometrical and material parameters as well as position and is consistent with a schematic coupled dipole picture. The sensitivity per electric field unit in which the analyte is placed is greater when the field is parallel to the metal surface than for perpendicular alignment. However, due to a large enhancement factor observed for the latter arrangement shifts in this configurations are greater.