The spectral shift between near- and far-field resonances of optical nano-antennas

Christoph Menzel; Erik Hebestreit; Stefan Mühlig; Carsten Rockstuhl; Sven Burger; Falk Lederer; Thomas Pertsch

doi:10.1364/OE.22.009971

1. Introduction

Optical nano-antennas that resonantly interact with an external illumination at a single or multiple frequencies are of paramount importance for many applications [1, 2, 3, 4]. They promise to provide significantly enhanced electromagnetic fields at the nanoscale with tailorable and adjustable properties. The majority of optical nano-antennas are made of noble metals such as gold or silver. There, the resonant interaction is mediated by the excitation of localized surface plasmon polaritons [5]. However, they can be equally made of high permittivity dielectric materials where the excitation of Mie-type resonances, dominated by individual multipolar moments, are exploited [6, 7, 8].

Regardless of the actual implementation in terms of materials and structural geometry, a carefully designed optical nano-antenna permits the achievement of a huge field enhancement near the respective nano-antenna. This may occur at a single frequency, but eventually also at multiple frequencies or even bands. These resonant nano-antennas are then of interest to act as sensors or as sensing substrates that explicitly exploit this resonant field enhancement. Referential examples are Surface-Enhanced Raman Spectroscopy [9, 10] and Surface-Enhanced Infrared Absorption Spectroscopy [4, 11, 12]. With appropriate structures it became possible to probe for individual molecules with the far-reaching goal to even observe entire chemical reactions of isolated molecules [13]. Alternatively, these nano-antennas can be used to study the excitation of dipole-forbidden electronic transitions by strongly enhancing not just the electric field but also its gradient tremendously [14]. Eventually, the coupling of the nano-antenna eigen-modes and attached molecules beyond the weak coupling regime is possible [15], being useful for perspective applications in the context of novel single photon or entangled light sources. Such devices are essential to pave the way in many fields associated to quantum information processing and quantum computing.

In all these fields of research and the related applications it is obvious that an optimal performance requires the spectral position of the resonances sustained by the optical nano-antenna to be as close as possible to the transition frequencies of the attached atomic or molecular entities. Otherwise systems can be perceived as being optimized but their eventual performance is not as expected. The electromagnetic field close to an optical nano-antenna can, in principle, be probed by instruments such as a scanning near-field optical microscope (SNOM) [16]. But to collect the entire spectral and spatial characteristics remains challenging and in many cases only single frequencies are considered. This is naturally driven by the desire to preserve an experimental simplicity but it is not possible to identify the resonances in the near-field. The use of computational means certainly is an asset to quantify the dispersive nature of electromagnetic fields at the nanoscale. But its predictive power is only as strong as the information that are available on the nanostructures of interest. Although quite sophisticated efforts can be put in place nowadays to fully characterize the geometry of an optical nano-antenna and to take it into account [17] many uncertainties remain; e.g. those with respect to material properties at the nanoscale. Therefore and for all these reasons, it would be highly desirable to have a strategy available to derive the spectral response of an optical nano-antenna in the near-field from the measurable far-field response.

Although the spectral coincidence is very often implicitly assumed, at least to some lowest order, pioneering studies from Messinger et al. in 1981 explored already a frequency shift between the near-field and the far-field resonance in investigating scattering and extinction cross sections [18]. Recently the topic attracted renewed interest because nano-antennas of quite arbitrary shape are now at hand for many specific applications. In 2011 Zuloaga et al. [19] presented a simple model based on driven harmonic oscillators to estimate this frequency shift. In this work a formalism was developed to predict the shift between near- and the far-field resonance frequency for electric dipolar antennas by fitting the oscillator’s Lorentzian response to the extinction cross section. Very recently, for the first time Alonso-González et al. [20] presented measurements of this frequency shift by comparing the measured scattering cross sections of resonant electric dipolar nanowire antennas to aperture-less SNOM measurements of the near-field. They compared their measurements with the predictions of the simplified model [19]. Although this model has a reasonable predictive power its application is quite involved and might impede a direct and quick analysis of a specific nano-antenna.

Here, we intend to solve this issue in establishing a strategy that permits the immediate prediction of the spectral position of near-field resonances from measured far-field data. Moreover, it remains an open question how the analysis has to be extended to cope with more complicated nano-antennas that are not just characterized by an electric dipolar response.

Indeed, in this contribution we will show that there exists a surprisingly simple relation between the response of an optical nano-antenna in the near-field and the far-field; at least for optical nano-antennas where the resonance of interest is dominated by an electric dipole moment. By measuring the spectral dependency of the scattering cross section [21, 22] that is proportional to the intensity in the far-field, one can easily predict the intensity enhancement in the near-field, i.e. close to the antenna, by normalization of the measured scattering cross section by ω⁴. However, a detailed analysis shows that a variety of subtleties have to be kept in mind when relying on this simple analysis. At first, the shift depends on the actual distance of the probed molecule to the optical nano-antenna. At second, for any optical nano-antenna, except a spherical one operating in the quasi-static regime, higher-order electromagnetic multipolar contributions to the scattered field are no longer negligible. Then, the suggested simple functional dependency ceases to hold and a more involved dependency of the frequency shift on the actual position r arises. At third, higher-order multipolar moments might dominate the near-field although they exhibit just spurious features in the far-field, where their intensity maxima occur at frequencies different to those of the electric dipole. To address these issues we are going to present a detailed analysis here, where different degrees of complexity are used to elaborate all details.

The paper is structured with the desire to provide a concise treatment of the entire effect; starting from analytical considerations towards full numerical simulations required for more complicated structures. Therefore, first we derive the basic results beginning with analytical considerations for the field of an electric dipole. In a next step we consider a plasmonic sphere in dipole approximation and then compare exemplarily the frequencies shift for a gold and a silver sphere rigorously employing the Mie theory [23]. In a last step we rigorously analyze and compare the shift for realistic nanowire-type antennas made of gold or silver by means of full-wave simulations [24]. The paper concludes with a summary of important findings.

Eventually, we would like to mention that the problem we explore here has been previously discussed, albeit from a different perspective, by F. Moreno et al. [25]. However, this interesting and highly relevant work was brought to our attention during production review.

2. A Lorentzian dipole - analytical considerations

We start with the electric field of an electric dipole where the dipole moment in frequency domain is derived from an electric field E₀ driven-damped harmonic oscillator:

p (ω) = \frac{f}{ω_{0}^{2} - ω^{2} - i γ ω} \frac{E_{0}}{| E_{0} |}

with the oscillator strength f and the damping constant γ. The resonance frequency of the oscillator itself is ω₀. However, the maximum of |p(ω)| as well as of ℑ[p(ω)] is shifted to lower frequencies while increasing the damping parameter γ. This is accompanied by a considerable broadening of the resonance [for an illustration see Fig. 1(a)].

Fig. 1 Lorentzian dipole with resonance frequency ω₀ = 500THz. (a) Normalized modulus |p(ω)|/max{|p(ω)|} vs. frequency and damping constant γ. The dashed blue line indicates the maximum of |p(ω)| for each γ. (b) Resonance shift in THz between the local field intensity at distance r and the near-field intensity vs. the damping constant γ and the inverse distance 1/r.

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In frequency domain the electric field of an electric dipole is given by [see Ref. [26], p. 475, Eq. (9.18)]:

E (r, ω) = \frac{1}{4 π ε_{0}} {ω^{2} (n \times p) \times n \frac{e^{i k r}}{c^{2} r} + [3 n (n \cdot p) - p] (\frac{1}{r^{3}} + \frac{i ω}{c r^{2}}) e^{i k r}}

The far-field is essentially governed by the first term which is proportional to ω²/r whereas the near-field follows the second term proportional to 1/r³. The last term proportional to ω/r² contributes mainly in the intermediate region.

With regard to the enhancement of the interaction of light with other nanoscopic systems such as quantum dots, molecules, or nitrogen-vacancy centers in diamond, the intensity of the electric field is of primary interest; at least in what is frequently called the electric dipole approximation. In the near-field, at a fixed position r, the frequency-dependent intensity is given by

I^{nf} (r, ω) \propto {| E^{nf} (r, ω) |}^{2} \propto {| p (ω) |}^{2}

being proportional to the dipole moment such that the maximum of the intensity occurs always at the maximum of the dipole moment. This frequency is, in general, different to the resonance frequency ω₀ of the dipole.

For the far-field intensity we have

I^{ff} (r, ω) \propto {| E^{ff} (r, ω) |}^{2} \propto ω^{4} {| p (ω) |}^{2} .

The intensity and, hence, the radiated power as well as the scattering cross-section are proportional to ω⁴|p|². Thus their maxima will be shifted towards higher frequencies compared to the maximum of the dipole moment itself. The shift is a simple result of weighting the dipole moment by ω², where the ω⁴-dependency of the scattering cross section is known as Rayleigh scattering - blue light is scattered more strongly than red light.

Although these results may be found in any textbook on electrodynamics, a few important remarks are in order:

The shift between near- and far-field maxima depends, of course, on the particular dispersive nature of the dipole moment, i.e. p = p(ω). For very sharp resonances the emerging shift will be negligible. But for very broad resonances the shift of the maximum will be large. For a Lorentzian dipole the shift can be analytically calculated. However, the corresponding expressions are quite lengthy and do not provide an easily accessible insight.
Due to the interference of the near- and the far-field as well as the additional presence of what was called the intermediate field, the maximum of the intensity will depend on the distance r = |r| between dipole and observation point, i.e. there will be a notable transition region between near- and far-field.
The shift in the near-field is only visible in such quantities as the intensity and energy density. But of course, it does not emerge in such quantities as the Poynting-vector due to the non-radiative nature of the near-field.
Assuming that the response of a given particle is electrically dipolar only, the maxima in the near-field can be easily calculated from the measured scattering cross section by a simple division by ω⁴.

To visualize these analytical findings we determined the frequency dependent intensity from Eq. (2) for different damping constants γ and distances r along a certain direction, calculated the maximum of the intensity as a function of the frequency and plotted the distance resolved frequency shift

ω_{max}^{r} - ω_{max}^{nf}

in Fig. 1(b), where

ω_{max}^{r} = {ω : max_{ω} [I (ω, r)]}

is the frequency of maximum intensity at the distance r and

ω_{max}^{nf} = {ω : max_{ω} [I (ω, r \to 0)]}

is the frequency of maximum intensity in the near-field. The frequency of maximum intensity in the far-field

ω_{max}^{ff}

is defined analogously by taking r → ∞.

Obviously, the shift increases considerably with increasing damping constant γ. The transition between near- and far-field is accomplished on small length scales.

Here the basic features of the frequency-shift between near- and far-field are analytically revealed. The reason for the shift is the ω⁴-dependency of the scattering cross section.

3. Spherical systems - Mie theory based considerations

In the following the scattering problem for an isolated sphere is rigorously solved by means of the Mie theory. There, the incident and the scattered fields are expanded into the eigenfunctions of the Helmholtz equation in spherical coordinates, the so-called vector spherical harmonics (VSHs). For the scattered field E_sca this expansion reads as

E_{sca} (r, ω) = \frac{ω^{2}}{c^{2}} \sum_{l = 1}^{\infty} \sum_{m = - l}^{l} E_{l m} [a_{l m} (ω) N_{l m} (r, ω) + i b_{l m} (ω) M_{l m} (r, ω)],

where N, M are the VSHs, a_lm, b_lm are the complex expansion coefficients and E_lm is a prefactor. Analytical formulas for the VSHs can be found elsewhere [27]. In passing we draw the attention of the reader to the fact that the prefactor E_lm is quite arbitrarily chosen in the literature. Here, we rely on a definition that results in a simple relation between the expansion coefficients and the scattering cross section later on and define it as

E_{l m} = {(- 1)}^{m} \frac{| E_{0} |}{2 \sqrt{π}} \sqrt{(2 l + 1) \frac{(l - m)!}{(l + m)!}},

where |E₀| is the magnitude of the incident field that is set to unity in the following. In Mie theory the incident field is usually assumed as a plane wave which is expanded into the VSHs similar to Eq. (7). The well-known Mie coefficients (a definition is given in [23]) connect the expansion coefficients of the incident field to the expansion coefficients of the scattered field a_lm, b_lm. The scattering cross section of the sphere is obtained as

C_{sca} = \frac{ω^{2}}{c^{2}} \sum_{l, m} l (l + 1) [{| a_{l m} |}^{2} + {| b_{l m} |}^{2}] = \sum_{l, m} C_{sca} (a_{l m}) + C_{sca} (b_{l m}) .

Importantly, it can be shown that the expansion coefficients a_lm, b_lm are identical, except some prefactors, to the multipole moments in spherical coordinates [28]. For the first order, i.e. l = 1, the expansion coefficients a_1m are the electric dipole and b_1m the magnetic dipole moments, respectively, in spherical coordinates; l = 2 denotes the quadrupole moments and so on. Therefore, the above equation of the scattering cross section allows to identify which multipole moments contribute to the scattering cross section at a given frequency.

The transformation between spherical and Cartesian dipole moments is obtained by comparing Eq. (7) for the first order with Eq. (2). For example, the dipole moment in z-direction is given by the spherical multipole moment a₁₀

p_{z} = - \frac{c}{ω} ε_{0} \sqrt{12 π} i a_{10} .

The resulting scattering cross section of the p_z moment is given by using Eq. (9) as

C_{sca} (p_{z}) = \frac{ω^{4}}{c^{4} ε_{0}^{2} 6 π} {| p_{z} |}^{2}

The expected ω⁴|p|²-dependency of the far-field intensity [cp. Eq. (4)] can be clearly recognized.

In what follows this Mie theory-based approach is applied to investigate the frequency shift between near- and far-field intensities for isolated spheres.

3.1. Spheres in dipole approximation

We start with a sphere with radius R = 90nm in vacuum. The permittivity of the gold sphere is modeled according to Ref. [19] by

ε = ε_{\infty} - \frac{ω_{B}^{2}}{ω (ω + i γ)}

with ω_B = 8.9488eV, ε_∞ = 9.5 and γ = 0.06909eV. By using the Mie code we determined the spherical multipole coefficients and the scattering cross sections. The latter are shown in Fig. 2(a). The full scattering cross section C_sca (black solid line) is identical to the electric dipolar contribution C_sca(a_1m) (red solid-dotted line) almost everywhere. Only around 620THz an additional electric quadrupolar contribution C_sca(a_2m) (blue dashed line) becomes significant. Anyway, the response is dominated by the electric dipole such that we can focus on the electric dipolar contribution in the following.

Fig. 2 (a) Normalized scattering cross sections (black solid - full scattering cross section, red dotted - electric dipolar contribution, blue dashed - electric quadrupolar contribution, blue solid - scattering cross section normalized to ω⁴ displaying the expected near-field intensity), (b) Normalized intensity at different distances to the origin of the fictitious electric dipole.

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Solely based on the electric dipolar contribution we calculated the scattered electric field and intensity for different distances |r| between the origin of the conceptual dipole and the observation point. Considering the intensity of the electric field here is analogous to the use of the generalized cross sections as defined in [18] or [29]. We rely on the intensity as a measure for the interaction with probe molecules.

In Fig. 2(b) we have plotted these intensities normalized to their maxima. Clearly, for different distances a shift between the maxima is observed. For distances far away from the origin (≥ 5 μm) the frequency dependent intensity coincides with the scattering cross section C_sca(a_1m) shown in Fig. 2(a). Note, that at a distance of 0.5 μm the response is already far-field-like. Very close to the origin, i.e. in the near-field, the intensity [blue solid line in (b)] coincides with the scattering cross section divided by ω⁴ [blue solid line in (a)]. This supports the statement, that a purely electric dipolar near-field response is given by division of the far-field spectrum by ω⁴, whereas the near-field intensity follows |p(ω)|².

Two important remarks need to be made. First, upon expanding the scattered field into a series of multipole contributions, one conceptually replaces the response of a certain optical nano-antenna by multipoles located at the origin. Such an expansion is physically meaningful only outside the actual optical nano-antenna, i.e. here outside a spherical surface that fully encloses the scattering object. This means for the given example a radius of r ≥ R = 90nm. The response at such a reasonable distance, however, is about half the near- and half the far-field response [see e.g. red solid line in Fig. 2(b)]. This clearly suggests that a point at the surface of the sphere does not, so to say, strictly belong to the near-field. The direct application of the scaling of a far-field quantity to get the near-field quantity, therefore, would overestimate the shift, at least in the present example. Consequently, the frequency of maximum intensity cannot be evaluated solely by normalizing the scattering cross section by ω⁴. Therefore, it is an important consequence that we have, strictly speaking, to distinguish between the physical near-field by means of kr ≪ 1 and the colloquial use of the term near-field for points close to the scattering object.

Second, we neglected the influence of higher order multipoles. Even when they might be negligible for the scattering cross section, their influence might become significant at small distances. This involved issue will be addressed in the next subsection.

3.2. A rigorous treatment of spheres

In a next step, we will further increase the complexity, eventually approaching real systems by taking into account the contributions of higher-order multipoles beyond the electric dipolar one, too. We consider the response of gold and silver spheres with large radii of R = 110nm, where tabulated data for the silver and gold permittivity are used [30]. Where the frequency dependency for pure multipolar fields at a fixed distance r is independent of the spherical angles φ and ϑ such that we could choose an arbitrary angle for the pure electric dipolar fields, the situation changes dramatically when higher-order multipoles have to be taken into account, i.e. whenever different multipoles occur simultaneously. Then, the interference of the fields from different multipolar contributions leads to a strong angular dependency. Moreover, different multipolar contributions with different frequency behavior may eventually dominate the field in different directions. For example, in those directions where the field of the electric dipole vanishes or at least becomes small, the frequency dependence is obviously determined by higher-order multipole moments. To take this complication into account but with the purpose to maintain it traceable, we calculate the overall scattered field and integrate the intensity over spherical surfaces. With this procedure we obtain the averaged intensity as function of distance r and frequency ω. These averaged intensities are then normalized for each distance r by the maximum intensity of the electric dipolar contribution. The results for gold and silver sphere are shown in Figs. 3(a) and 3(b).

Fig. 3 Normalized, angularly averaged intensities and scattering cross sections for gold (a) silver (b) spheres with R = 110nm as function of frequency ω and inverse distance 1/r. The surface is normalized to the maximum intensity evoked by the dipole moment and indicated by the black dotted line. At large distances, i.e. small inverse distances, the normalized scattering cross sections and their respective multipolar contributions are shown as black solid (full scattering cross section), blue solid (electric dipolar), blue dashed (electric quadrupolar), blue dotted (electric octupolar) and red solid (magnetic dipolar) lines, where the overall scattering cross section nicely coincides with the angular averaged intensity.

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The complex figures must be read in the following way: The x-axis shows the frequency dependency. The y-axis shows the dependency on the inverse distance. Such scale is preferable to accommodate the different length scales in a reasonable manner. Thus, it is evident that a reciprocal distance of zero corresponds to the far-field. And the largest value of 1/r ≈ 9 μm⁻¹ corresponds to the surface of the sphere (radius R = 110nm). The colored plotted surfaces represent the averaged intensity, where each line of constant r is normalized to the maximum intensity of the dipolar resonance. These maxima (normalized to unity) are indicated by the black solid-dotted lines for both spheres. Clearly observable is the shift of the maxima towards larger frequencies upon transition between near- and far-field. In the far-field, i.e. at 1/r ≈ 0, we have additionally plotted the normalized full scattering cross section as black solid lines. They coincide perfectly with the averaged intensity. Further lines represent the electric dipolar (blue solid), the electric quadrupolar (blue dashed), electric octupolar (blue dotted) and magnetic dipolar (red solid) contributions to the scattering cross section. This ultimately allows to distinguish the type of resonance.

The results for the gold sphere, shown in Fig. 3(a), indicate an almost pure electric dipolar resonance for the scattering cross section. Slight deviations for larger frequencies are attributed to a magnetic dipolar and a small electric quadrupolar contribution. At small distances the electric dipolar contribution still dominates the response. However, at the surface of the sphere the electric quadrupolar contribution increases considerably and might be important for certain observation angles.

The results for the silver sphere, as shown in Fig. 3(b), indicate a more involved response. The electric quadrupolar contribution is large even in the far-field. In addition, also the electric octupolar contribution must not be neglected. The shift of the electric dipolar resonance is again clearly observable. The resonances of higher-order multipoles show a similar shift, which is, however, hardly distinguishable from interference effects. Quite importantly, the electric quadrupolar as well as the octupolar contributions are strongly increasing in the near-field compared to the electric dipolar one, which is a result of the different r-dependencies for different multipole orders. At least in the very near-field the electric field for a multipole of order l is proportional to 1/r^l+2, such that higher order multipolar contribution become increasingly important for smaller distances.

The most important issue, however, is the angular dependence of the maximum intensity. At different positions (with fixed distance r) of a probe molecule with respect to the sphere, a different spectral behavior can be expected. This feature represents a real difficulty that will be addressed for realistic nano-antennas in the succeeding section.

4. Cylindrical nano-antennas

In a last step of increasing complexity we consider the scattering response of realistic, cylindrical nanowire-type antennas made of both silver and gold. They consist of a 100nm long straight cylindrical nanowire of 50nm radius with spherical caps, having an overall length of 200nm. Such nano-antennas are used for increasing the interaction between light and molecules by local field enhancement [31]. We calculated the scattering cross section and the total field in the vicinity of the nano-antenna upon normal incident y-polarized plane wave excitation. For both purposes we rely on the full-wave Maxwell solver JCMsuite based on a finite element approach [24]. This method is highly suitable since it combines high-order convergence of FEM with geometrical flexibility of unstructured and adaptive spatial discretization.

The scattering cross sections of the individual multipolar contributions are shown in Figs. 4(a) and 4(d) for silver and gold nano-antennas, respectively. Both antennas are essentially electric dipole-like in the investigated frequency range with C_sca ≈ C_sca(a_1m). In fact, for non-spherical, elongated particles and nano-antennas the resonances, dominated by multipoles of different orders, are well separated in frequency domain [31, 32]. Additionally we plotted the total scattering cross section C_sca normalized to ω⁴ as solid black lines such that we can estimate the expected shift between near and far-field maxima to 32THz and 17THz for the silver and gold nano-antenna, respectively. These shifts are compared to the actual shift of the intensity maxima in the vicinity of the nano-antennas. The frequency shift $ω_{max}^{ff} - ω_{max}^{r}$ between the maximum in the scattering cross section (equivalent to the far-field intensity) and the actual maximum in the local intensity are plotted in Figs. 4(b) and 4(e) for silver and gold, respectively. A positive frequency shift designates a shift of the near-field maxima towards smaller frequencies compared to the far-field maxima. Furthermore, the common logarithm of the intensity $I (ω_{max}^{r}, r)$ is shown in Figs. 4(c) and 4(f). Obviously and as expected, the shift strongly depends on the actual position. But most importantly the shift near the nano-antenna and, in particular, in the important regions of high intensity, i.e. large interaction enhancement, is very well estimated by those values derived from the normalized scattering cross section. Hence, estimating the shift by the normalized scattering cross section is the method of choice to predict the frequency of maximum near-field enhancement. Note, that the slight asymmetry in Figs. 4(b) and 4(e) with respect to the x-axis is due to the small contribution of higher multi-poles. In fact, for a pure electric dipole these figures would exhibit a rotational invariance with respect to the center. Close to the line y = 0, i.e. along the propagation direction, the shift even changes sign. However, the intensity is quite small there.

Fig. 4 Cylindrical silver (upper row) and gold (lower row) nano-antennas. The left figures (a) and (d) show the normalized electric dipolar contribution to scattering cross section C_sca(a_1m) (blue solid line), the full scattering cross section C_sca normalized by ω⁴ (black solid line), the electric quadrupolar (red solid line), the electric octupolar (green solid line) as well as magnetic dipolar contributions (blue dash-dotted line) in arb. units. The central plots (b) and (e) show the spatially resolved frequency shift of the local (near-)field with respect to the far-field in THz, i.e. $ω_{max}^{ff} - ω_{max}^{r}$ . Note, that the maximum frequency shift close to the antenna agrees almost perfectly with the shift predicted in (a) and (d). The right plots (c) and (f) show spatial distribution of maximum local intensity (common logarithm scale) near the nano-antenna and at the frequencies indicated in (b) and (e), i.e. $I (ω_{max}^{r}, r)$ representing a map of maximum interaction enhancement.

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5. Conclusion

In conclusion, we have comprehensively studied the differences in the spectral position of resonances in the near- and far-field of optical nano-antennas. For small optical nano-antennas, whose scattering response can be accurately described in the electric dipolar limit, an easy to apply formula has been given that links a resonance frequency in the far-field to its counterpart in the near-field. We shed light on the issue of what should be actually considered as a near-field. Subtle but important differences exist to the field that can be described in the near-field approximation and the actual field close to the very surface of the optical nano-antenna. We extended our analysis towards more complicated optical nano-antennas the optical response of which is strongly affected by higher-order multipolar moments. It was shown that, spatially dependent, strong deviations exist between near- and far-field resonances. Surprisingly, these differences can be nearly as large as 1/10 of the pertinent resonance frequency; suggesting that it is not just a minor deviation but rather profound.

Our findings are important for nearly all applications that intend to exploit the tremendously enhanced near-field close to optical nano-antennas and which require a carefully spectral tuning. Important applications are in the field of life sciences where molecules are attached to optical nano-antennas to enhance their fluorescence signal which greatly improves the sensitivity of imaging. But also surface enhanced Raman scattering or up-conversion processes can be better controlled. Exactly the same holds for applications where other nanoscopic systems are coupled to optical nano-antennas, e.g. quantum dots or diamond nanocrystals containing nitrogen vacancies. These plexitonic hybrid devices are very important ingredients for a future architecture of integrated quantum optical circuits being the basis of quantum information devices.

Finally and with the purpose of avoiding any misunderstandings, we would like to stress that we do not intend to argue that the peculiarities of the near-field were insufficiently considered in previous studies. But it is rather our aim to accurately predict the spectral position of near-field resonances for an actual structure. In experiments, spatially and spectrally dependent information on the total electric field is required. This has to be provided by a measurements scheme that, in addition, shall not affect the quantity to be measured. It is easy to imagine that this is a complicated task. Moreover, numerical simulations that, in principle, can provide routinely information on the near-fields, suffer from the requirement to know every detail of the fabricated optical nano-antenna. Therefore, ideally quantities measured in the far-field should provide information on near-field quantities. This is at the heart of our contribution. Although being immediately applicable for electric dipole dominated optical nano-antennas, our approach can be further sophisticated by taken higher-order multipolar moments from optical nano-antennas into account. They are also potentially experimentally accessible with devoted experimental measurements schemes, i.e. those that measure angularly resolved the scattering response. Our work, therefore, is an important contribution towards the design of optical nano-antennas that shall mediate light-matter interaction at the nanoscale.

Acknowledgments

This work was supported by the German Federal Ministry of Education and Research(PhoNa), by the Thuringian State Government(MeMa) and Deutsche Forschungsgemeinschaft (DFG Research Center Matheon). C. Menzel gratefully acknowledges financial support by the Carl Zeiss Foundation. C. Rockstuhl would like to acknowledge the German Science Foundation for support within the project RO 3640/3-1.

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The spectral shift between near- and far-field resonances of optical nano-antennas

Abstract

1. Introduction

2. A Lorentzian dipole - analytical considerations

3. Spherical systems - Mie theory based considerations

3.1. Spheres in dipole approximation

3.2. A rigorous treatment of spheres

4. Cylindrical nano-antennas

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (12)

Optics Express