An analysis of the peculiar features of optical wire antennas with displaced terminals is presented. Full-wave simulations and a semi-analytical technique based on Pocklington’s equation are used in order to systematically study the behavior of input impedance and field enhancement at the antenna terminals when the feed-gap region is shifted with respect to the center of the wire. A simple analytical model based on the evaluation of the effective wavelength of the propagating surface wave is also suggested for the interpretation of the results.
©2010 Optical Society of America
In the last years, plasmonics has been one of the hottest research topics in the field of nanotechnology, due to the tremendous potential of surface-plasmon devices for the realization of photonic circuitry characterized by integration of electronic and optical functions on a single chip . In this context, optical antennas  are key components for coupling of light from the external world to nanoscale circuits and viceversa [3, 4]. As a matter of fact, unique properties of optical antennas, such as the possibility of achieving light localization beyond the diffraction limit and molding light behavior at the nanoscale, have already demonstrated to be essential for applications in spectroscopy and microscopy, photovoltaics, and for novel light sources .
Research on optical antennas is still in an embryonic stage, nevertheless some interesting studies have already been reported in the literature. In particular, the scientific community has focused the attention on the simplest geometry, i.e. the dipole antenna. Indeed, optical dipole antennas were fabricated, and the near-field distribution has been mapped through two-photon induced luminescence (TPL) microscopy , or scanning near-field optical microscopy (SNOM) in the mid-IR [6, 7]. For what concerns antenna theory and modeling, linear scaling of the effective wavelength (which is related to the antenna length) has been reported . An approach based on concepts borrowed from circuit theory has been proposed for the analysis of the effects due to the presence of the feed-gap region [7, 9], and tuning of the scattering response with nanocircuit loads has been demonstrated . Moreover, a semi-analytical procedure based on Pocklington’s equation [10, 11] has been applied in order to determine the current distribution along wire antennas; in the same work, antenna equivalent circuits that are commonly used in antenna theory at RF  were exploited to provide an accurate estimate of both field enhancement at the antenna terminals and scattering response . Last, but not least, a thorough analysis of the optical properties of isolated and coupled nanorods has been reported, with great emphasis on the physical mechanisms that are responsible for field enhancement .
In this work an analysis of the behavior of optical dipole antennas with displaced terminals is presented, focusing the attention on the effects due to the gap shift on input impedance and field enhancement at the antenna terminals. It is worth noting that optical antennas characterized by a displacement of the feed-gap region with respect to the center of the structure have already been described in the literature: in particular, it was shown that the introduced asymmetry has the effect of splitting the resonance peaks . Nevertheless, to the best of our knowledge, a systematic study as a function of the gap shift has never been reported.
2. Antenna structure and modeling
As a reference example, we have studied the behavior of a cylindrical rod made of silver, with length L and radius R equal to 110 and 5 nm, respectively. The air gap region is placed with a s-nm shift with respect to the rod center, and its thickness is fixed to g = 3 nm [see Fig. 1(a)]. Silver is described by exploiting the Drude model with ε ∞ = 5, plasma frequency fp = 2.175 PHz and collision frequency γ = 4.35 THz [9, 11].
Results obtained through different modeling procedures were compared in order to evaluate input impedance and field enhancement in the feed-gap region in an accurate and efficient way. In particular, we carried out full-wave simulations by using the finite-element method: we exploited the axial symmetry of the problem in order to reduce the problem from 3D to 2D, and we excited the antenna through a delta-gap source (in the transmitting mode), or a plane-wave (in the receiving mode) . A semi-analytical model based on the solution of Pocklington’s equation was also used. The validity of this procedure has already been demonstrated in the case of optical wire antennas with centered feed-gap , and its extension has been quite straightforward. A Thévenin equivalent circuit, directly borrowed from antenna theory at radio-frequency , can be used in order to better understand the behavior of the antenna in the receiving mode. It is worth noting a key feature of the analyzed structure: a shift of the feed-gap region from the center of the rod to one of the ends permits to tune the antenna equivalent-circuit parameters, whereas load impedance is unaffected by the shift. In fact, the gap region is simply moved along the wire, therefore the lumped capacitance associated with the gap  is constant. Figure 1(b) depicts the resulting equivalent circuit, wherein the arrows indicate the circuit parameters that can be tuned by shifting the gap region.
3. Optical properties of wire antennas with displaced feed-gap
In this section we will report the main findings emerging from the numerical analysis of the reference structure. First, we will compare the behavior of optical wire antennas with centered and shifted gap region, by emphasizing the effect of the gap shift on the parameters that mainly characterize the antenna performance, i.e. input impedance and field enhancement at the terminals. In Figs. 2(a)–2(c) we show input resistance and reactance (calculated in the transmitting mode), and field enhancement (evaluated in the receiving mode) for the nanoantenna with centered feed-gap and with a gap region shifted by 20 nm with respect to the center of the cylindrical rod. The input impedance is calculated as the parallel combination between the dipole impedance Zdip, which can be tuned by varying the gap position, and the constant load impedance ZL due to the gap region, that behaves as a plane capacitor . Results obtained from the solution of Pocklington’s equation are also reported, and they exhibit excellent agreement with simulations performed by exploiting the finite-element method.
Interesting effects due to the displacement of the terminals can be noticed. In the case of the center-fed dipole antenna two resonances are visible: the first one (at 264 THz) is the first short-circuit resonance, whereas the second one (at 353 THz) is the first open-circuit resonance. As well known, near each open-circuit resonance we have a peak of the input resistance and a large field enhancement . The behavior of the antenna significantly varies when the feed-gap is displaced, and in detail four different resonances can be identified: the first one (at 262 THz) corresponds to the first short-circuit resonance of the center-fed dipole; the second one (at 321 THz) is an open-circuit resonance, shifted to a lower frequency with respect to the case of the center-fed dipole; the third one (at 442 THz) is another short-circuit resonance, whereas the fourth one (at 488 THz) is a further open-circuit resonance, at higher frequency. Field enhancement is large near both the open-circuit resonances, nevertheless the radiation efficiency η = Rrad/(Rrad + Rloss) (where Rrad and Rloss are the radiation and loss resistances) decreases with increasing frequencies (6% at the second open-circuit resonance, with respect to 23% at the first one), and this causes a reduction of field enhancement at the second peak. As we have already shown in Ref. , the peaks of radiation and loss resistances perfectly match with the peak of the input resistance, independently from the value of the gap shift. It is worth noting that at the second short-circuit resonance the electric field into the gap region tends to zero. This result can also be justified by resorting to the plasmon hybridization model . Indeed, we have verified through calculation of the surface charge density that the lower- and the higher-frequency short-circuit resonances correspond to bonding and anti-bonding interactions between the segments, respectively. As a consequence, the higher-frequency mode appears only when the symmetry of the system is broken by displacing the feed-gap region, where we have a near-zero electric field .
4. Tailoring the antenna response through displacement of the feed-gap
In this section we will study the current distribution I(z) along the antenna in the transmitting mode [Figs. 3(a)–3(d)], with an eye to the frequency position of the four resonance frequencies [see Fig. 4(a)]. In the following we describe a collection of results wherein we considered gap shifts from 0 to 50 nm. Figure 3(a) shows I(z) at the first short-circuit resonance; it is possible to note that the shape of the curve (which is the typical stationary wave we have in a half-wavelength dipole) is barely affected by the gap shift, and as a consequence the related resonance frequency only slightly moves around 260 THz. In Fig. 3(b) we report I(z) at the first open-circuit resonance. In this case, the structure behaves as it is composed of two coupled half-wavelength dipoles, and the shape of the curve within each segment is basically preserved. Indeed, the position of the minimum value of current moves with the gap region. Moreover, it is interesting to note that the more the feed-gap is shifted toward one of the ends, the more magnitude of the current in the long segment grows with respect to current in the short segment. As a result, current distribution in the long segment is dominant, and the resonance moves toward longer wavelengths as the gap shift is increased. In Fig. 3(c) we have I(z) at the second short-circuit resonance and we can note that, as in the case of Fig. 3(a), a shift of the gap changes the amplitude but it preserves the shape of the curve, which is a one-wavelength stationary wave. In fact, the resonance frequency is around 440 THz and it is practically independent from the gap shift. Finally, in Fig. 3(d) we show the results at the second open-circuit resonance. The analysis of I(z) reveals that the short segment resembles a half-wavelength dipole, whereas in the long segment we can see the formation of a one-wavelength stationary wave. Notice that the point of minimum current moves with the feed-gap, therefore the two segments behave as coupled nanorods.
The treatment reported above is the starting point for a systematic study of the behavior of optical wire antennas with displaced terminals. In Fig. 4(a) we show the four resonance frequencies as a function of the gap shift. Solid lines and circles refer to solutions of Pocklington’s equation and full-wave simulations, respectively. Agreement between the two modeling techniques is excellent, as we have already shown in the previous section for a specific case (s = 20 nm). The dashed lines indicate an estimate of the asymptotic behavior of the resonance frequencies obtained through the application of a simple analytical model based on the evaluation of the dispersive properties of the cylindrical rod. Indeed, we calculated the effective wavelength as λeff = λ/neff - 4R/N, being N the order of the resonance and neff the effective index of the principal wave of the cylindrical rod . The five dashed curves correspond to the following equations: with increasing frequency, we have λeff = 2L (independent from the gap shift), λeff = 2L 1 (decreasing with the gap shift), λeff = L (independent from the gap shift), λeff = 2L 2 (increasing with the gap shift), and λeff = L 1 (decreasing with the gap shift).
These results are coherent with the analysis of the current distributions. In particular, we have shown that the first and the second short-circuit resonances are characterized by current patterns which resemble half- and one-wavelength stationary waves, respectively [see Figs. 3(a), 3(c)]; in fact, the resonance frequencies are not dependent on the gap shift and can be predicted by imposing that λeff is equal to integer multiples of the rod length L. In the case of the first open-circuit resonance, the current distribution in the long segment dominates [see Fig. 3(b)], therefore the corresponding frequency is located near the half-wavelength resonance of the segment with length L 1; notice that when the feed-gap is centered the capacitive load is responsible for the discrepancies between analytical model and full-wave simulation, whereas the two curves tend to overlap if the gap is moved toward one of the ends. Finally, the behavior of the second open-circuit resonance deserves a careful analysis. Indeed, the resonance frequency increases with the gap shift for small values of s, but the curve bends and starts decreasing for a value of s around 25 nm. It is worth noting that, as we have previously observed, for small values of the gap shift the resonance frequency tends to be close to the half-wavelength resonance of the short segment, whereas for larger values of s the curve bends because the current distribution in the long segment, which resembles a one-wavelength stationary wave, dominates.
In Fig. 4(b) we show field enhancement at the resonance frequencies as a function of the gap shift. As expected, the open-circuit resonances exhibit large enhancement: the first resonance is characterized by decreasing enhancement with increasing gap shift, whereas the second one exhibits a maximum enhancement around s = 20 nm. This is in agreement with the results reported in Fig. 4(a), wherein we show that the resonance frequency is determined by two competing effects with different asymptotic behavior. It is worth noting that the input resistance is comparable in the two cases [see Fig. 2(a)], but the second resonance is affected by larger losses than the first one. The two short-circuit resonances exhibit smaller field enhancement. In particular, the first resonance is characterized by an appreciable increase of the enhancement (up to 100) only for large values of the gap shift, whereas the electric field into the gap is about zero at the second resonance due to the one-wavelength stationary-wave current pattern.
A numerical study of the properties of optical wire antennas with displaced terminals has been reported, with particular emphasis on the effect of the feed-gap shift on input impedance and field enhancement at the antenna terminals. The reported treatment demonstrates that the possibility of tailoring the antenna response by shifting the feed-gap region can supply a further degree of freedom for the design of devices conceived for light localization at the nanoscale.
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