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Silver folded dipoles consisting of two parallel nanowires may operate as efficient transmitting and receiving nanoantennas in the optical domain in both cases of silver-nanowire-terminated arm-edges and open-terminated arm-edges, in contrast to their conventional radio-frequency (RF) counterparts that only operate efficiently when they are short-wire-terminated arm-edges. The mode decomposition analysis with the equivalent circuit reveals that the difference of the wave numbers between the common and the differential modes allows this feature for the optical folded dipole nanoantenna under both arm-edge conditions. The analysis also estimates the efficiency of the folded dipoles via the equivalent radius of nanowires for the common mode. These folded nanostructures may exhibit the enhanced efficiency with the maintained radiation patterns, where the efficiency of folded dipoles is the same as that of the effective single dipole at resonance.
©2011 Optical Society of America
1. Introduction
Plasmonic dipole nanoantennas are among the key devices to convert optical radiation into localized fields and vice versa. They have been studied for wireless link nanocircuits [1], manipulating the laser emission [2,3], photodetectors [4], microscopy [5,6], and molecule vibrational detection [7], to name a few. Yagi-Uda antennas composed of several cylindrical nanowires for the driven element, reflector, and some directors were demonstrated numerically [8] and experimentally [9] to enhance the directivity. They were also studied when nanoparticles were considered as directors and reflectors [10,11].
The effective wavelength scaling for plasmonic dipoles has been discussed with an elegant explanation of the relationship between the dipole length and the operating wavelength [12]. A nanocircuit loading concept [13] was applied to the resonance tuning for plasmonic dipoles [14]. The loading effect was experimentally considered and validated by the near field microscopy [15]. The current distribution of an Ag dipole was calculated by a surface-impedance integration equation [16,17]. The effects of the width [18,19] and the location [17,20] of the gap along the dipole were also investigated. An impedance matching of a nanoantenna to the transmission line was discussed [21]. A plasmonic patch [22] for the Purcell factor enhancement and a cross antenna [23] for any polarization state have also been studied recently.
In terms of efficiency, a nanodimer antenna was theoretically studied to provide an efficiency of ~90% [24]. In Ref [12], a collection of nanowire arrangement made of a center-driven dipole in proximity of two other nanowires showed a 1.65 times efficiency as compared to a single nanowire. In the radio frequency (RF) and microwave domains, folded dipoles have generally short-wire-terminated edges [25–27]. They would not work as efficient radiators in the case of open-terminated arm-edges due to near-zero input impedance at resonance. A question arises; what is the arm-edge condition for optical folded dipole nanoantennas? To the best of the authors’ knowledge, plasmonic folded dipole nanoantennas with Ag nanowire-terminated arm-edges have not been reported yet. In other words, no investigation of collections of nanowire dipole antennas has been reported from the folded dipole view point.
In this paper, collections of parallel nanowires are investigated theoretically using the mode decomposition analysis [25–27] from the folded dipole point of view for the first time, to the best of our knowledge. In Section 2, arm-edge conditions of Ag folded dipoles are discussed, and the mode decomposition analysis used in the RF and microwave domains is reviewed. In Section 3, the dispersion characteristics are discussed in the nanowire transmission lines with analytical and numerical results. The strong dispersion is one of the specific distinguishing features of these optical nanoantennas as compared and contrasted with the conventional antennas in the RF and microwave domains. The validity of the mode decomposition analysis is confirmed in the impedance and radiation efficiency via the effective single dipole. The concluding remarks are then given in Section 4.
2. Plasmonic folded dipole nanoantennas
2.1 Terminal conditions at both arm-edges
Figure 1 (a), (b) show the snap shots (in time) of the electric field distributions of the Ag folded dipole nanoantennas that were simulated by the Finite-Integration-Technique-based simulator, CST Microwave StudioTM [28]. A Drude model was used for the Ag permittivity, εAg = ε 0{ε ∞-fp 2/[f(f + iγ)]}, with ε 0 = 8.85 × 10−12 [F/m], ε ∞ = 5, fp = 2175 THz, γ = 4.35 THz, and f is the operating frequency [29]. The folded dipole consists of the two parallel cylindrical Ag nanowires; one is driven at the central feed gap having the gap width g, and the other is coupled to the first nanowire via close proximity. The central feed gap has an optical voltage source, and the input impedance is evaluated, as defined in [14], as the ratio of the driving optical voltage difference across the gap to the total flux of induced optical displacement current flowing from the feeding source into the nanoantenna. Transmitting nanoantennas driven by optical voltage sources are investigated throughout the paper. In Fig. 1 (a), the two nanowires have “open-ended terminals”, while in Fig. 1 (b), the Ag nanowire-terminated (“shorted”) folded dipole has the two nanowires ended at the edges by Ag nanowires. In both cases, the dipoles have the strong electric fields at the edges, as shown in Fig. 1. Unlike the case of conventional antennas in the RF and microwave domains, here both the open-terminated and the Ag nanowire-terminated conditions provide the efficient performance of the plasmonic folded dipole nanoantennas as transmitting and receiving antennas.
Fig. 1 Snap shots (in time) of electric field distributions of Ag folded dipole nanoantennas obtained using the CST Microwave Studio™ simulation. (a) Open-terminated arm-edges at 304 THz and (b) Ag nanowire-terminated arm-edges at 301 THz. (dimensions; L = 110 nm, r = 5 nm, d = 15 nm, and g = 3 nm)
2.2 Decomposition into the common and differential modes
The displacement currents I 1 and I 2 on the two nanowires of the Ag folded dipole (Fig. 2(a) ) is decomposed into the common mode currents (Fig. 2(b)) and the differential mode currents (Fig. 2(d)) by the mode decomposition analysis commonly used in the RF and microwave domains [25–27]. The common mode (Fig. 2 (b)) has the same directional currents, and can be expressed by the “effective” single dipole having an equivalent radius (Fig. 2(c)). The differential mode (Fig. 2(d)) is expressed with the parallel connection of the two stubs having the differential directional currents. The equivalent circuit of Fig. 2(e) has the three parallel impedances; Zcap for the capacitor of the feed gap, vi −2 times stepped up Zeff_dip for the effective single dipole (common mode), where vi = I 1/(I 1 + I 2) is the current dividing ratio, and twice of Zstub for the half nanowire length stub (differential mode). Referring to [25–27], those are presented as
where Zsim is the impedance directly obtained with the simulation models, as shown in Fig. 1, Zfold is the “de-embedded” impedance [14] that is the “intrinsic” impedance of a pure folded dipole nanoantenna excluding the capacitive air gap, and Zc is the characteristic impedance of the stub, Zedge is the edge impedance of open-edge or Ag nanowire-edge, and βdif is the wave number for the differential mode of the stub. The impedance step-up ratio vi −2 becomes four in case of I 1 = I 2. The resonance frequency is defined as the frequency at which the imaginary part of impedance becomes zero throughout the paper. Moreover, the efficiency is defined as the ratio of the radiation resistance to the summation of the radiation resistance and the ohmic-loss resistance. For example, for the effective single dipole, the efficiency is defined as Rrad_eff_dip /(Rrad_eff_dip + Rloss_eff_dip). This analysis can be applied to Ag nanowire-terminated folded dipoles, although it has been discussed for open-terminated folded dipoles. The mode decomposition analysis has an assumption in which the parallel nanowire structure in the folded dipole works as a parallel nanowire transmission line. In the RF and microwave domains, the mode decomposition analysis gives accurate estimations when the wire-to-wire separation is much shorter than the free space wavelength λ0, e.g., typically an order of 0.01 λ0 [25].Fig. 2 (a) Relationship of currents and feed voltages between (a) folded dipole, (b) common mode, (c) effective single dipole, and (d) differential mode. The equivalent circuit is also presented in (e).
3. Performance
3.1 Wave numbers of nanowire transmission lines
In order to obtain the radius of the effective single dipole, the complex wave numbers βsing for the single nanowire transmission lines are investigated theoretically at 300 THz, as shown in Fig. 3 . The guided wavelength becomes shorter (i.e., the real part of βsing increases) and the loss (i.e., the imaginary part of βsing) increases as the single nanowire becomes thinner. The simulation results using the CST Microwave StudioTM (represented by the circles) agree well with the analytical results (solid curves) that were based on the work of Takahara, et al. Ref [30]. The validity of the simulation models is therefore confirmed. In addition, the CST simulation is performed on the two parallel nanowire transmission line, and the results are shown in Fig. 3. The common mode of the parallel nanowire transmission line having a 5-nm radius and a 15-nm center-to-center separation, represented by a cross, shows a real part of the wave number of Re(βcom) = 3.6 k 0 that corresponds to a single nanowire transmission line having a radius of r = 6.5 nm. Consequently, an equivalent radius of 6.5 nm is set for the effective single dipole in the mode decomposition analysis for the common mode of the parallel nanowire transmission line. On the other hand, the differential mode of the parallel nanowire transmission line has Re(βdif) = 7.1 k 0 that is nearly twice of that of the common mode case. This is a distinct difference between the characteristics of the optical folded dipole nanoantennas and their RF and microwave counterparts where the wave numbers for the common and differential modes are nearly the same.
Fig. 3 Wave numbers (real part (a) and imaginary part (b)) of the single nanowire, and the common and differential modes of the parallel nanowire as a function of the nanowire radius. The dots (shown as “circles” for the single nanowire, “crosses” for the common mode and “triangles” for the differential mode of the parallel nanowire transmission line) are the results of the CST Microwave Studio™ simulation. The solid lines for the single nanowire are the results based on the analysis of Takahara, et al. in Ref [30]. The insets show the magnetic field distributions in the cross sections at r = 5 nm.
Figure 4(a), (b) show the frequency characteristics of the complex wave numbers βsing for the single nanowire transmission line having a 6.5-nm equivalent radius and βcom for the common mode of the parallel nanowire transmission line having a 5-nm radius and a 15-nm center-to-center separation. It can be seen that the single nanowire and the common mode have nearly the same wave numbers in the frequency range from 100 THz to 500 THz. It is confirmed that the equivalent radius of 6.5-nm works effectively in this whole frequency range.
Fig. 4 Frequency variation of wave numbers (real part (a) and imaginary part (b)) of the single nanowire having a 6.5-nm equivalent radius and the common mode of the parallel nanowire. The dots (shown as “circles” for the single nanowire and “crosses” for the common mode of the parallel nanowire transmission line) are the results of the CST Microwave Studio simulation. The solid lines for the single nanowire are the results based on the analysis of Takahara, et al. in Ref [30].
3.2 Validity of mode decomposition analysis
The effective single dipole nanoantenna and nanowire stubs (half of the folded dipole nanostructures in Fig. 1(a), (b)) were simulated by the CST Microwave StudioTM. Figure 5(a) shows the de-embedded impedance of the effective single dipole nanoantenna, where the capacitance of the air gap is excluded from the simulated input impedance using Eqs. (1), (2). A “short-circuit” resonance occurs at 316 THz when the dipole has a half-wavelength resonance. Around this resonance the real and imaginary parts of this impedance increase with frequency. The inset of Fig. 5(a) shows the feeding structure that consists of the optical voltage source across the gap of the perfect-conductor circular capacitor having a 6.5-nm radius that is inserted at the center of the Ag dipole nanoantenna having the same radius. Figure 5(b) shows the input impedance of the open-terminated stub having the dimensions of a 5-nm radius and a 15-nm center-to-center separation. A quarter- and a half-wavelength resonances are seen at 194 THz for the “short-circuit” resonance and 361 THz for the “open-circuit” resonance, respectively. For the Ag-terminated stub, we have a quarter-wavelength “open-circuit” resonance at 207 THz and a half-wavelength “short-circuit” resonance at 381 THz, as shown in Fig. 5(c). The stub-edge conditions swap the order of short-circuit and open-circuit resonance frequencies in the open- and Ag-terminated stubs. Instead of obtaining Zedge, L, and Zc in Eq. (5) analytically, the stub impedances are directly obtained here by the ratio of the reflected wave to the incident wave at the input port, where no perfect-conductor is utilized as the insets of Fig. 5(b), (c) show. Simulated stub impedance includes the fringing effect at the open-edge, i.e., some electric fields outside the physical stub edge in Fig. 5(b), or the capacitive impedance at the Ag nanowire-edge based on the nanocircuit concept [13] in Fig. 5(c). The discrepancies in the 1st and 2nd resonance frequencies, 194 THz / 207 THz and 361 THz / 381 THz, are seen between the two stubs (Fig. 5(b), (c)) that correspond to 6.3% and 5.2% smaller resonance frequencies of the open-terminated stub comparing with the Ag-terminated stub. The discrepancies may be due to the fringing effect for the open-terminated stub and the capacitive impedance for the Ag nanowire-terminated stub. Both stubs have large absolute impedances in the imaginary part, as compared with the impedance of the effective single dipole around 300 THz. This is due to the difference in the wave numbers for the common and differential modes of Fig. 3, as mentioned before. The “wavelength” here is the effective wavelength for the single nanowire having a 6.5-nm radius in Fig. 5(a) and for the differential mode of the parallel nanowire in Fig. 5(b), (c) when half- and quarter-wavelength resonances are discussed.
Fig. 5 Simulated impedances of (a) the effective single dipole with a 6.5-nm equivalent radius, (b) the open-terminated stub, and (c) the Ag nanowire-terminated stub. The stubs have the dimensions of a 5-nm nanowire radius, a 15-nm center-to-center separation, and the half nanowire length. (solid lines: real part, dashed lines: imaginary part) The insets show the enlarged feeding structures in which (a) the dipole has the circular capacitor having a 6.5-nm radius and (a), (b) the stubs have no capacitors.
The “de-embedded” input impedance of the open-terminated folded dipole is calculated using the CST Microwave StudioTM simulation of Fig. 1(a) and Eqs. (1), (2), and is shown in Fig. 6(a) . Incorporating the input impedance of the effective single dipole (Fig, 5(a)) with the impedance of the open-terminated stub (Fig. 5(b)) using the equivalent circuit model (Fig. 2(e)), we can find the equivalent impedance shown in Fig. 6(b). Analogously, the Ag nanowire-terminated folded dipole impedance is presented in Fig. 7 (a), (b) , using the simulation model (Fig. 1(b)) for Fig. 7(a) and using the equivalent circuit (Fig. 2 (e)) with the impedances of the effective single dipole (Fig. 5(a)) and the Ag nanowire-terminated stub (Fig. 5(c)), shown in Fig. 7(b). The agreement between the direct simulation results and the equivalent impedances using the equivalent circuit model is good in both terminal conditions, and the validity of the mode decomposition analysis is therefore confirmed in the impedances.
Fig. 6 Impedances of the open-terminated folded dipole having the dimensions of r = 5 nm and d = 15 nm. (a) Simulation result of the folded dipole model (Fig. 1(a)); and (b) the calculated result using the equivalent circuit (Fig. 2(e)) incorporating impedances of the effective single dipole (Fig. 5(a)) and open-terminated stub (Fig. 5(b)). (solid lines: real part, dashed lines: imaginary part) The inset of (a) shows the enlarged feeding structure that has a 5-nm radius capacitor.
Fig. 7 Impedances of the Ag nanowire-terminated folded dipole having the dimensions of r = 5 nm and d = 15 nm. (a) Simulation result of the folded dipole model (Fig. 1(b)); and (b) the calculated result using the equivalent circuit (Fig. 2(e)) incorporating impedances of the effective single dipole (Fig. 5(a)) and Ag nanowire-terminated stub (Fig. 5(c)). (solid lines: real part, dashed lines: imaginary part) The inset of (a) shows the enlarged feeding structure that has a 5-nm radius capacitor.
Some discrepancies between the direct simulation results and the equivalent impedances are seen as resonance frequency shifts in Figs. 6 and 7. The shift of “open-circuit” resonances, e.g., the peak in the real part around 400 THz in Fig. 6, may be mainly due to the effect of the different sizes in the feeding structures. The folded dipole models have 5-nm radius parallel circular-plate capacitors in the feeding structures while the effective single dipole model has a 6.5-nm radius capacitor (The de-embedded input impedances and feeding structures are shown in Fig. 6(a), Fig. 7(a), and Fig. 5(a), respectively). The capacitance in Zcap for the effective single dipole is larger than that for the folded dipoles as written in Eq. (2). The equivalent circuit of Fig. 2(e) implies that the Zcap in the parallel connection affects on the antenna input impedance as the resonance frequency shift in “open-circuit” resonances. In other words, the Zcap itself is de-embedded, but the effect of the feeding size remains. On the other hand, in “short-circuit” resonances this paper focuses on, the Zcap does not cause the shift in the resonance, as the equivalent circuit of Fig. 2(e) indicates. Consequently, the mode decomposition analysis shows good estimates in “short-circuit” resonances.
Discrepancies less than 5% in the “short-circuit” resonance frequency shifts are seen between direct simulation results and equivalent impedances, e.g., 304 THz / 316 THz (Fig. 6(a), (b)) for the open-terminated arm-edges and 301 THz / 316 THz (Fig. 7(a) (b)) for the Ag-terminated arm-edges at resonance. That resonance frequency of 316 THz of both folded dipoles in the equivalent impedances comes from the same resonance frequency of the effective single dipole impedance (Fig. 5(a)) because of 2 Zstub >> vi −2 Zeff_dip. The discrepancies may be due to the fact that the arm-edge shapes and the fringing effects are not considered in the length of the effective single dipole that was set at 110 nm for the sake of simplicity. In other words, the optical length in the dipole is equivalently longer than the physical length. When the effective single dipole length L is set at 116 nm where the fringing effect is equivalently included, the estimated “short-circuit” resonance frequency via the equivalent circuit reduces from 316 THz to 304 THz in both arm-edge conditions. This fringing effect also affects on the “open-circuit” resonance frequency shifts by the same order, e.g., a 6-nm increase in the effective dipole length reduces an “open-circuit” resonance of 414 THz (L = 110 nm, Fig. 6(b)) to 405 THz (L = 116 nm) in the equivalent impedances while the direct simulation result has a 380 THz resonance (L = 110 nm, Fig. 6(a)) in the open-terminated condition. It was pointed out in [14] that the discrepancy between the physical dipole length L and the estimated length (βsing /k 0) λ0/2 is due to reactive fields at the arm-edges in the single nanodipoles. It can be said that a longer effective nanodipole considering the fringing effect improves the estimation accuracy in the “short-circuit” resonance but this compensation is not enough for the “open-circuit” resonances.
Another factor of the frequency discrepancies may come from an assumption in the mode decomposition analysis in which the parallel nanowire structures in the folded dipoles work as parallel nanowire transmission lines. A 15-nm nanowire separation is no longer too negligible for highly accurate estimations.
The radiation efficiency, the resistances, and the resonance frequency are summarized in Table 1 for four dipoles; (i) original single dipole, (ii) effective single dipole, (iii) open-terminated folded dipole, and (iv) Ag nanowire-terminated folded dipole. The parentheses in columns (iii) and (iv) represent estimated values from the mode decomposition analysis that has the same efficiency and four times resistances as those of (ii), assuming vi −2 = 4 and 2 Zstub >> vi −2 Zeff_dip. The radiated power is obtained by integrating the radiated power flux over the 4π solid angle in the simulations. The efficiency is the ratio of the radiated power to the input power. The radiation resistance is obtained by multiplying the efficiency times de-embedded input resistance. The efficiency and resistances including de-embedded resistance, radiation resistance, and ohmic-loss resistance in (iii) and (iv) agree with the estimated values. The validity of the mode decomposition analysis is also confirmed in the efficiency. (iv) has the physical connections between the two nanowires at the arm-edges, resulting in vi −2 = 4, while (iii) has vi −2 ~4 due to the proximity coupling between the two nanowires. This gives slightly smaller resistances in (iii). The (iii) and (iv) also have nearly the same resonance frequency. With regard to the single dipole comparison between (i) and (ii), the efficiency and the resonance frequency increase from 18.8% to 37% and from 262 THz to 316 THz as the single dipole becomes thicker from r = 5 nm to 6.5 nm. This can be understood from the behavior of the wave number in the single nanowire transmission line shown in Fig. 3. The efficiency agreement between (ii) 37%, (iii) 37.9%, and (iv) 37.9% confirms the validity of the equivalent radius of 6.5-nm because the dipole radius is sensitive to the efficiency. This also indicates that not the dipole radius but the dipole length should be adjusted for accurate estimations for the “short-circuit” resonance, as discussed before. While in the RF and microwave domain, it is well known that the resonance frequency slightly decreases as the single dipole becomes thicker.
Table 1. Efficiency, Resistances, and Resonance Frequency for Four Dipoles
From the equivalent circuit, it must be noted that the folded dipole has the same efficiency and four times larger input resistance as those of the effective single dipole when vi −2 = 4 and 2 Zstub >> vi −2 Zeff_dip at resonance. One of the distinct distinguishing features in the optical folded dipole nanoantennas as contrasted with the conventional RF and microwave antennas is that the efficiency of a single dipole largely changes with the diameter of nanowires in the optical domain due to the strong dispersion of nanowires that was implied by the wave number characteristics of the nanowire transmission lines of Fig. 3.
It should be mentioned that (iii) the open-terminated and (iv) Ag-terminated folded dipole nanoantennas having an inter-wire distance of 15-nm maintain the omnidirectional nature of the radiation pattern in the yz-plane while can enhance the efficiency. This inter-wire distance is shorter than the distance required for the directivity enhancement in the case of Yagi-Uda dipoles that have typically a λ0/4 separation between nanowires. This is a distinct difference between folded nanodipoles and Ygi-Uda nanodipoles in their functionalities.
3.3 Discussion
In the RF and microwave domains, metal dipoles and parallel-wires can be treated with the perfect-conductor assumption. They have transverse electromagnetic field distributions and those wave numbers are the same as that in the free space wave number, i.e., βsing ~βcom ~βdif ~k 0. Namely, the single dipole length is L ~λ0/2 at the 1st resonance, and the wave number of the stub is βdif ~2π/λ0. In the transmission-line theory, the stub impedance Zstub in Eq. (5) is given as ~j Zc tan(βdif L/2) ~ + ∞ for the short-wire-edge of Zedge = 0, or ~Zc /(j tan(βdif L/2)) ~0 for the open-edge of Zedge ~ + ∞ with βdif L/2 ~π/2. A quarter-wavelength transmission line is well known as a short-open transformer. The RF and microwave folded dipoles can be understood in the equivalent circuit of Fig. 2(e) and Eq. (5) as the following. The short-terminated stub transforms the impedances from Zedge = 0 to Zstub ~ + ∞. As a result, the folded dipole impedance Zfold becomes ~vi −2 Zeff_dip because of 2 Zstub >> vi −2 Zeff_dip in the parallel connection. It is well known that short-terminated folded dipoles operate efficiently with an impedance step up ratio of vi −2 = 4 comparing with single dipoles. Analogously, the open-terminated folded dipole has the stub impedance of Zstub ~0 and that parallel connection indicates the antenna impedance Zfold ~0. That is why the RF and microwave folded dipoles only operate efficiently when they are short-terminated arm-edges. The optical folded dipole nanoantennas, however, have the antenna length of L ~(βcom /k 0) λ0/2 at the 1st resonance and the stub has the wave number of βdif. The wave number difference between βcom and βdif yields a phase difference of βdif L/2 that is far from π/2, i.e., the stub length is no longer a quarter effective wavelength for βdif. Thus, Zstub has large absolute impedances in the imaginary part, e.g., 153 + j2470 Ω for the open-terminated stub (Fig. 4(b)) and 167-j1780 Ω for the Ag-terminated stub (Fig. 4(c)) at 300 THz. As a result, 2 Zstub >> vi −2 Zeff_dip ( = 59.6 Ω in Table 1) can be assumed, and Zfold is ~vi −2 Zeff_dip for open- and Ag-terminated folded dipole nanoantennas. That is why the optical folded dipole nanoantennas operate efficiently in both arm-edge conditions.
The nanostructures studied in this work have an important advantage from the nanofabrication point of view in that no physical connections are needed at the arm-edges in the case of open-terminated folded dipole nanoantennas in the optical domain. This provides the possibility of having folded dipole nanoantennas as stacked dipoles, e.g., the driven element on the bottom quartz substrate and the coupled parasitic element on the SiO2 film layer. As a result, the degrees of freedom for design of folded dipole nanoantennas can indeed be extended since folded dipoles can be fabricated vertically as well as horizontally with current nanofabrication technologies [2–7,9,15,18,19]. A stacked folded nanoantenna with the liquid inter-layer may suggest the possibility for a design of a sensor for detecting the thickness of the layer and the variation of the thickness of the inter-layer between the dipole elements, by observing variation of the emission spectrum since the resonance peak can be shifted as the inter-dipole-elements are varied (d < ~50 nm). This is a distinct difference from the RF and microwave folded dipole antennas in that physical connections, e.g., via holes, are needed at the arm-edges in vertically implemented folded dipoles.
There is also a distinct difference between the optical domain and the RF and microwave domains in the method of calculation for the equivalent radius of the “effective” single dipole in the mode decomposition analysis. An equivalent radius of 6.5 nm was obtained by comparing the wave numbers between the common mode of the parallel transmission line and the single nanowire transmission line in the optical domain, as shown in Fig. 3, while in the RF and microwave domains, the equivalent radius is expressed as (rd)1/2 with the perfect conductor assumption [25–27]. It should be mentioned that an equivalent radius of (rd)1/2 = 8.66 nm calculated using the dimensions of Fig. 1, which is analogous to the RF calculation, would not provide an agreement between the mode decomposition analysis results and direct simulation results in antenna input impedances and efficiencies in the optical domain, whereas our method of analysis has resulted in a good agreement.
With regard to similarity between the optical nanoantennas and the RF and microwave antennas, a folded structure has an advantage in increasing the antenna input impedances. In [21], the arm length and width were adjusted to achieve an impedance matching for a nanoantenna to the parallel transmission line having a characteristic impedance of an order of 200 Ω. A folded structure would be a promising matching technique, particularly, when optical transmission lines have higher characteristic impedances.
4. Conclusions
Ag folded dipoles have been theoretically investigated using the mode decomposition analysis. The analysis has shown that both open- and Ag-terminated arm-edge conditions work in the optical domain, due to the wave number difference between the common and differential modes. The efficiency was enhanced in both arm-edge conditions maintaining the omnidirectional pattern in the plane normal to the folded dipole axis. Not only the efficiency but also the impedance can be controlled by the folded structure. In the RF and microwave domains, folded dipole antennas have been widely used to enhance the performance of the antenna systems, e.g., the driven elements in Yagi-Uda antennas. When optical folded dipole nanoantennas are considered for such performance enhancement, one should address the question about the arm-edge conditions. This new finding with the physical insights allows us to design plasmonic folded optical dipole nanoantennas properly and effectively in future nanocircuit and nanophotonic systems.
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