1. Introduction

A well-established approach towards the theoretical development of optical plasmonic nanoantennas relies on adapting to the nanoscale design methodologies that have been originally applied to radio frequency (RF) and microwave antennas. Therefore, it is not surprising that thus far researchers have investigated dipole [1–8], loop [9], nanorod [10–14], Yagi-Uda [15,16], and bowtie type nanoantennas [7,8]. The design process of an antenna primarily involves the determination of the shape of a radiating element that can achieve the desired response. Moreover, in order to increase the functionality or to meet certain design specifications, the baseline antenna is appropriately modified usually by introducing some type of tuning mechanism. For conventional RF antennas one way to realize these tuning mechanisms is via lumped circuit loads. It has been demonstrated that similar loading schemes can also be implemented in the case of optical nanoantennas provided that the necessary nanocircuit configuration can be devised [17,18]. Note here that the nanoscale optical circuits are not readily available in the conventional form of lumped elements. For this reason, whenever some specific nanocircuit functionality is required the recommended approach is to design a nanodevice that can effectively mimic the desired admittance/impedance response.

In this paper we expand on a proven RF antenna design concept to theoretically investigate the feasibility of devising an optical trapped nanodipole. Conventional trapped dipole antennas are very popular among the radio amateur community and they typically operate in the High-Frequency (HF) range [19]. Trapped dipoles are loaded along their two arms with resonating parallel LC (inductive/capacitive) circuits, usually referred to as “traps”. The appealing feature of a trapped dipole is that by judiciously choosing the resonant frequency of these traps as well as the location where they are attached along the wire, a dual- or even multi-band operation can be enabled. Note that the multi-band operation of an antenna is in general a highly desirable feature because it permits the use of the same antenna to cover different bands, rather than having to employ several different antennas (one for each band). This property becomes particularly important in the case of nanoantennas since, as already mentioned, there are no standardized methodologies available for devising the required tuning mechanism based on nanocircuit elements. So far the most prominent methodology for introducing some nanocircuit functionality into a nanoantenna configuration was proposed in [17] and [20]. The approach requires filling a loading volume defined in between the two arms of an optical nanodipole, which can be used to tune its input impedance and therefore its radiation properties. This type of loading scheme has been experimentally demonstrated in [21,22], and [23]. In the first two studies the loading of the antenna is realized by removing small portions of material from around the center of a nanobar. This local geometrical modification creates an effect equivalent to that of a lumped capacitor loading the impedance of the nanoantenna. In [23] the response of a nanodimer comprised of circular disks is tuned by loading the volume defined between the two disks with different material combinations. Moreover, an alternative type of tuning mechanism has been documented in [24]. In this case the loading volume of a plasmonic nanodipole is filled with photoconductive material; the non-linear switching response of this material allows the two nanodipole arms to either couple or decouple. The two states of the photoconductive material essentially result in a dual-mode operation.

Consequently, the question becomes: what is required in order to replicate the multi-band response of a trapped dipole at the nanoscale? Any attempt to answer this question must address the following design considerations: how can the traps be realized, and what baseline antenna should be chosen? Given the similarities in the radiation properties of an RF wire antenna and a plasmonic nanorod, the latter was chosen as the baseline radiating element. For the realization of the optical traps, cylindrical dielectric blocks loaded with plasmonic core-shell particles were employed. The properties of this loading configuration have been analyzed in [18] in the context of developing tunable nanocircuit loads for optical dipole nanoantennas. It was demonstrated that this type of load can be characterized by an effective permittivity governed by two successive mixing rules. This is of paramount practical importance because it permits the effective material properties of these loads to be customized by simply choosing the volume fraction and the constitution of the mixed material.

As mentioned previously, the role of the optical loads is to replicate the admittance resonance of a parallel LC circuit. But we recall that the admittance of a cylindrical capacitor, or more generally of a radial waveguide, is proportional to the permittivity of the dielectric material it is filled with. Therefore, if the permittivity of this dielectric goes to zero then the corresponding admittance of the cylindrical load ought to also go to zero at the same frequency. Consequently, in what follows it will be demonstrated that the effective material properties of the loads can be engineered so as to exhibit zero real permittivity at a specific frequency. This property creates an effective admittance resonance and permits replication of the dual-mode operation of a conventional trapped dipole at the nanoscale. Throughout this paper the $e^{j\omega t}$time convention is adopted.

2. The trapped dipole antenna

In this section the principles of operation of a trapped dipole antenna are briefly presented. Without loss of generality the analysis is laid out with respect to a specific example. Let us first consider two separate wire type antennas, 40 m and 24 m long, respectively. The wire radius is assumed infinitesimal while its material constitution is as a perfect electric conductor (PEC). It is well known that this type of wire antenna exhibits its first admittance resonance, which corresponds to its first and most efficiently radiating mode, when the length is equal to λ/2. For the two antennas considered, this resonant length corresponds to 3.75 MHz and 6.25 MHz, respectively.

The concept behind the operation of a trapped dipole antenna dictates that given the 40 m long wire, its λ/2 resonance as well the λ/2 resonance corresponding to the shorter 24 m wire antenna, can be excited, if we symmetrically load it along its length by a judiciously chosen set of parallel LC (inductance/capacitance) circuits. These LC circuits are usually referred to as “traps” and thus the name “trapped dipole”. These traps are high Q resonators and the values of their lumped elements are chosen such that their resonance occurs around the λ/2 resonance of the shorter wire dipole that is being targeted for excitation. The behavior of the trap is revealed by examining its admittance response, which is represented by:

In order to better illustrate the operation of the trapped dipole under consideration, the structure was modeled using FEKO. The schematic of the 40 m long PEC wire loaded with two parallel LC circuits is shown in Fig. 1(a). The structure is excited at its center. Note that the two traps are symmetrically placed with respect to center of the wire while their relative distance is set equal to 24 m. Based on numerical experimentation, the values of the lumped elements for the two LC circuits were set equal to C = 300 pF and L = 2270 nH, which correspond to a resonant frequency equal to 6.1MHz.

 

Fig. 1 (a) Computational model for the trapped dipole antenna. (b) Input admittance. (c) RCS. (d) Current distribution.

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The structure’s input admittance as computed by FEKO is shown in Fig. 1(b), where it can be clearly seen that apart from the expected λ/2 and 3λ/2 resonances showing up around 3.5 MHz and 11.5 MHz, respectively, there is an additional resonance at approximately 6.0 MHz. This corresponds to the additional λ/2 type resonance introduced in the radiating system due to the resonating parallel LC circuits. In order to further verify the nature of this additional resonance, Fig. 1(c) shows the backscattered radar cross-section (RCS) of the structure. Evidently it exhibits three resonant peaks, which occur at the frequencies where the input admittance is resonant. Finally, Fig. 1(d) plots the current distribution along the wire computed at the frequencies where the RCS exhibits its peaks. Obviously, at 3.55 MHz the current magnitude exhibits the well-known λ/2 (half-sinusoid) profile. As expected, at 6.10 MHz the λ/2 current distribution is confined within the 24 m long wire segment between the two traps. Outside this section the magnitude of the current on the wire is minute. Finally, at 11.50 MHz the current magnitude distribution exhibits the profile of a 3λ/2 resonance. Before proceeding with the remainder of our analysis, it should be emphasized that this additional λ/2 radiating mode at 6.10 MHz has the typical characteristics of a Fano resonance as evidenced by the sharp switching behavior exhibited in both the admittance and the RCS plots of Fig. 1.

3. Optical trapped nanorod antenna

In this section the previous design methodology is adapted to optical frequencies with the ultimate objective being the realization of an optical trapped nanodipole and the demonstration of its dual-mode operation. The two critical design parameters that would determine its success are the choice of the optical analogue of the PEC wire dipole antenna, and the realization of the optical equivalent of a parallel LC resonant circuit. For the former we recall that RF wire type antennas to a certain extent exhibit similar radiation properties with plasmonic nanorod antennas [1,2]. Therefore, the plasmonic nanorod is chosen as the baseline structure for the proposed nanoantenna system. Regarding the realization of an optical parallel LC circuit, it is well known that such nanostructures are neither readily available at the nanoscale nor are there standardized methodologies to devise them, at least in the lumped element sense. For this reason, whenever some specific nanocircuit functionality is desired the recommended approach is to design a nanodevice which is characterized by an equivalent admittance/impedance response. For the nanoantenna system under study, the optical traps should mimic the electric response of a parallel LC resonant circuit. Here it is proposed and demonstrated how the targeted circuit functionality can be enabled using plasmonic core-shell particle loads.

For the design of the proposed antenna it is more convenient to begin by defining first the geometric characteristics of the shorter nanodipole section (analogous to the 24 m long wire discussed in the RF antenna example from the previous section). For this reason, a 135 nm long nanorod is considered. Its radius is set equal to a = 15 nm and its material constitution is defined as silver which is modeled as a Drude dielectric whose permittivity, after fitting experimental data [25], is described by

 

Fig. 2 (a) Computational model for the short nanorod. (b) Input admittance. (c) Extinction and scattering efficiency. (d) Electromagnetic field distribution at resonance.

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Note that in the preceding formula the value of H_φ is probed at the center of the z-directed edge of the excitation voltage gap. The numerically computed admittance of the structure is plotted in Fig. 2(b). The displayed admittance resonance corresponds to the λ/2 radiating mode of the structure. This is also clearly revealed when the nanorod is illuminated by a plane wave polarized parallel to its long dimension. The corresponding extinction and scattering efficiencies are plotted in Fig. 2(c), while Fig. 2(d) provides the electric and magnetic field distributions computed at the resonant frequency. Both of these results were numerically computed using CST-MWS. Note throughout this study we chose to plot both the extinction and the scattering efficiency of the structures under study since the ratio of the two is an indication of how efficiently a resonant mode radiates. Furthermore, these efficiencies were obtained after dividing the extinction and scattering cross section by the nanorod’s geometric cross section. Clearly, the electromagnetic field distribution at resonance exhibits the characteristic profile of a λ/2 type resonance.

At the next step of the design process the optical traps are developed. Provided that the previously studied nanorod resonates at around 405 THz, the design methodology of a trapped dipole dictates that the optical traps should resonate around that frequency as well. Towards this end, and given the cylindrical symmetry of the structure under study, we recall that the admittance of a cylindrical capacitor, or more generally of a radial waveguide, is proportional to the permittivity of the dielectric material it is filled with [26]. Therefore, if the permittivity of this dielectric has a zero crossing around the desired frequency then the corresponding admittance should exhibit a zero crossing at the same frequency as well. The ideal candidate material for this application would be a Drude dielectric with a monotonically increasing real permittivity branch that goes to zero at 405 THz. An alternative approach is the one followed in this study which is based on engineering the desired permittivity properties via an effective medium approach.

It should be emphasized here that the ultimate advantage that mixtures offer, which makes them ideally suited for the problem under study, is that if one of their constituents is a Drude dielectric then the resulting effective permittivity exhibits a Lorentzian or even a multi-Lorentzian response. The real part of this effective permittivity is characterized by multiple monotonically increasing branches and therefore it exhibits multiple zero crossings. More importantly, the frequencies at which the zero crossings occur can be engineered by judiciously choosing the volume fraction and the material constitution of the associated mixture.

Towards this goal the cylindrical capacitor configuration studied in [18] is employed. A 3D view of the capacitor configuration is shown in Fig. 3(a) while its longitudinal cross-section is shown in Fig. 3(b). The radius and height of the cylindrical volume are fixed and set equal to a = 15 nm and h = 25 nm, respectively. Its material constitution is a dielectric with relative permittivity ε_h. It is further assumed that at the center of this cylindrical volume there exists a homogenous dielectric sphere with permittivity ε_f and a fixed radius equal to a₂ = 10 nm. If this system is excited by a z-polarized electric field (i.e. parallel to the cylinder’s axis of symmetry) then it can be characterized by an effective permittivity ε_e given by

 

Fig. 3 (a) Cylindrical capacitor loaded with a core-shell particle. 3D view. (b) Cylindrical capacitor loaded with a homogeneous dielectric sphere. (c) Core-shell geometry that is substituted for the homogeneous dielectric sphere in (b).

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An additional degree of freedom that permits further manipulation of the effective permittivity response can be introduced into the system if the homogeneous dielectric particle is replaced by the core-shell particle shown in Fig. 3(c) Electrically small spherical core-shell particles exhibit internal mixture properties and, as a result, their polarizability is equivalent to that of an effective homogeneous dielectric sphere. The dielectric properties of this effective homogeneous sphere are defined as the effective permittivity of a binary mixture governed by the Claussius-Mossoti mixing rule, where the host permittivity ε₂ and the filler permittivity ε₁ are the shell and the core dielectrics, respectively [27–30]. Furthermore, the volume fraction of this mixture is defined as the ratio of the core volume over the particle volume, or p ≡ (a₁/a₂)³. Subsequently, the effective core-shell permittivity is given by:

For the application under study numerical experimentation revealed that an effective permittivity with a zero real part around 405 THz may be achieved, if the inner radius of the core-shell particle is set equal to a₁ = 8 nm. Note that, according to our formulation, although the length of the inner radius a₁ may vary, the length of the outer radius a₂ is fixed and set equal to 10 nm. Furthermore, for the realization of the targeted trap it is required to set the permittivity of the core material equal to ε₁ = 5, while the shell material is defined as silver with permittivity ε₂ given in Eq. (2). Finally the permittivity of the cylindrical volume host material is defined as ε_f = 5.

Following the geometrical arrangement of the trapped dipole, the aforementioned core-shell loaded cylindrical capacitor system is placed at the two ends of the 135 nm long nanorod. Finally, in order to complete the trapped nanorod antenna design, the structure is extended beyond the ends where the two capacitors have been placed by adding two 42.5 nm long nanord sections. These added sections have a radius equal to 15 nm while their material constitution is defined as silver. They are also tapered on the ends by hemispherical caps with a radius of 15 nm.

We begin the analysis of the system performance by computing first the input admittance of the compound structure. Again due to its rotational symmetry, the nanorod is modeled using the 2D axisymmetric solver of COMSOL. The corresponding CAD schematic is shown in Fig. 4(a). The input admittance of the loaded nanorod is computed using Eq. (3) and the results are plotted in Fig. 4(b). As expected, the structure exhibits two admittance resonances indicating the existence of two distinct radiating modes. Notice that the higher frequency resonance occurs slightly above 400 THz which corresponds to the λ/2 resonance of the shorter 135 nm long nanorod section.

 

Fig. 4 (a) Computational model for the trapped nanodipole antenna. (b) Input admittance. (c) Real part of the complex effective permittivity of the cylindrical load. (d) Imaginary part of the complex effective permittivity of the cylindrical load.

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As mentioned previously, the effective permittivity models given in Eqs. (4) and (5) have been derived assuming that the structures are excited by a constant and uniform z-polarized electric field. However, given the successful demonstration of the dual-mode operation of a nanoantenna, as evident by the results in Fig. 4(b), it is of interest to examine how accurate the aforementioned models are in the case of a less ideal excitation, such as the localized gap excitation depicted in Fig. 4(a). For this reason, during the COMSOL simulation, the effective permittivity along the z-direction of the core-shell loaded dielectric cylindrical volume is estimated by using the following volume averaging formula:

In order to further investigate the nature of these resonances the scattering response of the trapped nanorod was studied when excited by a plane wave polarized parallel to its long dimension. The extinction and scattering efficiencies of the structure as computed by CST-MWS are shown in Fig. 5(a). It can be clearly seen that the loaded nanorod exhibits two distinct scattering peaks occurring at 272 THz and 402 THz, which correspond well to the admittance predictions. Figure 5(b) portrays the electromagnetic field distributions computed at these two frequencies. Obviously, at 272 THz the electric and magnetic field distributions follow the typical profile of a λ/2 resonance that extends across the total length of the nanorod. More importantly, at 402 THz the λ/2 resonance is confined within the 135 nm long nanorod segment. It should be noted that similar to what was demonstrated in the analysis of the conventional RF trapped dipole, in this case the scattering signature of the second λ/2 mode also exhibits the sharp switching characteristics indicative of a Fano type resonance.

 

Fig. 5 (a) Extinction and scattering efficiencies of the trapped nanodipole. (b) Electric and magnetic field distribution calculated at the first two resonance peaks as shown in Fig. 5(a). Top and bottom plots correspond to 272 THz and 402 THz respectively. (c) Extinction and scattering efficiencies of a solid nanodipole. (d) Electric and magnetic field distribution calculated at the resonance frequency shown in Fig. 5(c) occurring at 243 THz. (e) Extinction and scattering efficiencies of the free space loaded nanodipole. (f) Electric and magnetic field distribution calculated at the resonance frequency shown in Fig. 5(e) occurring at 395 THz.

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For comparison purposes we have also included in Fig. 5 the efficiencies and electromagnetic field distributions when the cylindrical load volumes are filled with silver and free space respectively. These correspond to the two most extreme loading scenarios. The former loading scenario corresponds to a solid silver nanorod, while in the latter case the structure is comprised by three separate silver nanorod sections with different lengths, which are electromagnetically coupled. In Fig. 5(c) it can be clearly seen that, as expected, the solid silver structure exhibits a single λ/2 resonance that occurs around 243 THz, which is also verified by the field distribution displayed in Fig. 5(d). The second scattering mode that shows up around 565 THz is simply the 3λ/2 mode of the nanorod. For the second loading scenario the structure is characterized by a single scattering resonance, which is demonstrated in Fig. 5(e). This resonance occurs at 395 THz and it is a red-shifted version of the middle section’s λ/2 mode, originally resonating at 405 THz. This red shift is the direct consequence of the ensuing coupling between the middle nanorod section with the two shorter ones. Note here that the higher order 3λ/2 modes shown in Figs. 5(a) and 5(c), occurring around 550 THz also exhibit the characteristic Fano profile. This is a well-known feature of plasmonic nanorod antennas that has been studied in [12] and [14].

We conclude this study by examining the structure’s scattering response when the total length of the nanorod is varied. Note, however, that the length variations are realized by changing only the length of the two outer nanorod segments. In other words, the length of the middle nanorod section is kept constant and equal to 135 nm. Additionally, the geometrical characteristics and the constitution of the traps are kept the same as well. Figure 6 shows the scattering efficiency, plotted on linear scale, of the trapped nanodipole for six different total length values, ranging from 240 nm to 340 nm, in increments of 20 nm.

 

Fig. 6 Scattering efficiency of the trapped nanorod antenna for different values of its total length. For better clarity the graphs are shifted by 20 units along the vertical axis.

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It can be clearly seen that the scattering response of all the examined structures exhibit the as engineered λ/2 resonant mode of the middle nanorod section. The increase in the total length of the nanorod, in the fashion explained previously, has an impact only on the frequency of the compound structure’s λ/2 resonance. Note also that for the last two lengths examined (320 nm and 340 nm) an additional resonance shows up at 586 THz and 556 THz. This occurs because the lengths of the outer nanorod sections are sufficiently long in order to support a λ/2 resonance within the frequency range of interest.

4. Conclusion

We have theoretically demonstrated the feasibility of devising a dual-mode plasmonic nanorod antenna. The proposed design methodology relies on adapting to optical frequencies the principles of operation of an RF trapped dipole antenna. The key ingredient for the successful realization of such a nanoantenna is the design of an appropriate nanocircuit, or optical trap, that would mimic the response of an electric parallel LC circuit at resonance. It was demonstrated that this can be achieved if we longitudinally load a plasmonic nanorod antenna with plasmonic core-shell particles. By properly selecting the geometry and constitution of the core-shell particles as well as their position along the nanorod, they can function as frequency dependent nanoswitches. As a result, when the two nanoswitches are at resonance they can electrically isolate the middle section of the nanorod. Consequently, a radiating mode may be established that corresponds to the λ/2 resonance of the isolated nanorod section. On the other hand when the two nanoswitches are not resonating then a λ/2 radiating mode may be excited that extends along the total length of the nanorod. This effectively permits the dual-mode operation of the nanorod antenna. The proposed methodology introduces a compact approach towards the realization of dual-mode optical sensors while at the same time it clearly demonstrates the inherent tuning capabilities that core-shell particles can offer.

As a final comment it should be emphasized that the proposed methodology constitutes a very compact approach to realizing dual-mode optical nanoantennas. Its main advantage is that it allows the designer the freedom to tune independently both of the structure’s modes. Additionally, although in this case the optical traps were realized through plasmonic core-shell particles, it is expected that any nanocircuit or optical metamaterial configuration that can provide an effective zero real permittivity at the desired frequency could be a potential candidate for the implementation of the optical traps. Here we have chosen to focus on core-shell particles primarily because the methods for fabricating them are relatively mature and there is a clear path forward to realizing practical multi-modal loaded nanoantennas.