1. Introduction

In the past decade there has been an extensive research effort focused on the theoretical and experimental development of plasmonics and nanophotonics [1,2]. In particular, the study and development of nanoantennas has received considerable attention from the nanophotonics community since by properly engineering their radiation properties one can control light at the nanoscale [3]. Among all possible types of nanoantennas, nanorods and nanodipoles have received the most attention due to the fact that their radiation properties closely resemble those of their radio frequency and microwave counterparts, and also due to their ease of fabrication[4–8]. To date, all of the traditional computational electromagnetics techniques have been successfully applied for the modeling of nanoantenna systems. In addition to full wave approaches several studies have been documented dealing with the development of circuit models that would permit a more qualitative interpretation of a nanoantenna’s radiation mechanism. At the same time, from an analysis and design standpoint, these circuit models would provide a reliable solution for the fast simulation and performance prediction of a radiating nanostructure [9–16].

In this paper we further investigate the circuit based approach for the modeling of a nanorod type plasmonic antenna. In particular, the objective is to develop a compact circuit based framework in order to describe the radiation properties of a loaded nanorod antenna. By the term “loaded” we refer to material inhomogeneities occurring along the length of the nanorod. According to our assumptions these inhomogeneities occupy small cylindrical volumes longitudinally placed across the nanorod. Our approach treats the loaded nanorod as a multi-port network where the number of ports is determined by the number of loads. It turns out that network theory, a mainstream methodology for the analysis of microwave devices, can be successfully applied for the nanoantenna geometry under consideration here. Of particular importance for the applicability of the proposed methodology is the definition of appropriate admittance expressions for the loading volumes. For this reason we employ dielectric waveguide theory and derive useful closed-form expressions for the load admittances.

The significance of the proposed methodology is that one can very efficiently compute the input admittance of the nanorod, for any loading combination. The methodology becomes even more appealing when the loading volumes are significantly smaller than the nanorod. In such cases, where only a small part of the geometry under study may change, it becomes extremely costly and inefficient in terms of computational resources to repetitively simulate, via full-wave methods, the entire structure in order to examine the effect of different loading combinations. In addition, knowledge of an antenna’s admittance response allows its scattering signature to be predicted when the structure is excited by an incident plane wave; the admittance resonances correspond to the frequencies where the scattered field exhibits its maximum value. Throughout this paper the $e^{j\omega t}$time convention is adopted.

2. Probing the nanorod and load admittance derivation

In this section we present the loaded nanorod geometry under study, and we further develop the necessary mathematical framework to characterize it in terms of the input admittance of the structure and its loads. For our study we are considering single- and double-loaded nanorods.

The corresponding layouts of these two structures are illustrated in Figs. 1(a) and 1(b). For both structures the length (tip-to-tip) and radius of the nanorod is 300 nm and a = 15 nm, respectively. For the single-loaded nanorod the height of the cylindrical load is equal to h = 25 nm and its center is located at z = 52.5 nm. For the double-loaded nanorod both cylindrical loads have a height of h = 25 nm while their centers are symmetrically positioned at z = 52.5 nm and z = −52.5 nm, respectively. For the modeling of these structures we exploit their symmetry and we utilize the finite element based body-of-revolution solver of COMSOL following the methodology introduced by Locatelli in [16]. The corresponding numerical models are shown in Figs. 1(c) and 1(d). For our experiments the length of the excitation gap was set equal to g = 2 nm. Along the Π-shaped boundary of the excitation gap a z-directed electric field is hard-sourced with a magnitude equal to 1 V/m. Now, in both cases the input admittance of the structure is defined with respect to the voltage and current measured at point A, and it is given by

 

Fig. 1 (a)-(b) Single- and double-loaded nanorod geometry and probing points. The model dimensions are only indicative. (c)-(d) Actual single- and double-loaded nanorod body-of-revolution computational model.

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In a similar manner we can determine an estimate of the loading volume’s input admittance. As a matter of fact, with respect to point B shown in Fig. 1(a), the input admittance of the load can be defined as

In order to better assess the accuracy and applicability of the expression in (2) for the problem under study, in what follows we derive a closed-form expression for the load admittance. For this reason we treat the load volume as a dielectric circular cross-section waveguide with permittivity${\epsilon }_{rL}$. It is further assumed that mode TM₀₁ ($E_{\rho }$,$E_z$,$H_{\phi }$) suffices to describe the electromagnetic field distribution within this waveguide. This is a reasonable assumption to make since upon illumination by a plane wave the radiating modes of a plasmonic nanorod are attributed to electric and magnetic field distributions that closely resemble those of a linear wire antenna. This will be further demonstrated later in our analysis.

The $z$-component of the electric and the $\phi$-component of the magnetic field for this mode such that $\rho \le a$are given by [17]

Now, in order to demonstrate the accuracy of the derived admittance expression we examine two characteristic loading scenarios for the single-loaded nanorod shown in Fig. 1(a). The constitution of the nanorod is assumed to be silver modeled as a Drude dielectric whose permittivity, after fitting experimental data [18], is given by

The load conductance and susceptance computed using (5) and (6) are displayed in Fig. 2. The two responses are denoted as “Dynamic Model’ and “Capacitor Model”, respectively, and we have included them both in order to demonstrate their applicability. These quantities are compared against the numerically predicted values, using the computational model shown in Fig. 1(b), and utilizing the expression in (2). It can be clearly seen how remarkably accurate the dynamic model is for these highly dispersive loading scenarios. It should be mentioned however that the capacitor model does not always fail to provide accurate admittance predictions. As a matter of fact our numerical experimentation revealed that for certain material loadings that do not necessarily result in highly dispersive wavenumbers, the capacitor model captures fairly accurately the admittance response of the load.

 

Fig. 2 Conductance and susceptance comparisons. (a) and (c) ε_rL = 1-j60. (b) and (d) ε_rL = −20-j.

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At this point it should be emphasized that since the length-to-radius ratio of the cylindrical loading volume is greater than one, it would be insufficient to assume a-priori that its admittance can be described by that of a cylindrical capacitor. For this reason it was required to apply a “higher order” capacitor model that would be valid beyond the quasi-static limit. A cursory examination of the available field configurations revealed that the TM₀₁ mode would be the most appropriate candidate where, as our analysis indicates, the capacitor admittance is simply the low frequency approximation of the transverse admittance corresponding to the TM₀₁ mode. From our numerical experimentation the validity of the TM₀₁ mode assumption is subject to three conditions. First, it is dependent upon the radius and length of the cylindrical load volume. In other words, the larger these dimensions are the more modes are required in order to accurately describe the admittance properties. Second, the nanorod permittivity should be as negative as possible in order to ensure that an effect equivalent to that of a radial waveguide or of a parallel plate capacitor is established. Third, the nanorod sections are extended long enough so that a cavity effect is created within the cylindrical load volume.

3. Y_ij-parameters extraction of the loaded nanorod

The calculation of a loaded multi-port network’s input admittance requires knowledge of its admittance matrix. Although for microwave frequencies the extraction of the latter is straightforward, this is not the case with nanodevices operating at optical wavelengths, such as the nanorod geometry under study. The difficulty stems primarily from the fact that it is not trivial to define “ports” across a nanostructure. In this case, a port is defined as a pair of terminals where the voltage difference across them as well as the current flowing through them can be probed. Given an N-port network its Y_ij-parameters may be defined as

 

Fig. 3 (a) Schematic of the single-loaded 2-port network. (b) Schematic of the double-loaded 3-port network.

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It turns out that an effective way to short-circuit the required ports is by imposing perfect electric conducting (PEC) conditions along the three edges that comprise the Π-shaped boundaries of the 3 ports. Essentially, this boundary condition nullifies the corresponding tangential electric field components, and from (14) this yields zero voltage, or a short-circuited port. For the two structures under study, one needs to perform 2 and 3 numerical simulations, respectively, (shorting the necessary ports each time) in order to derive the appropriate set of Y_ij parameters for each structure.

Once the preceding task has been completed and the Y-parameters have been determined, the next step is to define how the loaded nanorod’s input admittance is related to the admittance of the loads. For the derivation of such a relation we recall that if the second port of the 2-port network is terminated with a load of admittance$Y_L$, as shown in Fig. 1(a), then the port currents and port voltages are related as:

4. Numerical examples

4.1 Single loaded nanorod

First we examine the simplest loading scenario for the single-loaded nanorod corresponding to the case where the loading volume is filled with silver (the same material as the nanorod), with permittivity given by the expression in (7). In this case the resulting structure is representative of a homogeneous nanorod. This test case should be indicative of the methodology’s accuracy. This is because the proposed technique postulates that even if the current induced along the surface of the plasmonic nanorod is coarsely segmented, through the introduction of the cylindrical loads, then the electromagnetic response of the antenna can still be accurately recovered through the previously derived admittance matrix.

For this demonstration, the first step in the process is to numerically predict the radiation properties of the nanorod antenna. For this reason the loaded nanorod is illuminated by a plane wave polarized parallel to its long dimension. This simulation was performed using the full 3D finite element solver of CST-MWS and the corresponding extinction Q_ECS and Q_SCS scattering efficiencies are shown in Fig. 4(a). These efficiencies were obtained after dividing the extinction and scattering cross section by the nanorod’s geometric cross section defined as S = 300 × 30 nm². Note here that the ratio between these two efficiencies can be considered as an estimate of how effectively the structure scatters.

 

Fig. 4 (a) Extinction and scattering efficiencies of loaded nanorod when ${\epsilon }_{rL1}={\epsilon }_r^{Ag}$. (b) Input admittance of loaded nanorod. (c) Electric and magnetic field distribution calculated at the first four resonance peaks as shown in 4(a). Top to bottom plots correspond to 241 THz, 565 THz, 708 THz, and 776 THz respectively.

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The first four resonance peaks are observed at 241 THz, 565 THz, 708 THz, and 776 THz, respectively. In order to identify the nature of these resonances in Fig. 4(c) we have included the electric and magnetic field distributions computed at the aforementioned frequencies. From these surface plots it is evident that all four resonances correspond to odd multiples of the half-wavelength resonance. Note here that the wavelength based terminology stems from the resemblance of a nanorod’s resonances with those of a metallic wire type radio-frequency (RF) antenna. However, in contrast to RF wire antennas, the wavelength that these resonances occur is not directly proportional to the nanorod’s length, as described by Novotny in [22].

Now, for the derivation of the structure’s input admittance given that${\epsilon }_{rL1}={\epsilon }_r^{Ag}$, the complex wavenumbers $k_z$and $k_{\rho 1}$ were first computed from (8), and subsequently (5) was used to compute the admittance of the load. In Fig. 4(b) the orange vertical lines indicate the frequencies where the scattering resonances occur. Evidently, the proposed admittance model can predict very well the scattering peaks of the loaded nanorod. It should be emphasized that the line-shape of the scattering peaks corresponding to the higher order resonances, as shown in Fig. 4(a), clearly signifies their Fano character, a feature of high aspect ratio plasmonic objects that has also been described in [23]. The Fano character of these resonances is also reflected in their admittance representation. In particular, from Fig. 4(b) it can be clearly seen how much broader the conductance and susceptance of the first resonance are in comparison to the sharper admittance characteristics of the three higher order Fano-type resonances.

The second loading scenario to be examined corresponds to a dielectric material characterized by a relative permittivity of${\epsilon }_{rL1}=2.4$. According to (8) it turns out that the values of$k_z$lie below the reactive decay region of the structure where the solution of (8) has no meaning [24,25]. Therefore we set$k_{\rho 1}=\sqrt{{\epsilon }_{rL1}}{k}_0$, and substitute this into (5) to obtain the admittance of the load, which is displayed in Fig. 5(b). The structure’s extinction and scattering efficiencies are presented in Fig. 5(a). In this case the first four peaks occur at 319 THz, 511 THz, 569 THz, and 676 THz, respectively. These frequencies are also indicated for comparison purposes by the orange vertical lines in Fig. 5(c). Evidently, the admittance model can predict the loaded nanorod’s scattering response with a high degree of accuracy.

 

Fig. 5 (a) Extinction and scattering efficiencies of loaded nanorod when${\epsilon }_{rL1}=2.4$. (b) Input admittance of loaded nanorod. (c) Electric and magnetic field distribution calculated at the first four resonance peaks as shown in 5(a). Top to bottom plots correspond to 319 THz, 511 THz, 569 THz, and 676 THz respectively.

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The electric and magnetic field distributions computed at the aforementioned frequencies are shown plotted in Fig. 5(c). Note that in this case the structure is comprised by two separate and different in length silver nanorod sections, connected via the dielectric load. This becomes also evident upon examination of the structure’s scattering response: the first and second resonances correspond to the half-wavelength λ_e/2 resonances of the long and the short nanorod sections, respectively. The third resonance is the collective response created by the synchronization of a half-wavelength like resonance excited along the shorter nanorod and by the one-wavelength resonance along the longer nanorod. This combined effect creates an effective 3λ_e/2 resonant behavior. Finally, the forth resonance corresponds to a 3λ_e/2 resonance of the longer nanorod section. Note that the third and fourth resonances are Fano-type, a feature which is again reflected in the structure’s input admittance response.

4.2 Double loaded nanorod

Finally, in this last subsection, we present four indicative loading scenarios that demonstrate the accuracy of the admittance model in the case of a double-loaded nanorod. Similar to the previous subsection, the first case examined is the one where the two cylindrical loads are filled with silver so that the resulting structure corresponds to a homogeneous silver nanorod. For the second case the two loading volumes are filled with gold whose relative permittivity after fitting experimental data [26] is given by:

 

Fig. 6 Left column: extinction and scattering efficiencies. Right column: admittances. (a)-(b) Two silver loads. (c)-(d) Two gold loads. (e)- (f) Two free space loads. (g)-(h) Two dielectric loads with ${\epsilon }_{rL1}={\epsilon }_{rL2}=2.4$.

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5. Conclusion

We have demonstrated that the scattering response of a multi-loaded plasmonic nanorod can be accurately predicted by treating the structure as a multi-port network and deriving its admittance matrix. The key ingredient for the successful realization of this is the definition of proper ports along the nanorod that enable the corresponding voltages and currents to be effectively defined. Our analysis reveals that for the structure under consideration the short-circuit conditions required for the extraction of the Y_ij parameters can be trivially achieved by nullifying the tangential electric field components across the ports. Given the structure’s admittance matrix and any set of loads, one can then compute the input admittance of a loaded nanorod. From the resonances of the structure’s admittance one can predict its scattering response. This is because the scattering peaks of an elongated structure, excited by a plane wave polarized parallel to its long dimension, correspond to (2n + 1)λ_e/2 field configurations and therefore occur at the same frequencies where its input admittance resonates.

The proposed methodology to a certain extent is reminiscent of a simplified reduced order method of moments (MoM) type technique [27]. What differentiates the two approaches is that in the proposed methodology, the size of the admittance matrix is equal to the number of ports squared, as opposed to the conventional MoM technique where the size of corresponding admittance matrix is determined by the structure discretization. Furthermore, in the proposed approach the electromagnetic wave propagator is the finite element solver, and this completely eliminates the necessity to perform current integrations with respect to the problem specific Green’s function, which would be the case for a conventional MoM formulation.

As a final comment, it should be noted that the applicability of the proposed methodology is subject to certain design constraints. First, and in relation to the discussion included in the preceding paragraph, the proposed methodology is not expected to yield accurate results if a very large number of loads is defined across the nanorod. This is simply because the resulting configuration would correspond to a very coarse MoM discretization of the nanorod which would naturally result in erroneous predictions. The technique is expected to provide accurate results only when a small part of the nanorod’s volume is occupied by loading material pockets. Second, the proposed methodology assumes that the admittance of the loading volumes can be characterized by the TM₀₁ mode of a dielectric cylindrical waveguide. This is valid only when the radius of the nanorod, and therefore the loading volume, is small enough while at the same time the length of the loading volume is not significantly longer than its radius. Additionally, the developed admittance model can accurately describe the structure’s resonances when it is illuminated by a normally incident plane wave polarized parallel to its length. When the same structure is illuminated by an obliquely incident plane wave, then the admittance model will predict only the resonances excited by the longitudinal plane wave component. The additional resonances that may be excited due to the transverse component cannot be accurately predicted by the model introduced here.