1. Introduction

The development of nanoscale optical devices that operate in the sub-wavelength regime and at high frequencies are some promises of plasmonics. On the other hand, the efficient excitation of plasmonic waveguides from free space, have been a challenge that could be solved by means of optical antennas.

Optical antennas/nanoantennas [1, 2] are devices that have recently gained prominence with potential application in several branches of photonics, such as: optical sensors [3, 4], lasers [5], solid state lighting [6, 7], photovoltaics [8], among others. However, since their inception such antennas have been mostly operated as sensors and their design as free-space coupling elements to plasmonic transmission lines have not been fully addressed [9–14]. Also, nanoantennas properties, such as: directivity, gain, and radiation efficiency [15] can greatly exceed sensor functionalities. Here we propose a novel kind of nanoantenna operating in the optical regime. Through numerical simulations we devise its potential application to enhance the coupling of a free space optical beam to a nanostrip waveguide [16] (receiving mode - Rx), and also to emit power with a directive beam from a nanostrip waveguide (transmitting mode - Tx).

In general, DRs as ideally isolated electromagnetic devices may exhibit infinite resonant modes. Through an appropriate excitation of certain resonant modes, the DRs may be used as resonant cavities or efficient radiators. Thus with the knowledge of its resonant modes, the one can qualitatively predict the antenna behavior and estimate the produced far field, which is fundamental to the antenna design. For example, in the case of the circular cylindrical resonator, the TE_01δ, TM_01δ, and HE_11δ modes have been the most used ones in applications involving radiation and behave qualitatively, respectively as: a short vertical magnetic dipole, a short vertical electric dipole, and as a horizontal magnetic dipole [17]. Here, vertical and horizontal refers to the directions which are parallel and orthogonal, respectively, to the cylinder axis.

In microwaves, DRAs represent an antenna category whose resonant element is made by a dielectric (generally a ceramic with ε_r: 4 - 100) commonly placed above a printed circuit board (PCB) and fed by a coaxial probe or microstrip line [18]. The most simple and used DRAs geometries are cylinders of rectangular, circular, and elliptical cross-sections, and also, hemispherical shapes. Their resonance frequencies depend on the dimensions, geometry, and permittivity/losses of those ceramics. Moreover, the DRAs exhibit interesting characteristics for the design of antennas, such as: reduced dimensions, low losses, low profile, high density of integration, and high radiation efficiency. Furthermore, it has low sensitivity at resonance frequency due to external and near objects. Their resonance frequencies are well defined and tend to produce wideband and low return losses. Two of the most common feeding schemes for DRAs are based on the use of a microstrip transmission line and coaxial one [18]. In a similar way, the nano-scale version of the microstrip line is known as nanostrip waveguide [16], which will be used to feed our NDRA.

In photonics, DRs are commonly associated to disks or ring resonators and are used in the design of several applications in integrated optics. Thus, inspired by the above mentioned advantages of DRAs in the microwave domain, and taking into account the basic elements of photonics (DRs) and plasmonics (nanostrip waveguide) we numerically investigated, by observing the fundamental antenna parameters, the performance of DRAs at optical frequencies.

2. Design

The design took into account a NDRA operating at C-band as central frequency. The proposed geometrical configuration is shown in perspective, top, and lateral views, Figs. 1(a) , 1(b), and 1(c), respectively. Figure 1(a) illustrates the NDRA operating at Tx mode by means of the propagation vector orientation. Figure 1(b), illustrates the magnetic field vectors of the fundamental mode of the nanostrip waveguide and the DR’s HE_11δ mode, showing their compatibility for modal coupling. The metallic regions of the nanostrip are composed by silver (shaded in light gray in Fig. 1(c)); whose dispersive properties were described by the Drude model assuming ε_inf = 5, f_p = 2175THz, and γ = 4.35THz [19, 20]. The substrate (of the antenna) is made by SiO₂ (ε_r = 2.1, the relative permittivity was considered constant for this theoretical study). The geometry of the chosen radiator element has a circular-cylindrical shape and is based on silicon (ε_r = 11.56, the relative permittivity was considered constant for this theoretical study). In Figs. 1(b) and 1(c), the “O” point represents the Cartesian-coordinate-system’s origin.

 

Fig. 1 Views of the NDRA. (a) NDRA perspective view: the propagation vector, $\overrightarrow{k}$, depicts the optical power flowing along the y direction through the nanostrip and being transferred to the DR end from it being orthogonally radiated to free space (assuming the Tx mode). (b) NDRA top view: magnetic field lines showing the coupling compatibility between the fundamental nanostrip mode and the DR’s HE_11δ mode. (c) NDRA lateral view: the layers present in the NDRA feeding geometry and their respective thickness parameters.

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The effective index from the nanostrip was computed by means of a 2-D modal analysis, using the finite-element software package COMSOL Multiphysics, assuming an infinite nanostrip line and wavelength of 1.55 µm. Following that procedure we obtained: h₁ = 0.145µm, h₂ = 0.020µm, h₃ = 0.010µm, and w = 0.340µm; with the effective index given by n_eff = 1.66. The nanostrip’s magnetic field vectors (in white) are depicted as inset in Fig. 2(a) , which shows the effective index versus the nanostrip substrate’s thickness, h₁, varying around the optimum value.

 

Fig. 2 (a) Effective index versus nanostrip substrate’s thickness. Electric field vectors (in white) of the DR’s HE_11δ mode: (b) top and (c) lateral cross-section views.

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The resonance frequency of the DR’s HE_11δ mode in free space, can be estimated from the relation [21],

According to (1), that mode can resonate at 193.5 THz when d and h assume, for example, the values of 0.494µm and 0.285µm, respectively. Starting from these dimensions a 3-D numerical modal analysis was carried out using COMSOL, assuming just the DR in Fig. 1 (without the nanostrip). Figures 2(b) and 2(c) show the DR’s electric field vectors (in white) for top and lateral cross-section views confirming the excitation of the HE_11δ mode near the desired frequency.

Next, the entire antenna (Fig. 1) was analyzed through the time-domain finite integration technique (FIT) from the full-wave software package, CST Microwave Studio; which is widely used for microwave and photonics design either for industry and academy [14, 20, 22, 23]. The substrate and ground plane dimensions assumed on x-y plane have 4.5x4.6µm². The NDRA have been excited by a waveguide from a rectangular port waveguide. The symmetry of the problem was taken into account and only half of the entire geometry was considered in the simulations to reduce the computational cost. To this end, a perfect magnetic conductor (PMC) was positioned at the y-z plane, along the middle of the nanostrip and NDRA.

Starting from DR’s dimensions in free space, the DR’s dimensions were tuned in order to achieve a good impedance matching related to the scattering parameter, S₁₁. Through this procedure the dimensions d = 0.510µm and h = 0.325µm were obtained. The position of the DR above the nanostrip’s termination was also optimized, see parameter s shown in Fig. 1(c). The optimum value obtained was s = 0.120µm. The dielectric layers h₂ and h₃ (shown also in Fig. 1(c)) are, generally, absent in microwave designs of this kind, however, here they are necessary to be taken into account considering the characteristics of the planar process technology.

3. Results and discussion

The near fields generated by the antenna when operating at the HE_11δ are shown in Fig. 3 . The behavior of the magnetic field being transferred from the nanostrip to the DR (as illustrated in Fig. 1(b)) is shown in Figs. 3(a) and 3(b) through the contour plots taken at planes z = h₁/2 = 0.0725µm and z = h₁ + h₂ + h₃ + h/2 = 0.338µm, respectively. Figures 3(a) and 3(b) points out the coupling of the magnetic field coming from the nanostrip’s fundamental mode to the DR’s HE_11δ mode. The Figs. 3(c), 3(d) and 3(e) show the electric-field modulus behavior related to phase variations of 0°, 45° and 90°, respectively.

 

Fig. 3 Cross-section views of electric and magnetic field modulus (in decibel scale). (a) magnetic field between nanostrip and ground plane, and (b) magnetic field modulus at NDRA’s half height. Vertical cross-sections of the NDRA showing the electric field modulus dynamic behavior at 193.5THz related to y-component phase variation of (c) 0°, (d) 45°, and (e) 90°. The dashed line in each inset represents the plane of observation.

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The return loss (S₁₁) curve shown in Fig. 4(a) , exhibits a minimum resonant value at 193.5THz (λ₀ = 1.55µm) with pronounced deep around −32 dB, and a relatively wideband for values smaller than −10dB and −15dB, approximately 20THz and 13.4THz, respectively. The −10dB S₁₁ band covers practically three optical communication bands: L-band (1.565-1.625µm = 0.06µm), C-band (1.530-1.565µm = 35µm), and S-band (1.460-1.530µm = 70µm). Such wide bandwidth may be advantageous for nano-scale fabrication, because it could ensure the design’s robustness against fabrication tolerances. Along the −10dB S₁₁ band the radiation patterns maintain very similar shapes with a very directive principal lobe and small gain variation around 7.5dB. In our simulations we have observed that the HE_11δ mode has been excited between 185THz and 196THz, and the HE_12δ mode between 196THz and 205THz, within the −10dB S₁₁ band.

 

Fig. 4 Fundamental parameters of the antenna depicted in Fig. 1. (a) Return loss, S₁₁, in red solid line and gain curves in dark-blue solid line. (b) 3-D radiation pattern at 193.5THz (1.55µm) with broadside behavior.

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The NDRA proposed here exhibits a broadside radiation - as shown in Fig. 4(b) - i. e., the radiating power flows perpendicular to the feeder waveguide’s plane, or parallel to the z-axis, see Figs. 1(a) and 1(b). However, depending on the excited radiation resonating mode, NDRAs can radiate either end-fire or broadside; this property can be very useful for a wide range of applications such as flexible and effective on-chip [24, 25] or inter-chip [26] wireless interconnections in the optical regime, including infrared and terahertz technologies.

In this study, the proposed antenna was investigated in the transmission (Tx) mode, however, considering the reciprocity theorem [27], all the properties discussed here are equally valid to the reception (Rx) mode. This means that NDRAs can also be used to couple the far-field radiated by a distant optical source into plasmonic circuits or other structures that operate in the subwavelength regime.

4. Conclusion

In summary, we have reported a new category of nanoantenna, the NDRA, which exhibits an interesting and quite impressive performance in terms of bandwidth and realized gain, and can be efficiently used to transmit/couple optical energy from/into plasmonic circuits. The properties of the dielectric materials were assumed as constant and the Drude-model was considered for the metal used (Ag) in this first numerical demonstration. The NDRA shows itself as a very promising device for applications based on planar lightwave circuit technology, such as optical wireless communications and intra/inter chip communications.