Wireless Optical Nanolinks with Yagi-Uda and Dipoles Plasmonic Nanoantennas

In this work, we present a theoretical analysis of wireless optical nanolinks formed by plasmonic nanoantennas, where the antennas considered are Yagi-Uda and cylindrical nanodipoles made of Au. The numerical analysis is performed by the finite element method and linear method of moments, where the transmission power and the near electric field are investigated and optimized for three nanolinks: Yagi-Uda/Yagi-Uda, Yagi-Uda/dipole and dipole/dipole. The results show that all these case can operate with good transmission power at different frequencies by adjusting the impedance matching in the transmitting antennas and the load impedance of the receiving antennas.


Introduction
The electromagnetic scattering of metals in optical frequency region possesses special characteristics. At these frequencies, there are electron oscillations in the metal called plasmons with distinct resonant frequencies, which produce strongly enhanced near fields at the metal surface. This effect can be analyzed using Lorentz-Drude model of the complex dielectric constant. The science of the electromagnetic optical response of metal nanostructures is known as plasmonics or nanoplasmonics [1,2].
One subarea of nanoplasmonics is the field of optical nanoantennas, which are metal nanostructures used to transmit or receive optical fields [3][4][5]. This definition is similar to that of conventional radio frequency (RF) and microwave antennas. The main difference between these two regimes (RF-microwave and optical) is due to physical properties of the metals at optical frequencies where they cannot be considered as perfect conductors because of the plasmonic effects [2]. Comprehensive reviews on optical antennas have been presented in [6][7][8][9][10][11]. In these works, the authors described recent developments in calculation of such antennas, their applications and challenges in their design. In Figure 1, we present some examples of fabricated nanoantennas. Yagi-Uda and dipole. The numerical analysis is performed by the method of moments (MoM) [18] and the finite element method (FEM) through the software COMSOL Multiphysics [19]. In this analysis, the transmission power and the near electric field are investigated for three nanolinks: Yagi-Uda/dipole, Yagi-Uda/ Yagi-Uda and dipole/dipole. This work is organized as follow: Section 1 is the introduction, Section 2 presents the description of nanolinks, Section 3 presents the numerical model used in the analysis, Section 4 contain the numerical results, and Section 5 are the conclusions.

Description of nanolinks
In this work, three models of nanolinks are proposed and analyzed. The first is a nanolink formed by dipole/dipole antennas (Figure 2, without reflector and directors), the second by Yagi-Uda/dipole antennas (Figure 2) and the third by Yagi-Uda/Yagi-Uda antennas (Figure 2, with the receiving antenna equal to the transmitting antenna).
The geometry of the Yagi-Uda/dipole nanolink is presented in Figure 2, where a voltage source V S excites the left nanoantenna, which functions as a transmitter (Yagi-Uda) and the right nanoantenna that acts as a receptor (dipole), connected to load impedance Z C . The nanolink is located in the free space and is formed by cylindrical conductors of gold. The complex permittivity of this material is represented by the Lorentz-Drude model of Au [11]. The Yagi-Uda transmitting nanoantenna is composed of a dipole, a reflector and three directors (Figure 1 left). The dipole of the transmitter, located in the z = 0 plane along the x-axis and centered at the origin, has total length 2h dT + d dT , radius a dT and voltage gap d dT . The reflector has h r length and a r radius. The directors have the same length h d and radius a d . The parameters d hr and d hd are the distances between reflector and directors element to Yagi-Uda antenna, respectively (Figure 1 left). The receiver antenna is a dipole (Figure 1 right), located in the z = 0 plane and displaced at a d TR relative to the dipole axis of the transmitting antenna, with total length 2h dR + d dR , radius a dR , gap length d dR and load Z C connected to its gap.

Numerical model
In this section, we present the numerical methods used in the analysis of the wireless optical nanolinks described in the last section. The methods used here are MoM [18] and FEM [19].

Method of moments
The linear MoM presented here is based in the linear current approximation with an equivalent surface impedance model of the cylindrical conductors, with sinusoidal test and base functions [18]. The method will be explained for the particular example of a single dipole radiating in a free space, composed by plasmonic cylindrical elements made of gold. Figure 3 shows the geometry of the original problem, the equivalent MoM and circuit models of the nanodipole. In this figure, L is the length of the arms, d is the nanodipole gap and a is the dipole radius. The total length of this antenna is L t = 2 L + d. In present analysis, we do not take into account the capacitance generated by the air gap (C gap ) of the nanodipole. In this case, our input impedance is equivalent to that Z a presented in [20] without C gap .
The losses in metal are described by the surface impedance Z s . This impedance can be obtained approximately by considering cylindrical waveguide with the mode TM 01 . In this case, the surface impedance is given by [22]. being . The boundary condition for the electric field at the conductor surface is where a l is a unitary vector tangential to the surface of the metal, E s is the scattered electric field due to the induced linear current I on the conductor, E i the incident electric field from the voltage source (Figure 3), and I is the longitudinal current in a given point of the nanodipole.
The integral equation for the scattered field along the length l of the nanodipole is given by where g R ð Þ ¼ e Àjk 0 R =4πR is the free space Green's function, and R ¼ |r À r 0 | is the distance between source and observation points. The numerical solution of the problem formulated by (1)-(3) is performed by linear MoM as follows. Firstly, we divide the total length L t = 2 L + d in N = 2N a + 2 straight segments, where N a is the number of segments in L À 0.5d with the size ΔL = (L À 0.5d)/N a (white segments in Figure 3), and two segments in the middle with the size ΔL = d (gray segments in Figure 3). Later, the current in each segment is approximated by sinusoidal basis functions. The expansion constants I n are shown in Figure 3 where each constant defines one triangular sinusoidal current. To calculate these constants, we use N À 1 rectangular pulse test functions with unitary amplitude and perform the conventional testing procedure. As a result, the following linear system of equations is obtained: where Z mn is the mutual impedance between sinusoidal current elements m and n, Δm = 1/2[ΔL m + ΔL m + 1], and V m is non-zero only in the middle of the antenna (m = N/2), where V N /2 = V s . The solution of (4) gives the current along the dipole and the input current I s . For V s = 1 V, the input impedance is Z in = 1/ I s = (R r + R L ) + jX in = R in + jX in , where R r , R L , R in , and X in are the radiation resistance, loss resistance, input resistance, and input reactance, respectively. The total input power is calculated by P in = 0.5Re(V s I s *) = 0.5(R L + R r )|I s | 2 = 0.5R in | I s | 2 = P r + P L , P r is the radiated power, and P L the loss power dissipated at the antenna's surface which is calculated by The radiated power can be obtained by P r = P in À P L , and the resistances by R r = 2P r /|I s | 2 and R L = 2P L /|I s | 2 . The radiation efficiency is defined by e r = P r / P in = P r /(P r + P L ) = R r /(R r + R L ) = R r /R in .

Finite element method
The nanolinks of Figure 3 were also analyzed numerically by FEM. Figure 5a shows the mesh of the nanolink of Figure 2 modeled in the COMSOL, where the antennas are in a spherical domain of air, with scattering absorbing condition (PLM) applied at their ends. Figure 5b shows an enlarged image of the Yagi-Uda nanoantenna mesh and its surroundings.

Isolated antennas in transmitting mode
In this section, the transmitting antennas Yagi-Uda and dipole (Figure 2 without reflector and directors) are analyzed separately. For this analysis, the values of the antennas parameters are those shown in Table 1, where with these values the main resonances are in the frequency range of 100-400 THz considered. The parameters of the isolated dipole are based on [23] and those of the elements of the Yagi-Uda antenna were chosen so that the reflecting element was larger than the dipole and the smaller dipole directors. In Table 1, a = a r = a dT = a d = a dR and d = d hr = d hd . Figure 6 shows the input impedance (Z in ) for the Yagi-Uda antennas (Figure 6a) and dipole (Figure 6b). These input impedances were calculated by FEM with the COMSOL software, and by a linear MoM applied to cylindrical plasmonic nanoantennas [18], by coding the mathematical model in Matlab software [24]. We observe a good agreement between these two methods. Also, the input impedances of the antennas are calculated for two situations, the first with the antennas in the free space (without substrate) and the second on a SiO 2 substrate with a permittivity of 2.15.
Comparing the input impedance result between the Yagi-Uda and free-space dipole antennas, it is noted that the first two resonant frequencies are close, which shows that the directors and reflectors do not significantly affect the original resonant frequencies of the isolated dipole. The main differences between these two transmitting antennas are observed near the frequencies of 175 and 260 THz, which correspond physically to the dipole resonances of the reflector and directors, respectively. These resonances can be observed in the distributions of currents in these frequencies, which are not shown here. Figure 6 also shows the effect of the substrate in the input impedance and resonant properties of the antennas. It is observed that by placing the antennas on the substrate, their resonances are shifted to smaller frequencies in relation to the antennas in the free space. This effect of the substrate is similar to that observed in antennas in the microwave regime [25]. Figure 7 shows the results of directivity (D) and gain (G), radiation efficiency (e r ) and reflection coefficient (Γ), all in dB, versus frequency for the Yagi-Uda antennas (Figure 7a) and dipole (Figure 7b). The directivity and gain are calculated in the + y direction (Figure 2). In Figure 7b it is observed the conventional characteristic of the isolated dipole, where in a wide range of 150-300 THz one has approximately D ≈ 1.6, e r ≈ 0.6 e G ≈ 1 (G = e r D). In the case of Yagi-Uda (Figure 7a), there is a peak of D = 12, near F = 264 THz, but at this frequency the radiation efficiency is minimal e r ≈ 0.1, and the gain is (G = 0.86). However, if greater gain and efficiency are desired rather than high directivity, the frequency near F ≈ 240 THz is more adequate, where the maximum gain is approximately G max ≈ 1.6. The reflection coefficient of both antennas was calculated considering a transmission line with characteristic impedance of 50 Ω connected to antennas. With this result, it is observed that the best impedance matching for both antennas occurs around the first resonant frequency, however the maximum radiation efficiency occurs at higher frequencies. Figure 8 shows the 3D far field gain radiation diagrams of the Yagi-Uda and dipole antennas, calculated at the frequency of 240 THz. It is observed that the maximum gain of the Yagi-Uda (G max ≈ 1.6) is approximately 60% greater than the maximum gain (G max ≈ 1) of the dipole. For the case of the Yagi-Uda antenna, the maximum gain occurs in the +y direction with a small lobe in the Ày direction.

Nanolinks analysis
In this section, we present the results obtained in the analysis of the dipole/ dipole nanolinks (Figure 2, without reflector and transmitter directors), Yagi-Uda/ dipole (Figure 2) and Yagi-Uda/Yagi-Uda (Figure 2, with the receiving antenna equal to the transmitting antenna) for the frequency range of 100-400 THz. The parameters used for the receiving antennas are same as those of the transmitting antennas, with Z C = 50 and 1250 Ω for each nanolink model. Figure 9 shows the power transmission in dB (or power transfer function) for the three nanolinks, calculated by the ratio between the power delivered to the Z C load and the power delivered by the source V s at the transmitting antenna terminals. The results show that the Yagi-Uda/Yagi-Uda nanolink presents a small improvement in power transmission, at some frequency points, in relation to the dipole/dipole and Yagi-Uda/dipole nanolinks. In addition, it can be observed that the links can operate with good transmission power at the frequency points 170 and 240 THz, for Z C equal to 50 and 1250 Ω, respectively, where the power transmission are maximum. Figure 10 shows the magnitude and phase of the electric near field, which is defined by E = 20 log 10 (|Re (E x )|), in the plane z = 25 nm, of the dipole/dipole nanolinks (a, b), Yagi-Uda/dipole (c, d) and Yagi-Uda/Yagi-Uda (e, f). The receiver antennas are positioned at a distance d TR = 5 μm from the transmitting antennas, with F = 170 THz and Z C = 50 Ω for the fields of figures (a), (c) and (e), and with F = 240 THz and Z C = 1250 Ω for the cases of figures (b), (d) and (f). In all types of nanolinks shown in this figure, the radiated wave can be visualized by propagating from the transmitting antennas to the receiving antennas, with the appropriated wavelength. It is observed the amplitude decay of the electric field with the distance and the decrease of the wavelength with the increase of the frequency.

Conclusions
It was presented in this work, a comparative analysis of nanolinks formed by Yagi-Uda and dipole plasmonic nanoantennas, where was investigated the power transmission for Yagi-Uda/Yagi-Uda, Yagi-Uda/dipole and dipole/dipole nanolinks types. These nanolinks were numerically analyzed by method of moments and finite element method. The results show that the Yagi-Uda/Yagi-Uda nanolink presents a small improvement in power transmission, at some frequency points, in relation to the other cases. In addition, the three links can operate with good power transmit at different frequency points, varying the load impedance of the receiving antenna, which is of great importance for future applications in nanoscale wireless communication. In future work, we intend to feed these nanolinks by more realistic sources, such as gaussian beams, to verify their influence on power transmission results.