Results of parametric analysis of the composed nanoantenna.
An impedance matching analysis of two plasmonic nanocircuits connected to cylindrical nanoantennas is presented. In the first case, a bifilar optical transmission line (OTL) with finite length is connected between two nanodipoles, where one is illuminated by an optically focused Gaussian beam (receiving dipole) and the other radiates energy received from the OTL (emitting dipole). In the second case, the OTL is fed by a voltage source on one side and connected to a dipole‐loop composed antenna on the other side. These circuits are analysed electromagnetically by the linear method of moments (MoM) with equivalent surface impedance of conductors. Some results are compared using the finite element method. The results show the impedance matching characteristics of the circuits as a function of their geometries and the broadband response of the second circuit due the broadband dipole‐loop antenna.
- plasmonic circuits
- cylindrical nanoantennas
- impedance matching
- broadband nanoantennas
- method of moments (MoM)
Nanophotonics is the study of optical systems in the nanometre scale . A sub‐area of nanophotonics is the nanoplasmonics, which analyses the interaction of optical fields with metal nanostructures . With the development of nanoplasmonics, the concept of nanoantennas or optical antennas has emerged naturally as metal nanostructures that receive, transmit, localize and enhances optical fields [3–6]. This definition is similar to conventional RF‐microwave antennas, but the difference between these two regimes is that nanoantennas are not perfect conductors and we need to consider the finite conductivity of the metal. This characteristic leads to nanoantenna’s size smaller than the wavelength of the incident wave. In other words, nanoantennas are electrically small resonant structures that can manipulate optical fields in small regions beyond the diffraction limit of light. This property of wavelength scaling for optical antennas is discussed in [7–9].
The research studies in nanoantenna field have increased mainly due to the development in modern nanofabrication techniques, such as the colloidal lithography, that is a bottom‐up process, and the top‐down processes like focused‐ion beam (FIB) and electron‐beam lithography (EBL) . Some review papers on nanoantennas about theory, modelling, fabrication process and applications have been published [11–16].
The development in nanoantenna theory has also been increased due the important applications in different fields [17–28]. For example, the ability of metal nanoparticles to confine and enhance optical fields in nanometre regions is used in high‐resolution microscopy, where fluorescence emission from a single molecule can be strongly enhanced [17–22]. Also, this radiation of a single emitter can be highly directed by nanoantenna arrays . Other important applications are in nanobioimaging to analyse biological process , plasmonic photovoltaic cells , treatment of cancer in medicine , use of wireless at a nanoscale , plasmonic laser and optical data storage , and sub‐wavelength integrated optical circuit . In this work, we focus on the last application.
Examples of nanoantennas connected to plasmonic waveguide are presented in [29–31]. In this work, we make an alternative analysis and extend the results by using a different optical antenna with broadband characteristics. In particular, we consider two plasmonic nanocircuits connected to cylindrical nanoantennas. In the first circuit, a finite optical transmission line (OTL) is connected between two nanodipoles, where they are referred as receiving and emitting dipoles. In this circuit, the first dipole receives the energy from an optically focused Gaussian beam and delivers it to OTL, which connects to the second dipole to radiation. In the second circuit, the OTL is fed by a voltage source on one side and connected to a dipole‐loop composed antenna on the other side. The analyses of these circuits are made by the linear method of moments (MoM)  with equivalent surface impedance of conductors . We compare some results with the finite element method (FEM) . The results show the impedance matching characteristics of the circuits as a function of their geometries, and the frequency response of the second circuit connected to the broadband dipole‐loop antenna.
2. Theoretical model
In this section we present the geometries of the two nanocircuits, the linear method of moments (MoM) model used in the theoretical analysis, the Lorentz‐Drude permittivity model and the equivalent surface impedance of the conductors, and the Gaussian beam used to feed the one circuit.
2.1. Nanocircuit geometries
The geometries of the analysed nanocircuits are presented in Figures 1 and 2. In the first case as given in Figure 1, the geometry is composed by an OTL, of length
In the second case as given in Figure 2, the circuit is composed by an OTL and one dipole‐loop combined antenna. This circuit is fed on the left side by a voltage source of width
In both circuits, the distance between the conductor’s axis of OTL is
2.2. Method of moment model
There are different formulations of the linear approximation of MoM in the literature [32, 35–37]. Here we use the model given in Ref. , where the linear currents are expanded with sinusoidal basis functions. A brief description of this method is presented below, where we consider the scattering problem of Figure 1 to explain the method.
In the scattering problem of Figure 1, the background medium is free‐space and the conductors are in gold. The electrical permittivity of the gold conductors are represented by the Lorentz‐Drude model
where and ,
The integral equation of the scattering problem is obtained by the boundary condition of tangential electric field at the surface’s conductors , where is a unitary vector tangential to the surface of the metal, is the scattered electric field due to the induced linear current
where is the free‐space Green’s function, and is the distance between source and observation points.
The numerical solution of the problem formulated by the boundary condition and Eqs. (1)–(3) is performed by linear MoM as follows. First, we discretize the linear circuit as shown in the right side of Figure 1, where
2.3. Gaussian beam source
In the case of nanocircuit 1 (Figure 1), the incident field is a Gaussian beam. This kind of wave is obtained by solving the scalar Helmholtz wave equation with the paraxial approximation . The magnetic vector potential of a Gaussian beam polarized on the
The excitation beam used in Figure 1 is focused on the receiving dipole 1 with polarization along the dipole axis (
3. Analysis of first nanocircuit
In this section, we analyse the first nanocircuit of Figure 1. In this case, we fix the Gaussian beam source given in Figure 3 at
3.1. Numerical example
Based on the theoretical model presented above, we developed a MoM code in Matlab to analyse the nanocircuit shown in Figure 1. In this sub‐section, we present an example of simulation of the nanocircuit shown in Figure 1 fed by the Gaussian beam depicted in Figure 3. Figure 4 shows the geometry and discretization parameters used in this simulation and the result of the current distribution along the circuit. The near‐field distribution for this example is given in Figure 5. Note that we use the total length of dipole as
We observe in these results the stationary behaviour in the OTL, which is due to the mismatching in the impedances of nanodipole 2 and OTL. To make a quantitative measure of the impedance matching degree, we calculate approximately the voltage stationary wave ratio (
3.2. Impedance matching analysis
This section presents a parametric analysis of the impedance matching of the nanocircuit for different values of
Analysing these curves we come to some conclusions. We note that the nanocircuits possess in general a smaller degree of input impedance matching (higher |ΓV|) when the gap of the OTL
We also observe that in general the impedance matching is better when
All these results show that we have many situations of good matching for different values of
One way to choose the best geometric parameters of the circuit is to consider the case with better impedance matching and efficiency simultaneously. The efficiency of the circuit depends mainly on the attenuation of the current along the OTL, i.e. depends on the loss constant
To understand better the behaviour of the impedance matching and efficiency characteristic of the results presented in Figure 6, we plot in Figures 7 and 8 the current distributions for different geometric parameters. Figure 7 shows the currents for two cases with same
Figure 8 presents the current distribution for two cases with good impedance matching |ΓV| = 0.26 and 0.36, but with different loss constant of
Another analysis of this circuit was done varying the dimensions of the receiving dipole 1 and fixing the dimensions of dipole 2 . We observed that the dimensions of dipole 1 can modify the impedance matching and efficiency characteristics of circuit. The results show that good impedance matching does not necessarily mean a good efficiency in the receiving dipole, i.e. higher input current amplitude in dipole 1. For example, Figure 9 presents the total electric field distribution at plane
4. Analysis of second nanocircuit
This section presents the analysis of second nanocircuit of Figure 2, where a voltage source fed an OTL on the left side and the other side is connected to a dipole‐loop combined antenna. Due the broadband characteristic of this circuit, the analysis presents the spectral response of the circuit in the range of 100–400 THz for different values of the geometrical parameters (Figure 2). First, we analyse the isolated broadband dipole‐loop antenna, and then we consider the OTL connected to this antenna.
4.1. Analysis of isolated dipole‐loop
In this section, we analyse the isolated dipole‐loop antenna and its spectral response. Note that this case without the OTL is obtained when
The first result obtained is the input impedance (
Figure 11 shows the results of calculation of the radiation efficiency and the reflection coefficient obtained by MoM and Comsol for the isolated nanodipole (left) and dipole‐loop antenna. The reflection coefficient is given by Γ = |(
For the isolated nanodipole (Figure 11, left), the maximum radiation efficiency calculated by the MoM and the Comsol are -1.06 and -1.32 dB, respectively, occurring around the second resonant frequency. However, the best input impedance matching point occurs around the first resonant frequency, using
For the dipole‐loop antenna (Figure 11, right), we can see that insertion of the loop besides modifying the input impedance also changes the reflection coefficient causing an increase of the bandwidth of the nanoantenna to 37.1% by MoM and 35.1% by Comsol. The increased bandwidth occurs because in the compound antenna occur an overlapping of different resonances of loop and dipole, which produces a greater bandwidth. We can also observe that the resonance of the loop (near
Figure 12 shows the 3D far‐field gain radiation pattern for the isolated dipole and the dipole‐loop nanoantenna. The frequencies to which these diagrams were calculated correspond the bandwidth central frequencies (
Table 1 shows a parametric analysis of bandwidth of the composed nanoantenna for
|30 nm||40 nm||50 nm||60 nm||70 nm|
4.2. Analysis of dipole‐loop connected to OTL
In this section, we present the analysis of nanocircuit 2 of Figure 2. First, we present the variation of near‐ and far‐field distribution in function of frequency, then the voltage reflection coefficient versus frequency is presented and finally a parametric analysis is presented.
4.2.1. Near‐ and far‐field results
In this section, we present the variation of near and far‐field distribution for a given example of nanocircuit as a function of frequency.
Figure 13 shows the current distribution along the nanocircuit for the frequencies of 100, 200 and 300 THz, for the following parameters:
Figure 15 shows the 3D far‐field gain radiation diagram for this nanocircuit for
4.2.2. Voltage reflection coefficient
To analyse the impedance matching of the OTL with the antenna, it is necessary to calculate the voltage reflection coefficient as was done previously in Section 3.1. The results of calculations for the nanocircuit presented in Figures 13 and 14 are shown in Figure 16 for |ΓV| as a function of frequency for the nanocircuit with the loop and without it. In the figure, the minimum points |ΓV| are highlighted. They are -7.6 and -13.5 dB, at
For the frequencies corresponding to these minimum voltage reflection coefficients for the case of the nanocircuit with loop shown in Figure 16, the current distribution in Figure 17 and normalized electrical field distribution in the plane
It can be observed in Figures 17 and 18 that with increasing of frequency the attenuation of the current and the electric field in the OTL increases due to conduction losses. It may be noted that for the frequency of 383.4 THz there is a significant drop at the standing wave rate in relation to the frequency of 157.3 THz, which presents a decrease in reflection losses on the line. On the other side, for the frequency 157.3 THz one has a higher level of current in OTL (higher transmission efficiency) than for the frequency 383.4 THz (Figure 17). This means, again, that a better impedance matching does not imply a higher transmission efficiency along the OTL.
Figure 19 shows the 3D far‐field gain radiation pattern for the nanocircuit for
4.2.3. Parametric analysis
Finally, a parametric analysis of the voltage reflection coefficient is shown in Figure 20. For the simulations we fix the following parameters: the distance between the surfaces of the OTL (
Analysing these figures one comes to the following conclusions. In all simulated geometries of the circuit with the loop, there is an improvement in comparison with the circuit without the loop regarding impedance matching at some points as can be seen in the figures. The second conclusion is that, for smaller values of
An impedance matching analysis of two plasmonic nanocircuits with nanoantennas was presented. The first circuit is composed by an OTL connected between two nanodipoles, where one nanodipole is illuminated by a Gaussian beam. In the second circuit, a voltage source fed an OTL that is connected to a dipole‐loop broadband nanoantenna. The linear MoM with finite surface impedance was used for numerical calculations, and some results were compared with FEM.
In the analysis of first circuit, we concluded that good impedance matching and transmission efficiency depends not only on the OTL and the emitting dipole, but also on the receiving dipole. In other words, the electromagnetic behaviour depends on the whole circuit. Also, we verified that good impedance matching does not imply on good transmission efficiency. An example of these opposite situations is presented in Figure 9.
In the second analysis, it was showed that a dipole‐loop combined nanoantenna connected to an OTL can increase the operating bandwidth and improve the degree of impedance matching, when compared to the conventional isolated nanodipole. We obtained a best fractional bandwidth of 42%, for the dipole‐loop nanoantenna, and a minimum voltage reflection coefficient of –25dB for this second nanocircuit.