1. Introduction

Recently, the field of optical nano-metals and plasmonics has experienced an extensive growth in research activities, owing to the fascinating possibility of confining electromagnetic radiation to subwavelength spatial domains via surface plasmon polaritons (SPPs) [1]–[6]. These surface modes are established by the couplings of electromagnetic fields to conduction electrons at the boundary between a noble metal and a dielectric. This phenomenon offers unique opportunity for nanooptics physics enabling, areas such as plasmonic waveguides, nano-interconnects and plasmonic light sources [1]. Other newly proposed applications include plasmonic THz nano-antennas [7]–[14] and solar-cells energy-efficient guiding [15]–[19]. Patterning plasmonic particles can offer significant advantage for achieving desired optical properties.

One of the interesting applications of the plasmonic particles is to build antenna devices in optical frequencies featuring photonic communication. However, design and fabrication of nanoantennas is the most challenging part mainly due to the optical properties of metal. For instance, at optical frequencies metal is well described by a frequency dispersive negative permittivity material rather than a high-conductivity structure as what is observed in microwave spectrum. As a consequent, one can offer resonant nanoparticles using core-shell dielectric-plasmonic elements enabling nanoantenna radiation [14]. The goal of this article is to theoretically exploit the performance of arrays of plasmonic nanoparticles featuring enhanced radiation characteristics when placed over layered substrates.

At microwave frequencies, in order to achieve a high gain antenna performance, one needs to use the concept of array antennas where different radiators are tailored in unique arrangements to control the fields amplitudes and phases in the far-field spectrum enabling directive emission. This concept can be realized in optics with the use of plasmonic particles acting like small dipole radiators. Li et al. [7] used this concept for providing a working THz Yagi-Uda antenna in free-space. Similar performance is observed in [14]. From practical point of view, however, one needs to deposit the nano radiators on a substrate. In this case, the radiation performance changes drastically as for instance illustrated for an array of plasmonic rods located above a half-space medium through a numerical analysis [11].

It can be of great benefit if one establishes a theoretical formulation for performance analysis of nanoantennas composed of an array of core-shell spherical particles located above layered substrates. This will allow successful study of the complex structure via understanding the physics and obtaining the optimal radiation behavior and beam scanning for the application of interest.

To establish the theory of nanoantennas, array of plasmonic core-shells over a layered material is approximated with electric dipoles where their induced dipoles are calculated through the Green’s function analysis of dipoles over the layered material [20]–[22]. The obtained Sommerfeld integrals are efficiently calculated by defining a robust integral path [20]. We apply our theory-modeling to obtain the radiation performance of a dipole exciting an array of plasmonic nanoantennas located above a planar layered material. The induced dipole on each nano coreshell is the key parameter determining the antenna radiation characteristics. A full-wave finite difference time domain (FDTD) technique [23] is also applied to characterize the performance of array of plasmonic nanoantennas on layered substrate, validating our theoretical model. The behavior of nanoantennas array on finite-size substrate is also explored.

Optimal patterning of plasmonic particles along with engineering their layered substrate can successfully tailor the optical parameters for achieving a directive emission. It is illustrated that a focused-beam can be obtained for a 2D array of plasmonic particles-nanoantenna deposited on a layered materials and arranged in a Yagi-Uda type configuration. We show that by engineering the layered substrate one can control the radiation beam for focusing it in the desired direction, or suppressing the back radiation.

2. Theory and modeling

The configuration of a layered substrate plasmonic particle-antenna is shown in Fig. 1. The geometry consists of subwavelength nano core-shells implanted above an infinitely long (in x and y directions) semiconductor substrate where layer(s) of plasmonic or dielectric materials are coating the semiconductor surfaces. Here, the plasmonic particles are spherical core-shells with dielectric cores (k_c,ε_rc) and plasmonic coatings (k_s,ε_rs). The use of plasmonic material as the coating allows greater control on scattering characteristics of the nano core-shells [7].

At the plasmonic resonance the scattered field from a small spherical core-shell under the influence of an incident field is dominated by dipole terms. Thus, a subwavelength particle at the scattering resonance can be viewed as an induced dipole (p), which is related to the total electric field upon that particle by a polarizability factor (α).

 

Fig. 1. Array of plasmonic nanoantennas located above a layered substrate.

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where k ₀ is the wave number of the surrounding medium and, Γ^(e) ₁ is the first electric scattering coefficient of the plasmonic core-shell [7, 24]:

U ^(e) _n and V ^(e) _n are given by:

where r ₁ and r ₂ are the inner and outer radii respectively. It is well known that, the scattering resonance of a plasmonic concentric core-shell structure can be obtained at desired frequencies by tuning the (r ₁/r ₂) radii ratio [7]. Figures. 2(a) and (b) present the magnitudes and phases of normalized polarizability versus the r ₁/r ₂ ratio for a concentric core-shell with silver coating and dielectric core made of SiO ₂, wherein the total radius r ₂/0.1λ ₀ is taken to be constant and the core’s radius (r ₁) varies. As it is shown, by tuning the r ₁/r ₂ ratio, we can tailor the resonance frequency of the plasmonic nano-particle, for instance, the nano-sphere goes to resonance at f=650 THz (λ₀=461 nm) if r ₁/r ₂=0.71. Another interesting feature in these figures, is that at the resonance the phase of polarizability is close to π/2 (phase of i), which means that at the resonance the induced dipole is almost in phase with the incident field. Notice that, to tune the resonant frequency for a simple metallic nano-particle, one should change the total size of the particle, while in many applications it is required to maintain a desired size. Nevertheless, with the use of core-shell structures, the total size can be remained the same, whereas at the same time the resonant frequency can be tuned by changing the radii ratio.

 

Fig. 2. Magnitude and phase of normalized polarizability of a plasmonic core-shell sphere versus r ₁/r ₂ ratio for different frequencies for a structure with r ₂=0.1λ, ε_rc=2.2+0.01i and silver coating. (a) Polarization magnitude, and (b) Polarization phase, (c) Silver permittivity behavior.

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In order to analyze the frequency dependency of the scattering resonance, it is of value to consider the frequency dispersion of the plasmonic material. The permittivity of the silver in optical regime can be described with a Drude model, in which the numerical parameters are achieved by fitting the available experimental data of the Silver’s permittivity to the Drude model [25] (see also Fig. 2(c)).

In addition, note that silver is the material of choice, due to its small loss in optical frequencies.

Now, let us investigate the case of Fig. 1. If a non-periodic array of plasmonic core-shells are placed over a layered material, with the presence of an incident field, each nano-particle can be viewed as an induced dipole around the scattering resonance of that nano core-shell. From (1) it is clear that, the induced dipole on each nano core-shell is proportional to the total field upon that particle. In this scenario, the total field upon each nano core-shell can be expressed as the summation of three terms. The first part is associated with the total incident field in the absence of the nano core-shells (E ^total_inc), the second part is the electric field due to the couplings between the nano core-shells in the absence of the layered material (G̿^l _dipole(r _l,r _q)p ^q). Since this term represents the couplings between the nano core-shells, for the l^th nano core-shell we consider the fields of all other nano core-shell except itself (excluding the field of the l^th particle). The last term, is associated with the reflected fields from the layered substrate (G̿^l_reflected(r _l,r _q)p ^q). Note that for the computing the last two terms we approximate each nano core-shell with an electric dipole. Hence for the second term we calculate the dipolar couplings and the Green’s function analysis of dipoles over layered material is applied for evaluation of the final part [20]–[22]. Also, it is worth mentioning that the fields associated with every nano core-shells (both couplings and reflected field) is directly proportional to the induce dipole moment, thus the induced dipole moment for each particle is derived by solving the following linear system of equations. For l,q ∊ {1,2; …,N} with N being the total number of particles, we obtain:

where E^total_inc denotes the sum of the incident field and its reflection from the layered material in the absence of the nano-spheres. G̿^l _dipole is the dyadic Green’s function of the q^th nano core-shell evaluated at the position of the l^th particle. G̿^l _reflected is the reflected Greens function of the q^th nano-sphere (from the layered material) computed at the location of the l^th one, i.e.,

By defining r=r _l-r _q, the elements of the matrix in equation (6b) are derived as [20, 21]:

where for example, Ge^lz_rx is Green’s function for the z-directed electric field associated with a x-directed dipole. Let G̃ be the kernel of the integral, for w ∊ {x,y,z} and s ∊ {x,y} we have [20]:

where G^hlz_rw is the Green’s function for the z-directed magnetic field due to a w-directed dipole located above the layered material [20]–[22]. R̃^TM/TE _1,2 is the generalized reflection coefficient (that incorporates subsurface reflections) at the interface between region 1 and region 2 [20]. The generalized reflection coefficient at the interface of region i and i+1 is defined as [20]:

In the above equations, k_i is the wave number in the layer i and:

The integrals in (7) and (8) are calculated by defining a robust numerical path whereby the discretization of the integration steps is chosen to adapt to the integrand’s oscillation [20]. For the verification purposes, the integrals in (7) and (8) are computed in a homogenous medium (ε=ε ₀,µ=µ ₀) where their exact values are given by the response of a dipole in the same homogenous regime. Figure .3 shows the maximum percentage of error between all the computed integrals (nine integrals) in (7) and (8) for each observation point marked as (ρ,z ₀). As it is exhibited, the maximum error is less than 0:1% for different observation points, validating our method of integration.

 

Fig. 3. Maximum percentage of error in obtaining the integrals in (7) and (8) using our theoretical model compared to their exact values for different observation points. ρ and z0 denote the location of the observation point.

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It is worth noticing that, if instead of the array only one nano core-shell is implanted above a layered material then the induced dipole is achieved utilizing the following equation:

where G̿_reflected in this particular case simplifies to a diagonal matrix, meaning that there is no coupling between the components of the induced dipole. It is worth highlighting again, that in this case the induced dipole is only related to the reflected field of that nano-particle from the layered material and the incident field.

As mentioned earlier, regardless of the structure, we assume that each nano core-shell can be modeled as an electrically small dipole with the induced dipole p. Yet, the induced dipole for each nano core-shell is derived by using (5), in which the parameters of the G̿_reflected are determined by the geometry and materials of the layered structure. Moreover, the far-field radiation pattern is also affected by the presence of the substrate. Hence, to understand the behavior of the antenna located above a layered material, let us first analyze the radiation pattern of such antennas. Since each nano core-shell behaves like an electric dipole, the far-field radiation pattern of such a structure depicted in Fig. 1, consists of summation of the radiation patterns of electric dipoles located above a layered material.

The far zone electric field for a dipole deposited over a layered-substrate can be evaluated using the conventional steepest decent contour (SDC) technique with the transformation k_ρ=k sinθ, where the θ is the spherical angle from the z axis. So for each dipole (px,py,pz) located at (x ₀,y ₀,z ₀), the upper half-space and lower half-space far-field radiation pattern can be represented as [26]:

Where the index J ∊ [1;N] is to distinguish between the upper-half (ε ₁,µ ₁) and lower-half (ε_N,µ_N). The potential parameters are defined as [26]:

with n ₁ and n_N being the refractive indices for the upper and lower half-spaces respectively. The T^TM/TE is the generalized TM/TE transmission coefficient for a layered material [20]. And k_Jr and k_Jz are specified as:

also:

Notice that, from (13) a remarkable result is achieved. The first term in the potentials (Φ¹ _J-Φ³ _J) can be interpreted as the far zone radiation pattern of a dipole, where the second term can be identified as the radiation from a dipole located at the image plane (at the distance z0 beneath the d ₁) weighted by generalized reflection coefficients. It is also worth mentioning that, at θ=π/2, k _1z=0 and hence R^TM/TE _1,2 is zero, which leads to R̃^TM/TE _1,2=-1 and consequently, the electric field is zero at θ=π/2 direction. So clearly the radiation pattern of a nanoantenna located above a planar layered material, is much different from that in the absence of layered-substrate. By utilizing the dipole approximation for the nano-particles, in the next section we discuss the required steps for designing a narrow beamwidth antenna.

3. Antenna applications

In practice, it is often required to design an antenna system that will yield the prescribed far-field radiation pattern. For example, a very common request is to design an antenna whose pattern has a narrow beamwidth and small side lobes. At radio frequencies, an approach to achieve this, is based on the concept of Yagi-Uda antennas [27]. The high directivity is achieved by placing a reflector and several directors arranged in proper locations around a resonant feed element. The directors are designed to be more capacitive while the reflector is more inductive, resulting in a narrow radiation beam toward the direction of directors and a minimum toward the reflector. To take advantage of this principle in optical regime, it has been suggested to use plasmonic nano-particles as the antenna element and to put an optical dipole source as the feed element [7]–[14]. In reality the antenna must be placed on a substrate where surface wave couplings-interactions with the plasmonic particles antenna elements can drastically change the radiation performance. This phenomenon is addressed carefully in our work by applying the developed formulations. A composite of indium gallium arsenide (InGaAs)-silver (Ag)-silicon dioxide (SiO2) substrate is engineered to first suppress the surface waves propagation and second tailor the radiation beam in a desired direction enabling an on-chip nanoantenna.

Figure 4(a) illustrates a Yagi-Uda type optical nanoantenna constructed from a reflector core-shell sphere located at r _ref=-0.2λ ₀y and a director located at r _dir=0.3λ ₀y. Both particles have the core made of SiO ₂ (ε_rc=2.2+0.01i) and the shell made of silver (ε_rs=-8.06+0.175i). For the working frequency of f=650 THz, the outer radius of each particle is 0.1λ ₀ whereas the radii ratios for reflector and director are r _1r/r _2r=0.75 and r _1d/r_2d=0.65 respectively. The ratios of radii r ₁/r ₂ are chosen to set the nano spheres operate around their resonances. Further, under this design, the phase of the scattering coefficient for the reflector compared to the incident field is greater than π/2 (inductive), and that for the director is less than π/2 (capacitive). These conditions ensure successful Yagi-Uda antenna realization. Note that, unless otherwise mentioned, the optical dipole source throughout this work is perpendicular to the top surface of the multilayer substrate (p=z). The radiation patterns for the 2-element Yagi-Uda depicted in Fig. 4(a), are obtained using our theoretical model and are demonstrated in Figs. 4(b) and 4(c) in xy and yz planes, respectively. In this scenario, since the optical dipole is polarized in the z direction, the induced dipoles on the nano core-shells are also polarized in the z direction. The polarizabilities for the reflector and the director associated with the radii ratio are α_r=0.059e ^i0.77π ε _oλ30 and α_d=0.052e ^i0.21π ε _o λ ³ ₀ respectively. From the polarizabilities one can notice that the phases of the reflector and the director are shifted from π/2 indicating the inductive and capacitive behavior for the reflector and the director respectively. It is worth noticing that, all the radiation patterns plotted here are the radiated power normalized to the maximum value of the radiated power of the optical dipole source in the absence of the nano core-shells (and the layered substrate).

To validate the accuracy of our dipole-mode model, a full wave numerical analysis based on an in-house developed FDTD method [23] is applied for characterizing the plasmonic core-shell structures and determining the radiation patterns (as shown in Figs. 4(b) and (c) with the red dashed lines). The near-fields based on theory and numerical techniques are also compared in Figs. 5(a) and (b). Good comparisons are observed. It is worth highlighting that, FDTD characterizes the complete core-shell particles while in our theoretical model we approximate them with dipole elements located at the core-shells’ centers. Therefore, the near field characteristics of the antenna obtained by applying our theoretical approach show the field behavior of dipoles. This implies that, the electric field has local maxima at the location of the dipoles (centers of the core-shells). While the FDTD provides the dipole behavior outside the core-shells. Namely one does not observe the local maxima in the center of the core-shells. Nevertheless, considering the small size of particles and the fact that we are interested in the field performance outside the particles, this is a very good approximation as observed from the comparisons.

It must also be mentioned that due to the spherical complexity of the particles and our FDTD cubical meshing, a fine discretization with stair-casing around the boundary is used. The dispersive property of the plasmonic shells adds another difficulty for the FDTD analysis. The FDTD obtained resonant frequencies for the core-shell nanoantenna particles have almost 8% error in compared to the theoretical prediction. Using smaller cell-size can reduce the error to about 5%. Here, we scale properly the FDTD operating frequency to be around the resonance in accordance with our theory implementation for the particles. Notice that, unless otherwise mentioned, the near-field distributions are normalized to the maximum value of E_z in the y_z plane.

 

Fig. 4. (a) A Yagi-Uda type antenna constructed from a dipole source exciting two nano core-shells, operating at frequency f=650 THz. The reflector’s radii ratio is r _1r/r _2r=0.75 and the radii ratio of the director is r _1d/r _2d=0.65. Both core-shells have core made of SiO ₂(ε_rc=2.2+0.01i) and shell made of silver (ε_rs=-8.06+0.175i). The normalized radiated power obtained by theory and FDTD, (b) in xy plane, and (c) in yz plane. A good comparison is observed. Note that FDTD characterizes the actual plasmonic core-shell structure whereas in theoretical model we approximate them with dipole modes.

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Fig. 5. Magnitude of E_z(dB) for the nanoantenna in Fig. 4(a) in xy (z=0) plane and yz (x=0) plane. (a) Theory, (b) FDTD. It is worth highlighting again, that in our theory each nano core-shell is modeled with an electric dipole, where FDTD calculates the field for the actual structure. Outside the plasmonic core-shells one can expect the similar performance.

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The next step is to investigate the performance of the array antenna when it is located above an InGaAs substrate. This offers potential advantage for making on-chip nanoantennas. The 2-particle nanoantenna of the previous example is now located above an InGaAs slab of 0.35λ ₀, see Fig. 6(a) (in our theoretical modeling a refractive index of about n ~3.4 is assumed). The radiation pattern and the electric fields in xy and yz planes are shown in Figs. 6(b) and 7. As highlighted earlier, due to the surface wave propagation along the substrate and its strong interactions with the plasmonic antennas the radiation beams are degraded and the radiated power is split in different directions. Similar observation has been reported in [11, 26]. Far field behaviors reveals the antenna beam has a maximum in the xz plane. Although the source is polarized along the z, but because of the plasmonic particles locations along the y and their slab interactions, there are some y-polarized induced dipoles for the plasmonic particles, cf. Fig. 7(a). This can be seen from (8) where Ge^ly_rz is not zero, and hence the induced dipole have both y and z components. In addition, the near field in the xz plane in Fig. 7(b) shows that the electric field above and below the substrate are comparable verifying the field propagation through the substrate. The FDTD results for the near-fields are also illustrated in Fig. 8 validating our Green’s function base formulations and concluding discussions.

 

Fig. 6. (a) Nanoantenna configuration in Fig. 4(a) located above a slab of InGaAs with the thickness of 0.35λ ₀. (b) The 3D radiation pattern of the radiated power. The substrate considerably degrades the antenna radiation pattern.

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For the sake of completion, we also investigate the behavior of a nanoantenna, when the source feed is in parallel direction with the slab surface, cf. Fig. 9(a). The radiation pattern and the near-field (normalized to the maximum value of E_x) in yz plane are demonstrated in Figs. 9 (b) and (c), respectively. In this scenario, the feed dipole is polarized along x (p=x) and hence the induced dipoles on the nano core-shells are also x-polarized. Fig. 9 (b) exhibits that the surface wave generation has a dominant effect on the radiation performance.

 

Fig. 7. Near field distribution for the antenna in Fig. 6(a) obtained by using the theoretical approach of this paper, (a) The magnitude of E_y(dB) in xy plane, (b) The magnitude of E_z(dB) in yz plane. Field penetration and propagation through the substrate is illustrated.

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The back radiation can be suppressed by coating the silicon surface with a layer of silver material with ε=-8.06+0.175i and thickness of 0.2λ ₀. The two plasmonic radiators are now implanted at the top of the silver, refer to Fig. 10(a). Figures. 10(b) and 11 represent the far field radiation and the near field behavior of the antenna. As it is demonstrated, the back-radiation is stopped and the nanoantenna array radiates towards the up where the side lobe level gets smaller. Also the the near field distribution of the antenna proves that the magnitude of the field below the substrate is considerably smaller (around 50dB) than that in the forward region (upper half-space). That is to say, by adding a plasmonic layer we prevent the wave to transmit on the other side of the substrate.

 

Fig. 8. Near field distribution for the antenna in Fig. 6(a) obtained utilizing FDTD numerical technique, (a) The magnitude of E_y(dB) in xy plane, (b) The magnitude of E_z(dB) in yz plane.

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Fig. 9. (a) A nanoantenna array of a horizontal dipole, p_x, and two nano core-shells located above a InGaAs slab of 0.35λ0 thickness. (b) The 3D radiation pattern. (c) The magnitude of E_x(dB) in yz plane.

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From the above, and considering the strong effect of the layered materials on the plasmonic nanoantenna performance, one may engineer the substrate of an array antenna properly to achieve a desired radiation pattern characteristic. Basically, the substrate and the reflected wave from that can control the dipolar polarization of each plasmonic-particle array elements, shaping the radiation beam. To provide some physical understanding, it is valuable to study the performance of a single optical dipole located at the top surface of a planar layered material, refer to Fig. 12(a). Fig. 12(b) shows the angle of the maximum beam (in the yz plane) for an optical dipole located above an InGaAs slab (of 0.35λ ₀) which its top surface is coated with 0.2λ ₀ of silver. The dashed line exhibits the angle of maximum radiation for the antenna, versus the distance from the layered material (ε _r2=1). As observed, with changing the distance between the dipole and the multi layer substrate the angle of radiation changes from 30± to around 75°. Note that, as we increase the distance between the dipole and the layered material the number of side lobes increases (see also [26]) so the beam of the main lobe changes and a jump in the maximum radiation is exhibited. Adding another layer to this configuration will provide better opportunity for controlling the beam radiation angle. We consider a case where a third layer of SiO ₂ medium is grown over the silver and the dielectric slab (InGaAs). The solid line in Fig. 12(b) demonstrates that the beam angle (for maximum radiation) varies between 30° to around 90°. Evidently, by depositing the third dielectric layer, we have a better control on the angle of radiation.

 

Fig. 10. (a) Configuration of the nanoantenna in Fig. 4(a) where the nano core-shells are located above a InGaAs slab of 0.35λ ₀ which its top surface is coated with 0.2λ ₀ of silver. (b) The 3D radiation pattern. The silver layer suppresses the back radiation.

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Fig. 11. Near field distribution for the antenna in Fig. 10(a). (a) The magnitude of E_y(dB) in xy plane, (b) The magnitude of E_z(dB) in yz plane.

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A three-layers substrate can be engineered to manipulate the plasmonic particles dipolar modes and tilt the beam of a two-particles free-space Yagi-Uda antenna from endfire-type radiation to close broadside performance. The layered-substrate includes an InGaAs slab with thickness of 0.35λ ₀, silver with thickness of 0.2λ ₀ and a coating dielectric with ε_r=2.2+0.01i and thickness of 0.1λ ₀ (refer to Fig. 13(a)). The radiation performance is obtained in Fig. 13(b). It shows the main lobe around 16° with suppressed back radian (due to negative permittivity of the silver layer). This is an interesting result, because we started with a z-directed optical dipole source which has a maximum radiation around θ=90° and we end up with a maximum radiation around 16°, by tailoring the plasmonic particles array and the substrate. The near-field performance is plotted in Fig. 14(a) and (b).

As observed from Fig. 13, the two nano core-shells parasitic radiators array and its engineered substrate provides a narrow beam only in the yz plane. To control the 3D radiation performance in both planes (xz and yz) and achieve a pencil-beam broadside radiation one may change the distances of the nano core-shells and/or place four plasmonic nano core-shells around the dipole excitation. Figure 15(a) shows the configuration for an optimized design where four plasmonic nano core-shells are placed such that the antenna has a maximum radiation around θ=0. In this case by using the data from Fig. 2(a), the radii ratios for the nano particles are chosen to be closer to the 0.71 (which is the radii ratio of the resonance for f=650 THz). Hence the polarizability of the particles are larger and the antenna can have higher power transmission. Figure 15(b) demonstrates the 3D radiation pattern of the radiated power. As expected a pencil-beam around the z-axis is successfully achieved. The four plasmonic nanoantennas manipulate the induced dipole polarizations, and hence control the radiation characteristic to the application of interest.

 

Fig. 12. (a) A vertical dipole located at the top of a planar layered material. The layered substrate includes a slab of InGaAs with thickness of 0.35λ ₀ where its top surface is coated with silver of thickness 0.2λ ₀. A third layer of dielectric ε _r2 is deposited at top of the silver. (b) The angle of maximum radiation in the yz plane vs. the thickness of the dielectric layer. The jump in angle is because the maximum radiation occurs in another direction.

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Fig. 13. (a) Nanoantenna configuration in Fig. 4(a) located above a 3 layered substrate including InGaAs slab of 0.35λ ₀, silver with thickness of 0.2λ ₀ and a 0.1λ ₀ layer of SiO ₂. (b) The 3D radiation pattern of the radiated power. Adding a third layer controls better the radiation pattern.

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From practical points of view, it is challenging to precisely fabricate the core-shell antenna. Specially one might be concerned about the exact thickness of the plasmonic shell that goes beyond one nanometer. Therefore, it is important to investigate the effect of changing the radii ratio. Figure 16 shows the change in the far field behavior, if the thickness of the shell varies. As it is evident the direction of the maximum beam is not so sensitive to the plasmonic shell thickness. Nevertheless the power transmission of the antenna changes.

We need to highlight again that the designed structures presented in this paper are achieved for infinite-size substrates (along the x and y directions), where a finite substrate size antenna can have different performance due to the type of surface wave generation, the fields propagation in and along the substrate, and the edges diffractions. It will be instructive to investigate the performance of an array of four plasmonic core-shells located over a finite-size substrate with dimensions of 1.3λ ₀×1.3λ ₀, as depicted in Fig. 17(a). The FDTD results for the radiation patterns and near-fields are shown in Fig. 17(b) and (c). The finite size substrate has less effect on the radiation pattern of the 2D Yagi-Uda antenna compared to its counterpart studies in Fig.15 (infinite layers). This is because of the small size of the substrate where a surface wave cannot be supported much and instead the waves penetrate through the layers from the edges. Since the infinite case is the limit of the finite one, when the size of the structure approaches to infinity the field at the back can be rather small leading to a strong directional beam above, as the one we obtain from the theoretical computation. Although, in this case the edges diffractions can have some undesirable effects on the radiation pattern increasing the side-lobes. The optimum design for a plasmonic array nanoantenna located above a finite substrate is currently under investigation.

 

Fig. 14. Near field distribution for the antenna in Fig. 13(a), (a) The magnitude of E_y(dB) in xy plane, (b) The magnitude of E_z(dB) in yz plane.

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Fig. 15. (a) 4-particle nanoantenna array with an engineered substrate demonstrating a broadside radiation characteristic. The radii ratio for the particles are r ₁ r/r _2r=0.72 and r _1d/r _2d=0.69, and dref=0.25λ ₀ while d_dir=0.65λ ₀. The layered material includes 0.5λ ₀ of InGaAs, 0.2λ ₀ of silver coating and a dielectric layer (SiO ₂) with the thickness of 0.1λ ₀. (b) The 3D radiation pattern.

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4. Conclusion

In this paper, the radiation performance of an array of plasmonic core-shell nanoantennas located above engineered substrates are explored. We develop a theoretical-modeling framework base on the Green’s function analysis of dipole excitation above layered structures, to successfully characterize the antenna performance. The plasmonic core-shells are approximated with electric dipoles where their induced dipoles are calculated through the Greens function modeling of dipoles over the layered materials. The theoretical approach and developed formulations are verified through applying a full wave numerical analysis utilizing FDTD method. Both the radiation patterns and near-field characteristics are determined.

The effect of the dielectric substrate layers on the radiation performance is highlighted. It is obtained that a dielectric substrate can drastically change the radiation pattern of a plasmonic nanoantenna from what is obtained for that when is located in free space. Integrating a silver layer can reduce the antenna interaction with the dielectric substrate and suppress the back radiation. A composite substrate for an optimized 2D array of four-plasmonic nano core-shell radiators is engineered to tailor the dipolar modes of the nanoantennas and accomplish a pencil beam radiation.

 

Fig. 16. Radiation pattern sensitivity to the change of r _1r for the configuration shown in Fig. 13(a). The radiation performance in yz plane, where r _2r=r _2d=46 nm and r _1d=30 nm.

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Fig. 17. (a) 4-particle nanoantenna array located above a finite-size substrate. In this case the radii ratios are r _1r/r _2r=0.75 and r _1d/r _2d=0.65. (a) The radiation pattern in xy and yz planes, (b) The near-field distribution of E_z(dB) in yz plane. The FDTD technique demonstrates the effect of finite-size substrate on the field penetration through the substrate and the edge diffractions (as this will establish different radiation characteristic compared to what is obtained for the infinite-size substrate model).

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The FDTD is applied to manipulate the performance of an array of plasmonic nanoantennas located above a finite-size substrate. A small size substrate may not have much effect on the radiation performance, where increasing the size can tailor the plasmonic particles polarizations and control the radiation pattern of interest.

The results of this paper can be of great value for creating on-chip nanoantennas with desired high-performance radiation characteristics.