An optical reflectarray nanoantenna: The concept and design

Akram Ahmadi; Shabnam Ghadarghadr; Hossein Mosallaei

doi:10.1364/OE.18.000123

1. Introduction

Reflectarray antennas are a class of antennas that combine the features of reflectors and phased arrays providing a directive beam in a desired scanned angle. The most important advantages of reflect arrays over phased arrays are the elimination of complexity and losses of the feeding network and the higher efficiency [1]. Also they are easier for manufacturing in compared to reflector antennas. To design a reflectarray, the phase of the reflected wave should have a progressive variation over the whole surface such that the total phase delay from the feed to a fixed aperture plane is constant for all the elements. Then, a critical feature in the design of a reflectarray is the choice of elements for obtaining the required phase distribution.

In microwave, various potential reflectarray element designs are considered, which include variable size patches, patches with variable length slot, and patches with fixed slot fed by variable length stripline [1–4]. Introducing radiating elements that can work in optics is of great interest which will be explored in this paper for making a reflectarray nanoantenna. An optical reflectarray nanoantenna can enable successful wireless communication in optics.

Recently, Engheta et al suggested a method of realizing nanoantennas system at optical frequencies by using nanoparticles of concentric structures with cores made of ordinary dielectrics and shell of plasmonic materials [5]. In another study, subwavelength particles at plasmonic scattering resonance were suggested as antenna elements for Yagi-Uda antennas at optical frequencies [6,7]. The scattering resonance of these concentric structures can be tailored at different wavelength range by adjusting the core and shell radii or the material properties [5–8].

In this paper, we illustrate the concept of a reflectarray nanoantenna implemented in optics with the use of array of core-shell dielectric-plasmonic materials, each of them optimized properly to achieve the required phase shift. The concept and radiation performance are investigated. A 3-D finite difference time domain (FDTD) technique is applied to obtain the required reflection phase for a periodic array of a specific nanoparticle design. Then, the obtained result is integrated into the making a 6×6 array of nanoparticles, see Fig. 1, scanning successfully a narrow beam optical radiation. To remove the back radiation, the array is pinned on top of a dielectric-silver layer which removes wave penetration to the other side of the nanoantenna. The radiation pattern demonstrates a successfully designed optical reflectarray performance.

2. Array of core-shell nanoparticles

The general solution for the diffraction problem of a single sphere of arbitrary material with the frame of electrodynamics was first given by Mie in 1908 [9] and then refined by Stratton [10]. Mie applied Maxwell’s equations with appropriate boundary conditions in spherical coordinates using multipole expansions of the incoming electric and magnetic fields. Clusters can be composed of more than one element forming core-shell particles [11].

Fig. 1. Schematic of the system: (a) A concentric dielectric-plasmonic nanoparticle, and (b) Reflectarray nanoantenna structure.

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Engheta et al demonstrated the scattering performance of a sphere particle made of plasma and dielectric materials with the goal of making a Yagi-Uda antenna [6,7]. Here, we integrate the obtained characteristic into the design of a reflectarray antenna for optical frequencies.

The scattering resonant frequency of the core-shell structure depends on material properties and radii of core and shell. Then, by adjusting these parameters one can control the resonant performance. A small particle can be modeled as an induced electric dipole with polarizability α that relates the induced dipole to the incident field as p = αE. The scattered field by this particle is equivalent to the radiated field from the induced dipole. The polarizability can be related to the scattering coefficient of the dipolar term given by Engheta et al in [7,8]:

α = - \frac{6 π ε_{0}}{k_{0}^{3}} i c_{1}^{TM}

where k ₀ is the wavenumber in the surrounding medium and c^TM ₁ is the scattering coefficient of the TM mode of order 1 in the Mie scattering analysis. If one follows the closed form equation which has been obtained for c^TM ₁ in [7,8], it will be obtained that by adjusting the ratio b/a or the permittivities of core and shell, the scattering resonance can be tailored at different wavelength range. In this study, a Drude material is used to describe the frequency dependence of the permittivity of silver which is the shell material, i.e.

ε (ω) = ε_{0} (1 - \frac{ω_{p}^{2}}{ω (ω + i γ_{p})})

where ω_p is the bulk resonant frequency of the material, and damping factor γ_p represents the losses present. (In this study, ω_p = 2π × 2000THz and γ_p = 0.001ω_p)

Figure 2 shows the magnitude and phase of the polarizability α of a concentric nanoshell particle for different core materials and for different b/a ratios (see Fig. 1(a) for the geometry of particle). Operating wavelength is 357.1 nm and ε_shell = (-4.67 + .01i)ε ₀. The outer radius is a = 22.5 nm and in Fig. 2(a), b/a is assumed to be 0.533. Figure 2(b) shows the performance versus b/a when ε_core = 3ε ₀. As seen, due to the shift in resonant frequency, the phase performance changes and these particles can be used for a reflectarray antenna. In this paper we use the same b/a for all the elements and change the core permittivity to control the phase (as this will be easier for our FDTD analysis).

Fig. 2. Magnitude and phase of the polarizability α of a concentric nanoshell particle vs: (a) the permittivity of core when b/a = 0.533 and, (b) the ratio of radii b/a when ε_core = 3ε ₀. Operating wavelength is 357.1 nm and the shell is made of silver [ε_shell = (-4.67 + .01i)ε ₀].

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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. 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 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 [12,13]. Also, it is worth mentioning that the field associated with every nano core-shell (both couplings and reflected field) is directly proportional to the induced 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:

p^{l} = α_{l} (E_{inc}^{total} (r_{l}) + \sum_{q, q \neq l} {\overset{̿}{G}}_{dipole}^{l} (r_{l}, r_{q}) p^{q} + \sum_{q} {\overset{̿}{G}}_{reflected}^{l} (r_{l}, r_{q}) p^{q}),

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 Green’s function of the q^th nano-sphere (from the layered material) computed at the location of the l^th one, i.e.,

{\overset{̿}{G}}_{dipole}^{l} (r_{1}, r_{q}) = [\begin{matrix} (k_{1}^{2} + \frac{\partial^{2}}{\partial^{2} x}) & \frac{\partial^{2}}{\partial x \partial y} & \frac{\partial^{2}}{\partial x \partial z} \\ \frac{\partial^{2}}{\partial x \partial y} & (k_{1}^{2} + \frac{\partial^{2}}{\partial^{2} y}) & \frac{\partial^{2}}{\partial y \partial z} \\ \frac{\partial^{2}}{\partial x \partial z} & \frac{\partial^{2}}{\partial y \partial z} & (k_{1}^{2} + \frac{\partial^{2}}{\partial^{2} z}) \end{matrix}] \frac{e^{i k_{1} ∣ r_{l} - r_{q} ∣}}{4 π ε_{1} ∣ r_{l} - r_{q} ∣},

{\overset{̿}{G}}_{reflected}^{l} (r_{1}, r_{q}) = [\begin{matrix} G e_{rx}^{lx} (r_{l} - r_{q}) & G e_{ry}^{lx} (r_{1} - r_{q}) & G e_{rz}^{lx} (r_{l} - r_{q}) \\ G e_{rx}^{ly} (r_{l} - r_{q}) & G e_{ry}^{ly} (r_{l} - r_{q}) & G e_{rz}^{ly} (r_{l} - r_{q}) \\ G e_{rx}^{lz} (r_{l} - r_{q}) & G e_{ry}^{lz} (r_{l} - r_{q}) & G e_{rz}^{lz} (r_{l} - r_{q}) \end{matrix}] .

where for example, Ge^lz_rx is Green’s function for the z-directed electric field associated with a x-directed dipole [12,13].

By obtaining dipole polarizations from Eq. (3), one can comprehensively characterize the radiation performance of the array of core-shell nanoparticles antenna deposited over a layered structure. This needs the sommerfeld integrals calculations for the array of dipoles (with arbitrary polarization) located on a layered substrate. This has carefully been addressed in [13]. Hence, for each dipole (p_x, p_y, p_z) located at (x ₀, y ₀, z ₀) above a substrate instance, the upper half-space and lower half-space far-field radiation patterns can be represented as [14]:

E = ∣ \begin{matrix} E_{θ} \\ E_{φ} \end{matrix} ∣ = \frac{k_{J}^{2}}{4 π ε_{J}} \frac{e^{i k_{J} r}}{r} ∣ \begin{matrix} (p_{x} \cos φ + p_{y} \sin φ) \cos θ Φ_{J}^{2} - p_{z} \sin θ Φ_{J}^{1} \\ - (p_{x} \sin φ - p_{y} \sin φ) Φ_{J}^{3} \end{matrix} ∣ .

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 in [13,14]. Notice that, each potential is composed of two terms, whereas the first term can be interpreted as the far zone radiation pattern of a dipole, while the second term can be identified as the radiation from a dipole located at the image plane weighted by generalized reflection coefficients.

3. Optical reflectarray nanoantenna

To establish an optical reflectarray nanoantenna, one needs to design an array of nano-radiators where each of them is tailored properly to provide a desired reflection phase, where as a result the array can re-direct and scan the beam in a specific direction. Having an array of nanoradiators allows narrowing the radiation beam. Thus, the reflectarray nanoantenna can successfully scan a directive optical radiation beam. This will be of significant interest for optical far-field engineering.

Assuming the excitation is achieved by a feed located in the far-field of the array antenna, the phase of the incident wave at each particle is proportional to the distance d_l from that particle [see Fig. 1(b)]. Then, the required phase of the reflected field for this element to achieve a reflected beam in a given direction (θ ₀,ϕ ₀) is obtained by [1]

Φ_{l} = k_{0} [d_{l} - \sin θ_{0} (x_{l} \cos ϕ_{0} + y_{l} \sin ϕ_{0})]

where (x_l, y_l) is the coordinate of the center of element l and $k_{0} = ω \sqrt{μ_{0} ε_{0}}$ . The required phase can successfully be achieved by optimizing core-shell dielectric-plasmonic nanoparticles. Basically, one can change the radii of the configuration or its materials parameters to achieve this importance. Here, for the sake of simplicity in FDTD analysis, the material core is considered as the variable for obtaining the required reflection phase.

3.1 Reflection-phase synthesis

In order to characterize the radiating element of an array, the effect of the surrounding elements needs to be taken into account. To achieve this, one can envision a nanoparticle inside the array as a periodic configuration and then apply a full wave numerical analysis to demonstrate the reflection phase from the array configuration [1–3]. A 3-D finite difference time domain (FDTD) [15,16] approach with periodic boundary condition (PBC) is applied to characterize the performance of periodic array of nanoparticles.

Let us first highlight the scattering performance of a core-shell nanosphere. Figure 3 illustrates the FDTD simulation of the particle having a = 22.5nm, b/a = 0.533, silver as the shell material and ε_core = 3ε ₀. In the FDTD model, the computational domain is configured with cubical Yee cells with Δ = 0.75nm and the core-shell structure is illuminated by a plane wave having x-polarized electric field and traveling in the -z direction. The sphere is placed at the center of computational domain (0,0,0) and the scattered field is stored at (0,0,100Δ) in front of the concentric sphere. The scattered field is plotted in Fig. 3. The polarizability α of the same particle based on Mie theory is also shown in Fig. 3. These two graphs can be compared in regard of the position of resonances. A good comparison between the full wave numerical analysis and theoretical model is established. At resonance point (λ ₀ = 357.1nm) the error is less than 4% which can be smaller by choosing a smaller Yee cell.

Fig. 3. Resonance performance of a concentric nanoshell particle, b/a = 0.533, ε_core = 3ε ₀. Close comparison between Mie-theory and FDTD is illustrated.

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The array of nanoparticles is investigated next. The objective is to successfully tailor the reflection phase from a nanoparticle located inside the array to the value of interest. To accomplish this, the FDTD is applied to characterize one unit-cell of the periodic structure (array of particles in free-space) and determine the phase of the reflected field. The unit-cell size is considered to be 150nm × 150nm (in x and y directions) and the same Yee cell is used. The required phase is manipulated by changing the core material of the nanoparticle. Figure 4 shows the reflection amplitude and phase for different core materials (b/a is fixed at 0.533). As observed, the resonance performance depends on the material properties. Careful selection of the operating wavelength gives a better control over the reflected phase swing. If we choose operating wavelength λ ₀ = 357.1nm (f = 840 THz) as the design wavelength, we can realize a relatively large phase swing of about 115°. Figure 5 illustrates the reflection phase variation in terms of change in the material parameter of core. This is a very useful curve that will be used in next section to realize a reflectarray of interest.

Fig. 4. FDTD simulated results for different core materials and construction of phase design curve: (a) Reflection amplitude and, (b) Reflection phase.

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Fig. 5. Phase of reflection coefficient vs the core permittivity at λ ₀ = 357. 1nm.

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3.2 Scanned-beam characteristics

In this paper, two reflectarray designs (including 6 × 6 elements) for beam scanning at 15° and 30° are considered. The geometry of reflectarray is depicted in Fig. 1(b). The center of array is placed at (0,0,0). For the first design the feed is located at (x_f, y_f, z_f) = (-400nm, 0, 2250nm), and for the second design it is at (x_f, y_f, z_f) = (-400nm, 0, 4500nm). The feed excitation is modeled with an infinitesimal electric dipole which is polarized along the x-direction and operates at λ ₀ = 357. 1nm.

In order to ensure field radiation only in one side of the antenna, the array antenna is deposited on a silver layer coated by a dielectric material. The silver layer has a thickness of 35.71nm = 0.1λ ₀ and the same material as the shell (ε_rs = -4.67 + .01i). The dielectric material is made of a thin SiO ₂ film with ε_rd = 2.2 and thickness of 3.571nm. Considering the thin thicknesses of the substrate layers, they do not have much effects on the reflection phase, and they will only help to suppress the back radiation. Hence, one can still use the reflection phase curve demonstrated in Fig. 5 for an array of core-shell nanoparticles located in free-space. This will be validated later by both our theoretical and full-wave numerical models.

Let us first consider the 15° scan angle case (θ ₀ = 15°, ϕ = 0°). From Eq. (6), one can first determine the required reflection phases for the array elements. Then, from Fig. 5 the required material parameters for the nanoparticles cores are evaluated, as given in Table 1(a). The cores materials range from ε_r = 1.2 to ε_r = 3.9. The Mie theory formulations discussed in section 2 is used to determine the equivalent dipole modes for each of the nanoshells as illustrated in Table 2(a). Since the values of p_y s are much smaller than those of p_x and p_z dipoles, they are not shown in this table. The values are normalized to maximum of |p|. The radiation pattern for the array of dipoles elements located above the layered substrate is obtained in Fig. 6(a). This validates a successful 15° beam scanning for the reflected field (25° difference compared to the beam illumination). The half-power beamwidth is 22° which is an improvement of about 4 times compared to dipole excitation itself (which has a 90° beamwidth). Note that, the obtained beamwidth for reflectarray is in good comparison with the performance of a uniform array offering 19° beamwidth [17]. Thus, the 6 × 6 reflectarray nanoparticles antenna successfully scans a narrow beam optical emission. One can reduce the beamwidth even much more by simply increasing the number of array elements [17]. The magnitude (dB) and phase of the x-directed electric field in an x-y plane located at z = 0.5λ ₀ above the plane of nano core-shells are also depicted in Figs. 7.

Table 1. Core relative permittivity of nanoantenna array elements: (a) θ ₀ = 15°, (b) θ ₀ = 30°.

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The similar procedure can be followed to design a 6×6 array for 30° scan angle. Table 1(b) illustrates the required core material parameters where they range from ε_r = 1.2 to ε_r = 4.8. Induced dipole modes for the nanoparticles are summarized in Table 2(b). Figure 6(b) illustrates the radiation characteristic. The radiation pattern shows that the main beam is directed along θ ₀ = 30° whose half-power beamwidth is about 30°. Figures 8 show the distribution of the E_x in an x - y plane, 0.5λ ₀ above the nano particles’ plane.

Table 2. Induced dipoles, p_x s and p_z s: (a) θ ₀ = 15°, (b) θ ₀ = 30°.

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Fig. 6. Radiation pattern in the x-z plane at λ ₀ = 357.1nm : (a) θ ₀ = 15°, (b) θ ₀ = 30°.

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Fig. 7. Near-field ( E_x ) of the reflectarray for 15° beam scanning [Fig. 6(a)] in a plane located at 0.5λ ₀ above the nanoantenna: (a) Magnitude (dB), (b) Phase.

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Fig. 8. Near-field ( E_x ) of the reflectarray for 30° beam scanning [Fig. 6(b)] in a plane located at 0.5λ ₀ above the nanoantenna: (a) Magnitude (dB), (b) Phase.

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A full-wave FDTD numerical analysis is also applied to validate our dipole-modes modeling results, and further explore the effects of finite-size substrate on the radiation characteristics. To accurately model thin silver shells in FDTD, we need to use very small-size Yee cells, hence characterizing the whole 6×6 array would be very huge and time-consuming. Instead, we model the core-shell structures with their equivalent induced dipoles (using Mie analytical model), and then we integrate them with FDTD numerical technique. This is called hybrid FDTD-dipolar mode technique. In the FDTD simulation, each nanoparticle is modeled by its induced dipoles given in Table 2. There will be 6 × 6 sets of dipoles on top of the finite-size layered substrate. Since the p_y is about ten times smaller than p_x and p_z, in our simulation we ignored the p_y s. At operating wavelength, λ ₀ = 357. 1nm, the slab size in transverse plane is 3λ ₀ × 3λ ₀. The hybrid FDTD-dipolar modes is applied and the radiation patterns are demonstrated in Fig. 9. Very good comparisons in compared to the full theoretical model are presented. The side lobes are slightly increased due to the wave diffractions from the substrate edges. The FDTD results validate successfully the concept and radiation performance of the optical reflectarray nanoantennas investigated in this paper.

Fig. 9. FDTD radiation pattern in the x-z plane at λ = 357.1nm : (a) θ ₀ = 15°, (b) θ ₀ = 30°. Good comparisons compared to dipole-modes theoretical results (Fig. 6) are observed.

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Frequency sensitivity of the reflectarray design is also explored in this paper. The radiation patterns for scanning 30° at different frequencies f = 0.9 f ₀, 0.95 f ₀, 1.05 f ₀, 1.1 f ₀ are plotted in Fig. 10. By changing the frequency, both the size and material (for the silver coatings) parameters of the nanoparticles will be changed affecting the radiation pattern and degrading the performance.

Fig. 10. Radiation patterns in the x-z plane at different frequencies for 30° beam scanning: (a) f = 0.9 f ₀ , (b) f = 0.95 f ₀ : (c) f = 1.05 f ₀, and (d) f = 1.1 f ₀, (f ₀ = 840THz is the design frequency).

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4. Conclusions

This paper presents the concept of reflectarray nanoantenna implementation in optics for the first time, with the use of array of core-shell nanoparticles. Optimized geometry-material plasmonic nanoparticles determine successfully the required reflection phases for desired far-field manipulation. Efficient dipole-modes theoretical model and FDTD full-wave numerical method are applied to demonstrate the physics of array of nanoantennas and fully characterize the radiation characteristics. Successful narrow-beamwidth directive emission is demonstrated. The radiation pattern results illustrate that the reflectarray nanoantenna is able to very effectively shape the beam and scan desired directions. Increasing the number of array elements and optimizing the particles configurations will lead to an entirely new paradigm for efficient wireless communication in optics.

Acknowledgments

This work was supported in part by the U.S. Air Force Office of Scientific Research (AFOSR) and National Science Foundation (NSF). We would like to thank Prof. Nader Engheta at University of Pennsylvania for his great discussions and suggestions during this work.

References and links

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(a) θ = 15°							(b) θ = 30°
	m = 1	m = 2	m = 3	m = 4	m = 5	m = 6	m = 1	m = 2	m = 3	m = 4	m = 5	m = 6
n = 1	1.2	2.25	2.7	2.8	2.84	2.8	1.2	2.1	2.7	2.88	3.47	3.91
n = 2	1.8	2.67	2.87	3.11	3.3	3.04	1.6	2.53	2.77	3.04	3.76	4.62
n = 3	2.09	2.76	3.0	2.59	3.69	3.51	1.7	2.59	2.81	2.27	3.92	4.77
n = 4	2.09	2.76	3.0	2.59	3.69	3.51	1.7	2.59	2.81	2.27	3.92	4.77
n = 5	1.8	2.67	2.87	3.11	3.3	3.04	1.6	2.53	2.77	3.04	3.76	4.62
n = 6	1.2	2.25	2.7	2.8	2.84	2.8	1.2	2.1	2.7	2.88	3.47	3.91

(a) θ = 15°
	m = 1	m = 2	m = 3	m = 4	m = 5	m = 6
n = 1	0.57e ^-i0.99π	0.84e ^-i0.67π	0.63e ^-i0.41π	0.68e ^-i0.31π	0.67e ^-i0.01π	0.56e ^i0.18π
n = 1	0.30e ^-i0.3π	0.25e ^i0.06π	0.27e ^i0.45π	0.20e ^i0.8π	0.20e ^-i0.88π	0.07e ^-i0.65π
n = 2	0.79e ^-i0.97π	0.66e ^-i0.74π	0.55e ^-i0.57π	0.56e ^-i0.48π	0.5e ^-i0.07π	0.5e ^i0.07π
n = 2	0.39e ^-i0.19π	0.40e ^i0.14π	0.39e ^i0.45π	0.30e ^i0.87π	0.27e ^-i0.87π	0.08e ^-i0.59π
n = 3	0.76e ^-i0.95π	0.61e ^-i0.78π	0.56e ^-i0.63π	0.94e ^-i0.61π	0.33e ^i0.01π	0.28e ^i0.15π
n = 3	0.26e ^-i0.04π	0.39e ^i0.29π	0.32e ^i0.5π	0.33e ^i0.97π	0.18e ^-i0.92π	0.03e ^i0.16π
n = 4	0.76e ^-i0.95π	0.61e ^-i0.78π	0.56e ^-i0.63π	0.94e ^-i0.61π	0.33e ^i0.01π	0.28e ^i0.15π
n = 4	0.26e ^-i0.04π	0.39e ^i0.29π	0.32e ^i0.5π	0.33e ^i0.97π	0.18e ^-i0.92π	0.03e ^i0.16π
n = 5	0.79e ^-i0.97π	0.66e ^-i0.74π	0.55e ^-i0.57π	0.56e ^-i0.48π	0.5e ^-i0.07π	0.5e ^i0.07π
n = 5	0.39e ^-i0.19π	0.40e ^i0.14π	0.39e ^i0.45π	0.30e ^i0.87π	0.27e ^-i0.99π	0.08e ^-i0.59π
n = 6	0.57e ^-i0.99π	0.84e ^-i0.67π	0.63e ^-i0.41π	0.68e ^-i0.31π	0.67e ^-i0.01π	0.56e ^i0.18π
n = 6	0.30e ^-i0.3π	0.25e ^i0.06π	0.27e ^i0.45π	0.20e ^i0.8π	0.20e ^-i0.88π	0.77e ^-i0.65π
(b) θ = 30°
	m = 1	m = 2	m = 3	m = 4	m = 5	m = 6
n = 1	0.74e ^i0.86π	0.82e ^-i0.61π	0.74e ^-i0.16π	0.41e ^i0.17π	0.55e ^i0.55π	0.24e ^-i0.93π
n = 1	0.58e ^-i0.4π	0.39e ^-i0.05π	0.33e ^i0.62π	0.25e ^-i0.89π	0.12e ^-i0.38π	0.04e ^i0.46π
n = 2	0.77e^i0.9π	0.37e^-i0.55π	0.68e^-i0.24π	0.30e^-i0.01π	0.48e^i0.55π	0.12e^-i0.95π
n = 2	0.88e^-i0.34π	0.57e ^-i0.02π	0.41e ^i0.57π	0.33e ^-i0.99π	0.14e ^-i0.42π	0.02e ^i0.46π
n = 3	0.78e^i0.9π	0.51e^-i0.5π	0.54e^-i0.26π	0.31e^-i0.04π	0.53e^i0.55π	0.10e^-i0.85π
n = 3	0.71e^-i0.34π	0.58e ^i0.02π	0.42e ^i0.54π	0.26e ^-i0.8π	0.12e ^-i0.19π	0.07e ^i0.84π
n = 4	0.78e^i0.9π	0.51e^-i0.5π	0.54e^-i0.26π	0.31e^-i0.04π	0.53e^i0.55π	0.10e^-i0.85π
n = 4	0.71e^-i0.34π	0.58e ^i0.02π	0.42e ^i0.54π	0.26e ^-i0.8π	0.12e ^-i0.19π	0.07e ^i0.84π
n = 5	0.77e^i0.9π	0.37e^-i0.55π	0.68e^-i0.24π	0.30e^-i0.01π	0.48e^i0.55π	0.12e^-i0.95π
n = 5	0.88e^i0.34π	0.57e ^-i0.02π1	0.41e ^i0.57π	0.33e ^-i0.94π	0.14e ^-i0.42π	0.02e ^i0.46π
n = 6	0.74e^i0.86π	0.82e ^-i0.61π	0.74e ^-i0.16π	0.41e ^i0.17π	0.55e ^i0.55π	0.24e ^-i0.93π
n = 6	0.58e^-i0.4π	0.39e ^-i0.05π	0.33e ^i0.62π	0.25e ^-i0.89π	0.12e ^-i0.38π	0.04e ^i0.46π

(a) θ = 15°							(b) θ = 30°
	m = 1	m = 2	m = 3	m = 4	m = 5	m = 6	m = 1	m = 2	m = 3	m = 4	m = 5	m = 6
n = 1	1.2	2.25	2.7	2.8	2.84	2.8	1.2	2.1	2.7	2.88	3.47	3.91
n = 2	1.8	2.67	2.87	3.11	3.3	3.04	1.6	2.53	2.77	3.04	3.76	4.62
n = 3	2.09	2.76	3.0	2.59	3.69	3.51	1.7	2.59	2.81	2.27	3.92	4.77
n = 4	2.09	2.76	3.0	2.59	3.69	3.51	1.7	2.59	2.81	2.27	3.92	4.77
n = 5	1.8	2.67	2.87	3.11	3.3	3.04	1.6	2.53	2.77	3.04	3.76	4.62
n = 6	1.2	2.25	2.7	2.8	2.84	2.8	1.2	2.1	2.7	2.88	3.47	3.91

(a) θ = 15°
	m = 1	m = 2	m = 3	m = 4	m = 5	m = 6
n = 1	0.57e ^-i0.99π	0.84e ^-i0.67π	0.63e ^-i0.41π	0.68e ^-i0.31π	0.67e ^-i0.01π	0.56e ^i0.18π
n = 1	0.30e ^-i0.3π	0.25e ^i0.06π	0.27e ^i0.45π	0.20e ^i0.8π	0.20e ^-i0.88π	0.07e ^-i0.65π
n = 2	0.79e ^-i0.97π	0.66e ^-i0.74π	0.55e ^-i0.57π	0.56e ^-i0.48π	0.5e ^-i0.07π	0.5e ^i0.07π
n = 2	0.39e ^-i0.19π	0.40e ^i0.14π	0.39e ^i0.45π	0.30e ^i0.87π	0.27e ^-i0.87π	0.08e ^-i0.59π
n = 3	0.76e ^-i0.95π	0.61e ^-i0.78π	0.56e ^-i0.63π	0.94e ^-i0.61π	0.33e ^i0.01π	0.28e ^i0.15π
n = 3	0.26e ^-i0.04π	0.39e ^i0.29π	0.32e ^i0.5π	0.33e ^i0.97π	0.18e ^-i0.92π	0.03e ^i0.16π
n = 4	0.76e ^-i0.95π	0.61e ^-i0.78π	0.56e ^-i0.63π	0.94e ^-i0.61π	0.33e ^i0.01π	0.28e ^i0.15π
n = 4	0.26e ^-i0.04π	0.39e ^i0.29π	0.32e ^i0.5π	0.33e ^i0.97π	0.18e ^-i0.92π	0.03e ^i0.16π
n = 5	0.79e ^-i0.97π	0.66e ^-i0.74π	0.55e ^-i0.57π	0.56e ^-i0.48π	0.5e ^-i0.07π	0.5e ^i0.07π
n = 5	0.39e ^-i0.19π	0.40e ^i0.14π	0.39e ^i0.45π	0.30e ^i0.87π	0.27e ^-i0.99π	0.08e ^-i0.59π
n = 6	0.57e ^-i0.99π	0.84e ^-i0.67π	0.63e ^-i0.41π	0.68e ^-i0.31π	0.67e ^-i0.01π	0.56e ^i0.18π
n = 6	0.30e ^-i0.3π	0.25e ^i0.06π	0.27e ^i0.45π	0.20e ^i0.8π	0.20e ^-i0.88π	0.77e ^-i0.65π
(b) θ = 30°
	m = 1	m = 2	m = 3	m = 4	m = 5	m = 6
n = 1	0.74e ^i0.86π	0.82e ^-i0.61π	0.74e ^-i0.16π	0.41e ^i0.17π	0.55e ^i0.55π	0.24e ^-i0.93π
n = 1	0.58e ^-i0.4π	0.39e ^-i0.05π	0.33e ^i0.62π	0.25e ^-i0.89π	0.12e ^-i0.38π	0.04e ^i0.46π
n = 2	0.77e^i0.9π	0.37e^-i0.55π	0.68e^-i0.24π	0.30e^-i0.01π	0.48e^i0.55π	0.12e^-i0.95π
n = 2	0.88e^-i0.34π	0.57e ^-i0.02π	0.41e ^i0.57π	0.33e ^-i0.99π	0.14e ^-i0.42π	0.02e ^i0.46π
n = 3	0.78e^i0.9π	0.51e^-i0.5π	0.54e^-i0.26π	0.31e^-i0.04π	0.53e^i0.55π	0.10e^-i0.85π
n = 3	0.71e^-i0.34π	0.58e ^i0.02π	0.42e ^i0.54π	0.26e ^-i0.8π	0.12e ^-i0.19π	0.07e ^i0.84π
n = 4	0.78e^i0.9π	0.51e^-i0.5π	0.54e^-i0.26π	0.31e^-i0.04π	0.53e^i0.55π	0.10e^-i0.85π
n = 4	0.71e^-i0.34π	0.58e ^i0.02π	0.42e ^i0.54π	0.26e ^-i0.8π	0.12e ^-i0.19π	0.07e ^i0.84π
n = 5	0.77e^i0.9π	0.37e^-i0.55π	0.68e^-i0.24π	0.30e^-i0.01π	0.48e^i0.55π	0.12e^-i0.95π
n = 5	0.88e^i0.34π	0.57e ^-i0.02π1	0.41e ^i0.57π	0.33e ^-i0.94π	0.14e ^-i0.42π	0.02e ^i0.46π
n = 6	0.74e^i0.86π	0.82e ^-i0.61π	0.74e ^-i0.16π	0.41e ^i0.17π	0.55e ^i0.55π	0.24e ^-i0.93π
n = 6	0.58e^-i0.4π	0.39e ^-i0.05π	0.33e ^i0.62π	0.25e ^-i0.89π	0.12e ^-i0.38π	0.04e ^i0.46π

An optical reflectarray nanoantenna: The concept and design

Abstract

1. Introduction

2. Array of core-shell nanoparticles

3. Optical reflectarray nanoantenna

3.1 Reflection-phase synthesis

3.2 Scanned-beam characteristics

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (10)

Tables (2)

Equations (7)

Optics Express