Approach to analysis of all-dielectric free-form antenna systems

Volodymyr I. Fesenko; Anton S. Kupriianov; Andrey Sayanskiy; Vitalii I. Shcherbinin; Alexander Trubin; Vladimir R. Tuz

doi:10.1364/OE.27.022363

1. Introduction

Resonant metasurfaces are of much current interest because of their ability to provide efficient manipulation of light, either by means of strong field confinement on the surface or through scattered far-field radiation [1, 2]. Typically such metasurfaces are made of periodic arrays of metallic or dielectric particles, although, according to the Babinet’s principle, they can be also realized in the form of metallic plates perforated with slits (open apertures) [3]. As a rule, arrays with subwavelength separation of elements and spatially varying geometric parameters are utilized in the resonant metasurfaces to form a spatially varying optical response. Proper arrangement of the resonators into arrays with desired spatial optical response, makes it possible to achieve unique properties of metasurfaces for various applications, including directional scattering [4], lasing [5, 6], bio-sensing [7], photo detection [8], polarization control [9, 10], light trapping [11, 12], nonlinear frequency conversion [13, 14], beam bending and focusing [15–17], optical switching [18, 19], optical signals processing and near-field optical spectroscopy [20].

In particular, the resonant metasurfaces are attractive as a platform for modern antenna systems [21, 22]. The metasurfaces for this application are typically in the form of resonant metallic particles arranged periodically on a flat surface. In the microwave range, such antenna systems demonstrate good performance, which inevitably deteriorates at higher frequencies due to high Ohmic losses in metallic components. To avoid this, high-refractive-index low-loss dielectric metasurfaces can be used in antenna design instead of their metallic counterparts [23, 24]. They are referred to as all-dielectric metasurfaces.

The advantageous operation of all-dielectric metasurfaces as a platform for antenna systems has been confirmed for both terahertz [25] and visible [22, 26] ranges. Their beneficial properties are compact size, high efficiency and ability to operate on various resonant modes possessing specific radiation patterns [27]. These modes are electric and magnetic multipole (Mie-type) modes of dielectric resonators, which compose the metasurface and produce strong resonant response to the field of incoming radiation. Spectral positions of corresponding resonances can be tuned independently by changing either the geometrical parameters or spatial orientation of the resonators [28].

Compared to metallic metasurfaces, which usually operate on electric modes associated with plasmons [29], all-dielectric metasurfaces support both electric and magnetic Mie-type modes. They are related to the polarization currents, which are induced in the resonators by the irradiating wave and can have a comparable strength for electric and magnetic modes [30]. In fact, an all-dielectric metasurface can operate either on pure electric or magnetic mode, aswell as on their overlapping. When electric and magnetic modes of the same strength are spectrally overlapped, the strong scattering directivity appears [31, 32]. The wide diversity of resonant conditions inherent in all-dielectric metasurfaces allows one to design modern antenna systems with high directivity.

In order to achieve high directivity of radiation, antenna systems often require non-planar reflectors (typical example is a parabolic antenna). For non-planar metasurfaces with arbitrary curved shapes, the design of optical elements of “free-form” metasurface becomes extremely complex and requires careful consideration of the substrate geometry [33, 34]. To facilitate this design, the new theoretical approach called “conformal boundary optics” has been recently proposed [35] with the aim to describe the boundary conditions at the surfaces of arbitrary shapes. This numerical technique relies on time-domain simulations and makes it possible to determine the reflection or refraction of light from the free-form metasurfaces. Along with numerical methods, simple analytical models are still highly required to facilitate design of free-form metasurfaces and enhance understanding of their optical properties. Such analytical models can be used as free, efficient and reliable alternative to cumbersome and time-consuming numerical methods. They are particularly useful for preliminary estimation of the optical characteristics of free-form metasurfaces.

Therefore, in the present paper we propose an analytical model for description of antenna systems based on free-form all-dielectric metasurfaces (reflectors). The model relies on the coupled mode theory and perturbation theory for the Maxwell’s equations. We apply our method to study antenna systems with different curved reflectors. This allows us to obtain the near-field and far-field characteristics of the antenna systems, as well as to reveal their spatial selectivity, beam steering and focusing features. The model is checked against numerical simulations performed with commercial ANSYS HFSS electromagnetic solver and experiments with several antenna prototypes prepared for their characterization in the microwave range.

2. Theoretical description

We start from the theoretical consideration of an antenna system constructed of an array of N dielectric resonators, which, in general, can be made of particles of different shape, volume and material. Particles can be arbitrary distributed in space, but are not far apart from each other. For simplicity, we suppose that all resonators operate on the same mode at some frequency $f_{0} = ω_{0} / 2 π$ .

The dielectric resonators used in the antenna system are open electromagnetic structures. Thus closely-spaced resonators are electromagnetically coupled by the near-fields [36]. Without loss of generality, we suppose further that all dielectric resonators of the antenna system are mutually coupled to each other, and one of them is an active resonator producing a primary field.

In order to determine eigen-oscillations for the array of mutually coupled dielectric resonators, as well as to solve the problem of electromagnetic wave scattering from the ensemble of particles, we start from Maxwell’s equations for a time-harmonic field

rot \vec{E} = - i \tilde{ω} μ_{0} \vec{H}, rot \vec{H} = i \tilde{ω} ε \vec{E},

where a time factor

exp (i \tilde{ω} t)

is assumed and omitted,

\tilde{ω} = ω^{'} + i ω^{″}

(with

ω^{'} = ω

) is the complex angular frequency of the overall structure,

ε = ε_{0}

and

ε = ε_{1} = ε^{'} - i ε^{″}

are permittivities of free space and resonator’s volume, respectively, ε₀ and μ₀ are the vacuum permittivity and permeability, respectively.

Let us assume that the complex-valued eigenfields ( ${\vec{e}}_{s}$ , ${\vec{h}}_{s}$ ) of individual (uncoupled) resonators used in the antenna array are given solutions of Maxwell’s equations subject to the boundary condition (at the air-dielectric interfaces) and radiation conditions at $z \to \pm \infty$ (in particular, these eigenfields can be evaluated by an approximate method described in [37]). The fields outside and inside s-th individual resonator can be presented as $({\vec{e}}_{s}^{0}, {\vec{h}}_{s}^{0})$ and $({\vec{e}}_{s}^{1}, {\vec{h}}_{s}^{1})$ , respectively.

Applying the Cauchy-Schwarz and Minkowski inequalities to the functions $({\vec{e}}_{s}^{0}, {\vec{h}}_{s}^{0})$ and $({\vec{e}}_{s}^{1}, {\vec{h}}_{s}^{1})$ , we obtain the next relationship (which is useful for the following consideration)

| \int_{V_{k}} ({\vec{e}}_{k}^{1}, {\vec{e}}_{s}^{0 *}) d v | \int_{V_{k}} | \vec{e_{k}^{1}} |^{2} d v ∽ | \int_{V_{k}} ({\vec{h}}_{k}^{1}, {\vec{h}}_{s}^{0 *}) d v | \int_{V_{k}} | \vec{h_{k}^{1}} |^{2} d v, (s \neq k),

here V_k is the volume of the k-th resonator, and the asterisk represents the complex conjugation.

Let us denote the complex resonant frequency of the s-th resonator as ${\tilde{ω}}_{s} = ω_{s}^{'} + i ω_{s}^{″}$ , ( $s = 1, 2, \dots, N$ ). We suppose that the $ω_{s}^{'}$ is the same for all resonators of the array and equals the resonant frequency ω₀ of uncoupled resonators. By contrast, $ω_{s}^{″}$ can vary slightly with s. As a result, the eigenfields ( ${\vec{e}}_{s}$ , ${\vec{h}}_{s}$ ) of individual resonators in the array differ from each other. Thus we can assume that there are N different solutions of the boundary-value problem for resonators located in different points of the array. Substituting all such solutions into Maxwell’s equations (1) and using the vector relationship $div [\vec{A}, \vec{B}] = (\vec{B}, rot \vec{A}) - (\vec{A}, rot \vec{B})$ , we obtain expressions, which describe fields ( $\vec{e}$ , $\vec{h}$ ) of the whole resonant system

div {[\vec{h_{s}^{*}}, \vec{e}] + [\vec{h}, \vec{e_{s}^{*}}]} = i (ω - ω_{0}) [ε_{1} (\vec{e}, \vec{e_{s}^{*}}) + μ_{0} (\vec{h}, \vec{h_{s}^{*}})] - (ω^{″} + ω_{s}^{″}) [ε_{1} (\vec{e}, \vec{e_{s}^{a} s t}) + μ_{0} (\vec{h}, \vec{h_{s}^{*}})],

and fields of individual resonators

div {[\vec{h_{s}^{*}}, \vec{e_{s}}] + [\vec{h_{s}}, \vec{e_{s}^{*}}]} = - 2 ω_{s}^{″} [ε_{1} | \vec{e_{s}} |^{2} + μ_{0} | \vec{h_{s}} |^{2}] .

We seek the total electromagnetic field ( $\vec{e}$ , $\vec{h}$ ) of the system of N coupled resonators (which also satisfies the boundary and radiation (at $z \to \pm \infty$ ) conditions) in the form of a linear superposition of the eigenfields ( ${\vec{e}}_{s}$ , ${\vec{h}}_{s}$ ) of the individual resonators

\vec{e} = \sum_{s = 1}^{N} b_{s} {\vec{e}}_{s}, \vec{h} = \sum_{s = 1}^{N} b_{s} {\vec{h}}_{s},

where the expansion coefficients b_s are as yet unknown. They are complex amplitudes of the electromagnetic field of a certain mode of individual resonators.

The system of equations with respect to the unknown amplitudes b_s can be derived as follows. At first, we substitute the field expansions (5) into (3) and use Eq. (4). Then we integrate the resulting expression over the volume V_s of the s-th individual resonator and take into account the relationship (2). This yields the desired system of equations with linear coupling operator $K = ∥ κ_{s n} ∥$

\sum_{s = 1}^{N} κ_{s n} b_{s} - λ b_{n} = 0, (n = 1, 2, \dots, N),

where λ and

κ_{s n}

are given by the following expressions:

λ = 2 (\frac{δ ω}{ω_{0}} + \frac{i ω^{″}}{ω_{0}}), κ_{s n} = \frac{i}{2 ω_{0} w_{n} (1 + δ_{s n})} \oint_{s_{n}} {[{\vec{e}}_{s}, {\vec{h}}_{n}^{*}] + [{\vec{e}}_{n}^{*}, {\vec{h}}_{s}]} \vec{n} d s,

Here $δ ω = Re (\tilde{ω} - ω_{0})$ , $κ_{s n}$ is the complex coefficient of mutual coupling between the s-th and n-th resonators, $\vec{n}$ is the normal to the surface s_n of the n-th resonator, $w_{n} = 0.25 \int_{V_{n}} (ε_{1} | \vec{e_{n}} |^{2} + μ_{0} | \vec{h_{n}} |^{2}) d v$ is the energy stored in the volume V_n of the n-th resonator, $δ_{s n}$ is the Kronecker delta function, which equals unity for $s = n$ and zero elsewhere. For each n the diagonal component $κ_{n n}$ of the matrix $∥ κ_{s n} ∥$ relates to the radiation quality factor (Q-factor) of the n-th resonator as $κ_{n n} = i Q_{n}^{- 1}$ , whereas the off-diagonal components $κ_{s n}$ define the mutual coupling between the s-th and n-th resonators.

System of equations (6) has nontrivial solution, once its determinant is zero. This condition yields an equation for N complex eigen-frequencies ${\tilde{ω}}_{s}$ , which can be used to calculate the total field ( ${\vec{e}}^{s}$ , ${\vec{h}}^{s}$ ) of the antenna array.

Next we suppose that the primary electromagnetic field has amplitudes ( ${\vec{E}}_{0}$ , ${\vec{H}}_{0}$ ). The solution of the problem of scattering the electromagnetic wave ( ${\vec{E}}_{0}$ , ${\vec{H}}_{0}$ ) from the array of the N coupled resonators is sought in the form

\vec{E} \approx {\vec{E}}_{0} + \sum_{s = 1}^{N} a^{s} {\vec{e}}^{s}, \vec{H} \approx {\vec{H}}_{0} + \sum_{s = 1}^{N} a^{s} {\vec{h}}^{s},

where a^s are new unknown complex amplitudes, (

{\vec{e}}^{s}

,

{\vec{h}}^{s}

) is the electromagnetic field defined above for the s-th eigen-oscillation of the antenna array.

A procedure similar to that used in the derivation of Eq. (6) leads to the following system of equations for unknown coefficients a^s:

\sum_{s = 1}^{N} a^{s} b_{t}^{s} Q_{s t} (ω) = - {(c_{0}^{t})}^{*} / P_{t}^{D},

where

Q_{s t} (ω) = 2 i \frac{ω - {\tilde{ω}}_{s}}{ω_{0}} Q_{t}^{D} + \frac{ω}{ω_{0}}, P_{t}^{D} = ω_{0} \frac{ε^{″}}{2} \int_{V_{t}} | \vec{e_{t}} |^{2} d v,

and

b_{t}^{s}

is the amplitude of the s-th coupled mode of the t-th resonator at the frequency

{\tilde{ω}}_{s}

,

P_{t}^{D}

is the power losses of material in the volume V_t of the t-th resonator,

\vec{e_{t}}

is the electric field inside the t-th resonator,

Q_{t}^{D} = ε^{'} / ε^{″}

, and

c_{0}^{t} = - \frac{1}{2} \oint_{s_{n}} {[{\vec{e}}_{t}, \vec{n}] {\vec{H}}_{0}^{*} + [\vec{n}, {\vec{e}}_{t}] {\vec{E}}_{0}^{*}} d s,

$c_{0}^{t}$ are the coefficients of expansion of the primary field in terms of eigenfields of individual resonator (one can find the explicit form of the coefficients $c_{0}^{t}$ in [37]).

The general solution of system (9) can be written as follows

a^{s} (ω) = - det B_{s} (ω) / det B (ω),

where matrices

B_{s} (ω)

and

B (ω)

are

B_{s} = | \begin{matrix} b_{1}^{1} Q_{11} & \dots & {(c_{0}^{1})}^{*} P_{1}^{D} & \dots & b_{1}^{N} Q_{N 1} \\ b_{2}^{1} Q_{12} & \dots & 0 & \dots & b_{2}^{N} Q_{N 2} \\ ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ b_{N}^{1} Q_{1 N} & \dots & 0 & \dots & b_{N}^{N} Q_{N N} \end{matrix} |, B = | \begin{matrix} b_{1}^{1} Q_{11} & \dots & b_{1}^{N} Q_{N 1} \\ b_{2}^{1} Q_{12} & \dots & b_{2}^{N} Q_{N 2} \\ ⋮ & ⋱ & ⋮ \\ b_{N}^{1} Q_{1 N} & \dots & b_{N}^{N} Q_{N N} \end{matrix} | .

Finally, the desired expression used to calculate of the far-field pattern of the overall antenna system can be written in the following matrix form

\begin{matrix} [\begin{matrix} \vec{E} (ω) \\ \vec{H} (ω) \end{matrix}] = \sum_{t = 1}^{N} A_{t} (ω) [\begin{matrix} {\vec{e}}_{t}^{\infty} \\ {\vec{h}}_{t}^{\infty} \end{matrix}], \end{matrix}

where

{\vec{e}}_{t}^{\infty}

and

{\vec{h}}_{t}^{\infty}

are the electric and magnetic fields of the t-th isolated resonator in the far-field region, and

A_{t} (ω) = \sum_{s = 1}^{N} a^{s} (ω) b_{t}^{s}

.

Thus we obtain the approximate solution of the problem of wave scattering from the all-dielectric antenna array composed of N coupled resonators. It is valid for dielectric resonators with high permittivity ( $ε_{r} = Re (ε_{1} / ε_{0}) ≫ 1$ ) regardless of their form and aspect ratio. We should note, that the developed analytical model can be extended to the case of resonators made of anisotropic materials with permittivity and/or permeability of tensor form. Moreover, this approach can also find use in analysis of plasmonic and metallic structures which are characterized by weak sensitivity of plasmons to external fields.

Fig. 1 Sketches and photos of prototypes of three different all-dielectric antennas composed of an active resonator (yellow) and array of passive resonators (blue); (a) Design ①, (b) Design ②, and (c) Design ③.

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3. Antenna prototypes

In order to verify the developed theoretical approach, several antennas have been fabricated as prototypes operated in the microwave frequency range. Fig. 1 shows schematic views and photos of prototypes of the all-dielectric resonant antennas under study. Antenna systems are made of dielectric nonmagnetic particles (cylindrical resonators). The Taizhou Wangling TP-series microwave polymer-ceramic composite was used as a dielectric material for particle fabrication. For the frequency of 10 GHz the relative permittivity of this material is $ε_{r} = 21.5$ and loss tangent is $\tan δ \leq 1.5 \times 10^{- 3}$ . The set of resonators was fabricated with the use of a precise milling machine. The diameter and thickness of the resulting resonators used in the prototypes are $d = 8.0$ mm and $h_{r} = 3.7$ mm, respectively.

To arrange the resonators into a lattice, we have milled an array of holes in a custom holder made of a rigid foam-based material (Styrofoam). The relative permittivity of Styrofoam is $ε_{s} = 1.05$ and the thickness of the holder is h_s = 50 mm. The total number of resonators used in the antenna reflector is $M = N - 1$ .

Next we perform a comparative study of three “free-form” configurations of the antenna array (reflector). These configurations include resonators arranged in a planar reflector (Design ①, $\sqrt{M} = 11$ ), parabolic cylindrical reflector (Design ②, $\sqrt{M} = 11$ ), and reflector made in the form of paraboloid of revolution (Design ③, $\sqrt{M} = 9$ ) (see Fig. 1). In the non-planar reflectors, z-coordinate of the center of the ( $i, j$ )-th resonator in the antenna array ( $i, j = 1, 2, \dots, \sqrt{M}$ ) is defined as $z_{i, j} = x_{i}^{2} / 4 F$ and $z_{i, j} = (x_{i}^{2} + y_{j}^{2}) / 4 F$ for Design ② and Design ③, respectively, where $F = 60$ mm is the focal distance. In all above-described designs, the projection of the lattice constant on the x-y plane is $l = l_{x} = l_{y} = 18$ mm. In the E-plane and H-plane, the aperture length of the array is $D_{x} = (\sqrt{M} - 1) l_{x} + d$ and $D_{y} = (\sqrt{M} - 1) l_{y} + h_{r}$ ., respectively.

The central (active) resonator is placed at the focal distance F in the front of the antenna array. In the prototypes, active resonator has a form a cylindrical resonator coupled to a 50 Ohm coaxial (feeding) line by the metallic wire loop [see Fig. 1(c)]. An advantage of this excitation method is that the active resonator can bexcited directly from a source and has no need of special matching. For the chosen resonator dimensions (the aspect ratio is η = h_r/d = 0.46), antenna system operates on the lowest TE $_{011}$ mode of the individual cylindrical resonators of the array. The corresponding operating frequency of this mode is $f_{0} = 8.57$ GHz.

4. Results and discussion

In contrast to metallic reflectors whose reflection characteristics are invariant in a wide frequency range, all-dielectric metasurfaces behave as frequency-selective surfaces. They strongly reflect incident radiation at the frequencies of the coupled oscillations supported by dielectric resonators and are almost transparent in the rest of frequency range [37]. This fact allows us to construct a new class of frequency-selective resonant antennas based on the all-dielectric metasurfaces. At the excitation frequencies of particular modes their characteristics resemble those of conventional reflector antennas.

In what follows, we compare characteristics of the far-field patterns for three above-described antenna designs. These characteristics are obtained from the analytical model, ANSYS HFSS simulations and experimental measurements (see Appendix for details of the experimental setup). For all antenna designs, the simulated and measured magnitude of the $| S_{11} |$ -parameter is below $- 25$ dB at the operating frequency f₀.

Fig. 2 Far-field radiation patterns and their cross-sections in the E-plane and H-plane for (a) Design ①, (b) Design ②, and (c) Design ③ of the antenna reflector.

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In order to simulate excitation conditions for the active resonator in the analytical model, the coupling operator K is written in the form [37]:

K = ∥ κ_{s n} + i k_{L} δ_{s 1} δ_{n 1} ∥ (s, n = 1, 2, \dots, N),

where

κ_{s n}

is given by Eq. (7), k_L is the coupling coefficient between the active resonator and the feeding line, which is related to the actual parameter

| S_{11} |

as

Γ = k_{L} / (1 + k_{L} + Q^{- 1})

, where

| S_{11} | = 20 \log (Γ)

and Q is the radiation quality factor of the active resonator.

4.1. Effect of array geometry on the antenna far-field radiation pattern

As a reference point, we first consider an antenna made of a flat reflector [Design ①, Fig. 1(a)]. The mutual coupling between dielectric resonators in the antenna system leads to the redistribution of the field radiated by active resonator. As a result, the radiation pattern of the whole antenna system changes drastically: a main lobe with radiation maximum directed orthogonally to the array surface (that is, in the direction of $θ = 0^{\circ}$ ) appears, together with some sidelobes [Fig. 2(a)].

One can readily see a good agreement between the results of our analytical model, numerical simulation and experimental data, whereas minor discrepancy between them can be explained by both the fabrication tolerances and the possible misalignment of the antenna prototype in experiment. Analytical model stronger deviates from measurements, since it ignores the presence of metallic wire loop connected to active resonator. To check the validity of this approximation, we have performed additional simulation and measurement of the far-field radiation pattern of the isolated active resonator coupled to the metallic wire loop (Fig. 3). It can be seen that the actual far-field radiation pattern is somewhat unbalanced and slightly differs from the ideal pattern of the magnetic dipole, which corresponds to the TE $_{011}$ mode of a cylindrical resonator (the correspondence between eigenwaves and Mie-type modes of a cylindrical dielectric resonator can be found in [38]).

A misalignment between the loop and resonator results in slight rotation of actual magnetic moment (about 6°) with respect to the x-z plane. In both the E-plane and H-plane, this effect leads to the symmetry breaking of the far-field radiation pattern. In actual conditions, this misalignment distorts the far-field pattern of the whole antenna system. In particular, it affects the width of the main lobe and gives rise to sidelobes.

Fig. 3 Far-field radiation pattern of the TE $_{011}$ mode of the isolated active resonator coupled to the metallic wire loop and its cross-sections in the E-plane and H-plane. Here $\vec{m}$ denotes the magnetic dipole moment. The operating frequency is $f_{0} = 8.57$ GHz.

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Despite the fact that the field beyond the flat antenna reflector is significantly suppressed, the directivity of such antenna is low (the beam width at the half-power is about 61° in both E-plane and H-plane) and the antenna radiation pattern is broad. In the most part, this is explained by the nonuniform distribution of spatial phase delay across the reflector aperture. In practice, the local phase error over the reflector aperture can be reduced either by changing parameters of resonators forming the metasurface (such as their size or permittivity) [39, 40] or by changing their spatial positions in order to attain the chirping array.

Alternative way to resolve these issues is to use a free-form metasurface instead of the flat reflector. Numerical and experimental results for the curved antenna arrays are presented in Figs. 2(b) and 2(c) for Design ∘led2 and Design ③, respectively. One can see, that there is a noticeable improvement in the far-field patterns, namely both the beam width and level of sidelobes are now noticeably reduced. This evidence confirms that the free-form reflectors can effectively control the beam shape in both planes. In particular, for the antenna system with the parabolic reflector the pencil-beam is achieved. In this case, the radiation pattern is compressed to 19° and 22° in the E-plane and H-plane, respectively.

4.2. Beam steering

Often it is necessary to control the angular direction of the main lobe, together with beam shape, to perform beam steering. Typically the beam steering is controlled by shifting the feed position from the focal point of a reflector. This leads to the phase aberration, which, in its turn, causes deformation of the far-field radiation pattern. For antennas with parabolic metallic reflector, the axial feed defocusing leads to the widening of the main lobe due to the quadratic phase error across the reflector aperture, while lateral feed displacement results in the beam steering [36].

In order to reveal the peculiarities of the beam steering mechanism in the all-dielectric antenna system (Design ③), we first investigate active and passive resonators with identical orientation (with axes along the y axis) [Fig. 4(a)].

Fig. 4 (a) Sketch of Design ③ of the antenna system when the axis of active resonator is oriented along the y axis. The pattern of the active resonator is shown in the inset. (b)-(m) Far-field radiation patterns for different lateral displacement of the active resonator.

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Fig. 5 The cross-sections of the far-field radiation pattern shown in the E-plane and H-plane for Design ③ of the antenna system, when the lateral displacement of the active resonator is Δx = −0.6l and Δy = 0.3l. The axis of active resonator is oriented along the y axis.

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When the active resonator is laterally displaced on the distance $Δ l = \sqrt{Δ x^{2} + Δ y^{2}}$ , $Δ x = m l$ , $Δ y = n l$ from the focal point, then the beam radiated from the antenna system can be steered in the opposite direction at the angle θ_B. The angle of the beam steering θ_B is relative to the offset distance $Δ l$ by the approximate formula $θ_{B} \approx BDF \cdot \overset{- 1}{\tan} (Δ l / F) = BDF \cdot ψ$ , where BDF is the beam deviation factor, and ψ is offset angle [36].

Several representative far-field radiation patterns are shown in Figs. 4(b)-4(m) for the different lateral displacement of the active resonator. The maximum of the beam steering angles is limited by the acceptable level of the far-field radiation distortion and is estimated as $θ_{B} \approx \pm 34^{\circ}$ ( $ϕ \in [0^{\circ}, 360^{\circ}]$ ) for the offset displacements $Δ x = \pm 2.5 l$ and $Δ y = \pm 2.5 l$ . This limitation is caused by the effect of cubic phase error, which is inherent in the sidelobes and induces their growing on the boresight side, while suppressing them on the other side. In addition to distinct increase in sidelobs, one can achieve a splitting of the main lobe, once the lateral displacement becomes too large. The latter effect can be clearly seen from Figs. 4(b) and 4(i).

If the beam steering is performed in the E-plane (that is, $Δ l = Δ x$ , $Δ y = 0$ and $ϕ = 0^{\circ}$ ), then the maximum beam steering ranges from $- 65^{\circ}$ (at $Δ x = 4 l$ ) to 65° (at $Δ x = - 4 l$ ). For the beam steering in the H-plane (that is, $Δ x = 0$ and $ϕ = 90^{\circ}$ ) the range is narrower and lies between $- 42^{\circ}$ (at $Δ y = 4 l$ ) and 42° (at $Δ y = - 4 l$ ).

Fig. 6 Same as in Fig. 4 but for active resonator is oriented along the z axis.

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Fig. 7 The cross-sections of the far-field radiation pattern shown in the E-plane and H-plane for Design ③ of the antenna system, when the lateral displacement of the active resonator is Δx = 0.22l and Δy = 0.1l. The axis of active resonator is oriented along the z axis.

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In order to check these analytical predictions we have performed a series of far-field measurements in both planes for active resonator with certain lateral displacement. The simulated and measured data are summarized in Fig. 5 for the offset displacement $Δ x = - 0.6 l$ and $Δ y = 0.3 l$ , which corresponds to the pattern presented in Fig. 4(j). One can conclude that there is a good agreement between the simulated and measured far-field characteristics of the chosen antenna system. A slight difference in the patterns (about 1°) has been found for lobe angular direction in both planes. It can be explained by some misalignment inherent in experimental prototype.

The above-discussed mechanism of beam steering can be applied to obtain several beams from the antenna system. In particular, it has been demonstrated that two-beam pattern can be achieved when the axis of active resonator is placed orthogonal to the axis of passive resonators of the antenna reflector. Corresponding far-field radiation patterns are summarized in Fig. 6.

For such an arrangement of the active resonator, the main lobe of the antenna system splits into two beams oriented along the $\approx \pm θ_{b}$ directions. One can conclude that, in this case, the steering of two split beams can be achieved in the wide range of angles by shifting position of the active resonator from the focal point of the reflector. The experimental validation of the beam splitting effect is presented in Fig. 7.

5. Conclusions

In conclusion, we have proposed the analytical model for description of antenna systems based on free-form all-dielectric metasurfaces. It relies on the coupled mode theory and the perturbation theory. The model has been applied to determine the eigen-oscillations of the array of mutually coupled dielectric resonators, as well as to describe the characteristics of the overall all-dielectric antenna system. In order to check the validity of the developed analytical model, three designs of the antenna system have been numerically simulated, fabricated and experimentally tested in the microwave range.

Our results show that the pencil-beam and two-beam patterns can be obtained in the all-dielectric antenna by varying the orientation of the active resonator with respect to the antenna reflector array. In addition, we have shown experimentally that the single and two beam steering can be achieved in a wide range of angles by shifting the active resonator from the focal point of the antenna array.

The proposed model is quite general and can be applied to study all-dielectric metasurfaces composed of resonators, which have arbitrary shape and are arranged in a non-flat array.

Appendix. Experimental setup

The sketch and photo of the experimental setup with a sample are shown in Fig. 8. The antenna prototype under study is placed on a rotating platform at the distance $L = 2.8$ m from the receiving dielectric-lens antenna (HD-100LHA250). This distance corresponds to the far-field zone $L \geq 2 {[(D_{x} + D_{y}) / 2]}^{2} / λ_{0}$ , here $λ_{0} = c / f_{0}$ . The active resonator is coupled to the wire loop via 50 Ohm coaxial cable. All measurements are performed in an anechoic chamber with the use of Keysight E5071C Vector Network Analyzer. For more details about the measurement setup and preparation of samples, see [41, 42].

Fig. 8 (a) Schematic view and (b) photo of the experimental setup with a prototype of an all-dielectric antenna system.

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Acknowledgments

This work was supported by Jilin University.

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Approach to analysis of all-dielectric free-form antenna systems

Abstract

1. Introduction

2. Theoretical description

3. Antenna prototypes

4. Results and discussion

4.1. Effect of array geometry on the antenna far-field radiation pattern

4.2. Beam steering

5. Conclusions

Appendix. Experimental setup

Acknowledgments

References

Cited By

Figures (8)

Equations (15)

Optics Express

Volodymyr I. Fesenko	https://orcid.org/0000-0001-9106-0858
Vitalii I. Shcherbinin	https://orcid.org/0000-0002-9879-208X
Vladimir R. Tuz	https://orcid.org/0000-0001-6096-7465