The design philosophy adopted rests on the use and integration of three basic elements that form the antenna design: the driver dipoles, the printed reflector and the parasitics. The antenna was brought into operation over an evolved sequential design following first the principles. In the following, the design is discussed for each of the three constituent basic elements.

### Driver dipoles

The substrate chosen for the printed antenna configuration has a dielectric constant *ε*_{r} = 3.55, loss tangent (tan *δ*) = 0.0027 and a thickness of 0.508 mm. The substrate area was chosen to mitigate the surface waves and the fringing edge effects to a considerable extent by maintaining a peripheral distance between the structure to the boundaries greater than a quarter wavelength. The substrate area was thus set to 12 mm × 11 mm. A λ/2 printed dipole centred at the chosen frequency of 14 GHz was designed. The arms of the dipole were chosen to be printed on opposite sides of the substrate to act as a self-balancing structure. Figure 1a shows the front view of the design of an arm of the dipole on the substrate and Fig. 1b shows the back view of the other arm. Initially, the length of each arm was selected as λ_{0}/4 at 14 GHz, equal to 5.35 mm. The length of the arm was later fine-tuned through full-wave simulations to 3.9 mm to have the band centred at 14 GHz. A parametric variation of the arm length and its effect on S11 is shown in Fig. 2. For this parametric variation of other parameters such as the transition length and width, the reflector length and width are kept constant and only the arm length is varied in steps of 0.25 mm, displaying the tuning behaviour over the 12–18 GHz range for an arm length variation from 3 mm to 5.25 mm. This tuning behaviour demonstrates the potential of the antenna to be reconfigured in frequency.

### The printed reflector

The antenna design requires a printed reflector to enhance the directivity and the front-to-back ratio. The design of the printed reflector is performed to use the reflector both as a ground and as a supporting structure to the SMA feeder connector. This requirement makes the reflector a stepped design with the width (tx) of the section facing the SMA near to *λ*_{0}/4 at 14 GHz and equal to 5.5 mm. The actual reflector length (ref_x) is *λ*_{0}/2 at 14 GHz equal to 10.7 mm. The final printed reflector, which is printed on the face opposite to the side where the SMA pin connects, is shown in Fig. 3 with its optimized dimensions.

A parametric study was performed over the transition width (tx), transition length (tz) and the reflector length (ref_x). As shown in Fig. 4, the reflector length variation (ref_x) does not affect the frequency behaviour excessively. This strengthens the dependence of the frequency behaviour of the antenna mainly on the arm length as shown in Fig. 2 with the reflector aiding in enhancing the front-to-back ratio. A redesign of the reflector is needed for each tuning frequency. It was observed that the transition dimensions affect the amount of realizable cross-polar levels. A parametric study of this dependence was performed by varying the transition length (tz) and transition width (tx) vs the observable cross-polar levels at Phi0 and Phi90 principal cut planes, and results are shown in Figs. 5, 6, 7 and 8. It was seen that the effect of varying the transition width (tx) is not as pronounced as the effect of varying the transition length (tz). Variation of tz controls the observable cross-polar levels. As seen from Fig. 8, a substantial increase in the cross-polar levels is observed around boresight for variations over tz. The length before the transition step is a crucial factor that controls the cross-polar levels of the antenna and therefore has been deduced to be equal to 1.66 mm both by considering the surface current distribution as shown in Fig. 9 and the cross-polar levels of the pattern. Here the surface current distribution is stronger on the dipole arms for the depicted cases of absolute surface current phase of 0°, 90° and 180°, showing that the reflector performs the field enhancement through reflection. The full-wave solver calculated values of the cross-polar levels after the full design of the antenna helps also in deciding the transition step length of tz = 1.66 mm and plays a critical role in the design.

### The parasitics

In Fig. 9, the surface current distribution shows the currents strongly distributed on the dipole arms. It is intended to couple the electric field due to this onto the parasitics in a constructive way to enhance the far-field gain. To achieve this, a systematic distribution of parasitics is proposed. The first parasitic that couples the dipole arm field is a rectangle-shaped parasitic as shown in Fig. 10. The electric field coupled onto this rectangle-shaped parasitic is distributed along two oblique directions, using two symmetrically placed square parasitic elements, whose dimensions are 0.05 λ × 0.05 λ that translates to 1 mm × 1 mm, each suitably spaced from each other to enhance coupling, while equally distributing the electric field along two symmetric oblique directions. Figure 10 shows the rectangle-shaped parasitic and the two square parasitics coupling onto the dipole. The capacitive coupling gap between them was set to maximize the coupling.

In the following, the design of the arc-shaped parasitics is discussed. The shape of the parasitics was chosen as an arc, to channel the electric field obliquely. In addition, it also maximizes the coupling by maximizing the amount of exposed surface area that couples onto the next parasitic. The fibre-optic principle was used to position the parasitics along the antenna plane in front of the square resonators. The design derives its inspiration by analogy to a graded-index fibre-optic cable, whose core can be considered analogous to a certain cuboidal volume region, lying directly below the endfire central axis of the antenna, where the graded real refractive index is set to a maximum. The outward displacement towards the edges can be considered to represent the cladding—where there is a gradient decrease of the real refractive index moving away from the centre. This is achieved by the methodical placement of the arc-shaped parasitics in front of the driver dipoles over the antenna plane. The procedure to arrive at this is discussed next.

The dielectric substrate used in the antenna can be considered a homogeneous slab for the frequencies under consideration. The refractive index is the reconfigurable parameter that is used to direct the mechanism of radiation. Then according to [9], the general procedure for the retrieval of the material parameters, in this case, the real part of the refractive index is as follows: the arc-shaped parasitics are placed so that they focus the electric field towards the boresight axis. A one-dimensional transfer matrix *T* relates the fields on one side of a slab to the other. For a homogeneous 1D slab, it takes the form (1):

$$ T=\left(\begin{array}{cc}\cos (nkd)& -\frac{z}{k}\sin (nkd)\\ {}\frac{k}{z}\sin (nkd)& \cos (nkd)\end{array}\right) $$

(1)

where *n* is the refractive index, *z* is the wave impedance, *k* is the wave number and *d* is the thickness of the slab. A scattering matrix relates the impinging field amplitude to the outgoing field amplitude. The elements of the *S* matrix can be related to the elements of the *T* matrix as in (2) [10]:

$$ {\displaystyle \begin{array}{l}{S}_{21}=\frac{2}{T_{11}+{T}_{22}+\left({ ik T}_{12}+\frac{T_{21}}{ik}\right)},\\ {}{S}_{11}=\frac{T_{11}-{T}_{22}+\left({ ik T}_{12}-\frac{T_{21}}{ik}\right)}{T_{11}+{T}_{22}+\left({ ik T}_{12}+\frac{T_{21}}{ik}\right)},\\ {}{S}_{22}=\frac{T_{22}-{T}_{11}+\left({ ik T}_{12}-\frac{T_{21}}{ik}\right)}{T_{11}+{T}_{22}+\left({ ik T}_{12}+\frac{T_{21}}{ik}\right)},\\ {}{S}_{12}=\frac{2\det (T)}{T_{11}+{T}_{22}+\left({ ik T}_{12}+\frac{T_{21}}{ik}\right)}\end{array}} $$

(2)

For a homogeneous slab from equation (1),

$$ {T}_{11}={T}_{22}={T}_S\kern0.24em \mathrm{and}\kern0.36em \det (T)=1 $$

And the *S* matrix is symmetric. Using the information in equation (1), (3) and (4) can be derived:

$$ {S}_{21}={S}_{12}=\frac{1}{\cos (nkd)-\frac{i}{2}\left(z+\frac{1}{z}\right)\sin (nkd)} $$

(3)

$$ {S}_{11}={S}_{22}=\frac{i}{2}\left(\frac{1}{z}-z\right)\sin (nkd) $$

(4)

Solving equations (3) and (4) for *n* and *z* yields (5) and (6) [9]:

$$ n=\frac{1}{kd}{\cos}^{-1}\left[\frac{1}{2{S}_{21}}\left(1-{S_{11}}^2+{S_{21}}^2\right)\right] $$

(5)

$$ z=\sqrt{\frac{{\left(1+{S}_{11}\right)}^2-{S_{21}}^2}{{\left(1-{S}_{11}\right)}^2-{S_{21}}^2}} $$

(6)

The design principle of the antenna revolves mainly around equation (5) which gives the expression for the refractive index as a function of the reflection and transmission *S* parameters and the slab length *d*. The application of equation (5) is first performed over a single-unit cell of the arc-shaped parasitics (ASP) as shown in Fig. 11. The unit cell is excited by waveguide ports excited on the two faces of the dielectric slab on which the ASP is printed. On running the simulation in a full-wave solver for the desired frequency of interest, the *S*_{11} and *S*_{21} values can be obtained. The length of the arc of the ASP is 1.7 mm, around *λ*/10 at 14 GHz.

Of the total 10 ASPs selected to enhance the endfire gain, five are on the left side of the endfire axis and the other five are symmetrically on to the other side. Equation (5) is used to calculate the refractive index as illustrated in Fig. 12 which depicts the way the refractive indices were calculated starting from the endfire central axis, towards the extreme left. Since the structure is symmetrical across the endfire axis, the same calculation of the refractive index holds for the right side as well.

From Fig. 12, starting from the extreme right, i.e. the central ASP, setting up the *S* parameter extraction and retrieval of the refractive index on the full-wave solver returns values specific to the band of interest and length of the slab (*d*) equal to 1 mm. This procedure is repeated for other four cases of *d* = 1.5, 2, 2.5 and 3 mm. The distance along the *x*-axis in Fig. 12 is not to scale, it is a pictorial depiction for clarity purposes of describing the setup. This methodology extracts the real part of the refractive index for this kind of unit cells in a sequential way. Their values as a function of frequency are plotted in Fig. 13, where the real part of the refractive index variation on the structure is shown, with different curves signifying—the distance between the arc-shaped parasitic element and the square resonator (d)—1 mm being the closest one near to the central axis and 3 mm for the one on the extreme flaring edges. The imaginary part of the refractive index in these cases is near zero signifying minimum loss in radiating energies.

Figure 14 shows gradient depiction of the variation of the refractive index from the centre towards the edges with dark blue representing the highest real part of refractive index and lighter shade depicting a decreased value. This distribution of the real part of the refractive index makes the antenna behave analogous to a graded-index optical fibre that focuses energy towards the centre increasing the gain of the antenna. The simulation results of the antenna have been presented in [11].

### Effect of antenna backplate

The antenna design described above in Sections 2.1, 2.2 and 2.3 is fed with a suitably designed 50-ohm coaxial SMA connector as in Fig. 15a. As the goal is to use the antenna in an array, a common mechanical support plate to connect and support multiple antenna elements is needed. The supporting plate is designed with a square hole in the centre as shown in Fig. 15b. The presence of the supporting plate with a hole has two functions: (1) It sufficiently isolates the ground of the designed antenna and maintains a purity in the tuning of the resonant frequency. (2) It enhances the obtained gain pattern by providing for improved cross polar levels. The 3D co-polar gain radiation pattern obtained out of the antenna without the backplate and with the holed backplate is shown in Fig. 16. The 3D cross-polar gain pattern for the same two cases is shown in Fig. 17.

The cross-polar discrimination (XPD) improvement for the case with the backplate is compared with the case without and is shown in the Cartesian plot in Fig. 18. It can be clearly seen that the XPD is improved considerably with the use of a holed backplate by 15 dB along the boresight direction for a beamwidth of 20°.