Analysis of tightly-coupled dipole phased array antennas with metasurface superstrate

Tightly coupled dipole arrays, including connected arrays, and capacitively coupled arrays, are one of the best solutions for wideband phased array antenna designs. However, to increase bandwidth and maximum scan angle, we can use a metasurface superstrate. We propose an analytical model to compute the scan impedance of a tightly coupled dipole array loaded with a metasurface. This analytical model helps us to simplify the calculation of the scan impedance of the array and speeds up the design process of these array antennas. It is shown that the metasurface superstrate improves the bandwidth and beam scanning angles of the array. Using the proposed general transmission line model, the computation of the scan impedance of tightly coupled dipole arrays is done very fast with minimum error. The semi-analytical model is about 7 times faster than a full-wave simulation.


Analytical model
The relatively broad bandwidth and low-profile structure of the dipole antenna makes it a popular element in the phased array antennas.We have closed-form relations for dipole array antennas' scan impedance 24 .These relations do not work for TCDAs.So, we use scan impedance relations developed for the connected arrays based on spectral Green's function 8,25 .

TCDA scan impedance
A top-down view of a TCDA is shown in Fig. 1.The antenna array is located in the xy-plane and the elements are x-polarized.It is assumed that the elements' thicknesses are infinitesimal and are coupled to each other with the impedance Z.This general assumption for the coupling impedance allows us to evaluate the scan impedance of both connected arrays (Z = 0) and capacitively coupled arrays ( Z = 1/jωC ).The dipole element size is d x length and w d width.The unit-cell sizes in x and y directions are d x and d y , respectively.The analysis of the TCDA is based on 26,27 and a brief review of the scan impedance derivation is stated in 8 .The current distribution of the array is derived by enforcing the E-field continuity boundary condition, and the scan impedance is found to be where δ d is the dipole feed gap and k xm and k ym are wavenumbers of the Floquet modes m x and m y respectively: θ 0 and φ 0 are scan angles, k 0 = 2π 0 is the free space wavenumber and G xx represents the x-x component of the dyadic spectral Green's function.J 0 is the Bessel function of the first kind and zero-order.The xx-component of the dyadic spectral Green's function of an electric source is stated as 26 : v TE/TM is the normalized voltage for an equivalent transmission line fed by the unit generator.
(1) For simple isotropic metasurfaces, the shunt equivalent impedance ( Z MS ) is the same grid impedance 28 .For example, for the patch array metasurface, shown in Fig. 3, grid impedance is represented as follows 28,29   www.nature.com/scientificreports/where

General model for metasurface
There is no analytical formula for the grid impedance of all metasurfaces.So, we must use a more general model for calculating metasurface impedance.Using a full-wave simulation of the metasurface unit-cell and obtaining its impedance matrix, we can calculate the proper impedance to use in the transmission-line model.Figure 4 shows a suitable general lattice network 30 .We can state the parameters used in the network versus impedance matrix parameters as Replacing Z MS in Fig. 2 with the model shown in Fig. 4, the term Z MS ||Z TE,TM 0 in ( 9) is replaced with for either TE or TM modes.
Using this model, we have a semi-analytical solution for the scan impedance of the TCDA.A full-wave simulation of the metasurface unit-cell is necessary, before the analytical calculation of the scan impedance.Using, scan angle-dependent impedance parameters derived from the full-wave simulation, this semi-analytical model can be used for the off-broadside scan impedance calculation easily.

Numerical validation
For evaluating the analytical relations and equivalent transmission-line model stated in "Analytical model" section, the results are compared with the full-wave simulation using CST MWS.

TCDA
Figure 5 compares the scan impedance of a connected array using analytical relations and full-wave simulation.A 5:1 frequency band (0.2-1 GHz) is used.The dipole is half-wave at the upper frequency and impedance is calculated for the broadside.
Figure 5 shows a good agreement between analytical and full-wave scan impedances.The average percent error is about 2% with a maximum of about 4%. ( 7) www.nature.com/scientificreports/

TCDA loaded with metasurface
Analytical and simulation results for a connected array loaded with a patch array metasurface can be seen in Fig. 6.The Patch array metasurface and its parameters are shown in Fig. 3.The metasurface unit-cell size and gap between patches are p x = p y = p = 0 10 and g = 0 50 respectively.The metasurface is located d = 0 5 above the dipole and the backplane reflector is placed h = 0 4 below the array plane.0 is the wavelength at the highest frequency ( f 0 ).
Excellent agreement is observed between the analytical model and simulation results.The average percent error is about 3% with a maximum of about 10% at high frequencies.

Generalized model
Figure 7 shows the scan impedance of the array loaded with metasurface as in Fig. 6 but the results of impedance derived from a generalized model introduced in Fig. 4 are compared to the full-wave and analytical results.
Another metasurface composed of a split-ring resonator (SRR) unit-cell, is used as the superstrate to show the performance of the semi-analytical model.A connected array unit-cell with an ultra-thin SRR metasurface is shown in Fig. 8.
Figure 9 shows the semi-analytical and full-wave (CST) simulation results with good agreement.Full-wave simulation using CST MWS 2019, takes 21 min and 41 s long while the semi-analytical calculation takes only 3 min and 10 s long.Analytical calculations are programmed in MATLAB 2020a on the same laptop (Intel® Core™ i7-2620 M CPU and 12 GB memory).www.nature.com/scientificreports/For evaluating the performance of the model for bianisotropic structures a wire and split ring bianisotropic Huygens' unit-cell is derived from 30 , as shown in Fig. 10.
Figure 11 shows the connected array loaded with the bianisotropic metasurface.
Figure 12 shows the semi-analytical and full-wave(CST) simulation results with good agreement.

Scan
To evaluate the analytical model and relations stated in the "General model for metasurface" section, the scan impedance of a connected array loaded with a patch array metasurface, is plotted versus the scan angle in Fig. 13.www.nature.com/scientificreports/ The array and metasurface parameters are the same as in "TCDA loaded with metasurface" section, and the frequency is 0.2 GHz.It can be seen that the analytical result conforms with the full-wave simulation.Similarly, to evaluate the semi-analytical model stated in "Scan" section, consider the SRR metasurface introduced in "Generalized model" section.Scan impedance in the E-plane is shown in Fig. 14 for three frequencies.
It can be seen that the model can predict the impedance variation of the connected array very well.

Discussion
The analytical model introduced in "Analytical model" section, and numerically validated in "Numerical validation" section, now can be used to investigate the wide-band wide-angle performance of tightly coupled array antennas loaded with metasurface superstrate.In this section, the ability of the metasurface to improve the bandwidth and maximum scan angle of the array is investigated briefly.All simulations and analytical calculations are done in a 5:1 frequency band such as the previous sections but an optimum design to achieve the maximum bandwidth is not done.

Impedance mismatch loss
Figure 15 shows the return loss or S11 of a connected array designed in 0.2-1 GHz, with and without metasurface superstrate.Unit cell size is half-wave at the upper frequency and dipole width is 0.1 of wavelength at this frequency.Bandwidth improvement is obvious and while the bandwidth of S11 lower than − 9.5 dB (equivalent to VSWR = 2) is 91.5% (0.35-0.94 GHz) for a connected array without metasurface superstrate, adding a metasurface superstrate improve it to 105.3% (0.31-1 GHz).We can use impedance mismatch loss for evaluating arrays' scan and bandwidth performances.The reflection coefficient ( Ŵ ) is calculated between the scan element impedance and reference impedance (source impedance).Having the reflection coefficient, |Ŵ| 2 is the relative reflected power, and T = 1 − |Ŵ| 2 is the relative transmitted power.The mismatch loss is defined as 10logT = 10log 1 − |Ŵ| 2 .Reference impedance is taken to be a constant real impedance, and by changing it, we choose the best value.This impedance is taken to be 350 Ω and 400 Ω for the array without metasurface and the array with metasurface, respectively.Figure 16 shows mismatch loss.www.nature.com/scientificreports/

Scan performance
Beam scanning changes the array scan impedance and thus for a constant reference impedance, we have a drop in the power transmitted or received.The reflection coefficient at the terminals of the dipoles is defined relative to broadside impedance.The generalized reflection coefficient is defined as where Z b = Z s (θ = 0, φ = 0) is the broadside scan impedance.To determine the scan performance of a TCDA loaded with a metasurface, the relative transmitted power (or transmittance), computed as is monitored in all scan planes.The results are shown in Fig. 17.It can be seen that the scan performance has improved considerably.For comparison, we define the maximum scan angle so that the transmittance is 0.79 (equivalent to 1 dB loss).So, the maximum scan angle is improved from 46°, 38° and 34° to 65°, 67 and 41° in the E-, D-and H-planes respectively.

Conclusion
A transmission-line model for a unit cell of an infinite phased array of tightly coupled dipole arrays backed by a ground plane and loaded with a metasurface has been presented.The model can be used for accurate analysis of the array with a superstrate.Using this model, significant improvement in loading the array with metasurface superstrate is seen, both in bandwidth and maximum scan angle.The results of the model fully conform with the full-wave simulations.This fast analytical mdelling makes it useful in designing tightly coupled dipole array antennas loaded with metasurfaces.

Figure 2 .
Figure 2. Transmission-line model of a TCDA backed by a ground plane and loaded with a metasurface.

Figure 5 .
Figure 5.The scan impedance of a connected array with a backing reflector 8 .

Figure 6 .
Figure 6.Scan impedance of a connected array loaded with a metasurface.

Figure 7 .Figure 8 .
Figure 7.The scan impedance of a connected array loaded with a metasurface.

Figure 9 .
Figure 9.The scan impedance of a connected array with an ultra-thin SRR metasurface.

Figure 12 .
Figure 12.The scan impedance of a connected array with a bianisotropic Huygens' metasurface.

Figure 13 .
Figure13.Scan angle dependence of a connected array loaded with a metasurface at the frequency f = 0.2 GHz.

Figure 14 .
Figure 14.Scan angle dependence of a connected array loaded with a metasurface in E-plane.