2D matrix representing an LWA

This paper presents the above-mentioned 2D planar LWA structure, designed unlike in paper [Reference Lai, Leong and Itoh7] in a compact integrated form, which includes eight-port SIW power dividers [Reference Hao, Hong, Li, Zhang and Wu13, Reference Shehab, Karmakar and Walker14] connected directly to the 2D CRLH mushroom structure [Reference Sievenpiper, Zhang and Yablonovitch15]. These cells are capacitively coupled to neighboring cells via slots in the top metallization. This structure radiates in the way controlled by the amplitudes of the signal fed to the particular edges of the matrix.

The beam steering in elevation is due to the change in frequency and represents the standard LWA behavior, as will be described by dispersion characteristics, see Fig. 5. The elevation angle θ is measured from axis z, which is perpendicular to the MTS, see Fig. 1.

Figure 1. A 2D CRLH transmission line composed of mushroom cells coupled by slots, represents a metasurface. Picture shows the construction of the internal leaky wave propagating under angle φ determined by the propagation vector β.

The beam steering in azimuthal angle is caused by the interaction of two excited internal leaky waves. These waves are quasi-transversal-electromagnetic (TEM) waves with electrical field directed along the z axis, magnetic field is rectangular to the direction of propagation, i.e., parallel to the y axis for the wave with power P ₁ and parallel to the x axis for the wave with power P ₂. The superposition of these two waves is the leaky wave propagating in the direction φ, see Fig. 1. Angle φ is determined by the direction of the final magnetic field vector. The beam direction in the azimuthal plane is naturally equal to this angle φ and is therefore controlled by changing powers P ₁ and P ₂ fed into adjacent ports represented by the matrix edges, see Fig. 1, and is determined by [Reference Lai, Leong and Itoh7]

(1)\begin{equation}tan\left( \varphi \right) = {P_2}/{P_1}.\end{equation}

The radiating structure is represented by the 2D matrix of mushroom cells. Following paper [Reference Yang, Zhang, Li, Jin, Cheng, Li and Li16], the equivalent lumped-element circuit of the rectangular unit cell shown in Fig. 2 is illustrated in Fig. 3. The structure has been analyzed as lossless, so that all included resistors are disregarded.

Figure 2. The mushroom cell geometry and dimensions.

Figure 3. Equivalent lumped-element circuit of the unit cell shown in Figure 2.

The elements of the circuit are given by formulas that have been derived based on ideas presented in paper [Reference Yang, Zhang, Li, Jin, Cheng, Li and Li16]:

(2)\begin{equation}{C_V} = \frac{{{\varepsilon _0}{\varepsilon _r}\left( {{l^2} - \pi {a^2}} \right)}}{d}\end{equation}

(3)\begin{equation}{C_F} = {\varepsilon _0}{\varepsilon _r}\frac{{2p}}{\pi }{\text{ln}}\left( {{\text{si}}{{\text{n}}^{ - 1}}\left( {\frac{{\pi \left( {p - l} \right)}}{{2p}}} \right)} \right)\,\end{equation}

(4)\begin{equation}{L_P} = \frac{{{\mu _0}{l_p}}}{{2\pi }}\left( {\ln \left( {\frac{{2{l_p}}}{{{w_p} + {t_p}}}} \right) + \frac{1}{2}} \right)\end{equation}

(5)\begin{equation}{L_G} = \frac{{{\mu _0}{l_g}}}{{2\pi }}\left( {\ln \left( {\frac{{2{l_g}}}{{{w_g} + {t_g}}}} \right) + \frac{1}{2}} \right)\,\end{equation}

(6)\begin{equation}{L_V} = \frac{{{\mu _0}d}}{{4\pi }}\left( {{\text{ln}}\left( {\frac{{{p^2}}}{{{a^2}}}} \right) + \frac{{{a^2}}}{{{p^2}}} - 1} \right)\end{equation}

where l_p, l_g, w_p, w_g are dimensions of the patch and the ground (equal to l, and p in Fig. 2), t_p and t_g are thicknesses of the patch and ground metallization. Next, the ABCD matrix of the circuit in Fig. 3 is determined. The generally complex propagation constant

(7)\begin{equation}\gamma = \alpha + j\beta \end{equation}

where α is the attenuation and β is the phase constant, is given in terms of the ABCD matrix elements as:

(8)\begin{equation}\gamma = \frac{{{\text{argcosh}}\left( {\frac{{A + D}}{2}} \right)}}{p}.\end{equation}

The characteristic (7) describes the propagation of a wave in the Γ–X direction of the 2D matrix composed of mushroom cells. The dispersion characteristic calculated by equation (8) is compared with the characteristic calculated by the CST Microwave Studio (CST MWS) [17] Eigenmode Solver in Fig. 4. This corresponds to the dispersion characteristic plotted for the designed antenna in Fig. 5.

Figure 4. Comparison of the dispersion characteristic calculated by equation (8) with the characteristic calculated by the CST MWS Eigenmode Solver. The phase constant is substituted by the elevation angle by using equation (10).

Figure 5. (a) Dispersion characteristics of the 2D antenna matrix calculated by the Eigenmode Solver of the CST MWS are represented by solid lines. Dots correspond to values read from radiation patterns calculated by the CST MWS. Red color represents the wave propagation under an azimuthal angle of 0° (fed at only one port, Γ–X direction). Black color represents the wave propagation under an azimuthal angle of 45° (fed at two adjacent ports, Γ–M direction). (G stands for Γ in the plot legend.) (b) Leakage constant calculated by the CST MWS for the wave propagating at azimuthal angle 0° and for particular pin radii.

The Bloch impedance is determined by [Reference Sabah, Urbani and Uckun18]

(9)\begin{equation}{Z_B} = \frac{{2B}}{{\left( {D - A} \right) \pm \sqrt {{{\left( {A + D} \right)}^2} - 4} }}.\end{equation}

The ± solutions correspond to the characteristic impedance for positive and negative travelling waves, respectively. For symmetrical networks, these impedances are the same except for the sign; the characteristic impedance for a negatively travelling wave turns out to be negative.

This radiating structure has been designed to show balanced behavior with broadside radiation at 22.25 GHz. At the same time, relatively small cell dimensions have been required. The cells have been designed using a substrate of 0.625 mm in thickness and permittivity equal to 10. The thickness of metallization is 35 μm. The cell dimensions are 5 × 5 mm, the slots between the top metal patches are 0.1 mm in width, and the metallic pins connecting the top patches with the ground are a = 0.6 mm in diameter.

The dispersion characteristic of this cell in the first Brillouin zone calculated by the Eigenmode Solver of the CST MWS is plotted in Fig. 5. Instead of the propagation constant β, the plots show the angle of the maximum radiation in elevation θ determined by

(10)\begin{equation}sin\left( \theta \right) = \beta /{k_o}\end{equation}

where k _o is the free space propagation constant. These values are plotted in solid lines, where red is for propagation in the Γ–X direction (propagation along the x or y axis) and black for propagation in the Γ–M direction (propagation in the diagonal direction).

Dots in Fig. 5 correspond to elevation angles of maximum radiation according to radiation patterns calculated by the CST MWS. These calculations and measurements were performed by analyzing the whole antenna array, see Fig. 7(b). Equation (10) describes the radiation of an ideal structure organized as an infinite array of individual cells and is not always valid for real structures. This is the reason for the imperfect fit between data in Fig. 5. The plotted dispersion characteristics indicate that the radiation of the 2D antenna matrix is similar in the vicinity of the Γ point (center of the Brillouin zone) both for direction of the wave propagation under 45° (which is the Γ–M direction) and for direction of the wave propagation under 0° or 90° (which is the Γ–X direction). The former refers to the matrix fed from two adjacent ports by equal powers, while the latter corresponds to feeding at only one port. The plot in Fig. 5 confirms that the antenna operates as a 2D CRLH transmission line in balanced regime, i.e., with zero frequency gap. Measurements show that by changing frequency from 21 to 23.8 GHz, the beam is steered in the elevation direction from −20° (backward radiation) to 20° in the forward direction when exciting the antenna at only one port. This interval of angles is reduced to the band from −15° to 15° when exciting the antenna at two ports at equal amplitudes. The dispersion characteristics plotted in Fig. 5 depend on the azimuthal direction of the wave propagation in the structure. From this follows that the radiating patterns defined in the elevation depend on the azimuthal angle. This will be shown in Figs. 14, 16, and 18. The radiation in elevation angles is determined by frequency dependences of the dispersion characteristics.

The leakage constant (real part of the complex propagation constant (7)) is plotted in Fig. 5(b) for the wave propagating at the Γ–X direction, i.e., wave excited by only one port. These values have been calculated by equation (8) as stated in paper [Reference Machac, Polivka and Zemlyakov19] using the cell ABCD matrix calculated by the CST MWS analyzing a transmission line that represents a single row or column of the radiating MTS matrix. Three values of the conducting pin radius are used 0.1 mm, 0.3 mm (value applied in the designed antenna), and 0.5 mm. The value of the leakage constant determines the antenna gain. There is an optimum value of the leakage constant to ensure maximal gain. In the case of the proposed antenna, the maximal gain has been obtained for the radius of the conducting pins equal to 0.3 mm, see Fig. 20. The radius equal to 0.5 mm gives substantially lower gain than the radius equal to 0.3 mm. For the radius equal to 0.1 mm, the gain is up to frequency 22.5 GHz nearly the same as for 0.3 mm, for higher frequencies it is lower.

Models of the presented antenna

The radiating structure – 2D matrix of mushroom cells has been discussed and analyzed in the previous paragraph. The matrix composed of 8 × 8 mushroom cells is fed through eight-port SIW power dividers [Reference Hao, Hong, Li, Zhang and Wu13, Reference Shehab, Karmakar and Walker14] connected directly to the 2D CRLH mushroom structure. The design of the divider has been done according to paper [Reference Hao, Hong, Li, Zhang and Wu13]. All SIW parts are of the same width equal to the dimension of the mushroom cell 5 mm. The final calculated scattering parameters are plotted in Fig. 6. The structure is symmetrical therefore only four ports adjacent to the particular cells are considered. The divider behavior is acceptable in the band between 22.5 and 23.5 GHz used in simulations and measurement in following text. This band is the working band of the designed antenna version B, see Fig. 13.

Figure 6. Calculated S _ij parameters of the eight-port SIW power divider, where j = 1–5, i = 1 stands for the feeding 50 Ω microstrip line.

The standard SIW divider structure has been used. The alternative structure is that one presented in paper [Reference Gómez-Guillamón Buendía, Podilchak, Comite, Baccarelli, Burghignoli, Gomez Tornero and Goussetis20] that occupies less space on the PCB structure.

Two versions of the proposed 2D LWA has been designed and fabricated. The first structure (antenna A) uses four dividers connected to four perpendicular edges of the 2D matrix (see Fig. 7(a)); the second structure (antenna B) uses only two dividers, while the two remaining matrix edges are terminated by resistors ensuring the matching of individual lines (see Fig. 7(b)). Each cell at these edges is terminated by two resistors connected in parallel. The antenna A structure enables steering the beam in dependence on feeding from particular ports in the azimuthal plane in the whole interval of 360°. In contrast, the simplified antenna B structure provides beam steering only within 90°. However, as will be shown later, the constructed specimen A shows worse symmetry compared to the constructed specimen B. The problem is in the imperfect fabrication of the power dividers in antenna A. Specimen B utilizes power dividers with improved behavior.

Figure 7. Sketch of the 2D antenna matrix. (a) Antenna A is fed by the power dividers at all four ports. (b) Antenna B is fed only at two orthogonal ports. The antenna matrix is terminated by lumped resistors (two in parallel to each individual cell) at the remaining edges. Dimensions are in mm.

The resistors terminating the edges of the individual mushroom cells have values of 30 Ω. So the termination of these edges is 15 Ω. This corresponds to the Bloch impedance of the mushroom arrays forming the individual rows and columns of the matrix. These impedances were calculated from the ABCD matrix by formula (9). The resistors do not significantly reduce the efficiency of the antenna because the field at the respective edges is due to the leakage relatively weak as the transmission of particular rows and or columns (S ₂₁) of the radiating metamaterial matrix are in the whole frequency band lower than 0.09. This was calculated by the CST MWS and corresponds to the value of the leakage constant from Fig. 5(b). This S ₂₁ can be reduced by increasing a radius of used pins in mushroom cells or by reducing the width of slots between cells. These changes in the cell structure however would detune the dispersion characteristic – frequency of broadside radiation and the band of the antenna matching.

Beam steering in the azimuthal plane is documented in Fig. 8, which shows the simulated distribution of electric field excited by antenna B, depending on the power fed into particular ports. The plots represent radiation patterns calculated by the CST MWS in the Cartesian coordinate system at the 23.4 GHz frequency, offering radiation in the forward direction. The relative feeding amplitudes applied in particular plots in Fig. 8 are as follows: (a) [1, 0], (b) [1, 0.414], (c) [1, 0.708], (d) [1, 1], (e) [0.708, 1], (f) [0.414, 1], and (g) [0, 1]. According to equation (1), the azimuthal angles correspond to 0°, 22.5°, 35.5°, 45°, 54.5°, 67.5°, and 90°. Fig. 8 clearly shows the rotation of the beam main lobe in the azimuthal plane within 90°. The radiation beam is widened when moved to the oblique direction as the antenna is fed by two waves. The beam is even wider at frequency 24 GHz at which the elevation angle is higher, see Fig. 5, than shown in Fig. 8(d). This is due to the superposition of the two particular beams that have maxima at greater distance.

Figure 8. Radiation patterns (directivity normalized to the same maximum, see the scale) of the version B antenna matrix for particular feeding conditions into two ports calculated by the CST MWS at 23.4 GHz. The given antenna ports are fed by the following amplitudes: (a) [1, 0], (b) [1, 0.414], (c) [1, 0.708], (d) [1, 1], (e) [0.708, 1], (f) [0.414, 1], and (g) [0, 1]. The inset shows the position of feeding ports valid for these plots together with the coordinate system.

The radiated field has a linear polarization. The electric field vector direction is parallel to the vector β defined in Fig. 1.

Measurement of 3D radiation patterns

The antenna matrix is generally excited at two ports supplying different power at the same phase. The measurement setup is presented in Fig. 9.

Figure 9. The 3D radiation pattern measurement setup.

The 3D radiation patterns of the antenna under test (AUT) have been obtained in a fully anechoic chamber. It is equipped with a roll-over-azimuth measuring system, where the AUT is mounted on the roll (φ) positioner, which is in turn attached to the elevation (θ) positioner. The AUT is scanned in a 200° azimuthal span (±100°) with a 1° step for the whole 180° φ range to obtain hemispherical radiation patterns. At each measured point, the receiving probe antenna (DRH40 [21]) is rotated to both orthogonal (0° and 90°) positions to obtain precise polarization information from the AUT. The AUT and the probe antenna are separated by a distance of 4 m.

Antenna patterns are measured and computed automatically, employing the NSI2000 far-field measuring software [22] for a predefined frequency span. The measuring system uses the two-port vector network analyzer (VNA) Rohde & Schwarz ZVA40 as a transmitter/receiver. In this setup, the AUT is connected to one port of the VNA, the test probe to the second one, and the S₂₁ transmission coefficient is measured. The gain was determined by the gain-transfer method using the antenna with known gain – DRH40 [21]. The antenna feeding is implemented by means of a two-way power divider (Pulsar PS2-50-450/8S [23]) with a very low phase imbalance (below 6°) and amplitude imbalance (below 0.5 dB). The antenna is connected to the divider by cables with identical (electrical) length. The appropriate amplitude difference at port 1 and port 2 is ensured by an attenuator attached to one port of the AUT, while on the second port, an adapter with same electric length as the used attenuator is mounted.

Experimental verification

Antenna A

The constructed 2D antenna matrix, version A, is shown in Fig. 10. Two ports are connected to a network analyzer, the remaining two are matched. The calculated and measured scattering parameter S _ii is plotted in Fig. 11 for i = 1−4. The drawback of the version A antenna is its lack of symmetry, caused by power dividers that are not identical. The reason for this shortcoming is the imperfect technology process, consisting in drilled holes for SIW sidewall pins. The antenna is well matched around 23.4 GHz. The version A antenna has the same results in simulations of radiated fields as the version B antenna. Therefore, all experiments have been conducted by using the version B antenna (see the next paragraph).

Figure 10. The constructed antenna matrix, version A.

Figure 11. Comparison of measured and calculated (black line) S _ii, antenna matrix version A.

Antenna B

The constructed 2D antenna matrix in version B is shown in Fig. 12. A comparison of the S ₁₁ parameter as measured and as simulated by the CST MWS for the antenna matrix B is provided in Fig. 13. The measurement shows the band of proper antenna behavior, where S ₁₁ < −10 dB between 22.3 and 23.4 GHz.

Figure 12. The constructed antenna matrix, version B.

Figure 13. Comparison of the measured and calculated S ₁₁ parameter, antenna matrix version B.

Fig. 14 shows the evolution of the 3D radiation pattern of the antenna matrix B measured at 23.4 GHz. The antenna is fed from two ports with equal amplitudes of [1, 1]. Individual lines in Fig. 14 are measured at increasing elevation angles, starting from 12° up to 20°, measured from the broadside direction. They represent the radiation pattern measured on a sphere as a function of the azimuthal angle φ taken between 0° and 90°. This evolution corresponds to the 2D radiation pattern from Fig. 8(d). The maximum of the radiation is at 45°, as expected.

Figure 14. Details of 3D radiation patterns measured at 23.4 GHz and at given elevation angles θ varying from 12° up to 20°. The antenna is fed from two ports with equal amplitudes [1, 1].

A comparison of radiation patterns of the antenna matrix B as measured and as simulated by the CST MWS is provided in Fig. 15. The frequency used is 23.4 GHz and the patterns are taken at the 45° azimuthal angle. The antenna is fed by equal powers into its two ports. The measurement corresponds to the simulation, apart from the sharp dip measured at 6°.

Figure 15. Measured and simulated radiation patterns of the antenna matrix version B at 23.4 GHz as a function of the elevation angle θ at an azimuthal direction of 45°. The antenna is fed from two ports with equal amplitudes.

The next experiment has been performed by attaching the −3 dB attenuator into the P1 port, so the antenna is now fed from two ports with amplitudes of [0.708, 1]. Here the azimuthal angle of the maximum radiation according to equation (1) is φ = 54.5°. The evolution of the measured radiation pattern at a feeding of [0.708, 1] with an increasing elevation angel is plotted in Fig. 16. The maxima of the individual patterns correspond to an azimuthal angle of 54.5°. The maximum of the radiation is between elevation angles 12° and 24°. This measured field distribution corresponds to the 2D radiation pattern from Fig. 8(e).

Figure 16. Details of 3D radiation patterns measured at 23.4 GHz and at given elevation angles θ varying from 12° up to 24°. The antenna is fed from two ports with amplitudes of [0.708, 1].

A comparison of radiation patterns as measured and as simulated by the CST MWS for the antenna matrix B is provided in Fig. 17. The frequency used is 23.4 GHz and the patterns are taken at an azimuthal angle of 54.5°. The antenna is fed from two ports with amplitudes of [0.708, 1].

Figure 17. Measured and simulated radiation patterns of the antenna matrix version B at 23.4 GHz as a function of the elevation angle θ at an azimuthal direction of 54.5°. The antenna is fed from two ports with amplitudes of [0.708, 1].

The final experiment has been performed by feeding the antenna from only one port, i.e., [1, 0]. Here the azimuthal angle of maximum radiation is naturally 0°. The evolution of the measured radiation pattern with an increasing elevation angel is plotted in Fig. 18. The maxima of individual patterns oscillate around an azimuthal angle of 0°. The maximum of the radiation at elevation is at around 20°. This evolution corresponds to the calculated 2D radiation pattern from Fig. 8(g).

Figure 18. Details of 3D radiation patterns measured at 23.4 GHz and at given elevation angles θ varying from 12° up to 24°. The antenna is fed from one port.

A comparison of radiation patterns as measured and as simulated by the CST MWS for the antenna matrix B is provided in Fig. 19. The frequency used is 23.4 GHz and the patterns are taken at an azimuthal angle of 0°. The antenna is fed from one port. The measurement corresponds to the simulation, apart from the wavy character of the measured pattern.

Figure 19. Measured and simulated radiation patterns of the antenna matrix version B at 23.4 GHz as a function of the elevation angle θ at an azimuthal direction of 0°. The antenna is fed from one port.

Plots in Figs. 14, 16, and 18 verify the capability to steer the beam by changing amplitudes of signals feeding the specific antenna matrix ports as predicted theoretically in Fig. 8 and according to equation (10).

Fig. 20 plots measured and calculated gain in the broadside direction versus frequency. The measured data fit sufficiently the data simulated by CST. Fig. 21 plots the gain measured at 22.45 and 23.4 GHz as a function of elevation angle, taken at an azimuthal angle equal to 0°. The antenna was in these two measurements excited from only one port, i.e., excitation [1, 0], as in Figs. 18 and 19. The radiation pattern plotted in Fig. 21 at frequency 23.4 GHz copies, except the scale, the patterns plotted in Fig. 19. Figure 21 shows as well the gain measured at frequency 22.45 GHz, where Fig. 20 shows the maximal value that is about 7 dBi.

Figure 20. Measured and simulated antenna gain at the broadside direction as the function of frequency. The antenna is fed from one port [1, 0].

Figure 21. Measured antenna gain at frequencies 22.45 and 23.4 GHz at an azimuthal direction of 0° as the function of the elevation angle. The antenna is fed from one port [1, 0].

The maximum antenna gain is approximately 7 dBi for the antenna fed to one port at frequency 22.45 GHz, see Fig. 21. The radiation efficiency of the antenna estimated from the simulation performed by CST MWS is approximately 35%. The relatively low gain and efficiency values are due to losses in the power dividers used. The other reason of the low gain is in the relatively small radiating area of the presented structure that is the matrix of 8 × 8 mushroom cells of dimensions 5 × 5 mm, i.e., 40 × 40 mm, that is 3 × 3 λ _o at frequency of the maximum gain 22.25 GHz, see Fig. 7(b). This dimension is shown in Table 1.

Table 1. Comparison of the proposed design and other 2D leaky-wave antennas (according [Reference Podilchak, Freundorfer and Antar24])

Table 1 compares the parameters of the presented antenna with data taken from the literature. The antenna array presented in paper [Reference Lai, Leong and Itoh7] is able to steer the beam in the azimuthal plane as well as our design. The paper [Reference Lai, Leong and Itoh7] did not present any parameters shown in Table 1. The paper [Reference Lai, Leong and Itoh7] only proposed the idea. The same is valid for paper [Reference Nguyen, Abielmona, Rennings and Caloz10]. For all the antennas listed in Table 1, the beam steering in elevation is controlled by changing the frequency. That is not the case of paper [Reference Guzmán-Quirós, Weily, Gómez-Tornero and Guo11] where the direction of the main beam in elevation is controlled by DC voltage setting varactors capacities. Our goal was to present a reconfigurable antenna with a radiation pattern controlled by external means, in this case by feeding the structure into individual ports with variable amplitude, not by changing the frequency, which is the feature of standard LWAs. The radiation pattern controlled in azimuth by switching into particular sectors as presented in paper [Reference Zhang, Zhang, Liu, Ma, Yang and Han12] does not however represent the continuous beam steering.

The measured side lobe level (SLL) of the antenna is around −5 dB in most of the presented results. The maximum difference of the side lobe from the main lobe was experimentally obtained around −11 dB, as shown in Fig. 16 at the elevation angle 20° (experiment performed with the attenuator inserted in port P1 – feed [0,708, 1]). The simulation shows the best SLL value of −13.5 dB for an antenna fed only to one edge – feed [1, 0]. The antenna radiation efficiency (simulated) is quite small, approximately 35%. This is due to losses in the power dividers.