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Abstract
An original liquid crystal (LC)-based substrate integrated waveguide (SIW) leaky-wave antenna is proposed. Inside the SIW, there is an embedded stripline sandwiched between an LC pool and another dielectric slab. The antenna couples the guided quasi-TEM mode into free space through a periodic set of complementary electric inductive-capacitive (cELC) resonators. Simulation results show that the antenna performs fixed-frequency continuous beam steering of 52° from backward −28° to forward 24° at 25.85 GHz. This relatively wide beam scan angle is achieved by tuning the LC permittivity through an applied quasi-DC bias voltage to the stripline. Simulation results show that the antenna has high realized gain through the entire scanning range (less than 1 dB degradation), relatively wide bandwidth, and good tolerance to frequency drift and fabrication errors.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Modern communication systems call for low-profile antennas featuring pattern reconfigurability without resorting to complex architectures such as phased arrays and those based on mechanical rotation [1]. For this purpose, simple low-profile beam-steerable antennas are now gaining increasing interest. A beam-steerable antenna focuses the radiated power in a narrow beam at a specific direction (angle of radiation) to increase the spatial resolution and the signal-to-noise ratio. It steers its beam angle to increase coverage and keep track of a larger area or different/moving targets [2]. Reducing the number of components, hardware complexity, and cost, beam-steerable antennas have become the core component in imaging, sensing, and communications, especially at high frequencies [1].
Leaky-wave antennas (LWAs) are inherently beam-steerable, as they feature the well-known frequency-scanning behavior. They radiate when the traveling wave in the guiding structure leaks power into free space and they scan their beam when frequency changes [3]. This simplicity in the feeding and beam-steering mechanism made them attractive for higher microwave and millimeter frequencies [4]. However, the frequency-scanning property may not be suitable for spectrally efficient applications and bandwidth of operation [5]. It also increases complexity in the front-end design in terms of matching [6]. Since frequency controls the traveling wave phase constant ($\beta _0$) in LWAs, it is sufficient to change the phase constant by continuously tunable materials, such as, e.g., liquid crystals (LC) [7–10], graphene [11,12], ferroelectric materials [13,14], piezoelectric actuators [15] or even lumped elements (see, e.g., [16–18] and refs. therein) to make them steerable at a fixed frequency. In this work, we will focus on those architectures based on LC technology.
LCs have low dielectric constants and moderate losses in the GHz range, and they need relatively low bias voltage to be tuned. The thickness of the LC determines the response time: the thinner its profile, the faster its response time [19]. Nevertheless, too thin LC can cause large surface loss, large reflections, and difficulty in impedance matching and power coupling [19]. There are also other techniques to improve the response time [19], such as polyimide alignment layers to anchor the LC molecules in one direction when no bias voltage is applied [20]. This method is especially effective when the LC thickness is below 150 $\mathrm {\mu m}$ [20]. In addition, the alignment layers help to exploit the whole range of LC anisotropy [19].
The LC anisotropy essentially depends on the LC mixture and, in principle, the higher the anisotropy is, the larger the beam angles and coverage will be. Since the anisotropy may not be sufficient to achieve the desired beam steering angle, one needs to consider other approaches. Researchers proposed several techniques to improve the available LC mixtures’ tunability [21–23]. One of the simplest and most versatile techniques for traveling-wave antennas is the meander phase shifter (delay line) due to its planar, and integrable structure [22,24,25].
Simple LC-based meander lines for beam steering microstrip antennas were introduced in [22,24,25] and have limited/moderate performance such as small scan angle, few details on gain (loss), low efficiency, limited impedance bandwidth or too thick LC for an acceptable response time. Reference [24] presents a high frequency (63 GHz) design with the LC thickness of 101 $\mathrm {\mu m}$ and reports a $14^{\circ }$ beam scan angle. In [22], the simulation results show $47^{\circ }$ beam scan angle obtained with delay lines, but losses were not taken into account whereas the performance of an LC-based LWA with delay lines is significantly dependent on the loss. Reference [25] reports an antenna scan angle of $41^{\circ }$ by simulation, and low antenna gain, despite using a relatively low-loss LC mixture. Both [22] and [25] have 250 $\mathrm {\mu m}$ thick LC which adversely affect the response time [19,20].
In this work, we present the theoretical analysis and simulation results of an innovative solution consisting of a substrate integrated waveguide (SIW) LWA based on the complementary electric inductive-capacitive (cELC) resonators [26] as apertures and original meander LC delay lines, which considerably improve the antenna performance with respect to state of the art (see, e.g., [22,24,25]). A careful design of the meander line is the key in our work to obtain unprecedented performance in terms of scanning sensitivity, impedance bandwidth, LC thickness (response time), loss, antenna efficiency, and gain. Besides, cELC shows high capability and flexibility in controlling the coupling strength which is crucial when additional losses are introduced by the LC-based meander line [27].
This article is structured as follows. Section 2 explains the LC electromagnetic model for the full-wave simulations. In Section 3, we comprehensively study the antenna design leveraging concepts from both leaky-wave theory and array theory. Both the unit-cell and its use in a five-elements linear array, as well as the design of delay lines are optimized to achieve an optimal radiating performance. Comparisons against the literature results are reported to corroborate the advancement with respect to the state of the art.
2. Liquid crystal
Nematic LCs are composed of polar rod-shaped molecules with a dipole moment. Therefore, an electric force can change the orientation of LC molecules. More interesting, different molecular orientations result in different electromagnetic characteristics [19]. The electromagnetic properties (behavior) of the nematic LC are modeled as a tensor [19,21,28]
In our design, we considered alignment layers above and beneath the LC slab in order to fix the director along the $\hat {x}$ direction at the boundaries (Fig. 1). By applying the bias voltage to the strip line, the LC director tilts along the $xz$-plane toward the $z$-axis with tilt angle $\psi$ from $x$-axis. Therefore, $\hat {n}$ is
The zero voltage (reference voltage) is applied to the waveguide bottom plate. Note that the voltage applied to the stripline can affect the LC molecules beneath the stripline or close to it, thanks to the fringe field effect. Therefore, in simulations, the LC permittivity is changed only under the stripline. Such an attempt is a good approximation due to the large stripline width compared to the LC thickness. Thus, most of the variation in the phase constant of the traveling wave is well approximated by considering the variation in the permittivity of LC only under the stripline.
Fig. 1. Antenna unit cell and the cELC aperture; the quasi-DC bias voltage is applied between the stripline and the waveguide wall; $h=523\,\mu {\rm m}$, $g=142\,\mu {\rm m}$, $s=533\,\mu m$, $w= 13.476$ mm, $w_m= 13$ mm, $p=5.8025$ mm, $d= 0.2$ mm, $D= 0.3$ mm, $x_1=1.867$ mm, $y_1=3.2$ mm, $x_2=1.333$ mm, $y_2=2.667$ mm, $x_3=y_3=x_4=0.267$ mm, $y_4=0.667$ mm. The alignment layers are not illustrated to have the figure more clear.
The LC mixture used in the design is w-1825 [19], which exhibits very high anisotropy ($\varepsilon _{\perp }=2.78,\varepsilon _{\parallel }=3.82$) and acceptable dielectric losses ($\tan (\delta _{\perp })=0.02, \tan (\delta _{\parallel })=0.01$). To simulate w-1825 in HFSS, we inserted permittivity and loss tangent tensors calculated using Eq. (1) for each specific tilt angle. To represent the antenna performance for different LC tilt angles, only performance for minimum, middle, and maximum LC tilt angles are shown for the sake of brevity. More detailed analyses of the director profile can be performed as discussed in [29,30].
There are mainly three challenges when applying LCs for their tunability in microwave frequencies. The first is the very limited anisotropy, which hinders the tunability of the design. The second is the LC loss which can easily lower the efficiency of the design. The last one is the LC thickness which determines the response time of LC. The thinner the LC, the faster the response time is. However, at microwave frequencies, a thickness in the order of a few hundred of micrometer is already thin enough to cause large surface loss, large reflections, and difficulty in impedance matching and power coupling. At the same time, it is thick enough to increase the response time to more than several seconds [19].
3. Design and simulation results
3.1 Antenna unit cell
The guiding structure of the LWA is a substrate integrated waveguide (SIW) which consists of a dielectric substrate sandwiched between two wide metal conductors that are connected by conductive metal vias at their sides (Fig. 1) [31]. SIWs have low profile, light-weight, low cost, and are easy to fabricate and integrate with planar circuit structures like printed circuit boards (PCBs) [31].
Here, the dielectric substrate in the SIW is a hybrid of an RT/duroid 5880 ($\varepsilon _{r}=2.2$ and $\tan \delta =0.0009$ [32]) slab on top of a w-1825 LC pool (slab), sandwiching a stripline where the RF signal and the bias voltage are applied (Fig. 1). Therefore, the traveling wave is a slow-wave TEM mode with orthogonal electric field intensity and parallel magnetic field intensity to the conductor plates. The phase constant of this traveling wave is $\beta =\omega \sqrt {\varepsilon _{\rm r,\,eff}}/c_0$, where $\omega$ is the angular frequency, $\varepsilon _{\rm r,\,eff}$ is the effective permittivity of the hybrid substrate, and $c_0$ is the speed of light in free space. The relative permittivity of the RT/duroid 5880 is less than the permittivity range of the LC. As a consequence, the LC permittivity has a greater contribution to the effective permittivity $\varepsilon _{\rm r,\ eff}$, thus allowing the phase constant $\beta$ to be more sensitive to the variation of the LC permittivity.
The aperture in the SIW is a complementary electric-LC (cELC) resonator (Fig. 1). The cELC couples the guided magnetic field parallel to the top metal plate of the waveguide to the radiation mode at its resonant frequency. The resonant complementary metamaterial apertures provide high design flexibility in terms of controlling the coupling strength, resonant frequency, and properties of the radiated fields [27]. Using such aperture for fixed-frequency LWAs is beneficial as the designer can easily control the radiation frequency and the coupling strength (leakage) per unit cell.
The electric and magnetic field intensities at the aperture are shown in Fig. 2. The H-field intensity is maximum at the center of the aperture (Fig. 2(b)). This time-varying $H$ field can be modeled as an inductance by Faraday’s law of induction. On the other hand, the electric field resonates at aperture arms (Fig. 2(a)), which can be modeled as a capacitance by the extended Ampère’s law. Therefore, the cELC aperture can be modeled by inductance ($L$), radiation resistance ($R$), and capacitance ($C$) [26]. By changing the geometric parameters of the aperture, $L$, $R$, and $C$ change, and so do the resonant frequency and coupling strength [26,27].
Fig. 2. (a) The electric field resonates at the arms of the aperture and can be represented by a capacitance (b) the time varying magnetic field intensity at the center of the aperture can be represented by an inductance.
The geometric parameters of the cELC are optimized for weak coupling at the operational frequency because strong coupling dissipates an excessive amount of power per unit cell and prevents the power from reaching all apertures in the array. Besides, it can perturb the guided mode, and increase reflections [33]. Since the geometric parameters do not affect the behavior of the cELC independently [27], they are optimized by full-wave numerical analysis.
To realize the design, the authors consider a $381\,\mu {\rm m}$ thick RT/duroid 5880 with an $18\,\mu {\rm m}$ copper cladding. This dielectric slab will be etched on top to realize the cELC apertures and at the bottom to realize the stripline. An alignment layer will be spin coated onto the stripline and mechanically rubbed on a velvet cloth. This dielectric slab will cover a cavity that is surrounded by spacers at the sides and a copper layer at the bottom. An alignment layer is also spin coated onto the copper layer at the bottom, and mechanically rubbed on a velvet cloth. The cavity thickness is $142\,\mu {\rm m}$. This cavity will be filled with liquid crystal and then sealed properly. The through vias will be added at the sides of the waveguide where the spacers are located. The design is indeed unconventional and does not require a thermal press with a high thermal load.
It is essential to keep $|S_{11}|$ as low as possible at each unit cell to avoid large internal reflections in the array. This is primarily important for broadside radiation of the array because the phase difference between antenna elements is $2n\pi$ with $n=0, \pm 1, \pm 2,\ldots$ at broadside. As a consequence, the reflections would add in-phase and degrade the radiation at broadside [34]; a phenomenon commonly known as ’open stop band (OSB) issue’ that notably affects periodic LWAs. The S-parameters of the antenna unit cell are shown in Fig. 3. The $|S_{11}|$ is lower than -13 dB for all LC permittivities (however, only minimum, middle, and maximum LC permittivities are shown in the figure for brevity), and $|S_{11}|$ maximums happen at the cELC resonant frequency (25.85 GHz), whereas the $|S_{21}|$ plots exhibit minima at the resonant frequency. Too low $|S_{21}|$ at resonant frequency means strong coupling and excessive radiation loss. We kept $|S_{21}|$ around -2 dB at resonance, which does not affect much the guided mode and makes it possible to feed multiple elements in the array thus realizing the weak coupling condition, typically used in cELC designs [27]. In this regard, it is worthwhile to note that the weak coupling condition can be considered an alternative technique to mitigate the OSB issue in periodic LWAs (see, e.g., [35]).
Fig. 3. S-parameters of the antenna unit cell for minimum, medium and maximum LC permittivities (LC tilt angles $\psi$s). At the cELC aperture resonance frequency, $|S_{11}|$ and $|S_{21}|$ are at their maxima and minima, respectively. The $|S_{21}|$ level at the resonance frequency represents the coupling strength.
The radiation patterns (realized gain) of the antenna unit cell for all LC permittivities look the same as the one shown in Fig. 4(a), with only slight variations in values. Also considering the same resonant frequency shown in S-parameters, we can say the design successfully maintains similar performance at a fixed frequency when tuning the LC permittivity.
Fig. 4. (a) The normalized realized gain (dB) of the antenna unit cell for LC tilt angle $\psi =0$. Other tilt angles lead to similar results, only slightly different in values, and are not shown for brevity. (b) The realized gain patterns in the E-plane and H-plane. The HPBW is $80^{\circ }$ which helps to maintain high radiation levels at the wide angles obtained by the array.
Figure 4(b) shows the symmetric E-plane (and H-plane) patterns with $80^{\circ }$ half-power beamwidth (HPBW). Such wide beamwidth helps to maintain a high radiation level at wide scan angles obtained by the array.
When the wave travels within the unit cell, its power will be attenuated by the dielectric loss $P_{\rm d}$, the conductor loss $P_{\rm c}$ and the radiation loss $P_{\rm rad}$ which are represented by the attenuation constant $\alpha$ obtained from the S-parameters according to
where $P_{\rm loss}$, $P_{\rm in}$, $P_{\rm ref}$ and $P_{\rm out}$ are the total loss, input, reflected and output power, respectively, and $\ell$ is the length of the unit cell. We simulated the antenna unit cell (LC tilt angle $\psi =0$) with lossless media to obtain the radiation attenuation constant $\alpha _{\rm rad}$ [31] and with all losses to obtain $\alpha _{\rm total}$ from Eq. (8).The attenuation constant for all losses has a maximum at the resonant frequency where the radiation loss is maximum, see Fig. 5. The radiation efficiency can be calculated as $\left ( \alpha _{\rm rad}/\alpha _{\rm total} \right )^{2}$ [36] which returns the $55.4\%$ efficiency at 25.85 GHz. This efficiency is close to the $57.06\%$ radiation efficiency calculated directly from HFSS. However, at frequencies between 25.9 to 26 GHz, results are not reliable as they would lead to radiation efficiencies greater than one. This inconsistency that occurs only in a very few points in frequency, is most likely caused by typical numerical issues of full-wave solvers based on finite element method, as HFSS, namely, the implementation of the perfectly matched layer, and accuracy of the S-parameters, the convergence criterion, etc. These issues can be mitigated at the expense of higher computation resources. With reference to the performance analysis needed in this Section 3.A, the trade-off between the accuracy we reached and the computational burden was satisfactory.
3.2 Antenna array
The unit-cell investigated and optimized in the previous Subsection 3.1. is here used in an array configuration (see Fig. 6). Radiation from this array can also be regarded as a periodic LWA. In particular, the periodicity allows describing the aperture field as a superposition of an infinite number of Floquet waves (space harmonics) with phase constants [4]
where $\beta _n$ and $\beta _0$ are the phase constants of the $n$-th Floquet wave and the fundamental harmonic, respectively, and $p$ is the period. Among all these harmonics, only those with a phase constant less than the free space wavenumber $k_0$ will radiate [4] with the horizontal and vertical wavenumbers satisfying the separation relation $k_0^{2}=k^{2}_y +k_z^{2}$ [4], being $k_x=0$ as we assume the fields to be invariant along the $x$-axis. To have a single-beam scan, we want only one radiating harmonic [4]. Also, the negative phase constant has negative phase velocity and makes backward radiation possible [4]. Hence $\beta _{-1}$ is chosen to be the only radiating harmonic by setting $p=\frac {\lambda _0}{2}$ at 25.85 GHz, which is the resonant frequency of the antenna unit cell. Note that when $p=\lambda _0/2$, the distance between harmonics is $2k_0$ Eq. (9) which equals the radiating condition interval in Eq. (10).Fig. 6. Schematic of the periodic LWA as an array of 5 elements. The coaxial cables, the sealable opening for LC injection and the terminating dielectric sections at each end to hold LC illustrate the potential feeding and fabrication mechanisms respectively. A commercial bias-Tee may be used to decouple the RF signal from the quasi-DC driving voltage.
Figure 7(a) shows the S-parameters of the periodic LWA. At least 3 GHz of matching bandwidth is achieved ($|S_{11}|$ below -10 dB). The $|S_{21}|$ plots show maximum radiation loss at or close to 25.85 GHz, thus confirming that the mutual coupling among apertures in the array is negligible, since the resonance frequency is the same as the frequency of the unit cell.
Fig. 7. (a) S-parameters of the periodic LWA. At least 3 GHz matching bandwidth is achieved. The same resonant frequency as the unit cell shows the negligible mutual coupling between apertures in the array. (b) Realized gain of the periodic LWA. The small beam scan angle of $10^{\circ }$, in the backward direction from $-20^{\circ }$ to $-10^{\circ }$ is achieved.
The beam angles of the antenna can also be obtained by [4]
which results in $-21.7^{\circ }$ and $-11.6^{\circ }$ respectively, close to those obtained from the full-wave analysis (Fig. 7(b)). A summary of the antenna parameters at different beam angles is provided in Table 1.Equation (11) shows that to increase the overall beam angle, we need to increase the phase constant difference ($\Delta \beta _{0}$) achievable by tuning LC. One way is to increase the LC anisotropy ($\varepsilon _{\parallel }-\varepsilon _{\perp }$). However, this property is strictly limited by the LC mixture. Therefore, we chose a different approach based on the array factor [37]:
Table 1. Antenna parameters for the array. $\theta _0$, Max RG, HPBW, SLL, and $e_0$ represent beam angle, maximum realized gain, $-3\,\mathrm {dB}$ beamwidth, sidelobe level, and radiation efficiency, respectively.
Figure 8 shows the patterns calculated using the pattern multiplication principle. There is good agreement with the full-wave results at the main lobe, but the values obtained for the sidelobe levels are lower, which is mainly due to reflections that are considered in the full-wave analysis. These results also prove that the radiating elements are weakly coupled. When the phase constant of the traveling wave varies due to the LC tuning, $\tau$ changes, and so does the AF pattern and beam angle. In Fig. 8, the overall beam angle of $10^{\circ }$ is achieved when $\tau$ changes from $66.46^{\circ }$ to $36.19^{\circ }$ ($\Delta \tau =30.27^{\circ }$) by tuning LC. However, a larger $\Delta \tau$ will result in a larger overall beam scan angle.
Fig. 8. Excellent agreement between HFSS and the pattern multiplication principle at the main lobes. The difference in the side lobes is due to the reflection and backward propagation wave. The results also prove that the radiating elements are weakly coupled or nearly uncoupled.
In Fig. 9, patterns for different $\tau$ values are shown, all calculated with Eq. (12), and the unit cell realized gain pattern for LC tilt angle $\psi =0^{\circ }$ (Fig. 4(b)). This figure gives us an excellent estimate of the array behavior when changing $\tau$. As expected, the main lobe is attenuated when $\tau$ goes beyond $\pm 120^{\circ }$. This occurs because the beam angle approaches $\pm 40^{\circ }$, i.e. the angles corresponding to the half-power beamwidth of the unit cell pattern. Furthermore, when the beam angle goes beyond $\pm 22^{\circ }$ ($\tau =\pm 75^{\circ }$), the sidelobe levels get considerably larger. Note that the side lobes are even larger according to full-wave results, due to the inclusion of reflections. Therefore, the $\tau$ variation from $-75^{\circ }$ to $75^{\circ }$ is the design target to produce a phase variation that steers the beam angle from $22^{\circ }$ to $-22^{\circ }$.
Fig. 9. Patterns calculated by Eq. (12) and the unit cell realized gain pattern for LC tilt angle $\psi =0^{\circ }$. Since the results are symmetric with respect to $\theta =0$, only $\tau \geq 0$ are shown.
3.3 Phase difference booster (PDB)
The phase difference between two adjacent elements in the array is $\tau =-\beta _0 d$, where $d$ is the spectral distance the wave travels between those elements. Therefore, by increasing $d$, $\Delta \tau$ caused by LC tuning will increase. Since for $p>\lambda /2$ grating lobes may appear, we packed more striplines in the same period to create a phase difference booster (PDB) (Fig. 10).
When increasing $d$, the design goals for the PDB are threefold: (i) the change of $\tau$ ($S_{21}$ phase) from $75^{\circ }$ to $-75^{\circ }$ by means of LC reorientation. (ii) the minimization of the reflections ($|S_{11}|$). To keep reflections low, the corners are designed to act like a $45^{\circ }$-tilted mirror and guide the wave at $90^{\circ }$-twists with minimum reflections. (iii) the loss per unit cell ($|S_{21}|$). As the travel length of the wave $d$ increases, the dielectric and conductor losses will increase, and excessive loss per unit cell will exhaust the power and prevent feeding the last elements in the array.
Fig. 10. The top view schematic of the phase difference booster unit cell (PDB). $d_1= 5.4$ mm, $d_2=2.5$ mm, $d_3=1.601$ mm, $d_4=1.7$ mm.
All these parameters for the optimized design (Fig. 10) are shown in Fig. 11(a) and Fig. 11(b). $S_{21}$ phase changes from $75^{\circ }$ to $-75^{\circ }$ by LC tuning at 25.85 GHz. When increasing $d$ and/or LC tilt angle $\psi$, the $S_{21}$ phase reduces periodically at each frequency. We increased $d$ untill the $S_{21}$ phase for $\psi = 0^{\circ }$ was around $75^{\circ }$ and by increasing $\psi$ to $90^{\circ }$, $S_{21}$ phase decreases continuously to $-75^{\circ }$. The PDB also shows negligible reflections ($|S_{11}|$ below $-20$ dB) and maximum power loss ($|S_{21}|$) of $2.3$ dB per unit cell for $\psi = 0^{\circ }$.
Fig. 11. (a) $d$ is increased till the $S_{21}$ phase for LC tilt angle $\psi = 0^{\circ }$ was around $75^{\circ }$ and by increasing $\psi$ to $90^{\circ }$, $S_{21}$ phase decreases continuously to $-75^{\circ }$. (b) S-parameters of the optimized PDB. $|S_{11}|$ is below $-20$ dB which means negligible reflections, and $|S_{21}|$ shows maximum power loss of $2.3$ dB per unit cell for $\psi = 0^{\circ }$.
The PDB design helped the antenna array (Fig. 12) to increase its overall beam angle from $10^{\circ }$ to $52^{\circ }$ (see 3D patterns in Fig. 13(a) and E-plane patterns in Fig. 13(b)). The antenna can steer its beam quasi-symmetrically from backward ($-28^{\circ }$) to forward ($24^{\circ }$). This shows great agreement with the approximation made by Eq. (12) in Fig. 9 and the successful implementation of the idea by PDB. Compared to the design without PDB, the maximum realized gain is reduced by an average of 2.5 dB (compare Fig. 7(b) and Fig. 13(b)) because the wave is experiencing more media loss by traveling a longer distance.
Fig. 13. (a) 3D normalized radiation pattern of the periodic LWA with PDB. (b) Realized gain of the periodic LWA with PDB at E-plane. Minimum of 7 dB and maximum of 7.7 dB is obtained; less than 1 dB decrease during the whole beam scan. The minor sidelobes are in the order of 8 dB and the 3-dB beamwidth of the antenna is $20^{\circ }$.
The maximum realized gain of the antenna changes from a minimum of 7 dB to a maximum of 7.7 dB, which shows less than a 1 dB decrease during the whole beam scan. The minor sidelobes are in the order of 8 dB and should be mitigated when finalizing an application for this antenna. The HPBW of the antenna is $20^{\circ }$ which helps the antenna to exhibit a wider bandwidth [4] because minor displacement from the operational frequency will not change significantly the beam angle. A summary of the antenna parameters at different beam angles is provided in Table 2.
Table 2. Antenna parameters for array with PDA.
The simulated antenna with PDB is expected to have different gain values from the theoretical multiplication principle due to the following reasons: first, the PDB loss is included in the simulation but not in Fig. 9. Second, 1-D periodic LWAs as those studied here have the largest reflections at broadside due to the presence of an open stopband. This is not considered in the theoretical multiplication principle. Third, the simulation takes coupling between elements and reflections into account, but the theoretical calculation does not.
When at broadside, the reflection is maximum (LC tilt angle $\psi =48^{\circ }$ in Fig. 14(a)) because the cELC apertures are in phase, and therefore reflections at each unit cell will be added constructively. This lowers the efficiency of the antenna when compared to other beam angles (Table 2). However, the design is successful in keeping this maximum reflection low enough to maintain high realized gain. In fact, broadside radiation degrades only by 0.6 dB from the maximum radiation at $24^{\circ }$, thus confirming that the open stopband issue has been effectively mitigated. In addition, the matching bandwidth is at least 3 GHz, which is rather wide for typical LWA designs.
Fig. 14. (a) Reflections of the periodic LWA with PDB. The matching bandwidth is at least 3 GHz. (b) Losses of the periodic LWA with PDB. The falls are not sharp and the antenna should be able to radiate well from 25.7 GHz to 26 GHz.
The dips in $|S_{21}|$ plots (Fig. 14(b)) show radiation loss at and around 25.85 GHz. The lower $|S_{21}|$ for LC tilt angle $\psi =0^{\circ }$ at non-radiating frequencies show higher LC loss when no bias is applied. This is expected due to dichroism, which lowers the antenna efficiency compared to the maximum bias state ($\psi =90^{\circ }$) (Table 2). Another fact is that the dips are not sharp, and the antenna radiates well in the -3 dB gain bandwidth (operating bandwidth) which is about 500 MHz (25.55-26.05 GHz).
We also calculated $\alpha _{\rm rad}$ and $\alpha _{\rm total}$ by simulation of the antenna without material losses and with total losses respectively and Eq. (8) (see Fig. 15). The radiation efficiency $\alpha _{\rm rad}/\alpha _{\rm total}=33\%$ [36] at 25.85 GHz is obtained for LC tilt angle $\psi =0^{\circ }$, which is close to the efficiency of $35.15\%$ computed by HFSS. Such an agreement further corroborate the effectiveness and validity of the leaky-wave interpretation of this structure, in spite of the high values featured by the attenuation constants. Indeed, the leaky-wave theory is supposed to be accurate as long as the leakage rate is lower than 0.2, approximately [38], whereas it reaches values as high as 0.5 due to the non-negligible losses of the LC. We also note that such high values of the leakage rate contribute to having a relatively short LWA made of five elements (a few wavelengths at 26 GHz) compared to conventional designs which typically feature antenna lengths of several wavelengths. Still, the aperture efficiency of the investigated structure, i.e. $\eta =1-e^{-2\alpha _{\rm total}\ell }$ is greater than 99.99%. In this sense, the structure is long enough to be used as an LWA whose size is usually set to have at least 90% of aperture efficiency [4].
Fig. 15. The normalized attenuation constants for the antenna array with PDB. The results belong to the simulations with LC tilt angle $\psi =0^{\circ }$.
Figure 16(a) shows the traveling E-field magnitude at the $xy$ plane located in the middle of the LC slab ($z=g/2$). The field is calculated at 25.85 GHz when $\psi =90^{\circ }$ which has the least LC loss. The concentration of the field under the stripline shows low perturbation for the guided wave, and the decaying magnitude shows that the field is weak at the last aperture.
Fig. 16. (a) E-field magnitude at the $xy$ plane located at the middle of the LC slab ($Z=g/2$); frequency is 25.85 GHz and $\psi = 90^{\circ }$. (b) Surface current density at the apertures show the weak dipolar resonant nature of this subwavelength radiation aperture.
In addition, the surface current density vectors at the apertures in Fig. 16(b) show the weak dipolar resonant nature of the subwavelength radiation apertures.
The antenna also shows tolerance to errors in frequency. Patterns at different frequencies for the LC tilt angle $\psi =0^{\circ }$ are shown in Fig. 17(a). Accepting a 2 dB decrease in maximum realized gain, a 300 MHz frequency tolerance is obtained for all three LC states (only LC tilt angle $\psi =0^{\circ }$ is presented for brevity). Plus, the antenna radiation is considerably attenuated at larger frequency differences; therefore, the antenna beam will maintain approximately its designed and intended angle or will be considerably attenuated when drifting from the operational frequency.
Fig. 17. (a) The frequency error tolerance for LC tilt angle $\psi = 0^{\circ }$ is 300 MHz. Radiation is significantly attenuated for more than 1 GHz of frequency difference. (b) 10% error tolerance for LC thickness.
In addition, the antenna shows tolerance to errors in thickness. Patterns of a design with 10% larger LC thickness, for three different LC states are shown in Fig. 17(b). The patterns are approximately similar to those with smaller thickness.
Table 3 compares our results with the state of the art which utilizes LC-based delay lines to steer the beam of the LWA [22,24,25]. We included the LC $\Delta \varepsilon _r$, LC $\tan \delta$ and the frequency in the table as the effective parameters in the beam steering angle, the realized gain and the response time of the LC layer ($\sim$ LC thickness) respectively. It is important to notice that the larger beam scan angle requires a longer delay line which increases the loss and decreases the realized gain. Our work shows significant improvement by making an excellent balance in this trade-off. For example, our beam scan angle is nearly 4 times the beam scan angle in [24] with a higher realized gain and our matching fractional bandwidth is $11.6\%$ vs. $3.33\%$ in [24]. In [22] the beam steering angle was increased by 56% (from $30^{\circ }$ to $47^{\circ }$) by adding a meander line. In our work, thanks to the meander line, the beam steering angle improves by 420%. However, loss and gain are not discussed in [22] to be able to do the comparison. In addition, our impedance bandwidth is 5 times wider than that reported in [22]. Also in comparison with [25], our larger beam scan angle is illustrated with higher realized gain although the LC mixture in our work is lossier. We applied the same LC as in [25] to our design and obtained a realized gain of 9 dB at beam angle $22^{\circ }$ and 9 dB at $-22^{\circ }$ at 25.9 GHz (3 dB to 4 dB more than [25]). Besides, [22] and [25] use 250 $\mu {\rm m}$ thick LC which escalates the response time or bias voltage and power consumption considerably. Please note Refs. [22] and [25] use different values for the LC modeling in simulations and measurements and we compare our results with their simulation results (their best results), and even in this case we achieve better performance. As mentioned before, our work also shows tolerance to frequency drifts and fabrication errors.
Table 3. Comparison with other LC-based LWAs that utilized LC delay lines in their simulations.
4. Conclusion
An original liquid crystal (LC)-based SIW-LWA is proposed. A stripline is embedded in the SIW and is sandwiched between an LC pool and another dielectric slab. The stripline guides a slow wave TEM mode which leaks into free space through a periodic set of complementary electric LC resonators (cELCs). The cELCs provide high flexibility in radiation characteristics and hence help promoting the fixed-frequency beam-steering performance by LC. The antenna continuously scans a wide angle of $52^{\circ }$ (from backward $-28^{\circ }$ to forward $24^{\circ }$), including broadside, at 25.85 GHz by tuning the LC permittivity through an applied quasi-DC bias voltage to the stripline. The antenna keeps high realized gain through the whole scan, even at broadside and extreme angles, wide matching bandwidth, and tolerance to frequency and LC thickness errors. Our work proposes a state-of-the-art design by making an excellent balance among beam angle, loss, and response time trade-off in LC-based LWAs and can be exploited for applications to 5G networks, radar, and autonomous vehicles, just to name but a few.
Funding
King Abdullah University of Science and Technology (CRG8-4056.2).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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