Magnetic field assisted beam-scanning leaky-wave antenna utilizing one-way waveguide

We propose a Leaky-Wave Antenna (LWA) based on one-way yttrium-iron-garnet (YIG)-air-metal waveguide. We first analyze the dispersion of the LWA, showing the one-way feature and the radiation loss. Owing to the unique one-way dispersive property, the beam radiated from the LWA can have very narrow beam width, at the same time having large scanning angle. The main beam angle obtained by full-wave simulation is consistent with our theoretical prediction with the aid of the dispersion. For a given frequency, we can realize continuous beam scanning by varying the magnetic field, where the 3 dB beam width is much narrower than previously demonstrated. Our results pave a new way to realize continuous angle scanning at a fix frequency for modern communications.


Results
Dispersion properties of one-way waveguide and LWA. In order to investigate the proposed LWA, we first analyze the dispersion property of the YIG-air-metal structure, as illustrated in Fig. 1(c). The thickness of the air layer is denoted by d, and the thickness of the YIG is assumed to be semi-infinite. With the static magnetic field applied in the −z direction, the YIG in the waveguide is gyromagnetic anisotropic with the relative permittivity ε =15 where ω is the angular frequency, ω m is the characteristic circular frequency, ω πγ = 2 H 0 0 (γ = 2.8 × 10 6 rad/s/G is the gyromagnetic ratio) is the precession angular frequency, and = γ ω ∆ v H 2 (ΔH is the resonance linewidth) is the damping coefficient 28 . Such a two-dimensional (2D) waveguide can support both SMPs and the regular mode, and their dispersion relations are given by 2 for the regular mode 23,26 . The linear term with respect to k in Eq. (2) indicates that both modes are non-reciprocal. The dispersion relations for SMPs and the regular mode in the YIG-air-metal structure are numerical calculated, and the results are shown the dashed lines and solid line in Fig. 1(a), respectively. Here, we assume the YIG medium to be lossless (Δ = H 0 G) with ω π = × 10 10 m 9 rad/s ( = f 5 GHz m ), and ω 0 is set at ω m , which corresponds to = H 1785 G 0 . The thickness of the air layer is fixed as = d 1 mm to gain large band for the one way propagation. As seen in Fig. 1(a), there exists a one-way propagation band (the middle shaded area) in the bandgap of the magnetized YIG, whose bulk modes (the uppest and lowest shaded areas) are given by ε μ < k k m v 2 0 2 . The one-way region ranges from ω sp to ω ω + m 0 , equivalent to the frequency region . is the asymptotic frequency of SMPs at → −∞ k . In this region, the modes are allowed to propagate only in the forward direction due to the group velocity of > dw dk / 0 . To further verify its one-way guiding property, we perform the simulation of wave transmission with the finite element method (FEM) using COMSOL Multiphysics. In the simulation, the metal in the system was assumed to be a perfect electric conductor, and a linear magnetic current source located on the center of the air layer was used to excite wave. Figure 1(b) shows the simulated electric field amplitudes for = f 9 GHz. Evidently, the electromagnetic wave can only propagate in the forward direction as expected.
Then, we analyze the dispersion property of the proposed LWA, which is formed by the YIG-air-metal waveguide with periodic holes in the metal layer, as illustrated in the left panel of Fig. 2(a). The period of the unit cell www.nature.com/scientificreports www.nature.com/scientificreports/ is denoted by p, and the width and depth of the hole are denoted by w and h, respectively. This 2D LWA can support the transverse electric (TE) mode whose electric field is polarized along the z direction. With respect to a practical device, a 3D LWA, see the right panel in Fig. 2(a), is proposed with the finite width of the antenna, sandwiched by two metal slabs in the z direction. By solving the eigenfrequency problem, the dispersion relation of the LWA was calculated with FEM. Note that the performance of the LWA is strongly dependent on its structural parameters. In this work, our interest focuses on analyzing the dependence of continuous scanning property on the non-structural parameters, such as the frequency and applied magnetic field. As an example, the parameters of the periodic unit are = w 6 mm, = h 1 mm, and = p 12 mm, respectively. The leaky mode lies within the light cone (indicated by the dot-dashed lines). In the whole first Brillouin zone, the modal group velocity ( ω β d d / ) is always positive, which indicates the one-way propagation behavior. This is confirmed by the results shown in Fig. 4. The dispersion for the leaky mode lies in the one-way region for the YIG-air-metal waveguide, see the middle shaded area in Fig. 2(b). We also calculate the dispersion relation for the 3D system with the waveguide width of = L 5 mm, and the obtained results (see circles) agree well with those for the 2D system. The loss, associated with the radiation, for the leaky mode is illustrated in Fig. 2(c). Note that in our calculation we do not take the material loss into account. The radiation loss changes significantly when tuning the frequency, and shows the maximum around β = 0. The electric field distribution in the unit cell at β = 0 is also displaced in the inset of Fig. 1(b), indicating the leaky feature. For the given structure parameters, the frequency range of the LWA is .
. [8 2, 9 35] GHz. The two intersections between the light lines and the dispersion curve imply that we can control the beam radiation angle from −90° to 90° when changing the frequency. Especially, at the frequency of = .
f 8 64 GHz, β becomes almost zero, meaning that this mode can radiate at broadside. As illustrated in Fig. 2(b), the phase of the leaky mode can be continuously changed by tuning the frequency. For our proposed LWA, compared to high-order harmonic, the radiation by the 0th fundamental harmonic is predominant, and the beam angle for the 0th harmonic is given by peak 0 According to Eq. (3), when β − ≤ < k 0 0 , the backward beam scanning ( ϕ −°≤ <°90 0 peak ) can be obtained; whereas the forward beam scanning ( ϕ°< Frequency scanning LWA. To verify the continuous frequency scanning feature of the LWA, we calculate the electric field distributions and far-field radiation patterns by the full-wave simulation. In the simulation, wave www.nature.com/scientificreports www.nature.com/scientificreports/ is first coupled into the YIG-air-metal waveguide at its left side, see Fig. 2(a), and when it travels forward within the waveguide, it gradually radiates to the free space through the periodic holes, and the residual wave exits at the right side. The number of the periodic hole, denoted by N, usually needs to be sufficiently large to achieve high directivity, and here it is fixed as = N 50. Figure 3 , and it gradually decreases in the forward and backward direction when tuning the frequency. Figure 3(b) shows the radiation efficiency of LWA. This result agrees well with the result for the radiation loss shown in Fig. 2(c), and the maximum radiation energy obtained at ϕ =°0 peak is also consistent with the result that the radiation loss is largest when β = 0 at = .   f 9 21 GHz. Moreover, we also calculate the values of ϕ peak with Eq. (2) for various frequencies, as seen the circles in Fig. 3(c). Obviously, this analytic result is in good agreement with those obtained by our full-wave simulations.
Besides, we systematically analyze the radiation pattern of LWA on different losses ΔH. It can be found from Fig. 3(d) that with the increase of ΔH from 0 to 20 G, the radiation angle almost remains unchanged and the 3 dB beam width increases slightly; meanwhile, the gain of LWA decreases from 25 dBi to 12.3 dBi. As an example, ϕ =°0  Fig. 4(a-c), respectively. Evidently, the electromagnetic wave can only propagate forward as expected when placing a line current source in the periodic structure of LWA, which is in a good agreement with the one-way property of the leaky mode in Fig. 2(b). More importantly, the wave radiates toward different directions from backward to forward when increasing the frequency, as shown in Fig. 4. Therefore, the LWA exhibits continuous frequency-scanning from the backward to forward directions with the large scanning angle and narrow 3 dB beam width. It should be noted that in the absence of the applied magnetic field, the LWA will lose the continuous scanning property because it does not support any guided wave.
Fixed-frequency scanning LWA. For the LWA discussed above, the value of the magnetic field is fixed at Compared to the frequency-scanning LWA, the frequency-independent LWA is preferable for applications in modern communication systems. Here, the property of the proposed LWA strongly depends on the applied magnetic field H 0 due to the usage of the magneto-optical material. To investigate the influence of H 0 on the dispersion of the LWA, we calculate the dispersion relations for the leaky modes at various H 0 , as shown in Fig. 5(a). It can be seen that the dispersion curves shift up when increasing H 0 . We emphasize that when tuning H 0 the dispersion for the leaky mode always lie in the one-way region for the waveguide. Here we choose a fixed frequency of = f 9 GHz 0 as an example, see the horizontal dashed line in Fig. 5(a). When tuning the magnetic field from 1646 G to 2108 G, we can control β changing from β = k 0 to −k 0 , see the two circles in Fig. 5(a). This implies that the beam angle for the LWA in principle can be realized from −90° to 90° by tuning the magnetic field for the given frequency. Especially, when = H 1934 G 0 , β becomes zero, meaning that this mode can radiate at the broadside for the fixed-frequency LWA.
To verify the continuous fixed-frequency scanning feature of the LWA, the far field radiation patterns at different H 0 values for = f 9 GHz 0 are shown in Fig. 5(b). It is clearly seen that the continuous scanning behavior of www.nature.com/scientificreports www.nature.com/scientificreports/ the LWA can be realized by tuning the magnetic field from 1650 G to 2100 G. The antenna gain is found to be maximum at = H 1934 G 0 with the beam angle of ϕ =°0 peak , and it gradually decreases in the forward and backward direction when tuning the magnetic field, which is similar to that for the frequency scanning shown in Fig. 3(a). ϕ peak and ϕ Δ dB 3 versus H 0 are illustrated as the solid and dashed lines in Fig. 5(c), respectively. The results for ϕ peak are in a good agreement with those obtained by Eq. (3) for various H 0 , as seen the circles in Fig. 5(c). The LWA has a narrow 3 dB beam width of ϕ Δ ≤°5 . Besides, we also calculate the S-parameters of the LWA for two different N values: = N 50 (the solid line) and 100 (the dashed line), as illustrated in Fig. 5(d). The transmission coefficient S 21 is found to be minimum around = H 1934 G 0 , which is consistent with the results shown in Fig. 5(b). Note that in our proposed LWA, S 21 (dB) decreases linearly with N and S 11 is always zero due to the suppression of the reflected waves. We also evaluate the radiation pattern for the 3D model, see the right panel of Fig. 2(a), when we take the loss (Δ = H 5 G) into account. The width of 3D waveguide is 5 mm in the z direction. It can be observed from Fig. 6 that the normalized far-field radiation patterns of 2D and 3D LWAs in the xy plane are in good agreement. Table 1 shows the comparison of performances between the proposed one-way waveguide based LWA and several published LWAs. Liu et al. 7 , and Paulotto et al. 11 reported different kinds of frequency scanning LWAs, whose maximum beam-scanning ranges are 40° and 89°, respectively. Different fixed-frequency scanning LWAs are reported in 12,14 , whose maximum beam-scanning ranges are 21° and 104°, respectively. Compared with these LWAs, the proposed LWAs exhibit larger scanning angle (>107°) and smaller 3 dB beam width (<5°) for both frequency scanning and fixed-frequency scanning. Moreover, this structure is relatively easy to be realized in practice, when comparing to the previous ferrite-loaded LWA 27 .

Conclusions
In conclusion, we have proposed a LWA based on one-way YIG-air-metal waveguide with periodic holes under a static external magnetic field. The dispersion and radiation properties of the LWA have been analyzed, showing that the leaky mode supported by the LWA exhibits one-way feature. With the aid of one-way waveguide, the radiated beam by the LWA can have narrow 3 dB beam width, while at the same time having large scanning angle. The main beam angle obtained by the full-wave simulations are confirmed by our theoretical prediction. More importantly we have realized continuous beam scanning by varying the magnetic field at a fixed frequency. Our results demonstrated here show a promising way to realize continuous angle scanning for modern communications.