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Abstract
We propose to collimate the reflected electromagnetic fields from convex reflectors by using meta-surfaces. More specifically, both umbrella-shape and polygon prism convex reflectors are shown to be capable of generating highly directive pencil-beams and fan-beams, respectively, when integrated with meta-surfaces offering additional gradient phase distributions over the reflecting boundaries. Steerable beams from both reflectors are also verified from the full-wave simulations. Finally, we practically implement such meta-surface based convex reflectors loaded with properly devised subwavelength ring patches and demonstrate their functionalities of tuning reflected electromagnetic beams with greatly enhanced reflections. Our approach of generating well collimated electromagnetic fields from convex reflectors would be applicable to conformal components on the non-planar platform for the highly directive radiations.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Parabolic reflectors have widely been used to collimate electromagnetic fields and generate highly directive radiations due to the intrinsic geometry characteristics, where all rays from the focal point would obtain the equal travelling paths when reflected [1]. On this basis, metamaterial-based planar reflectors with comparable electromagnetic properties to the traditional parabolas have been proposed and applied to conformal applications, offering immediate benefits of flat profile and small volume [2–4]. Convex reflectors, on the other hand, generally diverging lights and reflecting the incoming waves outwards, are usually employed in different scenarios with specific applications including the subreflector in the cassegrain reflector system and wide angle imaging mirrors in optical area. However, there is always demands from convex shaped platforms to collimate electromagnetic fields, whereas the classic parabolic and planar reflectors turn out to be hardly appropriate to mount on.
Conformal antenna arrays have thus been intensively studied with such a concern in the electromagnetic area since the 1950s, where Harrington et al. proposed a general solution to estimate the radiation patterns of the ideal continuous current sheets on the cylindrical reflector [5]. The microstrip phase array is then practically implemented on the cylinder as radiators with the benefits of low profile and easy installation [6], and further been extended to the conformal antennas mounted on multilayered cylindrical structures with arbitrary oriented patches to synthesize the desired radiations [7]. Recently, meta-surface has proved itself as a promising candidate for building up novel electromagnetic devices, and offered a straightforward design to manipulate electromagnetic fields simply at surfaces with advanced applications. For example, meta-surfaces with abrupt phase discontinuities enable us to redirect propagating waves while transforming the wavefront, thus fulfill the imagination of human beings to become invisible when the scatters are cloaked [8–15], anti-reflected with perfect absorptions [16,17], or camouflaged by creating illusions [18,19]. Moreover, meta-surface has demonstrated the capacity of decoupling the geometry profile of the antennas and their radiation performances, and initiated the quest for tangible applications in the antenna community with anomalous reflections and refractions [20–28]. In this paper, we propose to use an umbrella-shape and a polygon prism convex reflectors to generate the highly directive pencil-beams and fan-beams respectively. We will demonstrate that the meta-surface coating with properly devised subwavelength ring patches will equip the convex reflectors with functionalities of collimating the electromagnetic fields rather than diverging, and the present approach should pave the ways to construct novel conformal antenna arrays.
2. Design and numerical results
Let us start with the ray analysis for the convex reflecting surface as shown in Fig. 1(a). The bare convex reflector diverges the incident wave outwards at the metallic boundary, leading to the fact that conventional convex surfaces cannot normally be used to collimate the electromagnetic beams. However, if we could modify the reflecting phase of the lights from the convex boundary to specific values using the meta-surface, the rays could be well collimated. For instance, light SP1 and SP2 could be redirected as light P1A and P2B with anomalous negative reflections, respectively, whereas light SP3 arriving at the bare ground would be directly transformed as light P3C with the specular reflection. In order to obtain equal electrical light path on the reference plane, we can have:
where the φref and φs refer to the constant phase of the reference plane and the source at the certain moment respectively, and φ(P1) and φ(P2) refer to the abrupt phase changes from the meta-surface on the convex boundary based on the generalized Snell’s law [20]. Since the φref − φs is a constant and points P1 and P2 are the generalized points on the meta-surface, we could obtain the phase distributions of the point P with coordinate value of (x, y, z) as where k refers to the wave number in free space, r refers to the radius of the curvature, f refers to the focal length, and z component can be represented by x, y, and r. Fig. 1(b) and Fig. 1(c) thus illustrate the phase distributions on the reflecting boundaries of two types of meta-surfaces. One is spherical shape with the focal length of 90 mm, and the other is cylindrical shape with the focal length of 60 mm. Both surfaces are based on the ideal circular curvature with the radius of 195 mm. The phase distributions over the spherical surface have six sets of periodic concentric circles from center point to the border on the reflecting aperture of the meta-surface, while the phase distributions over the cylindrical surface are similar, and have three groups of strip-like cycles. The incoming fields from the focal points should be well collimated with such phase distributions over the convex reflectors, and the spherical and the cylindrical meta-surfaces are thus generating highly directive pencil-beams and fan-beams, respectively.
Fig. 1 Ray analysis from the convex reflectors and phase distributions over the corresponding meta-surfaces to generate highly directive radiations at 20 GHz. (a) Ray tracing from bare convex ground and the convex curvature loaded with meta-surface. (b) The phase distributions over the spherical meta-surface with the focal length chosen as 90 mm. (c) The phase distributions over the cylindrical meta-surface with the focal length chosen as 60 mm.
In order to practically implement these two reflectors, we discrete the ideal convex surfaces to form an umbrella-shape reflector and a polygon prism reflector, respectively, and devise the meta-surfaces of subwavelength ring patches to cover the proposed reflectors. We expect to make the discrete structures closer to the ideal curvature so that to sample the obtained phase distributions in Fig. 1 from both of the convex reflectors. As shown in Fig. 2(a) and Fig. 2(b), the umbrella-shape reflector consists of an octagon in the center and eight groups of bending surfaces with two planar isosceles trapezoid faces, while the polygon prism reflector has nine identical rectangle surfaces connecting to each other. We employ the subwavelength circular ring patches as the coating meta-surface to cover the umbrella-shaped reflector due to the symmetric profile along both the x-axis and the y-axis, while for the polygon prism reflector, we choose the rectangle ring patches instead. Considering the fact that the electromagnetic responses of the designed ring patches vary with the incidence angles, we divide every discrete surface of the reflectors into sub-areas as shown in Fig. 2(a) and Fig. 2(b), and obtain five sub-areas with the average incident angle of 0°, 13°, 32°, 50°, and 67° for the umbrella-shaped reflector and of 0°, 18°, 33°, 46°, and 57° for the polygon prism reflector. Fig. 2(c) and Fig. 2(d) demonstrate the unit cell of the subwavelength circular and rectangular ring patches and the corresponding phase simulation models. The subwavelength ring patches are printed on the FR4 (εr = 4.4 and loss tangent 0.02) grounded slabs of 0.5 mm thick and the unit cell dimensions are 3.0 mm × 3.0 mm and 1.8 mm × 3.6 mm for the circular and for the rectangle ring patches, respectively. The phase simulations are performed with the unit cell of a single circular or rectangle subwavelength ring patch under the classical Floquet mode simulator using the periodic boundary condition. The structural parameters R and W1 of the subwavelength circular ring patch and D and W2 of the subwavelength rectangular ring patch are functioning as the tuning parts to regulate the reflecting phases when under the illuminations of plane waves with different incident angles. More specifically, the subwavelength ring patch would produce the specific electromagnetic responses that can be retrieved from reflection coefficients when we modify the structural parameters of the rings. Figure 2(e) and Fig. 2(f) thus demonstrate the phase distributions of the two types of ring patches under different structural dimensions at 20 GHz when considering the different incident angles. As we can observe, the reflecting phases are able to cover the required phase distributions of 2π range for the convex reflectors when the structural dimensions of the rings are changed. Therefore, we can finally build up the umbrella-shape and polygon prism convex reflectors with meta-surfaces of subwavelength ring patches with specific structural dimensions according to the ideal phase distributions shown in Fig. 1(b) and Fig. 1(c).
Fig. 2 Configurations of the meta-surface based convex reflectors and the relationships between the reflecting phases and the structural dimensions of the subwavelength ring patches at 20 GHz. (a) The umbrella-shape convex reflector with meta-surface of subwavelength circular ring patches. (b) The polygon prism convex reflector with meta-surface of subwavelength rectangular ring patches. (c) The unit cell of the subwavelength circular ring patch and the corresponding phase simulation model. (d) The unit cell of the subwavelength rectangular ring patch and the corresponding phase simulation model. (e) The relationships between the reflecting phases and the subwavelength circular ring patches with structural parameters R and W1 when considering the different incident angles. (f) The relationships between the reflecting phases and the subwavelength rectangular ring patches with structural parameters D and W2 when considering the different incident angles. We choose L=2D for simplification.
We now perform the full-wave simulations (CST Microwave Studio) to verify the proposed designs. Fig. 3 illustrates the radiation performances of the convex reflectors with and without meta-surfaces, fed by the standard WR51 waveguide at the focal points. We can observe that the highly directive pencil-beam and fan-beam are generated from the meta-surface based umbrella-shape and polygon prism convex reflectors respectively, while the bare convex reflectors only diverge the electromagnetic fields. The maximum gains of the umbrella-shape and polygon prism convex reflector are 22.5 dBi and 15.6 dBi, which are 21.8 dBi and 12.3 dBi higher than the two bare convex reflectors respectively. The umbrella-shape reflector has the 3-dB beamwidth of 6.2 degrees in H-plane and 7.1 degrees in E-plane, while the polygon prism reflector has the 3-dB beamwidth of 6.8 degrees in H-plane and 30.7 degrees in E-plane. We can also observe that both the umbrella-shape and polygon prism convex reflectors obtain the satisfactory reflection coefficients, which are both less than −10 dB from 19 GHz to 21 GHz. In the meanwhile, both of the meta-surface based convex reflectors maintain highly directive radiations, although there are some fluctuations in gains over the examined bandwidth, and this is reasonable because the phase responses from the ring-patch based meta-surfaces vary with the frequencies. The gains of the umbrella-shape reflector have the peak value of 22.9 dBi at 20.2 GHz, while the gains of the polygon prism reflector have the peak value of 15.78 dBi at 20.2 GHz.
Fig. 3 Radiation performances of the convex reflectors with and without meta-surfaces, feeding by a standard WR51 waveguide. (a) Far-fields of the umbrella-shape reflector with meta-surface. (b) Far-fields of the polygon prism reflector with meta-surface. (c) Far-fields of the umbrella-shape reflector without meta-surface. (d) Far-fields of the polygon prism reflector without meta-surface. (e) Bandwidth properties of the umbrella-shape reflector with meta-surface. (f) Bandwidth properties of the polygon prism reflector with meta-surface.
Fig. 4 continues to demonstrate the beam-steering performances of the umbrella-shape and polygon prism convex reflectors with meta-surfaces when we move the feeding waveguide away from the focal point to acquire the off-axis feeding. The near-field distributions are shown from Fig. 4(a) to Fig. 4(h) with the two proposed reflectors both having the 0 mm, 10 mm, 20 mm, and 30 mm off-axis displacement from the focal point in H-plane. We can observe that the umbrella-shape reflector has the steering beams in 0°, 5°, 10°, and 15°, while the polygon prism reflector has the steering beams in 0°, 7°, 15°, and 22°, respectively. The corresponding far-field performances are demonstrated in Fig. 4(i) and Fig. 4(j). The proposed reflectors are capable of generating highly directive beams with nicely steering performances. For the off-axis displacement of 0 mm, 10 mm, 20 mm, and 30 mm, the maximum gains of the umbrella-shape reflector are 22.5 dBi, 22.0 dBi, 20.8 dBi, and 19.3 dBi, while the maximum gains of the polygon prism reflector are 15.6 dBi, 14.3 dBi, 13.8 dBi, and 13.7 dBi. The constant off-axis feeding values would obtain the proportional change in the beam-steering angle. Every 10 mm displacement feeding would result in 5° beam-steering for the umbrella-shape reflector and approximately 7° beam-steering for the polygon prism reflector.
Fig. 4 Beam steering performances from the meta-surface based convex reflectors with off-axis feeding. Normalized E-field distributions through the proposed reflectors at 20 GHz with the WR51 waveguide source at the focal point [(a), (e)], 10 mm [(b), (f)], 20 mm [(c), (g)], and 30 mm [(d), (h)] off-axis displacement from the focal point. The E-fields are normalized by 400 V/m for the umbrella-shape reflector [(a), (b), (c), and (d)] and 600 V/m for the polygon prism reflector [(e), (f), (g), and (h)], respectively. (i) Far-field radiations for the umbrella-shape reflector with off-axis feeding. (j) Far-field radiations for the polygon prism reflector with off-axis feeding.
3. The fabrication and experimental results
Finally, we fabricate the proposed convex reflectors loaded with meta-surfaces and carry out the experiments to verify our proposed designs as demonstrated in Fig. 5. The convex reflectors are constructed on the supporting aluminum plate bases with ring-patch based meta-surfaces on the FR4 grounded slabs. We can observe from the experimental results that both the meta-surface based umbrella-shape and polygon prism convex reflectors show the ability to generate the highly directive radiations as we designed. The pencil-beam from the umbrella-shape reflector has the 3-dB beamwidth of 5.32 degrees in H-plane and 7.14 degrees in E-plane, while the fan-beam from the polygon prism reflector has the 3-dB beamwidth of 7.29 degrees in H-plane and 29.77 degrees in E-plane. The pencil-beam and the fan-beam from the umbrella-shape and the polygon prism reflectors have the measured gains of 20.53 dBi and 13.75 dBi at 20 GHz respectively. In the meanwhile, the reflection coefficients of the two proposed reflectors turn out to have the similar trend to the simulations, and are both less than −10 dB from 19 GHz to 21 GHz. There are a few degradations of the measured maximum gains with slightly raised sidelobes, and these mainly attribute to the fabrication tolerance of the meta-surfaces. However, the overall radiation performances from such convex reflectors have fully demonstrated the functionalities of generating highly directive radiations that agrees quite well with our designs.
Fig. 5 The manufactured photos and the measured radiation performances of the umbrella-shape and polygon prism convex reflectors. (a) The umbrella-shape reflector with metasurface. (b) The polygon prism reflector with meta-surface. (c) The measured far-fields of the umbrella-shape reflector with meta-surface. (d) The measured far-fields of the polygon prism reflector with meta-surface. (e) The measured reflection coefficients of the umbrella-shape reflector with meta-surface. (f) The measured reflection coefficients of the polygon prism reflector with meta-surface.
4. Conclusion
In conclusion, we have demonstrated highly directive convex reflectors with meta-surfaces. The umbrella-shape and polygon prism convex reflectors, coating with a proper arrangement of subwavelength ring patches offering the desired phase distributions, have been shown to generate the highly directive pencil-beams and fan-beams respectively. Our approach of generating well collimated electromagnetic fields from convex reflectors should be applicable to wide electromagnetic areas, especially for the design of conformal components on the non-planar platform with the highly directive radiations.
Funding
National Natural Science Foundation of China (61671344, 61301072); the Fundamental Research Funds for the Central Universities from China (JB160206, JBG160216).
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