Abstract
In this paper, a conical antenna array capable of radiating terahertz vortex beams was designed. Specifically, circular microstrip antenna with coaxial feeding was taken as an element to place on concentric rings at equal intervals, independently constructing a phased planar microstrip antenna array and a phased conical microstrip antenna array, which were then simulated and optimized by the simulation software Ansoft HFSS. The results showed that the conical microstrip antenna array can not only radiate a narrow beam with stable gain, but also generate a vortex-beam radiation pattern with orbital angular momentum, which was demonstrated as a conical antenna array that can generate pseudo-Bessel beams.
Keywords: Terahertz; Conical Antenna Array; Vortex Beam; Pseudo-Bessel Beam
Introduction
Research on key technology in 5G mobile communications comes to the fore in the field of mobile communication worldwide, exhibiting intense competition among different counties. After the entrance into the era of digital communication, mobile communication generates a large quantity of data along with its rapid development, imposing huge demanding on the high Baud rate. Thus, various multiplexing technologies were brought in communication field for improving the communication capacity. At the same time, a new multiplexing method based on Orbit Angular Momentum (OAM) is of great interest to scientists due to the new degrees of freedom it brought about. L. Allen et al. [1] stated that each photon carrying OAM has an azimuthal angular dependence of, where is topological charge and is the azimuth angle, demonstrating that OAM is an attribute that can generate the beam with helical phase distribution. In 2007, Thide [2] generated electromagnetic waves with OAM eigenstates orthogonal to each other through array antennas, revealing that electromagnetic waves also have the characteristics of vortex-waves and promoting the research of vortex electromagnetic waves in wireless frequency band to a new era. In 2011, through OMA coding with a transceiver antenna, Tamburini [3] realized the transmission of multi-channel signals on the same frequency band without mutual interference due to the information carried by the angular momentum. In 2013, Bennis [4] generated vortex wave with topological charge number by metasurface. Dr. Zelenchuk [5] realized circularly polarized vortex wave radiation by controlling the phase delay at different positions in each ring unit. In 2015, Dr. Zhou Shouli has successively published two papers [6],[7] on vortex waves, which describe how to utilize antenna arrays to realize vortex waves in C and Ku bands independently. In terahertz band, Li Yao [8] implemented antenna array with metasurface to realize vortex wave beam. Chen Yanan et al. [9] discussed the imaging technology of vortex waves and found that the antenna needs to concentrate its energy in a very narrow space to radiate, which is eligible for the requirement imposed by the reliability for targets finding and tracking. Particularly, an array antenna with narrow beam and low sidelobe is preferred to enhance directivity and gain. Lemaitre-Auger P et al. [10],[11] proposed that pseudo-Bessel beams with finite energy, limited lateral dimensions and eventual divergence can be realized through a regular hexagonal antenna array composed of 91 antenna elements, where the divergence starts from the outermost sidelobe, followed by the divergence from the second outer sidelobe and eventual divergence from the main lobe. In this paper, an antenna array was designed to generate pseudo-Bessel beam with spiral phase distribution. Specifically, circular microstrip antenna elements were simulated by HFSS to obtain the antenna element size operating at about 113 GHz after global optimization, followed by a separate construction of planar and conical phased microstrip antenna arrays with ninety antenna elements designed previously. Afterwards, a comparison analysis was performed to reveal that the designed conical microstrip antenna array can radiate narrow beams with stable gain and generate pseudo-Bessel beams with vortex properties.
Theoretical Framework
The plane wave irradiates to the bottom of the axicon and subsequently undergoes the axicon transformation to form two cone waves, i.e. the outgoing cone wave and the incoming cone wave, as shown in Figure 1. The two waves can be represented by zeroth-order Hankel function of the first kind and zeroth-order Hankel function of the second kind, respectively [12]:
Figure 1: Schematic diagram for the generation of Bessel beam by axicon
${H}_{0}^{(2)}(x)\mathrm{exp}(-i{k}_{z}z)=[{J}_{m}(x)-i{N}_{m}(x)]\mathrm{exp}(-i{k}_{z}z)\text{(2)}$
Combine equation (1) and equation (2) to obtain:
$[{H}_{0}^{(1)}({k}_{\rho}\rho )+{H}_{0}^{(2)}({k}_{\rho}\rho )]\mathrm{exp}(-i{k}_{z}z)=2{J}_{m}({k}_{\rho}\rho )\mathrm{exp}(-i{k}_{z}z)\text{(3)}$Assume that
$U(\rho ,\phi ,z)\equiv {J}_{m}({k}_{\rho}\rho )\mathrm{exp}(-i{k}_{z}z)\text{(4)}$Where ρ is the radical coordinate, ϕ is the azimuth angle, k is the wave number, N_{m} is m -order Neumann function, J_{m} is m -order Bessel function, k^{2}_{ρ} = k^{2} − k^{2}_{z} . The intensity along the propagation axis can be expressed as:
$I(\rho ,\phi ,z)=|U(\rho ,\phi ,z){|}^{2}={J}_{m}^{2}({k}_{\rho}\rho )\text{(5)}$Equation (5) reveals that the intensity distribution is independent of the propagation distance z, i.e. no variation of the intensity along the cross section, which is the reason why the radiation is called diffraction-free beam, also known as Bessel beam. Therefore, by superposing the resulting outgoing cone wave and incoming cone wave after passing through the axicon, Bessel beams can be generated.
Antenna Design
Antenna Element Design
Coaxial feeding was implemented to reduce the mutual interference among the large quantity of antenna array elements involved and circular microstrip antennas were selected for convenience of adjustment, with the structure as shown in Figure 2. The antennas elements, from top to bottom, are circular patch, dielectric substrate, coaxial feed and ground plate, respectively.
Figure 2: Circular microstrip antenna array element
Through the formula related to the microstrip antenna, the size of the circular microstrip antenna element was calculated and optimized by HFSS to obtain the specific parameters of the array element satisfactory for the design, as shown in Table 1. In the table, a is the radius of the circular patch, LD is the length of the square ground, Ls is the length of the substrate, H is the thickness of the dielectric substrate with FR4 as the material, F is the distance of the coaxial line from the origin, the inner diameter of the coaxial line is R with pec as its material. Figure 3 is the simulation result of an antenna array element. From the figure, the central working frequency of the antenna is f_{r}=113GHz, the minimum return loss value is S_{11}= -18.74dB, where its -10dB bandwidth is about 3.97GHz, demonstrating that the antenna element has a good radiation function suitable for the construction of antenna array.
Table-1: Microstrip antenna sizes working at 113 GHz
a |
LD |
Ls |
H |
F |
r |
465nm |
1600nm |
1200nm |
80nm |
150nm |
12.5nm |
Figure 3: Return loss S11 of antenna array element
Antenna Array Design
A planar array and a conical array are independently constructed through the aforementioned array elements, as shown in Figure 4. The center frequency f_{r}=113GHz corresponds to wavelength λ=2.654mm, and the radius of the i-th circle of the circular ring array is R_{i}=2.8×i with equal-interval distribution of antenna array elements on its circumference. Afterwards, the coordinate of element on the x axis is taken as (R_{i}, 0) to form a 90-element planar antenna array, as shown in Figure 4(a). The conical antenna array is constructed by setting the tilt angle as α=4.9587° so that the height of i-th layer (in the z propagation direction) is sin cos h_{i} = R_{i} × sinα × cosα , making the coordinate of element as (R_{i} × cos^{2}α, h_{i}). Therefore, a 90-element conical antenna array is constructed by taking the radius of the i-th circle from the axis (i.e. z axis) as R'_{i}= R_{i}×cos^{2}α×i with equally spaced 6×i antenna array elements around the circumference, as shown in Figure 4(b). It is worth noting that the antenna element can be regarded as a point array element relative to far field, and the resulting antenna array can be treated as an axicon-like structure.
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Figure 4: Antenna arrays with two different structures. (a) 90-element planar antenna array; (b) 90-element conical antenna array.
Results and Discussion
The radiated electromagnetic wave, if fed with constant phase difference and equal amplitude on all elements of the uniform circular array antenna, will rotate around the array axis for one or multiple circles, generating vortex wave with spiral phase wave front. Particularly, if all array elements of the antenna array are fed with the same amplitude and phase, the electromagnetic wave beam with OAM value l = 0 will be radiated. Otherwise, if following the setup aforementioned and prescribing the phase difference with equal amplitude as ±60^{o}, ±30^{o}, ±20^{o}, ±15^{o}, ±12^{o} respectively (i.e. the innermost ring is ±60^{o}, the second ring is ±30^{o}, etc.), the electromagnetic wave beam with OAM value l = ±1 will be radiated. Besides, under the same condition but the phase difference is replaced with ±120^{o}, ±60^{o}, ±40^{o}, ±30^{o}, ±24^{o} the electromagnetic wave beam with OAM value l = ±2 will be radiated. Figure 5, Figure 6 and Figure 7 show the diagrams of vectored electric field with OAM in different modes. When mode l = 0 , the antenna array radiates electromagnetic waves without spiral phase wave front. If l =+1, +2, a clockwise spiral phase wave front is obtained. Similarly, if l =−1, −2 , a counterclockwise spiral phases wave front is obtained. Furthermore, there are 2 lobes in the vectored electric field when l = ±1, while there are 4 lobes in the vectored electric field diagram when l = ±2 , showing consistency with the conclusions reported previously [6, 7]. It is worth noting that, as shown in these figures, the aperture of the antenna array becomes larger with the increase of the number of array elements, leading to larger distribution area and easier outward divergence of the vectored electric field.
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Figure 5: Vectored electric field diagram of 36-element conical antenna array. (a) l = 0 ; (b) l = ±1; (c) l = ±2 .
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Figure 6: Vectored electric field diagram of 60-element conical antenna array. (a) l = 0 ; (b) l = ±1; (c) l = ±2
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Figure 7: Vectored electric field diagram of 90-element conical antenna array. (a) l = 0 ; (b) l = ±1; (c) l = ±2
Figure 8 shows the far-field gain of the planar array as well as the conical array when l = 0 and the direction angles are ϕ = 0° and ϕ = 90° independently. Furthermore, the energy of the conical array antenna, compared with the planar array antenna, concentrates towards the main lobe with smaller level of the side lobe. Besides, the gain of 36-elements conical antenna array is 18 dB in contrast to approximate 21.5 dB stable gain obtained from the 60- and 90-element antenna arrays due to the relatively narrow beam.
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Figure 8: The gain diagram of planar and conical antenna array when l = 0 (a) 36-elements planar antenna array; (b) 36-elements conical antenna array; (c) 60-elements planar antenna array; (d) 60-elements conical antenna array; (e) 90-elements planar antenna array; (f) 90-elements conical antenna array
Figure 9 shows the gain diagram of the conical antenna array when l =1 and l = 2, respectively. Specifically, when l =1, the gain of the antenna array is 20 dB, 22 dB, and 24 dB, respectively, for 36-, 60- and 90-elements antenna array independently. When, the counterpart is 15.5 dB, 17.8 dB, and 18 dB, respectively, for 36-, 60- and 90-elements antenna array independently. Therefore, it can be concluded that the gain of antenna array decreases when more modes are involved. Furthermore, more antenna array element will result in increased gain, but ending with stable gain and lower side lobe level when the array element continues to increase. As determined by the vectored electric field and gain diagram, the electromagnetic wave radiated by the conical antenna array is a pseudo-Bessel beam, which is a quasi-diffraction-free beam.
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Figure 9: The gain diagram of three different conical antenna array. (a) gain of 36-elements antenna array when l = ±1; (b) gain of 36-elements antenna array when l = ±2 ; (c) gain of 60-elements antenna array when l = ±1 (d) gain of 60-elements antenna array when l = ±2 ; (e) gain of 90-elements antenna array when l = ±1; (f) gain of 90-elements antenna array when l = ±2
Conclusion
In this paper, a conical antenna array working at terahertz frequency with circular microstrip antennas as its elements was designed, which is similar to an axicon and can radiate pseudo-Bessel electromagnetic wave beams with diffraction-free characteristics. In particular, conical antenna array radiates electromagnetic wave without spiral phase distribution when l = 0 and it, otherwise, radiates vortex wave with 2l lobes. Vortex wave radiated by this antenna array is expected to be implemented in terahertz communication, terahertz imaging as well as measurement, and other fields.
Acknowledgement
This work was supported by the National Natural Science Foundation of China (Grant No. 61571271), the Natural Science Foundation of Fujian Province of China (No. 2018J01646; No. 2016J01760)
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