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Abstract
In this paper, a nondiffracting beamforming scheme employing 3-bit coding metasurfaces to adjust the beam lengths is proposed. The 3-bit coding is implemented with a four-layer phase control unit whose phase can be adjusted by tuning the side length of the metal square. Simulation and measured results show that the electric field distributions generated by the metasurface is independent of the polarization of the incident wave due to the central symmetry of the structure. Experimental results are provided which verifies the feasibility of the proposed scheme in adjusting the nondiffracting beam lengths. A relative error as small as 6.0% in the beam lengths is observed, and all results prove the proposed 3-bit coding metasurfaces can be applied to generate different nondiffracting beam lengths.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In 1987, a scalar expression was derived from the wave equation by Durnin, which proves the existence of nondiffracting beams in the microwave range [1]. The theoretical results prove that the magnitude of the electromagnetic fields do not decay during propagation. In recent years, many research efforts have been devoted to the potential applications of nondiffracting beams in areas such as long-distance wireless power transmission [2], optics [3], detection [4] and communication [5].
However, ideal nondiffracting beams cannot be generated due to the fact that they contain infinite energy. In reality, only quasi-nondiffracting beams can be generated. Many types of quasi-nondiffracting beamformer have been proposed including the leaky wave antenna [6], radial line slot array antenna (RLSA) [7], axicon [8] and gradient index metamaterial [9] and so on. In addition, the holograms method has also been proposed to generate nondiffracting beams [16]. All the work in these literatures focus on how to generate a quasi-nondiffracting beam effectively, but none of them discusses how to adjust the nondiffracting length of the generated beams.
In this paper, we employ metasurface (MS) as the beamformer due to its low profile, low cost, and convenience for integration. In addition, MS has been successfully used to manipulate electromagnetic waves and beams, and applied to realize reflectors [10], absorbers [11], cross-polarization converters [12], near-field beam focusing [13], and radar cross section (RCS) reducing [14].
Reported works in Refs. [6–9] show that the phase distribution over the radiating aperture of the beamformer is the critical factor in forming a nondiffracting beam. In this work, we propose a scheme of adjusting the nondiffracting beam length by generating different phase distributions with a 3-bit coding MS. Simulation and experimental results are provided demonstrating that the proposed scheme is able to generate the nondiffracting beams with desired lengths. Moreover, with a symmetrical topology of the MS and its phase control unit, the nondiffracting beamforming performance is insensitive to the polarization incident waves.
2. Metasurface design
As mentioned in section 1, the phase distribution over the MS is the determining factor of nondiffracting beamforming. Here, a centrosymmetric unit of the MS is proposed, which consists of four identical F4B substrates (ɛr = 2.55, tanδ = 0.002) as shown in Fig. 1. A metal ring and a metal square are etched on top side of each substrate with a thickness of 0.035 mm. In addition, there is an air gap (g) between two adjacent substrates to obtain a wide phase range coverage. The parameters of the phases control unit are: g = 2 mm, h = 2 mm, p = 20 mm, l = 19 mm, w = 0.5 mm as shown in the figure. The wanted phases can be obtained by adjusting d, the side length of the metal square.
In this work, we build a 3-bit coding MS with eight different unit designs providing eight discrete transmitting phases. The eight discrete transmitting phases are equally spaced with a difference of 45° as given by (1).
In the expression, θi’ is the discrete phase and θi represents the original phase. To obtain these desired phases, the phase unit is built in the electromagnetic simulation software CST. By adjusting the length of metal square side, eight phase control units are obtained. The transmission magnitude (Tm) and transmission phase (Tp) curves of these phase units are shown in Fig. 2. From picture (a) to (h), it can be found that the transmission phase curve moves from the bottom to the top, and the move step is close to 45°. Furthermore, it is worth noting that, besides the transmission phase, the transmission magnitude of the phase control unit is also very important in some application fields. For example, in an MS based microwave wireless power transmission system, the transmission magnitude of phase control unit will significantly influence the system’s efficiency. The greater the transmission magnitude, the more microwave power can be transmitted to the receiving terminal. Therefore, the transmission magnitude curves of these units is also presented in Fig. 2. From these curves, it is easy to observe that most of the magnitudes are greater than −1.8 dB, and the minimum magnitude of the transmission coefficient is also greater than −2.7 dB, which means that more than half of the power can be transmitted to the receiving terminal side in the microwave power transmission system. In addition, the topology of 3-bit coding units are shown in Fig. 3, and the parameters of these units are depicted in Table 1.
Fig. 2. Transmission magnitude (Tm) and transmission phase (Tp) curves of the 3-bit phase control unit.
Table 1. The Parameters of Code Units
To generate the nondiffracting beams, the phase units are arranged with a specific distribution over the MS. In this work, a phase calculation expression is derived from our previous work and applied to realize the wanted phase distribution [15]. The derived formula is shown as (2). The expression is used in our previous work to determine the transmission phase θ given the nondiffracting beam length Zmax. In this work, (2) is used to adjust Zmax by designing the distribution of transmission phase θ, as elaborated in the follow section.
A nondiffracting beam-forming diagram is shown in Fig. 4. Here, F is the distance between the phase center of feeding horn and the MS, ρ is the distance between the phase control unit and the MS center, D is the MS aperture size, and Zmax is the length of the generated nondiffracting beam. Due to the centrosymmetric structure of the phase control unit and designed MS, the electric field vector in the beam propagation direction will be enhanced ($\mathop {{n_1}}\limits^ \to + \mathop {{n_2}}\limits^ \to $). As for the electric field in the radial direction, two vectors will be offset ($\mathop {{n_3}}\limits^ \to + \mathop {{n_4}}\limits^ \to = 0$), then lead to most of the transmission power will be concentrated in a narrow channel (the red area in the Fig. 4) and formed the nondiffracting beam.
Here, three nondiffracting beams with lengths (Zmax) of 400 mm, 500 mm, and 600 mm are built and simulated in CST. The parameters of simulation environment setting as: D = 530 mm, F = 150 mm, 0≤ρ≤D/2, and the working frequency is 5.8 GHz. According to formula (2), it is easy to calculate the phase distribution on the MS under different nondiffracting length conditions, and the original phase distribution and 3-bit discrete phase distribution are calculated and shown in Fig. 5.
Fig. 5. The phase distribution on the MS, (*−1) and (*−2) represent the original phase distribution and 3-bit discrete phase distribution, respectively. (a-*), (b-*) and (c-*) represent the phase distribution when nondiffracting length are 400 mm, 500 mm and 600 mm, respectively.
3. Simulation results
Three MSs are built in the CST software based on the 3-bit phase distributions in Fig. 5, and a corrugated horn is used to act as the feeding horn. The centrosymmetric structure of the phase control unit and designed MSs lead to a similar electric field distribution when the designed MS illuminated by two mutually perpendicular linearly polarized waves (x-polarization and y-polarization). The normalized electric field distribution on the beam propagates path is shown in Fig. 6 under different conditions.
Fig. 7. The normalized electric field when the beam lengths are 400 mm, 500 mm and 600 mm, respectively.
From Fig. 6, it is easy to observe that the generated electric fields are similar in both cases, which means that the designed MS can be applied to different application scenarios. From the simulation results, we can find that the electric field in the middle area is stronger than the electric field on either side, which means that most power is restricted in a narrow channel in the propagation direction, which makes it potentially useful for microwave wireless power transmission. In addition, it is worth noting that the width of the channel does not widen during propagation, which indicates that the nondiffracting beam is formed.
In order to further expand the function of the MS for the nondiffracting beam controlling, three MSs are designed based on the 3-bit discrete phase distribution as showing the Fig. 5(a-2), (b-2) and (c-2) to realize the different nondiffracting length adjusting purpose. The finally simulated electric field distribution is shown in Fig. 7 when beam lengths are 400 mm, 500 mm and 600 mm, respectively.
From these electric field distributions, it is easy to observe that a narrow channel appears in the center area under all conditions. It proves that, instead of spreading out, most of the power is confined to these narrow areas. Besides that, from the picture, we can find that, the greater the value of Zmax, the longer the length of the narrow area. The simulation results demonstrate that these designed 3-bit coding MS can be applied to adjust the nondiffracting length of the generated beam.
To reveal the electric field change on the propagation path, the normalized electric field are shown in Fig. 8. Although the nondiffracting beam generator is applied, the electric field will still attenuate on the propagation path. Therefore, it is usually just a matter of decreasing the rate at which the electric field decays. This work defines the nondiffracting length as the length where the electric field magnitude is greater than −3.0 dB. From Fig. 8, it is easy to find that the electric field intensity decreases sharply with the increase of the propagation distance under without MS condition. After loading the proposed MSs, the electric field curves maintain a slight slop and the electric field decays significantly slower than without MS case.
According to the curves in Fig. 8, we can measure that the nondiffracting length of the generated beams are 370 mm, 530 mm and 640 mm, respectively. Here, the formula (3) is used to evaluate the relative error between the simulated length and the preset length, where l1 and l0 are the simulated nondiffracting length and preset length, respectively. According to the expression, we can obtain the relative errors are 7.5%, 6.0% and 6.7% when the preset length are 400 mm, 500 mm and 600 mm, respectively. The main reason results in the simulated value deviates from the set value is that, all 3-bit MSs consist of only eight phases. More phases may lead to a better result.
4. Experiment results
Here, according to the phase distribution shown in Fig. 6, three MSs are fabricated, and they are measured in a near-field measurement system. The MSs and measurement system are shown in Fig. 9. In the measurement system, the working frequency is 5.8 GHz, a corrugated horn acts as the feeding horn, and it is fed by a coaxial line. A fabricated MS is fixed in front of the feeding horn at a distance of 150 mm. Besides that, a near-field probe is applied to measure and record the generated electric field in the measurement surrounding. In the x and y directions, the scanning ranges of the probe are 530 mm and 530 mm, respectively. In the z direction, the feeding horn and MS, are moved from 100 mm to 800 mm.
Fig. 9. The fabricated MSs with different beam lengths, a. Zmax = 400 mm, b. Zmax = 500 mm, c. Zmax = 600 mm; d. the measurement environment.
The MSs are measured in turn by the near-field probe. To exhibit the features of the generated beams, the electric field distribution at different distances is recorded and drawn in Fig. 10. In the figure, the electric field distribution on the first, second and third row present the electric field is generated by the a, b and c MS as showing in Fig. 10, respectively.
In Fig. 10, it is easy to find that the strong electric field appears at the central area in all electric field distribution, which means most of the energy is concentrated in the inner are. As proving by the simulation results, the different phase distribution lead to different nondiffracting lengths. From the first row it can be found that, compared with the first four sub-images, the range of strong electric field in the center area of the next four sub-images is larger, which indicates that the beam is spreading. The same thing happens at 500 mm on the second row and 600 mm on the third row, respectively. As expected, the nondiffracting length of three generated beam are 400 mm, 500 mm and 600 mm, respectively.
To further show the merits of the MSs, the electric field distribution on the propagation path is displayed in Fig. 11 when the fabricated MSs is illuminated by the x-polarization and y-polarization incident waves. Where the x-* and y-* represent incident waves are x and y polarizations, respectively. And number 600, 500 and 400 donate the preset nondiffracting lengths are 600 mm, 500 mm and 400 mm, respectively.
From the picture, it is easy to find that the generated beam is very similar in the first and second rows when the MS is irradiated by the different polarization incident waves. As for the third row, due to the generated beam is so close to the MS location, results in the electric field is easily affected by the MS. Hence, we can find the generated beams are slight difference when MS is irradiated by x and y polarization waves.
As an ideal nondiffracting beam does not exist, all nondiffracting beams have a finite propagation distance before significantly decayed. The nondiffracting beam length is usually defined as the distance at which −3 dB decay is observed.
Figure 12 depicts the normalized electric field distribution generated by the three MSs in this paper. From the curves, it is easy to find that, when the value of the black curve is equal to −3.0 dB, the beam propagates a distance is 380 mm, that is to say, the nondiffracting length is 380 mm. As for the red and black curves, the same thing happens when the beam propagates distances are 530 mm and 610 mm, respectively. Obviously, the measured lengths are also different from the preset value of the nondiffracting length, and the maximum relative error calculated by Eq. (3) is 6.0%. The main factor for this result is that, due to the limitation of the phase control unit, only few phases can be applied to composed of the MS. In addition, the MS surface also slightly impacts the generated beam quality. From the measured results, we can conclude that, although the designed MS cannot precisely control the nondiffracting length of the generated beam, it can change the preset value of Zmax to adjust the 3-bit phase distribution and generate the beam with different nondiffracting length.
5. Conclusion
In summary, this work demonstrates the proposed method can be applied to achieve the nondiffracting beam length adjustment. The simulation and measurement results show that, a stronger electric field can be generated and limited in a narrow area after loading a proposed MS. Moreover, the results also prove the MSs can be applied to different linearly polarized waves, making it suitable for multiple application environments. Finally, it can be found that the relative error between the measured nondiffracting length and preset length is only 6.0%. All results demonstrate the proposed method can be applied to adjust the nondiffracting beam length.
Funding
Chengdu-Chongqing Dual City Economic Circle Construction Technology Innovation Project under Grant KJCXZD2020001; Chongqing Technology Innovation and Application Development under Grant cstc2019jscx-dxwtBX0001; National Natural Science Foundation of China under Grant U22B2095.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.
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