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Abstract
This paper proposes a phase-only modulation method to improve the main lobe's efficiency of the Airy beam based on the truncated asymptotic expression of the Airy function. As the truncated asymptotic expression can greatly reduce the range of amplitude modulation, it can be applied to the equal-amplitude modulation method with less amplitude error, which simplifies the design difficulty of the device and improves the beam's efficiency. A transmissive metasurface with a relative bandwidth of 10% is further adapted to generate the Airy beam with characteristics of non-diffraction, self-bending, and high-efficiency, which benefits the practical application of the Airy beam.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The diffraction characteristics of electromagnetic waves make the electromagnetic energy gradually diverge during propagation, which reduces the transmission distance and efficiency. For this reason, nondiffracting beams that can suppress diffraction to a certain extent have been widely studied [1–3]. Among them, zero-order and high-order Bessel beams have been extensively researched [4–7], showing the unique characteristics and novel functions of non-diffraction beams [8]. Meanwhile, the Airy beam has the characteristics of both non-diffraction and self-bending, which enables the beam to reduce the influence of obstacles during the propagation, showing a wider application range [9–12].
Since the realization of Airy beams with finite energy has been achieved and observed [13,14], researches on Airy beams have been carried out, and various optical devices have realized the generation of Airy beams [15–17]. In comparison, the research on Airy beams in the microwave band started relatively late, and the generating devices using a metasurface with a two-dimensional structure can fully simplify the complexity of the device [18–22]. Each unit on these metasurfaces is modulated mainly based on the Airy function, which requires the modulation of the amplitude or phase of the unit. At the same time, the research is currently mainly limited to the generation of Airy beams, and the performance improvement is mainly achieved through the design of metasurface units [23], while there is less research on the perspective of the Airy beam's generation mechanism to achieve a certain breakthrough. Therefore, it has not been able to fully simplify the generation device's design or improve the Airy beam's performance.
Based on the comparative study of the Airy function and its asymptotic expression, this paper theoretically shows that the truncated asymptotic expression of the Airy function requires a smaller amplitude modulation range, so it is more suitable for the generation method of phase-only modulation. Meanwhile, the theoretical analysis of different generation methods is implemented, and the results prove for the first time that the beam realized by the method based on the truncated asymptotic expression can significantly improve the main lobe energy, namely the beam efficiency while maintaining the unique characteristics of the Airy beam. Furthermore, a metasurface with a relative bandwidth of 10% is constructed based on the proposed phase-only modulation method, thereby realizing the Airy beam that can achieve high-efficiency energy or information transmission. The proposed method realizes the high-performance Airy beam and simplifies the device based on the generation mechanism, which provides a new solution for generating and applying Airy beams.
2. Theoretical analysis and verification
When generating Airy beams, the modulation of the metasurface unit is mainly based on the amplitude and phase distributions required by the Airy function in the microwave band. If the co-modulation of amplitude and phase is used, the carefully designed unit must control the amplitude and phase independently. Since amplitude modulation reduces the contribution of many units to the beam's energy, the main beam's efficiency will decrease. If the amplitude modulation is simplified, the phase-only modulation can be adopted, and the energy of the main beam will be increased while the side lobes’ energy will also increase due to the large amplitude error. If the beam's efficiency needs to be improved on the premise of ensuring the beam's performance, a generation method with a smaller amplitude variation range should be used. Considering the expression of the Airy function is [24]:
where Ai represents the Airy function. Correspondingly, its asymptotic expression is [25,26]: where B is an arbitrary constant, and i is the imaginary unit. The comparison of the amplitude changes of the above two functions can be obtained as shown in Fig. 1:It can be seen from Fig. 1 that the asymptotic expression can be used as the envelope of the Airy function when Bx < -1.2. Moreover, when -10 < Bx < -1.6, the amplitude change of the asymptotic expression is within 0.3 - 0.5, while the amplitude change of the Airy function is within 0 - 0.5. So the amplitude variation range is 60% lower when using the truncated asymptotic expression, which can approximate the Airy function and greatly reduce the amplitude variation range. Therefore, the corresponding phase distribution of the truncated asymptotic expression can be used to perform equal-amplitude modulation to improve the beam's efficiency while ensuring the Airy beam's characteristics.
To verify the above scheme, a linear array with 25 elements can be expanded to a two-dimensional array with 25 × 20 elements located on the xoy plane to generate Airy beams and compare the beams’ performances based on the above generation methods. Then the phase-only modulation based on the truncated asymptotic expression (denoted as “PM_TAE Method”), the amplitude-phase modulation based on the Airy function (denoted as “APM_AF Method”), and the phase-only modulation based on the Airy function (denoted as “PM_ AF Method”) is separately used to generate Airy beams. The corresponding normalized E-field distributions are shown in Fig. 2(a)–(c). It can be seen that the beams achieved by the three methods have similar trajectories, while the first side-lobe of the beam achieved by PM_ AF Method is most pronounced compared to its main lobe because this method has the largest magnitude error. At the same time, although PM_TAE Method also uses phase-only modulation, its side-lobe energy is much smaller than the main lobe energy due to its smaller amplitude error. Furthermore, the comparison of the E-field magnitude values without normalization of the three beams is correspondingly shown in Fig. 2(d)–(f).
Fig. 2. The comparison of the normalized E-field distributions using (a) PM_TAE Method, (b) APM_AF Method, and (c) PM_ AF Method. And (d), (e), and (f) are the corresponding E-field magnitude distributions without normalization using the above three generation methods.
It can be seen that the main beam's energy achieved using PM_TAE Method is stronger. To quantitatively explain the efficiency of each beam, we use different reference planes parallel to the array with dimension three times of the array to completely contain the main and side lobes of the Airy beam, and the total power of the main and side lobes on these reference planes can be calculated. The total power on the reference plane at a distance of 1 mm from the array when using PM_TAE Method is used as the benchmark to calculate the whole beam's efficiency of different beams at different distances, and the results are shown in Table 1. As we calculate the efficiency at different propagating distances, the results can reflect the characteristics of the generated Airy beams and further demonstrate whether there is sufficient electromagnetic energy along the propagation for the practical application of Airy beams.
From Table 1. we know that the efficiency of the two phase-only modulation methods is similar near the array, while the efficiency after amplitude modulation is greatly reduced, which is caused by the suppression of the energy radiated by array elements. Moreover, the efficiency of the whole beam generated by PM_TAE Method is much greater than that of PM_AF Method at different propagation distances, which proves the effectiveness of the proposed method. Further, the efficiency of the main lobe in the PM_TAE Method can be analyzed. The main lobe's energy is calculated using the reference plane that can contain the main lobe and compared with the energy on the reference plane containing the main and side lobes, and the ratio of the main lobe's energy using the PM_TAE Method is shown in Table 2.
Table 1. Efficiency comparison of the whole beams using different methods at different propagation distances
Table 2. The ratio of the main lobe's energy to the whole energy of the Airy beam using the PM_TAE Method
It can be seen that the ratio of the main lobe's energy gradually increases as the distance increases, which is caused by the rapid decay of the side lobe's energy as the distance increases. Meanwhile, we can find that the changing trend of the ratio of the main lobe’ energy is consistent with the E-field distribution in Fig. 2. Therefore, the proposed PM_TAE Method is a scheme that can achieve high-efficiency Airy beam, and its main lobe still has high energy after propagating a certain distance, which shows the high-performance of the generated Airy beam.
3. Implementation of the high-efficiency Airy beam
3.1 Simulation results
Based on the proposed generation method, a metasurface is used to generate the Airy beam with high efficiency. The plane wave was often used to be the feed of metasurfaces to realize the Airy beam in previous works. In this case, the horn antenna needs to be far enough away from the metasurface so that the beam radiated from the horn can be approximated as the plane wave when it reaches the metasurface. This kind of feeding method will cause a large amount of path loss to the energy radiated by the horn, which does not meet the requirements of this work for high-efficiency Airy beams. As a result, we make the metasurface can directly use the horn antenna for short-range feeding through phase compensation.
The metasurface unit adopted is shown in Fig. 3(a), its period is 10 mm, and it consists of four layers of the same substrate with a dielectric constant of 2.65 and a thickness of 1.5 mm. A square patch with side length Ls is printed on the upper surface of the uppermost substrate, and a square ring with an outer side length of 10 mm and a width of 1.76 mm is printed on the lower surface. The lower surface of each underlying substrate is sequentially printed with the same square patch, square ring, and square patch. Using HFSS to simulate the unit with the infinite period boundary and the Floquet port, the normalized magnitude of the S21 parameter and transmission phase of the unit versus the length of Ls can be obtained as shown in Fig. 3(b). Within 9.5 - 10.5 GHz, with the change of Ls from 0.5 mm to 7.5 mm, the transmission phase can cover more than 300°. Meanwhile, the normalized magnitude of the unit in this frequency band basically remains above 80%, while it is greater than 90% in most dimensions of the unit. Therefore, the metasurface unit has a good characteristic within 10% relative bandwidth.
Fig. 3. (a) Geometry of the metasurface unit and (b) the normalized magnitude of S21 and transmission phase versus the length of Ls.
Using the proposed generation method and metasurface unit, the compensation phase required by each unit can be calculated. The compensation phase required to compensate for different distances between the feed horn with the coordinate $({x_f},{y_f},{z_f})$ in the coordinate system and each unit with the coordinate $({x_{ij}},{y_{ij}},0)$ on the xoy plane is:
Then the final compensation phase required for each unit is:
The metasurface formed with 25 × 20 mentioned units is located on the xoy plane, and its geometric center is located at the coordinate origin. The distance between the feed horn and the metasurface is 200 mm and their geometric centers are aligned with each other. The compensation phase distribution of the metasurface can be obtained as shown in Fig. 4(a)–(c). Based on the relationship between Ls and the transmission phase, the metasurface can be formed and its top view is shown in Fig. 4(d).
Fig. 4. The compensation phase distribution of (a) φF, (b) φA, (c) φC, and (d) the top view of the metasurface.
For the convenience of understanding, the relative positions of the metasurface and different observation planes during simulation are shown in Fig. 5(a). Here the observation plane parallel to the metasurface (named horizontal observation plane) and the observation plane perpendicular to the metasurface (named vertical observation plane) are used to observe the E-field distributions. It can be seen that each horizontal observation plane with a size of 400 mm × 400 mm is parallel to the xoy plane, and it is located on the positive semi-axis of the x-axis, while is symmetrical about the y-axis. The first horizontal observation plane is 100 mm away from the metasurface, and each horizontal observation plane is separated by 100 mm. The vertical observation plane is located on the xoz plane. Since the trajectory of the Airy beam is toward the positive semi-axis of the x-axis, the area of the vertical observation plane on the positive semi-axis of the x-axis is larger to facilitate the observation of the beam's trajectory. When simulating the model in HFSS, the feed horn, the metasurface, and each observation plane need to be surrounded by vacuum boxes that can completely wrap them, and the boundary of the vacuum boxes is set to the FE-BI boundary for simulation. When the simulation frequency is set to 10 GHz, the normalized E-field distribution on the observation plane can be obtained as shown in Fig. 5(b). Due to the radiation characteristics of the feed horn, the E-field's magnitude of the beam is enhanced near the metasurface and the propagation trajectory of the Airy beam is consistent with its self-bending characteristics. Meanwhile, the energy is mainly concentrated on the propagation path of the main lobe, which exhibits the characteristics of non-diffraction and aligns with the design requirements. The normalized E-field distributions on the horizontal observation planes are shown in Fig. 5(c)–(h). Each distribution shows that the energy of the Airy beam at different distances is fully concentrated in a certain area, and since the positions of the energy-concentrated area on different horizontal observation planes gradually change, the Airy beam's self-bending characteristic is further proved.
Fig. 5. (a) The schematic diagram of the simulation model with different observation planes. And the normalized E-field distributions on (b) the vertical observation plane, and the horizontal observation planes with the distances of (c) 100 mm, (d) 200 mm, (e) 300 mm, (f) 400 mm, (g) 500 mm, and (h) 600 mm away from the metasurface.
Further observing the Airy beam at different frequency points, the E-field distributions on the vertical observation plane can be obtained as shown in Fig. 6(a)–(c). It can be seen that within 9.5 – 10.5 GHz, the Airy beam maintains good non-diffraction and self-bending characteristics.
Fig. 6. The normalized E-field distributions on the vertical observation plane at (a) 9.5 GHz, (b) 10 GHz, and (c) 10.5 GHz.
To analyze the efficiency of the Airy beam, we can integrate the Poynting vector on different reference planes to obtain the energy on these planes using the field calculator of HFSS. Firstly, set two reference planes with the same dimension as the metasurface located in the front and back of the metasurface, considering the direction of the Poynting vector is the direction from the feed horn towards the metasurface, so the energy on the front reference plane is the energy incident on the metasurface minus the energy reflected by the metasurface. As the average value of the metasurface unit's reflection coefficient is about -15 dB, the metasurface can reflect about 3.16% of the energy radiated onto it. Considering there is still a certain distance between the front reference plane and the metasurface which also causes some path loss, the reflected energy at the front reference plane will be less than 3.16%. It can be calculated that the ratio of the incident energy to the energy of the feed horn is about 59.16%, which is much larger than the plane wave used as the feed. The energy on the back plane can represent the energy after being modulated, and the ratio of it to the energy incident to the metasurface is the conversion efficiency, which is 84.5%, embodying the high efficiency of the metasurface. Then set some reference planes of the same dimension as the metasurface for efficiency calculation. Compared with the simulation model with observation planes, the position of the feed horn antenna, the metasurface, and the coordinate system remain unchanged, and the distances between different reference planes and the metasurface are consistent with the distances between different observation planes and the metasurface. During the simulation, the feed horn, the metasurface, and each reference plane need to be wrapped by vacuum boxes with the FE-BI boundary. The position of each reference plane can be determined after the Airy beam's E-field distributions on the observation planes are obtained, and the geometric center of each reference plane lies on the propagation path of the Airy beam's main beam. The ratio of the energy on these reference planes to the energy radiated from the metasurface is the efficiency of the beam at these transmission distances, which can be obtained as 82%, 76%, 71%, 65%, 62%, and 59%, respectively. The efficiency of the full-wave simulation is lower than that of the theoretical calculation, which is because the smaller reference planes are used and the actual factors such as coupling between units are considered. However, the realized Airy beam remains high efficiency with propagation, so the simulation results confirm that the generation method and device used can realize the high-efficiency Airy beam.
3.2 Experimental measurement results
Based on the simulation results, the metasurface is fabricated and experimentally measured. The measurement scene in a microwave anechoic chamber is shown in Fig. 7. The normalized E-field distributions on different horizontal observation planes at different distances from the metasurface can be obtained by near-field scanning of the metasurface.
Meanwhile, it is also sufficient to illustrate the measured E-field distribution along the propagation direction and the characteristics of the Airy beam by comparing the simulated and measured results of the E-field distributions on the horizontal observation planes. To this end, the simulation results in Fig. 5 can be compared with the measured results at 10 GHz and displayed as shown in Fig. 8. It can be seen that simulation results are highly consistent with measured results in the positions of the energy-concentrated area so that the E-field distribution of the measured beam along the propagation direction is also consistent with the simulation results.
Fig. 8. The comparison of the simulated and measured normalized E-field distributions of the fabricated metasurface on the horizontal observation planes with distances of 100 mm, 200 mm, 300 mm, 400 mm, 500 mm, and 600 mm, respectively at 10 GHz.
The measured results at different frequency points and distances are shown in Fig. 9, and the ratio of the main lobe's energy to the total energy on the observation plane is shown in Table 3. It can be seen the beam's energy is always concentrated in a certain range, showing the non-diffraction characteristic within 10% relative bandwidth. And the ratio of the main lobe's energy is increasing with the increase of propagation distance, which is consistent with the changing trend in Table 2. Compared with the simulation results, the energy around the main lobe has increased and the ratio of the main lobe's energy decreases, mainly due to the closeness of the metasurface and the scanning bracket and the inevitable measurement disturbances and errors. Meanwhile, the main energy is concentrated on the main lobe, which is in line with the characteristics of high efficiency. Analyzing the E-field distributions at different distances, the energy-concentrated area at different frequency points is consistent with the simulated results in Fig. 5, reflecting the self-bending characteristics. Therefore, the measurement results prove that the generated Airy beam has good characteristics.
Fig. 9. The normalized E-field distributions of the fabricated metasurface on the horizontal observation planes with the distances of 100 mm, 200 mm, 300 mm, 400 mm, 500 mm, and 600 mm, respectively, at (a) 9.5 GHz, (b) 10 GHz, and 10.5 GHz.
Table 3. The ratio of the main lobe's energy in the observation planes at different propagation distances
4. Conclusion
This paper starts from the generation mechanism of the Airy beam and compares the amplitude distribution of the Airy function and its asymptotic expression to find the generation method which can simplify the generation device and improve the performance of Airy beams. We realize a high-efficiency Airy beam based on this generation method and a transmissive metasurface with 10% relative bandwidth. Theoretical analysis, simulation, and measurement results prove that the modulation method based on the truncated asymptotic expression is more suitable for phase-only modulation and can improve the beam's efficiency. Meanwhile, the design can be further combined with research such as the abruptly autofocusing waves [27], which gives Airy beams a good application prospect in wireless communication and wireless power transfer.
Funding
National Natural Science Foundation of China (62001342, 62288101); National Key Research and Development Program of China (2021YFA1401001); Key Research and Development Projects of Shaanxi Province (2021TD-07); Fundamental Research Funds for the Central Universities (20103224952).
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 may be obtained from the authors upon reasonable request.
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