Open Access
Add to BibSonomy
Abstract
Bandwidth, orbital-angular momentum (OAM) divergence, and mode purity are the three critical issues for the practical terahertz orbital angular momentum manipulation, especially in the next sixth-generation (6G) communication system. Here we propose the broadband high-order Bessel vortex beam carrying multiple OAM modes reflective metasurface in the terahertz domain. The simulation results agree with the theoretical expectation, and the diffracting divergence of OAM vortex beam characteristics has been alleviated. The research on the relationship between the varieties of lattice type and mode purity is also relatively scarce. Henceforth, a comparison study has been conducted between three lattice types, i.e., square lattice, triangular lattice, and concentric ring lattice. And corresponding results of the relationship of mode purity with those lattice types show that the concentric ring lattice has the best performance.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Due to more intelligent services playing a significant role in our daily life situations, e.g., self-driving cars, Artificial Intelligence (AI), Augmented Reality (AR), Internet of Things (IoT), and smart cities, a surge of higher quality requirements of wireless communication has been proposed in aspects such as data rate, latency, and capacity [1–9]. It’s well known that mm-Wave and sub-terahertz spectrum have been used in the fifth-generation (5G) communication system. And for the next sixth-generation (6G) era, it’s no doubt that the terahertz (THz) wave will play a vital role as its excellent advantages. THz can provide an extensive continuous unexplored bandwidth range from 0.1 to 10 THz. With its narrow beam feature, the THz wave has better directivity and excellent anti-interference and anti-interception capabilities than the microwave. Furthermore, some emerging technologies and new functionalities related to the effective and ultra-fast regulation of THz waves have been the subject of intensive research [5,10,4,3,11]. The Orbital Angular Momentum (OAM) is characterized by a doughnut intensity profile and a helical wavefront with a phase proportional to $e^{il\phi }$, where $l$ is an integer known as the topological charge. Since the discovery of light’s OAM 30 years ago [12], the investigation of OAM has led from fundamental theories to applications [13–17]. Due to the OAM beams theoretically can contain infinite orthogonal eigenstates, which provide additional freedom to increase the channel capacity and spectral efficiency in wireless communication, many ordinary OAM wave generation methods have been proposed, e.g., uniform circular array antenna (UCA) [18,19], spiral phase plate (SPP) [20,21], holographic plate [22] and angular gratings [23], etc.
Past two decades, two-dimensional metasurface, a class of artificially engineered materials composed of an electromagnetic particle with length scales much smaller than the wavelength of the incident light, has attracted much research attention. Due to its low profile, lightweight, and high-performance characteristics, several approaches for generating OAM using metasurface have been investigated until now, such as the angular metalens [24], polarization-encrypted OAM multiplexed metasurface [25], quasi-perfect vortices Pancharatnam-Berry phase metasurface [26], high-efficiency spin-related vortex metalenses [27], holographic leaky-wave metasurface [28], suspended metasurface [29], and spin-decoupled metasurface [30], etc [31–42]. However, those studies mainly concentrated on the microwave and optical domains, and in contrast, rather seldom researches are studied in the THz field. Additionally, vortex beams gradually spread along the radial direction on the transverse plane during propagation, limiting further practical application in long-range wireless transmission [43,44]. Henceforth, generating spatially stable THz vortex waves carrying OAM in free space without changing their initial field distribution at any plane orthogonal to the direction of propagation becomes an urgent issue that merits further study. Recall that the wave equation of the theory of the electromagnetic field Bessel waves are one of the beam-like solutions to the Helmholtz equation in the cylindrical coordinate system [45]. Zero-order Bessel Beams were first studied and physically realized by Durnin et al. in 1987, named as the diffraction-free beams, using an annular aperture at the front focal plane [46,47]. Hereafter, the axicon lens, proposed by Herman and Wiggins [48], radial leakage waveguide [49], etc., has been demonstrated to achieve non-diffraction Bessel beams in optics and microwaves domain. Moreover, with the rapid development of the metasurface, the novel generation method of Bessel beams carrying OAM has further progressed [50–57]. However, the frequency bands of most works are still limited due to the intrinsically multi-resonant mechanism [58,59]. Furthermore, improving OAM mode purity is critical in high-performance OAM transmissions [60]. In contrast, the research on the relationship between the varieties of structure lattice type and mode purity is still relatively scarce. For the development of the high-performance OAM generation device both in scientific research and industry application, it is indispensable to examine whether mode purity is affected by the structure lattice type.
In this paper, we proposed in the THz domain the generation of vortex beams with the quasi-non-diffraction property using the reflective metasurface based on the feature that high-order Bessel modes inherently having high-order OAM modes. Three different lattice types, including square lattice, triangular lattice, and concentric ring lattice, are selected for the numerical analysis. Both of the field intensities and phase distributions simulation results verify the theoretical predictions. As a crucial factor in evaluating the characteristics of diffracting divergence of the OAM mode, the inner radius of the vortex beam also has been studied. The instant comparison of reduced divergence of three lattice types is shown in the paper. And corresponding relationship of mode purity with those three lattice types also has been demonstrated.
2. Principle and design of quasi non-diffraction OAM reflective type metasurface
According to the theoretical analysis of interpretation of the non-diffraction Bessel beam, one of the approaches to the generation of the Bessel beam is the superposition of traveling conic wave [45,47]. Considering the cylindrical symmetry, the exact solution of the scalar wave equation of electromagnetic field after using the separation of variables method is as follows in Eq. (1),2
Fig. 1. Schematic diagram of the proposed OAM generation metasurface. (a) Top view of the unit cell and the geometries relationship: $l_{1}$= 400$\mu$m, $l_{2}$= 330$\mu$m, $l_{3}$= 335$\mu$m, $w_{1}$= 42$\mu$m, $w_{2}$= 33$\mu$m, $w_{3}$= 80$\mu$m, $r$= 100$\mu$m, $p$=800$\mu$m. (b) Side view of the unit cell, the substrate is 200$\mu m$ thick lossy polymide ($\varepsilon$= 3.5, tan$\delta$ = 0.0027). (c) The Floquet port and periodic boundary condition (PBC) configurations are used for the numerical analysis of the unit cell. (d) Superposition of the traveling conic waves to generate the Bessel beam.
Figure 1(a) shows the unit cell of the metasurface used for the OAM beam generation, which has been detailed studied in our previous research [59]. Both of the top pattern and bottom ground is $200nm$ thick aluminum ($\sigma =3.56\times 10^{7} S\cdot m^{-1}$). The detail geometries and the numerical analysis configurations of the unit cell shows in the Fig.1(a,b,c). Figure 1(d) is the schematic diagram of the proposed superposition of the traveling conic wave to generate the Bessel beam, which is origin from the conventional axicon lens principle as shown in Fig. 2. The $\gamma$ is the cone angle, where tan$\gamma$ = $k_{r}/k_{z}$ [45].
Spectrum response under the oblique incidence has been firstly analyzed in order to characterize the functionality of the metasurface angular tolerance. To increase the operation bandwidth of the metasurface and quickly realize the frequency-independent devices, the Pancharatnam-Berry (P-B) phase mechanism has been adopted to achieve the complete phase control of the reflection wave. In the meta-atom unit cell structure simulations using the commercial Maxwell’s equation solver CST studio, the periodic boundary conditions are used in the $x$ and $y$ directions, Floquet boundary conditions are used in the $z$ direction, and perfect electric conductor boundary conditions are used in the -$z$ direction to extract only the reflection coefficient of the metasuraface. The corresponding numerical analysis results of the unit cell under the different incident angles from $0^{\circ }$ to $40^{\circ }$ is depicted in Fig. 3(a). We can see that in the interested frequency range (0.13THz-0.19THz) the magnitude of the reflective coefficients is higher than the 0.88 even under the big oblique incident angle up to the $40^{\circ }$. The corresponding phase degree also demonstrates the robust angular tolerance in the whole frequency range. For an anisotropic unit cell in metasurfaces, which rotates an angle of $\phi$, and considering a circularly polarized terahertz wave incident in the normal direction, the corresponding reflection wave phase can cover the full 2$\pi$ range. Furthermore, the phase modulation presents a suitable linearity property with respect to the rotation angle $\phi$. It is noteworthy that these super characteristics including the broadband should be attributed to the excellent P-B phase modulation ability.
Fig. 3. (a) Magnitude and phase of reflection coefficients of the proposed unit cell under the oblique incidence of the circular polarization terahertz wave. (b) Reflection phase of the anisotropic unit cell concerning the rotation at different frequencies.
In order to successfully transform the incident wave to the Bessel beam, the reflective type metasurface needs the essential sophisticated phase distribution. We use the Jones matrix in Eq. (3), (4) to represent the transformation between the incident and reflection wave, where $\phi _{u}$ and $\phi _{v}$ is the propagation phase of two principal directions in the reference coordinates shown in Fig. 3(b), and 2$\phi$ is the so-called P-B phase [61]. Furthermore, the provided constraints, both $A_{u}$ = $A_{v}$ = 1 and $\phi _{u}$-$\phi _{v}\approx \pi$, are needed to keep the high reflection coefficients of the co-polarization while depressing the cross-polarization reflection coefficients. Hence, the detailed procedure of manipulation of reflected co-polarized field using the proposed metasurface shows as follows in Eq. (3)–(8), where $M(\phi )$ is the rotation matrix, and $C$ is the transformation matrix between linear and circular polarization states.
Theoretically, the phase distribution of a vortex beam carrying OAM mode with topological charge $l$ can be described by
where the $\varphi$ is the azimuthal angle, the position of the specific $m$-th unit cell element at the metasurface are assumed as $(x_{m},y_{m})$. Considering the coordinate system detailed in Fig. 4, the progressive phase distribution on the metasurface that produces a beam in the direction ($\theta _{a}$, $\varphi _{a}$), as known from array theory, is expressed as in Eq. (10), where $k_{0}$ is the propagation constant in vacuum.Fig. 4. Coordinate system of the typical metasurface unit cell and relationship with beam direction.
Furthermore, for generation the quasi-Bessel beam exhibiting the non-diffraction characteristic, the semi-apex angle of the cone is assumed to be $\gamma$ to mimick the planar axicon lens to the superposition of the traveling conic wave. Hence the phase pattern can be defined as shown in Eq. (11).
Thus, according to the theory above, the total phase pattern $\eta$ introduced by the meta-atom at $(x_{m},y_{m})$ to produce a bessel beam carrying orbital angular momentum (OAM) propagating along the direction ($\theta _{a}$, $\varphi _{a}$) can be described as in Eq. (12).
Without loss of the generality, Fig. 5 illustrates the final digitized phase profiles matrix calculated by Eq. (9)–(12), which is composed of 31$\times$31 meta-atoms. Fig. 5(a) shows the phase profile of the generation the Bessel beam carrying the OAM using the proposed reflective metasurface with topological charge $l=-1$ and $\gamma =16^{\circ }$. The cone angle $\gamma$ is set for generating the corresponding conic waves. Fig. 5(b) shows the phase profiles of the generation vortex beam carrying the topological charges $l=2$ along the direction $(\theta _{a},\varphi _{a})$=(20,0). Lastly, Fig. 5(c) shows the phase profiles of the Bessel beam carrying the OAM topological charges $l=2$ and cone angle $\gamma =12^{\circ }$ along the direction $(\theta _{a},\varphi _{a})$=(20,0). With all of those phase information in hand, the specific meta-atoms geometry characteristic, which is only related to the rotation angle, positioned at the metasurface array can be easily determined using the P-B phase mechanism.
Fig. 5. Schematic superposition of the phase profiles of the designed functional metasurface (31$\times$31 elements). (a) phase profile of the carrying OAM topological charge $l=-1$, beam in the direction ($\theta _{a}$, $\varphi _{a}$)=(0,0) and the cone angle $\gamma =16^{\circ }$. (b) phase profile of the carrying OAM topological charge $l=2$, beam in the direction ($\theta _{a}$, $\varphi _{a}$)=(20,0) and the cone angle $\gamma =0^{\circ }$. (c) phase profile of the carrying OAM topological charge $l=2$, beam in the direction ($\theta _{a}$, $\varphi _{a}$)=(20,0) and the cone angle $\gamma =12^{\circ }$.
3. Study of the relationship between the lattice types and corresponding OAM mode purity
Present studies on the possible effects of variation lattice types on the non-diffraction vortex beam performance are relatively scarce. Our main aim here is to gain new insights into this problem. Here, we demonstrated the numerical study of three variations of the lattice type, i.e., square lattice, triangular lattice, and concentric ring lattice, which assumes $N_{m}$ (integer) elements are arranged in the $m$-th concentric ring array [57], in the same radius circular disk metasurface. As shown in Fig. 6, each red dot represents the specific unit cell, and the distance of the neighbor unit cell along the blue line is the unit cell length $p$, which is consistent with the previous analysis shown in Fig. 1 for a fair comparison. The prototype of the designed reflective full metasurface array is numerically analyzed using the CST Microwave Studio based on the Finite Difference Time Domain (FDTD) method in the terahertz frequency regime. Each meta-atom is rotated with an angle of $\eta$/2 according to the Eq. (12). The working frequency is set at the 0.161THz, and the input right circular polarized plane wave was simulated to be incident onto the metasurface at an incidence angle of $0^{\circ }$, i.e., $-z$ direction. Without loss of the generality, the circular disk diameter is set as the $15p$=12mm, which means the max unit cell number near the axis of diameter is set as $14$, and the margin on each side is $0.5p$.
Fig. 6. Schematic of several variations of the lattice type in the same circular disk such as (a) square lattice, (b) triangular lattice, (c) concentric ring lattice(assuming $N_{m}$ (integer) elements are arranged in the $m$-th concentric ring array) [62].
Fig. 7(i) illustrates the metasurface of square lattice carrying OAM topological charges $l=2$ and corresponding far-field absolute power in decibel units. It can be clearly observed that the hollow shape with which little energy in the center. The simulation results in Fig. 7(a) and (b) plot the near field E-field intensity real part distributions and absolute distributions in the $zox$-plane. Moreover, in Fig. 7(a), (b) inset three black dot lines show the three $xy$-plane sectional views along the $z$-axis, i.e., 18$mm$, 22$mm$, 26$mm$, respectively. The observation planes are perpendicular to the OAM radiation beam with an area of 7.2$\lambda \times$7.2$\lambda$. Furthermore, as shown in Fig. 7(c)-(h), the doughnut-shaped field intensity and helical phase front with two twists in the phase patterns illustrate that the $2\pi$ phase change in the azimuth direction, i.e., the topological charge of 2. The inner radius of the vortex beam is a crucial factor in demonstrating the characteristics of diffracting divergence of the OAM vortex beam [57]. And Fig. 7(b) inset at 9.3$mm$ shows the inner ring diameter corresponding to the lattice type with $l=2$ is 1.1171$\lambda$, where $\lambda$ is the working wavelength at the 0.161THz.
Fig. 7. Near field of the square lattice carrying OAM metasurface with $l$=2, (a) real part and (b) absolute of E-field. The corresponding normalized abs(E) and phase magnitude mag(E) at different plane along z-axis, (c) and (d) at 26mm, (e) and (f) at 22mm, (g) and (h) at 18mm, respectively. (i) The far field absolute power of the corresponding metasurface in decibel units.
For comparison, numerical simulations are also conducted to study the high-order Bessel beam carrying the same OAM mode number 2 but with different conical angle $\gamma$, e.g., equals 12 and 14, respectively, as shown in Fig. 8–9. Fig. 8(a), 8(b), 9(a) and 9(b) present intensity of the E-field concentrated around the central axis along the propagation direction. Furthermore, as shown in Fig. 8(g) and Fig. 9(g) E-field intensity in the inset $xy$-plane sectional views at 18$mm$ converge more than the case in Fig. 7(g), where conical angle $\gamma$ is zero. In addition, Fig. 8(b) and Fig. 9(b) inset at the same 9.3$mm$ illustrates that the inner ring diameter is 0.6703$\lambda$ and 0.5710$\lambda$, respectively. So the characteristic of the beam diffracting divergence can be clearly reduced to some extent as the feature of introducing a quasi-Bessel beam with cone angle $\gamma$, which indicates successful achieving the quasi non-diffraction functionality. As for the instant comparison, the bigger $\gamma$, the better performance of the reduced divergence. Similarly, other metasurfaces with different lattice types, triangular lattice in Fig. 10–12 and concentric ring lattice-type in Fig. 13–15, have also been conducted using the same analysis procedure as previously. For the convenience of comparison, we adopt the same position at 9.3$mm$, which is the distance between the place of the inset arrow and the metasurface. Both two lattice-type metasurfaces exhibit characteristics of a converging Bessel beam. But the concentric ring lattice performs best on the reduced OAM mode divergence among the three lattice types. And all of the simulation results are excellent and consistent with the expectation. Finally, the mode purity of generated OAM with the different lattice type metasurface is calculated based on the complex Fourier transform [63]. The linking Fourier relationship can be read as follow
where azimuthal angle $\varphi$ is a periodic function, $A_{l}$ is the spectrum component of the expected OAM mode number $l$. Using Eq. (13) the corresponding three lattice type metasurface OAM spectrum purity are depicted in Fig. 16–18. The results show the OAM spectrum purity is relatively dominant in the OAM mode number 2, which agrees well with our expectation in all cases as plotted in Fig. 16–18 regardless of the $\gamma =0$, $\gamma =12$, $\gamma =16$. However, there is a discrepancy between Fig. 16(a), Fig. 17(a), and Fig. 18(a), which were all in the same sectional view plane $z=18mm$. The three lattice types corresponding to OAM purity are 0.93, 0.95, and 1, respectively, which concludes that the OAM mode purity generated by the concentric ring lattice type metasurface has a better performance in the same radius circular disk than the OAM purity value generated by both the square and triangular lattice type. The reason can be attributed to the circular symmetry of the lattice type being a more favorable vortex phase representation. Furthermore, in the same square lattice type shown in Fig. 16(a) and (b), the Bessel beam carrying OAM mode has a higher OAM purity value than carrying the vortex beam only. It indicates the superposition of phase profile of spiral phase and conical Bessel phase exhibit characteristics of an improving mode OAM purity functionality.Fig. 8. Near field of the square lattice Bessel beam carrying OAM metasurface with $l$=2 and $\gamma =12$, (a) real part and (b) absolute of E-field. The corresponding normalized abs(E) and phase magnitude mag(E) at different plane along z-axis, (c) and (d) at 26mm, (e) and (f) at 22mm, (g) and (h) at 18mm, respectively. (i) The far field absolute power of the corresponding metasurface in decibel units.
Fig. 9. Near field of the square lattice Bessel beam carrying OAM metasurface with $l$=2 and $\gamma =16$, (a) real part and (b) absolute of E-field. The corresponding normalized abs(E) and phase magnitude mag(E) at different plane along z-axis, (c) and (d) at 26mm, (e) and (f) at 22mm, (g) and (h) at 18mm, respectively. (i) The far field absolute power of the corresponding metasurface in decibel units.
Fig. 10. Near field of the triangular lattice carrying OAM metasurface with $l$=2, (a) real part and (b) absolute of E-field. The corresponding normalized abs(E) and phase magnitude mag(E) at different plane along z-axis, (c) and (d) at 26mm, (e) and (f) at 22mm, (g) and (h) at 18mm, respectively. (i) The far field absolute power of the corresponding metasurface in decibel units.
Fig. 11. Near field of the triangular lattice Bessel beam carrying OAM metasurface with $l$=2 and $\gamma =12$, (a) real part and (b) absolute of E-field. The corresponding normalized abs(E) and phase magnitude mag(E) at different plane along z-axis, (c) and (d) at 26mm, (e) and (f) at 22mm, (g) and (h) at 18mm, respectively. (i) The far field absolute power of the corresponding metasurface in decibel units.
Fig. 12. Near field of the triangular lattice Bessel beam carrying OAM metasurface with $l$=2 and $\gamma =16$, (a) real part and (b) absolute of E-field. The corresponding normalized abs(E) and phase magnitude mag(E) at different plane along z-axis, (c) and (d) at 26mm, (e) and (f) at 22mm, (g) and (h) at 18mm, respectively. (i) The far field absolute power of the corresponding metasurface in decibel units.
Fig. 13. Near field of the concentric ring lattice carrying OAM metasurface with $l$=2, (a) real part and (b) absolute of E-field. The corresponding normalized abs(E) and phase magnitude mag(E) at different plane along z-axis, (c) and (d) at 26mm, (e) and (f) at 22mm, (g) and (h) at 18mm, respectively. (i) The far field absolute power of the corresponding metasurface in decibel units.
Fig. 14. Near field of the concentric ring lattice Bessel beam carrying OAM metasurface with $l$=2 and $\gamma =12$, (a) real part and (b) absolute of E-field. The corresponding normalized abs(E) and phase magnitude mag(E) at different plane along z-axis, (c) and (d) at 26mm, (e) and (f) at 22mm, (g) and (h) at 18mm, respectively. (i) The far field absolute power of the corresponding metasurface in decibel units.
Fig. 15. Near field of the concentric ring lattice Bessel beam carrying OAM metasurface with $l$=2 and $\gamma =16$, (a) real part and (b) absolute of E-field. The corresponding normalized abs(E) and phase magnitude mag(E) at different plane along z-axis, (c) and (d) at 26mm, (e) and (f) at 22mm, (g) and (h) at 18mm, respectively. (i) The far field absolute power of the corresponding metasurface in decibel units.
Fig. 16. OAM spectrum purity of the vortex beam with mode $l=2$ generated by square lattice metasurface in the sectional view plane $z=18mm$. (a) $\gamma =0$, (b) $\gamma =12$.
Fig. 17. OAM spectrum purity of the vortex beam with mode $l=2$ generated by triangular lattice metasurface in the sectional view plane $z=18mm$. (a) $\gamma =0$, (b) $\gamma =16$.
Fig. 18. OAM spectrum purity of the vortex beam with mode $l=2$ generated by concentric ring lattice metasurface in the sectional view plane $z=18mm$. (a) $\gamma =0$, (b) $\gamma =16$.
4. Conclusion
In summary, quasi non-diffraction vortex beams carrying OAM modes generation in the terahertz regime have been numerically studied using the proposed reflective metasurface based on the high-order Bessel inherently having high-order OAM modes. The Jones matrix and Pancharatnam-Berry phase have been used to help understand the whole broadband transformation characteristic. The unit cell and array simulation results agree well with the theoretical expectation. The diffracting divergence of OAM vortex beam characteristics has been alleviated. The three types, i.e., square lattice, triangular lattice, and concentric ring lattice, have been conducted to compare the performance of generating the OAM purity. The results show that the concentric ring lattice has more advantages in developing high purity OAM under the same radius circular disk condition. Furthermore, this study also indicates high-order Bessel vortex beam has an improving OAM purity performance than the case carrying OAM beam only. The insights of this study are helpful for the development of the high-performance OAM generation device both in scientific research and industry application.
Funding
Natural Science Foundation of Chongqing (cstc2021jcyj-bsh0198); National Natural Science Foundation of China (51877015, 62001068, U1831117).
Acknowledgments
X. Wu thanks the Project supported by the graduate research and innovation foundation of Chongqing, China (Grant NO. CYS17039), Project NO. 2018CDYJSY0055, NO. 106112017CDJXSYY0002 Supported by the Fundamental Research Funds for the Central Universities and supports from the China Scholarship Council (NO.CSC201806050092).
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.
References
1. Z. Zhang, Y. Xiao, Z. Ma, M. Xiao, Z. Ding, X. Lei, G. K. Karagiannidis, and P. Fan, “6G Wireless Networks: Vision, Requirements, Architecture, and Key Technologies,” IEEE Veh. Technol. Mag. 14(3), 28–41 (2019). [CrossRef]
2. M. Z. Chowdhury, M. Shahjalal, M. K. Hasan, and Y. M. Jang, “The Role of Optical Wireless Communication Technologies in 5G/6G and IoT Solutions: Prospects, Directions, and Challenges,” Appl. Sci. 9(20), 4367 (2019). [CrossRef]
3. A. Sawant, I. Lee, B. C. Jung, and E. Choi, “Ultimate Capacity Analysis of Orbital Angular Momentum Channels,” IEEE Wireless Commun. 28(1), 90–96 (2021). [CrossRef]
4. A. Minoofar, X. Su, H. Zhou, R. Zhang, F. Alishahi, K. Zou, H. Song, K. Pang, S. Zach, M. Tur, A. F. Molisch, H. Sasaki, D. Lee, and A. E. Willner, “Experimental Demonstration of Sub-THz Wireless Communications Using Multiplexing of Laguerre-Gaussian Beams When Varying two Different Modal Indices,” J. Lightwave Technol. 40(10), 3285–3292 (2022). [CrossRef]
5. A. J. Ali, M. Khalily, A. Araghi, S. E. Hosseininejad, and R. Tafazolli, “A Circular Reflectarray for OAM Generation at Terahertz Regime for 6G Applications,” in 2021 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (APS/URSI), (2021), pp. 1599–1600.
6. A. Sawant, I. Lee, and E. M. Choi, “Amplitude Non-uniformity of Millimeter Wave Vortex Beams Generated by Transmissive Structures,” IEEE Trans. Antennas Propag. 70(6), 4623–4631 (2022). [CrossRef]
7. S. Tian, X. Zhang, X. Wang, J. Han, and L. Li, “Recent advances in metamaterials for simultaneous wireless information and power transmission,” Nanophotonics 11(9), 1 (2022). [CrossRef]
8. A. A. A. Solyman and K. Yahya, “Key performance requirement of future next wireless networks (6G),” Bull. Electr. Eng. Informatics 10(6), 3249–3255 (2021). [CrossRef]
9. A. L. Imoize, O. Adedeji, N. Tandiya, and S. Shetty, “6G Enabled Smart Infrastructure for Sustainable Society: Opportunities, Challenges, and Research Roadmap,” Sensors 21(5), 1709 (2021). [CrossRef]
10. C. X. Liu, F. Yang, X. J. Fu, J. W. Wu, L. Zhang, J. Yang, and T. J. Cui, “Programmable Manipulations of Terahertz Beams by Transmissive Digital Coding Metasurfaces Based on Liquid Crystals,” Adv. Opt. Mater. 9(22), 2100932 (2021). [CrossRef]
11. C. Chaccour, M. N. Soorki, W. Saad, M. Bennis, P. Popovski, and M. Debbah, “Seven Defining Features of Terahertz (THz) Wireless Systems: A Fellowship of Communication and Sensing,” IEEE Commun. Surv. Tutorials 24(2), 967–993 (2022). [CrossRef]
12. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]
13. M. J. Padgett, “Orbital angular momentum 25 years on [Invited],” Opt. Express 25(10), 11265–11274 (2017). [CrossRef]
14. R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity, optical spin and related quantities in electromagnetic theory,” New J. Phys. 14(5), 053050 (2012). [CrossRef]
15. G. Xie, L. Li, Y. Ren, H. Huang, Y. Yan, N. Ahmed, Z. Zhao, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, “Performance metrics and design considerations for a free-space optical orbital-angular-momentum–multiplexed communication link,” Optica 2(4), 357 (2015). [CrossRef]
16. A. Luski, Y. Segev, R. David, O. Bitton, H. Nadler, A. R. Barnea, A. Gorlach, O. Cheshnovsky, I. Kaminer, and E. Narevicius, “Vortex beams of atoms and molecules,” Science 373(6559), 1105–1109 (2021). [CrossRef]
17. D. M. Fatkhiev, M. A. Butt, E. P. Grakhova, R. V. Kutluyarov, I. V. Stepanov, N. L. Kazanskiy, S. N. Khonina, V. S. Lyubopytov, and A. K. Sultanov, “Recent Advances in Generation and Detection of Orbital Angular Momentum Optical Beams—A Review,” Sensors 21(15), 4988 (2021). [CrossRef]
18. L. Kang, H. Li, J. Zhou, S. Zheng, and S. Gao, “A Mode-Reconfigurable Orbital Angular Momentum Antenna With Simplified Feeding Scheme,” IEEE Trans. Antennas Propag. 67(7), 4866–4871 (2019). [CrossRef]
19. J.-U. Yoo and H.-W. Son, “Quad-mode radial uniform circular array antenna for OAM multiplexing,” IET Microwaves, Antennas & Propag. 14(8), 728–733 (2020). [CrossRef]
20. M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature 464(7289), 737–739 (2010). [CrossRef]
21. R. Niemiec, C. Brousseau, K. Mahdjoubi, O. Emile, and A. Menard, “Characterization of an OAM Flat-Plate Antenna in the Millimeter Frequency Band,” Antennas Wirel. Propag. Lett. 13, 1011–1014 (2014). [CrossRef]
22. P. Genevet, J. Lin, M. A. Kats, and F. Capasso, “Holographic detection of the orbital angular momentum of light with plasmonic photodiodes,” Nat. Commun. 3(1), 1278 (2012). [CrossRef]
23. X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012). [CrossRef]
24. G. Ruffato, “OAM-inspired new optics: The angular metalens,” Light: Sci. Appl. 10(1), 96 (2021). [CrossRef]
25. H. Zhou, B. Sain, Y. Wang, C. Schlickriede, R. Zhao, X. Zhang, Q. Wei, X. Li, L. Huang, and T. Zentgraf, “Polarization-Encrypted Orbital Angular Momentum Multiplexed Metasurface Holography,” ACS Nano 14(5), 5553–5559 (2020). [CrossRef]
26. Z. Xiang, Z. Shen, and Y. Shen, “Quasi-perfect vortices generated by Pancharatnam-Berry phase metasurfaces for optical spanners and OAM communication,” Sci. Rep. 12(1), 1053 (2022). [CrossRef]
27. W. Wang, R. Zhao, S. Chang, J. Li, Y. Shi, X. Liu, J. Sun, Q. Kang, K. Guo, and Z. Guo, “High-Efficiency Spin-Related Vortex Metalenses,” Nanomaterials 11(6), 1485 (2021). [CrossRef]
28. H. Oraizi and H. Emamian, “Generation of orbital angular momentum modes via holographic leaky-wave metasurfaces,” Sci. Rep. 10(1), 7358 (2020). [CrossRef]
29. L. Wang, H. Shi, J. Yi, L. Dong, H. Liu, A. Zhang, and Z. Xu, “Suspended Metasurface for Broadband High-Efficiency Vortex Beam Generation,” Materials 15(3), 707 (2022). [CrossRef]
30. Y. Guo, S. Zhang, M. Pu, Q. He, J. Jin, M. Xu, Y. Zhang, P. Gao, and X. Luo, “Spin-decoupled metasurface for simultaneous detection of spin and orbital angular momenta via momentum transformation,” Light: Sci. Appl. 10(1), 63 (2021). [CrossRef]
31. X. Qi, Z. Zhang, X. Zong, X. Que, Z. Nie, and J. Hu, “Generating Dual-Mode Dual-Polarization OAM Based on Transmissive Metasurface,” Sci. Rep. 9(1), 97 (2019). [CrossRef]
32. S. Zhang, P. Huo, W. Zhu, C. Zhang, P. Chen, M. Liu, L. Chen, H. J. Lezec, A. Agrawal, Y. Lu, and T. Xu, “Broadband Detection of Multiple Spin and Orbital Angular Momenta via Dielectric Metasurface,” Laser Photonics Rev. 14(9), 2000062 (2020). [CrossRef]
33. H. Ren and S. A. Maier, “Nanophotonic Materials for Twisted-Light Manipulation,” Adv. Mater. 1, 2106692 (2022). [CrossRef]
34. H. Ren, X. Fang, J. Jang, J. Bürger, J. Rho, and S. A. Maier, “Complex-amplitude metasurface-based orbital angular momentum holography in momentum space,” Nat. Nanotechnol. 15(11), 948–955 (2020). [CrossRef]
35. A. H. Dorrah, N. A. Rubin, M. Tamagnone, A. Zaidi, and F. Capasso, “Structuring total angular momentum of light along the propagation direction with polarization-controlled meta-optics,” Nat. Commun. 12(1), 6249 (2021). [CrossRef]
36. S. Chen, K. Li, J. Deng, G. Li, and S. Zhang, “High-Order Nonlinear Spin–Orbit Interaction on Plasmonic Metasurfaces,” Nano Lett. 20(12), 8549–8555 (2020). [CrossRef]
37. A. H. Dorrah and F. Capasso, “Tunable structured light with flat optics,” Science 376(6591), eabi6860 (2022). [CrossRef]
38. J. Cheng, X. Sha, H. Zhang, Q. Chen, G. Qu, Q. Song, S. Yu, and S. Xiao, “Ultracompact Orbital Angular Momentum Sorter on a CMOS Chip,” Nano Lett. 22(10), 3993–3999 (2022). [CrossRef]
39. R. Feng, B. Ratni, J. Yi, H. Zhang, A. de Lustrac, and S. N. Burokur, “Versatile metasurface platform for electromagnetic wave tailoring,” Photonics Res. 9(9), 1650 (2021). [CrossRef]
40. K. Zhang, Y. Yuan, X. Ding, H. Li, B. Ratni, Q. Wu, J. Liu, S. N. Burokur, and J. Tan, “Polarization-Engineered Noninterleaved Metasurface for Integer and Fractional Orbital Angular Momentum Multiplexing,” Laser Photonics Rev. 15(1), 2000351 (2021). [CrossRef]
41. K. Zhang, Y. Wang, S. N. Burokur, and Q. Wu, “Generating Dual-Polarized Vortex Beam by Detour Phase: From Phase Gradient Metasurfaces to Metagratings,” IEEE Trans. Microwave Theory Tech. 70(1), 200–209 (2022). [CrossRef]
42. K. Zhang, Y. Wang, Y. Yuan, and S. N. Burokur, “A Review of Orbital Angular Momentum Vortex Beams Generation: From Traditional Methods to Metasurfaces,” Appl. Sci. 10(3), 1015 (2020). [CrossRef]
43. M. J. Padgett, F. M. Miatto, M. P. J. Lavery, A. Zeilinger, and R. W. Boyd, “Divergence of an orbital-angular-momentum-carrying beam upon propagation,” New J. Phys. 17(2), 023011 (2015). [CrossRef]
44. Y. Zhang, X. Liang, W. Qi, H. Wu, Q. Chen, J. Geng, and W. Wang, “Realization of Multimode OAM Beams with Almost the Same Divergence Angles,” Int. J. Antennas Propag. 2021, 1–10 (2021). [CrossRef]
45. S. Chavez-cerda, “A new approach to bessel beams,” J. Mod. Opt. 46, 923–930 (1999). [CrossRef]
46. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef]
47. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987). [CrossRef]
48. R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8(6), 932 (1991). [CrossRef]
49. M. Ettorre, S. M. Rudolph, and A. Grbic, “Generation of Propagating Bessel Beams Using Leaky-Wave Modes: Experimental Validation,” IEEE Trans. Antennas Propag. 60(6), 2645–2653 (2012). [CrossRef]
50. C. Pfeiffer and A. Grbic, “Controlling Vector Bessel Beams with Metasurfaces,” Phys. Rev. Appl. 2(4), 044012 (2014). [CrossRef]
51. J. Han, L. Li, H. Yi, and W. Xue, “Versatile orbital angular momentum vortex beam generator based on reconfigurable reflective metasurface,” Jpn. J. Appl. Phys. 57(12), 120303 (2018). [CrossRef]
52. S. Jiang, C. Chen, J. Ding, H. Zhang, and W. Chen, “Alleviating Orbital-Angular-Momentum-Mode Dispersion Using a Reflective Metasurface,” Phys. Rev. Appl. 13(5), 054037 (2020). [CrossRef]
53. Q. Feng, Y. Lin, M. Shan, Y. Mu, and L. Li, “Generation and Measurement of a Bessel Vortex Beam Carrying Multiple Orbital-Angular-Momentum Modes through a Reflective Metasurface in the rf Domain,” Phys. Rev. Appl. 15(6), 064044 (2021). [CrossRef]
54. X. Meng, X. Chen, R. Chen, H. Li, T. Qu, and A. Zhang, “Generation of Multiple High-Order Bessel Beams Carrying Different Orbital-Angular-Momentum Modes through an Anisotropic Holographic Impedance Metasurface,” Phys. Rev. Appl. 16(4), 044063 (2021). [CrossRef]
55. X. He, Y. Yang, L. Deng, S. Li, and B. Feng, “3D Printed Sub-Terahertz All-Dielectric Lens for Arbitrary Manipulation of Quasi-Nondiffractive Orbital Angular Momentum Waves,” ACS Appl. Mater. Interfaces 13(17), 20770–20778 (2021). [CrossRef]
56. J. Lin, C. Chen, J. Ding, S. Wang, and W. Chen, “Dual-frequency multiple compact vortex beams generation based on single-layer Bi-spectral metasurface,” Appl. Phys. Lett. 119(8), 081905 (2021). [CrossRef]
57. K. Zhang, Y. Yuan, D. Zhang, X. Ding, B. Ratni, S. N. Burokur, M. Lu, K. Tang, and Q. Wu, “Phase-engineered metalenses to generate converging and non-diffractive vortex beam carrying orbital angular momentum in microwave region,” Opt. Express 26(2), 1351 (2018). [CrossRef]
58. H.-H. Hsiao, C. H. Chu, and D. P. Tsai, “Fundamentals and Applications of Metasurfaces,” Small Methods 1(4), 1600064 (2017). [CrossRef]
59. X. Wu, H. Cao, Z. Meng, and Z. Sun, “Ultra-broadband Pancharatnam-Berry phase metasurface for arbitrary rotation of linear polarization and beam splitter,” Opt. Express 30(9), 15158 (2022). [CrossRef]
60. D. Isakov, Y. Wu, B. Allen, P. S. Grant, C. J. Stevens, and G. J. Gibbons, “Evaluation of the Laguerre–Gaussian mode purity produced by three-dimensional-printed microwave spiral phase plates,” R. Soc. Open Sci. 7(7), 200493 (2020). [CrossRef]
61. Y. Gou, H. F. Ma, L. W. Wu, Z. X. Wang, P. Xu, and T. J. Cui, “Broadband Spin-Selective Wavefront Manipulations Based on Pancharatnam–Berry Coding Metasurfaces,” ACS Omega 6(44), 30019–30026 (2021). [CrossRef]
62. G. Ding, K. Chen, T. Jiang, B. Sima, J. Zhao, and Y. Feng, “Full control of conical beam carrying orbital angular momentum by reflective metasurface,” Opt. Express 26(16), 20990 (2018). [CrossRef]
63. B. Jack, M. J. Padgett, and S. Franke-Arnold, “Angular diffraction,” New J. Phys. 10(10), 103013 (2008). [CrossRef]
