Phase-engineered metalenses to generate converging and non-diffractive vortex beam carrying orbital angular momentum in microwave region

Kuang Zhang; Yueyi Yuan; Dawei Zhang; Xumin Ding; Badreddine Ratni; Shah Nawaz Burokur; Manjun Lu; Kun Tang; Qun Wu

doi:10.1364/OE.26.001351

1. Introduction

With the rapid progress of wireless communication, the spectral efficiency and channel capacity of communication links are already approaching the Shannon limit [1]. Although high-order modulation technologies and coding methods may increase the spectral efficiency, the effect is only incremental [2–4]. Orbital angular momentum (OAM) describes orbital part of momentum carried by electromagnetic waves, which is associated with spatial distribution and characterized by a helical transverse phase structure possessing information about topological charge of OAM mode. Hence, OAM owns theoretically an infinity of modes and all modes working at the same frequency are mutually orthogonal [5], providing another spatial degree of freedom for enhancing system capacity by several numerical orders in physical level.

The generation and application of OAM modes has been a hot topic in the domain of optics [6–20] and electron beams [21–23]. Recently, studies of OAM have been extended to radio frequency and microwave domain [24–26]. As common and efficient radiating components in low frequency region, versatile antennas and antenna arrays have been explored to introduce azimuthal phase factor into reflected or transmitted waves in order to generate single or multiple OAM mode in far field region [27–31]. However, they both hold specific inherent limitations. For antennas based on twist reflector, the system is usually bulky [32], and the physical structure of the twist reflector can hardly be made electrically tunable, which means the OAM mode generated by antennas based on twist reflector is fixed and cannot be tuned quickly. For antenna arrays, complex feeding network is a must for OAM generation due to acquired phase difference between array elements [27,33–36], which can be a great obstacle when integrated with other equipment. To overcome these drawbacks, planar lenses based on frequency selective surface structure were proposed to mimic spiral phase plate [37,38]. More recently, metasurface with ultra-low profile has also been applied to illuminate OAM mode. Following design procedure of planar reflection array, reflective metasurfaces were designed to generate single and multiple OAM modes in far-field region [39,40]. Double-layered complementary metasurface was proposed to produce OAM mode in transmission manner [41].

The inner radius of vortex beam carrying OAM mode is a key factor to properly receive and identify OAM mode, which is characterized by a doughnut-shaped field intensity with little energy in the center. For application in optical communications, laser beams with high directivity and lenses can be used to overcome the limitation of enlargement of beam radius resulting from diffraction [10–12,18,19]. Nevertheless, in microwave region, current researches are mainly focused on the generation of OAM modes. There is no result showing how to control the radius of beam carrying OAM mode, no matter in near field or far field region. In this paper, two types of vortex beam carrying OAM mode with limited beam radius are demonstrated by metalenses with phase discontinuities at microwave frequencies. Theoretical design is firstly verified by full-wave simulation. Then the prototypes of metalenses are fabricated and measured to experimentally validate the theoretical approach. Both simulations and measurements show that the proposed designs can focus vortex beam carrying OAM mode at specific position and fulfill non-diffracting transmission upon a distance of the 23λ₀, constituting an important result for OAM multiplexing in wireless communication systems.

2. Converging vortex beam carrying OAM mode

The procedure of polarization conversion and manipulation of transmitted cross-polarized field obtained by the proposed metalens is schematically shown in Fig. 1(a). Under normal illumination of incident circularly polarized plane wave, cross-polarized transmitted wave from the metalens can realize specific spatial phase pattern by gathering the transmission electric field of different unit cells, which can be described by Eq. (1) [42]:

| {\vec{E}}_{o u t} 〉 = \sqrt{η_{E}} | {\vec{E}}_{i n} 〉 + (\sqrt{η_{R}} e^{\pm i \cdot 2 θ} | \vec{R} 〉 + \sqrt{η_{L}} e^{\mp i \cdot 2 θ} | \vec{L} 〉)

where

η_{E} = {| \frac{1}{2} (t_{x} + t_{y} e^{i ϕ}) |}^{2}

,

η_{R} = {| \frac{1}{2} (t_{x} - t_{y} e^{i ϕ}) 〈 {\vec{E}}_{i n} | \vec{L} 〉 |}^{2}

and

η_{L} = {| \frac{1}{2} (t_{x} - t_{y} e^{i ϕ}) 〈 {\vec{E}}_{i n} | \vec{R} 〉 |}^{2}

are the polarization order coupling efficiencies,

〈 \cdot | \cdot 〉

represents the inner product and

| \vec{R} (L) 〉

represents the right-handed (left-handed) circular polarized components. t_x and t_y are amplitudes of the transmission coefficients for two linear polarizations, and ϕ is the phase difference between the transmission coefficients. It can be observed in Eq. (1) that the transmitted electric field includes two components, one keeps the original helicity, and the other is transformed into the cross-polarized state. Following the tracing on Poincare sphere, when circularly polarized incident wave is transformed into the cross-polarized wave in the transmitted field, there will be a phase factor when following different routes, which is a geometrical phase called Pancharatnam-Berry (P-B) phase [42,43]. Utilizing this phase factor, arbitrary phase difference between 0 and 2π can be achieved by rotating the unit cell with angle of half of the phase factor according to Eq. (1).

Fig. 1 (a) Schematic principle of the metalens generating focusing vortex beam with OAM mode. The inset shows the unit cell structure where the lattice period a = 11.1 mm, the thickness of substrate with relative permittivity ε_r = 2.2 is w = 2 mm, and θ is the rotation angle of unit cell. (b) Simulated transmission coefficients of the cross-polarized component. (c) Phase changes of the unit cells with different rotation angles under circularly polarized incidence.

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In this work, the unit cell with single-layered microstrip structure is proposed with period a, and all other geometrical parameters of the unit cell are given proportional to a, as shown in the inset of Fig. 1(a). Under left-handed circularly polarized (LCP) incidence, the transmission coefficient of the cross-polarized wave is around 0.5 between 9.5 GHz and 10.5 GHz (Fig. 1(b)), approaching the theoretical limit for single-layered ultra-thin metasurface [44,45]. Meanwhile, the phase factor in the transmitted cross-polarized field shown in Fig. 1(c) can be tailored in the range from 0 to 2π, while the value of the phase factor is twice of rotating angle of the unit cell, which is in accordance with Eq. (1).

To generate a vortex beam carry OAM mode with l, where l is an arbitrary integer representing the topological charge, a phase factor is introduced in azimuth direction by the metalens, which can be described by:

φ_{l} (x, y) = l \cdot arc \tan (y / x)

According to P-B phase principle, this phase factor can be obtained by rotating unit cells with $θ (x, y) = 0.5 φ_{l} (x, y)$ . Then a metalens is constructed by using the unit cell shown in Fig. 1(a), and the topological charge is set to 3. Full-wave simulations are conducted at 10 GHz (λ₀ = 30 mm), based on the finite difference time-domain technique (FDTD), in order to verify the performance of the designed metalens. Figure 2(a) shows the energy distribution along the propagation direction. It can be clearly observed that the pure vortex beam with OAM mode is diverging seriously during the transmission due to the diffraction. Meanwhile, the doughnut-shaped intensity maps can be recognized in xoy plane with different transmitting distance (z = 5λ₀ and z = 1.67λ₀) in Figs. 2(b) and 2(c). The inner diameter of energy ring is enlarged by about two times (2.33λ₀ in z = 1.67λ₀ plane and 4.33λ₀ in z = 5λ₀ plane), which further verifies the characteristic of vortex beam divergence. Moreover, as it can be clearly observed in the insets of Figs. 2(b) and 2(c), there are three twists in the phase patterns, indicating that the phase change in the azimuth direction equals to l*2π. All these results reveal that vortex beam carrying OAM mode with l = 3 is indeed generated by the proposed ultra-thin metalens.

Fig. 2 Simulation and measurement results of the transmitted cross-polarized wave emitted from the metalens generating pure vortex beam and converging vortex beam with topological charge of 3, and photograph of measurement setup. (a) Simulated distribution of energy in xoz plane and xoy plane for (b) z = 5λ₀ and (c) z = 1.67λ₀ for pure vortex beam. (d) Simulated distributions of energy at xoz plane and xoy plane for (e) z = 5λ₀ and (f) z = 1.67λ₀ for converging vortex beam. The insets show the simulated phase distribution in corresponding xoy planes. (g) Measured distributions of energy in xoz plane and xoy plane for (h) z = 5λ₀ and (i) z = 1.67λ₀ for pure vortex beam. (j) Measured distribution of energy in xoz plane and xoy plane for (k) z = 5λ₀ and (l) z = 1.67λ₀ for converging vortex beam. The insets show the measured phase distribution in corresponding xoy plane. (m) The top view and (n) front view of the measurement setup, the distance between horn antenna and fabricated metalens d > 10λ₀.

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As shown in Fig. 2(a), vortex beam carrying OAM mode would expand along propagation direction, which is an obstacle for receiving ends in communication systems. Therefore, it is necessary to control the radius of the vortex beam along the propagation path. A direct method is to focus the vortex beam at a specific position, as schematically shown in Fig. 1(a). In order to achieve this result, the function of ultrathin metalens should be equivalent to an interference pattern between a converging lens and a spiral phase plate, so that the cross-polarized fields emerging from the metasurface can constructively interfere at specific focal plane to produce the focused OAM with specific topological charge. This can be fulfilled by the superposition of a spiral phase plate phase profile and that of a converging lens [10]. The required rotating angle $θ_{O A M + C o n v} (x, y)$ of any unit cell at the position of (x, y) can then be expressed as:

θ_{O A M + C o n v} (x, y) = \frac{1}{2} [l \cdot arc \tan (y / x) + π (\sqrt{f^{2} + (x^{2} + y^{2})} - | f |) / λ_{0}]

where f represents the focal length. Simulations are performed at 10 GHz (λ₀ = 30 mm) on the metalens which is composed of 35 unit cells both in x and y directions and able to generate focusing vortex beam with topological charge of 3 and focal length of 5λ₀ (f = 150 mm). It can be seen in Fig. 2(d) that the energy distribution approaches the center axis during propagation, and is closest to central axis in the presupposed focal plane z = 5λ₀. It can also be observed in Figs. 2(e) and 2(f) that the clear doughnut-shaped energy intensity pattern are obtained, and inner diameter of the energy ring in focal plane (z = 150 mm) is λ₀, while that of energy ring at z = 1.67λ₀ (z = 50 mm) plane is 2.33λ₀, indicating that the vortex beam have responded the particular character of converging lens with focal length of 5λ₀. The insets of Figs. 2(e) and 2(f) show three twists in the phase patterns in corresponding xoy planes, illustrating the topological charge of 3.

For demonstration, prototypes are fabricated using printed circuit board (PCB) technique to verify theoretical method and simulation results. The 2mm-thick substrate used has relative permittivity ε_r = 2.2 and the metallic layer is 0.035 mm thick copper, similar to the settings in simulation models. Field mappings are measured at 10 GHz (λ₀ = 30 mm) in xoy and xoz planes, in order to plot the field intensities and phase distributions. Measurements are also done on vortex beam carrying OAM mode with topological charge of 3 for comparison. The measurement setup is shown in Figs. 2(m) and 2(n). In the measurement system, the dual-polarized horn antenna is used to launch the quasi-plane wave (left-handed circular polarization, LHCP in this paper), which approaches to the plane wave configuration used in simulations. A fibre optic active antenna is used as field probe and is capable to measure both amplitude and phase of the transmitted electric field through the network analyzer. The probe shown in Fig. 2(n) is used as the receiving end whose position is controlled by a motion controller. Both the horn antenna and the field probe are connected to the vector network analyzer, which is adopted to measure the complex S₁₁ and S₂₁ parameters including the amplitude and phase. Here S₁₁ and S₂₁ parameters represent the reflection and transmission coefficient of the electric field, respectively. The orientation of the probe antenna here is set to be vertical and horizontal in order to obtain the two components of the electric fields, and then the RHCP field at one fixed pixel can be obtained, including both amplitude and phase. With the variation of the position of the field probe via the motion controller, the xoy and xoz planes can be covered and the experimental field intensities and phase profiles can be measured.

For pure vortex beam, transmitted vortex beam diverges from the central axis along the propagation direction (Fig. 2(g)), and the measured inner diameters of the energy ring in xoy planes is 4.67λ₀ for z = 5λ₀ and 2.5λ₀ for z = 1.67λ₀ as shown in Figs. 2(h) and 2(i). For the converging vortex beam, the transmitted wave is focused in the focal plane z = 5λ₀ along the propagation direction (xoz plane) as shown in Fig. 2(j). Inner diameters of energy rings in different xoy planes is 1.1λ₀ (z = 5λ₀ plane, Fig. 2(k)) and 2.63λ₀ (z = 1.67λ₀ plane, Fig. 2(l)), respectively. Compared to single OAM mode with the same topological charge, vortex beam carrying the combined phase profile of converging lens is clearly more constrictive, while the diameter of energy ring in xoy plane with z = 5λ₀ is about one fourth of that of pure vortex beam in the same plane, which agrees with simulation results. The phase distributions shown in the inset of Figs. 2(k) and 2(l) also agree with simulations, which validate the theoretical design. The proposed metalens with phase profile of spiral phase plate and converging lens can produce preferable converged vortex beam.

3. Non-diffraction vortex beam carrying OAM mode

Another method to control the incremental radius of vortex beam is to generate a non-diffracting vortex beam which can achieve propagation with highly concentrated energy distribution. High-order Bessel beam, which can be interpreted as zero-order Bessel beam carrying orbital angular momentum [46], can perfectly satisfy this requirement. There are several methods to generate quasi-Bessel beam, such as axicon lenses [47], holographic approach [48] and conical diffraction [49]. Here a metalens, which integrates the function of a spiral phase plate generating vortex beam and an axicon generating zero-order Bessel beam, is proposed in order to obtain high-order Bessel beam and realize non-diffractive vortex beam. Following the operating principle of Bessel beam generated through an axicon lens, maximum non-diffraction distance of transmitted beam is proportional to the radius of the surface of axicon, and inversely proportional to the base angle β of the axicon. Equivalent to the creation of an interference pattern between a vortex beam carrying OAM mode and a Bessel beam, the required rotating angle $θ_{O A M + B e s s e l} (x, y)$ of unit cell at the position of (x, y) can then be calculated based on the phase distribution pattern on the metalens and can be described as:

θ_{O A M + B e s s e l} (x, y) = \frac{1}{2} [l \cdot arc \tan (\frac{y}{x}) + \sqrt{x^{2} + y^{2}} \sin β]

Simulation of the electromagnetic characteristics of metalens that generates non-diffractive vortex beam carrying OAM mode is conducted. The metalens is composed of 25 × 25 unit cells, and the topological charge is set to 1. The simulation results of transmitted cross-polarized energy and phase distributions are shown in Fig. 3. It can be seen that a hollow wave with non-diffractive propagation up to 25λ₀ can be obtained. Energy is concentrated around the central axis along the transmitting direction in the xoz plane (Fig. 3(a)), which results from the interference of cross-polarized field illuminated by the metalens. Figures 3(b), 3(c) and 3(d) present the energy distribution of cross-polarized wave in xoy plane for z = 10λ₀ (z = 300 mm), z = 16.7λ₀ (z = 500 mm), z = 23.3λ₀ (z = 700 mm). The inner diameters of energy rings in the different planes are about 0.83λ₀, 1.13λ₀, and 1.33λ₀ respectively, showing only slight increase and indicating that vortex beam keeps centralized transmission with little diffraction or divergence.

Fig. 3 Simulation and measurement results of the transmitted cross-polarized wave emitted from metalens generating non-diffracting vortex beam with OAM mode having a topological charge of 1. (a) Simulated energy distribution in the xoz plane and xoy plane for (b) z = 10λ₀, (c) z = 16.7λ₀ and (d) z = 23.3λ₀. The insets show simulated phase distribution in the corresponding xoy planes. (e) Measured energy distribution in xoz plane and xoy plane for (f) z = 5λ₀, and (g) z = 20λ₀. The insets show measured phase distribution in the corresponding xoy planes.

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Meanwhile, measured energy intensity along the center axis achieves centralized distribution in xoz plane and non-diffraction transmission feature can be maintained up to a propagation distance of 23λ₀ (Fig. 3(e)). Due to the non-diffraction characteristic of Bessel beam, there is little change in the inner radius of doughnut-shaped energy ring in the xoy plane for z = 5λ₀ (z = 150 mm, Fig. 3(f)), and z = 20λ₀ (z = 600 mm, Fig. 3(g)), which are about 1.17λ₀ and 1.67λ₀, respectively. From the corresponding phase twist in the azimuth direction shown in the insets of Figs. 3(b)–3(d) and Figs. 3(f)–3(g), it can be concluded that non-diffractive vortex beam with topological charge of 1 originating from cross-polarized fields carrying phase profile of the axicon and spiral phase plate is generated by the metalens. The measurement results of the Bessel-like vortex beam are in good agreement with simulation results, which validate the theoretical method.

Furthermore, numerical simulations are conducted to compare the divergence of this Bessel-like vortex beam with that of a Laguerre-Gaussian (L-G) beam emitted by metalenses with same diameter upon the propagating distance of 100λ₀. The simulation result of normalized energy intensity along x-axis changing with propagating distance is shown in Fig. 4. Here the aperture at half-power, a parameter frequently used in microwave region, is considered to represent the divergence of the two beams. It can be seen that beyond the non-diffraction distance, the Bessel-like OAM beam will show divergence upon propagation, but the half-power aperture of Bessel-like OAM beam is always smaller than that of the L-G beam. For the transmission distance of 100λ₀, the half-power aperture of the Bessel-like OAM beam is 55% of that of the L-G beam. This means that, actually, the vortex beam superimposed with the phase pattern of Bessel beam can transmit more constrictively than L-G beam, and it could be an advantage in the adoption of this class of beams for long-range applications.

Fig. 4 Normalized energy intensity of Bessel-like OAM beam and Laguerre-Gaussian beam generated by metalenses with the same diameter in xoy planes for (a) z = 26λ₀, (b) z = 50λ₀ and (c) z = 100λ₀.

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4. Efficiency of the metalenses

The efficiency of the metalens is a key factor for practical applications. Here the efficiency is defined as the ratio of the energy transformed into the cross-polarized vortex beam carrying OAM mode by the metalens to the total incident energy. The efficiency of metalens generating converging vortex beam and the efficiency of metalens generating non-diffraction vortex beam carrying OAM mode are measured and shown in Fig. 5. It can be seen that in the frequency range from 9.6 GHz to 10.4 GHz, the efficiency of both metalenses is basically around 20%, approaching the theoretical limit of 25% for single-layered metasurfaces [44,45]. Here it should be noticed that the efficiency is calculated in term of energy, so the results are in accordance with the simulated transmission coefficient in term of electric field shown in Fig. 1(b). It may pave the way for applications of metalenses in generating vortex beam carrying OAM mode with limited inner radius in microwave region.

Fig. 5 Efficiency of metalenses generating converging vortex beam and non-diffraction vortex beam.

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5. Conclusion

In summary, converging and non-diffractive vortex beam carrying OAM modes with different topological charges have been experimentally demonstrated based on metalenses with phase discontinuities in microwave region. Based on the superposition of phase profile of spiral phase plate and that of a converging lens or an axicon, the inner radius of the vortex beam carrying OAM mode can be limited at specific point or distance, which is equivalent to the generation of an interference pattern between the vortex beam carrying OAM mode and the focusing beam or Bessel beam. Furthermore, the measured efficiency of the metalens is basically around 20% in the frequency range from 9.6 GHz to 10.4 GHz, approaching the theoretical limit of 25%. The non-diffractive distance and the topological charges can be designed to meet different requirements within the limit of Bessel-like OAM beam generated by the metalens. The OAM mode merged with characteristics of focusing beam or Bessel beam constitutes a promising candidate in future microwave communication systems.

Funding

National Natural Science Foundation of China (NSFC) (61771172, 61401122, 61701141), Open project of State Key Laboratory of Millimeter Waves (K201828, K201709), China Postdoctoral Science Foundation (2016M600248), Heilongjiang Post-doctoral Financial Assistance (LBH-Z16065).

Acknowledgements

The authors thank Dr. Yinghong Guan for the valuable discussions.

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Phase-engineered metalenses to generate converging and non-diffractive vortex beam carrying orbital angular momentum in microwave region

Abstract

1. Introduction

2. Converging vortex beam carrying OAM mode

3. Non-diffraction vortex beam carrying OAM mode

4. Efficiency of the metalenses

5. Conclusion

Funding

Acknowledgements

References and links

Cited By

Figures (5)

Equations (4)

Optics Express