## 3.2 Multi-functional full-space wavefront manipulation

In this section, we mainly study the proposed ultrathin single-layer MS by geometric phase principle for full-space multi-functional wavefront manipulation applications. The geometric phase also called Pancharatnam-Berry (PB) phase, which is usually introduced to manipulate the wavefront of the CP wave [32]. The geometric phase MS can always impart equal and opposite phase shifts on the two orthogonal CP states [27], which also shows dispersion less response, and solely depends on the orientation angle of the resonator structure. The geometric phase MS can allow a full coverage of 2π phase shift just by rotating the resonator structure with a certain orientation angle, while the amplitude of the orthogonal CP wave can be fixed [27].

If the unit-cell structure of the designed MS has a local orientation angle denoted by *α*, in which a CP wave will be converted to its orthogonal component or flipped its helicity after reflection or transmission, and an abrupt phase shift of ± 2*α* is imparted. Thus, various CP wavefront manipulation can be realized using the designed MS through a special arrangement of unit-cells with different orientation angle (*α*). To achieve the full-space CP wavefront manipulation of the proposed MS structure, we should firstly exam that if the 2π phase shift is full covered by rotating the *α* of the unit-cell structure while the high amplitudes of the orthogonal CP wave is kept for both reflection and transmission modes.

Figures 4 (a, c) present the simulated amplitudes of the both reflected and transmitted orthogonal CP wave of the proposed structure with eight different orientation angles (*α*) under the normal incident LCP/RCP waves propagating along the -z axis direction. It can be observed that the simulated amplitudes of the reflected and transmitted orthogonal CP wave of the unit-cell structure with different *α* by a step of 22.5° are near equal and always over 0.3, and are also consistent well with the experimental ones (see Figs. 2 (a, c)). The corresponding phase of the both reflected and transmitted orthogonal CP wave of the designed MS structure with difference *α* by a step of 22.5° are depicted in Figs. 4(b,d). It is clearly that the phase shift of the both reflected and transmitted orthogonal CP wave between two adjacent *α* is about 45° in broadband frequency range from 5 GHz to 15 GHz. In addition, the phase variation of the both reflected and transmitted orthogonal CP wave is almost linearly dependent on the change of *α* in above broadband range. Obviously, the full coverage of 2π phase-shift can be realized by varying *α*. Thus, the proposed structure could achieve the broadband CP conversion and phase control of the both reflection and transmission. As proof of the proposed single-layer MS for full-space wavefront manipulation, anomalous reflection and refraction, full-space vortex beam generation and planar focusing effect are demonstrated numerically.

**A. Anomalous reflection and refraction**

Firstly, we study the anomalous reflection and refraction effect of the proposed structure for the incident CP wave. Figure 5 presents the schematic diagrams of a MS supercell composed of eight unit-cells with different *α*, and corresponding amplitudes and phases of the reflected and transmitted orthogonal CP wave at 9 GHz. A linear gradient phase covering 0 to 2π for both reflection and transmission can be realized by a MS supercell. The amplitudes of the orthogonal CP wave for both reflection and transmission from the Unit1 to Unit8 are near the same and close to 0.47, and the corresponding 2π phase coverage can be achieved at 9 GHz. These results indicate that our designed MS supercell can realize the abnormal reflection and refraction by introducing geometric phase principle.

Considering the vacuum environment and normal incident CP wave (*n**i* =1 and *θ**i* = 0°), and from the generalized Snell’s law, the deflection angles of the reflection and transmission can be expressed as [36]:

$${\theta _r}={\sin ^{ - 1}}\left( {\frac{{{\lambda _o}}}{{pN}}} \right)$$

1

$${\theta _t}={\sin ^{ - 1}}\left( {\frac{{{\lambda _o}}}{{pN}}} \right)$$

2

here *θ**r*, *θ**t* and *θ**i* is the reflection, refraction and incident angle of the CP wave, *n**i* is the refractive index of the vacuum environment, *λ*0 is the operation wavelength, *p* is the lattice length of a unit-cell, and *N* is the number of unit-cell within the operation wavelength range. In this design of the MS for the anomalous reflection and refraction, the set *N* = 8, *p* = 15mm, the typical operation frequency *f*1 = 7GHz, *f*2 = 9GHz and *f*3 = 11GHz. Thus, the theoretical calculation *θ**r* and *θ**t* of the anomalous reflection and transmission of the proposed the single-layer MS for the normal incident LCP wave is about 20.92°, 16.12°, and 13.13° at 7 GHz, 9 GHz and 11 GHz, respectively.

To further demonstrate the function of the both anomalous reflection and refraction of the proposed MS for the normal incident CP wave, we present the simulated electric field distributions and normalized intensity of the both reflected and transmitted orthogonal CP (RCP) wave for the normal incident LCP wave propagating along the z-axis direction at 7 GHz, 9 GHz and 11 GHz, respectively, as shown in Figs. 6 and 7. It can be observed that both the reflected and transmitted waves have some oblique directions with respect to the z-axis, revealing that the normal incident LCP wave is converted to its orthogonal CP component and deflected to some anomalous directions. In addition, deflection angles will decrease with the increase of the operation frequency for both reflection and transmission. The deflection angles of the both reflected and transmitted wave are 21°, 16° and 13° at 7 GHz, 9 GHz and 11 GHz, respectively, approaching to the theoretical predictions (20.92°, 16.12°, and 13.13°). As shown in Figs. 6(d) and 7(d), the normalized intensity of both the reflected and transmitted waves are close to unity when the deflection angles are 21°, 16° and 13° at 7 GHz, 9 GHz and 11 GHz, respectively, which are also consistent well with the theoretical ones. It means that the deflection angles of both the reflected and transmitted waves for the proposed single-layer MS under normal incident LCP wave are 21°, 16° and 13° at 7 GHz, 9 GHz and 11 GHz, respectively. Therefore, the proposed single-layer MS can be functioned as a CP wave beam deflector for both reflection and transmission modes.

**B. Vortex beam generation**

Vortex beam carried with orbital angular momentum (OAM) plays a very important in the fields of wireless communications, which can be generated by MS under the illumination of CP wave [20, 28, 41, 42]. A vortex beam carrying OAM generated by the designed MS has a phase distribution of *e*− *ilφ* at the transverse plane, where *φ* is the azimuthal angle and *l* is the topological charge. OAM is related to spiral phase front of space distribution, which is the orbital part of momentum of the vortex beam. In this section, vortex beam carried OAM for both reflection and transmission are demonstrated numerically by the designed MS. To get the expected vortex beam with a spiral phase profile, the unit-cell should be arranged in a spiral shape in the MS. Thus, the required phase distribution of each unit-cell position (*x, y*) should meet the relationship with the azimuthal angle *φ* around the center point (0,0) as follows[42]:

$$\varphi \left( {x,y} \right)=l \cdot \arctan \left( {\frac{y}{x}} \right)$$

3

where *φ*(*x*, *y*) denotes the required phase distribution in designative position, and the *l* presents the desired OAM topological charges, which determines the *l-*th order vortex beam by changing the phase arrangements in MS. The topological charge *l* equals any integer (± 1, ± 2, ±3.. .), and the corresponding azimuthal phase variant can covers phase range of *l*×2π through the special designed MS.

We can get the desired vortex beams with different topological charges through different phase arrangements. In this study, to confirm the full-space vortex beam generations and simplify the proposed single-layer MS design, we divide the MS into eight triangular regions and only consider *l* = + 1 order vortex beam. Figure 8 presents the phase distributions of the proposed MS for both the reflection and transmission modes for the generated vortex beam carried OAM with topological charge *l* = + 1, in which the adjacent regions maintain a phase gradient of π/6. Thus, the phases are ranging from 0 to 2π for a full turn around the beam axis both the reflection and transmission modes. For the generated vortex beam in both reflection and transmission modes, the proposed MS is composed of a 14 × 12 unit-cells and over a total area of 210 × 180 mm2.

The result of reflected and transmitted CP vertex beams from the proposed single-layer MS is numerically simulated based on the finite integration technique (FIT). In practical simulation, the open boundaries in the *x* - and *y -* directions are applied for the whole 2D MS array. In addition, the LCP Gaussian beams as the excitation source are used to illuminate on the *xoy* plane of the designed MS along the -*z* direction to eliminates the truncation effect produced by the edges of the MS [20].

Figures 9 and 10 present the simulated near-field phase and intensity distributions of the reflected and transmitted vertex beams with topological charge *l* = + 1 at 7GHz, 9GHz and 11G Hz, respectively. It is clearly that there is a pair arm of near-field spiral phase and the rotatory directions of the spiral arms are inverse for both reflection and transmission modes at the above three different frequencies as shown in Figs. 9 (a.c.e) and Figs. 10 (a.c.e). These typical features of the near-field spiral phase distributions exists with OAM topological charge *l* = + 1, further revealing that both the reflected and transmitted vertex beams are generated at a broadband frequency range from 7GHz to 11GHz. As shown in Figs. 9 (b.d.f) and Figs. 10 (b.d.f), there is one near zero intensity area generated by the near-field phase singularity in the central region of the vortex beam with topological charge *l* = + 1, which corresponds to the both reflected and transmitted vortex beam at a broadband frequency range. In addition, it is conjectured that the designed single-layer MS can create a series of vortex beams with the higher order topological charge (e.g., *l* = ± 2, ± 3, ±4,....) for the both reflection and transmission in a broadband frequency range (not shown).

To demonstrate the performance of the designed MS for both reflection and transmission vortex beams, we calculated the OAM mode purities of the topological charge *l* = + 1 at three different frequencies based on discrete Fourier transform (DFT) algorithm [20]. The main and secondary mode power of the generated vortex beams with different topological charges can be illustrated through the spectral analysis of Fourier when electric field was projected into the spiral harmonics *e**ilϕ* at three different frequencies [27]. For the normal incident LCP Gaussian wave, the calculated the OAM mode purities of the reflected and transmitted vortex beams with topological charge *l* = + 1 at 7GHz, 9GHz and 11GHz are depicted in Fig. 11. It can be seen that the main OAM mode purities of the reflected vortex beam with the topological charge *l* = + 1 are up to 69.4%, 62.9%, and 55.2%, and the transmission ones are up to 64.9%, 60.1%, and 59.7% at above frequencies. It means that the preset OAM modes are relatively dominant at three different frequencies, revealing that the generated vortex beams have the high quality for both reflection and transmission modes. In addition, the purity of the other OAM modes is less 15% at above three frequencies, revealing that some relative small phase noises are generated inevitably. Thus, it is indisputable that the proposed MS can generate broadband vortex beams for both reflection and transmission modes.

**C. Planar focusing effect**

Meta-lens based on planar focusing effect is regarded as the most promising EM device and practical application of MS, and the function and performance are unavailable in conventional lens [29, 32, 36, 43]. The employ of planar focusing effect of the MS-based meta-lens can significantly simplify many focusing components, which can accelerate EM device integration in modern system. The various MS-based meta-lens can be achieved by special MS with discontinuous phase gradient [43].

In this section, as a proof, we achieved a planar focusing effect for both reflection and transmission modes based on the designed MS structure. In order to realize the purpose of full-space focusing effect for the incident CP wave, the orientation rotation angle of each unit-cell on the *xoy* plane are required to calculate specially to compensate for the phase delay. Generally, the phase distribution for the meta-lens in *xoy* plane can be calculated by equation[43, 44]:

$$\varphi \left( {x,y} \right)=\frac{{2\pi }}{\lambda }\left( {\sqrt {{x^2}+{y^2}+{F^2}} - F} \right)$$

4

where *φ*(*x, y*) represents the phase compensation at special point (*x,y*) of the *xoy* plane, *F* is the focal length and λ is the operation wavelength.

Figure 12 presents the schematic diagram of the special designed MS and phase distributions in *xoy* plane for the both reflection and transmission. As depicted in Fig. 12(a), 13 × 13 unit-cell structures are arranged to form a MS-based meta-lens capable of realizing focusing effect for both reflection and transmission simultaneously. As shown in Fig. 12(b), it can be seen that the distributed phase in *xoy* plane of the proposed single-layer MS can achieve full coverage of the 2π phase in both the *x*- and *y*-axis directions. To demonstrate the full-space focusing effect of the proposed MS-based meta-lens for both reflection and transmission, the theoretical value of focal length is set as 80 mm in advance, and the operation frequency is 7 GHz, 9GHz and 11GHz, respectively. In simulation, the open boundary conditions were applied, and a LCP wave propagating along -z direction was used as excitation source.

Figures 13 and 14 present the simulated electric field (|*E*|) distributions in the *x-z* plane and *x-y* plane, and electrical field intensity (|*E**RCP*|2) profile on focal plane of the reflected and transmitted orthogonal CP wave at 7 GHz, 9GHz and 11GHz, respectively. As shown in Fig. 13(a1-c1), when the incident LCP pane wave is propagating along the z-axis direction, the electric field of the reflected orthogonal CP wave is mainly distributed and concentrated on the position of the z = 74.13 mm, 76.66 mm and 75.27 mm of the *x-z* plane and form an obvious bright spot at 7 GHz, 9GHz and 11GHz, respectively. It means that the practical value of focal length *F**r* of the reflected orthogonal CP wave is about 74.13 mm, 76.66 mm and 75.27 mm at above three frequencies, which is approximately close to the theoretical value of 80 mm. As shown in Figs. 13(a2-c2), there is are obvious bright spot located on the center position of the focal planar at above frequencies, revealing that the EM energy of reflected orthogonal CP wave through the designed MS-based metal-lens are indeed well converged to a focal point in a broadband frequency range. As shown in Figs. 13(a3-c3), the calculated full width half maximums (FWHM) of the electric field intensity (|*E**LCP*|2) along the *x*-axis direction of the reflected orthogonal CP wave is about 33.22 mm, 26.18 mm and 21.49 mm at 7 GHz, 9GHz and 11GHz, respectively, revealing a obvious reflective subwavelength focusing effect for the designed meta-lens in a broadband frequency range. It should be noticed that such a FWHM value strongly depends on the aperture size of our MS-based meta-lens, and can be further reduced by increasing the total number of the unit-cells.

We then numerically characterize the focusing effect of our design at the transmission mode, which is illuminated by an LCP plane wave. Similarly to the reflection mode, the simulated 2D electric field distributions on *x-z* plane and *x-y* plane at the operation frequency clearly reveal the excellent focusing effect at the transmission mode. As shown in Fig. 14(a1-c1), the focal length *F**t* of the transmitted orthogonal CP wave is about 77.64 mm, 77.77 mm and 79.98 mm at 7 GHz, 9GHz and 11GHz, respectively, which is much more close to theoretical value of 80 mm. As shown in Figs. 14(a2-c2), the obvious bright spot located on the center position of the focal planar also can be clearly observed at above frequencies, revealing that the EM energy are indeed well converged to a focal point after transmission through the designed meta-lens in a broadband frequency range. As shown in Figs. 14(a3-c3), the calculated FWHM of the electric field intensity (|*E**LCP*|2) along the *x*-axis direction of the reflected orthogonal CP wave is about 35.56 mm, 26.18 mm and 22.27 mm at 7 GHz, 9GHz and 11GHz, respectively, revealing a obvious transmitted focusing effect for the our meta-lens in a broadband frequency range. In addition, the maximal intensity of the transmitted CP wave is about 22, 62 and 53 times more than the incident one at above three frequencies, revealing a prominent transmittive focusing effect of the designed meta-lens.