Prephase-based Equivalent Amplitude Tailoring for Low Sidelobe Levels of 1-Bit Phase-Only Control Metasurface under Plane Wave Incidence

A prephase synthesis method is proposed for side-lobe level (SLL) suppression of a 1-bit phase-only control metasurface under plane wave incidence. The array factor of the metasurface with N × N unit cells shows that controlling the number of prephases with varying values over the reflective surface can equivalently control the amplitude. Different from optimizing the prephase distribution, selection of the 0 and π/2 prephases in specific N regions is used to suppress the SLLs. Therefore, the number of parameters in the optimization can be dramatically reduced from N2 to N. The prephase distribution is then designed based on the optimized number of prephases and a symmetric matrix for SLL suppression in the entire space. The SLLs are further suppressed by optimizing some of the unit cell states based on similar equivalent-amplitude tailoring. Simulations and measurements of a set of 1-bit reflective metasurfaces with 20 × 20 unit cells verify that the phase-only control metasurface realizes SLL suppression to -13 dB for multiple beam directions from -30 to 30 degrees with a 10-degree step under normal plane wave incidence.

Compared with 2-bit unit cells, the 1-bit unit cell can reduce the number of switches (such as the PIN) in the unit cell design and simplify the circuit control. Therefore, the loss and cost caused by the switches and circuits can be reduced, especially in a high-frequency band, for example, the mmWave band.
For mmWave communications, systems focus electromagnetic wave energy in desired directions and suppress the energy in other directions [7]. In this case, a pencil beam with low sidelobe levels (SLLs) is preferred. Pencil beams with a high gain can be realized using reflective metasurfaces excited by spherical waves. The spherical waves introduce an intrinsic pseudorandom distribution of the phase quantization error to form a single beam [8], [9]. Furthermore, the spherical wave incidence results in a nonuniform amplitude distribution on the reflective surfaces, which helps suppress SLLs. For example, a 1-bit reflective metasurface based on a linear polarizer can achieve a fixed beam with an SLL of -14 dB [10], and a 1-bit reconfigurable reflective metasurface can achieve beam scanning with an SLL of -10 dB [11]. One-bit phase-controlled reflective metasurfaces under spherical wave incidence can usually realize single-beam patterns with SLLs from -20 to -10 dB [10]- [15].
Reflective surfaces also work under plane wave incidence [16], such as when controlling the propagation paths of electromagnetic (EM) waves for mmWave indoor and outdoor communications [17], [18]. Unlike spherical wave incidence, plane wave incidence brings a uniform amplitude distribution on the reflective surface. Moreover, the initial phases on the reflective surfaces are the same without a gradient for normal plane wave incidence. The same initial phases usually cause symmetric beams for single-beam forming when using 1-bit phase compensation. Even if oblique wave incidence brings a gradient initial phase distribution, the SLLs are usually very high because of quasiperiodic phase quantization errors [9]. Therefore, 1-bit single-beam design and SLL suppression are more challenging under plane wave incidence than under spherical wave incidence. Although 1-bit reflective metasurfaces with randomized phase quantization errors can effectively suppress symmetric beams [19], the worst SLL is larger than -8 dB for beam scanning [20]. One reason is the amplitude distribution on the reflective surfaces is uniform under the plane waves. Another reason is the fixed pseudorandom uniform prephase distribution cannot ensure SLL suppression in all beam directions.
An SLL suppression method for 1-bit single-beam reflective metasurfaces under normal plane wave incidence is proposed in this work. First, a theoretical analysis proves that control of the prephase distribution can be to some degree equivalent to amplitude control. By controlling the number of two different prephases rather than optimizing the prephase distribution, the optimization can be reduced from N 2 parameters to N parameters. Next, the phase design and optimization of the metasurface are described in detail, both for the prephase distribution and the 1-bit states of unit cells. Finally, a set of 1bit single-beam reflective metasurfaces with the same prephase distribution for different beam directions with suppressed SLLs are designed, fabricated, and measured.
Notations: Upper (Lower) bold-face letters are used to denote matrices (vectors). The (m, n)-th element of the matrix X is denoted as X m,n , and the k-th element of the vector x is denoted as x k .

II. THEORETICAL ANALYSIS FOR PREPHASE-BASED EQUIVALENT AMPLITUDE TAILORING
Assume that the 1-bit reflective metasurface has N ×N unit cells with a uniform interelement spacing (IES) of d, as shown in Fig. 1. The unit cells are simplified as point sources. That is, only the array factor is considered in the theoretical study. The prephase is introduced to each unit cell for single beamforming [20]. Under normal plane wave incidence, all unit cells have a uniform amplitude and the same initial phase, and the array factor AF(θ, φ) can be derived as where θ and φ represent the elevation angle and the azimuth angle, respectively, and s, k = 1, 2, . . . , N . The total phase of the unit cell Φ s,k (θ, φ) in the s-th row and k-th column is the sum of the 1-bit phase compensation ϕ (s,k) , the prephase ψ (s,k) , and the progressive phase ζ (s,k)(θ,φ) , that is, where and λ is the wavelength of the carrier frequency in a vacuum.
To maximize the energy in the desired beam direction, the ideal phase compensation ϕ ′ (s, k) for each unit cell is usually calculated by where (θ i , φ i ) is the designed beam direction (i = 1, 2, 3, . . .) and constant ϕ i is the reference phase [8]. Without loss of generality, the states of the unit cell can be determined: For normally incident plane waves, once the prephase distribution and designed beam direction (θ i , φ i ) are given, the states of the unit cells are determined by (4) and (5). In this work will focus on single-beam in the xoz plane (φ i = 0) and yoz plane (φ i = π/2). For different i, the unit cell states usually change but with the same prephase. The SLL can be obtained from AF(θ, φ).
where {p 1 ,p 2 ,p 3 , · · · } is a set of local maxima of AF(θ, φ), The SLL depends on the prephase distribution ψ s,k and 1bit phase compensation ϕ s,k (states of unit cells). Because ϕ s,k can be calculated when ψ s,k and the beam direction (θ i , φ i ) are given, the optimization of g(ψ s,k , ϕ s,k ) can be presented as arg max The SLLs are determined by the prephase distribution ψ s,k with N 2 parameters (assuming that the reference phase is optimal). In the xoz plane, φ = 0, so (3) can be simplified as and ζ s,k is the same for the same column (k) because of the orthogonality of the x-direction and y-direction. Furthermore, according to (4), the states of unit cells for the same prephase and column are the same if the designed beam is in the xoz plane. As discussed in [20], at least two kinds of prephases should be introduced for single-beam forming, and a π/2 phase difference can achieve a noneven function of AF(θ, φ) and break the symmetry. Thus, 0 and π/2 are chosen as the two kinds of prephases in this study. If there are x k (x k ∈ {0, 1, 2, ..., N }) prephases of 0 and (N − x k ) prephases of π/2 in the k-th column, then AF(θ, 0) can be derived as This article has been accepted for publication in IEEE Transactions on Antennas and Propagation. This is the author's version which has not been fully edited and content may change prior to final publication. where ψ k1 and ψ k2 ∈ (0, π) represent the 1-bit phase compensation of the unit cells. The SLLs in the xoz plane only depend on parameters x k , and the optimization of (7) can be simplified as arg max where g xoz (x k ) represents the SLL in the scanning plane (xoz plane) with φ i = 0 and φ = 0. Let |AF k (θ, 0)| represents the amplitude contribution of the k-th column and (11) Considering 1-bit phase compensation, (ϕ k1 − ϕ k2 ) can be chosen only among −π, 0, and π. Therefore, (11) can be further simplified to where |AF k (θ, 0)| can range from √ 2N/2 to N with different x k . Therefore, if only the 2D patterns of the xoz plane are considered, the equivalent amplitudes |AF k (θ, 0)| can be controlled using different numbers of prephases of 0 and π/2. For example, when all prephases in the k-th column are the same, |AF k (θ, 0)| will obtain the maximum value; when the number of 0 prephases is the same as the number of π/2 prephases, |AF k (θ, 0)| will obtain the minimum value. That is, the SLLs can be improved by controlling the equivalent amplitudes, |AF k (θ, 0)|, and the vector x = [x 1 , x 2 , · · · , x k ] determines the SLLs in the xoz plane.
However, because of 1-bit phase compensation, in the design of the prephase distribution, suppression of symmetric beams should also be considered. The equivalent phases of AF k (θ, 0) are usually different for different k. Some classical amplitude distributions, such as the Chebyshev distribution, usually default that the unit cells can provide perfect phases (0 to 2π) and amplitudes (0 to 1). Therefore, these classical amplitude distributions may not be suitable for this design. The phase design and SLL suppression are discussed in detail in the next section.

III. SLL OPTIMIZATION FOR A PLANAR ARRAY
In the design of the phase-only control metasurface, the number of unit cells and the interelement spacing also influence the SLLs. For comparison with the results in [20], the same setup with N = 20 and d = λ/2 is discussed as an example. The designed scanning range is from -30 to 30 degrees with a 10-degree step, and all unit cells have a uniform amplitude.
The workflow of SLL suppression is illustrated in Fig.  2. For the desired beam scanning range, a fixed prephase distribution should be designed for different beam directions (θ i , φ i ). Based on the theory in Section II and inspired by the concept of dimensionality reduction (DR) [21], [22], the optimization of ψ s,k is transformed into the optimization of x k , and a 2D planar array is converted into a 1D linear array. The method considers the equivalent amplitude and retains the pattern messages after DR. Then, a symmetric matrix is utilized to recover the planar array from the linear array to ensure beam scanning with the suppressed SLLs in the xoz and yoz planes. Next, the states of the 1-bit unit cells for the desired beam directions are designed considering the gain and SLL. Finally, SLL suppression is achieved by controlling the phase distributions (prephases and states) on the reflective metasurface. The detailed principles and methods are presented as follows.

A. Prephase Optimization for a Planar Array
To obtain the maximal radar cross-section (RCS) in the desired directions, the states of the unit cells can be calculated using (4) and (5) when the prephase, reference phase and beam direction are given. For different directions, the fixed prephase distribution dramatically influences the symmetric beams and SLLs. Although a pseudorandom uniformly distributed prephase can effectively suppress symmetric beams in single-beam forming, it cannot ensure SLL suppression for each beam direction. For example, the worst SLL of a prephased planar array is -7.7 dB for scanning from -30 to 30 degrees with a 10-degree step (N = 20, d = λ/2), as shown in Fig. 3(a). Although reference phases can be introduced in every beam direction to improve the patterns, the worst SLL is still larger than -8.0 dB, as shown in Fig. 3 For a reflective metasurface with N × N unit cells, N 2 prephases can be optimized, which may lead to considerable complexity. According to the analysis in Section II, for the patterns in the beam scanning plane (xoz plane and yoz plane), the control of the prephase distribution can provide equivalent amplitude tailoring, and optimization of the prephase distribution can be transformed into optimization of the number  of 0 and π/2 prephases in each column. Therefore, only N parameters are required for x k optimization, rather than the N 2 prephases for ψ s,k optimization.
In other words, if the designed beams are in the xoz plane, then equivalent amplitude control can be realized when using different numbers of 0 and π/2 phases in each column. The patterns depend on the equivalent modulus and the equivalent phase of AF k (θ, 0). Therefore, x k is optimized first to improve the patterns in the scanning plane, i.e., the xoz plane. The k-th column contains x k elements with 0 prephase and (N − x k ) elements with π/2 prephase (k = 1, 2, 3, . . . , N ). Thus, only N parameters need to be optimized, and the prephase design can be simplified. According to (9), the SLLs can be obtained and optimized based on different combinations of x k . Using the differential evolution algorithm (DEA), x = [8,10,5,10,13,11,2,13,9,12,19,6,2,14,11,10,11,9,12,9] is obtained for SLLs less than -13 dB in the scanning plane. Here, DEA is only used as an optimization tool. Other optimization tools can also be used to find x k . This result approaches the theoretical SLLs of phase-controlled point Start Descend the order: x find first x k and indexes a i (i = 1, 2, …, xk) For i = 1:xk Pk, ai = 0 and Pai, k = 0 xai = xai-1 Step 1 Step 2 Step 3 … Step k Step N source arrays with ideal phase compensation. The worst SLL is improved by nearly 5 dB compared with the random prephase distribution cases.
After optimizing x k , the number of 0 and π/2 prephases in each column can be obtained. The main issue for the design is arranging the prephases in each column to suppress SLLs and symmetric beams in the entire space (3D patterns). A symmetric matrix is utilized for the design to solve this issue to support beam scanning in the xoz plane and yoz plane. The symmetric matrix can break the symmetry of the array factor, so the symmetrical beam can be suppressed in 3-D space. Furthermore, for a symmetric matrix, the number of 0 and π/2 prephases is the same for the k-th column and the k-th row. Therefore, the equivalent amplitude control supports beams scanning in both the xoz plane and the yoz plane.
Here, we present a method to generate a symmetric matrix based on x. As shown in Fig. 4, the symmetric matrix can be generated line by line. The initialized matrix P of the prephase distribution is an N × N matrix, and all elements are equal to 1. The design begins with the first row and first column (k = 1). The details are given below: (S1). Find maximal values (first x k ) and the corresponding indexes a i , i = 1, 2, . . . , x k . (S2). Let P k,ai = P ai,k = 0 and x ai = x ai − 1. Additionally, let x k = 0. (S3). If x ai < 0, the vector x cannot be used to generate a symmetric matrix P. Another vector x should be found. (S4). If x ai ≥ 0 and k = N , then the final P is obtained. If x ai ≥ 0 and k < N , then k = k + 1 and return to (S1). The result of the symmetric matrix based on x is shown in Fig. 5. Using the 1-bit phase compensation and the symmetric prephase distribution in Fig. 5, the simulated SLLs are lower than -13 dB in the scanning plane and lower than -12 dB in the entire space for the beam scanning range of -30 to 30 degrees (10-degree step).
Finally, note that the -30 to 30-degree scanning range is analyzed as an example; the method supports other scanning ranges. For a larger scanning range in the xoz and yoz planes, the prephased-based equivalent amplitude tailoring method can also suppress SLLs physically. However, the unit cell needs support phase compensation and scattering in large reflective angles for the metasurface design. Because the proposed method focuses on SLL suppression for beam scanning in the xoz and yoz planes, the designed prephase distribution can support single-beam scanning in other azimuth planes but with a bit higher SLL.

B. 1-Bit State Optimization for Further SLL Suppression
As discussed in Section II, the states of unit cells can be calculated using (4) and (5). The principle is to obtain the maximum energy in the desired direction. In this part, we  1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  0 0 0 0 1 0 1 0 1 1 1 0 0 1 1 0 0    control some of the unit cell states to suppress the SLLs rather than to obtain the maximum power. Therefore, the SLLs can be further suppressed at the price of reducing the maximum RCS in the desired direction. Theoretically, the states of all unit cells can be optimized to improve the SLL suppression but doing so is very timeconsuming. Based on the priori knowledge, a taper amplitude can help improve the SLL performance, and phase control of the unit cells in a planar array realizes equivalent amplitude control for the scanning plane. So, the idea is to keep the maximize gain at center area of the metasurface and optimize the cells at four corners to sacrifice the gain for the SLL suppression. The final unit cell states for 30-degree beam direction are presented in Fig. 6. The array synthesis is used to calculate the states of the unit cells in the white area, and   Fig. 7. The simulated SLLs in the entire space are better than -14.95 dB for beam scanning after optimization of the states of some of the unit cells.

IV. DESIGN OF A 1-BIT PREPHASE UNIT CELL
To verify the proposed method for designing 1-bit phaseonly control metasurfaces, a 1-bit prephase unit cell is designed. As shown in Fig. 8, two substrate layers of Taconic TLY with ε r1 = 2.2 are utilized, and the FR-28 laminate with ε r2 = 2.8 is used to bond them together. Vias are used for the metallic cavity design, and the unit cell is in the cavity and etched on the second substrate layer. The structure and working principle of the unit cell are similar to those in [23]. Unit cells without four small stubs are State 0, and those with four small stubs are State 1.
Similar to the physical mechanism for the structure in [23], the current distributions on the unit cells for State 0 and State 1 are opposite. Therefore, the metallic cavity reduces the influence between adjacent unit cells, especially for those with opposite states. Furthermore, the via height of the cavity is optimized. The influence between unit cells cannot be suppressed if the height is not sufficient. However, the unit cell performance deteriorates for excessive cavity heights.
Various patch sizes are utilized for the prephase design, which leads to two prephases with a nearly 90 degree phase F r e q u e n c y ( G H z ) 0 d e g . p r e . , S t a t e 0 9 0 d e g . p r e . , S t a t e 0 0 d e g . p r e . , S t a t e 1 9 0 d e g . p r e . , S t a t e 1    Table I. In other words, the stubs of the unit cells control 1-bit phase compensation, and the unit cells with two different sizes realize a 90-degree prephase difference. In contrast to using the propagation phase difference to design the prephases of the metasurface [20], the proposed method enables all unit cells to be on the same layer. Fig. 8 shows the simulated reflective phases of the 1-bit prephased unit cells. Over the frequency range from 36.5 to 38.5 GHz, the unit cells can achieve an approximately 180 degree phase difference for the same prephase and different states and a nearly 90 degree phase difference for different prephases and the same state.

V. RESULTS AND DISCUSSIONS A. Simulation Results
A set of 1-bit prephased reflective metasurfaces with suppressed SLLs are designed according to Sections III and IV. The reflective metasurface with a 30-degree beam direction is presented in Fig. 10 as an example, and the unit cells are placed on the middle copper layer. Fig. 10(a) shows a 3D view of the metasurface, and Fig. 10(b) shows the details of the unit cells based on Figs. 5, 6 and 8. The CST simulation results are presented in Fig. 11 and Fig. 12 for patterns in the scanning plane and 3-D uv-plane, respectively. The simulated SLLs range from -13.8 dB to -12.7 dB for beam scanning from -30 to 30 degrees. One main reason why the SLLs differ from the results in Section IV is that the unit cells are not independent of each other.

B. Measurement Results and Discussion
Three metasurfaces with 10-degree, 20-degree, and 30degree beam directions are chosen for fabrication and testing from the set of designed metasurfaces. The measurement setup is presented in Fig. 13. To obtain approximate incident plane waves, the Tx horn is placed at a distance of 2200 mm away from the sample, and the radius of the motorized rotation stage is 1700 mm. The aperture size of the metasurface is 80 ×80 mm 2 . The horn antenna Rx is aimed at the center of This article has been accepted for publication in IEEE Transactions on Antennas and Propagation. This is the author's version which has not been fully edited and content may change prior to final publication.    the metasurface and rotates around the metasurface. A vector network analyzer (VNA) is used to record the electrical field level of Rx from -80 to 80 degrees in the horizontal plane, and a control PC is utilized to operate the motion controller and record the data from the VNA. A copper plate with the same size as the metasurface is also tested as a reference. Table II presents the simulated and measured RCSs and SLLs for different metasurfaces and the copper plate. Fig. 14 shows the simulated and measured patterns at the design frequency of 37.5 GHz. The measured patterns agree with those for the designed directions, and the measured SLLs in the beam scanning plane are suppressed to below -11 dB. This value is slightly higher than that for the simulated SLLs because of fabrication and measurement setup tolerances. Table III compares the proposed 1-bit reflective metasurface with the reported 1-bit reflective metasurfaces. Under normal plane wave incidence, the SLLs for 1-bit phase-control metasurfaces are usually larger than -10 dB, especially for the worst SLL in the beam scanning range. Additionally, a symmetric beam usually exists in space for single beamforming. The introduction of a fixed random prephase can avoid symmetric beams but cannot suppress SLLs for every beam direction. When the prephase distribution and unit cell states are well optimized and arranged, the SLLs can be effectively suppressed for all designed beam directions. Compared with [20], the proposed SLL suppression method can achieve better SLLs in a larger scanning range. Moreover, the proposed method can significantly reduce the complexity through optimization of the prephase number (numbers of 0 and π/2 prephases in special regions) rather than optimization of the prephase distribution. The symmetric matrix also brings a compact prephase design for 2-D beam scanning.

VI. CONCLUSION
A prephase synthesis method for SLL suppression of 1-bit phase-only control metasurfaces under plane wave incidence has been proposed. The main idea is to design a fixed prephase distribution for the desired beam scanning range, and the critical issue is optimizing the number of 0 and π/2 prephases in special regions rather than directly optimizing the prephase distribution. Optimizing the number of the two prephases leads to equivalent and limited amplitude tailoring for the patterns in the scanning plane, and only N parameters need to be optimized for a metasurface with N × N unit cells. Then, a symmetric matrix can be used to transform the optimized results into a prephase distribution. The 1-bit states of unit cells are calculated and optimized for beamforming and SLL suppression. For the 1-bit reflective metasurface design, various sizes are used to realize a single-layer prephase structure. As a demonstration, a set of 1-bit metasurfaces with a fixed prephase distribution has been designed, fabricated, and measured. The simulation and measurement results have shown that the SLLs can be effectively suppressed by optimizing and arranging the prephases to realize equivalent amplitude tailoring. Compared with conventional 1-bit metasurfaces under plane wave incidence, the proposed metasurface can achieve lower SLLs in the designed scanning range.