Convenient Conformal Transmissive Metasurface Antenna Embedded on Arbitrary Cylindrical Surface

Conformal metasurface antenna embedded on the platform with an arbitrary shape is of great significance for communication systems in practical applications. Herein, a convenient design strategy for conformal transmissive metasurface antenna with arbitrary cylinder surface is presented. First, a novel‐phase compensation concept for curved surface is inspired by ray‐tracing technique. Then, by using arc differentiation and numerical integration, the meta‐atoms’ coordinates calculation method is analyzed for arbitrary cylindrical surface. Next, the electromagnetic responses of planar and curved transmissive meta‐atoms are discussed, and conformal metasurfaces in single‐quadratic surfaces are simulated to validate the feasibility of the design strategy. Finally, for potential applications, three combined quadratics conformal metasurface antennas with capabilities of beam focusing, beam deflecting, and vortex generation are designed and measured. It is indicated in the results that the three conformal metasurface antennas with combined quadratic cylinder surfaces can transform the spherical wave efficiently into the plane wave, deflected beam, and vortex beam, respectively, verifying the validity of the designs. The convenient strategy has good generality and flexibility because it is applicable to arbitrary cylindrical metasurface with a specific alignment function, which has promising perspective in practical applications, and offers a novel approach to design conformal antennas without changing aerodynamic shapes of platforms.


Introduction
[3][4][5] Thanks to their anomalous characteristics, such as super-thin, flexible, and low loss, planar metasurfaces are widely studied and many fascinating phenomena have been demonstrated, such as wave front manipulation, [6][7][8][9] vortex-beam generation, [10][11][12] polarization conversion, [12,13] holograms, [14] radar cross section (RCS) reduction, [15] invisible cloak, [16][17][18][19][20] novel antenna, [21][22][23][24] and so on.Despite the aforementioned progress toward planar devices, conformal techniques have attracted many attentions.[35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51] Ref. [33] proposes glass cylinders that can focus light to a point in near infrared, behaving like aspherical lenses.Luo et al. present a strategy for backward-scattering reduction of wavelength-scale cylindrical target; [34] furtherly, by modulating the p-i-n diodes in meta-atoms, the dynamic control and wideband RCS reduction of the active cylindrical metasurface can be achieved. [35]In ref. [36], the arc metasurface carpet cloak is designed, which can suppress the scattering efficiently and has good stealth performance.In ref. [37], a low-reflection and Conformal metasurface antenna embedded on the platform with an arbitrary shape is of great significance for communication systems in practical applications.Herein, a convenient design strategy for conformal transmissive metasurface antenna with arbitrary cylinder surface is presented.First, a novel-phase compensation concept for curved surface is inspired by ray-tracing technique.Then, by using arc differentiation and numerical integration, the meta-atoms' coordinates calculation method is analyzed for arbitrary cylindrical surface.Next, the electromagnetic responses of planar and curved transmissive meta-atoms are discussed, and conformal metasurfaces in single-quadratic surfaces are simulated to validate the feasibility of the design strategy.Finally, for potential applications, three combined quadratics conformal metasurface antennas with capabilities of beam focusing, beam deflecting, and vortex generation are designed and measured.It is indicated in the results that the three conformal metasurface antennas with combined quadratic cylinder surfaces can transform the spherical wave efficiently into the plane wave, deflected beam, and vortex beam, respectively, verifying the validity of the designs.The convenient strategy has good generality and flexibility because it is applicable to arbitrary cylindrical metasurface with a specific alignment function, which has promising perspective in practical applications, and offers a novel approach to design conformal antennas without changing aerodynamic shapes of platforms.
scattering coding metasurface in THz is proposed, and it still performs well even wrapped around a cylinder.In addition to RCS reduction, scattering enhancements of conformal metasurfaces have also been studied.In ref. [38], dual-beam super-scatterer based on Pancharatnam-Berry cylinder metasurface is achieved, which can realize strong scattering enhancement toward two directions.In ref. [39], scattering enhancement cylinder metasurfaces is designed whose electromagnetic scattering is similar to flat conducting plates.Moreover, several works on polarization conversion and transmissive conformal metasurfaces are studied.In ref. [40], a linear-to-circular polarization metasurface is studied, which is utilized in omnidirectional circular-polarization antenna application.In ref. [41], a transmissive conformal metasurface by using Huygens element is designed which can produce dual beams.Li et al. [42] propose a circular cylinder conformal metasurface lens with radius of 200 mm which can achieve scanning coverage of 60°.Popov et al. [43] discuss how numerical calculation of Green's function can be employed to design conformal spare metasurface, and design semicylindrical spare metasurfaces with radius of 100 mm.Furthermore, Gregoire et al. propose a conformal ellipsoidal holographic antenna by laminating a set of 2D bands. [44]ots of progress have been achieved in conformal metasurface design, but there are still many challenges worthy of further researches.[50] The applicable conformal structures of those methods are seriously limited, and the conformal metasurface in arbitrary cylindrical surface is barely reported.For many application scenarios in practice, such as wireless platform in unmanned aerial vehicles, aircrafts, and so on, contours of them are various considering aerodynamic performance. On the ther hand, numerous works mainly focus on scattering manipulation of curved objects in reflection geometry such as RCS reduction [34][35][36]48] and scattering enhancement.[38,40,51] However, many wireless communication platforms call for conformal high gain antenna, thus conformal transmissive metasurface antennas are in great demands due to their capability to meet aerodynamic requirements.In addition, there are several researches are focusing on high-transmission mechanism, [52][53][54][55][56] such as Huygens' surface [53] and Fabry-Perot resonance, [54] which present the basis of the design for conformal metasurface.Therefore, it is necessary to design high-performance conformal antenna without changing aerodynamic shapes.Generally, contour of aircraft is convexity preserving to achieve good aerodynamic performance, and quadric curves or the combination of them is widely utilized in the design of aircraft outlines.[57] Since the fuselage of the platforms works as the transmitting apertures with antenna located inside, conformal metasurfaces in transmissive schemes following the shapes of mounting platforms are particularly suitable for this application as the part of the aircraft surface, which can improve the performance of antenna placed inside of the aircraft and realize applications of lens and special beam generators.Unfortunately, the phase compensation theories reported [29,38,[42][43][44][45][46][47] are only based on circular cylinder surface and derived from the geometric formula of circular arc, which only apply to the circular cylinder and are not applicable for metasurface with arbitrary cylindrical surface.Consequently, it is of great significance to provide a strategy to design conformal transmissive metasurface with the arbitrary cylindrical surface, especially for quadric cylinder surface.
In this article, a novel arc differential conformal metasurface antenna design strategy for the arbitrary cylindrical surface is proposed and investigated, which greatly increases the profile flexibility of conformal metasurface antenna.First, considering the phase difference caused by the curved profile, the conformal phase compensation method is proposed and discussed by using ray-tracing technique, and the phase compensation problem is transformed into the problem of coordinates calculation.Then, to obtain the coordinates of meta-atoms on curved surface, arc differentiation and numerical integration are utilized to solve uniform point arrangement problem on curve line, and a novel meta-atom coordinate calculation strategy for cylindrical conformal metasurface is proposed and analyzed which applies to arbitrary cylindrical surface.Third, the transmissive meta-atom with orthogonal grating and half wave plates are analyzed; at the same time, electromagnetic responses of planar and conformal meta-atoms are simulated respectively, and the influence of meta-atoms curvature radius on electromagnetic responses is discussed, which provides basis for precise phase compensation.Next, conformal metasurfaces lens antennas with elliptical, hyperbolic, parabolic, and combined quadratic cylinder profiles are designed and simulated.By designing each meta-atom on the basis of calculated position and phase compensation, the wave fronts can be tailored arbitrarily.Finally, considering the combined quadratic outline of aircrafts, combined quadratic metasurfaces lens antennas for beam-focusing, beam-deflecting, vortex-beam generation are fabricated and measured to validate the effectiveness of design strategy.We start from proposing a universal phase compensation method inspired by the ray-tracing technique for arbitrary cylinder surface, which is the key to design convenient conformal metasurface antenna.First, the definition of cylindrical surface should be introduced.Cylinder is a curved surface formed by parallel movements of straight line moving along a curve, in which the curve is the alignment of cylinder.Second, to have a better analysis for the phase compensation theory, principle schematic is depicted in Figure 1, which demonstrates the space geometrics relationships between cylinder surface and feed source in space rectangular coordinate system.C is defined as the arbitrary cylinder conformal metasurface with the alignment of z = f(x), S is a feed point on z axis with coordinates of (0, 0, F), where F is the focal distance of the metasurface.For the convenience of calculation, M is the reference plane parallel with xoy and the expression of M is z = L (L < f(x)).P is an arbitrary point on cylinder surface C with coordinates of (x, y, z), and P 0 is the projecting point of P on reference plane M with coordinates of (x, y, L).Next, for the design of conformal metasurface, the phase compensation should consider not only the light paths between the metasurface and the feed point, but also the distance between C and M caused by the curved structure of surface.Therefore, based on ray-tracing technique, the derivation of phase compensation is as following.

Results and Discussion
Emitted from the feed point, spherical electromagnetic waves pass through the curved surface C and reference plane M successively.Supposing the phase of the wave arriving at M is Θ, and phase compensation value of arbitrary point on metasurface is Ф P , the relationship between Ф P and light path should satisfy the following equations k 0 = 2π/λ 0 refers to the wave number in free space, jSPj (jPP 0 j) refers to the distance between S and P (P and P 0 ), where jSPj and jPP 0 j can be described as According Equation ( 1)-( 3), the phase compensation of the arbitrary point on metasurface can be written as Due to that the L is a constant, Equation (4) can be simplified as When the position of feed source S is fixed, the phase compensation only relates to Θ and coordinates of points on conformal metasurface.Θ and F are known parameters which can be configured in advance according to the functionality of the metasurface.Therefore, once the coordinates of arbitrary points on curved surface are given, the phase distribution of conformal metasurface can be obtained by Equation (5), which is only related to coordinates of meta-atoms.The calculation of coordinates is based on the alignment function f(x) and the metasurface dimension, of which the solution will be discussed in Section 2.1.2.

Meta-Atoms Coordinates Calculation
For conventional planar metasurface, meta-atoms are extended directly on 2D plane with intervals of meta-atom periodic along x-and y-direction.However, different from the planar structure, the bending profile for conformal metasurface brings challenges for meta-atoms coordinates calculation.Before calculating meta-atoms coordinates on metasurface, arrangement strategy of meta-atoms on curved surface should be discussed first.Currently, there are two meta-atoms arrangement strategies for conformal metasurface, as illustrated in Figure 2a,b, a red dot is plotted instead of a meta-atom and p is the periodic of meta-atom.On the one hand, meta-atoms are distributed along the profile of the conformal metasurface with projection point on coordinate axis with equal interval as shown in Figure 2a.On the other hand, meta-atoms are arranged with the equal interval of periodic p along curve surface profile as shown in Figure 2b.For the former one, the x-coordinate of meta-atom can be represented by i•p (i is the number of meta-atom), once the alignment function of the cylinder is given, meta-atoms coordinates can be obtained easily, but the interval of adjacent meta-atoms is not constant along curved surface profile, which will affect metaatom electromagnetic characteristics.Considering the uniform distribution of meta-atoms on curved surface, the meta-atoms arrangement strategy with the equal interval along curved surface profile is adopted in this work.
Thus, meta-atoms arrangements on circular and arbitrary cylinder surface profiles are shown Figure 2c,d, respectively.Circular cylinder is a special cylinder whose alignment is circular arc.Thanks for the same curvature of circular arc, the meta-atoms can be obtained based on the geometry feature.As depicted in Figure 2c, meta-atoms are arranged by equal  interval along circular cylinder surface profile, and based on geometry relation, it can be deduced that x 1 = ÀRsin(α), z 1 = Rcos(α), where r is the radius of circular cylinder, and α is the corresponding central angle.The relationship between central angle α and arc length l is α = l/R.Taking the meta-atom located in arc center as the basis element, coordinates of ith element in clockwise direction are x i = Rsin(ip/R), z i = Rcos(ip/R); similarly, coordinates of ith element in counterclockwise direction are x i = ÀRsin(ip/R), z i = Rcos(ip/R).Therefore, meta-atoms coordinates can be easily obtained if only the element number i is given.The coordinates calculation solution for circular cylinder metasurface is similar with that of planar metasurface; however, the solution is not applicable to arbitrary cylinder surface with different curvature which limits the solution adaptability.
In the majority, curvatures of carrier profiles are not constant in practical application, so it's necessary to investigate coordinates calculation solution for arbitrary cylinder surface profile which adapts to more circumstances.As illustrated in Figure 2d, the curvature is not a constant at different positions on the profile curve for arbitrary cylinder, elements coordinates cannot be deduced by geometrical features directly.To address this issue, arc differential combined with numerical integration is taken as the solution to meta-atoms coordinates calculation along arbitrary curved surface profile.
Along the y-axis, y-coordinates of meta-atoms vary linearly with the serial number that can be obtained easily.Therefore, the analyses on coordinates of x and z for meta-atoms become the essential factors in subsequence.For sake of simplifying the calculation, the problem is abstracted into a mathematical problem.For a straight line L 0 on x-axis with length np, the points in [p/2, npþp/2] with the equal interval of p are marked to compose the point set.For the neighbor points in S , | s jþ1-s j|2 = p.Keeping the line length as it is, the straight line L 0 is bent into a curve L c in terms of f(x), in which the coordinates of all points in S are shifted.In this circumstance, the mathematical problem is the calculation of new points coordinates in the curve L c when the x-coordinate of the beginning point is p/2.
First, the arc differential formula can be written as Then, the curvilinear integral of L c can be described as Second, because the x-coordinate of the start point s 0 on L c is = p/2, the integral upper limit function can be written as Points on L c divide the curve into n segments, thus it can be obtained that where x s j represent the x-coordinate of point s j in the curve L c .Analytical solution of definite integration g(q) cannot be obtained directly.Then, we use numerical integration method to calculate the coordinates of points in S, in which the x-coordinates are marked as X ¼ fx s 0 , x s 1 , x s 2 , x s 3 , : : : , x s n g.Finally, according to z = f(x), the corresponding z-coordinates of point s j in L c can be calculated, which are marked as Z ¼ fz s 0 , z s 1 , z s 2 , z s 3 , : : : , z s n g.The new coordinates of points in the point set S represent the meta-atoms coordinates.As a result, meta-atoms coordinates on curved surface are obtained.

Mate-Atom Characteristics
The meta-atom is composed of three flexible printed circuit boards (PCB), and two 3D-printed supporting layers, as depicted in Figure 3a.For PCB layers, the electric pattern with thickness of 0.018 mm is etched on the dielectric layer, and dielectric layers are made up of flexible material with a permittivity of ε r = 2.65, loss tangent of 0.003, and thickness of 0.127 mm, which are soft enough to be attached on the curved surface.The supporting layer is 3D-printed material ABS-M30 engineering plastics with permittivity of ε r = 2.7, loss tangent of 0.005, and thickness of 2 mm.The PCBs on top/bottom of the meta-atom are two orthogonal gratings, which can transmit x-polarized/y-polarized wave and reflect y-polarized/x-polarized wave.The middle PCB, with "I"-shaped electric pattern etched, can convert the polarization of incident wave into the cross-polarized one, and by tuning the structure parameters, the phase shift of transmitted wave can be controlled flexibly in broadband.Supposing an x-polarized wave is incident in the þz direction normally, four scattering components obey The bottom layer prevents the reflection of y-polarized wave (r yx ), and the top layer prevents the transmission of x-polarized wave (t xx ).And, the I-shaped structure functioning as a half-wave plate realizes cross-polarization conversion, thus the components of t yx is very high.The interference of polarization coupling in the multireflection process will enhance the reflected field of co-polarization and reduce cross-polarization reflection field.
To find out the electromagnetic characteristics, the meta-atom is simulated under the illumination of x-polarized plane wave propagating along þz direction, and the phase and amplitude of the transmission coefficient t yx are plotted in Figure 3c,d.It can be found that by tuning the arc angle α and rotation angle β (the variation range of α is 33-88°, the value of β is AE45°), the phase of t yx can realize 360°phase coverage and amplitude |t yx | is higher than 0.9 in 7.7-15.3GHz.Curves of phase shift have good parallelism when α and β take different value, and the phase changes almost linearly with structure parameters, indicating good phase manipulation ability of the meta-atom.In addition, to explore the electromagnetic response of conformal meta-atom, the meta-atom is bent on the cambered surface as the simulation setup drawn in Figure 3e.Five curved meta-atoms with interval of 5.8 mm are bent on the cylinder surface with radius of R, periodic boundaries are set in y-direction, and the discrete ports are set at the center of cylinder surface.The transmission amplitude and phase shift of the conformal meta-atom in the center are extracted and depicted in Figure 3f.With the increase of curvature radius, the phase of t yx increases gradually and finally fluctuates around À117°, while amplitude |t yx | remains nearly constant, indicating that phase shift is more sensitive to the variation of radius.Therefore, for the point with different curvature radius on the surface, the electromagnetic responses of different conformal meta-atoms with different radius can be obtained respectively, which guarantees the precise phase manipulation.

Quadratic Conformal Transmissive Metasurface
Thanks for the convexity preserving of quadratic cylinder, quadric curve are usually utilized in the design of aircraft profile.In this part, conformal metasurfaces with surface profile of typical quadratic cylinder surface, such as elliptic, hyperbola, and parabolic cylinder surface, are designed, modeled, and simulated, which can be extended to arbitrary cylinder.

Meta-Atoms Coordinates Calculation
The alignment function is the premise of calculating meta-atoms' coordinates, and in this article, alignment functions of elliptic, hyperbola, and parabolic cylinder surface are given as First, supposing the number of meta-atoms along the alignment is 24 and taking the meta-atom as a point, it can be found that points divided the alignment L c into 23 segment and points are arranged along L c with equal interval of p = 5.8 mm.Position of points in f(x) are plotted in Figure 4a-c, where the black "o" represents the points in straight line L 0 with equal interval of p, and red "*" presents the points in curve L c with equal interval of p.Then, taking the meta-atoms center at the position of "*", the x-coordinate of meta-atoms belongs to X ¼ fx s 0 , x s 1 , x s 2 , x s 3 , : : : , x s n g, and the corresponding z-coordinate of meta-atom is z s j ¼ f x s j .Third, the cylinder surface is subdivided into 24 Â 24 meshes with interval of p along alignment L c and y-axis, and a mesh represents a meta-atom, as shown in Figure 4d-f; the coordinate of meta-atom in ith row and jth column can be recorded as where the i and j (i, j = 1, 2, …, 24) are the order number of meta-atoms along x-and y-direction.Moreover, x s j and z s j can be calculated based on the numerical integration mentioned earlier respectively.Therefore, for the metasurface composed of 24 Â 24 meta-atoms, element coordinates on elliptic, hyperbola, and parabolic cylinder surfaces are demonstrated as Figure 4d-f.In addition, considering the phase deviation caused by curved structure, curvature radius of each meta-atom on the cylinder surface should be obtained, so we can engineer the conformal metasurface based on different curvature radius, and the phase can be manipulated more precisely.The curvature radius R is calculated as follows R is the curvature radius related with meta-atom's position, and z 0 and z 00 represent the first and second derivative of x.
In the case of knowing coordinates and alignment function, the curvature radius of every point on the curved surface can be obtained.

Quadratic Conformal Metasurface Modeling
In this part, elliptic, hyperbola, and parabolic cylinder metasurfaces composed of 24 Â 24 meta-atoms with function of beam focusing are taken as examples to illustrate the modeling of quadratic conformal metasurface.The vertex of quadratic surface is set at the origin, the focal distance is selected as 100 mm, so feed source is placed at (0,0,100) and the operating frequency is 10 GHz.For beam focusing, the conformal metasurface can convert the spherical wave into plane wave, so the phase of wave arriving at M should be constant and the value of Θ is set to 0.
Based on method mentioned earlier, the phase compensation for elliptic, hyperbola, and parabolic cylinder metasurfaces are calculated and depicted in Figure 5a,d,g.According to curvature radius and phase shifts, the value distributions for parameter α and β can be obtained as plotted in Figure 5b,e,h and c,f,i, respectively.Therefore, the quadratic conformal metasurface  (take parabolic cylinder as an example) can be built and configured, as the model and decomposition structure demonstrated in Figure 5j,k.

Numerical Results
To verify the theoretical analysis and the proposed design strategy, performances of quadratic cylinder-focusing metasurfaces are simulated by CST Microwave Studio.Vivaldi antenna with bandwidth of 8-18 GHz is selected as the feed source for its good radiation performance in broadband, and the x-polarized spherical wave emitted by the feed source is set to illuminate focusing metasurfaces along z direction.It should be noted that the phase center of Vivaldi antenna locates at the distance of 23 mm away from the front edge at 10 GHz.Therefore, to place the phase center at the focus point, the antenna is placed with front edge at the distance of 77 mm away from the origin, as depicted in Figure 6.
To provide a deep insight of the physical mechanism of the quadratic conformal metasurfaces antenna, the near-field distribution of Re(E x ) and Re(E y ) on xoz plane and yoz plane at 10 GHz are simulated and demonstrated in Figure 7. Before passing through the metasurface, E x -field is the main part while Re(E y ) is barely distributed in the incident field, and the Re(E x ) distribution exhibits spherical wave front, indicating the x-polarized spherical wave is emitted from Vivaldi antenna.After passing though the metasurface, E y -field is the main part while Re(E x ) is barely distributed in the transmission side of conformal meta-lens, and the flat wave front of y-polarized wave at the transmission side indicating the good focusing effects of the metasurfaces.E x -field is strong in the incident field and barely distributed in the transmission side of the conformal meta-lens, and opposite results can be obtained for E y -field.Near-field results suggest that the quadratic conformal metasurfaces can transform spherical wave into plane wave and realize polarization  conversion simultaneously, which are consistent with the theoretical design.
Figure 8 depicts the simulated far-field radiation patterns of quadratic conformal metasurfaces antennas.As demonstrated in Figure 8a,c,e, radiation-focusing beams have been formed by quadratic conformal metasurfaces lens antennas with directivity larger than 17.1 dB in 8.5-13 GHz.To gain straightforward understanding of the beam-focusing capability, the 3D and 2D far-field radiation patterns at 10 GHz on xoz plane and yoz plane are plotted in Figure 8b,d,f.It is obvious that at 10 GHz conformal meta-lens can radiate pencil beams, and the peak co-polarization (y-polarization) is at least 19.6 dB higher than that of cross-polarization (x-polarization) in all cases.For elliptic cylinder conformal meta-lens, the sidelobe level is at least 15.7 and 17.9 dB lower than that of main-lobe on xoz and yoz plane, as depicted in Figure 8b.For hyperbola cylinder conformal metalens, the sidelobe level is at least 18.4 and 21.7 dB lower than that of main-lobe on xoz and yoz plane, as plotted in Figure 8d.For parabolic cylinder conformal meta-lens, the sidelobe level is at least 17.8 and 18.3 dB lower than that of main-lobe on xoz and yoz plane, as depicted in Figure 8f.The near-field and far-field simulation results confirm each other and verify the validity of the conformal metasurface design strategy.

Combined Quadratic Conformal Transmissive Metasurface
In practice, profiles of aircrafts are the combinations of diverse quadratic cylinder surfaces rather than a single kind.Thus, we would like to discuss the design strategy for combined quadratic conformal metasurface, and three conformal metasurfaces with functionalities of beam focusing, beam deflecting, and vortex generating are designed, simulated, and fabricated.

Combined Quadratic Conformal Metasurface Model and Design
The alignment of combined quadratic surface is the combination of hyperbola and parabolic curves, and the alignment function f(x) is a piecewise function as depicted in Figure 9a.When x ≤ 0, the function is a parabolic curve with f(x) = g(x) = 0.03x 2 .When x > 0, the function is a hyperbola curve with q .The curve's projection on x axis is marked as X min X max .
Then we calculate the coordinates of meta-atoms.The curve is divided into 23 segments, in which there are 24 corresponding points.The points are arranged along f(x) with an equal interval of p = 5.8 mm, by using the method proposed in Section 2, meta-atoms coordinates are calculated with x-coordinate belonging to X = {x 1 , x 2 , x 3 , …, x 24 } and the z-coordinates belonging to Z = {z 1 , z 2 , z 3 , …, z 24 }.As depicted in Figure 9b, the red "*" represents the center of meta-atom; o 0 is the middle point of X min X max ; and x min and x max represent the lower and upper limits of domain, respectively.It should be noted that the phase compensation calculation method before is discussed with feed source located in z axis.But when the surface is the combined quadratic surface, the alignment projection X min X max on x axis is not symmetrical about x axis.The feed point will deviate from surface center if located at (0,0,F) as the yellow point plotted in Figure 9b, which will lead to performance deterioration.To solve this problem, the feed source is moved along x-axis to x min þx max 2 , 0, F À Á and the projection of point is located at, as the blue point drawn in Figure 9b.Therefore, the phase compensation calculation formula Equation ( 5) should be modified into Using Equation ( 13), it can be found that when the feed point is fixed, the phase distribution only relates to meta-atoms coordinates.
The phase distribution can be calculated based on Equation ( 13), and the functionality of the metasurface can be configured by setting the parameter Θ.For beam focusing, the metasurface can transform the spherical wave into plane wave, so the phase of wave arriving at M should be constant and the value of Θ is set to 0, as plot in Figure 10a.
For beam deflection, supposing the transmitted wave deflects in x-direction, linear gradient phase distribution should be added to realize beam deflecting, and the value of Θ can be written as where n = À11, 10,…, þ12 is the order number of meta-atoms along x-direction; p is periodic of meta-atom; and ξ is phase difference between two adjacent columns meta-atoms.According to generalized Snell's law, the relationship between ξ and deflection angle θ t of the transmitted wave can be described as When the deflection angle θ t is set to be 30°and operating frequency is selected as 10 GHz, according to Equation (15), the corresponding phase difference ξ is 0.6074 rad or 34.8°, and Θ d is plotted in Figure 10e.
For vortex wave generation, the key factor is to introduce azimuth phase dependence e Àjlφ into the transmitted wave; thus, the phase distribution should satisfy where n, m = À11, À10, …, þ12 present the order number of meta-atoms along x-and y-direction, respectively; l is the topological charge of vortex wave which is chosen as l = 1 in this article, and Θ v is drawn in Figure 10i.The compensation phase distribution for beam focusing, beam deflecting, and vortex wave generation can be described as In addition, focal length to aperture diameter (F/D) is one of factors that influence aperture efficiency.According to the relation between F/D and aperture efficiency, [58] three focusing conformal metasurface antennas with F/D in the range of 0.4-0.6 are designed and simulated.The gains of metasurface antennas are 20.2,21, and 20.6 dB, respectively, with focal length of 79, 100, and 120 mm.Thus, the focal length to aperture diameter F/D = 0.51 (F = 100 mm) is chosen.
Therefore, with F = 100 mm, l = 1, ξ = 34.8°,and curvature radius of each conformal meta-atoms calculated by Equation ( 12), distributions of Θ, compensation phase, α and β for beam-focusing, beam-deflecting, and vortex-wave-generation conformal metasurfaces composed of 24 Â 24 meta-atoms are calculated and illustrated in Figure 10.Distributions of Θ determine the metasurfaces' functions, and the Θ of focusing metasurfaces is constant as plotted in Figure 10a, while that of deflecting and vortex wave generation Θ is linear gradient phase profile and spiral phase profile respectively, as depicted in Figure 10e,j.Phase distributions shown in Figure 10f for beam-deflecting metasurface are combinations of Figure 10b,e; similarly, phase distributions for the vortex wave generation one shown in Figure 10j are combinations of Figure 10b,i.Once the curvature radius is The position of feed point should be modified to make the projection of feed point located at the center of X min X max .X 1 and X 24 presents the 1st and 24th meta-atom, X min and X max represent the projection of X 1 and X 24 on x axis.
determined, values distributions of α and β can be obtained according to the relationships between the phase shift and parameters of meta-atoms.

Numerical Results
Performances of the combined quadratic conformal metasurfaces are simulated in CST Microwave Studio.Illuminating an x-polarized electromagnetic wave emitted from the Vivaldi antenna onto three combined quadratic conformal metasurfaces, respectively, the far-field and near-field results can be obtained.
We first simulate the performance of the beam-focusing conformal metasurface antenna.As 3D radiation patterns depicted in Figure 11a-g, it can be found that the high-gain emissions and narrow beams are formed in 8.5-13 GHz, indicating the good beam-focusing performance of the conformal metasurface antenna in a broadband.Moreover, the near E-field distributions at frequency of 10 GHz are plotted in Figure 11h-k.The spherical wave front of E x shown in Figure 11i,k indicates the spherical x-polarized wave emitted from Vivaldi antenna.Referring to Figure 11h,j, the flat wave front of E y demonstrates that our focusing combined quadratic conformal metasurface can emit excellent plane wave and realize cross-polarization conversion Figure 10.The distribution of Θ, compensation phase, α and β for a-d) beam focusing, e-h) beam deflecting, and i-l) vortex-generation combined quadratic conformal metasurfaces.a) Constant Θ distributions for beam-focusing conformal metasurfaces, e) Θ distributions for beam-deflecting conformal metasurfaces with 34.8°phase gradient at x-direction, i) Θ distributions for vortex-generation conformal metasurfaces is vortex plate.n and m, with range from À11 to 12, stand for the order number of meta-atoms along x-and y-direction, respectively.b,f,j) Phase distributions for beam-focusing, beamdeflecting, and vortex-generation conformal metasurfaces calculated by Equation (17a-c), respectively.c,g,k) and d,h,l) Value distributions of α and β for beam-focusing, beam-deflecting, and vortex-generation conformal metasurfaces.
at the transmission side of the metasurface.The far-field and near-field results confirm each other.
Then, the beam-deflecting metasurface antennas are simulated with 3D radiation patterns drawn in Figure 12.The transmitted wave is deflected to an anomalous angle, and narrow beams in the oblique direction can be observed clearly in broadband.
For the vortex-generation metasurface antenna, the radiation patterns with amplitude nulls in the normal direction in 8.5-13 GHz can be obtained, as 3D far-field radiation patterns depicted in Figure 13a-g, indicating that the designed conformal metasurface can generate vortex beams in a bandwidth.Then, to obtain the mode of the transmitted vortex wave, an observed plane with 200 mm distance from xoy plane is set up to record the E y component, as shown in Figure 13h.The real part of E y distribution shown in Figure 13i has one spiral arm indicating the topological charge = 1, and reveals that the vortex wave with spiral wave front is produced successfully.Moreover, the doughnut-like E y magnitude drawn in Figure 13j demonstrates the magnitude null in the center of the beam, which is consistent with 3D far-field results.

Experimental Results
To further validate our design, all samples are fabricated with a standard PCB and 3D-printed technologies.Orthogonal gratings and half wave plates for beam-focusing, beamdeflecting, and vortex-generation metasurfaces antennas are   shown in Figure 14a-e.The flexible PCB are soft enough to be attached on the curved 3D-printed supporting layers, they are assembled with plastic screws as the assembled structure shown in Figure 14g, and Vivaldi antenna is set as a feed point with front edge at the distance of 77 mm away from the tangent surface of the metasurface.The fabricated and assembled combined quadratic conformal metasurfaces antennas are then measured in an anechoic chamber to avoid unwanted interference from surroundings.The metasurface is placed on a turntable so that it can be freely rotated and a broadband double-ridged horn antenna working in 1-18 GHz is located in the far-field region to record the radiation patterns, as depicted in Figure 14h.
For the beam-focusing conformal metasurface antenna, the simulated and measured 2D radiation patterns at 10 GHz in two radiation planes are plotted in Figure 15a-b.The measured gain of conformal metasurface-focusing antenna has increased about 12.2 dB, compared with that of the bare Vivaldi antenna, and the cross-polarization levels are lower than 23.2 dB on the main lobe.The measured (simulated) sidelobe level is 15.2 dB (17.4 dB) and 18.1 dB (21.7 dB) lower than that of the main lobe on xoz and yoz planes, respectively.In addition, the gain and aperture efficiency of conformal metasurface antenna in the bandwidth is also tested and drawn in Figure 15i.The measured gain of beam-focusing metasurface at 10 GHz is 20.9 dB.Calculated by Gλ 0 2 /4πS, where G is the gain and S = 139.2Â 139.2 mm 2 denotes the surface area, the antenna efficiency can be obtained.The maximum aperture efficiency is 45.5% at 10 GHz, and the measured À3 dB gain bandwidth covers 8.5-14 GHz.The measured and simulated results are consistent with each other, and demonstrate that the beam-focusing conformal metasurfaces can transform the x-polarized spherical wave emitted from the Vivaldi antenna into the y-polarized plane wave.
For the beam-deflecting conformal metasurface antenna, as the 2D radiation patterns on xoz plane shown in Figure 15c, the measured and simulated deflection angle is 30°at 10 GHz which shows good agreement with the theoretical prediction.
The cross-polarization levels are lower than 27.1 dB on the main lobe.We measured the power distribution for the transmitted wave in Figure 15d, and most of the transmitted waves are deflected to an anomalous angle in 8-13 GHz, which coincides well with generalized Snell's law (white stars in Figure 15d).As plotted in Figure 15i, the simulated and measured gain and aperture efficiency show a good matched bandwidth from 8 to 14 GHz.The measured gain of deflecting metasurface at 10 GHz is 19.5 dB.The maximum aperture efficiency is 32.9% at 10 GHz and the measured À3 dB gain bandwidth covers 8.5-13.5 GHz.The measured gain is a little bit lower than that of simulated, possibly caused by fabrication, assembling and measurement errors in experiments.
For vortex-generation conformal metasurface antenna, the farfield radiation patterns on xoz and yoz plane are drawn in Figure 15e-f ).It is clear that there is an energy null in the direction of φ = 0 and θ = 0, demonstrating the capability of generating well-defined vortex beams of the device.And the center null of the transmitted vortex wave is 22.8 dB lower than the maximum gain, and the cross-polarization level is lower than À21 dB in the maximum directions.In addition, to obtain a deep insight of the generated vortex wave, the near-field performance of the antenna is measured in an anechoic chamber.Illuminating the metasurface by an x-polarized wave, we detect the E y field distribution at the transmission space by a test probe and record the local electric field (E y ) by a vector-field network analyzer, as the schematic shown in Figure 14i.The test probes scan the xoy plane zone occupying 200 Â 200 mm 2 with step of 2 mm to measure the plane E-field under the control of stepping motor, as the near-field measurement configuration depicted in Figure 14j.The transmitted vortex wave in y-polarization is detected by the test probe at 10 GHz.The measured Re(E y ) shown in Figure 15g has one spiral arm indicating the vortex wave with topological charge l = 1 and spiral wave front is produced successfully.The typical fingerprint of the field magnitude, a doughnut-like E-field magnitude, is shown in Figure 15h.The measured near-field doughnut-shaped intensity with singularity in the center, indicating the radiation energy in the normal direction is zero, which satisfies the far-field characteristics of vortex beam.In addition, the efficiency is one of the most important indexes for vortex-beam generator, which is defined as the ratio of the energy carried by vortex wave to the total energy of incident wave. [59]When the x-polarized wave emitted by the feed passes through the conformal metasurface, the incident power will be converted into four parts, transmission of x-polarized wave T xx and y-polarized wave T yx , and reflection of x-polarized wave R xx and y-polarized wave R yx .Since the y-polarized wave is the vortex wave converted from the metasurface; therefore operating efficiency of vortex generator is defined as the ratio of transmitted y-polarized wave to the total power of incident wave, calculated by Thus, the efficiency of vortex-generation conformal metasurface is 84.2% at 10 GHz, and is higher than 80% in the range of 9.7-12.8GHz.The high working efficiency indicates that the proposed device is a good candidate in wireless communication system.
Generally, for transmissive combined quadratics conformal metasurface antennas analyzed earlier, the experimental results agree well with the simulated ones, and good performances verify validity and feasibility of the arbitrary cylinder conformal metasurface antenna design strategy.

Conclusion
We theoretically propose and experimentally investigate a novel strategy to design conformal metasurface antennas with arbitrary cylinder surface, which is more general and can extend the dimension of metasurface antennas into arbitrary cylindrical surfaces.We propose the conformal phase compensation method by ray-tracing technique, and coordinates calculation method on curved surface by using arc differential and numerical integration.Based on the meta-atom with orthogonal gratings and half-wave plate, focusing quadratic conformal metasurface antennas in elliptic, hyperbola, and parabolic cylinder surface are simulated respectively to verify the validity of the design strategy.For potential applications, combined quadratic conformal metasurfaces antennas with functionalities of beam focusing, beam deflecting, and vortex generation are designed, fabricated, and measured.Numerical and experimental results indicate that the proposed beam-focusing/beam-deflecting/vortex-generation conformal metasurface in arbitrary cylinder surface can transform the spherical wave into the plane wave/deflected beam/ vortex beam, indicating the unprecedented wave front manipulation capability of devices.The design strategy can be extended to arbitrary cylinder surface once the alignment function of cylinder is given.Our work proposes a general strategy to achieve wave front manipulation in transmission mode on curved platforms such as aircraft and vehicles, which increases flexibility and practicability of metasurface in application of communication, detection, and so on, and offers us a novel approach to design conformal antennas without changing the aerodynamics shapes of platforms.

Figure 2 .
Figure 2. Conceptual illustration of meta-atoms distribution strategies.Two kinds of meta-atoms distribution strategy: a) projection points of meta-atoms with equal interval of p on x-axis, b) meta-atoms with equal interval of p along cylinder profile.Elements distribution along c) circular cylinder and d) arbitrary curved cylinder surface profile of conformal metasurface.

Figure 3 .
Figure 3.The prototype and electromagnetic response of meta-atoms.a) Structure and b) middle printed circuit board (PCB) of the meta-atom.α is the arc angle of I-shaped particle, β is the angle between "I" structure and y-axis which takes the value of À45°or þ45°(anticlockwise is defined as positive direction).The geometrical parameters are listed as w 1 = 1.3 mm, w 2 = 1.6 mm, d = 0.4 mm, r l = 2.6 mm, and p = 5.8 mm.The c) phase and d) amplitude of x-to-y transmission coefficient t yx .e) Simulation setup for conformal meta-atoms.f ) Amplitude and phase of t yx with curvature radius variation for the meta-atom with α = 60°and β = þ45°.

Figure 4 .
Figure 4. Illustration of meta-atoms position on alignment and cylinder surface.The comparison of meta-atoms position on straight line L 0 and curve alignment L c for a) elliptic, b) hyperbola, and c) parabolic cylinder.The meta-atoms position on d) elliptic, e) hyperbola, and f ) parabolic cylinder surface.

Figure 5 .
Figure 5. Phase compensation, parameters distributions, and structure model for quadratic conformal metasurface.a-c) For the phase compensation, α and β distributions for elliptic cylinder metasurfaces.d-f ) For the phase compensation, α and β distributions for hyperbola cylinder metasurfaces.g-i) For the phase compensation, α and β distributions for parabolic cylinder metasurfaces.j) The conformal metasurface model.k) The decomposition structure of conformal metasurface model, which contains two 3D-printed supporting layers, two orthogonal grating and half wave plates layers.

Figure 7 .
Figure 7.The simulated near-field distribution on a-f ) xoz-plane and g-l) yoz-plane.a,b)/g,h) The distribution of Re(E x ) and Re(E y ) on xoz/yoz plane for elliptic cylinder metasurface.c,d)/i,j) The distribution of Re(E x ) and Re(E y ) on xoz/yoz plane for hyperbola cylinder metasurface.e,f )/k,l) The distribution of Re(E x ) and Re(E y ) on xoz/yoz plane for parabolic cylinder metasurface.

Figure 8 .
Figure 8. Simulated far-field radiation patterns of quadratic conformal metasurfaces lens antennas.The 3D far-field radiation patterns of a) elliptic, c) hyperbola, and e) parabolic cylinder metasurfaces lens antennas in 8.5-13 GHz.The 3D far-field radiation patterns, and 2D radiation patterns on xoz and yoz plane of b) elliptic, d) hyperbola, and f ) parabolic cylinder metasurfaces lens antennas at 10 GHz.

Figure 9 .
Figure 9.The alignment of combined quadratic surface is the combination of hyperbola and parabolic curves.a) The alignment function f(x) is piecewise function.When x ≤ 0, f(x) is parabola, when x > 0, f(x) is hyperbola.b)The position of feed point should be modified to make the projection of feed point located at the center of X min X max .X 1 and X 24 presents the 1st and 24th meta-atom, X min and X max represent the projection of X 1 and X 24 on x axis.

Figure 11 .
Figure 11.Simulation results of combined quadratic conformal metasurface antenna for beam focusing.a) The 3D far-field radiation patterns at 10 GHz and the structure of combined quadratic conformal metasurfaces.b-g) The 3D far-field radiation patterns in broadband.h-k) Near-field results at 10 GHz.Re(E y ) distribution on (h) xoz plane and (j) yoz plane, Re(E x ) distribution on (i) xoz plane and (k) yoz plane.

Figure 12 .
Figure 12.Simulation results of combined quadratic conformal metasurface antenna for beam deflecting.a) The 3D far-field radiation patterns at 10 GHz and the structure of combined quadratic conformal metasurfaces.b-g) The 3D far-field radiation patterns in broadband.

Figure 13 .
Figure 13.Simulation results of combined quadratic conformal metasurface antenna for vortex generation.a) 3D far-field radiation patterns at 10 GHz and the structure of combined quadratic conformal metasurfaces antenna.b-g) The 3D far-field radiation patterns in broadband.h) The E y distribution on the observed plane at 200 mm from xoy plane and the metasurface structure in perspective.i) The Re(E y ) distribution and j) magnitude distribution of the transmitted y-polarized wave on the observed plane at 10 GHz.

Figure 14 .
Figure 14.Fabricated samples, far-field and near-field experimental setup.Pictures of half wave plates for a) beam-focusing, b) beam-deflecting, and c) vortex-generation combined quadratic conformal metasurfaces antennas.d) Horizontal and e) vertical grating.f ) Fabricated 3D-printed supporting layers.g) Assembled prototype of the combined quadratic conformal metasurface.Inset of (g): picture of Vivaldi antenna.h) Far-field experimental setup in the anechoic chamber.i) Schematic diagram and j) picture of the near-field experimental setup in the anechoic chamber.

Figure 15 .
Figure 15.Measurement results of combined quadratic conformal metasurfaces.Far-field radiation patterns of beam-focusing metasurface on a) xoz plane and b) yoz plane at 10 GHz.Far-field radiation patterns of beam-deflecting metasurface on c) xoz plane at 10 GHz.d) Measured 2D radiation patterns intensity map of y-polarized waves as function of frequency and detection angle for the beam-deflecting metasurface.The white stars infer to the theoretical values predicted by generalized Snell's law.Far-field radiation patterns of vortex-generation metasurface on e) xoz plane and f ) yoz plane at 10 GHz.Measured g) Re(E y ) distribution and h) magnitude distribution of the transmitted y-polarized wave at 10 GHz.i) Broadband gain and aperture efficiency of beam-focusing and beam-deflecting metasurface antenna.