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Addition Theorem for Digital Coding Metamaterials Rui Yuan Wu, Chuan Bo Shi, Shuo Liu, Wei Wu, and Tie Jun Cui* Recently, 2D versions of metamaterials, metasurfaces, have attracted more attention,[11] due to their advantages of low cost, low profile, and strong abilities to manipulate spatial and surface waves. Novel generalized sheet transition condition method[12] and transverse resonance method[13] were first presented to analyze the EM performance of metasurfaces. Then generalized Snell’s law[14] was proposed to introduce the concept of abrupt phase when designing metasurfaces. By changing the size, shape, or orientation of unit cells, the abrupt phase provided by the metasurface can be tailored accordingly, and the outgoing EM waves are engineered arbitrarily. Metasurfaces have offered more convenience and freedom for manipulating EM wavefronts, and have been widely applied in the microwave,[15–20] terahertz,[21–24] visible,[25–28] and even acoustic[29,30] frequencies. Metamaterials and metasurfaces described by continuously effective medium parameters and phase distributions have powerful capabilities in controlling EM waves,[1–30] but in static ways. That is to say, once a metamaterial or metasurface is fabricated, its function will be fixed. In order to reach real-time controls to EM waves, digital coding characterization has been proposed to describe metamaterial, resulting in the concepts of coding, digital, and programmable metamaterials.[31] The binary 1-bit digital codes “0” and “1” are adopted to indicate the reflection phases of 0° and 180°, from which one can manipulate EM waves using different coding sequences. The digital codes have been extended to 2-bit and more to bring more freedom for controlling scattering beams. The digital states “0.” “1,” “2,” and “3” represent the reflection phases of 0°, 90°, 180°, and 270°, respectively. By designing a unit cell controllable by a diode to achieve either “0” or “1” state, the digital and programmable metamaterials have been realized to reach real-time manipulations to EM waves.[31] The digital coding representation links the traditional metamaterials to information theory, giving us an opportunity to control EM performance through discrete digital states. Based on these concepts, many kinds of functions such as beam steering[31–33] and reduction of radar crosssections[34] have been achieved by switching coding sequences on coding metamaterials in microwave and terahertz regions. Recently, the concept of anisotropic coding metamaterials has been demonstrated, which can achieve two independent coding behaviors for different polarizations.[35] Furthermore, convolution operations on coding metasurfaces were presented to

Coding representation of metamaterials builds up a bridge between the physical world and the digital world, making it possible to manipulate electromagnetic (EM) waves by digital coding sequences and reach field-programmable metamaterials. Here, the coding space is extended to complex domain and proposed complex digital codes to provide closer essence of EM-wave propagation. Based on the analytic geometry and complex variable functions, an addition theorem on complex coding is established, which reveals inherent connections among digital codes with different bits and enables all higher-bit digital codes to be represented by the 1-bit complex codes. According to the complex coding and addition theorem, multifunctional metamaterials can be directly designed and realized without considering mutual coupling. When two different coding patterns with different functions are added together via the addition theorem in complex form, the combined coding pattern will directly generate the two functions simultaneously without any perturbations. A series of realistic coding metasurfaces is presented to demonstrate the powerful and flexible performance of the complex coding and addition theorem for independent controls of EM waves to reach multiple functions. Good agreements between numerical simulations and experimental results prove the feasibility of the proposed concept and theorem in practical applications.

1. Introduction For a long time, how to control electromagnetic (EM) waves flexibly has been a very important subject. Metamaterials provide a concise and efficient method to manipulate EM waves.[1] One of the main features of metamaterials is the ability in tailoring effective medium parameters (e.g., electric permittivity, magnetic permittivity, refractive index, and impedance).[2] Due to the flexible and elaborate designs of meta-atoms and their arrangements, the effective medium parameters can be arbitrarily controlled to achieve strange EM phenomena such as negative refraction,[3,4] perfect imaging,[5] and invisible cloaking.[6–10]

Dr. R. Y. Wu, Dr. C. B. Shi, Dr. S. Liu, Dr. W. Wu, Prof. T. J. Cui State Key Laboratory of Millimeter Waves Southeast University Nanjing 210096, China E-mail: [email protected] Dr. R. Y. Wu, Dr. C. B. Shi, Dr. S. Liu, Dr. W. Wu, Prof. T. J. Cui Synergetic Innovation Center of Wireless Communication Technology Southeast University Nanjing 210096, China The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adom.201701236.

DOI: 10.1002/adom.201701236

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improve the control ability and allow arbitrary radiation angles in the upper space using 2-bit digital coding.[36] Because of the simple design, convenient manipulation of EM waves and realtime status switch, the coding metamaterials have shown great potentials in novel interdisciplinary of the electromagnetics and information science.[37–41] In the existing work on coding metamaterials,[31–41] however, all digital codes come from the absolute values of scattering phase ϕ. Here, according to the expression of EM waves from Maxwell’s equations and wave equations, we propose a novel principle for digital codes, which relates to the complex phase term e jϕ instead of ϕ itself. Because e jϕ is a complex number, we name it as complex code. The complex code involves entire phase information of EM waves and introduces many fantastic properties for digital coding metamaterials. We further propose addition theorem on the complex codes, which can be treated as microscopic manipulations on every digital states. More importantly, the complex codes and their addition operations establish links among arbitrary-bit codes, i.e., we can achieve arbitrary-bit codes from the 1-bit complex codes. It is shown that the complex codes help realize multifunctional coding metamaterials, in which several functions of coding metamaterials could be superposed together flexibly through the addition theorem. The correctness and robustness of the complex codes and addition theorem are confirmed theoretically and numerically with several examples, which are also verified experimentally by two fabricated samples in the microwave frequencies.

2. Definition of Complex Digital Codes theory,[42]

In EM uniform planar sinusoidal wave is a basic field mode. Complicated EM waves are superposition of many planar sinusoidal waves. In order to reduce the complexity of analysis, we consider a transverse electromagnetic wave propagating along the +z direction. The electric-field expression can be written as      E = E 0 e − jkz + jϕ = E 0 e − jkz e jϕ (1)

in which k is the propagation constant. There are three parts in   the electric-field expression: the first is E 0 , indicating the polarization direction and amplitude of electric field; the second part is e−jkz with the information of propagation direction, and the third part is ejϕ that contains whole details of the phase. According to the general Snell’s laws, when EM waves illuminate a metasurface with local phase change Δϕ, the phase of scattered waves becomes ϕ + Δϕ and hence the scattered beams can be manipulated flexibly.[14] Thus the phase part ejϕ plays an important role in controlling the EM waves. In coding metamaterials or metasurfaces, the phase shift is digitalized to several digital codes (e.g., “0” and “1” for 1-bit coding, “00,” “01,” “10,” and “11” for 2-bit coding).[31] In such coding regulations, however, only the absolute phase value ϕ is used for coding. Here, in order to retain all information of phases, we propose complex digital codes, in which the whole phase term ejϕ is used to define the digital states. The absolute value of complex codes is always 1, while the argument is similar to that in the traditional coding method.[31] Hence, two types of unit cells with 0 and π phase responses are adopted to realize 1-bit

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Figure 1.  The complex digital codes in “coding circle.” a) Geometric representation of a complex number. b–d) The 1-bit, 2-bit, and 3-bit complex codes and their corresponding arguments in the coding circle.

complex digital codes ‘0 ’ and ‘1’, which have different formats from the traditional scalar digital codes “0” and “1” since the top dots indicate the complex property. This regulation can be extended to 2-bit, 3-bit, and higher-bit complex codes. The complex plane is introduced to exhibit the properties of complex digital codes intuitively. An arbitrary complex number corresponds to a sole point Z (see Figure 1a) in the complex plane, and is represented by a vector from the original point O to Z, in which the absolute value equals to the distance between O and Z and the argument is the angle from the real axis to OZ. Hence, all complex digital codes are located on a unit circle, which is named as “coding circle.” Shown in Figure 1b–d, arbitrary-bit digital states can be denoted by unit vectors in the coding circle. To simplify the representations of complex digital codes, their abbreviations are presented in Table 1, in which the main body is the absolute value of a complex code, the top point means a complex number, and the index at the bottom right corner indicates the bit of digital state. Such representations provide great convenience in various operations on the complex coding sequences.

3. Addition Theorem on Complex Digital Codes The addition operation of two complex digital codes leads to e jϕ1 + e jϕ 2 = Ae jϕ 0 (2)

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Table 1.  Abbreviations of complex digital codes. The scale of codes

Corresponding digital states

1-bit complex codes

•

0

2-bit complex codes

00

3-bit complex codes

000

•

•

•

100

•

01 •

001 •

101

Abbreviations of digital states

•

1 •

10 •

010 •

110

0 1

11

•

0 2

1 2

2 2

3 2

•

0 3

13

2 3

3 3

•

4 3

5 3

6 3

7 3

11 011 111

in which ϕ1 and ϕ2 are arguments of the two complex digital codes. After addition, we get a new complex number with argument ϕ0 and magnitude A. Since the complex digital codes are always normalized, we only focus on variations of the augment. Based on the coding circle, the addition operation can be realized by the vector superposition principle in the traditional Euclidean geometry. As examples, two typical 2-bit complex-code addition processes are illustrated in Figure 2a,b, in which 0 2 + 1 2 results in a phase of 45°, corresponding to the complex digital state 1 3 in 3-bit coding, and 0 2 + 3 2 results in a phase of 315°, corresponding to the complex digital state 7 3 in 3-bit coding. Hence, we demonstrate the physical meaning of addition operation from two aspects of microcosmic and macroscopic. On the unit-cell level, it means information

addition of two complex codes; on the metasurface-system level, it means superposition of two coding patterns, implying that two far-field scattering patterns will appear simultaneously with bifunctions (see Figure 2c). The above results demonstrate that 2-bit complex codes turn to 3-bit complex codes after the addition. Thus we can use 3-bit complex codes to form the addition of two kinds of 2-bit complex codes. A more universal theory for addition operation is deduced that two N-bit complex codes produce an (N + 1)-bit complex code. That is to say, the addition operations set up connections between two different coding bits. If we start from 1-bit complex codes, we can get all higher-bit complex codes after several times of addition operations. We need to set regulations for addition operations. Since 1-bit, 2-bit, and 3-bit complex codes are most commonly employed, we will focus on the regulations among these three coding sets. However, there exists a situation of “indefinite coding addition,” in which the two complex codes have reverse directions with 180° phase difference, as shown in Figure S1 of the Supporting Information. In this situation, based on the parallelogram rule, the addition result is zero. In the other word, the resulted phase can be taken as any value. We name the coding elements related to the indefinite coding addition as “indefinite elements.” The indefinite addition will result in an effect that the information of two added complex codes cannot be recorded precisely and cannot be inverted properly. If there are many indefinite elements on coding metasurfaces, their final performance will be severely affected. Here, we propose the concept of indefinite-element rate, which is the ratio of numbers of indefinite elements and all elements on the metasurface, to characterize the influence from the indefinite elements. In common situations when designing the coding metamaterials, the coding sequences are arranged periodically on a metasurface and the probability of each Figure 2. Intuitive display for addition theorem on unit-cell level and metasurfcace-system code is the same, and the indefinite-element rate is related to the scale of complex codes. level. Two typical addition processes of 2-bit complex codes a) 0 2 + 1 2 = 13 and b) 0 2 + 3 2 = 7 3 When N-bit complex codes are undergoing are shown on the coding circle. c) The system-level performance of addition theorem.

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the addition operations, the number of digital states will be 2N. Then the total number of additive combinations is 22N if the order of addition is taken into consideration. From the coding circle, each state must have a corresponding state to form the indefinite elements. That is to say, the inherent number of indefinite elements is 2N. Hence, the indefinite-element rate of N-bit complex codes is estimated as 2N /22N = 1/2N . For a designed coding metasurface, the probability of each state is the same, and thus the indefinite-element rate of the metasurface is identical no matter how the coding sequence is chosen. In this way, 1-bit complex codes and corresponding addition operations suffer the biggest indefinite-element rate of 0.5. When the probability of every code is different in some special cases, the indefinite-element rate may become even bigger. According to the above analysis, we consider two identical complex coding sequences 0011 0011··· (Note: the top dot and subscript on digits are omitted for convenience in coding sequences, it is the same for the following discussions) along the x- and y-directions, Px and Py, to explore the influence of indefinite additions, in which the period of unit cell is 11 mm and the wavelength is 33.3 mm at 9 GHz. The scattering patterns of the two coding sequences Px and Py are illustrated in Figure 3a(ii),b(ii), showing two pencil beams in the x = 0 and y = 0 planes, respectively. The ideal scattering pattern of the additive coding pattern Px + Py should be addition of the two scattering patterns of Px and Py. However, we notice a huge beam at θ = 0°, as displayed in Figure 3c(ii), which is beyond our expectation. This phenomenon can be explained by the presence of massive indefinite elements on the additive coding metasurface Px + Py. From Figure 3c(i), we observe that the digital state of all indefinite elements in the addition coding scheme is 1 2, i.e., 0 1 + 1 1 = 1 1 + 0 1 = 1 2 , and there is no digital state 3 2 . In other words, when normally incident plane waves arrive on metasurface, the phase difference among these elements is zero, which results in normally outgoing waves, producing the wrong beam. Although the occurrence probability of the indefinite elements decreases and their digital states become varied when the coding bit increases from 1 to 2 and higher, which may reduce the influence from the indefinite elements, we still need to solve the problem fundamentally. Hence, we need to set regulations to break this consistency of identical codes distribution on the aperture caused by indefinite additions, as well as to maintain the rationality. For this purpose, we focus on rebuilding the regulation when meeting the indefinite situation. We note that the indefinite additions appear especially during the addition process of two 1-bit codes, in which only three digital states “0,” “1,” and “2” are acquired after addition. Therefore, the original edition of the addition operation results in the loss of digital states owing to the indefinite elements. There is a regular phenomenon in the process of addition operations on complex codes. That is to say, there are two included angles between two complex codes and the addition result is located on the angle bisector of the smaller one, which can also be regarded that the result is achieved by rotating the first complex code to the second one with the angle of half of the smaller included angle orderly. However, for the indefinite additions, the two angles are both π, as shown in Figure S1 of the Supporting Information. Hence, once the indefinite additions happen, we propose to use a sequential

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order of two codes to determine the addition result: we choose the addition result at the angle bisector of the angle from the first complex code to the second complex code. Namely, the 1-bit complex code additions 0 1 + 1 1 = 1 2 and 1 1 + 0 1 = 3 2 have different results. In this regulation, all digital states will be achieved and the code distribution is uniform. Figure S2 of the Supporting Information exhibits the solutions for all indefinite additions in 1-bit and 2-bit addition operations. According to the new regulations, we get the additive coding pattern as presented in Figure 3d(i). The corresponding far-field scattering patterns are illustrated in Figure 3d(ii),d(iii). We clearly observe that, under the new regulations, the scattering pattern of the additive coding pattern Px + Py is the ideal addition of the two scattering patterns of Px and Py. Therefore, we propose the final regulations of addition operations on complex digital codes for 1-bit and 2-bit cases, as illustrated in Table 2. This regulation can be extended to 3-bit and more under the same rule. All addition operations need to obey the regulations, which are easy to program in designing multifunctional coding metasurfaces.

4. Performance Characterization Based on the theory of complex digital codes and their addition theorem, we can realize a lot of peculiar EM phenomena which are difficult to achieve by the traditional metasurfaces. First, we consider a coding metasurface aimed at specially designed fourbeam scattering. Generally, the four-beam performance can be produced by chessboard-coding pattern, and the corresponding four beams are scattered to the directions of ϕ = 45°, 135°, 225°, and 315° with the deflection angle θ.[30] However, we wish the coding metasurface to radiate four beams to predesigned directions of ϕ = 0°, 90°, 180°, and 270°. More difficultly, the deflection angles can be manipulated in pair, i.e., the deflection angle of the beams at ϕ = 0° and 180° is different from that of the beams at ϕ = 90° and 270°. We realize this coding metasurface by using two 1-bit coding sequences 010101··· with different coding periods and the addition theorem. The deflection angle θ of the 010101··· coding sequence is determined by

θ = sin −1 ( λ /Γ) (3) in which λ is the wavelength at the working frequency, and Γ is the period of coding sequence. When we double the period of 0 and 1, the scattering patterns of the additive coding sequence (00110011··· along the x-direction plus 00110011··· along the y-direction, see Figure 3d(i)) are shown in Figure 3d(ii),d(iii), in which the four beams are directed to ϕ = 0° and 180° (the x-direction) and ϕ = 90° and 270° (the y-direction), and the deflection angles along the x- and y-directions are both 22.23°. However, if we define a new additive coding sequence (000111000111··· along the x-direction plus 00110011···along the y-direction, see Figure S3a of the Supporting Information, the corresponding scattering patterns are shown in Figure S3b–e of the Supporting Information. In this case, the four beams are still directed to ϕ = 0°, 90°, 180°, and 270°, but the deflection angles along the x- and y-directions are 22.23°

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Figure 3.  Coding schemes and their 3D and 2D scattering patterns calculated by FFT to illustrate the influence and solutions of indefinite coding addition. a) Coding scheme of the coding sequence 00110011··· along the x-direction. b) Coding scheme of the coding sequence 00110011··· along the y-direction. c) Coding scheme by direct addition of the two initial coding sequences, in which the colors from light to dark represent the digital states from 0 to 3, showing the influence of indefinite coding addition. d) New coding scheme by correcting the indefinite coding elements. (i) Coding schemes. (ii) 3D far-field scattering patterns. (iii) 2D far-field scattering patterns.

and 49.18°, respectively. We clearly observe that the simulation results have excellent agreements with the ideal addition theorem. We continue to study the other additive coding metasurface. As mentioned above, the coding sequence 010101··· can produce symmetrical dual pencil beams, which however cannot be controlled independently. In radar applications, to track dif-

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ferent targets simultaneously, the two beams are required to be manipulated independently. Here we propose the novel dualbeam radiator by 3-bit complex codes and the addition theorem, realizing flexible manipulations of two beams without any interference. We start from two coding sequences 012301230123··· and 3322110033221100··· along the x-direction, respectively. The

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Table 2.  The final regulations for addition operations on 1-bit and 2-bit complex digital codes. 1-bit

2-bit

0 1 + 0 1 = 0 2

0 1 + 11 = 1 2

11 + 0 1 = 3 2

11 + 11 = 2 2

0 2 + 0 2 = 0 3

1 2 + 0 2 = 13

2 2 + 0 2 = 6 3

3 2 + 0 2 = 7 3

0 2 + 1 2 = 13

1 2 + 1 2 = 2 3

2 2 + 1 2 = 3 3

3 2 + 1 2 = 0 3

0 2 + 2 2 = 2 3

1 2 + 2 2 = 3 3

2 2 + 2 2 = 4 3

3 2 + 2 2 = 5 3

0 2 + 3 2 = 7 3

1 2 + 3 2 = 4 3

2 2 + 3 2 = 5 3

3 2 + 3 2 = 6 3

former could produce a single deflected beam at angle 49.18°, while the latter will produce a deflected beam at angle 22.23°, both of which are with respect to the z axis. After addition operations, the additive coding metasurface (see Figure S4a, Supporting Information) generates two independent pencil beams simultaneously on the same plane (yoz plane), as illustrated in Figure S4b–e of the Supporting Information. Clearly, the two beams have little interference during the addition operations. We further consider produce two independent pencil beams on different planes: one on the xoz plane and the other on the yoz plane. In this case, we need to perform addition operations of two coding sequences along different directions. For example, the addition of coding sequence 0011223300112233··· along

the y-direction and coding sequence 01230123··· along the x-direction will generate an additive coding metasurface, as shown in Figure 4a. The far-field scattering pattern and its sectional view of the additive coding metasurface are presented in Figure 4b–e, respectively, from which we clearly observe two independent pencil beams on two different planes. These two examples reveal that the addition operations of complex coding sequences enhance the manipulating abilities of the coding metasurfaces significantly and provide a fast and flexible design method for such complicated manipulations. The addition operations of complex coding sequences can be combined with the convolution operations on coding metasurfaces[36] to realize multibeam radiations to arbitrary directions. For coherence, we extend the digital convolution operation to the complex-code domain, in which the key step is the following relation

ϕ 1 + ϕ 2 ⇔ e jϕ1 ⋅ e jϕ2 = e j (ϕ1 +ϕ2 ) (4) Equation (4) builds up a bridge between the digital convolution operation and the complex codes. The convolution operation acts on coding sequences rather than a single code, and can be used for the macroscopic controls of coding sequences, while the complex codes aim at the microcosmic digital states. As an example, we demonstrate how to achieve two arbitrarily oriented scattering beams by performing the addition and convolution operations on complex codes. For simplicity, we name the two beams as B1 and B2, respectively. B1 is produced by the convolution operation of two coding

Figure 4. Numerical performance of out-of-plane dual-beam coding metasurface by complex digital codes. a) The addition operation of coding sequences 01230123··· along the x-direction and 0011223300112233··· along the y-direction, in which the colors from light to dark represent the digital states from 0 to 7. b,c) The corresponding far-field scattering patterns. d,e) The E- and H-plane patterns prove that the two beams are out-of-plane.

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sequences S1 (012301230123···) along the y-direction and S2 (0011223300112233···) along the x-direction; while B2 is generated by the convolution of another two coding sequences S3 (333222111000333222111000···) along the y-direction and S4 (31203120···) along the x-direction. According to the convolution theorem,[35] the beam’s elevation angle θ and azimuthal angle ϕ can be calculated as

(

 −1 sin 2 θ1 ± sin 2 θ 2  θ = sin   sin θ 2   ϕ = tan −1    sin θ1  

)

(5)

in which θ1 and θ2 are elevation angles of the two convoluted coding sequences along the x- and y-directions, respectively. In this manner, the convolutions of S1 and S2 as well as S3 and S4 produce two designed pencil beams independently. When these two convolved patterns are superimposed together by using the complex addition operation, we will obtain the two predesigned pencil beams simultaneously. Figure S5 of the Supporting Information brings out the excellent effect that the two ideal beams B1 and B2 occur at the same time.

Some other peculiar performances which cannot be realized by traditional coding metasurfaces are exhibited in Figure S6 of the Supporting Information, indicating the powerful control abilities of complex digital codes and their addition operations, as well as the connection among complex digital codes with different bits. In this case, the addition of 1-bit coding sequence 010101··· along the x-direction and 2-bit coding sequence 01230123··· along the y-direction will generate a 3-bit coding metasurface (see Figure S6a, Supporting Information), which provides a final performance of three-pencil-beam radiations, as illustrated in Figure S6b–e of the Supporting Information. Theoretically speaking, any controllable multibeam radiations can be superimposed by using the addition theorem, which produces any multifunctional coding metamaterials.

5. Conclusions In summary, we introduced the concepts of complex digital codes and their addition theorem, which extend the coding metamaterial to the complex-number domain, improving the manipulation ability of coding metamaterials significantly. In fact, the complex digital codes are closer to the nature of EM

Figure 5.  Unit cells for 1-bit, 2-bit, and 3-bit complex digital codes. a) The sketch of adopted unit cell. b) The transmission amplitude and phase responses of the eight elements. c) The schematic diagrams of the eight elements and their corresponding 1-bit, 2-bit, and 3-bit complex codes.

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waves. The addition theorem makes multifunctional coding metamaterials a reality as well as reveals the inherent connections among complex digital codes with different bits. We proposed a regulation for addition operations on 1-bit and 2-bit coding metamaterials to eliminate the influence from indefinite additions. Several examples were presented to demonstrate

the controlling abilities of complex digital codes. Two typical coding metasurfaces were designed, fabricated and measured to validate the concept and theorem experimentally. We remark that the complex digital codes are not only limited in the microwave band, but can be easily extended to the terahertz and visible bands. The proposed method can find applications

Figure 6.  Photographs of the fabricated samples in experiment environment and corresponding experimental results. a) The schematic of experiment environment. b) The fabricated samples. c) 2D scattering patterns in the E-plane. d) 2D scattering patterns in the H-plane. (i) The four-beam coding metasurface P1. (ii) The dual-beam coding metasurface P2.

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in multifunctional devices and information systems, especially when combined with the programmable coding metasurfaces. For example, simultaneous and real-time communications of multitargets can be achieved by the complex digital codes and addition operations through several independent radiated beams.

6. Experiments and Measured Results To verify the beam manipulation ability of complex digital codes and their addition theorem, two different samples of coding metasurfaces are designed, fabricated, and measured. The proposed unit cell is composed of round-corner squares, as shown in Figure 5a, which is a multilayer structure for building up the transmission-type coding metasurfaces. There are three dielectric substrates and four identified square metallic patches with round corners. The size of the unit cell is 11 mm, about λ/3 at the working frequency 9 GHz, which is the same as that in numerical simulations. F4B is adopted as the substrate with the thickness of 1.5 mm, relative permittivity of ε = 2.65, and loss tangent of 0.001. When changing the size of the round-corner patch, the transmission phase will vary correspondingly. We select eight elements to form 3-bit complex digital codes, which enjoy the high transmittance elaborately. Their amplitude and phase responses can be clearly observed in Figure 5b, in which all amplitudes are better than 0.75 and the phase differences between every two adjacent states are about 45°. The 1-bit and 2-bit coding states are included in the 3-bit complex codes. For easy observations, the geometrically schematic diagrams of the eight elements and their corresponding 1-bit, 2-bit, and 3-bit complex digital codes are demonstrated in Figure 5c. Because of the limitation of measurement system, only scattering beams appearing in the x–z and y–z planes are measured. We choose two typical designs of the coding metasurfaces for more accurate full-wave simulations and experimental verifications. One design is the four-beam coding metasurface P1 shown in Figure 3d(i), while the other design is the independent-dual-beam coding metasurface P2 presented in Figure 4a. Both P1 and P2 samples include 24 × 24 unit cells, covering an area of 264 × 264 mm2. In full-wave numerical simulations, an array of lump ports is applied to form plane waves and the distance from the samples to the source array is about two wavelengths. The far-field scattering patterns are simulated using the commercial software, CST Microwave Studio, and the numerical results are presented in Figure S7 of the Supporting Information, which have very good agreements with the numerical simulations in Matlab shown in Figures 3d(ii) and 4b, respectively. From the two kinds of simulation results, we also note that the scattering beams have slight deteriorations. This phenomenon is attributed to two reasons: (1) the full-wave simulated array is smaller than that in the Matlab simulations and (2) the incident waves set in the full-wave simulations are quasi-plane waves. The experiments are carried out in anechoic chamber and the photograph of the measurement setup of far-field scattering patterns is shown in Figure 6a and the fabricated samples of the structured coding metasurfaces are exhibited in Figure 6b.

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An X-band rectangular horn antenna with working bandwidth from 8.2 to 12.5 GHz is employed as the feeding antenna. The customary measurement method[41] has been used, in which the feeding antenna and samples are both coaxially mounted on a woody board that could automatically rotate 360° precisely in the horizontal plane. We remark that the woody board and other supporting equipment have little influence on the final performance because the reflection can be ignored. To generate the quasi-plane waves for the coding metasurfaces, the distance between the feeding antenna and the measured sample is set as 1.8 m. During experiments, the samples P1 and P2 are measured separately. Because the samples are transmission type, the influence caused by leaky waves affects the measurements, which is an inherent problem of this kind of measurement system. The distance between the feeding antenna and the sample is much larger than the diameter of sample, resulting in the fact that massive waves arrive at the receiving antenna without passing the sample. This will produce a huge beam at θ = 0°. To reduce the impact and retain precise results, a metal plate with the same size is employed as the control. We can acquire relatively accurate far fields after subtracting the experimental results of the metal plate. The measured 2D scattering patterns of the two samples P1 and P2 are described in Figure 6 after normalization. Here, the scattering patterns of coding metasurface P1 in E- and H-planes are shown in Figure 6c(i),d(i), respectively. We clearly observe two in-plane scattering beams appearing at the angles of ±47.5°, which are in very good agreements with the simulation results ±49.18°. The difference between the amplitudes of the two beams is caused by the asymmetry of the measurement environment. In a similar manner, the scattering patterns of coding metasurface P2 in E- and H-planes are illustrated in Figure 6c(ii),d(ii), in which two out-of-plane scattering beams are observed. One appears at the angle of 46.8°, while the other appears at 22°, which have good matches to the simulated values 49.18° and 22.23°.

Supporting Information Supporting Information is available from the Wiley Online Library or from the author.

Acknowledgements R.Y.W., C.B.S., and T.J.C. conceived the idea. R.Y.W. conducted the numerical simulations and theoretical analysis. R.Y.W., S.L., and T.J.C. wrote the manuscript. All authors participated in the experiments and data analysis and read the manuscript. This work was supported by the National Natural Science Foundation of China (61631007, 61571117, 61501112, 61501117, 61071107, and 61071108), 111 Project (111-2-05), and Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX17_0092).

Conflict of Interest The authors declare no conflict of interest.

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Keywords addition theorem, coding metamaterials, complex coding, multifunctional metamaterials Received: November 16, 2017 Revised: December 11, 2017 Published online:

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