Control of the Radiation Patterns Using Homogeneous and Isotropic Impedance Metasurface

We propose to control the radiation patterns of a two-dimensional (2D) point source by using impedance metasurfaces. We show that the radiation patterns can be manipulated by altering the surface impedance of the metasurface. Full-wave simulation results are provided to validate the theoretical derivations. The proposed design enjoys novel properties of isotropy, homogeneity, low profile, and high selectivity of frequency, making it potentially applicable in many applications. We also point out that this design can be implemented with active metasurfaces and the surface impedance can be tuned by modulating the value of loaded elements, like resistors, inductors, and capacitors.

Recently, the concept of metasurface has been very hot.It is two-dimensional metamaterial rather than the previously proposed bulk metamaterials.Also, it enjoys the properties of low profile and low loss, making it very flexible and easy to implement in the real applications [1][2][3][4][5].Metasurface can be considered as a patterned thin metallic film, whose electromagnetic behavior may be effectively described by averaged surface impedance [6].This kind of impedance metasurface has various engineering applications [7].In 2009, Alù proposed that the impedance surface could be used in the cloaking techniques and gave the name "mantle cloak" [8].It is ultrathin and can be fabricated easily by printing designed metallic patterns [9].Later, Zhu et al. successfully realized a metasurface with 360-degree reflection phase turning by controlling the surface impedance [10].
The impedance metasurface also has various applications in the field of antenna.Oliner and Hessel first put forward the comprehensive analysis of propagation of leaky waves on impedance surface [11].Using this theory, Fong et al. suggested a two-dimensional holographic antenna [12].This surface can scatter a known input wave and transform it into a desired output by varying the surface impedance.In 2012, the anisotropic impedance metasurface was suggested to control the polarization of the radiated field [13].More recently, impedance surface was also employed in the threedimensional form [14].And Hunt et al. extended this surface to microwave imaging by designing far-field radiation mode [15].These designs of radiation pattern require inhomogeneous or even anisotropic surface impedance, which makes it difficult to implement compared with homogeneous and isotropic one.Here, we exploit the homogeneous and isotropic surface impedance in the control of a radiation pattern.The fact that mantle cloak enjoys many scattering modes motivates us to control the radiation pattern by magnifying a specific mode.
In this work, we propose a very simple model, in which a metasurface radiates under the excitation of a 2D point source (or a line source).With proper design of surface impedance of the metasurface, a specific pattern can be magnified and becomes dominant in the total field; hence, the model behaves like the radiation of this pattern.Moreover, the radiation pattern can be manipulated by employing different surface impedances.To demonstrate the validity of the theory, full-wave simulation results are given, including those mimicking the radiations of monopole, dipole, and quadripole.In order to demonstrate the flexibility of our design, 2 International Journal of Antennas and Propagation the simulation for two metasurfaces is also made.These simulation results agree very well with the theoretical calculations.Throughout the paper, a time-harmonic factor   has been used.
Let us first consider a cylindrical metasurface in free space.A 2D point source (i.e., a line source) is placed near the metasurface, as shown in Figure 1(a).In this situation, the cylindrical metasurface is radiating under the excitation of the line source.The corresponding 2D geometry is illustrated in Figure 1(b), in which  represents the radius of the cylinder, and  represents the current of line source.In analyzing the problem, we set up a cylindrical coordinate system, where the axis of the cylindrical metasurface coincides with the -axis.The coordinate center is set at the center of the metasurface and the coordinates of the line source are (  ,   ).Then the incident electric field   can be written as in which  =    0 ,  0 is the wave number in free space and   (⋅) and  (2)   (⋅) are cylindrical Bessel function and Hankel function of the second kind, respectively.In our analysis, the excitation is a line source with infinite length, making the electric field parallel to the -axis.
In the presence of the metasurface, the induced scattered field    and    in the interior and exterior regions of metasurface are expressed as in which the common factors − 0  0 /4 in ( 1)-( 3) are used in order to make the following expressions compact in form.Let us use the boundary condition [8]: where   denotes the surface impedance.It is obvious that   =   +   , in which   and   represent the resistance and reactance of surface impedance, respectively.Using (1)-( 4), we can easily derive the scattering coefficients as In order to magnify a specific pattern for radiations, the corresponding scattering coefficient is set to infinite, which can be easily achieved by setting the denominator of (5) to zero.This leads to the expression of the surface impedance as which is used in the calculation of surface impedance for the corresponding radiation of a monopole, a dipole, a quadripole, and so forth.This is easily obtained by setting the order  in the above equation to 0, 1, 2, . .., respectively.Using Huygens' principle, the electric field in the far-field region is calculated to determine the scattering width .Since the problem in this work is 2D, we only need to figure out  2D [16], which is given as follows: where  ⇀   and  ⇀   are the EM field on the integration contour , r0 is the unit vector of the scattering direction,  ⇀   is the position vector on the contour , and  0 is the free space wave impedance.So we can calculate the radar cross section based on this equation.Note that the integration contour  can be chosen in near-field when there is no source in exterior region of this contour.To generally measure the overall scattering width versus frequency, the total scattering width is used [9,17]: It is defined as the radio of the energy across a closed surface that encloses the scatterer to the incident irradiance.
Next we perform full-wave simulations to verify the proposed theory, in which the working frequency is set as 10 GHz.Using (6), we first consider the case of a monopole radiation.In our simulation, the radius of the cylindrical metasurface is set to  0 /5 ( 0 represents the wavelength in free space) and the polar coordinates of the excitation source (  ,   ) are ( 0 , 0).The values of  0 and   are both set as 1.The calculation result shows that the surface impedance for the monopole radiation is simply   = (−307.2+ 125.3)Ω. Figure 2 demonstrates the simulation result of the monopole radiation, in which Figure 2(a) illustrates the electric field distribution in the - plane.It is obvious that the scattered field by the metasurface looks like a monopole radiation.Moreover, it is clear that the incident field of line excitation becomes negligible when the monopole pattern is magnified.
In order to quantify our simulation result, the electric field on a centered circle with radius 0.1 m is extracted from the simulation result.Then the scattering width  2D is calculated using the proposed method, and the far-field radiation pattern is given in the polar diagram in Figure 2(b).Obviously, the radiation pattern is nearly a circle, which quantitatively verifies the correctness of the theoretical analysis.Note that a little discrepancy exists in the far-field radiation pattern, which is due to some disturbance from the remaining patterns.However, these disturbances are ignorable when compared to the designed radiation pattern, making the metasurface behave like a monopole.Then we choose  = 1, and the surface impedance of metasurface is calculated as   = (−195.2− 220.9)Ω, which corresponds to the case when the dipolar pattern is magnified.The performance for this metasurface is presented in Figure 3. Similarly, both the near electric field distribution and far-field radiation pattern are studied.The simulation result illustrated in Figure 3(a) shows that the radiation of this metasurface looks like a dipole, in which the excitation source also becomes ignorable.The quantitative depiction of far-field radiation pattern is given in Figure 3(b).Note that the curve of radiation pattern is not a rigid "∞" shape because some other patterns have also been magnified, affecting the total radiation field.But these patterns are overwhelmed by the dominant dipole field, making the total radiation pattern behave like a dipole.
We continue to consider the case of quadripole, and the corresponding value of surface impedance is obtained by setting  = 2, which yields   = (−22.2− 152.2)Ω. Figure 4 shows the simulation results of the quadripole case, in which Figure 4   radiate like a quadripole, whose far-field pattern has a typical petal shape.Likewise, other radiation patterns, like sextupole and octupole radiations, can also be achieved by setting the order  to 3 and 4, respectively.And from the calculation results, we can easily reach the conclusion that the proposed metasurface is active due to the fact that surface resistance has a negative value.
However, the proposed metasurface can be regarded as a basic unit.And we can put several units to create many other kinds of radiation patterns.In the following part, the simulations of radiation pattern when there are two cylindrical metasurfaces with same parameters are given, in which three cases mean mode  equals 0, 1, and 2, respectively.The simulation results are shown in Figure 5. Additionally, more attainable radiation patterns can be realized by different combination of these units.
Using (8), the variations of total scattering widths versus frequency are calculated, as shown in Figure 6, including the cases of monopole, dipole, and quadrupole.This figure demonstrates that the scattering width is very sensitive to the deviation of frequency from 10 GHz.In other words, this metasurface is selective in scattering frequency, which can avoid the disturbance of other frequencies in the real applications.
Recently, active devices have been suggested to realize some negative electromagnetic parameters, which may not be attainable in nature.In our work, a negative resistor model was put forward to meet the negative conductivity requirement of an exterior cloak [18].This model consists of a source and two resistors in order to make the input current have a negative linear relation with voltage.Also, loading of negative impedance converter (NIC) elements has been proposed as a way to realize an active metasurface [19].These loading elements can provide an effective negative capacitance.Based on these ideas, it can be expected that a metasurface with negative resistance can also be realized in a similar manner.
With regard to control of surface impedance, we point out that its value can be changed by modulating the parameter of some loaded elements, like resistors, inductors, or capacitors.For example, in Liu's work, the surface impedance is tuned by voltage-modulation of a varactor diode [20].
In summary, we have shown that a designed radiation pattern can be achieved by using a line source to excite a metasurface.By selecting proper surface impedance, the radiation pattern of total field can be a magnified monopole, dipole, quadripole, and so forth.This makes it possible to build up a multipolar antenna system with the simple configuration.The most important property of the model is that its radiation pattern is controlled by simply changing the surface impedance rather than modifying the geometrical shape.And this surface impedance is homogeneous and isotropic in contrast with previous holographic impedance surface.Moreover, this model is ultrathin and selective in the radiation frequency.

Figure 1 :Figure 2 :
Figure 1: Principle of the proposed model with a cylindrical metasurface and a line source (color online).(a) The three-dimensional depiction.(b) The two-dimensional geometry of the cross section.

Figure 3 :Figure 4 :
Figure 3: The simulation result for a dipole radiation (color online).(a) The electric field distribution in the - plane.(b) The far-field radiation pattern with a "∞" shape.
(a) is the near-field distribution and Figure 4(b) gives the corresponding far-field radiation pattern.Through the change of surface impedance, the metasurface can truly