Adiabatic Floquet-Wave Expansion for the Analysis of Leaky-Wave Holograms Generating Polarized Vortex Beams

Increasing the channel capacity of wireless systems is an essential demand in the implementation of new generations of telecommunication networks. An attractive way to meet this need without increasing the frequency bandwidth is to utilize photon orbital angular momentum (OAM) in the microwave regime thide2007 . Electromagnetic waves with helical spatial phase profiles can support different orthogonal OAM modes. Theoretically, the number of such modes is infinite. Consequently, combining these new multiplexing dimensions with other existing schemes such as time, frequency and polarization multiplexing can significantly increase the channel capacity chen2020 . In recent years, various methods have been proposed to generate the radio-frequency waves carrying OAM modes. Utilizing phased array antennas with circular arrangements gao2014 , spiral phase plates hui2015 , all-dielectric transformers yi2019 ; yi2_2019 , transmit-arrays jiang2018 and reflect-arrays karimipour2019 are among the most common methods for synthesizing OAM vortex waves. In phased array antennas the complexity of the feed network and the destructive effects of mutual coupling among adjacent elements are important limiting factors. Note that, eliminating the mutual couplings among elements in phased arrays drastically increases the complexity of realization process. To avoid these issues, all-dielectric transformers are proposed in the literature. In addition, reducing the divergence effects of vortex waves can be achieved by these transformers yi2_2019 . However, all-dielectric transformers and spiral phase plates are relatively bulky, which are unsuitable for use in low profile integrated systems. Recently, the advent of metasurfaces yu2011 -yu2014 has paved the way for the realization of integrated optical devices. These structures can manipulate electromagnetic wave front to obtain the desired properties by introducing abrupt phase shifts on the incident wave. Reflect/transmit arrays can be realized by metasurfaces. The use of metasurface-based reflect/transmit arrays has several advantages, including high gain, low divergence angles, low manufacturing cost and simple fabrication processes. However, in reflect/transmit arrays the feeding systems need to be mounted outside the structure, which makes them bulky. This issue contradicts the integrability of metasurfaces. An alternative solution to achieve the advantages of reflect/transmit arrays without the need for bulky feed systems is utilizing the leaky-wave meta-holograms, in such a way that the feed can be integrated with the metasurface plane. The application of holographic technique in antenna engineering was first proposed by Checcacci checcacci1970 and a practical example of leaky-wave hologram was presented in fong2010 . In the holographic technique, the information obtained from the interference of the surface wave (as the reference wave) and the desired space wave (as the object wave) is utilized for the synthesis of metasurface. In oraizi2020 the isotropic hologram is used to generate radiations capable of carrying OAM modes. In amini_2020 -bodehou_OAM_2019 anisotropic holograms are designed as polarized vortex beam generators. A serious obstacle in designing and optimizing leaky-wave holographic antennas is their large size. Generally, the dimensions of such structures to achieve a high gain radiation pattern should be selected about $10\lambda$ to $20\lambda$ ($\lambda$ as the free-space wavelength). On the other hand, due to the small size of unit cells in terms of wavelength ($\approx\lambda/6$), the number of meshes in full-wave simulations may increase significantly. This issue greatly increases the computational complexity and CPU time consumption. Various theoretical models have been reported for the analysis and synthesis of leaky-wave holograms. The aperture field estimation (AFE) method is proposed as an accurate synthesis method for realizing leaky-wave holograms with circular casaletti_2017 and vertical polarizations amini_2020 . In ovejero2015 ; bodehou2019 , the Method of Moments (MoM) framework is developed for the analysis of anisotropic holographic antennas, which is accurate and fast for treating large size structures. This method has been used to explain the radiation mechanism of holograms with circularly polarized shaped beams minatti_2015 , multiple beams ovejero2017 ; bodehou2020 and multi band radiations faenzi2019 . Another analytical method that can be considered is the adiabatic Floquet-wave expansion (AFW) method, which is relatively simpler than MoM in terms of mathematical complexity. This method is proposed for the first time by Minatti et al. minatti_2016 ; minatti_2_2016 for the shaping of far-zone patterns with different polarizations.

In this paper, the combination of aperture field estimation (AFE) technique (as a synthesis method) and adiabatic Floquet-wave (AFW) expansion method (as an analysis approach) is utilized to get deep insight into the radiation mechanism of OAM wave radiators enabled by leaky-wave holograms. Using AFE technique, the aperture field (consisting of both phase and amplitude information) and then impedance distribution for obtaining the polarized vortex beam have been estimated. Furthermore, AFW method determines the leakage properties of the surface wave. This method is much faster than the full-wave method, which makes it suitable for synthesizing and optimizing complex vortex beams.

The adiabatic generalization of the Floquet-wave (FW) theorem is a well known method for the analysis of leaky-wave holograms generating pencil beams in the microwave regime, which was initially proposed by Minatti et al. minatti_2016 . The advantage of using this method is the precise analysis of anisotropic holograms without the need for full-wave simulations, which significantly reduces the synthesis time consumption. A conceptual structure of anisotropic hologram is shown in Fig.1(a), including the modulated metasurface (hologram) and vertical monopole as surface wave generator. The monopole excites a cylindrical magnetic surface wave, which can be represented by moeini_scirep_2019 :

where $H_{1}^{(2)}$ is the Hankel function of second kind and first order. Also, $k^{(0)}$ is the complex surface wave number.

The metasurface consists of pseudo-periodic patches printed on the grounded dielectric substrate (as shown in Fig.1(b)). If the dimensions of patches are small enough compared to the wavelength ($\approx\lambda/6$), the metasurface effectively acts as an impedance surface. By proper patterning of surface impedance, the surface wave can be converted to leaky mode with the desired direction, polarization and topological charge. The main purpose of using adiabatic Floquet-wave analysis is to provide a systematic approach for calculating the distribution of leakage parameter ($\alpha$) across the aperture, that can be used to synthesize radiation patterns with the desired specifications minatti_2_2016 . In this paper, this method is used for the analysis of anisotropic holograms with the ability to generate vortex waves carrying OAM modes.

In the analysis of radiative apertures, the Fourier transformation and stationary phase theory can be applied to determine the far-zone fields. Thus, for the $\theta$ and $\phi$ components of the far-zone field we have balanis_2016 :

where $\tilde{E}_{ax}$ and $\tilde{E}_{ay}$ are the Fourier transforms of $E_{ax}$ and $E_{ay}$ respectively. Hence, in the cylindrical coordinate system we can write

To validate the analytical approach in the previous section, we present two examples of leaky-wave holograms generating OAM modes. In the first section, an isotropic unit cell is used to realize hologram with a broadside beam. In the second, anisotropic hologram with circular polarization is designed. Without loss of generality, the operation frequency of antennas is selected to be 18 GHz. Rogers RO4003 with dielectric constant 3.55, loss tangent $\tan\delta=0.0027$ and thickness 1.524 mm is used as the substrate. The square patch printed on the grounded dielectric has been chosen for realization of isotropic (scalar) impedance. The period of unit cell is chosen as 2.8 mm which is equal to $\lambda/6$ at the operating frequency. Fig.3 shows the proposed unit cell and reactance curve for variation of square width (namely $a$), which is obtained by Eigen-mode solver in CST microwave studio cst .

To implement the anisotropic (tensorial) impedances, we utilize asymmetric rectangular patches as the composition pixels. Fig.4(a) shows the proposed anisotropic unit cell. The width of the patch (namely $b$) and its orientation angle (namely $\psi$) are considered for changing the impedance. To extract the impedance tensor, first the reactances $X_{1}$ and $X_{2}$ are calculated for the surface wave propagating along the x and y directions, respectively. In both directions, the parameter $\psi$ is kept equal to zero and the parameter $b$ is varied from 0.2 to 2.4 mm. The reactance curves for $X_{1}$ and $X_{2}$ are plotted in Figs 4(b) and 4(c), respectively. For the retrieval of the impedance tensor the method proposed in werner_2014 is used, which can be expressed as

where

and $R$ is the rotation matrix:

Note that $T$ indicates the transpose operator.

Fig.5 shows the impedance maps of proposed anisotropic unit cell versus the width and orientation angle of patch.

To clarify the advantage of theoretical model over the full-wave simulation, in this section, the computational performance of the two methods are compared. Generally, leaky-wave holograms can be considered as traveling wave radiators. However, in order to have an appropriate radiation performance, they must be relatively large. The dimensions of these antennas to achieve high gain beams are about $10\lambda$ to $20\lambda$. On the other hand, the dimensions of unit cells to realize the hologram must be small enough ($<\lambda/5$) so that the surface can act as a homogenized impedance boundary condition. Therefore, the full-wave simulator applies dense meshing to be able to distinguish each inclusion, which greatly increases the simulation time. Table 1 compares the CPU time for both full-wave and theoretical methods. The allocated RAM for the theoretical model is 12 GB. Also, the processor dedicated to it is Core i7 6500U (Ultra Low Voltage) with two real cores. The processor used in full-wave simulations is the Core i7 6850K model with 6 real cores and the dedicated RAM is 64 GB. The results of Table 1 show that the analysis time for the proposed method is significantly shorter than the full-wave simulations. Observe too that the proposed method can be easily developed on conventional computers, where the full-wave simulation is impractical.

In this work, an analytical method based on adiabatic floquet-wave expansion and aperture field estimation technique is proposed for the implementation of vortex beam radiators. To determine the aperture field, three points must be considered: 1- The amplitude of the field specifying the shape of the pattern; 2- The phase of the aperture field that determines the direction and topological charge of OAM wave; 3- The relationship between the x and y components of the aperture field that controls the polarization of radiation beam. According to the above considerations, two structures based on isotropic and anisotropic unit cells is proposed. For both structures, the aperture fields are estimated to achieve the desired patterns, polarizations and topological charges and are synthesized by the relevant unit cells. It has been shown that the anisotropic unit cell has the distinct advantage of polarization control over that of the isotropic one. The advantage of using the analytical method is its high speed and less resource requirements and consumption. Furthermore, it has very good accuracy, which makes it suitable for analyzing and designing holograms with very large dimensions.

This appendix describes the calculation procedure of $\delta\alpha$ in detail. In (17) the tensor $\bar{\bar{\chi}}$ is obtained from the following equation minatti_2016 :

Note that $\bar{\bar{Z}}_{GF}^{(0)}$ is the tensorial Green’s function of grounded dielectric slab evaluated at $\vec{k}=\beta_{um}^{(0)}\hat{\rho}$ and can be obtained as follows:

where

In (52) the tensor $\bar{\bar{X}}^{(0)}$ denotes the average reactance in the case of $m_{\rho}=m_{\phi}=0$ and is defined as:

Also, $\bar{\bar{z}}^{(+1)}$ and $\bar{\bar{z}}^{(-1)}$ have the following forms:

where

Note that, In (57) and (58), the tensors $\bar{\bar{Z}}_{GF}^{(+1)}$ and $\bar{\bar{Z}}_{GF}^{(-1)}$ are the Green’s Functions of the dielectric slab at $\vec{k}=\beta^{(+1)}\hat{\rho}$ and $\vec{k}=\beta^{(-1)}\hat{\rho}$, respectively. Using (53)-(55) and substituting $\beta_{um}^{(0)}$ with $\beta^{(+1)}$ and $\beta^{(-1)}$ we can determine the tensors $\bar{\bar{Z}}_{GF}^{(+1)}$ and $\bar{\bar{Z}}_{GF}^{(-1)}$.