Holographic Beamforming

Abstract

This chapter presents the fundamentals of the holography technique to form the beam in electromagnetic (EM) structures. The application of holography in leaky-wave antennas, metasurface reflectors, and reconfigurable intelligent surfaces (RISs) is explained. Consequently, different methods to analyze and realize an EM hologram are presented. A comparison is made between forming the beam via holographic-based radiators, phased-array antennas, and MIMO systems. The thing which is common between these three is that all of them can contain a number of elements that are repeated in a fashion. However, the functionality of these elements in the three mentioned structures is totally different from each other. This concept is explained in detail in this chapter.

1. Introduction

Consider a case where an electromagnetic (EM) aperture with an arbitrary geometry is placed on the $\mathit{xy}$ plane as shown in Figure 1(a). To have the ability to form the constructed beam, it is required to control both phase constant $\left(\beta \right)$ and amplitude $\left(\alpha \right)$ of the EM waves at different segments of the aperture. The beam tilt angle can typically be controlled by regulating $\beta$ whereas $\alpha$ distribution over the aperture controls side-lobe-level (SLL). Four segments are marked in Figure 1(a) as an illustrative example. In the case of a conventional phased-array antenna, these four segments represent four physical elements, i.e. antennas, where $\beta$ and $\alpha$ of each can be controlled in straightforward approaches by using phase shifters and attenuators respectively, or with a fully passive custom-built feeding network with proper delay lines and by applying the power-splitting technique. All these elements make the final EM aperture together with an engineered beam as $\beta$ and $\alpha$ are governed at different positions of the aperture.

Figure 1.

Aperture formation. (a) The case where four elements make a large aperture altogether and (b) the case where four elements create four separated apertures.

Now consider the case where these four elements are located in such a way that they are not able to have a sensible impact on each other to build up a large EM aperture. Under this circumstance, each element acts as an individual aperture as shown in Figure 1(b) with its specific radiation properties. This configuration of elements can be applied in multiple input multiple output (MIMO) systems. With a dedicated port for each element, this configuration represents space diversity provided that the cross-correlation between the ports is kept low. It is also possible to use a single element and connect more than one port to it. In this case, each port belongs to a specific radiation state or the so-called mode. Having orthogonal modes in this scenario will lead to a low cross-correlation between the ports, making the structure a good candidate for MIMO systems. This orthogonality can be obtained in radiation patterns (pattern diversity) or polarization (polarization diversity) or a combination of both.

Let us move back to the large EM-aperture of Figure 1(a). The envision of such a large aperture is not limited to just phased-array antennas and can be obtained by several means including but not limited to leaky-wave structures, reflectarrays, and transmitarrays. Forming the beam in such structures is also fulfilled by regulating $\beta$ and $\alpha$ over the aperture but in approaches different from conventional phased arrays. One approach is to employ the holography technique [1] to govern $\beta$ on the structure and to correspondingly control the tilt angle of beam(s) which is known as “holographic beamforming”.

This chapter presents the principles of the holography technique and then explores its capability to form the beam. To this end, some background information on leaky-wave structures and reflectarrays is required.

2. Holographic-based antennas

2.1 Holographic-based leaky-wave antennas

Let us start with the application of holography in leaky-wave antennas. A holographic leaky-wave antenna is a type of antenna that utilizes the principles of holography and leaky-wave propagation to construct the beam and achieve beam scanning capabilities [2]. A leaky-wave antenna (LWA) operates by “leaking” EM energy along its length, which leads to the formation of a propagating wave [3]. Unlike conventional resonating antennas that typically radiate energy perpendicular to the antenna’s length, LWAs emit energy at an angle ${\theta }_m$ along their length. This tilt angle can be controlled by regulating the phase constant $\beta$ of the guided waves across the structure and formulated as [4]:

$${\theta }_m\approx {sin}^{-1}\left(\frac{\beta }{k_0}\right),$$E1

where $k_0$ being the free-space wavenumber. Considering (1), ${\theta }_m$ will have a real answer if and only if $\mid \beta \mid <k_0$. This means that LWAs support fast waves on the guide.

Holography, which originates from optics, is a technique to achieve a desired $\beta$ on the structure by governing the phase distribution on the structure. As $\beta$ is controlled in an LWA, the tilt angle of the constructed beam can be specified by (1). This technique is summarized below:

Having a dielectric slab on the $\mathit{xy}$ plane to design the aperture on, the first step is to define two field distributions on the structure known as reference wave $E_{\mathrm{ref}}$ and object wave $E_{\mathrm{obj}}$, generated by two hypothetical sources. To be more explicit, a source should be defined somewhere within the slab where it is aimed to place an actual surface-wave launcher (SWL). For a lossless structure, an ideal hypothetical source located at $\left(x=0,y=0\right)$ will generate a radially expanded field distribution on the slab for both TM and TE surface waves which can be formulated as below [5]:

$$E_{\mathrm{ref}}={\mathit{Ae}}^{-j{\beta }_{r}r},$$E2

where $r=\sqrt{x^2+y^2}$ and $A$ is the wave amplitude. This is schematically shown in Figure 2(a). It is clear that the location and type of this source can be in a variety of forms. For example, it is possible to place a number of ports at one end of the slab to generate parallel phase lines as shown in Figure 2(b) with the formulated reference wave of $E_{\mathrm{ref}}={\mathit{Ae}}^{j{\beta }_{y}y}$. It is also possible to locate the source somewhere out of the slab as presented in Figure 2(c). Under this circumstance, the final structure will not recognize as an LWA; this case is explained more in the next section.

Figure 2.

Different forms of reference wave on the slab. (a) Radial reference wave by an ideal single source at the center of the slab, (b) parallel reference wave formed by a number of sources at one edge of the slab and (c) induced reference wave from a source located outside the slab.

The next step is to define an object wave $E_{\mathrm{obj}}$ on the slab. To this end, another hypothetical source should be defined far from the slab toward the direction of the desired constructed beam. The slab is illuminated by this source and the corresponding induced waves should be calculated which represents $E_{\mathrm{obj}}$. For example, for a beam desired to be formed toward $\left({\theta }_m,{\varphi }_m\right)$, the induced $E_{\mathrm{obj}}$ is obtained by the mapping as below:

$$E_{\mathrm{obj}}={\mathit{Be}}^{{\mathit{jk}}_0\left\{sin\left({\theta }_m\right)cos\left({\varphi }_m\right)x+sin\left({\theta }_m\right)sin\left({\varphi }_m\right)y\right\}},$$E3

where $B$ is the amplitude of the object waves.

The next step is to calculate the superposition of $E_{\mathrm{s}}=E_{\mathrm{ref}}\times E_{\mathrm{obj}}$ as an interference pattern where $\angle E_{\mathrm{s}}$ defines the desired EM hologram.

The aforementioned steps of calculating $E_{\mathrm{ref}}$, $E_{\mathrm{obj}}$, and $E_{\mathrm{s}}$ make the “recording” process altogether which means to record the impact of the influencing parameters on the slab. For example, consider a case where an ideal source generates $E_{\mathrm{ref}}$ as presented in Figure 3(a) on the slab at a specific frequency. With a defined object wave toward $\left({\theta }_m=\pi /4,{\varphi }_m=2\pi /3\right)$, the obtained $E_{\mathrm{obj}}$ on the slab is shown in Figure 3(b) for the corresponding $k_0$. In this case, the pattern of the EM hologram is derived as presented in Figure 3(c).

Figure 3.

Recording process: (a) $E_{\mathrm{ref}}$ with an ideal source at the center of the slab, (b) $E_{\mathrm{obj}}$ in case of $\left({\theta }_m=\pi /4,{\varphi }_m=2\pi /3\right)$, and (c) the obtained EM hologram.

To embody a real-world structure from the calculated EM hologram, it is required to apply an SWL, exactly at the location where the hypothetical source has been placed in the recording process to be able to generate a field distribution as much similar to the derived $E_{\mathrm{ref}}$ as possible. Then, a quasi-periodic pattern of scatterers, with the geometry and lattice inspired by $\angle E_{\mathrm{s}}$ must be applied on the slab to locally sample the generated field distribution of the SWL. The scatterers can be printed metal-strips, sub-wavelength patches of arbitrary shape, dielectric cubes, or any other component that can scatter the launched surface waves on the slab. The process of applying the appropriate SWL and pattern of scatterers on the slab is called “reconstruction”.

When the structure in hand is excited by its SWL, the induced surface waves will be leaked out to the open environment toward the predefined tilt angle of $\left({\theta }_m,{\varphi }_m\right)$ which makes a holographic-based LWA (HLWA).

As an example, an open-ended coaxial cable presented in Figure 4(a) can be applied on a grounded dielectric slab to generate TM surface wave distribution similar to Figure 3(a). The surface-wave sampling can be performed by printing metal strips on the local maxima of the calculated EM hologram in Figure 3(c) which is presented in Figure 4(b). When this structure is excited, the simulated normalized radiation pattern is obtained as presented in Figure 4(c). This shows that the constructed beam is pointed well to the predefined angle of interest at the very first steps of the design which is $\left({\theta }_m=\pi /4,{\varphi }_m=2\pi /3\right)$.

Figure 4.

Reconstruction process: applying (a) surface-wave launcher, and (b) metal strips on the slab. (c) The constructed normalized radiation pattern.

2.2 Holographic-based reflectors

As briefly pointed out in Figure 2(c), the holography technique can be expanded to the case where the initial source is located outside the slab’s body. In this case, the obtained structure will be a holographic-based reflector (HR) [6].

This time, let us sample the EM hologram by using a number of printed sub-wavelength squared-shape patches. These patches will form a quasi-periodic structure where their size is modulated based on the holography technique. In periodic structures, the smallest geometry that is repeated in a fashion is called a unit cell. In this case, the unit cell is a small portion of the dielectric slab with a single printed patch on one side and a full ground plane on the other side as shown in Figure 5(a). The analysis of structure requires characterizing the surface impedance $Z_{\text{surf}}=E_t/H_t$ with $E_t$ and $H_t$ representing the tangential electric and magnetic fields respectively. The obtained structure is then an artificial impedance surface, commonly referred to as a metasurface.

Figure 5.

(a) The applied unit cell, a schema of (b) $E_{\mathit{ref}}$ and (c) $E_{\mathit{obj}}^{\kappa =1}+E_{\mathit{obj}}^{\kappa =2}$, (d) $Z_{\mathit{\text{surf}}}$ versus patch size variation and (e) the obtained $Z\left(x,y\right)$ on the surface [7].

The recording process in Section 2.1 is needed to be modified at the outset to reflect the location of the initial source, i.e. the feeder, on (2).

With an ideal feed located at $\left(x_f,y_f,z_f\right)$ and the slab on the $\mathit{xy}$ plane in a standard right-handed coordinate system, $E_{\mathrm{ref}}$ is modified as

$$E_{\mathrm{ref}}={\mathit{Ae}}^{-{\mathit{jk}}_{0}r},$$E4

where $r=\sqrt{{\left(x-x_f\right)}^2+{\left(y-y_f\right)}^2+z_f^2}$.

Figure 5(b) shows the obtained $E_{\mathit{ref}}$ when the feeder is placed at $\left(x_f,y_f,z_f\right)=\left(\mathrm{0,0,2.5}\mathrm{m}\right)$ for a $1.65$ m $\times$$1.25$ m large dielectric sheet at $f=3.5$ GHz [7].

In the holography technique, it is possible to define more than one main beam for the final constructed radiation pattern. Under this circumstance, a summation of the respective object waves will define the final distribution of $E_{\mathit{obj}}$ on the slab. Each object wave is derived by (3) toward the angle of interest. It is aimed in this structure to obtain two reflected beams to $\left({\theta }_{\kappa =1},{\varphi }_{\kappa =1}\right)=\left({45}^{\circ },0\right)$ and $\left({\theta }_{\kappa =2},{\varphi }_{\kappa =2}\right)=\left(-{45}^{\circ },0\right)$. The calculated $E_{\mathit{obj}}=E_{\mathit{obj}}^{\kappa =1}+E_{\mathit{obj}}^{\kappa =2}$ is presented in Figure 5(c).

In order to derive the EM hologram, it is now required to conduct a study on $Z_{\text{surf}}$ regarding the unit cell of the structure. This can be calculated by sweeping the phase delay (${\varphi }_D$) across the unit cell with periodicity of $p$ as follows [8]:

$$Z_{\text{surf}}={\mathit{jZ}}_0\sqrt{{\left(\frac{{\varphi }_D}{k_{0}p}\right)}^2-1},$$E5

where $Z_0$ is the free-space impedance. This can be fulfilled by using the eigenmode solver of a full-wave simulator for a specific size of the square patch. Then, the size of the patch must be varied and the calculation repeated to determine the span range of surface impedance $\mathrm{\Delta }Z$ with the mean value $Z_{\text{mean}}$ over the range of patch size variation. This is schematically shown in Figure 5(d).

Having all the above-mentioned information, it is possible to define the EM hologram pattern based on the impedance distribution as below [1]:

$$Z\left(x,y\right)=j\left(Z_{\text{mean}}+\frac{\mathrm{\Delta }Z}{2m}Re\left[\left(\sum _{\kappa =1}^m{E}_{\mathrm{obj}}^{\kappa }\right)E_{\mathrm{ref}}^{\ast }\right]\right),$$E6

where $m$ is the number of beams which equals $2$ in this case study.

It is shown that $Z_{\text{mean}}=428.16j\mathrm{\Omega }$ and $\mathrm{\Delta }Z=566.51j\mathrm{\Omega }$ for the studied range of patch-size variation [7]. This results in an impedance distribution of Figure 5(e) as the EM hologram.

The reconstruction process is to use Figure 5(d) and (e) to modulate the size of patches on each unit cell and print them on the slab. This will lead to a structure shown in Figure 6(a). When this metasurface reflector is illuminated by a feed horn located at the position defined during the recording process, the reflected beams from the surface are formed as presented in Figure 6(b) which is in line with the defined object waves.

Figure 6.

(a) The metasurface reflector and (b) simulated and measured normalized radiation patterns [7].

3. Reconfigurable intelligent surface (RIS)

Another possible scenario is the case when the initial source is located outside the surface, but far from the structure. Assume an ideal initial source located at the angle of $\left({\theta }_s=\pi /6,{\varphi }_s=\pi /4\right)$ with respect to the surface normal far from the structure. Following the routine explains in Section 2.1 and 2.2, $E_{\mathrm{ref}}$, $E_{\mathrm{obj}}$, and the EM hologram patterns are calculated as shown in Figure 7(a), (b), and (c) respectively provided that the reflected beam is aimed to be pointed to $\left({\theta }_m=\pi /3,{\varphi }_m=\pi /6\right)$.

Figure 7.

Recording process: (a) $E_{\mathrm{ref}}$ with an ideal source at the angle of $\left({\theta }_s=\pi /6,{\varphi }_s=\pi /4\right)$ with respect to the surface normal located far from the structure, (b) $E_{\mathrm{obj}}$ in case the reflected beam is aimed to be pointed to $\left({\theta }_m=\pi /3,{\varphi }_m=\pi /6\right)$, and (c) the obtained EM hologram.

To translate this mechanism into a practical format, this is the case when the surface reflects the incoming waves from a far-located source to a desired direction. This is not a simple mirror reflection where there is no control over the angle of reflection; indeed, the reflection angle can be engineered in this case using the holography technique. This brings a new idea for the future generation of cellular networks.

Consider a case where there is a blind spot within the area under the coverage of a base station (BS). The conventional approach to providing coverage for this blind spot is to add a new BS in the network. However, this method can be expensive and sometimes very challenging regarding the environmental barriers in an area. The idea is to locate a surface in the line of sight (LoS) of the BS so that it can be illuminated by the BS. Then the receiving EM waves reflect back to that blind spot to recycle the waves and provide coverage without adding a new BS. It is possible to reconfigure the response of the surface by applying some components like Varactor or PIN diodes on each unit cell and recalculating the EM hologram for each state of reflection. Under this circumstance, the obtained structure is called a reconfigurable intelligent surface (RIS) [9]. This scenario is schematically shown in Figure 8(a).

Figure 8.

(a) The coverage provisioning via reconfigurable intelligent surface (RIS) for a blind spot and (b) a prototype example of RIS [11].

Note that holography is not the only technique to regulate the response of the RIS. The generalized Snell’s law of reflection (GSR) can also be used in this regard [10]; a GSR-based RIS prototype [11] is shown in Figure 8(b).

One of the biggest problems for RIS to be industrialized and practically applied in real-world networks is its very low aperture efficiency $\left({\eta }_a\right)$. This will be more challenging for the uplink scenario when the user attempt to connect BS via RIS. To have a more clear idea about this problem, consider a rectangular reflecting surface with a planar dimension of $X\times Y$, illuminated by a feed horn, distanced by $R$ as presented in Figure 9(a). The feeder’s radiation pattern can be expressed by ${cos}^{q_E}\left(\theta \right)$ and ${cos}^{q_H}\left(\theta \right)$ at the E-plane and H-plane respectively. Product of the illumination $\left({\eta }_{\mathit{ill}}\right)$ and spillover $\left({\eta }_s\right)$ efficiencies is then defines ${\eta }_a$. In case of rectangular surfaces, ${\eta }_{\mathit{ill}}$ can be calculated by [12]:

Figure 9.

(a) The overall geometry of a reflective surface illuminated by a feeder, (b) the theoretical aperture efficiency when the feeder is not far from the aperture, (c) a schema of using reflective surfaces to provide the coverage for the user in the blind spot region of the BS, and (d) the theoretical aperture efficiency when the feeder is located relatively far from the aperture.

$${\eta }_{\mathit{ill}}=\frac{I^2}{\mathit{sII}}.$$E7

where $s=X\times Y$ and

$$\begin{array}{l}I={\int }_{x=-X/2}^{X/2}{\int }_{y=-Y/2}^{Y/2}\left\{\frac{1}{\sqrt{R^2+x^2+y^2}}\left[{\left(\frac{R}{\sqrt{R^2+x^2+y^2}}\right)}^{q_E+2}\frac{y^2}{x^2+y^2}\right.\right.\\ \left.\left.+{\left(\frac{R}{\sqrt{R^2+x^2+y^2}}\right)}^{q_H+1}\frac{x^2}{x^2+y^2}\right]\right\}\}\mathit{\text{dydx}},\end{array}$$E8

with

$$\begin{array}{l}\mathit{II}={\int }_{x=-X/2}^{X/2}{\int }_{y=-Y/2}^{Y/2}\left[{\left(\frac{R}{\sqrt{R^2+x^2+y^2}}\right)}^{2q_E}\frac{y^2}{x^2+y^2}\right.\\ \left.+{\left(\frac{R}{\sqrt{R^2+x^2+y^2}}\right)}^{2q_E}\frac{x^2}{x^2+y^2}\right]\frac{R}{{\left(\sqrt{R^2+x^2+y^2}\right)}^3}\mathit{\text{dydx}}.\end{array}$$E9

Under this circumstance, ${\eta }_s$ reads:

$${\eta }_s=\frac{\mathit{II}}{\mathit{III}},$$E10

with

$$\mathit{III}=\pi \left(\frac{1}{1+2q_E}+\frac{1}{1+2q_H}\right).$$E11

Now consider the rectangular aperture of Figure 6(a). Recall that the physical size of the aperture is $s=1.65$ m $\times$$1.25$ m at 3.5 GHz. With a symetrical radiation pattern at the feeder, $q=q_E=q_H$ and ${\eta }_{\mathit{ill}}$ and ${\eta }_s$ are obtained by (7) and (10) respectively, followed by calculation of ${\eta }_a={\eta }_{\mathit{ill}}\times {\eta }_s$. The result is shown in Figure 9(b) for different values of $q=10\sim 80$ and $R=0.5\sim 3$ m. This shows that it is possible to realize optimum values of $q$ and $R$ to use a specific feed horn and locate it at a specific distance from the surface to obtain the maximum possible ${\eta }_a$. This is a routine step of designing reflectarray antennas and metasurface reflectors. However, in the case of RIS where there is no control on $q$ and $R$, it is not possible to customize the structure to reach the optimum ${\eta }_a$.

Figure 9(c) shows a schema of applying the surface of Figure 6(a) for coverage provisioning purposes. Note that this surface has a very large size comparing to the operating wavelength which can potentially be a positive factor for ${\eta }_a$. We repeat the same calculation, but this time the distance range is expanded to $R=0.5\sim 50$ m. The result is shown in Figure 9(d). As it is clear, a massive region of this plot shows a very low ${\eta }_a$ which can make the structure impractical for real-world applications. This will be more challenging when we pay attention to two factors, 1st: in cellular networks, the cell radius is much longer than $50$ m, this means that the situation is even worse than Figure 9(d); 2nd: for the uplink connection, the user equipment (UE) will have a very low gain (or low $q$) which will make the connection very challenging if not impossible (see Figure 9(d) for low values of $q$ and high $R$). Finally, it should be noted that these are all theoretical calculations; when it comes to practice, the obtained ${\eta }_a$ is expected to be relatively lower than the theory. This is also true even for reflectarray antennas where the feeder is optimized.

4. Comparison between holographic, MIMO, and phased-array beamforming

Consider a case when printed patches are used in three different structures, i.e. a holographic-based metasurface, a phased-array antenna, and a MIMO system. The only common thing between these three is the repeating geometry of patches across the structure. The detailed functionality of these patches is as below:

• Holographic-based metasurface: in this case, the printed patches are used to sample an induced/launched surface wave on the structure. The size of these patches is in the sub-wavelength scope and varies. This variation in size is known as modulation which is governed based on the holography technique¹. The obtained structure will be an HLWA or an HR with respect to the location of the initial feeder. It is possible to form more than one beam by applying the superposition technique and tilting each of them to the direction of interest. Figure 10(a) schematically shows a multi-beam HR. With respect to Figure 1, this structure is a single aperture.

• Phased-array antenna: the size of patches is relatively bigger in the scopes of half-wavelength. Therefore, each patch is a resonating element. The element spacing is more than the case of HR metasurface, but should not be more than a wavelength or half a wavelength to stop forming unwanted grating lobes. Each element has typically a port where the corresponding amplitude and phase can be controlled to not only control the tilt angle but also to regulate the obtained SLL. A 4-element planar phased-array antenna with a tilted beam is schematically shown in Figure 10(b). Regarding Figure 1, these elements form a single aperture altogether.

• MIMO system: the size of patches is again in the scopes of half-wavelength to make a resonating element. However, the spacing between elements is managed to decrease the coupling between elements. When there is a port for each element, this low mutual coupling can decrease the cross-correlation between the ports which can be a good candidate to be used in MIMO systems. As each element is functionally separated from the others, it is common to be called an embedded element. In this case, each embedded element has its own radiation characteristics. A schema of a 4-element MIMO system is presented in Figure 10(c). This structure has four separate apertures considering Figure 1.

Figure 10.

(a) A multibeam metasurface reflector, (b) a tilted-beam phased-array antenna and (c) a 4-element MIMO system.

5. Conclusions

The holography technique and its application in forming the beam in electromagnetic structures are explained in this chapter. Leaky-wave antennas, metasurface reflectors, and reconfigurable intelligent surfaces (RISs) are assessed in this regard. The aperture efficiency of RIS is also studied which is one of the main barriers to industrializing this component in future cellular networks. A comparison is made between the functionality of the smallest building blocks in holographic-base metasurfaces, phased array antennas, and MIMO systems.

Conflict of interest

The authors declare no conflict of interest.