**1. Introduction**

Consider a case where an electromagnetic (EM) aperture with an arbitrary geometry is placed on the *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 ð Þ *β* and amplitude ð Þ *α* of the EM waves at different segments of the aperture. The beam tilt angle can typically be controlled by regulating *β* whereas *α* 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 *β* and *α* 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 *β* and *α* are governed at different positions of the aperture.

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

#### **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.*

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 *β* and *α* over the aperture but in approaches different from conventional phased arrays. One approach is to employ the holography technique [1] to govern *β* 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 leakywave 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 *θ<sup>m</sup>* along their length. This tilt angle can be controlled by regulating the phase constant *β* of the guided waves across the structure and formulated as [4]:

$$
\theta\_m \approx \sin^{-1}\left(\frac{\beta}{k\_0}\right),
\tag{1}
$$

where *k*<sup>0</sup> being the free-space wavenumber. Considering (1), *θ<sup>m</sup>* will have a real answer if and only if ∣*β*∣ < *k*0. This means that LWAs support fast waves on the guide.

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

*Holographic Beamforming DOI: http://dx.doi.org/10.5772/intechopen.112467*

Having a dielectric slab on the *xy* plane to design the aperture on, the first step is to define two field distributions on the structure known as reference wave *E*ref and object wave *E*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 ð Þ *x* ¼ 0, *y* ¼ 0 will generate a radially expanded field distribution on the slab for both TM and TE surface waves which can be formulated as below [5]:

$$E\_{\text{ref}} = Ae^{-j\beta\_r r},\tag{2}$$

where *<sup>r</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> <sup>p</sup> and *<sup>A</sup>* 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 *<sup>E</sup>*ref <sup>¼</sup> *Ae<sup>j</sup>βyy* . 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.

The next step is to define an object wave *E*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*obj. For example, for a beam desired to be formed toward ð Þ *θm*, *ϕ<sup>m</sup>* , the induced *E*obj is obtained by the mapping as below:

$$E\_{\rm obj} = B e^{jk\_0 \left\{ \sin(\theta\_m) \cos(\phi\_m) \mathbf{x} + \sin(\theta\_m) \sin(\phi\_m) \mathbf{y} \right\}\_{\mathbf{y}}},\tag{3}$$

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

The next step is to calculate the superposition of *E*<sup>s</sup> ¼ *E*ref � *E*obj as an interference pattern where ∠*E*<sup>s</sup> defines the desired EM hologram.

The aforementioned steps of calculating *E*ref, *E*obj, and *E*<sup>s</sup> 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*ref as presented in **Figure 3(a)** on the slab at a specific frequency. With a defined object wave toward ð Þ *θ<sup>m</sup>* ¼ *π=*4, *ϕ<sup>m</sup>* ¼ 2*π=*3 , the obtained *E*obj on the slab is shown in

#### **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.*

**Figure 3.**

*Recording process: (a) E*ref *with an ideal source at the center of the slab, (b) E*obj *in case of* ð Þ *θ<sup>m</sup>* ¼ *π=*4, *ϕ<sup>m</sup>* ¼ 2*π=*3 *, and (c) the obtained EM hologram.*

#### **Figure 4.**

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

**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)**.

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*ref as possible. Then, a quasi-periodic pattern of scatterers, with the geometry and lattice inspired by ∠*E*<sup>s</sup> 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 ð Þ *θm*, *ϕ<sup>m</sup>* 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 ð Þ *θ<sup>m</sup>* ¼ *π=*4, *ϕ<sup>m</sup>* ¼ 2*π=*3 .

#### **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 subwavelength 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*surf ¼ *Et=Ht* with *Et* and *Ht* representing the tangential electric and magnetic fields respectively. The obtained structure is then an artificial impedance surface, commonly referred to as a metasurface.

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 *xf* , *yf* , *zf* � � and the slab on the *xy* plane in a standard right-handed coordinate system, *E*ref is modified as

$$E\_{\rm ref} = A e^{-jk\_0 r},\tag{4}$$

#### **Figure 5.**

*(a) The applied unit cell, a schema of (b) Eref and (c) E<sup>κ</sup>*¼<sup>1</sup> *obj* <sup>þ</sup> *<sup>E</sup><sup>κ</sup>*¼<sup>2</sup> *obj , (d) Zsurf versus patch size variation and (e) the obtained Z x*ð Þ , *y on the surface [7].*

**Figure 5(b)** shows the obtained *Eref* when the feeder is placed at *xf* , *yf* , *zf* � � <sup>¼</sup> ð Þ 0,0,2*:*5m for a 1*:*65 m � 1*:*25 m large dielectric sheet at *f* ¼ 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 *Eobj* 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 *θκ*¼1, *ϕκ*¼<sup>1</sup> ð Þ¼ <sup>45</sup><sup>∘</sup> ð Þ , 0 and *θκ*¼2, *ϕκ*¼<sup>2</sup> ð Þ¼ �45<sup>∘</sup> ð Þ , 0 . The calculated *Eobj* <sup>¼</sup> *<sup>E</sup><sup>κ</sup>*¼<sup>1</sup> *obj* <sup>þ</sup> *<sup>E</sup><sup>κ</sup>*¼<sup>2</sup> *obj* is presented in **Figure 5(c)**.

In order to derive the EM hologram, it is now required to conduct a study on *Z*surf regarding the unit cell of the structure. This can be calculated by sweeping the phase delay (*ϕD*) across the unit cell with periodicity of *p* as follows [8]:

$$Z\_{\rm surf} = jZ\_0 \sqrt{\left(\frac{\phi\_D}{k\_0 p}\right)^2 - 1},\tag{5}$$

where *Z*<sup>0</sup> 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 Δ*Z* with the mean value *Z*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]:

$$Z(\mathbf{x}, \mathbf{y}) = j \left( Z\_{\text{mean}} + \frac{\Delta Z}{2m} \operatorname{Re} \left[ \left( \sum\_{\kappa=1}^{m} E\_{\text{obj}}^{\kappa} \right) E\_{\text{ref}}^{\ast} \right] \right), \tag{6}$$

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

It is shown that *Z*mean ¼ 428*:*16 *j*Ω and Δ*Z* ¼ 566*:*51 *j*Ω 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

**Figure 6.**

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

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.
