# Anisotropic MM Lenses for Directive Radiation

In addition to broadening the impedance bandwidth of an omnidirectionally radiating wire antenna, MMs with certain anisotropy can also be utilized to change the radiation pattern of an antenna over a wide bandwidth. More specifically, the MMs are designed to change the angular distribution of electromagnetic energy radiated from an antenna to generate highly directive beams. Several examples are presented in this section, including AZIM lenses with engineered electric and/or magnetic properties for generating a single as well as multiple highly directive beams from a simple antenna feed.

## Low-Profile AZIM Coating for Slot Antenna

### Dispersion of grounded AZIM slab

First, we exploit the leaky modes supported by a grounded AZIM slab to produce a single unidirectional beam at broadside over a wide bandwidth. Let us consider a grounded slab that is infinite in the у-direction, as shown in Fig. 1.6a. The slab has a thickness of *t *and is composed of an anisotropic medium with its permittivity and permeability tensors expressed as [45]

The у-component of the complex propagation constant (к_{у}) of the leaky wave is studied under the following conditions. A time- harmonic dependence *e ^{jat}* is assumed and used throughout the paper. Since no field variation is assumed in the x-direction, the two-dimensional nature of the configuration allows the problem to be separated into

*TE*and

_{z}*TM*modes. For the

_{z}*TE*mode, the set of material tensor parameters

_{z}*(e*and

_{rx}, (x_{ry},*n*are pertinent, while for the

_{rz})*TM*mode, the other set of material tensor parameters (ц

_{z}*, ?*

_{rx}_{ry}, and ?

_{rz}) are responsible for the wave propagation properties. For simplicity, only the

*TM*mode is considered here since a low-profile slot antenna will be employed, which represents a quasi-

_{z}*TM*source. The dispersion equation can be obtained using the transverse equivalent network model, as shown in Fig. 1.6b [46], which is given by

_{z}**Figure 1.6 **(a) Configuration of the grounded anisotropic slab with defined

geometrical dimensions and material properties. (b) Equivalent transverse network for both TE and TM modes of the grounded anisotropic slab.

The propagation constant *k™ = fi™ + ja™* of the leaky modes supported by the grounded anisotropic slab can be calculated numerically using Eq. (1.3). Figures 1.7a,b show the real and imaginary parts of *k™* (normalized to k_{0}), as a function of frequency for different *? _{rz}* values. The values of

*E*and

_{ry}*л*are fixed to be 2.4 and 1, respectively, while

_{rx}*f*denotes the frequency at which the thickness of the slab (t) is equal to one wavelength. The ordinary isotropic case is also considered where

_{0}*?*=

_{rz}*E*= 2.4. Multiple solutions can be found, including the physical surface wave (SW), leaky wave, and improper nonphysical SW (IN), all of which are labeled in the figures. For the isotropic grounded slab, the leaky-wave band is narrow and highly dispersive. The value of is only slightly smaller than that of

_{ry}*k*When the value of

_{0}.*?*decreases, the upper band improper SW mode disappears and the bandwidth of leaky-wave region is greatly extended. As the value of

_{rz}*?*is further decreased, the grounded anisotropic slab can be referred to as a grounded anisotropic low- index slab, where both

_{rz}*ft™*and

*a™*of the leaky modes become smaller and their profiles also get flatter. This is beneficial for maintaining a stable radiated beam near broadside across a wide frequency range. Theoretically, when

*?*is extremely close to zero,

_{rz}*k™*is forced to be zero in order to satisfy the dispersion relation in Eq. (1.4), which possesses an elliptical isofrequency curve with a near-vanishing short axis [47].

**Figure 1.7 ***TM _{z}* dispersion curves for a grounded anisotropic slab with

*s*2.4,

_{ry}=*n*= 1, and varying

_{rx}*?*as a function of frequency. (a)

_{rz}*fi™/k*(b)

_{0}.*a™/k*Reprinted, with permission, from Ref. 48, Copyright 2014, IEEE.

_{0}.