# Electromagnetic Specifications and Prototype Designs of Software Defined Surfaces

A metasurface is an electrically thin composite material layer, designed and optimized to function as a tool to control and transform electromagnetic waves [55,63,78,167,192]. The layer thickness is small and can be considered as negligible with respect to the wavelength in the surrounding space. Recently, the emerging topic of tunable metasurfaces for wave control [38, 40, 45, 68, 69, 82,106,119,177] has been actively advanced thanks to new possibilities for versatile and powerful control over the propagation of electromagnetic (EM) waves. Among the many available tuning mechanisms, the lumped-element enabled tunable metasurface provides the most powerful functionalities. However, the existing solutions are only using diodes or varactors, which tune only the effective surface impedance in a limited way, such as discrete values or a line in the complex impedance plane. A more general solution is to control both the real and imaginary parts of the surface impedance, i.e., tuning both the *R* and *C* parts of tuning components.

This book discusses the design, fabrication, implementation, and performances of a tunable metasurface that can perfectly absorb the incoming electromagnetic wave from different incidence angles, realize anomalous reflection, and control polarization of reflected waves. In this chapter, we present the electromagnetic design of the unit cell for the tunable metasurface. The content of this chapter is as follows. Sections 3.1 to 3.3 present basic information on metasurfaces and tunable metasurfaces. We discuss the effective-parameter modeling of metasurfaces and their electromagnetic properties in Section 3.1, compare metasurfaces with other sheet materials in Section 3.2, and review the tuning mechanisms in Section 3.3. Then, in Sections 3.4 and 3.5, we discuss the design flow of a switch-fabric prototype and its electromagnetic performance. In Section 3.6, we present designs of graphene-based prototypes. And, finally, we summarize the chapter in Section 3.7.

## ELECTROMAGNETIC MODELING OF METASURFACES

In this section we present electromagnetic models of generic metasurfaces and basic information about their electromagnetic properties.

### Unit Cell, Polarizability, and Interaction Constant

We consider probably the simplest metasurface: just one layer of small electrically polarizable unit cells (elements), as illustrated in Fig. 3.1. There is no ground plane, no substrate, and the array elements are in free space. The array is periodical and infinite, and the period is smaller than half wavelength. Moreover, we assume that the unit cells are electrically small and each unit cell can be considered as an electric dipole. We denote the dipole moment of one unit cell as p. Let us also assume that the metasurface is planar and the unit cells are planar (metal) patches in the plane of the metasurfaces. In this case all dipole moments of all unit cells are oriented in the plane of the metasurface.

The dipole moment of a unit cell is proportional to the electric field at the point where this particular unit cell is located, measured in the absence of this unit-cell patch. The electric field which excites each unit cell is called local field Ei_{oc}. Assuming linearity, for each unit cell of the metasurface we can write

Figure 3.1 Upon surface averaging of currents in unit cells, the metasurface can be considered as a sheet of electric and magnetic surface currents. In the simple example presented in this picture, the unit cells are only electrically polarizable, and the induced magnetic current J_{m} = 0.

where parameter *a* is called polarizability. The polarizability of one unit cell (one dipole particle) can be computed or measured if we take one single unit cell out of the array and study it isolated (in free space). The local field at the position of a particular unit cell is the sum of the incident field and the fields scattered by all other elements in the array. This model is illustrated in Fig. 3.2.

Let us consider excitation of metasurfaces bv external fields. For example, we illuminate the array by a plane wave, whose field we call the incident field E_{iuc}. If we would know how dipole moments of unit cells depend on the incident field, that is, if we would know the coefficient d (called collective polarizability) in the relation

we would be able to find the reflection and transmission coefficients in terms of d and then control them varying d. The concept of the individual and collective polarizabilities is illustrated in Fig. 3.3.

Let us simplify our problem even further and assume that the excitation is a normally incident plane wave. In this case all the unit cells feel the same exciting field, and all the dipole moments are the same. Now, assuming that we know d, we can find the induced electric surface current density J. Indeed, the surface density of electric polarization is simply the dipole moment per unit area, and we find it by dividing the dipole moment of one unit cell p by the unit-cell area *S.* Next, we note that the time derivative of the polarization density vector is the surface current density, which gives

We note that this relation has different physical meaning than the differential Ohm’s law, because Ohm’s law relates the current density and

Figure 3.2 Periodically arranged unit cells (a) and their model as interacting electric dipoles (b).

the *total* (not the *incident*) electric field. A two-dimensional analogue of Ohm’s law is derived in Section 3.1.2.

This electric current sheet radiates plane waves in both forward and back directions, and the amplitude of the electric field created by this electric current sheet is

where 77 is the free-space impedance. This result is obtained by equating the circulation of magnetic field around a current source to the current, which gives 2*H* = J, and the relation between electric and magnetic fields in a plane wave, *E* = *r)H.* Then, the reflection coefficient *R* of the metasurface reads

However, we do not know the collective polarizability d, because the unit cells are excited by the local field, which is not equal to the incident field! The local field is the sum of the incident field and the field created by the currents induced in all the other unit cells, which is called the **interaction field Ej _{nt}, **as illustrated in Fig. 3.3(a).

In general, we can write Ecp (3.1) as

If we know the incident field and the properties of one unit cell, we still need to find the interaction field, in order to find out what will be the metasurface response to this excitation. In the assumption of

Figure 3.3 (a) The local field is the sum of the incident field and the interaction field, (b) The unit-cell dipole moment can be expressed in terms of the individual polarizability *a* or the collective polarizability *a.* (c) The plane-wave reflection and transmission properties are defined by the surface-averaged electric current density.

excitation by a normally incident plane wave, all the dipoles are the same, and the interaction field is proportional to the dipole moment (of each unit cell) p:

Here, parameter *,3* is called the interaction constant. It depends on the array period, the frequency, the properties of the surrounding space (the presence of a substrate, for instance), but it does not depend on the unit cell polarizability a.

If we know the interaction constant /3, we can find the induced dipole moment in terms of the incident field. Combining Eqs. (3.6) and (3.7), we find

Now the problem is solved, since we know the reflection coefficient in terms of the unit-cell properties, and we can control it by tuning the polarizablity of unit cells.

We note that this approach works only for plane-wave excitations of infinite periodical (uniform) arrays. For normal incidence on an infinite array all the dipoles are the same, and we can write (3.7). For oblique incidence of plane waves all the dipoles have the same amplitude, and we know the phase shift from one unit cell to the next one. We can still use Eq. (3.7), where p is the dipole moment of one particular unit cell, which is at the origin of the coordinate system. In this case, calculations become more difficult, but still possible. However, as soon as the excitation is not a single plane wave or the array has a finite size (or it is not periodical), we need to solve the problem globally, because all the dipoles are in general different and we just have a linear system of equations containing all unknown dipole moments of each unit cell, which all interact with each other. In fact, if the array size is finite but large (compared to the period), an infinite-array approximation works well, because only 2-3 edge unit cells will “feel” the array edge.