## 5 Passive Dual-Band Phase-Gradient Metasurfaces  
Metamaterials are artificially engineered materials created using sub-wavelength elements, known as meta-atoms or unit cells [80]. Metasurfaces are the planar alternative to metamaterials and are sometimes referred to as 2D electromagnetic surfaces [81]. Metasurfaces are used for various applications, such as polarization conversion, filtering, designing phase-shifting surfaces and phase-gradient metasurfaces [22; 24]. Passive PGMs are free-standing surfaces with a saw-tooth phase profile in a specific direction while maintaining a constant phase in the orthogonal direction. An array of distinct cells (periodically repeated in one of the directions, either \(x\) or \(y\), while aperiodic in the orthogonal direction) determines this phase gradient, with each cell designed to provide a predetermined transmission phase shift to a normally incident electromagnetic wave. In some applications, the amplitude of the near-electric field can also be manipulated using these cells. We can refer to the antenna array theory to understand the working principle behind the PGMs. As per the antenna array theory [82], if several discrete identical radiating elements are arranged in a straight line, the field radiated by such an array can be given by Equation (1) in its simplest form. \[E_{\text{array}}=E_{\text{single element at reference point}}\times\text{Array factor} \tag{1}\]  
where \(E_{\text{array}}\) represents the total electric field. The array factor depends on the number of elements (N), the geometrical properties of the elements, their spacing (d), their relative magnitudes and their relative phase. When all the array elements are identical, with equal magnitude and a progressive phase delay among the adjacent elements, the array is called a uniform linear array. For a uniform linear array, the array factor (AF) is given by Equation (2). \[AF(\theta)=\frac{\sin\Bigl{(}\frac{Nkd}{2}\sin\theta\Bigr{)}}{\sin\Bigl{(}\frac{ kd}{2}\sin\theta\Bigr{)}}, \tag{2}\]  
where \(d\) is the spacing between the elements, \(k\) is the free space wave-number, and \(N\) is the total number of elements. The maxima of the beam occurs at \(kdsin\theta=2m\pi\), where \(m=0,\pm 1,\pm 2,\pm 3,\ldots\). Secondary maxima, known as the grating lobes, occur in array antennas when the inter-element spacing is large relative to the operating wavelength. These lobes occur at an angle away from the main beam, and their angular position can be calculated using Equation (3). \[\theta_{g}=\sin^{-1}\Bigl{(}\frac{m\lambda}{d}\Bigr{)} \tag{3}\]  
The following condition must be satisfied to avoid the grating lobes in the visible region (\(\theta=\pm 90^{\circ}\)) given in Equation (4). \[\frac{d}{\lambda}\leq 1-\frac{1}{N} \tag{4}\]  
To direct the radiated beam away from the broadside direction, all the antenna elements are fed with a signal that has been phase-shifted. This is peformed so that the phase shift between adjacent antenna elements remains constant, denoted by \(\phi_{step}\). For example, if we want to tilt the radiated beam at an angle \(\theta_{o}\) away from the broadside direction, \(\phi_{step}\) between adjacent elements can be set as \(\phi_{step}=kd\sin\theta_{o}\). One method to achieve a constant phase shift among adjacent elements is directly exciting the feed with phase-shifted signals, often utilizing active phase shifters. Alternatively, the feed length (microstrip or strip-line length) can be adjusted so that each element receives a phase-shifted input. Similarly to antenna arrays, in the case of passive PGMs, if an array of cells on the surface is arranged such that the phase difference between adjacent cells remains constant, denoted by \(\phi_{step}\), the incident beam on the surface can be tilted to an angle \(\theta_{o}\) away from the broadside direction, depending on this phase gradient.