Electromagnetic (EM) metasurfaces are quasi two-dimensional structures that consist of artificial “atoms”, typically referred to as meta-atoms. These meta-atoms, also known as unit cells, are small compared to the operational wavelength. The whole metasurface consists of many unit cells; thus, although the size of the unit cells is small compared to the wavelength of operation, the total size of the metasurface is a few wavelengths. Based on their properties, the unit cells can locally affect the amplitude, phase, and polarization of the EM wave that impinges on them. When properly designed and arranged, the accumulated effect of the unit cells allows the metasurface to alter the incident wave at will through transmission, reflection and absorption of the EM wave. Thus, EM metasurfaces can be used as a systematic means for beam shaping.

The design of a metasurface includes two main steps: macroscopic and microscopic design steps. The macroscopic design aims to determine the required surface properties of the unit cells, such as surface polarizabilities or surface susceptibilities, to support the desired field transformation. Once the macroscopic properties are determined, the microscopic design focuses on the practical design of the unit cells to physically realize the required surface properties.

This thesis focuses on the macroscopic design of metasurfaces. Based on the generalized sheet transition conditions, the tangential fields on the two sides of the metasurface are required for macroscopic design. In practice, these tangential fields, especially the transmitted fields, might not be directly known. To handle this, an inversion algorithm based on the nonlinear conjugate-gradient (CG) method was previously developed to infer the tangential fields. This thesis builds on this prior work and extends it in the following ways. Firstly, it provides an analytical expression for the nonlinear CG step length. Secondly, this inversion framework is extended to the 3D configuration, allowing the user to provide the desired power patterns on two perpendicular far-field cuts. Thirdly, a brief discussion on the existence of the surface waves in the formulation is presented which can assist with satisfying the local power conservation constraint in the design of passive and lossless metasurfaces.