Contents I. Introduction
The propagation of electromagnetic waves and in particular that of the light has been in the scope of scientist for centuries. Lenses have an important place in manipulating such waves. Besides the conventional lenses (curve shaped lenses) that are very well established, flat GRIN lenses have also received great deal of attention. The latter are based on a variation of the refractive index along the radius of the lens. Metamaterials or artificially engineered materials are often used to realize such varying indices. Recently, an all-dielectric flat lens, based on GRIN approach, has been fabricated by means of 3D printing technology [1]. In order to obtain the gradient-index, this new technology enables one to build the flat lens from a single material by introducing holes in the material in a one-step process. By changing the holes/material volume ratio, the desired refractive index or relative permittivity is obtained. Thus, the effective parameters obtained determine the behavior of the lens. However, as for all manufacturing processes, 3D printing technology can only be implemented with some uncertainties in the design parameters, for instance the hole dimensions, thickness of the lens, etc. As we demonstrate in this work, these parameter variabilities may influence the output response of the system. Therefore, the parameters such as the focal length may be affected and the gain may be compromised. In this work, we propose the use of Polynomial Chaos Expansion (PCE) analysis technique [2] to assess the effect of such a design parameter variability on the performance of the lens. Although Monte Carlo (MC) simulations are often used for this purpose, such MC analysis can be very time consuming task, because it requires a large volume of data in order to yield results with high confidence.
