IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 67, NO. 9, SEPTEMBER 2019 5997 Efficient Evaluation of Gradient Transmit-Arrays Through an Equivalent Dispersive Dielectric Description Parinaz Naseri , Student Member, IEEE, Sérgio A. Matos , Member, IEEE, Eduardo B. Lima , Student Member, IEEE, Jorge R. Costa , Senior Member, IEEE, Carlos A. Fernandes , Senior Member, IEEE, and Nelson J. G. Fonseca , Senior Member, IEEE Abstract—The growing popularity of transmit-arrays (TAs) for various antenna applications is calling for effective analysis and optimization methods. TAs are, usually, electrically large, comprising thousands of unit-cells formed by subwavelength metallic scatterers. Full-wave optimization cycles needed to meet stringent specifications in terms of gain, cross-polarization, band- width, scan-loss, etc., may be impaired by unrealistically required computational time and memory resources. To overcome this, we propose a modified homogenization method that, unlike other approaches, captures the internal reflections in the unit-cells and its resonances for each polarization, thus, correctly describing unit-cells’ frequency response in the band of interest. We define equivalent dispersive anisotropic media for gradient TAs. These surrogate models enable fast analysis and optimization of TAs without compromising the accuracy. As an example, we analyze a TA composed of phase rotation (PR) unit-cells. PR unit-cells present wideband low axial ratio for a TA but challenge the validation of existing homogenization methods. Detailed general description of the method is provided so that it can be applied to other unit-cells and avoid training time and resources required for machine learning-based methods. Using the surrogate cells, the full-wave analysis time and memory of the TA reduces 13 and 4 times, respectively. Index Terms— Anisotropy, dispersive material, effective media, flat-lens, gradient array, transmit-arrays (TAs). Manuscript received March 3, 2019; accepted May 9, 2019. Date of publication May 20, 2019; date of current version September 4, 2019. This work was supported in part by the Fundação para a Ciência e a Technologia under Grant UID/EEA/50008/2019 and Grant PTDC/EEI-TEL/30323/2017. (Corresponding author: Parinaz Naseri.) P. Naseri is with The Edward S. Rogers Sr. Department of Electrical & Computer Engineering, University of Toronto, Toronto, ON M4Y 2H9, Canada (e-mail: parinaz.naseri@utoronto.ca). S. A. Matos and J. R. Costa are with the Instituto de Telecomuni- cações, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal, and also with the Instituto Universitário de Lisboa (ISCTE-IUL), Departamento de Ciências e Tecnologias da Informação, 1649-026 Lisbon, Portugal. E. B. Lima is with the Department of Wireless Communications, Instituto de Telecomunicações, 1049-001 Lisbon, Portugal. C. A. Fernandes is with the Instituto de Telecomunicações, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal. N. J. G. Fonseca is with the Department of Antenna and Sub-Millimeter Wave Section, European Space Agency, 2200 Noordwijk, The Netherlands. Color versions of one or more of the figures in this article are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2019.2916761 I. I NTRODUCTION P LANAR arrays such as transmit-arrays (TAs) and reflect- arrays (RAs) have been an attractive solution to real- ize low-cost, low-profile, high-gain steerable pencil beam in many point-to-point and satellite communication applica- tions [1], [2]. These arrays comprise a collection of phase shifting unit-cells, which can provide beam collimation for the transmitted (TA) or reflected (RA) radiation that is coming from a primary feed. It is worth mentioning that a TA, which acts as a lens, does not suffer from feed blockage, as occurs in RA antennas. The most common unit-cell configuration is based on a stack of different metallic layers embedded in a dielectric host medium, resulting in thin planar arrays with high radiation efficiencies [2]–[10]. The phase shifts of the unit-cells are tuned by the careful design of sub-wavelength metallic fea- tures in each layer. Performing a full-wave analysis of these electrically large antennas, properly accounting the unit-cell fine sub-wavelength details is computationally challenging. Even with proper full-wave methods [9], the required hardware resources and simulation times can limit the design and iterative optimization of these antennas. Various approaches have been proposed and employed [13]–[17], to tackle this problem. The generalized sheet transition condition-based method [11] is a popular approach to analyze metasurfaces, including TAs and RAs, in terms of their electric and magnetic susceptibilities. The method accurately models a metamaterial unit-cell over a wide frequency range with the constraint that w, the period of the unit-cells should be less than half of the wavelength and l , its scatterer size, should not exceed 0.7 w [12]. However, after the surface is described with electric and magnetic susceptibilities, a numerical method is required to analyze the surface composed of the scatterers. Moreover, there is no report on how to employ general sheet transition condition (GSTC) for unit-cells composed of closely spaced layers of scatterers when interlayer coupling might exist. Physical optics/geometrical optics (PO/GO)-based methods are often used to obtain a fast (in the order of minutes) initial estimation of the TA’s performance. For example, in [13], the fields at the aperture are computed by multiplying the 0018-926X © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.