IEEE JOURNAL ON MULTISCALE AND MULTIPHYSICS COMPUTATIONAL TECHNIQUES, VOL. 2, 2017 159 Transformation-Optics-Based Design of a Metamaterial Radome for Extending the Scanning Angle of a Phased-Array Antenna Massimo Moccia, Giuseppe Castaldi, Giuliana D’Alterio, Maurizio Feo, Roberto Vitiello, and Vincenzo Galdi, Fellow, IEEE Abstract—We apply the transformation-optics approach to the the initial focus on EM, interest has rapidly spread to other design of a metamaterial radome that can extend the scanning an- disciplines [4], and multiphysics applications are becoming in- gle of a phased-array antenna. For moderate enhancement of the creasingly relevant [5]–[7]. scanning angle, via suitable parameterization and optimization of the coordinate transformation, we obtain a design that admits a Metamaterial synthesis has several traits in common technologically viable, robust, and potentially broadband imple- with inverse-scattering problems [8], and likewise poses mentation in terms of thin-metallic-plate inclusions. Our results, some formidable computational challenges. Within the validated via finite-element-based numerical simulations, indicate emerging framework of “metamaterial-by-design” [9], the an alternative route to the design of metamaterial radomes that “transformation-optics” (TO) approach [10], [11] stands out does not require negative-valued and/or extreme constitutive pa- rameters. as a systematic strategy to analytically derive the idealized material “blueprints” necessary to implement a desired field- Index Terms—Metamaterials, phased-array antennas, radomes, manipulation interpreted as a local distortion of the coordinate transformation optics (TO). reference frame. Several extensions have also been proposed in order to accommodate effects (e.g., nonlinear, nonreciprocal, I. INTRODUCTION bianisotropic, magnetoelectric, artificial moving, space time, nonlocal, non-Hermitian, topological) and observables (e.g., HE past two decades have witnessed an exponentially resonances, optical forces, density of states) not encompassed growing interest in electromagnetic (EM) “metamateri- T by the original formulation (see, e.g., [12]–[26]). Moreover, a als” [1]–[3]. These are artificial materials, typically consisting variety of mechanisms can be exploited in order to reconfigure of electrically small inclusions in a host medium, engineered the metamaterial response [27]. The reader is also referred to so as to attain unconventional EM responses, not necessarily [4], [28], and [29] (and references therein) for recent reviews of limited by the material properties available in nature. EM applications as well as extensions to other disciplines. From the computational viewpoint, the analysis and design In this paper, we apply the TO approach to the design of a of metamaterials represent quintessential multiscale problems, metamaterial radome capable of extending the scanning angle of characterized by several characteristic sizes spanning orders of a phased-array antenna. It is well known that, in typical phased magnitude: from the electrically large size of many operational arrays, the scanning angle is practically limited to ∼±60o from scenarios of practical interest, through the wavelength-sized spa- the broadside direction [30]. In this context, a suitably designed tial variations of the effective constitutive parameters required metamaterial radome appears as an attractive alternative to typi- in typical designs, up to the deeply subwavelength sizes of in- cal mechanical-augmentation systems, in terms of size, weight, clusions utilized for practical implementations. Moreover, after and complexity. A first metamaterial-based radome for extend- ing the scanning angle was proposed and successfully realized Manuscript received March 3, 2017; revised May 2, 2017; accepted May 27, by Lam et al. [31], [32]. Such design is heavily based on brute- 2017. Date of publication June 8, 2017; date of current version October 25, 2017. This work was supported in part by the Italian Ministry of Education, force numerical optimization, which results in a nonlinear (and University and Research (MIUR) through the Campania Aerospace District difficult to control) relationship between the input and output within the framework of the TELEMACO project (PON03PE-00112-1) “En- angles. Moreover, it relies on negative-index media [33], whose abling Technologies and Innovative Electronic Scanning Systems in Millimeter and Centimeter Bands for Avionic Radar Applications.” (Corresponding author: metamaterial implementations [34] are known to be highly dis- Vincenzo Galdi.) persive and prone to losses; this inherently curtails bandwidth M. Moccia, G. Castaldi, and V. Galdi are with the Fields and Waves and efficiency. In [35], Sun et al. proposed a different, TO-based Laboratory, Department of Engineering, University of Sannio, I-82100 Ben- evento, Italy (e-mail: [email protected]; [email protected]; approach to design 2-D arbitrarily shaped metamaterial radomes [email protected]). yielding a desired (e.g., linear) relationship between the input G. D’Alterio, M Feo, and R. Vitiello are with MBDA Italia s.p.a., and output angles. To overcome the significant complexity of the I-80070 Bacoli, Italy (e-mail: [email protected]; maurizio.feo@ mbda.it; [email protected]). resulting (anisotropic, inhomogeneous) transformation medium Digital Object Identifier 10.1109/JMMCT.2017.2713826 to be implemented, the same authors subsequently put forward 2379-8793 © 2017 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. 160 IEEE JOURNAL ON MULTISCALE AND MULTIPHYSICS COMPUTATIONAL TECHNIQUES, VOL. 2, 2017 confined to a finite region nearby the aperture. Even though arbitrary shapes are possible, for simplicity, we assume such region to be the annular domain R1 <r<R2 , −π/2 <θ< π/2. The aforementioned mapping belongs to the general class of “finite embedded coordinate transformations” introduced in [39]. As conceptually illustrated in Fig. 1(b), it locally distorts the energy flow so that the impinging beam [cf., Fig. 1(a)] is gradually steered and exits the annular region with a different direction θo = αθi (2) with α>1 denoting a desired enhancement factor. Fig. 1. Schematic of our approach. (a) In an auxiliary vacuum space, the beam Following the TO approach [11], and invoking the form- radiated by a phased array (indicated by a thick red arrow) is directed along invariance properties of Maxwell’s equations, the aforemen- an angle θi . (b) In the transformed space, a coordinate mapping embedded in R <r<R −π/2 <θ<π/2 tioned effects can be equivalently obtained by filling the annular the annular region 1 2 , (delimited by the blue- R <r<R −π/2 <θ<π/2 dashed contours) locally distorts the metric so that the same impinging beam is domain 1 2 , with an anisotropic, steered toward a different angle θo >θi . inhomogeneous medium characterized by relative permittivity and permeability tensors a modified approach [36], whose implementation relies on the ε (r)=μ (r) = det Λ (r) Λ−1 (r) · Λ−T (r) previously introduced “optical null medium” [37]. Though sim- (3) plified, this implementation still requires material constituents −1 −T where det, and denote the determinant, inverse, and in- with extreme parameters: one with very high permittivity and verse transpose, respectively, whereas Λ ≡ ∂F/∂r is the Ja- permeability, and the other with near-zero permittivity and per- cobian matrix associated with the coordinate transformation in meability. (1). From the aforementioned discussion, it appears that a prac- In what follows, we address the judicious choice of the coor- tical, broadband implementation is not at hand, and there dinate transformation, as well as the practical implementation is still considerable room for research. Accordingly, we of the TO-based constitutive “blueprints” in (3). propose here an alternative TO-based approach that yields constitutive parameters without extreme values (with possibly III. METAMATERIAL RADOME DESIGN nonmagnetic character) and with moderate anisotropy, which admit a potentially broadband implementation in terms of in- A. Coordinate Transformation clusions made of thin metallic plates [38]. More specifically, in To implement the manipulation schematized in Fig. 1, we Section II, we describe the problem geometry and outline its con- choose a 2-D coordinate transformation ceptual formulation. In Section III, we detail the metamaterial radome design, from the analytically derived, TO-based consti- r = Fr (r) (4a) tutive blueprints to the actual implementation. In Section IV, θ = Fθ (r) θ (4b) we illustrate some representative results, and validate our ap- proach via rigorous full-wave numerical simulations. Finally, in z = z (4c) Section V, we provide some closing remarks and perspectives. with the following boundary conditions: II. PROBLEM GEOMETRY AND FORMULATION Fr (R1 )=R1 (5a) The geometry and general idea underlying our approach Fr (R2 )=R2 (5b) are schematized in Fig. 1. Throughout this paper, we assume F (R )=1 an implicit exp (−iωt) time-harmonic excitation, and a 2-D θ 1 (5c) scenario, with all fields and quantities independent of z, and 1 F (R )= . transverse-magnetic (TM) polarization (i.e., z-directed mag- θ 2 α (5d) netic field). We begin considering an auxiliary vacuum space In particular, (5a) and (5c) ensure that the transformation re- with coordinates r ≡ (x,y,z) [and associated