## 1. Introduction

[2] Conformal antennas are desirable when integration of the antenna with a platform is required, e.g. antennas integrated in the skin of an aircraft reduce the aerodynamic drag and lower the fuel consumption. Array antennas can conform to a given platform and they offer the possibility of electronic beam steering and pattern synthesis. Important examples can be found in mobile communications where the base station antenna is required to have 360° coverage, which can be achieved by an array antenna that conforms to a circular cylinder. (Frequency-invariant pattern synthesis is also feasible for circular array antennas [*Steyskal*, 1989].)

[3] The design and usage of conformal array antennas involve several aspects and, here, we mention (1) the design of the array antenna's geometry and (2) the selection of excitation for the individual antenna elements. Given a supporting structure for the array antenna, the number of antenna elements must be chosen together with their positions and orientations, see *Josefsson and Persson* [2006] for a number of studies and further references to the literature. In practice, this process is subject to a number of additional constraints such as the feeding of the antenna elements and the mechanical strength of the supporting structure. Another aspect of the geometrical design relates to the actual shape (and materials) used for the individual antenna elements. For planar arrays, it is common to design the antenna element when used in an infinite periodic array [*Chio and Schaubert*, 2000], where the design must fulfill requirements on the active reflection coefficient and beam steering capabilities for a given frequency interval. Planar array antennas for operation over broad frequency-bands can be constructed by Vivaldi antenna elements [*Chio and Schaubert*, 2000] and similar solutions [*Holter*, 2007].

[4] In this article, we conduct a study of array antennas conformal to circular cylinders. The antenna elements are horn shaped and excited by the fundamental mode of a feeding parallel plate waveguide. We solve the electromagnetic field problem to high accuracy with a stable hybrid [*Rylander and Bondeson*, 2002] between the finite-difference time-domain (FDTD) scheme [*Taflove and Hagness*, 2005] and the finite element method (FEM) [*Jin*, 2002]. Given one time-domain computation, we compute the embedded element pattern and the full scattering matrix for a large frequency-interval. We perform an extensive parameter study with respect to the shape of the horn antenna elements and, for phase-mode excitations, we characterize the results in terms of (1) the active reflection coefficient and (2) the distortion of the far-field with respect to the excited phase-mode. This allows us to choose a design of the antenna element that is optimal in the Pareto sense [*Tamaki et al.*, 1996] (provided the sampled parameter space that we use) for a given frequency band and a given range of phase modes. Given a good design of the antenna element, we perform pattern synthesis by gradient based optimization of the phase-mode amplitudes such that a linear combination of two goals is minimized: (1) the directive gain outside an upper and lower mask; and (2) the power reflected at the array antenna aperture. This approach allows us to only use the phase-mode amplitudes that radiate well (given the frequency at hand) as design variables in the pattern synthesis. Alternatively, all phase-mode amplitudes can be included together with an additional term in the goal function that penalizes the reflected power, which typically stems from the upper range of phase modes.

[5] This article contains three substantial contributions: (1) an attempt to co-optimize the geometry and excitation to achieve a desired radiation pattern and a good antenna efficiency; (2) gradient based pattern synthesis that incorporates requirements on the efficiency of the array antenna; and (3) a design process for conformal array antennas that is motivated by means of phase-mode analysis for uniform array antennas conformal to circular cylinders. In addition, we consider a range of questions for the specific case of two-dimensional array antennas that conform to a circular cylinder: (1) extensive parameter studies by means of accurate and unbiased computational models for the mutual coupling and the embedded element pattern; (2) Pareto optimal geometries and excitations; and (3) comparisons with conventional pattern synthesis techniques such as the Dolph-Chebyshev synthesis [*Dolph*, 1946; *Lau and Leung*, 2000; *Vescovo*, 1999].