## 1. Introduction

[2] Reflector antennas normally involve struts that hold the microwave head. Such struts may cause significant degradation of the antenna's performance [*Kildal et al.*, 1988] in terms of losses, sidelobes and cross polarization. In turn, such effects may also increase the noise temperature of the reflector antenna [*Moreira et al.*, 1996].

[3] The design of the strut's cross section can be crucial to fulfill the overall requirements on the performance of reflector antennas that are used in modern communication systems and radio telescopes. In particular, it is advantageous to find designs that yield a low total scattering cross section [*Kildal et al.*, 1988]. For the transverse electric (TE) case, the magnetic field is parallel to the cylinder axis of the strut and satisfies the homogeneous Neumann boundary condition at the surface of a perfect electric conductor (PEC). For this case, it is feasible to achieve a relatively small total scattering cross section by making the strut's cross section oblong in the direction of propagation of the incoming wave [*Rusch et al.*, 1976; *Kildal et al.*, 1988; *Moreira et al.*, 1996]. For the transverse magnetic (TM) case, the electric field is parallel to the cylinder axis and it satisfies the homogeneous Dirichlet boundary condition at the surface of a PEC. This type of boundary condition at the surface of a strut yields substantial scattering and changing the shape of the metal strut's cross section is not a competitive way to reduce the total scattering cross section. However, it is feasible to construct surfaces that yield, at least in a narrow frequency band, an artificial magnetic conductor [*Kildal*, 1990] and this effectively gives a homogeneous Neumann boundary condition for the electric field in the TM case. Equipped with such an artificial magnetic conductor surface for the strut in the TM case, it is feasible to reuse the oblong shapes that yield a low total scattering cross section for the TE case. For example, the total scattering cross section can be reduced by coating the metal strut with a dielectric material [*Lin and Yaqoob*, 1992; *Kildal et al.*, 1996], which effectively yields an artificial magnetic conductor for a narrow frequency band [*Kildal*, 1990]. To reduce the total scattering cross section for the TE and TM case simultaneously, metal strips that circumvent the strut can be added to a metal cylinder coated with a dielectric. This creates an anisotropic boundary condition that allows for reduction of the total scattering cross section for both the TE and TM case simultaneously [*Kildal et al.*, 1996; *Kishk*, 2003, 2004; *Wang*, 1985].

[4] Shape optimization is an important tool for design in engineering. In particular, efficient methods can be formulated if the sensitivities are expressed in terms of the field solution to the original field problem combined with the solution to an adjoint field problem [*Pironneau*, 1984]. Thus, only two field computations yield the sensitivity with respect to an arbitrary number of design variables. For Maxwell's equations, shape optimization has been applied to several classes of problems: (1) magnetostatics [*Park et al.*, 1991], (2) steady state eddy current problems [*Park et al.*, 1994], (3) passive microwave circuits [*Lee and Itoh*, 1997; *Nair and Webb*, 2003], (4) monostatic radar cross section [*Bondeson et al.*, 2004], and (5) optical resonators [*Kao and Santosa*, 2008]. In other areas, such as structural design, shape and topology optimization [*Neves et al.*, 2002] are widely used tools and a popular way of describing the geometry is to use a level-set method [*Allaire et al.*, 2004a, 2002; *Osher and Fedkiw*, 2001]. Optimization using gradient-based methods that exploit an adjoint problem [*Dorn et al.*, 2000; *Burger*, 2001] or heuristic methods is also an important tool in shape reconstruction in inverse problems. The list given here is not complete and the interested reader is referred to books and review articles on this topic [e.g., *Haslinger and Mäkinen*, 2003; *Allaire and Henrot*, 2001; *Sokolowski and Zolesio*, 1992].

[5] In this article, we propose and test a gradient-based optimization algorithm for the total scattering cross section of infinitely long cylinders. It changes the shape that describes the cross section of the cylinder iteratively and, for the tests presented in this article, a significant reduction of the goal function is attained in approximately 10–20 iterations. We present a continuum sensitivity analysis for the total scattering cross section with respect to the shape of the cross section for both dielectric and PEC cylinders, which forms the basis of our optimization algorithm. With the optical theorem as a basis, the detailed derivations are presented for the TM case and the final results for both polarizations (see the work by *Bondeson et al.* [2004], where the related but different problem for the monostatic radar cross section is studied). The continuum formulation of the sensitivity expressions decouples the optimization procedure from the field solver and, here, we solve Maxwell's equations with a stable hybridization [*Rylander and Bondeson*, 2002] of the finite element method (FEM) [*Jin*, 2002] and the finite difference time domain (FDTD) scheme [*Taflove*, 2005]. It is formulated in the time domain and one computation solves the scattering problem for more than four octaves in frequency. We test the optimization algorithm on metal struts for the TE case and find different types of solutions: (1) conventional oblong shapes; and (2) oblong shapes with parts covered by corrugations that are parallel to the cylinder axis.

[6] This article features a couple of main contributions: (1) the sensitivity expressions for the total scattering cross section with respect to changes in the shape of the cylinder for both penetrable and impenetrable scatterers, where the sensitivities are given for the TE and TM case separately; (2) optimized shapes that feature conventional oblong cross sections; and (3) struts that have corrugations parallel to the cylinder axis which lower the total scattering cross section for higher frequencies.