## 1. Introduction

[2] The first rigorous studies of the diffraction by a perfect electric conducting (PEC) half plane immersed in a homogeneous isotropic medium are by *Poincaré* [1892] and *Sommerfeld* [1896]. Noticeable progress in the solution of these problems is attributed to the introduction and use of the Sommerfeld-Malyuzhinets (SM) technique and the Wiener-Hopf (WH) technique [*Senior*, 1978; *Hurd and Luneburg*, 1985; *Budaev*, 1995; *Senior and Volakis*, 1995; *Lüneburg and Serbest*, 2000; *Daniele*, 2003; *Antipov and Silvestrov*, 2006; *Lyalinov and Zhu*, 2006; *Daniele and Lombardi*, 2006]. In particular, we observe that the WH technique can deal with the most general problem of a half plane immersed in an anisotropic or bianisotropic medium, whereas the SM method cannot deal with it, as yet.

[3] The purpose of this paper is to establish a general WH theory to solve the electromagnetic problem of an imperfect half plane immersed in an arbitrary linear medium. Our theory yields to the factorization of matrices of order 2 and 4 in case of perfect and imperfect half plane, respectively. The solution of this kind of problems is known in closed form only for PEC or perfectly magnetic conducting (PMC) half planes surrounded by rather simple media; the perfectly conducting case is simpler because it requires the factorization of scalar kernels [*Seshadri and Rajagopal*, 1963; *Jull*, 1964; *Przezdziecki*, 2000] or of kernel matrices of order 2 [*Hurd and Przezdziecki*, 1981, 1985].

[4] To deal with the most general case, it is therefore convenient to introduce and use approximate techniques. For example, a general approximate factorization method has been introduced by *Daniele* [2004a, 2004b] and *Daniele and Lombardi* [2007], and the impenetrable half plane and wedge problems have been solved very efficiently by approximate factorization by *Daniele and Lombardi* [2006]. In this paper, this factorization method is applied to effectively solve a previously unsolved problem, thereby showing new results for a PEC half plane in a gyrotropic medium.

[5] In this connection we observe that problems involving very complex algebraic manipulations, such as those required by the problems of this paper as well as by those of *Graglia et al.* [1991], can nowadays be solved because powerful algebraic manipulator codes are readily available. The results of this paper were obtained by intensive use of the computing software Mathematica®. In fact, although the matrix formulas of this paper may look rather simple, the general explicit expression of each matrix coefficient in terms of the electromagnetic parameters usually occupies several pages.

[6] To facilitate the reading and the comprehension of the material presented in this paper, section 2 considers in detail the simpler cases of a PEC and of a PMC half plane; the most general case of an imperfect half plane is reported in Appendix A. The characteristic impedances and admittances of the bianisotropic medium that surrounds the half plane are defined in section 3, where we also provide two different methods for their evaluation. Special cases of PEC and PMC half planes amenable to closed-form solution are then discussed in section 4, whereas the general method to numerically factorize the WH kernels is given in section 5. One example of application of our numerical factorization method is also discussed in section 5, thereby showing with numerical results that the diffracted field contribution can be obtained by the saddle point integration method.