## 1 Introduction

[2] A perfect electromagnetic conductor (PEMC) is a nonreciprocal generalization of both a perfect electric conductor (PEC) and a perfect magnetic conductor (PMC), which was introduced by *Lindell and Sihvola* [2005a]. It is shown that fields cannot convey energy into a PEMC medium; hence, the PEMC is considered as an ideal boundary given by the following condition [*Lindell and Sihvola*, 2006a, 2009]:

where *M* is a scalar real parameter denoting the PEMC admittance, denotes the unit vector normal to the boundary, and subscript *t* stands for tangential components of the fields (i.e., electric * E* and magnetic

*). Obviously, for*

**H***M*→ ±∞ and

*M*= 0, the boundary condition (1) reduces to ×

*= 0 and ×*

**E***= 0, corresponding to PEC and PMC boundaries, respectively. The PEMC boundary can be also considered as a special type of surface impedance boundary. The general form of surface impedance boundary condition (SIBC), which is a linear relation between time-harmonic electric and magnetic field components tangential to the boundary surface, can be written as [*

**H***Wallen et al*., 2011]

where is the two-dimensional (2-D) surface impedance dyadic. For an isotropic impedance boundary (regular impedance boundary), , while for the PEMC boundary condition (1)

where is the two-dimensional (2-D) identity dyadic. Equation (2) clearly shows creation of nonreciprocal reflection, which is the most motivating property of the PEMC boundary. Due to this property, the PEMC boundary has many rich potential applications in electromagnetics, particularly for polarization transforming purposes [*Lindell and Sihvola*, 2005b, 2006b; *Sihvola and Lindell*, 2006; *Nayyeri et al*., 2012a, 2012b]. Hence, some approaches were introduced for the realization of the PEMC boundary [*Lindell and Sihvola*, 2005c, 2006c]. Recently, it has been practically realized by a grounded ferrite slab (as shown in Figure 1), where the effect of Faraday rotation, supported by a magnetically biased ferrite, was used [*Shahvarpour et al*., 2010].

[3] The electromagnetic wave interaction with the PEMC boundary is investigated in many studies such as *Lindell and Sihvola* [2005b], *Ruppin* [2006], *Komijani and Rashed-Mohassel* [2009a], *Rasouli Disfani et al*. [2011], and *Nayyeri et al*. [2012a]; however, due to the complexity of the boundary, analytic solutions are available only for relatively simple problems. On the other hand, numerical evaluation of such problems was considered only in *Sihvola et al*. [2007], where scattering by a small PEMC sphere was addressed. Thus, implementation of the PEMC boundary using numerical techniques that can treat an arbitrary problem is valuable.

[4] The finite difference time domain (FDTD) method [*Yee*, 1966; *Taflove and Hagness*, 2005; *Elsherbeni and Demir*, 2009; *Sullivan*, 2000; *Kunz and Luebbers*, 1993] is one of the most powerful computational electromagnetics methods, particularly for solving electromagnetic wave interaction with complex media and boundaries [*Young*, 1994; *Schuster and Luebbers*, 1996; *Teixeira*, 2008; *Nayyeri et al*., 2011a; *Nayyeri et al*., 2013]. The FDTD method is based on decomposition, discretization, and simultaneous solution of the Maxwell curl equations in the time and space domains using updating equations for * E* and

*. In problems involving boundary conditions like PEC, PMC, SIBC, etc., an updating equation for*

**H***or*

**E***at the boundary is required. For instance, at a PEC boundary, values of tangential*

**H***are set to zero at all time steps. On the other hand, for a PEMC, the boundary condition (1) or (2) is a nonreciprocal mixed one, involving both*

**E***and*

**E***. Thus, the most challenging part in FDTD implementation is the fact that*

**H***same components*of both

*and*

**E**_{t}*need to be colocated at both time and space, not realizable with the algorithm of*

**H**_{t}*Yee*[1966]. Of course, this is unlike the implementation of the (regular) SIBC (first proposed by

*Maloney and Smith*[1992] and

*Beggs et al*. [1992]), where

*=*

**E**_{t}*Z*[×

_{s}*] and the*

**H**_{t}*cross components*of

*and*

**E**_{t}*need to be colocated, more easily realizable with Yee's algorithm. In the latter (implementation of SIBC),*

**H**_{t}*at the interface is commonly approximated by its value at a half a cell size in front of the interface. Previously, in a conference paper [*

**H**_{t}*Nayyeri et al*., 2011b], which addressed plane wave reflection from a PEMC planar interface, we used the same approximation for implanting a PEMC planar interface in the one-dimensional (1-D) FDTD method. However, this technique could not be applied for modeling of the PEMC boundary in the 2-D and 3-D FDTD algorithms. For clarity, using this approximation couples updating equations for all the boundary grids to each other, which is, of course, not applicable.

[5] Basically, in the present paper, we propose an approach for modeling of an arbitrary PEMC boundary in the FDTD method. The PEMC boundary condition (1) is implemented in the FDTD method by a new formulation based on modifying the original Yee cell and using the backward difference scheme (BDS)/forward difference scheme (FDS) for the approximation of the field components at the boundary. The abstract of using the BDS for this purpose is briefly presented by the authors in *Nayyeri et al*. [2012c]. The proposed formulation is applied for modeling of PEMC planar interface, corner, and wedge. Full validation of the method is also presented.