## 1. Introduction

[2] The concept of artificial hard surfaces, commonly used in acoustic applications, has been introduced in electromagnetics in the last two decades [*Kildal*, 1990]. The artificial hard surface represents a surface with anisotropic impedance. For a planar artificial hard surface, the relation between the tangential electric field and tangential magnetic field is given by, *E*_{t} + *E*_{l} = *Z*_{t}*H*_{l} − *Z*_{l}*H*_{t}, where *Z*_{t} and *Z*_{l} are the transverse and longitudinal anisotropic surface impedances, respectively, with respect to the propagating direction [*Ruvio et al.*, 2003]. This results in the characterization of the ideal hard surface by imposing conditions on the surface impedances such that, *Z*_{l} = 0 and ∣*Z*_{t}∣ → ∞. Because of this property, ideal hard surfaces can be created by introducing narrow longitudinal alternating perfect electric conductor (PEC) and perfect magnetic conductor (PMC) strips along the propagating direction and allowing the strip width to approach zero. However, in practice, there are no microwave materials which behave similarly to a PMC. One way to realize an artificial PMC is to use a grounded dielectric slab of certain thickness. Another realization is the corrugated surface with dielectric-filled longitudinal corrugations along the propagating direction. Compared with the strip-loaded dielectric slab, the corrugated surface has more weight and higher manufacturing cost [*Sipus et al.*, 1997].

[3] Hard surfaces can also be implemented in guided-wave structures, for example, rectangular or circular waveguides. Modal characteristics of an ideal hard surface circular waveguide have been recently investigated by *Klymko et al.* [2005]. In this paper, we study a rectangular waveguide with an ideal hard surface boundary conditions, which is modeled by alternating longitudinal PEC and PMC strips with vanishing widths. The geometry is depicted in Figure 1, where *l* is the longitudinal direction of wave propagation, *n* is the inward normal direction to the waveguide boundary, and *t* is the direction tangential to the waveguide boundary and transverse to the longitudinal strips. The longitudinal PEC strips enforce *E*_{l}∣_{S} = 0, so that *Z*_{l} satisfies the longitudinal impedance condition, *Z*_{l} = 0. The longitudinal PMC strips require *H*_{l}∣_{S} = 0, resulting in the transverse impedance condition, ∣*Z*_{t}∣ → ∞ [*Ruvio et al.*, 2003]. When the hard surface behaves like a PEC conductor, *E*_{t} and the normal derivative of *E*_{n} are zero on the waveguide surface *S*_{e}, that is, and , and when the hard surface behaves like a PMC conductor, *E*_{n} and the normal derivative of *E*_{t} are zero on the waveguide surface *S*_{m}, that is, and .

[4] Compared to PEC and PMC rectangular waveguides, which can support only transverse magnetic (TM) and transverse electric (TE) modes, the most important feature of this “ideal hard surface rectangular waveguide” is that it allows a propagation of the arbitrarily polarized uniform transverse electromagnetic (TEM) mode with a zero cutoff frequency, and can be used as a miniaturized feeding network in dual-polarized antenna arrays. This waveguide will support only the TEM mode if the frequency is below the cutoff of any of TE and TM modes. Whereas the TM and TE modes “see” the waveguide as a PEC and PMC waveguide, respectively, the TEM mode “sees” the ideal hard surface waveguide as a combination of PEC and PMC waveguides modeled by alternating PEC/PMC strips with vanishing widths, and it satisfies the boundary conditions on the ideal hard surface partially contributed by PEC and PMC waveguides, that is, ∂*E*_{n}/∂*n*∣_{S} = 0 and ∂*E*_{t}/∂*n*∣_{S} = 0 (as shown in Figure 1).

[5] Different analytical and numerical techniques have been used for the analysis of hard surfaces. A Green's function approach has been used to model different realizations of open boundary planar hard surfaces, including ideal PEC/PMC strip models, corrugated surfaces, and strip-loaded grounded slabs [*Sipus et al.*, 1997]. Image theory has been applied to study hard surfaces with a time-harmonic source [*Lindell*, 1995]. Another effective and simple approach to analyze hard surfaces is based on the asymptotic boundary condition, which has been used to analyze a corrugated surface [*Kishk et al.*, 1998], grids of metal strips [*Kishk and Kildal*, 1997], circular waveguide with strip-loaded walls [*Kishk and Morgan*, 2001], and hard surfaces of two-dimensional structures [*Kishk*, 2003].

[6] The most common use of artificial hard surfaces is the hard horn antenna, which can support a TEM wave at the design frequency. The hard boundary results in a uniform field distribution over the horn aperture with zero cross polarization [*Kildal*, 1988]. The advantage of using a hard horn antenna is that it can increase the directivity and decrease the beam width for given aperture dimensions [*Lier and Kildal*, 1988]. Also, an infinite planar array of open-ended rectangular waveguides, whose *E* plane walls are dielectric-loaded hard walls, can achieve better matching of the array to free space and increase the aperture and gain to remove the scan blindness [*Skobelev and Kildal*, 1998]. The performance of an open-ended circular waveguide with strip-loaded dielectric hard walls used as the element of an antenna array is also studied by *Skobelev and Kildal* [2000].

[7] Recently, a rectangular waveguide with sidewalls covered by a printed dipole frequency selective surface (FSS) has been designed to obtain the quasi-TEM mode [*Cucini et al.*, 2004; *Maci et al.*, 2005]. Also, an electromagnetic band gap (EBG) structure implemented on the sidewalls of a rectangular waveguide has been used to create the TEM mode propagation [*Yang et al.*, 1999]. Miniaturized dielectric-loaded quasi-TEM rectangular waveguides have been proposed for dual-band array applications [*Kehn et al.*, 2004; *Kehn and Kildal*, 2005].

[8] The purpose of the present paper is to develop an electric dyadic Green's function for the modal analysis of an ideal hard surface rectangular waveguide excited by an arbitrarily oriented electric current source. A procedure for deriving the Green's function in terms of solenoidal and irrotational parts is presented, wherein the solenoidal part of the Green's function is obtained in the eigenmode expansion form as a superposition of three terms associated with the TM, TE, and TEM modes of the ideal hard surface waveguide. A term corresponding to the TEM mode is obtained analytically as the solution of vector Helmholtz's equation in the zero cutoff limit subject to the boundary conditions for the electric field on the ideal hard surface. Numerical results for field distributions are demonstrated for the TEM mode and a few representative TM and TE modes propagating in a rectangular waveguide with ideal hard surface boundary conditions due to an arbitrarily oriented electric dipole source. A uniform field distribution of the TEM mode is verified by modelling a realistic hard surface square waveguide using High-Frequency Structure Simulator (HFSS) commercial software [*Ansoft*, 2004].