## 1. Introduction

[2] The development of artificial materials by constructing lattice structures has gained considerable attention in recent years; in particular, the stop-band phenomenon associated with lattice structures has found many applications. For example, an antenna substrate etched with two-dimensionally (2-D) periodic holes has been utilized to suppress the surface waves introduced by printed antenna [*Yang*, 1996; *Gauthier et al.*, 1997; *Papapolymerou et al.*, 1998; *Lubecke et al.*, 1998]. The 2-D periodic layers in conjunction with planar structures have been investigated for both optical and microwave applications; one example is a high impedance surface that will not support a surface wave in any direction [*Sievenpiper et al.*, 1999; *Yang et al.*, 1999]. A 2-D periodic impedance surface has been employed as a simplified model to study its scattering and guiding characteristics, especially for its stop-band behaviors in bound-wave and leaky-wave regions [*Hwang and Peng*, 1999a, 1999b]. A 2-D periodic array of dielectric rods in a uniform surrounding has been shown to exhibit many interesting phenomena, such as spontaneous emission and localization of electromagnetic energy. The large pixels (square rods of dielectric material) have been proved to be able to obtain a very large absolute band gap [*Shen et al.*, 2002]. In addition, they also provided a fast plane-wave expansion method to speed up the computation for the band structure. Specifically, the periodic arrays of dielectric materials were employed as a novel waveguide to mold the flow of electromagnetic energy or as a novel cavity to store the energy [*Mekis et al.*, 1999; *Maystre*, 1994; *Joannopoulos et al.*, 1995; *Noponen and Turunen*, 1994; *Vardaxoglou et al.*, 1993]. Although the phenomenon of waveguiding in such a class of structures has been demonstrated by means of numerical and experimental studies, the purpose of this work is to gain a clear physical picture of wave processes involved and to develop design rules for practical considerations.

[3] The basic concept of this class of applications can be traced back to the early work of *Larsen and Oliner* [1967], who had used one-dimensionally (1-D) periodic dielectric slabs to form waveguide walls that are operated in their stop-band or below-cutoff condition. In this paper, we extend the structure to the two-dimensional case; that is, we replace the waveguide walls by finite stacks of 1-D periodic layers rather than uniform ones. For the purpose of comparison, the guiding characteristics of waveguides with uniform periodic dielectric layers are also investigated [*Hwang and Peng*, 2002].

[4] Specifically, the structure under consideration is a waveguide with 2-D periodic walls made of rectangular dielectric rods array immersed in a uniform surrounding, such as air. The 2-D periodic array is composed of a finite number of one-dimensionally periodic layers that are stacked with equal spacing between two neighboring ones. Each periodic layer is composed of an infinite number of rectangular dielectric rods of infinite length. In addition, we may displace every second row by a fractional part of the period to have any 2-D lattice pattern, so that the effect of the array pattern can be systematically investigated with ease. In this work, we shall employ the transverse resonance technique to obtain the dispersion relation; thus, the first step is to study the scattering characteristics of 2-D periodic dielectric rods array with finite thickness.

[5] The scattering characteristics of such a structure can be easily analyzed as a multilayer boundary-value problem. Here, we take the building block approach, such that the overall 2-D periodic structure can be regarded as a stack of unit cells, each consisting of a 1-D periodic layer in junction with a uniform one. By the rigorous method of mode matching, the input-output relation of a unit cell and the field distributions therein can be determined in a straightforward manner. It is noted that the present method offers a flexible approach to the analysis of 1-D periodic layer with arbitrary profile by making use of the staircase approximation to model it as a stack of cascaded 1-D periodic layers with different sizes of dielectric rods. In the absence of any incident wave, the existence of a non-trivial solution defines the condition of resonance in the transverse direction of the waveguide; in turn, this determines the dispersion relation of the waveguide.

[6] Based on the exact approach described above, we have carried out extensive numerical results to identify and explain the physical phenomena associated with the type of waveguides with 2-D periodic walls of finite thickness. Their dispersion characteristics are displayed in terms of both phase and attenuation constants. In particular, the contour plot for electric and magnetic field components and distribution of Poynting vector are used to gain a better understanding of the physical processes involved in the waveguiding in such a type of structures. Besides, the strong coupling between the incident plane wave and guided modes has also been studied in detail, including the mutual verification with the field and Poynting-vector distribution in the structure; these results establish consistently the distinctive characteristics of the waveguide with 2-D periodic walls of finite thickness from different viewpoints.