## 1. Introduction

[2] The class of periodic structures has been a subject of continuing interest in the literature. The main effort in the past had been on the scattering and guiding of waves by one-dimensionally periodic structures [*Rotman*, 1951; *Elliott*, 1954]. Recently, considerable attention has been focused on the study of wave phenomena associated with two-dimensional (2-D) ones, particularly in conjunction with the properties of photonic band gap [*Fan et al.*, 1996; *Sigalas et al.*, 1995]. Since wave propagation is forbidden in the stop band, this allows us to mold the power flow or to inhibit spontaneous emission. Consequently, many novel dielectric (optical) waveguides or cavities were developed by using the photonic band gap material. For example, the waveguide with 2-D periodic structures as its walls was designed to make the waves bounce back and forth around the channel [*Mekis et al.*, 1999; *Hwang and Peng*, 2003].

[3] In addition to the properties of strong reflection in the stop band, the anomalous refraction, such as ultrarefraction (or negative refraction), was found to exist in such a class of 2-D periodic structures, especially in the vicinity of the band edge [*Enoch et al.*, 2003; *Boris et al.*, 2000; *Notomi*, 2000]. Many researchers took advantage of these properties to design lenses with a very short focal length or to confine emission in a narrow lobe [*Gralak et al.*, 2000; *Boris et al.*, 2000]. It is noted that the previous research works were made under the condition that the spatial periods of the photonic band gap materials are of the order of operation wavelength.

[4] In addition to the behavior of wave reflection in stop band associated with a photonic band gap structure, the negative and infinite group velocities were experimentally observed in bulk hexagonal two-dimensional photonic band gap crystals within the band gap in the microwave region [*Solli et al.*, 2003]. On the basis of their experimental studies, they found that the crystal exhibits anomalous dispersion within the band gap, passing through zero dispersion at the band edges. In addition, the negative phase and group velocities, along with positive group and negative phase velocities (i.e., backward waves), were theoretically investigated by using a simple model to characterize the property of negative refractive index (NRI) of a metamaterial [*Mojahedi et al.*, 2003]. Recently, the same group had extended the work to design a medium which not only possesses NRI properties but also exhibits negative group velocities (NGV). In their invention, a resonant circuit is embedded within each loaded transmission line unit cell, resulting in a region of anomalous dispersion for which the group delay is negative [*Siddiqui et al.*, 2003].

[5] In a recent publication [*Hwang*, 2004], we have investigated the relationship between the scattering characteristic and the band structure of a two-dimensionally electromagnetic crystal containing a metal and dielectric medium. Therein, we have demonstrated the relationship between the transmission spectrum and the band structure of a 2-D periodic structure. Specifically, two types of stop band were clearly identified: one is referred to as the vertical type and the other as the slanted type. The former consists of the commonly known stop bands that are due to the effect of one-dimensional periodicity; thus each vertical stop band has a constant phase over the entire stop band. On the other hand, the later consists of the stop bands that are slanted at an angle on the *k*_{o}-β diagram (a part of the standard Brillouin diagram) and that are attributed to the combined effect of the periodicities in two dimensions. Notably, the 1-D periodic structure can also support the slanted stop band. The dispersion analysis of the shielded Sievenpiper structure [*Elek and Eleftheriades*, 2004] has been proved to support the slanted stop band, caused by the contradirectional coupling between the fundamental backward-wave harmonic and an underlying forward parallel-plate mode.

[6] In this paper, we present a thorough investigation of a 2-D periodic structure that is composed of rectangular metallic cylinders immersed in a uniform medium. Since the shape of the metallic cylinders considered here is rectangular and the material is assumed to be a perfect electric conductor, the electric fields inside the metal cylinder are zero, and those outside the metal region are expressed in terms of the superposition of waveguide modes (parallel-plate waveguide modes). These waveguide modes inherently satisfy the electromagnetic boundary condition; therefore this could speed up the numerical convergence for the tangential electric and magnetic fields. Besides, in the numerical computation, all the mathematical procedures resort to the matrix operation; the dimensions of these matrices are proportional to the number of space harmonics (waveguide modes) truncated. Thus the speed of computation and required memory space directly relate to the number of space harmonics. To ensure the accuracy of numerical results, we have carried out convergence tests for both the scattering and dispersion analyses against the number of space harmonics (or waveguide modes). We found that a small number of space harmonics is needed to achieve the power conservation criterion.

[7] The mode-matching method utilized in this paper could have the advantages as described previously. However, for the metallic cylinders with curved profile, such as circular ones, the present method remains to be improved. Although the staircase approximation could be used to partition the curved profile into a stack of rectangular layers, this would make the mathematical formulation complicated, and the artificial edges caused by the piecewise approximation would result in extra edge diffraction (especially in higher-frequency operation). The method of Green's function based on lattice sums is more suitable for such type of problems [*Botten et al.*, 2000].

[8] Concerning the mathematical procedures for this research work, first, the scattering of a plane wave by a structure of finite thickness was analyzed with particular attention directed to the variation of the group velocity (index) in terms of the phase angle of the transmittance spectrum. Then, we calculated the dispersion characteristics of a structure of infinite extent, including the phase constant (real part) and attenuation constant (imaginary part), as plotted in the form of the *k*_{o}-β diagram. By comparison between the scattering and dispersion characteristics, we have observed the negative group delay to exist in the region of slanted stop bands but not in the vertical ones.

[9] This paper is organized as follows. In section 2, we first introduce the structure configuration and incident conditions for the 2-D periodic structure under consideration. In section 3, we outline the mathematical formulations to resolve such a 2-D boundary value problem. The method of mode matching and the Floquet solutions were employed to transform the electromagnetic field problem into a representation of transmission line network. Moreover, the generalized scattering matrix representation and the Bloch condition were utilized to obtain a generalized eigenvalue problem for determining the dispersion relation of waves propagating in such an infinite 2-D periodic medium. The scattering characteristics, including the reflectance and transmittance of each space harmonic, were also calculated. In section 4, we carried out numerous numerical calculations on the group delay via the transmittance of plane wave at an oblique incidence. Moreover, the dispersion relation of a 2-D periodic medium was calculated and was demonstrated to verify the negative group velocity in the slanted stop band region. In section 5, we conclude this paper by making some remarks.