A full-wave analysis of planar electromagnetic band-gap (EBG) on anisotropic slab, which is characterized simultaneously by both nondiagonal second rank [ɛ] and [μ] tensors, is presented. The analytical procedure is based on Galerkin's method applied in the spectral domain, and the spectral dyadic Green's function for the considered structure is formulated based on Maxwell's equations directly. A kind of Uniplanar Compact EBG (UC-EBG) on anisotropic slab is computed efficiently, and the influence of the principal axes' rotation in the transverse plane on their band structures are studied for several slab materials.
 Electromagnetic Band-Gap (EBG) structures [Yablonovitch, 1993] are made periodic structures artificially which possess the capability to forbid electromagnetic propagation in either all or selected directions to realize spatial filters. They originated from optical regime and have been applied extensively in the microwave region, including the suppression of surface waves [Gonzalo et al., 1999], the construction of Perfect Magnetic Conducting (PMC) planes [Zhan and Samii, 2000] and antenna gain enhancement [Shum et al., 2000].
 More electromagnetic band gap (EBG) structures have received considerable attentions in recent years, several kinds of planar EBG structures have been suggested, one of them is comprised of a square lattice of holes etched in the grounded plane of a dielectric slab [Radisic et al., 1998], and another is consisted of planar periodic metal patches over a grounded dielectric slab [Hung-Yu et al., 1998], Uniplanar Compact EBG (UC-EBG) structures introduced by Yang et al.  are realized with square metallic pads connected by narrow strips to form a distributed LC network mounted on a bare or grounded dielectric slab. All kinds of planar EBG structures are particularly attractive and investigated intensively because of their easy fabrication, low cost, and compatibility with standard planar circuit technology. UC-EBG are more interesting because of being smaller in terms of wavelength, and they have been demonstrated in a variety of applications including, for instance, broadband low-pass, spurious-free band-pass filters [Yang et al., 1999] and harmonic-tuned power amplifiers [Hang et al., 1998].
 The planar EBG structures mounted on isotropic dielectric slabs have been computed using MoM [Hung-Yu et al., 1998] or FDTD [Coccioli et al., 1999; Yang et al., 2005], while these structures with anisotropic slab are analyzed infrequently. The purpose of this paper is investigating the effect of electrical and magnetic anisotropy on the band structure of planar EBG. A full-wave analysis is presented for planar EBG mounted on an anisotropic slab which is characterized by tensors [ɛ] and [μ] simultaneously, and the off-diagonal elements of the tensors also exist because of the misalignment of the material and structure coordinates systems in the transverse plane. Present analysis is performed in the spectral domain, and the periodic dyadic Green's function of the considered structure is obtained from Maxwell's equations directly. The Galerkin's procedure along with Parseval's theorem is applied in succession to obtain the band structure of these planar EBG structures. UC-EBG as one kind of complicated planar EBG structure is analyzed numerically. As the numerical results, the band structure of an UC-EBG with different medium parameters are computed efficiently, and the effects of the material axes' rotation in the transverse plane on the band gap are examined.
2. Analytical Formulation
 The schematic of an UC-EBG structure, shown in Figure 1, is assumed to be planar and infinite. The computation is executed in one cell of the UC-EBG lattice, and the cell is regarded as Jerusalem crosses apertures etched in the metallic plane which is mounted on a bare or grounded anisotropic slab with thickness d, as shown in Figure 1. In the slab region (region: 0 < z < d), the relative permittivity tensor [ɛr] and permeability tensor [μr] are assumed to be
where τ1, τ2 and τ3 are the principal values of [ɛ] in the material coordinate system (ξ, η, ζ) or those of [μ] in the material coordinate system (ξ′, η′, ζ′), and the misalignment angle for [ɛ] and θ for [μ] are shown in Figure 1.
 Now a Galerkin's procedure is applied numerically to set up a characteristic equation of the structure. The formulation is carried out in the spectral domain, a set of spectral discrete variables (αn, βm), which are of the Floquent mode (n, m), are defined as
where βx and βy are the fundamental propagation wave numbers in the x and y directions respectively.
 For each of the Floquet modes (n, m), the vector wave equation for the Fourier-transformed electric field vectors within the anisotropic substrate, by using differential matrix operators [Chen and Beker, 1993], can be written as
where the ⌊×⌋ operator is expressed as
 From (5) two coupled second order differential equation can be readily obtained, its matrix form is stated as
where coefficients a2, a0, b2, b0, c2, c0 and d2, d0 are defined in Appendix A.
 The general solution of both x and y components of the electric field to equation (7) can be written as eγz, and the characteristic equation to equation (7) is given by
Extracting the solutions γ1,2 of the characteristic equation (8), the general solution of x and y are expressed in terms of four elementary waves as
where A1 to A4 are unknown amplitude coefficients, and e1, e2 are given by
Now the remaining components of and fields in the anisotropic region can be obtained from Maxwell's equations directly. According to equation (11)
where the coefficients f1 to f4 are given by
 Now all the components of the magnetic field can be found from
the tangential components x and y are expressed as
where the coefficients g1 to g4 and h1 to h4 are given in Appendix A.
 Combining (9) and (15), all tangential field components within the anisotropic substrate can be expressed in a matrix form
 Now using the property of the matrix U(z)
the transition matrix which relates the tangential fields at “z = 0+” and “z = d−” interfaces may be obtained as
where the matrix T is defined as below and is partitioned into four 2 × 2 quadrant submatrices in succession
Now we introduce the wave admittance matrices for the air region which relate the transverse and field vectors at “z = 0−” and “z = d+” interfaces
where the components of the admittance matrices is given in Appendix A.
 Combining (22) and (24), an important equation (25) can be obtained by enforcing the boundary conditions (26) at the air-metal-slab interface (z = d):
where y and x is the Fourier-transformed current densities on the metallic plane, and the admittance matrices is given by
where is nether unilateral admittance matrix, for the two different cases of UC-EBG on a bare or grounded slab, it is respectively given by
Now according to equation (25), the dyadic Green's function of electric currents on the metallic patch can been expressed as −−1. When in the equation (24) are considered as magnetic currents in the aperture, the unilateral admittance matrices and can be regarded as the dyadic Green's functions of the nether and upper unilateral magnetic currents ±. After the dyadic Green's function has been determined, the Galerkin's procedure can be applied to diversified planar EBG on anisotropic slab.
 For the considered UC-EBG structure, the entire-domain basis functions to represent the electric current on the metallic patch cannot be presented because of the complexity of the metallic pattern, while the magnetic currents are distributed over the Jerusalem crosses aperture in space domain, it can be expressed in terms of a set of entire-domain type “junction basis functions” which have been expressed by Tsao and Mittra . After the appropriate substitutions of () in the equation (25) using the basis functions, the Galerkin's procedure along with Parseval's theorem is applied to obtain a matrix equation. The existence of this equation's nonzero solution requires the matrix determinant to be zero, which results in a characteristic equation of the UC-EBG, and the eigen-frequencies under the given fundamental wave numbers βx and βy can be obtained accordingly.
 When the UC-EBG are computed under different fundamental wave numbers βx and βy which are adopted between the symmetric points Γ(0, 0), X(π/W, 0), Y(0, π/L) and M(π/W, π/L) in the Brillouin zone, the band structure of this UC-EBG on anisotropic slap can be educed effectively.
3. Results and Discussion
 In order to verify the formulation of the problem and its numerical implementation, an UC-EBG for one special case, where the grounded dielectric slab is taken to be isotropic with ɛr = 10.2 and μr = 1.0, is computed. The physical dimensions in this work are: L = W = 120mil, h = 110mil, l = 65mil, w1 = 20mil, w2 = 10mil, the dielectric slab is 25-mil thick. As the numerical results, the band structure is displayed in Figure 2. Two complete band gaps are found, they span the frequency range from 11.0 GHz to 13.6 GHz and from 17.7 GHz to 22.1 GHz, respectively. Compared to the existing data available in the literature [Coccioli et al., 1999], the agreement is very good.
 Now all physical dimensions of the UC-EBG structure are kept the same, but the grounded isotropic dielectric slab is replaced by an electrically anisotropic material: sapphire with (ɛξξ, ɛηη, ɛζζ) = (11.6, 9.4, 9.4) in the material coordinate system (ξ, η, ζ). When the rotation angle θ is taken as “0°”, the band structure of the UC-EBG is shown in Figure 3, there are also two band gaps existed, one is between the first and second mode, and the other is between the fourth and fifth mode.
 In Figure 3, it is seen that both the lower and upper edges of these band gaps are at X(π/L, 0), Y(0, π/L) and M(π/L, π/L) points. To observe the alteration of band gaps as the principal axes rotates, the eigen-frequencies of these modes at X, Y and M points are computed as functions of the rotation angle θ. Now the eigen-frequencies of the first and second modes, which restrict the first band gap, are given in Figure 4a, while those of the fourth and fifth modes are given in Figure 4b.
 The numerical results in Figure 4 show that the axes rotation has little influence on the eigen-frequencies, and the location of the band gaps (the zone of crease in the Figure 4) change slightly as axes rotate. These can be attributed to the low axes ratio of the sapphire (ɛξξ/ɛηη = 1.23), so we take a suppositional electrically anisotropic material with the axes ratio 2.0 as the dielectric slab of the UC-EBG in succession. In this way, the eigen-frequencies versus the rotation angle θ are plotted in Figure 5. Now a larger alteration of these eigen-frequencies can be seen.
 Finally, in addition to the electrical anisotropy, the grounded dielectric slab is also magnetically anisotropic and is simultaneously characterized by (μξ′ξ′, μη′η′, μζ′ζ′) = (2.0, 1.0, 1.0) in the material coordinate system (ξ′, η′, ζ′), whose the rotation angle is expressed as θ′.
 The band structure of the UC-EBG is computed when the rotation angles are taken as θ = 0° and θ′ = 90°. Numerical results shown in Figure 6 indicate that the presence of [μr] has more influence on the band structure of UC-EBG. The first band gap is much narrower, and the second band gap no longer exists.
 In addition, the eigen-frequencies at X, Y and M points versus the rotation angle θ and θ′ are plotted in Figures 7 and 8, respectively. It is shown that the eigen-frequencies of the UC-EBG are more sensitive to the axes rotation of [μr]. This can be attributed to the fact that the tangential component of magnetic field while the longitudinal component of electric field are stronger in the grounded dielectric slab because of the grounded perfect conductor plane.
 Now according to the alteration of the eigen-frequencies as the axes rotate, the primary polarization of these modes can be speculated. For example, the eigen-frequency of the fifth mode at X point is going up as the rotation angle θ rises, as shown in Figure 7b, while it is going down rapidly as the rotation angle θ′ rises, as shown in Figure 8b. It is shown that the longitudinal component of magnetic field in this mode is in the direction mainly, while that of the electric field is in the direction. Because it is at X(π/L, 0) point, which denotes that the propagation is in the direction, the fifth mode can be speculated as a TM like mode.
 In this paper, the spectral-domain method is used to analyze electromagnetic characteristics of EBG on electrically and magnetically anisotropic slab. The method for the problem is developed and presented in view of the dielectric anisotropy, and the spectral Green's function for the considered structure is formulated based on Maxwell's equations directly. An Uniplanar Compact EBG on anisotropic slab was computed efficiently, and the influence of misalignment of material and structure coordinates systems were observed to be significant, particularly for the principal axes rotation of [μ], In addition, according to the alteration of the eigen-frequencies as axes rotates, the basic polarization for some modes can be speculated.
 The coefficients of coupled second order differential equations (7) are given by