Anisotropic Propagation of Electromagnetic Waves

Abstract

This chapter will analyze the properties of electromagnetic wave propagation in anisotropic media. Of particular interest are positive index, anisotropic, and magneto-dielectric media. Engineered anisotropic media provide unique electromagnetic properties including a higher effective refractive index, high permeability with relatively low magnetic loss tangent at microwave frequencies, and lower density and weight than traditional media. This chapter presents research including plane wave solutions to propagation in anisotropic media, a mathematical derivation of birefringence in anisotropic media, modal decomposition of rectangular waveguides filled with anisotropic media, and the full derivation of anisotropic transverse resonance in a partially loaded waveguide. These are fundamental theories in the area of electromagnetic wave propagation that have been reformulated for fully anisotropic magneto-dielectric media. The ensuing results will aide interested parties in understanding wave behavior for anisotropic media to enhance designs for radio frequency devices based on anisotropic and magnetic media.

1. Introduction

Recently engineered materials have come to play a dominant role in the design and implementation of electromagnetic devices and especially antennas. Metamaterials, ferrites, and magneto-dielectrics have all come to play a crucial role in advances made both in the functionality and characterization of such devices. In fact, a movement towards utilizing customized material properties to replace the functionality of traditional radio frequency (RF) components such as broadband matching circuitry, ground planes, and directive elements is apparent in the literature and not just replacement of traditional substrates and superstrates with engineered structures. A firm theoretical understanding of the electromagnetic properties of these materials is necessary for both design and simulation of new and improved RF devices.

Inherently, many of these engineered materials have anisotropic properties. Previously, the study of anisotropy had been limited mostly to the realm of optical frequencies where the phenomenon occurs naturally in substances such as liquid crystals and plasmas. However, the recent development of the aforementioned engineered materials has encouraged the study of electromagnetic anisotropy for applications at megahertz (MHz) and gigahertz (GHz) frequencies.

For the purposes of this chapter, an anisotropic electromagnetic medium defines permittivity ($\underset{¯}{\underset{¯}{{\epsilon }_r}}$) and permeability $\underset{¯}{\underset{¯}{{\mu }_r}}$ as separate tensors where the values differ in all three Cartesian directions (ε_x≠ε_y≠ε_z and μ_x≠μ_y≠μ_z). This is known as the biaxial definition of anisotropic material which is more encompassing than the uniaxial definition which makes the simplifying assumption that ε_x = ε_y = ε_t and μ_x = μ_y = μ_t. The anisotropic definition also differs from the traditional isotropic definition where ε_r and μ_r are the same in all three Cartesian directions defining each by a single value. For the definition of the tensor equations see Section 3.1. Anisotropic media yield characteristics such as conformal surfaces, focusing and refraction of electromagnetic waves as they propagate through a material, high impedance surfaces for artificial magnetic conductors as well as high index, low loss, and lightweight ferrite materials. The following sections aim to discuss in more detail some RF applications directly impacted by the incorporation of anisotropic media and also give a firm understanding of electromagnetic wave propagation as it applies to anisotropic media for different RF applications.

2. Applications of anisotropy in radio frequency devices

Traditionally, the study of anisotropic properties was limited to a narrow application space where traditional ferrites, which exhibit natural anisotropy were the enabling technology. These types of applications included isolators, absorbers, circulators and phase shifters [1]. Traditional ferrites are generally very heavy and very lossy at microwave frequencies which are the two main limiting factors narrowing their use in RF devices; however, propagation loss is an important asset to devices such as absorbers. Anisotropy itself leads to propagation of an RF signal in different directions, which is important in devices such as circulators and isolators [1]. For phase shifters and other control devices the microwave signal is controlled by changing the bias field across the ferrite [1, 2]. However, newer versions of some of these devices, utilizing FETs and diodes in the case of phase shifters, rely on isotropic media to enable higher efficiency devices.

As early as 1958, Collin showed that at microwave frequencies, where the wavelength is larger, it is possible to fabricate artificial dielectric media having anisotropic properties [3]. This has led some to investigate known theoretical solutions to typical RF problems, such as a microstrip patch antenna, and extend them utilizing anisotropic wave propagation in dielectric media [4, 5]. The anisotropic dielectric antenna shows interesting features of basic antenna applications featuring anisotropic substrates. While these solutions establish a framework for electromagnetic wave propagation in anisotropic media, they simplify the problem by necessarily setting μ_r to 1 and only focusing on dielectric phenomena of anisotropy.

The concept of artificial media is also exemplified by the proliferation of metamaterials research over the last few decades. Metamaterials incorporate the use of artificial microstructures made of subwavelength inclusions that are usually implemented with periodic and/or multilayered structures known as unit cells [6]. These devices operate where the wavelength is much larger than the characteristic dimensions of the unit cell elements. One characteristic feature of some types of metamaterials is wave propagation anisotropy [7]. Anisotropic metamaterials are used in applications such as directive lensing [8, 9], cloaking [10], electronic beam steering [11], and metasurfaces [12] among others.

Finally, a class of engineered materials exists that exhibits positive refractive index, anisotropy, and magneto-dielectric properties with reduced propagation loss at microwave frequencies compared to traditional ferrites. These materials show the unique ability to provide broadband impedance matches for very low profile antennas by exploiting the inherent anisotropy to redirect surface waves thus improving the impedance match of the antenna when very close to a ground plane. Antenna profile on the orders of a twentieth and a fortieth of a wavelength have been demonstrated using these materials with over an octave of bandwidth and positive realized gain [13, 14].

3. Plane wave solutions in an anisotropic medium

The recent development of low loss anisotropic magneto-dielectrics greatly expands the current antenna design space. Here we present a rigorous derivation of the wave equation and dispersion relationships for anisotropic magneto-dielectric media. All results agree with those presented by Meng et al. [15, 16]. Furthermore, setting $\underset{¯}{\underset{¯}{{\mu }_r}}$ = $\underset{¯}{\underset{¯}{I}}$, where $\underset{¯}{\underset{¯}{I}}$ is the identity matrix, yields results that agree with those presented by Pozar and Graham for anisotropic dielectric media [4, 5]. This section and the following section expand on the results presented by Meng et al., Pozar and Graham. Incorporating a fully developed derivation of anisotropic properties of both $\underset{¯}{\underset{¯}{{\epsilon }_r}}$ and $\underset{¯}{\underset{¯}{{\mu }_r}}$ expands upon the simplification imposed by both Pozar and Graham that uses an isotropic value of μ_r = 1. An expansion on the results of Meng et al. given in Section 4 develops the waveguide theory including a full modal decomposition utilizing the biaxial definition of anisotropy versus their simplified uniaxial definition. The derivation of anisotropic cavity resonance in Section 4 differs from that of Meng et al. by addressing the separate issue of how the direct relationship of an arbitrary volume of anisotropic material will distort the geometry of a cavity to maintain resonance at a given frequency. This property is especially important for the design of conformal cavity backed antennas for ground and air-based vehicle mobile vehicular platforms. Furthermore, the analysis of anisotropic properties is not restricted to double negative (DNG) materials, which is the case for both of the Meng et al. studies.

3.1. Source free anisotropic wave equation

In order to solve for the propagation constants, we will need to formulate the dispersion relationship from the anisotropic wave equation. This allows us to solve for the propagation constant in the normal direction of the anisotropic medium. We start with the anisotropic, time harmonic form of Maxwell’s source free equations for the electric and magnetic fields E and H

$$\nabla x\underset{¯}{E}=j{\mathit{\omega \mu }}_o\underset{¯}{\underset{¯}{{\mu }_r}}\cdot \underset{¯}{H},$$E1

$$\nabla x\underset{¯}{H}=-j{\mathit{\omega \epsilon }}_o\underset{¯}{\underset{¯}{{\epsilon }_r}}\cdot \underset{¯}{E},$$E2

where ω is the frequency in radians, ε_o is the permittivity of free space, μ_o is the permeability of free space, E = x_oE_x + y_oE_y + z_oE_z and H = x_oH_x + y_oH_y + z_oH_z. We define $\underset{¯}{\underset{¯}{{\mu }_r}}$ and $\underset{¯}{\underset{¯}{{\epsilon }_r}}$ as

$$\underset{¯}{\underset{¯}{{\epsilon }_r}}=\left[\begin{array}{lll}{\epsilon }_x& 0& 0\\ 0& {\epsilon }_y& 0\\ 0& 0& {\epsilon }_z\end{array}\right],$$E3

$$\underset{¯}{\underset{¯}{{\mu }_r}}=\left[\begin{array}{lll}{\mu }_x& 0& 0\\ 0& {\mu }_y& 0\\ 0& 0& {\mu }_z\end{array}\right].$$E4

Applying Eqs. (3) and (4) to Eqs. (1) and (2) yields the following

$${\underset{¯}{x}}_o\left(\frac{{\mathit{dE}}_Z}{\mathit{dy}}-\frac{{\mathit{dE}}_Y}{\mathit{dz}}\right)+{\underset{¯}{y}}_o\left(\frac{{\mathit{dE}}_X}{\mathit{dz}}-\frac{{\mathit{dE}}_Z}{\mathit{dx}}\right)+{\underset{¯}{z}}_o\left(\frac{{\mathit{dE}}_Y}{\mathit{dx}}-\frac{{\mathit{dE}}_X}{\mathit{dy}}\right)=-j{\mathit{\omega \mu }}_o\left({\mu }_x{H}_X{\underset{¯}{x}}_o+{\mu }_y{H}_Y{\underset{¯}{y}}_o+{\mu }_z{H}_Z{\underset{¯}{z}}_o\right),$$E5

$${\underset{¯}{x}}_o\left(\frac{{\mathit{dH}}_Z}{\mathit{dy}}-\frac{{\mathit{dH}}_Y}{\mathit{dz}}\right)+{\underset{¯}{y}}_o\left(\frac{{\mathit{dH}}_X}{\mathit{dz}}-\frac{{\mathit{dH}}_Z}{\mathit{dx}}\right)+{\underset{¯}{z}}_o\left(\frac{{\mathit{dH}}_Y}{\mathit{dx}}-\frac{{\mathit{dH}}_X}{\mathit{dy}}\right)=j{\mathit{\omega \epsilon }}_o\left({\epsilon }_x{E}_X{\underset{¯}{x}}_o+{\epsilon }_y{E}_Y{\underset{¯}{y}}_o+{\epsilon }_z{E}_Z{\underset{¯}{z}}_o\right).$$E6

Using the radiation condition, we assume a solution of E(x, y, z) = E(x, y)e^−jk_zz [17]. Now isolate the individual components of (5) by taking the dot product with x_o, y_o, and z_o respectively. This operation yields the following equations

$$\left(d/\mathit{dy}\right)E_z-{\mathit{jk}}_z{E}_y=-j{\mathit{\omega \mu }}_o{\mu }_x{H}_x,$$E7

$${\mathit{jk}}_z{E}_x-\left(d/\mathit{dx}\right)E_z=-j{\mathit{\omega \mu }}_o{\mu }_y{H}_y,$$E8

$$\left(d/\mathit{dx}\right)E_y-\left(d/\mathit{dy}\right)E_x=-j{\mathit{\omega \mu }}_o{\mu }_z{H}_z.$$E9

Assuming a solution of H(x, y, z) = H(x, y)e^−jk_zz for (6), the same procedure yields [17]

$$\left(d/\mathit{dy}\right)H_z-{\mathit{jk}}_z{H}_y=j{\mathit{\omega \epsilon }}_o{\epsilon }_x{E}_x,$$E10

$${\mathit{jk}}_z{H}_x-\left(d/\mathit{dx}\right)H_z=j{\mathit{\omega \epsilon }}_o{\epsilon }_y{E}_y,$$E11

$$\left(d/\mathit{dx}\right)H_y-\left(d/\mathit{dy}\right)H_x=-j{\mathit{\omega \mu }}_o{\mu }_z{H}_z.$$E12

Using (7)–(12) allows for the transverse field components of the electric and magnetic fields in terms of the derivatives of H_z and E_z as

$$E_x=-\frac{j}{k_o^2{\mu }_y{\epsilon }_x-k_z^2}\left({\mathit{\omega \mu }}_o{\mu }_y\left(d/\mathit{dy}\right)H_z+k_z\left(d/\mathit{dx}\right)E_z\right),$$E13

$$E_y=\frac{j}{k_o^2{\mu }_x{\epsilon }_y-k_z^2}\left({\mathit{\omega \mu }}_o{\mu }_x\left(d/\mathit{dx}\right)H_z-k_z\left(d/\mathit{dy}\right)E_z\right),$$E14

$$H_x=\frac{j}{k_o^2{\mu }_x{\epsilon }_y-k_z^2}\left({\mathit{\omega \epsilon }}_o{\epsilon }_y\left(d/\mathit{dy}\right)E_z-k_z\left(d/\mathit{dx}\right)H_z\right),$$E15

$$H_y=-\frac{j}{k_o^2{\mu }_y{\epsilon }_x-k_z^2}\left({\mathit{\omega \epsilon }}_o{\epsilon }_x\left(d/\mathit{dx}\right)E_z+k_z\left(d/\mathit{dy}\right)H_z\right).$$E16

The relationships for the transverse field components, applied to (1) and (2), yield the following solutions for H and E, respectively

$$\underset{¯}{H}=-\left({\underset{¯}{\underset{¯}{{\mu }_r}}}_{}^{-1},/,j,{\mathit{\omega \epsilon }}_o\right)\cdot \left(\nabla x\underset{¯}{E}\right),$$E17

$$\underset{¯}{E}=\left({\underset{¯}{\underset{¯}{{\epsilon }_r}}}_{}^{-1},/j{\mathit{\omega \epsilon }}_o\right)\cdot \left(\nabla x\underset{¯}{H}\right).$$E18

Taking the cross product of both sides and substituting (1) and (2) for the right hand side of (17) and (18) yields

$$\nabla x{\underset{¯}{\underset{¯}{{\epsilon }_r}}}_{}^{-1}\cdot \left(\nabla x\underset{¯}{H}\right)=k_o^2\underset{¯}{\underset{¯}{{\mu }_r}}\cdot \underset{¯}{H},$$E19

$$\nabla x{\underset{¯}{\underset{¯}{{\mu }_r}}}_{}^{-1}\cdot \left(\nabla x\underset{¯}{E}\right)=-k_o^2\underset{¯}{\underset{¯}{{\epsilon }_r}}\cdot \underset{¯}{E}.$$E20

Equations (19) and (20) represent the vector wave equations in an anisotropic medium [12].

3.2. Dispersion equation for H_z

We expand (19) in terms of (13)–(16)

$$\begin{array}{l}\nabla x\left\{\frac{{\underset{¯}{x}}_o}{{\epsilon }_x},\left[\left(d/\mathit{dy}\right)H_Z-\left(d/\mathit{dz}\right)H_y\right]+\frac{{\underset{¯}{y}}_o}{{\epsilon }_y}\left[\left(d/\mathit{dz}\right)H_x-\left(d/\mathit{dx}\right)H_z\right]\right.\\ \left.+\frac{{\underset{¯}{z}}_o}{{\epsilon }_z}\left[\left(d/\mathit{dx}\right)H_y-\left(d/\mathit{dy}\right)H_x\right]\right\}=k_o^2\underset{¯}{\underset{¯}{{\mu }_r}}\cdot \underset{¯}{H},\end{array}$$E21

Evaluating the remaining cross product of (21) yields the final form of the expanded wave equation

$${\underset{¯}{x}}_o{\mathrm{\Pi }}_x+{\underset{¯}{y}}_o{\mathrm{\Pi }}_y+{\underset{¯}{z}}_o{\mathrm{\Pi }}_z=k_o^2\underset{¯}{\underset{¯}{{\mu }_r}}\cdot \underset{¯}{H},$$E22

$${\mathrm{\Pi }}_x=\left[\left(d^2/\mathit{\text{dxdy}}\right)H_y-\left(d^2/{\mathit{dy}}^2\right)H_x\right]/{\epsilon }_z-\left[\left(d^2/{\mathit{dz}}^2\right)H_x-\left(d^2/\mathit{\text{dxdz}}\right)H_z\right]/{\epsilon }_y,$$E23

$${\mathrm{\Pi }}_y=\left[\left(d^2/\mathit{\text{dydz}}\right)H_z-\left(d^2/{\mathit{dz}}^2\right)H_y\right]/{\epsilon }_x-\left[\left(d^2/{\mathit{dx}}^2\right)H_y-\left(d^2/\mathit{\text{dxdy}}\right)H_x\right]/{\epsilon }_z,$$E24

$${\mathrm{\Pi }}_z=\left[\left(d^2/\mathit{\text{dxdz}}\right)H_x-\left(d^2/{\mathit{dx}}^2\right)H_z\right]/{\epsilon }_y-\left[\left(d^2/{\mathit{dy}}^2\right)H_z-\left(d^2/\mathit{\text{dxdy}}\right)H_x\right]/{\epsilon }_x.$$E25

Taking the dot product of (22) with z_o allows the isolation of H_z on the right hand side of the equation in terms of (265 on the left hand side

$$\left[\left(d^2/\mathit{\text{dydz}}\right)H_y-\left(d^2/{\mathit{dy}}^2\right)H_z\right]/{\epsilon }_x-\left[\left(d^2/{\mathit{dx}}^2\right)H_z+\left(d^2/\mathit{\text{dxdz}}\right)H_x\right]/{\epsilon }_y=k_o^2{\mu }_z{H}_z,$$E26

By keeping in mind that d/dz = −jk_z, setting E_z = 0, and differentiating (15) and (16) by d²/dxdz and d²/dydz, produces the following result

$$\left[\frac{k_z^2}{{\epsilon }_y\left(k_z^2-k_o^2{\epsilon }_y{\mu }_x\right)}-\frac{1}{{\epsilon }_y}\right]\left(d^2/{\mathit{dx}}^2\right)H_z+\left[\frac{k_z^2}{{\epsilon }_x\left(k_z^2-k_o^2{\epsilon }_x{\mu }_y\right)}-\frac{1}{{\epsilon }_x}\right]\left(d^2/{\mathit{dy}}^2\right)H_z=k_o^2{\mu }_z{H}_z.$$E27

Combining the d²H_z/dx² and d²H_z/dy² terms in (27) gives the following second order differential dispersion equation for H_z

$$\frac{k_o^2{\mu }_x}{k_o^2{\mu }_x{\epsilon }_y-k_z^2}\left(d^2/{\mathit{dx}}^2\right)H_z+\frac{k_o^2{\mu }_y}{k_o^2{\mu }_y{\epsilon }_x-k_z^2}\left(d^2/{\mathit{dy}}^2\right)H_z+k_o^2{\mu }_z{H}_z=0.$$E28

3.3. Dispersion equation for E_z

Expanding the $\nabla x\underset{¯}{E}$ term of (18) in terms of (13)–(16) yields

$$\begin{array}{l}\nabla x\left\{{\underset{¯}{x}}_o\left[\left(d/\mathit{dy}\right)E_Z-\left(d/\mathit{dz}\right)E_Y\right]/{\mu }_x+{\underset{¯}{y}}_o\left[\left(d/\mathit{dz}\right)E_X-\left(d/\mathit{dx}\right)E_Z\right]/{\mu }_y\right.\\ \left.+{\underset{¯}{z}}_o\left[\left(d/\mathit{dx}\right)E_Y-\left(d/\mathit{dy}\right)E_X\right]/{\mu }_z\right\}=k_o^2{\underset{¯}{\epsilon }}_r\cdot \underset{¯}{E}.\end{array}$$E29

Evaluating the remaining cross product of (29) gives the final form of the expanded wave equation

$${\underset{¯}{x}}_o{\xi }_x+{\underset{¯}{y}}_o{\xi }_y+{\underset{¯}{z}}_o{\xi }_z=k_o^2{\underset{¯}{\epsilon }}_r\cdot \underset{¯}{E},$$E30

$${\xi }_x=\left[\left(d^2/\mathit{\text{dxdy}}\right)E_Y-\left(d^2/{\mathit{dy}}^2\right)E_x\right]/{\mu }_z-\left[\left(d^2/{\mathit{dz}}^2\right)E_x-\left(d^2/\mathit{\text{dxdz}}\right)E_z\right]/{\mu }_y,$$E31

$${\xi }_y=\left[\left(d^2/\mathit{\text{dydz}}\right)E_z-\left(d^2/{\mathit{dz}}^2\right)E_y\right]/{\mu }_x-\left[\left(d^2/{\mathit{dx}}^2\right)E_y-\left(d^2/\mathit{\text{dxdy}}\right)E_x\right]/{\mu }_z,$$E32

$${\xi }_z=\left[\left(d^2/\mathit{\text{dxdz}}\right)E_x-\left(d^2/{\mathit{dx}}^2\right)E_z\right]/{\mu }_y-\left[\left(d^2/{\mathit{dy}}^2\right)E_z-\left(d^2/\mathit{\text{dydz}}\right)E_y\right]/{\mu }_x.$$E33

Taking the dot product of (30) with z_o allows isolation of the E_z component on the right hand side of the equation in terms of (33) on the left hand side

$$\left[\left(d^2/\mathit{\text{dxdz}}\right)E_x-\left(d^2/{\mathit{dx}}^2\right)E_z\right]/{\mu }_y+\left[\left(d^2/\mathit{\text{dydz}}\right)E_y-\left(d^2/{\mathit{dy}}^2\right)E_z\right]/{\mu }_x=k_o^2{\epsilon }_z{E}_z,$$E34

Keeping in mind that d/dz = −jk_z, setting H_z = 0, and differentiating (15) and (16) by d²/dxdz and d²/dydz produces the following result

$$\left[\frac{k_z^2}{{\mu }_y\left(k_z^2-k_o^2{\mu }_y{\epsilon }_x\right)}-\frac{1}{{\mu }_y}\right]\left(d^2/{\mathit{dx}}^2\right)E_z+\left[\frac{k_z^2}{{\mu }_x\left(k_z^2-k_o^2{\mu }_x{\epsilon }_y\right)}-\frac{1}{{\mu }_x}\right]\left(d^2/{\mathit{dy}}^2\right)E_z=k_o^2{\epsilon }_z{E}_z.$$E35

Combining the d²E_z/dx² and d²E_z/dy² terms in (35) gives the following second order differential dispersion equation for E_z

$$\frac{k_o^2{\epsilon }_x}{k_o^2{\mu }_y{\epsilon }_x-k_z^2}\left(d^2/{\mathit{dx}}^2\right)E_z+\frac{k_o^2{\epsilon }_y}{k_o^2{\mu }_x{\epsilon }_y-k_z^2}\left(d^2/{\mathit{dy}}^2\right)E_z+k_o^2{\epsilon }_z{E}_z=0.$$E36

3.4. Transmission and reflection from an anisotropic half-space

Birefringence is a characteristic of anisotropic media where a single incident wave entering the boundary of an anisotropic medium gives rise to two refracted waves as shown in Figure 1 or a single incident wave leaving gives rise to two reflected waves as shown in Figure 2. We call these two waves the ordinary wave and the extraordinary wave. To see how the anisotropy of a medium gives rise to the birefringence phenomenon, Eqs. (28) and (36) will yield a solution for k_z in the medium.

Figure 1.

A plane wave incident from free space on an anisotropic boundary.

Figure 2.

A plane wave incident from an anisotropic medium on a free space boundary.

Equations (28) and (36) yield the following solutions in unbounded anisotropic media restricted by the radiation condition in all three dimensions

$$E_z\left(x,y,z\right)=E_o{e}^{-j\left(k_{x}x+k_{y}y+k_{z}z\right)},$$E37

$$H_z\left(x,y,z\right)=H_o{e}^{-j\left(k_{x}x+k_{y}y+k_{z}z\right)}.$$E38

Plugging (37) into (36) (equivocally we could substitute (38) into (19)) allows the generation of a polynomial equation whose solutions give the values of k_z in the anisotropic medium. Noting that $d^2/{\mathit{dx}}^2=-k_x^2$ and $d^2/{\mathit{dy}}^2=-k_y^2$, (36) simplifies as

$$k_o^2{k}_x^2{\epsilon }_x{E}_z/\left(k_o^2{\mu }_y{\epsilon }_x-k_z^2\right)+k_o^2{\epsilon }_y{k}_y^2{E}_z/\left(k_o^2{\mu }_x{\epsilon }_y-k_z^2\right)-k_o^2{\epsilon }_z{E}_z=0.$$E39

Dividing out the $k_o^2{E}_z$ term and multiplying through by both denominators gives us the following factored polynomial

$$\left(k_o^2{\mu }_y{\epsilon }_x-k_z^2\right)\left(k_o^2{\mu }_x{\epsilon }_y-k_z^2\right){\epsilon }_z-k_x^2{\epsilon }_x\left(k_o^2{\mu }_x{\epsilon }_y-k_z^2\right)-{\epsilon }_y{k}_y^2\left(k_o^2{\mu }_y{\epsilon }_x-k_z^2\right)=0.$$E40

Finally, multiplying out (40) yields a fourth order polynomial whose roots yield the four values of k_z describing the ordinary wave and extraordinary wave in the positive and negative propagation directions

$$k_z^4{\mu }_z+\left[k_x^2{\mu }_x+k_y^2{\mu }_y-\left({\epsilon }_x{\mu }_y+{\epsilon }_y{\mu }_x\right)k_o^2{\mu }_z\right]k_z^2+\left[k_o^4{\epsilon }_x{\epsilon }_y{\mu }_x{\mu }_y{\mu }_z-k_o^2{k}_x{\epsilon }_x{\mu }_y{\mu }_x-k_o^2{\epsilon }_y{k}_y{\mu }_x{\mu }_y\right]=0.$$E41

Equation (41) is directly responsible for the existence of the extraordinary wave that is characteristic of the birefringence phenomenon. In an isotropic medium, the resulting polynomial for k_z is a second order polynomial, which yields only the values for the positive and negative propagation of the single ordinary wave.

4. Anisotropic rectangular waveguide

Electromagnetic wave behavior of waveguides is well understood in the literature. The mode within a waveguide that are based on the voltage and current distributions within the waveguide make up the basis for the electric and magnetic field calculations. This section derives similar formulations for a rectangular waveguide uniformly filled with an anisotropic medium as shown in Figure 3. Figure 3 shows propagation in the z_o-direction along the length of the waveguide. Rectangular waveguides are most commonly used for material measurement and characterization, and therefore understanding how electromagnetic waves propagate in an anisotropic waveguide is important for material characterization purposes. Furthermore, this section shows how the anisotropic derivation of waveguide behavior parallels that of a typical waveguide, and therefore how anisotropy may be applied to other waveguide geometries.

Figure 3.

Cross section of a closed rectangular waveguide filled with anisotropic metamaterial and surrounded by PEC walls.

4.1. Anisotropic mode functions

Assume source free Maxwell’s equations in the same form as (1) and (2). Then the transverse electromagnetic fields are defined

$${\underset{¯}{E}}_{\mathit{T\upsilon }}\left(x,y,z\right)=\sum _m\sum _n\left[V{\text{'}}_{\upsilon }\left(z\right)\underset{¯}{e}{\text{'}}_{\upsilon }\left(x,y\right)\right.\left.+V{\text{'}\text{'}}_{\upsilon }\left(z\right)\underset{¯}{e}{\text{'}\text{'}}_{\upsilon }\left(x,y\right)\right],$$E42

$${\underset{¯}{H}}_{\mathit{T\upsilon }}\left(x,y,z\right)=\sum _m\sum _n\left[I{\text{'}}_{\upsilon }\left(z\right)\underset{¯}{h}{\text{'}}_{\upsilon }\left(x,y\right)\right.\left.+I{\text{'}\text{'}}_{\upsilon }\left(z\right)\underset{¯}{h}{\text{'}\text{'}}_{\upsilon }\left(x,y\right)\right],$$E43

$$V_{\upsilon }\left(z\right)=V_o{e}^{-{\mathit{jk}}_{z}z},$$E44

$$I_{\upsilon }\left(z\right)=I_o{e}^{-{\mathit{jk}}_{z}z},$$E45

where E_T and H_T are the transverse electric and magnetic fields, V(z) and I(z) are the voltage and current at point z, e and h are the waveguide mode equations, and υ є [m, n] is the mode number defined by the two indices m and n.

4.1.1. Incident TE mode

Assuming only a TE type mode in the waveguide sets E_z = 0. Then (13)–(16) become

$$E_{\mathit{x\upsilon }}^{″}=-j{\mathit{\omega \mu }}_o{\mu }_y\left(d/\mathit{dy}\right)H_{\mathit{z\upsilon }}^{″}/\left(k_o^2{\mu }_y{\epsilon }_x-k_{\mathit{z\upsilon }}^2\right),$$E46

$$E_{\mathit{y\upsilon }}^{″}=j{\mathit{\omega \mu }}_o{\mu }_x\left(d/\mathit{dx}\right)H_{\mathit{z\upsilon }}^{″}/\left(k_o^2{\mu }_x{\epsilon }_y-k_{\mathit{z\upsilon }}^2\right),$$E47

$$H_{\mathit{y\upsilon }}^{″}=-{\mathit{jk}}_{\mathit{z\upsilon }}^2\left(d/\mathit{dx}\right)H_{\mathit{z\upsilon }}^{″}/\left(k_o^2{\mu }_x{\epsilon }_y-k_{\mathit{z\upsilon }}^2\right),$$E48

$$H_{\mathit{y\upsilon }}^{″}=-{\mathit{jk}}_{\mathit{z\upsilon }}^2\left(d/\mathit{dy}\right)H_{\mathit{z\upsilon }}^{″}/\left(k_o^2{\mu }_y{\epsilon }_x-k_{\mathit{z\upsilon }}^2\right).$$E49

To solve for $H_{z\upsilon }^{\text{'}\text{'}}$ we formulate the anisotropic wave equation from (1) and (2) where (52) resembles (19)

$$\nabla \times \nabla \times \underset{¯}{H}=j{\mathit{\omega \epsilon }}_o\underset{¯}{\underset{¯}{{\epsilon }_r}}\cdot \left(-j{\mathit{\omega \mu }}_o\underset{¯}{\underset{¯}{{\mu }_r}}\cdot \underset{¯}{H}\right),$$E50

$$\nabla \times \nabla \times \underset{¯}{H}=j{\mathit{\omega \epsilon }}_o\underset{¯}{\underset{¯}{{\epsilon }_r}}\cdot \left(\nabla \times \underset{¯}{E}\right),$$E51

$$\nabla \times {\underset{¯}{\underset{¯}{{\epsilon }_r}}}^{-1}\cdot \left(\nabla \times \underset{¯}{H}\right)=k_o^2\underset{¯}{\underset{¯}{{\mu }_r}}\cdot \underset{¯}{H}.$$E52

Expanding the curl of (52)

$$\left[\begin{array}{lll}\frac{1}{{\epsilon }_z}\frac{{\mathit{d\chi }}_{\mathit{z\upsilon }}^{″}}{\mathit{dy}}-\frac{1}{{\epsilon }_y}\frac{{\mathit{d\chi }}_{\mathit{y\upsilon }}^{″}}{\mathit{dz}}& 0& 0\\ 0& \frac{1}{{\epsilon }_x}\frac{{\mathit{d\chi }}_{\mathit{x\upsilon }}^{″}}{\mathit{dz}}-\frac{1}{{\epsilon }_z}\frac{{\mathit{d\chi }}_{\mathit{z\upsilon }}^{″}}{\mathit{dx}}& 0\\ 0& 0& \frac{1}{{\epsilon }_y}\frac{{\mathit{d\chi }}_{\mathit{y\upsilon }}^{″}}{\mathit{dx}}-\frac{1}{{\epsilon }_x}\frac{{\mathit{d\chi }}_{\mathit{x\upsilon }}^{″}}{\mathit{dy}}\end{array}\right]\left[\begin{array}{l}\begin{array}{l}1\\ \end{array}\\ \begin{array}{l}1\\ \end{array}\\ 1\end{array}\right]=k_o^2\left[\begin{array}{l}\begin{array}{l}{\mu }_x{\mathit{H}}_{\mathit{x\upsilon }}^{″}\\ \end{array}\\ \begin{array}{l}{\mu }_y{\mathit{H}}_{\mathit{y\upsilon }}^{″}\\ \end{array}\\ {\mu }_z{\mathit{H}}_{\mathit{z\upsilon }}^{″}\end{array}\right],$$E53

$${\mathit{\chi }}_{\mathit{x\upsilon }}^{″}=\left(d/\mathit{dy}\right){\mathit{H}}_{\mathit{z\upsilon }}^{″}-\left(d/\mathit{dz}\right){\mathit{H}}_{\mathit{y\upsilon }}^{″},$$E54

$${\mathit{\chi }}_{\mathit{y\upsilon }}^{″}=\left(d/\mathit{dz}\right){\mathit{H}}_{\mathit{x\upsilon }}^{″}-\left(d/\mathit{dx}\right){\mathit{H}}_{\mathit{z\upsilon }}^{″},$$E55

$${\mathit{\chi }}_{\mathit{z\upsilon }}^{″}=\left(d/\mathit{dx}\right){\mathit{H}}_{\mathit{y\upsilon }}^{″}-\left(d/\mathit{dy}\right){\mathit{H}}_{\mathit{x\upsilon }}^{″}.$$E56

Isolating the z_o-component of (53) gives the following relationship for $H_{z\upsilon }^{\text{'}\text{'}}$

$$\left[\left(d^2/\mathit{\text{dxdz}}\right){\mathit{H}}_{\mathit{x\upsilon }}^{″}-\left(d^2/{\mathit{dx}}^2\right){\mathit{H}}_{\mathit{z\upsilon }}^{″}\right]/{\epsilon }_y+\left(\left(d^2/\mathit{\text{dydz}}\right){\mathit{H}}_{\mathit{y\upsilon }}^{″}-\left(d^2/{\mathit{dy}}^2\right){\mathit{H}}_{\mathit{z\upsilon }}^{″}\right)/{\epsilon }_x-k_o^2{\mu }_z{\mathit{H}}_{\mathit{z\upsilon }}^{″}=0,$$E57

and substituting (48) and (49) for $H_{x\upsilon }^{\text{'}\text{'}}$ and $H_{y\upsilon }^{\text{'}\text{'}}$ yields the following differential equation that can be solved for $H_{z\upsilon }^{\text{'}\text{'}}$

$$k_o^2{\mu }_x\left(d^2/{\mathit{dx}}^2\right){\mathit{H}}_{\mathit{z\upsilon }}^{″}/\left(k_o^2{\mu }_x{\epsilon }_y-k_{\mathit{z\upsilon }}^2\right)+k_o^2{\mu }_y\left(d^2/{\mathit{dy}}^2\right){\mathit{H}}_{\mathit{z\upsilon }}^{″}/\left(k_o^2{\mu }_y{\epsilon }_x-k_{\mathit{z\upsilon }}^2\right)+k_o^2{\mu }_z{\mathit{H}}_{\mathit{z\upsilon }}^{″}=0.$$E58

Assuming a solution of the form

$$H_{\mathit{z\upsilon }}^{″}=H_o\text{cos}\left(k_{\mathit{x\upsilon }}x\right)\text{cos}\left(k_{\mathit{y\upsilon }}y\right)e^{-{\mathit{jk}}_{\mathit{z\upsilon }}z},$$E59

$$k_{\mathit{x\upsilon }}=\mathit{m\pi }/a,$$E60

$$k_{\mathit{y\upsilon }}=\mathit{n\pi }/b,$$E61

which meets the boundary conditions at the PEC walls of the waveguide, then plugging (58) into (53) imposes the following restriction on the values of the tensors in (3) and (4)

$${\mu }_x{k}_{\mathit{x\upsilon }}^2/\left(k_o^2{\mu }_x{\epsilon }_y-k_{\mathit{z\upsilon }}^2\right)+{\mu }_y{k}_{\mathit{y\upsilon }}^2/\left(k_o^2{\mu }_y{\epsilon }_x-k_{\mathit{z\upsilon }}^2\right)={\mu }_z.$$E62

Solving (62) for k_zυ gives the following equation which yields four solutions to the propagation constant for the ordinary and extraordinary waves described in Section 3.4

$$\begin{array}{l}{k}_{\mathit{z\upsilon }}^2=\left\{k_o^2\left({\mu }_x{\epsilon }_y+{\mu }_y{\epsilon }_x\right)-k_{\mathit{x\upsilon }}^2{\mu }_x/{\mu }_z-k_{\mathit{y\upsilon }}^2{\mu }_y/{\mu }_z\right.\\ \left.\pm \sqrt{4k_{\mathit{x\upsilon }}^2{k}_{\mathit{y\upsilon }}^2{\mu }_x{\mu }_y/{\mu }_z^2+{\left[k_o^2\left({\mu }_x{\epsilon }_y-{\mu }_y{\epsilon }_x\right)-k_{\mathit{x\upsilon }}^2{\mu }_x/{\mu }_z+k_{\mathit{y\upsilon }}^2{\mu }_y/{\mu }_z\right]}^2}\right\}/2.\end{array}$$E63

Equations (62) and (63) provide the criteria for determining the cutoff frequency for the propagation of modes inside the waveguide. Plugging (59) into (46)–(49) yields the following equations for the TE mode vectors in (42) and (43)

$$\begin{array}{l}{\underset{¯}{e}}_{\upsilon }^{″}\left(x,y\right)=j{\mathit{\omega \mu }}_o{H}_o\left[{\underset{¯}{x}}_o{\mu }_y{k}_{\mathit{y\upsilon }}\text{cos}\left(k_{\mathit{x\upsilon }}x\right)\text{sin}\left(k_{\mathit{y\upsilon }}y\right)/\left(k_o^2{\mu }_y{\epsilon }_x-k_{\mathit{z\upsilon }}^2\right)\right.\\ \left.-{\underset{¯}{y}}_o{\mu }_x{k}_{\mathit{x\upsilon }}\text{sin}\left(k_{\mathit{x\upsilon }}x\right)\text{cos}\left(k_{\mathit{y\upsilon }}y\right)/\left(k_o^2{\mu }_x{\epsilon }_y-k_{\mathit{z\upsilon }}^2\right)\right],\end{array}$$E64

$$\begin{array}{l}{\underset{¯}{h}}_{\upsilon }^{″}\left(x,y\right)={\mathit{jk}}_{\mathit{z\upsilon }}{H}_o\left[{\underset{¯}{x}}_o{k}_{\mathit{x\upsilon }}\text{sin}\left(k_{\mathit{x\upsilon }}x\right)\text{cos}\left(k_{\mathit{y\upsilon }}y\right)/\left(k_o^2{\mu }_x{\epsilon }_y-k_{\mathit{z\upsilon }}^2\right)\right.\\ \left.+{\underset{¯}{y}}_o{k}_{\mathit{y\upsilon }}\text{cos}\left(k_{\mathit{x\upsilon }}x\right)\text{sin}\left(k_{\mathit{y\upsilon }}y\right)/\left(k_o^2{\mu }_y{\epsilon }_x-k_{\mathit{z\upsilon }}^2\right)\right].\end{array}$$E65

4.1.2. Incident TM mode

Assuming only a TM type mode in the waveguide sets H_z = 0. Then (13)–(16) become

$${\mathit{E}}_{\mathit{x\upsilon }}^{\text{'}}=-{\mathit{jk}}_{\mathit{z\upsilon }}\left(d/\mathit{dx}\right){\mathit{E}}_{\mathit{z\upsilon }}^{\text{'}}/\left(k_o^2{\mu }_y{\epsilon }_x-k_{\mathit{z\upsilon }}^2\right),$$E66

$${\mathit{E}}_{\mathit{y\upsilon }}^{\text{'}}=-{\mathit{jk}}_{\mathit{z\upsilon }}\left(d/\mathit{dy}\right){\mathit{E}}_{\mathit{z\upsilon }}^{\text{'}}/\left(k_o^2{\mu }_x{\epsilon }_y-k_{\mathit{z\upsilon }}^2\right),$$E67

$${\mathit{H}}_{\mathit{x\upsilon }}^{\text{'}}=j{\mathit{\omega \epsilon }}_o{\epsilon }_y\left(d/\mathit{dy}\right){\mathit{E}}_{\mathit{z\upsilon }}^{\text{'}}/\left(k_o^2{\mu }_x{\epsilon }_y-k_{\mathit{z\upsilon }}^2\right),$$E68

$${\mathit{H}}_{\mathit{y\upsilon }}^{\text{'}}=-j{\mathit{\omega \epsilon }}_o{\epsilon }_x\left(d/\mathit{dx}\right){\mathit{E}}_{\mathit{z\upsilon }}^{\text{'}}/\left(k_o^2{\mu }_y{\epsilon }_x-k_{\mathit{z\upsilon }}^2\right).$$E69

Solving (66)–(69) for E_z and substituting (1) for $\nabla \times \underset{¯}{H}$ formulates the anisotropic wave equation for E where (72) resembles (20)

$$\nabla \times \nabla \times \underset{¯}{E}=-j{\mathit{\omega \mu }}_o\underset{¯}{\underset{¯}{{\mu }_r}}\cdot \left(\nabla \times \underset{¯}{H}\right),$$E70

$$\nabla \times \nabla \times \underset{¯}{E}=-j{\mathit{\omega \mu }}_o\underset{¯}{\underset{¯}{{\mu }_r}}\cdot \left(j{\mathit{\omega \epsilon }}_o\underset{¯}{\underset{¯}{{\epsilon }_r}}\cdot \underset{¯}{E}\right),$$E71

$$\nabla \times {\underset{¯}{\underset{¯}{{\mu }_r}}}^{-1}\cdot \left(\nabla \times \underset{¯}{E}\right)=k_o^2\underset{¯}{\underset{¯}{{\epsilon }_r}}\cdot \underset{¯}{E}.$$E72

Equation (72) represents the anisotropic wave equation for the time harmonic electric field. Expanding the curl of (72) and isolating the z_o component as we did for H_z in Section 4.1.1 yields the following solution for the E_z component

$${\mathit{E}}_{\mathit{z\upsilon }}^{\text{'}}=E_o\text{sin}\left(k_{\mathit{x\upsilon }}x\right)\text{sin}\left(k_{\mathit{y\upsilon }}y\right)e^{-{\mathit{jk}}_{\mathit{z\upsilon }}z}.$$E73

Plugging (73) into (66)–(69) yields the following equations for the TM mode vectors in (42) and (43)

$$\begin{array}{l}{\underset{¯}{e}}_{\upsilon }^{\text{'}}\left(x,y\right)=-{\mathit{jk}}_{\mathit{z\upsilon }}{E}_o\left[{\underset{¯}{x}}_o{k}_{\mathit{x\upsilon }}\text{cos}\left(k_{\mathit{x\upsilon }}x\right)\text{sin}\left(k_{\mathit{y\upsilon }}y\right)/\left(k_o^2{\mu }_y{\epsilon }_x-k_{\mathit{z\upsilon }}^2\right)\right.\\ \left.+{\underset{¯}{y}}_o\text{sin}\left(k_{\mathit{x\upsilon }}x\right)\text{cos}\left(k_{\mathit{y\upsilon }}y\right)/\left(k_o^2{\mu }_y{\epsilon }_x-k_{\mathit{z\upsilon }}^2\right)\right],\end{array}$$E74

$$\begin{array}{l}{\underset{¯}{h}}_{\upsilon }^{\text{'}}\left(x,y\right)=j{\mathit{\omega \epsilon }}_o{E}_o\left[{\underset{¯}{x}}_o{\epsilon }_y{k}_{\mathit{y\upsilon }}\text{sin}\left(k_{\mathit{x\upsilon }}x\right)\text{cos}\left(k_{\mathit{y\upsilon }}y\right)/\left(k_o^2{\mu }_x{\epsilon }_y-k_{\mathit{z\upsilon }}^2\right)\right.\\ \left.-{\underset{¯}{y}}_o{\epsilon }_x{k}_{\mathit{x\upsilon }}\text{cos}\left(k_{\mathit{x\upsilon }}x\right)\text{sin}\left(k_{\mathit{y\upsilon }}y\right)/k_o^2{\mu }_y{\epsilon }_x-k_{\mathit{z\upsilon }}^2\right].\end{array}$$E75

4.2. Anisotropic transverse resonance

This section describes the derivation of an anisotropic transverse resonance condition established between resonant walls of a rectangular waveguide. Assume an infinite rectangular waveguide partially loaded with an anisotropic medium, then w(z) represents the width of the anisotropic medium at any point z along the direction of propagation as shown in Figure 4. At any length z along the waveguide, the assumption that the horizontal distance between two resonant walls represented as a partially filled parallel plate waveguide is a valid presumption. We can calculate L_g(z) as the unknown distance between the edge of the anisotropic medium and the cavity wall based on a transverse resonance condition in the x_o-direction. However, we first need to derive the characteristic impedance of the anisotropic region in the transmission line model.

Figure 4.

Symmetrically loaded transmission line model with a short at either end.

4.2.1. Electromagnetic fields in free space regions

Calculating the fields in the free space region of the waveguide begins with Maxwell’s source free Eqs. (1) and (2) and the equations for the individual vector components of the electromagnetic fields (13)–(16). Using the standard derivation of the wave equation for H_z in free space from (1) and (2) shows

$${\nabla }_T\times {\nabla }_T\times {\underset{¯}{H}}_{\upsilon }=j{\mathit{\omega \epsilon }}_o\left({\nabla }_T\times {\underset{¯}{E}}_{\upsilon }\right)={\nabla }_T\left({\nabla }_T\cdot {\underset{¯}{H}}_{\upsilon }\right)-{\nabla }_T^2{H}_{\mathit{z\upsilon }},$$E76

$$j{\mathit{\omega \epsilon }}_o\left(-j{\mathit{\omega \mu }}_o{H}_{\mathit{z\upsilon }}\right)+{\nabla }_T^2{H}_{\mathit{z\upsilon }}=0,$$E77

$$\left[\left(d^2/{\mathit{dx}}^2\right)+\left(d^2/{\mathit{dy}}^2\right)+k_o^2\right]H_{\mathit{z\upsilon }}=0.$$E78

Utilizing (52) and setting k_zo = 0 due to the assumption of the transverse resonance condition in the free space region of the waveguide will lead to the solution to H_z. Assuming the dominate mode to be TE₁₀ because a ≥ 2b, then k_yo = 0 for the first resonance at cutoff [1]. With k_yo = 0 no variation of the fields in the y_o direction and (d²/dy²)H_z = 0 then (78) becomes

$$\left(d^2/{\mathit{dx}}^2+k_o^2\right)H_z=0.$$E79

Equation (80) is a standard differential equation with a known solution [17]

$$H_{\mathit{z\upsilon }}={\mathit{Ae}}^{-{\mathit{jk}}_{o}x}+{\mathit{Be}}^{+{\mathit{jk}}_{o}x},$$E80

where A and B are yet to be determined coefficients. Substituting (80) into (13)–(16) yields the expression for E_y

$$E_y=k_o{\mathit{\omega \mu }}_o\left({\mathit{Ae}}^{-{\mathit{jk}}_{o}x}-{\mathit{Be}}^{+{\mathit{jk}}_{o}x}\right)/k_o^2=Z_o\left({\mathit{Ae}}^{-{\mathit{jk}}_{o}x}-{\mathit{Be}}^{+{\mathit{jk}}_{o}x}\right).$$E81

Accounting for the restrictions imposed by the transverse resonance conditions on E_z, k_zo and k_yo, then E_x = 0, H_x = 0 and H_y = 0 as well.

4.2.2. Electromagnetic fields in anisotropic region

Starting with (1) and (2) for the source free Maxwell’s equations in an anisotropic medium, the vector components (13)–(16) led to the derivation of the dispersion Eqs. (26) and (38) for H_z and E_z, respectively.

The cutoff frequency or resonance of a rectangular waveguide is determined when the propagation constant in the direction of resonance, in the case the x_o-direction, is 0 [1]. By definition, when the waveguide’s dominant mode υ = 1 propagates, then k_x1 > 0 and the guide is resonant whereas when the mode attenuates then k_x1 < 0 and there is no resonance. Therefore, the resonance first manifests itself when k_x1 = 0. For the dominate mode to be TE₁₀ then a ≥ 2b and d²H_z/dy² = 0. Simplifying (28) with these substitutions produces a simpler form to solve for H_z

$$\left[\left(d^2/{\mathit{dx}}^2\right)+k_o^2{\mu }_z{\epsilon }_y\right]H_z=0,$$E82

$$\beta =k_o\sqrt{{\mu }_z{\epsilon }_y}.$$E83

Solving (82) for H_z and plugging the result into (13)–(16) yields

$$H_z={\mathit{Ce}}^{-\mathit{j\beta x}}+{\mathit{De}}^{+\mathit{j\beta x}},$$E84

$$E_y=Z_o\beta \left({\mathit{Ce}}^{-\mathit{j\beta x}}-{\mathit{De}}^{+\mathit{j\beta x}}\right)/k_o{\epsilon }_y=Z_o\sqrt{{\mu }_z/{\epsilon }_y}\left({\mathit{Ce}}^{-\mathit{j\beta x}}-{\mathit{De}}^{+\mathit{j\beta x}}\right),$$E85

We can see from (13)–(16) that based on our resonance conditions on E_z, k_z1 and k_y1 that E_x = 0, H_x = 0 and H_y = 0.

4.2.3. Characteristic impedances of the two regions

The first boundary condition exists at the perfect electric conductor (PEC) boundary when x = −a/2 and E(x, y, z) = 0

$${\left.E_y\right|}_{x=-a/2}=0\to {\mathit{Ae}}^{{\mathit{jk}}_{o}a/2}={\mathit{Be}}^{-{\mathit{jk}}_{o}a/2},$$E86

$$A={\mathit{Be}}^{-{\mathit{jk}}_{o}a}.$$E87

Plugging (87) into (80) and (81) yields

$$E_y=Z_o{\mathit{Be}}^{-{\mathit{jk}}_{o}a/2}\left[e^{-{\mathit{jk}}_o\left(x+a/2\right)}-e^{+{\mathit{jk}}_o\left(x+a/2\right)}\right],$$E88

$$E_y=-2Z_o{\mathit{Be}}^{-{\mathit{jk}}_{o}a/2}\text{sin}\left[k_o\left(x+a/2\right)\right].$$E89

Similarly,

$$H_z=2{\mathit{Be}}^{-{\mathit{jk}}_{o}a/2}\text{cos}\left[k_o\left(x+a/2\right)\right].$$E90

Equations (89) and (90) solve for the impedance of the free space region as Z = −E_y/H_z

$$Z_o=-E_y/H_z={\mathit{jZ}}_o\text{tan}\left[k_o\left(x+a/2\right)\right],$$E91

within the region 0 ≤ (x + a/2) ≤ (a−w)/2. The second boundary condition exists at x = −w/2 where the tangential fields at the boundary are equal. In this case, there are two tangential fields in E_y and H_z. At the boundary, we have the following three conditions

$${\left.E_y^{-}\right|}_{x=-w/2}={\left.E_y^{+}\right|}_{x=-w/2},$$E92

$${\left.H_z^{-}\right|}_{x=-w/2}={\left.H_z^{+}\right|}_{x=-w/2},$$E93

$${\left.Z_o\right|}_{x=-w/2}^{-}={\left.Z_1\right|}_{x=-w/2}^{+}.$$E94

Plugging Eqs. (80) and (81) into (92) and (93) yields the following set of equations

$$-2{\mathit{jBe}}^{-{\mathit{jk}}_o\left(a/2\right)}\text{sin}\left[k_o\left(a-w\right)/2\right]=\sqrt{\left({\mu }_z/{\epsilon }_y\right)}\left({\mathit{Ce}}^{-j{\beta }_1\left(w/2\right)}-{\mathit{De}}^{+j{\beta }_1\left(w/2\right)}\right),$$E95

$$2{\mathit{Be}}^{-{\mathit{jk}}_o\left(a/2\right)}\text{cos}\left[k_o\left(a-w\right)/2\right]={\mathit{Ce}}^{-j{\beta }_1\left(w/2\right)}-{\mathit{De}}^{+j{\beta }_1\left(w/2\right)}.$$E96

Equations (95) and (96) give two equations to solve for three unknowns. Match equation (91) to the impedance in the anisotropic region at x = −w/2 to solve for the third unknown. Now solve for Z = −E_y/H_z from (89) and (90)

$$\begin{array}{l}{Z}_1=Z_o\sqrt{\left({\mu }_z/{\epsilon }_y\right)}\left[\left({\mathit{De}}^{+j{\beta }_{1}x}-{\mathit{Ce}}^{-j{\beta }_{1}x}\right)/\left({\mathit{De}}^{+j{\beta }_{1}x}+{\mathit{Ce}}^{-j{\beta }_{1}x}\right)\right]\\ =Z_o\sqrt{\left({\mu }_z/{\epsilon }_y\right)}\left[\left(1-\rho e^{-j2{\beta }_{1}x}\right)/\left(1+\rho e^{-j2{\beta }_{1}x}\right)\right],\end{array}$$E97

$$\left|Z_1\right|=Z_o\sqrt{\left({\mu }_z/{\epsilon }_y\right)},$$E98

where ρ = C/D. Now apply boundary condition (94) to (91) and (97) in order to yield an expression for ρ

$$Z_o\sqrt{\left({\mu }_z/{\epsilon }_y\right)}\left[\left(1-\rho e^{j{\beta }_{1}w}\right)/1+\rho e^{j{\beta }_{1}w}\right]={\mathit{jZ}}_o\text{tan}\left[k_o\left(a-w\right)/2\right],$$E99

$$\left(1-\rho e^{j{\beta }_{1}w}\right)/1+\rho e^{j{\beta }_{1}w}=j\sqrt{\left({\epsilon }_y/{\mu }_z\right)}\text{tan}\left[k_o\left(a-w\right)/2\right]=j\mathrm{\Psi },$$E100

$$\rho =e^{-j{\beta }_{1}w}\left(1-j\mathrm{\Psi }\right)/\left(1+j\mathrm{\Psi }\right).$$E101

Substituting (101) into (97) yields the last equation along with (95) and (96) to solve for B, C and D.

4.2.4. Anisotropic transverse resonance condition

To simplify the calculation, consider Figure 4 as slice of Figure 3 in only one direction that is partially filled with an anisotropic medium. Figure 4 represents a transmission line representation that allows for a solution to L_g in terms of w for a given wavelength. Now use standard transmission line theory to calculate the input impedance Z_in at x = 0 from both directions. Transmission line theory says that as we approach the same point in a transmission line from either direction the input impedances should be equal. Then by symmetry the transverse resonance condition simplifies to Z_in = 0 from either direction.

Starting with the short located at x = a/2, calculate Z_in2 at x = w/2 as

$$Z_{\mathit{\text{in}}2}={\mathit{jZ}}_o\text{tan}\left(k_o{L}_g\right).$$E102

Now calculate Z_in1 at x = 0 as

$$Z_{\mathit{\text{in}}1}=Z_1\left[Z_{\mathit{\text{in}}2}+{\mathit{jZ}}_1\text{tan}\left({\beta }_{1}w/2\right)\right]/\left[Z_1+{\mathit{jZ}}_{\mathit{\text{in}}2}\text{tan}\left({\beta }_{1}w/2\right)\right].$$E103

The symmetric transverse resonance condition simplifies (103) to

$$Z_1+{\mathit{jZ}}_{\mathit{\text{in}}2}\text{tan}\left({\beta }_{1}w/2\right)=0.$$E104

Plugging (98) and (101) into (104) yields the following equation for L_g [14]

$$L_g=\lambda {\text{tan}}^{-1}\left[\sqrt{\left({\mu }_z/{\epsilon }_y\right)}/\text{tan}\left(\mathit{\pi w}\sqrt{{\mu }_z{\epsilon }_y}/\lambda \right)\right]/\left(2\pi \right).$$E105

where λ is wavelength. Importantly, the solution of (105) shows that the transverse resonance only depends on two of the six $\underset{¯}{\underset{¯}{{\epsilon }_r}}$ and $\underset{¯}{\underset{¯}{{\mu }_r}}$ components. This means that maintaining a constant resonance in a waveguide or cavity relies on the clever engineering of ε_y and μ_z and leaves designers free to adjust the other components as they see fit to enhance performance in other ways. Furthermore, if ε_y = μ_z = 1 then the resonance in the x_o direction will see the anisotropic substrate as air, while other tensor elements can be utilized to achieve performance attributed to materials with an arbitrarily high refractive index.

4.2.5. Suppression of birefringence in a rectangular waveguide

Section 3.4 discusses the phenomenon of birefringence in an unbounded anisotropic half-space by deriving the existence of a fourth order polynomial for the wavenumber in the propagation direction. However, for low order resonances, a rectangular waveguide suppresses the birefringence inherent to anisotropic media by suppressing propagation in the vertical direction of the waveguide. In other words, k_y = 0 and d²/dy² = 0 assuming the horizontal dimension of the waveguide is at least twice the size of the vertical dimension or a ≥ 2b in Figure 1 [3]. The bound on the waveguide geometry simplifies (39), the dispersion equation for E_z in an anisotropic waveguide, to

$$k_o^2{k}_x^2{\epsilon }_x{E}_z/\left(k_o^2{\mu }_y{\epsilon }_x-k_z^2\right)-k_o^2{\epsilon }_z{E}_z=0,$$E106

and results in the following second order polynomial for k_z

$$k_z^2={\epsilon }_x\left(k_o^2{\mu }_y-k_x^2/{\epsilon }_z\right).$$E107

The suppression of the k_y term in (39) yields a second order differential equation for the wave number in the propagation direction, thereby eliminating the property of birefringence for this case.

5. Conclusions

Recently engineered materials have come to play an important role in state of the art designs electromagnetic devices and especially antennas. Many of these engineered materials have inherent anisotropic properties. Anisotropic media yield characteristics such as conformal surfaces, focusing and refraction of electromagnetic waves as they propagate through a material, high impedance surfaces for artificial magnetic conductors as well as high index, low loss, and lightweight ferrite materials. This chapter analyzes the properties of electromagnetic wave propagation in anisotropic media, and presents research including plane wave solutions to propagation in anisotropic media, a mathematical derivation of birefringence in anisotropic media, modal decomposition of rectangular waveguides filled with anisotropic media, and the full derivation of anisotropic transverse resonance in a partially loaded waveguide.