1.

INTRODUCTION

Recent trends in left-handed metamaterials (LHM) have led to a renewed interest on retrieving their effective permittivity ε and permeability μ^1-4. Much of the LHMs realized so far are assumed to be biaxially anisotropic. Other important classes of man-made materials such as carbon-fiber reinforced plastics (CFRP) are characterized by uniaxial anisotropy⁵.

One of the most important methods for determining complex material parameters is the Nicolson-Ross-Weir (NRW) method. In 1970, Nicolson and Ross⁶ gave the formulas for extraction of the complex permittivity and the permeability of an isotropic sample from measurements of the scattering matrix elements S₁₁ and S₂₁. The material under test (MUT) was placed in a coaxial transverse electromagnetic (TEM) waves transmission line (TL). In 1974, Weir⁷ obtained analogous relations for the MUT placed in a rectangular waveguide. In both works the field reflection coefficient Γ at the interface of two half-spaces and the propagation coefficient P were used. They can be expressed by the above mentioned ε and μ.

If the sample is a solid, it is enough to consider the problem of the propagation through the slab. At least two additional layers should be considered for loose materials (e.g. synthetic granulates or grain seeds) and liquids. For this reason, the propagation in layered structure was considered.

The problem of propagation of electromagnetic (EM) waves in an anisotropic layered structure placed in the waveguide is not new or even unresolved for simple configurations. Its solution is more arduous than difficult and occurs fragmentarily in many publications. The key work seems to be article by Damaskos et al.^{8, 9}, describing the problem of propagation through the biaxially anisotropic slab placed in an empty rectangular waveguide. This work is surprisingly little cited in the scientific literature.

An extraction of the material parameters from scattering matrix elements S₁₁ and S₂₁ is called the inverse problem, while obtaining S₁₁ and S₂₁ from ε and μ is a simple problem. The purpose of this work is to present the solution of the latter for the anisotropic material.

In our work we first derived from basics relations described transverse electric (TE) and transverse magnetic (TM) waves propagating in a rectangular waveguide filled with an anisotropic medium. Field theory methods were used¹⁰. Although this derivation is not new, it highlights the analogy and differences with propagation in an empty waveguide. Second, we discuss the transmission matrix method in anisotropic layered media highlighting the mutual relationships between the field theory and the circuit theory. It is usually not shown in elementary textbooks^11-14. Third, we present scattering matrix elements S₁₁ and S₂₁ at the form known from the Nicolson-Ross-Weir model.

2.

DESCRIPTION OF THE CONFIGURATION

Consider a medium consisting of t + 2 layers placed inside a rectangular waveguide with cross-sectional dimensions a and b. The boundaries of the layers are perpendicular to the waveguide z-axis and marked as d_l, l = 0, 1, …, t (Fig. 1).

Figure 1.

A layered medium placed in a rectangular waveguide with cross-sectional dimensions a and b

Layers marked as 0 and t + 1 have vacuum parameters. In each layer we are looking for a solution in the form of monochromatic waves propagating along the z-axis

where E^(l) and H^(l) denotes the electric and magnetic field intensity respectively. The upper sign (minus) corresponds to the wave running in the positive and the lower sign (plus) in the negative direction of this axis. To simplify the notation, in further considerations we will skip the arguments of the function, remembering the exp(+jωt) time dependence of all fields.

Each layer has biaxial anisotropy with principal axes parallel to the axes of the Cartesian system. In this case the magnetic flux density B^(l) and the electric flux density D^(l) are described by the constitutive relations

and the matrices of relative permeability and magnetic permittivity tensors ε^(l) and μ^(l) for each layer are diagonal

If two of the three parameters are the same, the medium is uniaxially anisotropic. The another parameter can be anywhere, e.g.

At this case the optic axis is along the z axis. Maxwell’s equations in each layer are

We assume that the waveguide walls are made of a perfect electric conductor (PEC), then the boundary conditions on the inner surface of the waveguide S mean vanishing the normal component of B^(l) and the tangential components of E^(l).

At the interface z = d_l between the medium l and l + 1, the boundary conditions mean the continuity of the normal component of flux densities B^(l) and D^(l) and the continuity of the tangential components of field intensities E^(l) and H^(l).

Substituting the assumed form of the solution (1) and (2) into Maxwell’s equations, it can be shown that the longitudinal components of the fields satisfy the coupled system of equations. Decoupling of E_0z from H_0z occurs in two particular cases (see %Appendix A). In the first case TE_m0 i TE_0n modes can exist in the layered structure with biaxial material with ε^(l) and μ^(l) given by (5). In the second case the TE_mn modes (including TE_m0 and TE_0n) and TM_mn modes can propagate for specific configuration of material parameters. The latter case includes a particular form of an uniaxial anisotropy (so-called transverse isotropy) determined by (6). Wave modes result from the substitution of general solutions of decoupled wave equations into boundary conditions on the surface S (see Appendix B).

Our goal is to find elements S₂₁ and S₁₁ of the scattering matrix for the EM wave propagating from the left through the layer system.

3.

SCATTERING MATRIX ELEMENTS FOR LAYERED STRUCTURE

Let’s focus on the TE_mn mode transmission through a system of layers with uniaxial anisotropy defined by (6) located in the waveguide. The field in a medium l is the sum of waves running in the positive and negative direction of the z axis (see Appendix 2):

where

Boundary conditions at the interface z = d_l require continuity of the normal component of flux densities and the tangential components of field intensities. Relations between the amplitudes A^(l) and B^(l) can be obtained by using boundary conditions continuity of fields E_x and H_y only:

It may be shown that other three equations are not independent. Substituting (9) and (10) to (11) we get

The above system of equations can be written in the matrix form (see Appendix C)

Where

And

An Eq. (15), described the matrix W^(l,l+1), is a generalization of the Eq. (12) presented in the Rulf¹⁵ paper. That work concerned with the isotropic medium.

Let’s write the following expressions based on (14)

Substituting (16) to (17) we obtain

and finally

Let’s denote h₀ = d₀ and the widths of particular layers as h_l

One can then define

For a structure composed of t + 2 layers (see Fig. 1) we obtain

Where

To obtain the S₂₁ and S₁₁ scattering matrix elements, one must find the relations of the amplitudes A^(t+1) and B^(t+1) with the notation A_t+1 and B_t+1 used in the theory of TL for the same reference plane z = d_t. The amplitudes A₀ and B₀ at point z = 0 on a uniform TL are related to the amplitudes A_t+1 and B_t+1 at point z = d_t on the same line (see e.g. Rulf¹⁵, Eq. (13).

If d₀ = 0, then A⁽⁰⁾ = A₀ and B⁽⁰⁾ = B₀. Then d_t is the sum of the widths of all the layers. Hence

The expression (20) can then be written as

The scattering matrix elements S₂₁ and S₁₁ satisfy following relationships

If the wave is incident from the left, then B_t+1 = 0. We obtain

Next, if we denote elements of the W matrix as

we get the scattering matrix elements S₂₁ and S₁₁ of the layer system:

4.

AN EXAMPLE: SCATTERING MATRIX ELEMENTS OF A SINGLE SLAB

As an example, we examine the transmission through a single slab placed in an empty waveguide. We shall find a relation between the amplitudes on the two sides of an interface, first. We may assume without loss of generality that there is an interface z = d₀ = 0 between layer 0 (to the left) and 1 (to the right). Then

Assuming that in the area 1 there is no wave returning from infinity, then B⁽¹⁾ = 0. We define the reflection coefficient Γ and the transmission coefficient T at the interface as (see e.g. Balanis¹², p. 181):

After some manipulations we obtain

We will consider transmission through layer 1 assuming that media denoted by 0 and 2 have vacuum parameters. In addition, for simplicity we denote d₁ = d. From (19), formally

Taking into account that k⁽²⁾ = k⁽⁰⁾ and W^(1,2) = W^(1,0) then the above formula takes the form

or using (22)

The matrix W can be rewritten as

where the letter P indicates propagation factor⁷

After some manipulations matrices W^(1,0) and W^(0,1) can be expressed by the reflection coefficient at the interface Γ

Finally W takes the form

It can be shown that W is an unimodular matrix (see e.g. Born and Wolf¹³, p. 60). This means that the determinant of the matrix is equal to unity.

Repeating the arguments cited above B₂ = 0, B₀ = S₁₁A₀ and A₂ = S₂₁A₀. From (23)

or

The form (28) of obtained scattering matrix elements S₂₁ and S₁₁ of the layer with uniaxial anisotropy is consistent with the initial formulas of the NRW method^{6, 7} of the isotropic layer, derived using the method of graphs.

5.

REMARKS ON OTHER CONSIDERED WAVE MODES

It is not difficult to notice that equations (28) are also valid for other considered cases. In the expression for P, should be substituted the appropriate wave number k⁽¹⁾ and in the formula for Γ – the corresponding impedance quotient Z^(0,1). These parameters are summarized in the Table 1 and the Table 2. Material parameters affecting on EM wave propagation are given in brackets in the table head.

Table 1.

Parameters Γ and P of the TEmn and TMmn modes propagation through an uniaxially anisotropic slab

Table 2.

Parameters Γ and P of the TEm0 and TE0n modes propagation through a biaxially anisotropic slab

6.

CONCLUSION

We have derived a formalism and presented a solution of the boundary value problem of EM waves transmission through an anisotropic layered structure placed in a rectangular waveguide. The main axes of the crystallographic system and the Cartesian system were oriented parallel to the waveguide edges. Hybrid modes as well as TE and TM may exist in this configuration, with TM waves only for an uniaxial anisotropy of a special form (so-called transverse isotropy). For the most general biaxial anisotropy, only TE_m0 and TE_0n waves can exist⁸.

The TE_mn mode transmission through a system of uniaxially anisotropic layers was considered in detail. We found the elements S₂₁ and S₁₁ of the scattering matrix for the EM wave propagating from the left. The resulting matrix W (see Eq. (21)), derived in this paper, plays a crucial role in determining S₂₁ and S₁₁. As a matter of fact, the form of matrix W is the same as reported by other authors for the isotropic¹¹ and electrically anisotropic⁵ media. The novelty is obtaining new relations for wave impedances and wave numbers assigned to the individual layers. For other cases, considered in the Appendices A and B, these relationships are shown in Table 1 and 2. Appropriate parameters for isotropic media, known from the circuits theory¹¹, should be replaced by these expressions.

In the important case of transmission through a single slab, the solutions in the form of the S₁₁ and S₂₁ elements of the scattering matrix are obtained and presented in the same form as in the NRW method^6-7. This enables their direct application for the extraction of complex components of permittivity and permeability tensors.

In the general case it is possible to determine as many material parameters as many independent measurement results (in general, complex) we have. For the isotropy, the complex parameters S₂₁ and S₁₁ correspond to the complex permittivity ε and permeability μ. It is possible to extract both materials parameters using the scattering matrix elements by the only measurement in a specific configuration. In the case of anisotropy, it is clear that an additional one (see Table 1) or two (see Table 2) measurements should be made in a different configuration. For example, the MUT should be prepared differently (e.g. rotated⁵, placed in another waveguide¹⁶) or a different mode should be used⁸.

In addition to the LHM and CFRP structures mentioned earlier, the results can be utilized to study classic anisotropic materials with known orientation of the main axes and man-made material structures, in particular artificial dielectrics¹⁷, magnetodielectrics¹⁸ and other layered composites, characterized by transverse isotropy of effective parameters.