Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities Filled with Anisotropic Metamaterial

In this chapter, we present an extension of the rigorous analysis of the propagation of electromagnetic waves in magnetic transverse (TM) and transverse electric (TE) modes in a metallic circular waveguide partially filled with anisotropic metamaterial. In our analysis, the design of waveguide filters with uniaxial discontinuities is based on the determination of the higher-order modes, which have been analyzed and exploited. Below the cutoff frequency, the back backward waves can propagate in an anisotropic material. The numerical results with our MATLAB code for TM and TE modes were compared to theoretical predictions. Good agreements have been obtained. We analyzed a waveguide filters filled with partially anisotropic metamaterial using the mode matching (MM) technique based on the Scattering Matrix Approach (SMA), which, from the decomposition of the modal fields (TE and TM modes), are used to determine the dispersion matrix and thus the characterization of a discontinuity in waveguide. We extended the application of MM technique to the anisotropic material. By using modal analysis, our approach has considerably reduced the computation time compared to High Frequency Structure Simulator (HFSS) software.


Introduction
Guided modes in circular waveguides consist of metamaterials [1][2][3][4][5][6][7][8][9][10][11][12][13] have been studied in the literature. Many studies of propagation modes in this waveguides with isotropic media [14][15][16][17] or double negative metamaterials [18,19] have been presented in the literature. However, the rigorous study of the dispersion of anisotropic metamaterials in circular waveguides presents a lack in the literature. In this chapter, we present an extension of the rigorous analysis of the propagation of electromagnetic waves in magnetic transverse (TM) and electric transverse (TE) modes in the case of anisotropic circular waveguides, who take account of the spatial distribution of the permittivity and permeability of the medium. In this structure, the propagation modes are exploited. The effects of anisotropic parameter on cutoff frequencies and dispersion characteristics are discussed. Below the cutoff frequency, the back backward waves can propagate in an anisotropic material. The numerical results with our MATLAB code for TM and TE modes were compared to theoretical predictions. Good agreements have been obtained. We analyzed a waveguide filters filled with partially anisotropic metamaterial using the mode matching (MM) technique based on the Scattering Matrix Approach (SMA) which, from the decomposition of the modal fields, are used to determine the dispersion matrix and thus the characterization of a discontinuity in waveguide. We extended the application of MM technique to the anisotropic material.
This formulation can be a useful tool for engineers of microwave. The metamaterial is largely applied by information technology industries, particularly in the radio frequency devices and microwaves such as the waveguide antennas, the patch antennas, the circulators, the resonators and the filters.

Formulation
In the anisotropic diagonal metamaterials medium, the Maxwell equations are expressed as follows and Let consider a circular waveguide of radius R completely filled with anisotropic metamaterial without losses, as represented in the Figure 1. The wall of the guide is perfect conductor.
By considering the propagation in the Oz direction and manipulating Eqs. (1) and (2), we obtain the expressions of the transverse electromagnetic fields according to the longitudinal fields.
H r ¼ Àj When E and H are the electric and magnetic field respectively. ε and μ are the permittivity and permeability. k z is the propagation constant in z-direction.
In this chapter, we study rigorously the TE and TM modes in this anisotropic waveguide.

Transverse electric (TE) modes
From Eq. (1), the differential equation for z-component can be obtained as follows Using the separation of the variables (r,θ), the expression of the longitudinal magnetic field H z for the TE mn modes in the circular metallic waveguide completely filled with anisotropic metamaterial is necessary for the resolution of the differential Eq. (12). H z can be written as follows J n is the Bessel function of the first kind of order n (n = 0, 1, 2, 3, … ). The expressions (5) (17) With J 0 n is the derivative of the Bessel function of the first kind of order n (n = 0, 1, 2, 3, … ).
The boundary conditions are written as follows: Consequently, from Eq. (15), we obtain Where u 0 nm represents the m th zero (m = 1, 2, 3, … ) of the derivative of the Bessel function J 0 n of the first kind of order n. The constant H 0 is determined by normalizing the power flow down the circular guide.
Where * indicates the complex conjugate. Eq. (21) gives Finally, the propagation constant in TE mode is given by The cutoff frequency is written We can introduce the following effective permeability and effective permittivity to describe the propagation characteristics of the waveguide modes [6,7,13].
Further, it is apparent that: The sign of ε TE r,eff depends on the sign of μ rz . In the following, we will consider all cases that arise from the different sign of μ rz .
Therefore, the relative permeability μ rz below the cutoff frequency determines the sign of the relative effective permittivity of the anisotropic metamaterial in the circular waveguide. And the sign of the product μ rr :μ rz of the metamaterial below the cutoff frequency determines the sign of the propagation constants of the waveguide studied.
The backward waves are obtained for μ rr < 0 and μ rz > 0 and the forward waves for μ rr > 0 and μ rz < 0 and. Therefore, the backward waves and the forward waves can propagate below the cutoff frequency.

Transverse magnetic (TM) modes
Similar to TE modes, TM modes can be derived as follows: From Eq. (2), the differential equation for z-component can be obtained Using the separation of the variables (r,θ), the expression of the longitudinal electric field E z for the TM nm modes in the circular metallic waveguide completely filled with anisotropic metamaterial is necessary for the resolution of the differential Eq. (34). E z can be written as follows In Eq. (41) u nm represents the m th zero (m = 1, 2, 3, … ) of the Bessel function J n of the first kind of order n.
The constant E 0 is determined by normalizing the power flow down the circular guide.
Eq. (42) gives: We can introduce the following effective permeability and effective permittivity to describe the propagation characteristics of the waveguide modes.
Similar to the previous discussion, we have three possibilities: Further, It is apparent that: Consequently, the sign of μ TM r,eff depends on the sign of ε rz . In the following, we will consider all cases that arise from the different sign of ε rz .
It is also seen that the relative permittivity ε rz which is independent of μ rθ determines the sign of the relative effective permeability μ TM r,eff of the anisotropic metamaterial in the circular waveguide. The forward wave propagates in the waveguide for ε rz < 0 and ε rr > 0, and backward wave propagates for ε rz > 0 and ε rr < 0.
Therefore from this analysis, it is found that both the backward waves and the forward waves can propagate in any frequency region. This is determined by the sign of ε rz and ε rr for TM modes and the sign of μ rz and μ rr for TE modes.

Analysis of uniaxial discontinuities in the circular waveguides
In this section, we analyzed a waveguide filters filled with partially anisotropic metamaterial using the extension of the mode matching technique based on the Scattering Matrix Approach which, from the decomposition of the modal fields, are used to determine the dispersion matrix and thus the characterization of a discontinuity in waveguide. The discontinuities are considered without losses.
In Figure 2 we consider a junction between two circular waveguides having the same cross section filled with two different media. a i and b i are the incident and the reflected waves, respectively.
The transverse electric and magnetic fields (E T , H T ) in the wave guides can be written in the modal bases as follows [20]: where H T and E T are the transverse magnetic and electric fields (T refers to the components in the transverse plane), h i m , e i m represent the m th magnetic and electric modal Eigen function in the guide i, respectively and A i m and B i m are complex coefficients which are determined by normalizing the power flow down the circular guides (m is the index of the mode and i = I, II).
At the junction, the continuity of the fields allows to write the following equations: Junction between two circular waveguides filled with two different media having the same cross section.
which can be written in matrix form:  where U is the identity matrix. M 1 and M 2 are defined as: The scattering matrix of the discontinuity is: The total scattering matrix is obtained by chaining the S scattering matrices of all the discontinuities in a waveguide having cascaded uniaxial discontinuities [21].

Propagating modes
We choose the radius of the circular metal guide R = 13.4 mm. In a first case, we study the TE modes of a circular guide completely filled with anisotropic metamaterials (see Figure 1) with negative μ rr or negative μ rz . The fundamental mode of the equivalent empty circular waveguide has a resonant frequency of 6.57 GHz. For the case of metamaterials with a permeability μ r ¼ À1 and permittivity ε r ¼ À4:4, the fundamental mode presents a resonance frequency of f TE c:11 ¼ 3:13 GHz.
In Figure 3 the curves of the propagation constant, for frequency range 1-10 GHz and for the first five TE modes with μ rr ¼ 1, μ rz ¼ À1 and ε rθ ¼ 4:4, are represented. We observe that all modes propagate without cutoff frequencies (forward waves). Figure 4 represents the same diagrams for μ rr ¼ À1, μ rz ¼ 1 and ε rθ ¼ 4:4. When n and m are small and ω is large, the waves stop propagating. So, these modes propagate at low frequencies and cutoff at high frequencies (backward waves).
It is interesting to see that both forward and backward waves can be obtained by controlling the signs of μ rz and μ rr . Our results agree well with the predicted ones.
In a second case, we study the TE modes of this circular waveguide. Figure 5 represents the curves of propagation constant for the frequency range 1-10 GHz and for the first five TM modes with ε rr ¼ 4:4, ε rz ¼ À4:4 and μ rθ ¼ 1. All modes propagate without cutoff (forward waves).
Calculated curves of propagation constant for the frequency range 1-10 GHz and for the first five TM modes with ε rr ¼ À4:4, ε rz ¼ 4:4, μ rθ ¼ 1 are presented. We notice that both forward wave and backward wave can be obtained by controlling the signs of ε rr and ε rz . Figures 5 and 6 show that our results agree well with the predicted ones.
We observe that the cutoff frequencies of lowest TE modes decreased with the respect increase of μ rz for μ rr ¼ À1 and ε rθ ¼ 4:4 (see Figure 7). In a same manner, the TM cutoff frequencies decreased with the respect increase of ε rz for ε rr ¼ À4:4 and μ rθ ¼ 1 (see Figure 8). Consequently, by varying the parameters of material the propagating mode can be controlled.  for TM mode of the circular waveguide completely filled anisotropic metamaterial with parameters ε rr ¼ 4:4, ε rz ¼ À4:4, μ rθ ¼ 1.
The transmission and reflection coefficients using our numerical method with MATLAB and HFSS are presented in Figure 10. We used 8 modes in the whole circuit for the modal method. The simulations results show that are in perfect agreement. However and especially if the number of discontinuities increases, our method is significantly faster than HFSS. Then, by using our approach, it could easy to design filters according to a given specifications.

Conclusion
Rigorous analysis of propagating modes in circular waveguides filled with anisotropic metamaterial has been developed. It was demonstrated that the   propagation constant of the waveguide are closely dependent on constitutive parameters of the metamaterial. Using our MATLAB code the dispersion curves of the fundamental mode and the first four higher order modes of the metamaterial waveguide are obtained.
We found that in different frequency ranges below and above the cutoff frequency both the forward and the backward waves can propagate. This is determined by the sign of ε rz and ε rr for TM modes and by the sign of μ rz and μ rr for TE modes. Our simulation results are in good agreement with the theoretical prediction.
Moreover, using the Scattering Matrix Approach we applied the extension of MM technique to determine the dispersion matrix and to analyze multiple uniaxial circular discontinuity in waveguide filled with anisotropic metamaterials. This introduced tool is applied to the modeling of large complex structures such as filters where its rapidity compared to the commercial simulation tools is verified.

Author details
Hedi Sakli 1,2 * and Wyssem Fathallah 1 1 MACS Research Laboratory, National Engineering School of Gabes, Gabes University, Gabes, Tunisia 2 EITA Consulting, Montesson France *Address all correspondence to: hedi.s@eitaconsulting.fr © 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.