Open access peer-reviewed chapter

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

By Hedi Sakli and Wyssem Fathallah

Submitted: November 15th 2019Reviewed: February 6th 2020Published: March 11th 2020

DOI: 10.5772/intechopen.91645

Downloaded: 146

Abstract

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.

Keywords

  • anisotropic metamaterials
  • forward and backward waves
  • MM
  • modal analysis
  • waveguides discontinuity

1. 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.

2. Formulation

In the anisotropic diagonal metamaterials medium, the Maxwell equations are expressed as follows

×E=μ¯¯.HE1
×H=ε¯¯.EE2

with

μ¯¯=μ0μrr000μ000μrz=μ0μrt00μrz.E3

and

ε¯¯=ε0εrr000ε000εrz=ε0εrt00εrzE4

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.

Figure 1.

Geometry of circular waveguide filled with metamaterial.

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.

Er=jKc.r2kzEzr+ωμ0μrHzθE5
Eθ=jKc.θ2kzrEzθ+ωμ0μrrHzrE6
Hr=jKc.θ2ωε0εrEzθ+kzHzrE7
Hθ=jKc.r2ωε0εrrEzrkzrHzθE8

with

Kc.r2=k02εrrμkz2E9
Kc.θ2=k02εμrrkz2E10
k02=ω2ε0μ0E11

When Eand Hare the electric and magnetic field respectively. εand μare the permittivity and permeability. kzis the propagation constant in z-direction.

In this chapter, we study rigorously the TE and TM modes in this anisotropic waveguide.

2.1 Transverse electric (TE) modes

From Eq. (1), the differential equation for z-component can be obtained as follows

2Hzr2+1rHzr+Kc.θh.μKc.rh.μrr21r22Hzθ2+μrzμrrKc.θh2Hz=0.E12

Using the separation of the variables (r,θ), the expression of the longitudinal magnetic field Hzfor the TEmnmodes in the circular metallic waveguide completely filled with anisotropic metamaterial is necessary for the resolution of the differential Eq. (12). Hzcan be written as follows

Hzh=H0sinKc.θh.μKc.rh.μrrn.θJnμrzμrrKc.θh.rejkzzE13

Jnis the Bessel function of the first kind of order n (n = 0, 1, 2, 3, …).

The expressions (5)(8) become

Erh=jωμ0μKc.r2.rKc.θh.μKc.rh.μrrn.θH0.cosKc.θh.μKc.rh.μrrn.θJnμrzμrrKc.θh.rejkzzE14
Eθh=jωμ0μrrKc.θ2μrzμrrKc.θhH0.sinKc.θh.μKc.rh.μrrn.θJn'μrzμrrKc.θh.rejkzzE15
Hrh=jkzKc.θ2μrzμrrKc.θhH0.sinKc.θh.μKc.rh.μrrn.θJn'μrzμrrKc.θh.rejkzzE16
Hθh=jkzKc.r2.rKc.θh.μKc.rh.μrrnH0.cosKc.θh.μKc.rh.μrrn.θJnμrzμrrKc.θh.rejkzzE17

With Jn'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:

Eθr=R=Ezr=R=0E18

Consequently, from Eq. (15), we obtain

Jn'μrzμrrKc.θh.R=0E19

This implies

unm'=μrzμrrKc.θh.RE20

Where unm'represents the mth zero (m = 1, 2, 3, …) of the derivative of the Bessel function Jn'of the first kind of order n.

The constant H0is determined by normalizing the power flow down the circular guide.

PTE=0R02πErhHθhEθhHrhrdrdθ=1E21

Where indicates the complex conjugate.

Eq. (21) gives

H0=Kc.r3ωμ0kzμrzμNnmhE22

With

Nnmh=1σn2.unm'2n21/2.Jnunm'E23
σn=2π,ifn=0πsin4πa.n4a.n,ifn>0E24
a=Kc.θh.μKc.rh.μrrE25

Finally, the propagation constant in TE mode is given by

kz.nmTE=±k02ε.μrrμrrμrzunm'R2E26

The cutoff frequency is written

fc.nmTE=c2π1εμrz.unm'R.E27

We can introduce the following effective permeability and effective permittivity to describe the propagation characteristics of the waveguide modes [6, 7, 13].

μr,effTE=μrr,E28
εr,effTE=ε11εμrzk02.unm'R2.E29

Further, it is apparent that:

  • kzTE=k0μr,effTE.εr,effTE>0,forμr,effTE>0andεr,effTE>0;

  • kzTE=k0μr,effTE.εr,effTE<0,forμr,effTE<0andεr,effTE<0;

  • kzTE=±jk0μr,effTE.εr,effTE,forμr,effTE.εr,effTE<0.

The sign of εr,effTEdepends on the sign of μrz. In the following, we will consider all cases that arise from the different sign of μrz.

2.1.1 First case μrz>0

For ε>0, we have.

εr,effTE=ε11εμrzk02.unmR2=ε1fc.nmTEf2<0,iff<fc.nmTEE30

And for ε<0, εr,effTEis rewritten as

εr,effTE=ε1+1εμrzk02.unmR2<0.E31

It can be seen that μrz>0leads toεr,effTE<0below the cutoff frequency whenever ε>0orε<0.

2.1.2 Second case μrz<0

Forε>0, εr,effTEis rewritten as

εr,effTE=ε1+1εμrzk02.unmR2>0.E32

And for ε<0, we obtain.

εr,effTE=ε11εμrzk02.unmR2=ε1fc.nmTEf2>0,iff<fc.nmTE.E33

Consequently, μrz<0leads toεr,effTE>0below the cutoff frequency whenever ε>0orε<0.

Therefore, the relative permeability μrzbelow 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 μrr.μrzof the metamaterial below the cutoff frequency determines the sign of the propagation constants of the waveguide studied.

The backward waves are obtained for μrr<0and μrz>0and the forward waves for μrr>0and μrz<0and. Therefore, the backward waves and the forward waves can propagate below the cutoff frequency.

2.2 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

2Ezr2+1rEzr+Kc.re.εKc.θe.εrr21r22Ezθ2+εrzεrrKc.re2Ez=0.E34

Using the separation of the variables (r,θ), the expression of the longitudinal electric field Ezfor the TMnmmodes in the circular metallic waveguide completely filled with anisotropic metamaterial is necessary for the resolution of the differential Eq. (34). Ezcan be written as follows

Eze=E0cosKc.θe.εrrKc.re.εn.θJnεrzεrrKc.re.rejkzzE35

The expressions (5)(8) become

Ere=jkzKc.rεrzεrrE0.cosKc.θe.εrrKc.re.εn.θJn'εrzεrrKc.re.rejkzzE36
Eθe=jkzKc.θ.Kc.rεrrεnrE0.sinKc.θe.εrrKc.re.εn.θJnεrzεrrKc.re.rejkzzE37
Hre=jωε0Kc.θ.Kc.rεεrr.nrE0.sinKc.θe.εrrKc.re.εn.θJnεrzεrrKc.re.rejkzzE38
Hθe=jωε0Kc.rεrrεrz.E0.cosKc.θe.εrrKc.re.εn.θJn'εrzεrrKc.re.rejkzzE39

The boundary condition (18) gives the following equation

Jnunm=0.E40

with

unm=εrzεrrKc.re.R.E41

In Eq. (41)unmrepresents the mth zero (m = 1, 2, 3, …) of the Bessel function Jnof the first kind of order n.

The constant E0is determined by normalizing the power flow down the circular guide.

PTM=0R02πEreHθeEθeHrerdrdθ=1E42

Eq. (42) gives:

E0=Kc.r2ωε0εrrkzNnmeE43

with

Nnme=1unm.Jn'unm.δn2E44
δn=2π, ifn=0πsin4πb.n4b.n, ifn>0E45
b=Kc.θe.εrrKc.re.εE46

Finally, the propagation constant in TM mode is given by:

kz.nmTM=±k02εrr.μεrrεrzunmR2E47

Obviously, the cutoff frequency is written

fc.nmTM=c2π1μεrz.unmR.E48

We can introduce the following effective permeability and effective permittivity to describe the propagation characteristics of the waveguide modes.

εr,effTM=εrr,E49
μr,effTM=μ11μεrzk02.unmR2.E50

Similar to the previous discussion, we have three possibilities:

Further, It is apparent that:

  • kzTM=k0μr,effTM.εr,effTM>0,forμr,effTM>0andεr,effTM>0;

  • kzTM=k0μr,effTM.εr,effTM<0,forμr,effTM<0andεr,effTM<0;

  • kzTM=±jk0μr,effTM.εr,effTM,forμr,effTM.εr,effTM<0.

Consequently, the sign of μr,effTMdepends on the sign of εrz. In the following, we will consider all cases that arise from the different sign of εrz.

2.2.1 Case when εrz>0

In this case, for μ>0, μr,effTMis rewritten as.

μr,effTM=μ11μεrzk02.unmR2=μ1fc.nmTMf2<0,iff<fc.nmTME51

And for μ<0, we have

μr,effTM=μ1+1μεrzk02.unmR2<0.E52

It can be seen that εrz>0leads to μr,effTM<0below the cutoff frequency whenever μ>0,orμ<0.

2.2.2 Case when εrz<0

In this case, for μ>0, we have

μr,effTM=μ1+1μεrzk02.unmR2>0.E53

and for μ<0, we obtain.

μr,effTM=μ11μεrzk02.unmR2=μ1fc.nmTMf2>0.,iff<fc.nmTME54

It is also seen that the relative permittivity εrzwhich is independent of μdetermines the sign of the relative effective permeability μr,effTMof the anisotropic metamaterial in the circular waveguide. The forward wave propagates in the waveguide for εrz<0and εrr>0, and backward wave propagates for εrz>0and εrr<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 εrzand εrrfor TM modes and the sign of μrzand μrrfor TE modes.

2.3 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. aiand biare the incident and the reflected waves, respectively.

Figure 2.

Junction between two circular waveguides filled with two different media having the same cross section.

The transverse electric and magnetic fields (ET,HT) in the wave guides can be written in the modal bases as follows [20]:

ET=m=1Amiami+bmiemiE55
HT=m=1BmiamibmihmiE56

where HTand ETare the transverse magnetic and electric fields (Trefers to the components in the transverse plane), hmi, emirepresent the mth magnetic and electric modal Eigen function in the guide i, respectively and Amiand Bmiare 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:

EtI=EtIIE57
HtI=HtIIE58

By postponing the Eqs. (55) and (56) in (57) and (58), we obtain:

m=1N1AmIamI+bmIemI=p=1N2ApIIapII+bpIIepIIE59
m=1N1BmIamIbmIhmI=p=1N2BpIIapII+bpIIhpIIE60

N1 and N2 are the number of considered modes in guides 1 and 2, respectively. By applying the Galerkin method, Eqs. (59) and (60), lead to the following systems:

m=1N1AmIamI+bmIemI|epII=ApIIapII+bpIIE61
BmIamIbmI=p=1N2BpIIapII+bpIIhpII|hmIE62

The inner product is defined as:

emep=SemepdSE63

The Eqs. (61) and (62) give:

apII+m=1N1AmIApIIamIemI|epII=bpIIm=1N1AmIApIIbmIemI|epIIE64
amI+p=1N2BpIIBmIapIIhpII|hmI=bmI+p=1N2BpIIBmIbpIIhpII|hmIE65

which can be written in matrix form:

UM1M2Ua1I.aN1Ia1II.aN2II=UM1M2Ub1I.bN1Ib1II.bN2IIE66

where U is the identity matrix. M1 and M2 are defined as:

M1ij=BjIIBiIhjIIhiIE67
M2ij=AiIAjIIeiIejIIE68

The scattering matrix of the discontinuity is:

S=UM1M2U1UM1M2UE69

The total scattering matrix is obtained by chaining the S scattering matrices of all the discontinuities in a waveguide having cascaded uniaxial discontinuities [21].

3. Numerical results and discussion

3.1 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 μrror negative μrz. 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 μr=1and permittivity εr=4.4, the fundamental mode presents a resonance frequency of fc.11TE=3.13GHz.

In Figure 3 the curves of the propagation constant, for frequency range 1–10 GHz and for the first five TE modes with μrr=1, μrz=1and ε=4.4, are represented. We observe that all modes propagate without cutoff frequencies (forward waves). Figure 4 represents the same diagrams for μrr=1, μrz=1and ε=4.4. When nand mare small and ωis large, the waves stop propagating. So, these modes propagate at low frequencies and cutoff at high frequencies (backward waves).

Figure 3.

Curves of propagation constant kzTE for TE mode of the circular waveguide completely filled anisotropic metamaterial with parameters μrr=1, μrz=−1, εrθ=4.4.

Figure 4.

Curves of propagation constant kzTE for TE mode of the circular waveguide completely filled anisotropic metamaterial with parameters μrr=−1, μrz=1 and εrθ=4.4.

It is interesting to see that both forward and backward waves can be obtained by controlling the signs of μrzand μrr. 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 εrr=4.4, εrz=4.4and μ=1. All modes propagate without cutoff (forward waves).

Figure 5.

Curves of propagation constant kzTM for TM mode of the circular waveguide completely filled anisotropic metamaterial with parameters εrr=4.4, εrz=−4.4, μrθ=1.

Calculated curves of propagation constant for the frequency range 1–10 GHz and for the first five TM modes with εrr=4.4, εrz=4.4, μ=1are presented. We notice that both forward wave and backward wave can be obtained by controlling the signs of εrrand εrz. Figures 5 and 6 show that our results agree well with the predicted ones.

Figure 6.

Curves of propagation constant kzTM for TM mode of the circular waveguide completely filled anisotropic metamaterial with parameters εrr=−4.4, εrz=4.4, μrθ=1.

We observe that the cutoff frequencies of lowest TE modes decreased with the respect increase of μrzfor μrr=1and ε=4.4(see Figure 7). In a same manner, the TM cutoff frequencies decreased with the respect increase of εrzfor εrr=4.4and μ=1(see Figure 8). Consequently, by varying the parameters of material the propagating mode can be controlled.

Figure 7.

The cutoff frequencies for the first five TE modes versus μrz with μrr=−1, εrθ=4.4.

Figure 8.

The cutoff frequencies for the first five TM modes versus εrz with εrr=−4.4, μrθ=1.

3.2 Filter design

We consider now, 12 discontinuities (see Figure 9) constituted by juxtaposing 13 circular waveguides having the same dimensions (R = 13.4 mm). The circuit is formed by alternation of empty guide (εr=μr=1) of width l = 10 mm and guide filled by anisotropic metamaterials (εrr=ε=εrz=4.4; μrr=μ=μrz=1) of width d = 0.2 mm (periodic structure). Figure 9 represents the geometry of the studied structure.

Figure 9.

Geometry of the circular waveguide with 12 discontinuities.

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.

Figure 10.

Reflection coefficient of the periodic structure with 12 discontinuities.

4. 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 εrzand εrrfor TM modes and by the sign of μrzand μrrfor 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.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite and reference

Link to this chapter Copy to clipboard

Cite this chapter Copy to clipboard

Hedi Sakli and Wyssem Fathallah (March 11th 2020). Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities Filled with Anisotropic Metamaterial, Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering, Patrick Steglich, IntechOpen, DOI: 10.5772/intechopen.91645. Available from:

chapter statistics

146total chapter downloads

More statistics for editors and authors

Login to your personal dashboard for more detailed statistics on your publications.

Access personal reporting

Related Content

This Book

Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering

Edited by Patrick Steglich

Next chapter

Long-Distance and Low-Radiation Waveguide Antennas for Wireless Communication Systems inside Tunnels

By Shotaro Ishino, Satoshi Denno, Narumi Yashiro and Seiichi Suzuki

Related Book

First chapter

Introductory Chapter: Fiber Optics

By Patrick Steglich and Fabio De Matteis

We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.

More About Us