A rigorous Wiener-Hopf approach is used to investigate the band stop filter characteristics of a coaxial waveguide with finite-length impedance loading. The representation of the solution to the boundary-value problem in terms of Fourier integrals leads to two simultaneous modified Wiener-Hopf equations whose formal solution is obtained by using the factorization and decomposition procedures. The solution involves 16 infinite sets of unknown coefficients satisfying 16 infinite systems of linear algebraic equations. These systems are solved numerically and some graphical results showing the influence of the spacing between the coaxial cylinders, the surface impedances, and the length of the impedance loadings on the reflection coefficient are presented.

1. Introduction

Coaxial discontinuity structures are widely used as an element of microwave devices, and in the permeability and permittivity measurement for materials [13]. Hence, the diffraction at discontinuities in coaxial waveguides is a very important topic in microwave theory and have been subjected to numerous past investigations. Most simple types of discontinuities such as steps in inner or outer conductors (see, e.g., [47]) and wall impedance discontinuities [8, 9] were analyzed and characterized. For example, in [8, 9] the scattering of a shielded surface wave in a coaxial waveguide by a wall impedance discontinuity in the inner cylinder has been analyzed. These classical results are related mostly with isolated discontinuities, and fail when there are several of them close enough to interfere with each other. The aim of the present work is to consider a new canonical scattering problem consisting of the propagation of the dominant TEM mode at the finite-length impedance discontinuities in the inner and outer conductors of a coaxial waveguide (see Figure 1). The contributions from the successive impedance discontinuities are accounted for through the solution of a system of coupled modified Wiener-Hopf equations. Notice that the present problem may also be thought as a first-order approximation for coaxial waveguides loaded with a dielectric filled shallow grooves, forming a band stop filter in the microwave region [10]. The band stop properties of such structures are exploited in multiple frequency circuits, such as parametric amplifiers. Constant surface impedance model may also be used to simulate corrugations, thin finite-length dielectric linings on the inner and outer metallic walls of a coaxial waveguide [1113]. The impedance representation is also applicable to ribbed or helical slow-wave structures if the repeat distance of the ribs or helix is small relative to the distance within which the field in the guide changes substantially. Hence the present problem is also of interest when a coaxial phase shifter is used. It consists of a section of coaxial waveguide with a helix cut in the inner rod, which can be screwed to a variable depth into the end of the smooth inner conductor. The length of this screwed section then provides the required phase shift [9].

A problem similar to the one considered in this work has been recently treated by the authors in the simpler case where the impedance loading is present on the outer conductor only [14]. The generalization consisting of assuming that both inner and outer conductors are loaded with different surface impedances of finite-length is not straightforward since we end up with a coupled system of modified Wiener-Hopf equations. It is well known that the classical Wiener-Hopf technique is applicable when the diffracting obstacle is infinitely thin and has a semi-infinite straight boundary. However, since the impedance loadings on the inner and outer conductors of the coaxial waveguide are of finite-length and are different from each other, the resulting Wiener-Hopf equations are modified and coupled, respectively [15].

The method adopted here is similar to that described in [16] where parallel plate waveguides with different impedance loading of finite-length on the upper and lower plates were dealt with. Indeed, by using the analytical properties of the functions that occur we were able to uncouple these modified Wiener-Hopf equations and obtain the exact formal solution through the factorization and decomposition procedures. The formal solution involves 8 sets of infinitely many constants satisfying 8 sets of infinite systems of algebraic equations. An approximate solution to these infinite system of equations is obtained numerically and the radiation characteristics of the impedance-loaded coaxial waveguide is studied.

2. Analysis

Consider a coaxial waveguide whose inner cylinder is of radius 𝜌=𝑎, while the radius of the outer cylinder is 𝜌=𝑏 with (𝜌,𝜙,𝑧) being the usual cylindrical coordinates. The part 0<𝑧<𝑙 of the inner and outer conductors are characterized by different constant surface impedances denoted by 𝑍1=𝜂1𝑍0 and 𝑍2=𝜂2𝑍0, respectively, with 𝑍0 being the characteristic impedance of the free space.

Let the incident TEM mode with angular frequency 𝜔 and propagating in the positive 𝑧 direction be given by 𝐻𝑖𝜌=0,𝐻𝑖𝜙=𝑢𝑖,𝐻𝑖𝑧=0, with 𝑢𝑖=exp(𝑖𝑘𝑧)𝜌,(2.1a) where an exp(𝑖𝜔𝑡) time factor is assumed and suppressed. 𝑘 is the propagation constant which is assumed to have a small imaginary part corresponding to a medium with damping. The lossless case can be obtained by letting Im𝑘0 at the end of the analysis.

2.1. Formulation of the Problem

The total field 𝑢𝑇(𝜌,𝑧) can be written as 𝑢𝑇=𝑢𝑖+𝑢1(𝜌,𝑧),𝜌(𝑎,𝑏),𝑧(,),(2.1b)𝑢1(𝜌,𝑧) appearing in (2.1b) is an unknown function which satisfies the Helmholtz equation 1𝜌𝜕𝜌𝜕𝜕𝜌+𝜕𝜕𝜌2𝜕𝑧2+𝑘21𝜌2𝑢1(𝜌,𝑧)=0,𝜌(𝑎,𝑏),𝑧(,),(2.2) and the following boundary conditions and continuity relations: 𝑢1𝜕(𝑏,𝑧)+𝑏𝑢𝜕𝜌1𝑢(𝑏,𝑧)=0,𝑧(,0)(𝑙,),(2.3a)1𝜕(𝑎,𝑧)+𝑎𝑢𝜕𝜌1𝜂(𝑎,𝑧)=0,𝑧(,0)(𝑙,),(2.3b)2𝑢11(𝑏,𝑧)𝜕𝑖𝑘𝑢𝜕𝜌1𝜂(𝑏,𝑧)=2𝑏𝑒𝑖𝑘𝑧𝜂,𝑧(0,𝑙),(2.3c)1𝑢11(𝑎,𝑧)+𝜕𝑖𝑘𝑢𝜕𝜌1𝜂(𝑎,𝑧)=1𝑎𝑒𝑖𝑘𝑧,𝑧(0,𝑙),(2.3d) where 𝜂1=𝜂1+1,𝜂𝑖𝑘𝑎2=𝜂21.𝑖𝑘𝑏(2.3e) To obtain the unique solution to the mixed boundary value problem stated by (2.2) and (2.3a)–(2.3d) one has to take into account the following edge and radiation conditions [17]: 𝑢𝑇𝑢(𝑎,𝑧)𝑇=𝒪(𝑏,𝑧)|𝑧|1/2𝒪||||,|𝑧|0,𝑧𝑙1/2𝑢,𝑧𝑙,1𝑒(𝜌,𝑧)=𝒪𝑖𝑘|𝑧|,|𝑧|.(2.3f)

The infinite-range Fourier transform of (2.2) yields 1𝜌𝜕𝜌𝜕𝜕𝜌𝜕𝜌+𝐾21(𝛼)𝜌2𝐹(𝜌,𝛼)=0,(2.4a) with 𝐹(𝜌,𝛼)=𝐹(𝜌,𝛼)+𝐹1(𝜌,𝛼)+𝑒𝑖𝛼𝑙𝐹+𝐹(𝜌,𝛼),(2.4b)(𝜌,𝛼)=0𝑢1(𝜌,𝛼)𝑒𝑖𝛼𝑧𝐹𝑑𝑧,(2.4c)1(𝜌,𝛼)=𝑙0𝑢1(𝜌,𝛼)𝑒𝑖𝛼𝑧𝐹𝑑𝑧,(2.4d)+(𝜌,𝛼)=𝑙𝑢1(𝜌,𝛼)𝑒𝑖𝛼(𝑧𝑙)𝑑𝑧.(2.4e) In (2.4a), 𝐾(𝛼) stands for 𝐾(𝛼)=𝑘2𝛼2.(2.5) The square-root function is defined in the complex 𝛼-plane, cut along 𝛼=𝑘 to 𝛼=𝑘+𝑖 and 𝛼=𝑘 to 𝛼=𝑘𝑖, such that 𝐾(0)=𝑘 (see Figure 2).

Owing to the analytical properties of Fourier integrals, 𝐹+(𝜌,𝛼) and 𝐹(𝜌,𝛼) are yet unknown functions which are regular in the half-planes Im(𝛼)>Im(𝑘) and Im(𝛼)<Im(𝑘), respectively. The function 𝐹1(𝜌,𝛼) defined by (2.4d) is an unknown entire function.

The solution of (2.4a) reads 𝐹(𝜌,𝛼)+𝐹1(𝜌,𝛼)+𝑒𝑖𝛼𝑙𝐹+(𝜌,𝛼)=𝐴(𝛼)𝐽1(𝐾𝜌)+𝐵(𝛼)𝑌1(𝐾𝜌).(2.6) Here 𝐴(𝛼) and 𝐵(𝛼) are spectral coefficients to be determined. By using the Fourier transform of the boundary conditions in (2.3a) and (2.3b), one can easily show that they are related to 𝑃1(𝑎,𝛼) and 𝑃1(𝑏,𝛼) through =𝐴(𝛼)𝐵(𝛼)𝐾(𝛼)𝑌0(𝐾𝑏)𝑎𝑇1(𝑎,𝑏,𝛼)𝐾(𝛼)𝑌0(𝐾𝑎)𝑏𝑇1(𝑎,𝑏,𝛼)𝐾(𝛼)𝐽0(𝐾𝑏)𝑎𝑇1(𝑎,𝑏,𝛼)𝐾(𝛼)𝐽0(𝐾𝑎)𝑏𝑇1𝑃(𝑎,𝑏,𝛼)1𝑃(𝑎,𝛼)1(𝑏,𝛼),(2.7a) with 𝑃1(𝜌,𝛼) being given by 𝑃1(𝜌,𝛼)=𝐹1̇𝐹(𝜌,𝛼)+𝜌1(𝜌,𝛼).(2.7b) In (2.7b), the dot () denotes the derivative with respect to 𝜌, that is, ̇𝐹1(𝜌,𝛼)=(𝜕/𝜕𝜌)𝐹1(𝜌,𝛼). Hence, (2.6) can be rearranged as 𝐹(𝜌,𝛼)+𝐹1(𝜌,𝛼)+𝑒𝑖𝛼𝑙𝐹+𝑇(𝜌,𝛼)=2(𝜌,𝑏,𝛼)𝑎𝑇1(𝑃𝑎,𝑏,𝛼)1𝑇(𝑎,𝛼)2(𝜌,𝑎,𝛼)𝑏𝑇1(𝑃𝑎,𝑏,𝛼)1(𝑏,𝛼),(2.8a) with 𝑇1(𝑎,𝑏,𝛼)=𝐾2𝐽(𝛼)0(𝐾𝑎)𝑌0(𝐾𝑏)𝐽0(𝐾𝑏)𝑌0,𝑇(𝐾𝑎)2𝐽(𝑎,𝑏,𝛼)=𝐾(𝛼)1(𝐾𝑎)𝑌0(𝐾𝑏)𝐽0(𝐾𝑏)𝑌1.(𝐾𝑎)(2.8b) Multiplying both sides of (2.3c) and (2.3d) by 𝑒𝑖𝛼𝑧 and then integrating from 0 to 𝑙 we obtain 𝜂2𝐹1𝜂(𝑏,𝛼)=2𝑒𝑖𝑏(𝑘+𝛼)𝑖(𝑘+𝛼)𝑙+11̇𝐹𝑖𝑘1𝜂(𝑏,𝛼),1𝐹1𝜂(𝑎,𝛼)=1𝑒𝑖𝑎(𝑘+𝛼)𝑖(𝑘+𝛼)𝑙11̇𝐹𝑖𝑘1(𝑎,𝛼).(2.9) Putting𝜌=𝑏 and then 𝜌=𝑎 in (2.8a) and its derivative with respect to 𝜌, and then using the relations (2.9) we end up with the following coupled systems of modified Wiener-Hopf equations valid in the strip Im(𝑘)<Im(𝛼)<Im(𝑘): 𝑖𝑘𝑎Ψ1(𝑎,𝛼)+𝑉1(𝛼)𝑃1(𝑎,𝛼)𝑖𝑘𝑎𝑒𝑖𝛼𝑙Ψ+1(𝑎,𝛼)=2𝑖𝑘𝜂1𝜋𝑏𝑇1𝑃(𝑎,𝑏,𝛼)1(𝑏,𝛼)𝑘𝜂1𝑒𝑖(𝑘+𝛼)𝑙1,𝑘+𝛼𝑖𝑘𝑏Ψ2(𝑏,𝛼)𝑉2(𝛼)𝑃1(𝑏,𝛼)𝑖𝑘𝑏𝑒𝑖𝛼𝑙Ψ+2(𝑏,𝛼)=2𝑖𝑘𝜂2𝜋𝑎𝑇1(𝑃𝑎,𝑏,𝛼)1(𝑎,𝛼)𝑘𝜂2𝑒𝑖(𝑘+𝛼)𝑙1,𝑘+𝛼(2.10) with, 𝑉1𝒳(𝛼)=1(𝛼)𝑇1,𝑉(𝑎,𝑏,𝛼)2𝒳(𝛼)=2(𝛼)𝑇1,𝒳(𝑎,𝑏,𝛼)(2.11a)1(𝛼)=𝑇1(𝑎,𝑏,𝛼)+𝑖𝑘𝜂1𝑇2𝒳(𝑎,𝑏,𝛼),2(𝛼)=𝑇1(𝑎,𝑏,𝛼)+𝑖𝑘𝜂2𝑇2Ψ(𝑏,𝑎,𝛼),1(𝛼)=𝜂1𝐹1(𝑎,𝛼)+̇𝐹𝑖𝑘Ψ(𝑎,𝛼),+1(𝛼)=𝜂1𝐹+1(𝑎,𝛼)+̇𝐹𝑖𝑘+Ψ(𝑎,𝛼),2(𝛼)=𝜂2𝐹1(𝑏,𝛼)̇𝐹𝑖𝑘Ψ(𝑏,𝛼),+2(𝛼)=𝜂2𝐹+1(𝑎,𝛼)̇𝐹𝑖𝑘+(𝑏,𝛼).(2.11b)

2.2. Solution of the Wiener-Hopf Equations

Now, let us rearrange the equations in (2.10) as 𝑅1(𝛼)𝑉1(𝛼)𝑃1(𝑎,𝛼)+𝑒𝑖𝛼𝑙𝑆+1(𝛼)=2𝑖𝑘𝜂1𝜋𝑏𝑇1𝑃(𝑎,𝑏,𝛼)1𝑅(𝑏,𝛼),(2.12a)2(𝛼)+𝑉2(𝛼)𝑃1(𝑏,𝛼)+𝑒𝑖𝛼𝑙𝑆+2(𝛼)=2𝑖𝑘𝜂2𝜋𝑎𝑇1(𝑃𝑎,𝑏,𝛼)1(𝑎,𝛼).(2.12b) with 𝑅1(𝛼)=𝑎𝑖𝑘Ψ1(𝛼)+𝑘𝜂1,𝑅(𝑘+𝛼)2(𝛼)=𝑏𝑖𝑘Ψ2(𝛼)+𝑘𝜂2,𝑆(𝑘+𝛼)+1(𝛼)=𝑎𝑖𝑘Ψ+1(𝛼)𝑘𝜂1𝑒(𝑘+𝛼)𝑖𝑘𝑙,𝑆+2(𝛼)=𝑏𝑖𝑘Ψ+2(𝛼)𝑘𝜂2𝑒(𝑘+𝛼)𝑖𝑘𝑙.(2.13) Notice that 𝑅1(𝛼) and 𝑅2(𝛼) are regular functions of 𝛼 in the lower half-plane Im(𝛼)<Im(𝑘) except at the pole singularity occurring at 𝛼=𝑘, while 𝑆+1(𝛼) and 𝑆+2(𝛼) are regular in the upper half-plane Im(𝛼)>Im(𝑘).

Now, consider the Wiener-Hopf factorization of the kernel functions 𝑉1,2(𝛼), defined in (2.11a) as 𝑉1,2(𝛼)=𝑉+1,2(𝛼)𝑉1,2(𝛼),(2.14a) where 𝑉+1,2(𝛼) and 𝑉1,2(𝛼) denote certain functions which are regular and free of zeros in the half-planes Im(𝛼)>Im(𝑘) and Im(𝛼)<Im(𝑘), respectively. Their explicit expressions can easily be found by using the method described in [18] 𝑉±1,2(𝛼)=𝑉1,21(0)1𝛼/𝜉1𝑚=11𝛼/𝛽𝑚(1,2)1𝛼/𝜉𝑚+1.(2.14b) Here ±𝛽𝑚(1) and ±𝛽𝑚(2),𝑚=1,2,, denote the symmetrical zeros of 𝑉1(𝛼) and 𝑉2(𝛼): 𝑉1,2±𝛽𝑚(1,2)=0,Im𝛽𝑚(1,2)>Im𝑘,(2.15) whereas ±𝜉𝑚,𝑚=1,2, are the zeros of 𝑇1(𝑎,𝑏,𝛼), that is, 𝑇1𝑎,𝑏,±𝜉𝑚=0,Im𝜉𝑚>Im𝑘.(2.16)

Multiplying both sides of (2.12a) by 1/𝑉1(𝛼) and 𝑒𝑖𝛼𝑙/𝑉+1(𝛼), and both sides of (2.12b) by 1/𝑉2(𝛼) and 𝑒𝑖𝛼𝑙/𝑉+2(𝛼), we obtain, after using the Wiener-Hopf decomposition procedure, the analytical continuation principle, and the Liouville theorem, the following results [19]: 𝑅1(𝛼)𝑉11(𝛼)2𝜋𝑖𝑆+1(𝜏)𝑒𝑖𝜏𝑙𝑑𝜏𝑉1(𝜏)(𝜏𝛼)2𝑖𝑘𝜂11𝜋𝑏2𝜋𝑖𝑃1(𝑏,𝜏)𝑑𝜏𝑇1(𝑎,𝑏,𝜏)𝑉1=(𝜏)(𝜏𝛼)𝑘𝜂1(𝑘+𝛼)𝑉+1,𝑉(𝑘)+1(𝛼)𝑃11(𝑎,𝛼)2𝜋𝑖+𝑆+1(𝜏)𝑒𝑖𝜏𝑙𝑑𝜏𝑉1(𝜏)(𝜏𝛼)2𝑖𝑘𝜂11𝜋𝑏2𝜋𝑖+𝑃1(𝑏,𝜏)𝑑𝜏𝑇1(𝑎,𝑏,𝜏)𝑉1(=𝜏)(𝜏𝛼)𝑘𝜂1(𝑘+𝛼)𝑉+1,𝑅(𝑘)2(𝛼)𝑉2(1𝛼)2𝜋𝑖𝑆+2(𝜏)𝑒𝑖𝜏𝑙𝑑𝜏𝑉2(+𝜏)(𝜏𝛼)2𝑖𝑘𝜂21𝜋𝑎2𝜋𝑖𝑃1(𝑎,𝜏)𝑑𝜏𝑇1(𝑎,𝑏,𝜏)𝑉2(=𝜏)(𝜏𝛼)𝑘𝜂2(𝑘+𝛼)𝑉+2(,𝑘)𝑉+2(𝛼)𝑃11(𝑏,𝛼)2𝜋𝑖+𝑆+2(𝜏)𝑒𝑖𝜏𝑙𝑑𝜏𝑉2(+𝜏)(𝜏𝛼)2𝑖𝑘𝜂21𝜋𝑎2𝜋𝑖+𝑃1(𝑎,𝜏)𝑑𝜏𝑇1(𝑎,𝑏,𝜏)𝑉2(=𝜏)(𝜏𝛼)𝑘𝜂2(𝑘+𝛼)𝑉+2(,𝑆𝑘)+1(𝛼)𝑉+1(1𝛼)=2𝜋𝑖+𝑅1(𝜏)𝑒𝑖𝜏𝑙𝑑𝜏𝑉+1(𝜏)(𝜏𝛼)2𝑖𝑘𝜂11𝜋𝑏2𝜋𝑖+𝑃1(𝑏,𝜏)𝑒𝑖𝜏𝑙𝑑𝜏𝑇1(𝑎,𝑏,𝜏)𝑉+1(,𝑉𝜏)(𝜏𝛼)1(𝛼)𝑒𝑖𝜏𝑙𝑃11(𝑎,𝛼)=2𝜋𝑖𝑅1(𝜏)𝑒𝑖𝜏𝑙𝑑𝜏𝑉+1(𝜏)(𝜏𝛼)2𝑖𝑘𝜂11𝜋𝑏2𝜋𝑖𝑃1(𝑏,𝜏)𝑒𝑖𝜏𝑙𝑑𝜏𝑇1(𝑎,𝑏,𝜏)𝑉+1,𝑆(𝜏)(𝜏𝛼)+2(𝛼)𝑉+21(𝛼)=2𝜋𝑖+𝑅2(𝜏)𝑒𝑖𝜏𝑙𝑑𝜏𝑉+2+(𝜏)(𝜏𝛼)2𝑖𝑘𝜂21𝜋𝑎2𝜋𝑖+𝑃1(𝑎,𝜏)𝑒𝑖𝜏𝑙𝑑𝜏𝑇1(𝑎,𝑏,𝜏)𝑉+2,(𝜏)(𝜏𝛼)𝑉2(𝛼)𝑒𝑖𝜏𝑙𝑃11(𝑏,𝛼)=2𝜋𝑖𝑅2(𝜏)𝑒𝑖𝜏𝑙𝑑𝜏𝑉+2+(𝜏)(𝜏𝛼)2𝑖𝑘𝜂21𝜋𝑎2𝜋𝑖𝑃1(𝑎,𝜏)𝑒𝑖𝜏𝑙𝑑𝜏𝑇1(𝑎,𝑏,𝜏)𝑉+2.(𝜏)(𝜏𝛼)(2.17) The positions of the integration lines ± are shown in Figure 2.

The above integrals can be evaluated by using Jordan’s Lemma and the residue theorem. The result is 𝑅1(𝛼)𝑉1(𝛼)2𝑖𝑘𝜂1𝜋𝑏𝑚=1𝑉+1𝛽𝑚(1)𝑃1𝑏,𝛽𝑚(1)𝒳1𝛽𝑚(1)𝛽𝑚(1)𝛼𝑚=1𝑉+1𝛽𝑚(1)𝑒𝑖𝛽𝑚(1)𝑙𝑆+1𝛽𝑚(1)𝑉1𝛽𝑚(1)𝛽𝑚(1)=𝛼𝑖𝑘𝜂1𝑖(𝑘+𝛼)𝑉+1𝑅(𝑘),(2.18a)2(𝛼)𝑉2+(𝛼)2𝑖𝑘𝜂2𝜋𝑎𝑚=1𝑉+2𝛽𝑚(2)𝑃1𝑎,𝛽𝑚(2)𝒳2𝛽𝑚(2)𝛽𝑚(2)𝛼𝑚=1𝑉+2𝛽𝑚(2)𝑒𝑖𝛽𝑚(2)𝑙𝑆+2𝛽𝑚(2)𝑉2𝛽𝑚(2)𝛽𝑚(2)=𝛼𝑖𝑘𝜂2𝑖(𝑘+𝛼)𝑉+2𝑆(𝑘),(2.18b)+1(𝛼)𝑉+1(𝛼)2𝑖𝑘𝜂1𝜋𝑏𝑚=1𝑉+1𝛽𝑚(1)𝑒𝑖𝛽𝑚(1)𝑙𝑃1𝑏,𝛽𝑚(1)𝒳1𝛽𝑚(1)𝛽𝑚(1)+𝛼𝑚=1𝑉+1𝛽𝑚(1)𝑒𝑖𝛽𝑚(1)𝑙𝑅1𝛽𝑚(1)𝑉1𝛽𝑚(1)𝛽𝑚(1)𝑆+𝛼=0,(2.18c)+2(𝛼)𝑉+2+(𝛼)2𝑖𝑘𝜂2𝜋𝑎𝑚=1𝑉+2𝛽𝑚(2)𝑒𝑖𝛽𝑚(2)𝑙𝑃1𝑎,𝛽𝑚(2)𝒳2𝛽𝑚(2)𝛽𝑚(2)+𝛼𝑚=1𝑉+2𝛽𝑚(2)𝑒𝑖𝛽𝑚(2)𝑙𝑅2𝛽𝑚(2)𝑉2𝛽𝑚(2)𝛽𝑚(2)+𝛼=0,(2.18d)2𝑖𝑘𝜂1𝜋𝑏𝑚=1𝑉+1𝛽𝑚(1)𝑃1𝑏,𝛽𝑚(1)𝒳1𝛽𝑚(1)𝛽𝑚(1)𝛼2𝑖𝑘𝜂1𝑉𝜋𝑏+1(𝛼)𝑃1(𝑏,𝛼)𝒳1(𝛼)+𝑉+1(𝛼)𝑃1(𝑎,𝛼)𝑚=1𝑉+1𝛽𝑚(1)𝑒𝑖𝛽𝑚(1)𝑙𝑆+1𝛽𝑚(1)𝑉1𝛽𝑚(1)𝛽𝑚(1)𝑉𝛼+1(𝛼)𝑒𝑖𝛼𝑙𝑆+1(𝛼)𝑉1(𝛼)=𝑘𝜂11(𝑘+𝛼)𝑉+1,𝑉(𝑘)(2.18e)+2(𝛼)𝑃1(𝑏,𝛼)+2𝑖𝑘𝜂2𝜋𝑎𝑚=1𝑉+2𝛽𝑚(2)𝑃1𝑎,𝛽𝑚(2)𝒳2𝛽𝑚(2)𝛽𝑚(2)+𝛼2𝑖𝑘𝜂2𝑉𝜋𝑎+2(𝛼)𝑃1(𝑎,𝛼)𝒳2(𝛼)𝑚=1𝑉+2(𝛽𝛼)𝑚(2)𝑒𝑖𝛽𝑚(2)𝑙𝑆+2𝛽𝑚(2)𝑉2𝛽𝑚(2)𝛽𝑚(2)𝑉𝛼+2(𝛼)𝑒𝑖𝛼𝑙𝑆+2(𝑎,𝛼)𝑉2(𝛼)=𝑘𝜂21(𝑘+𝛼)𝑉+2,𝑒(𝑘)(2.18f)𝑖𝛼𝑙𝑉1(𝛼)𝑃1(𝑎,𝛼)2𝑖𝑘𝜂1𝜋𝑏𝑚=1𝑉+1𝛽𝑚(1)𝑒𝑖𝛽𝑚(1)𝑙𝑃1𝑏,𝛽𝑚(1)𝒳1𝛽𝑚(1)𝛽𝑚(1)+𝛼2𝑖𝑘𝜂1𝑉𝜋𝑏1(𝛼)𝑒𝑖𝛼𝑙𝑃1(𝑏,𝛼)𝒳1(𝛼)𝑚=1𝑉+1𝛽𝑚(1)𝑒𝑖𝛽𝑚(1)𝑙𝑅1𝛽𝑚(1)𝑉1𝛽𝑚(1)𝛽𝑚(1)𝑉+𝛼1(𝛼)𝑒𝑖𝛼𝑙𝑅1(𝛼)𝑉1(𝛼)=0,(2.18g)𝑒𝑖𝛼𝑙𝑉2(𝛼)𝑃1(𝑏,𝛼)+2𝑖𝑘𝜂2𝜋𝑎𝑚=1𝑉+2𝛽𝑚(2)𝑒𝑖𝛽𝑚(2)𝑙𝑃1𝑎,𝛽𝑚(2)𝒳2𝛽𝑚(2)𝛽𝑚(2)++𝛼2𝑖𝑘𝜂2𝑉𝜋𝑎2(𝛼)𝑒𝑖𝛼𝑙𝑃1(𝑎,𝛼)𝒳2(+𝛼)𝑚=1𝑉+2𝛽𝑚(2)𝑒𝑖𝛽𝑚(2)𝑙𝑅2𝛽𝑚(2)𝑉2𝛽𝑚(2)𝛽𝑚(2)𝑉+𝛼2(𝛼)𝑒𝑖𝛼𝑙𝑅2(𝛼)𝑉2(𝛼)=0.(2.18h) Here the dash () denotes the derivative with respect to 𝛼𝑉1,2(𝛽𝑚(1,2))=(𝜕/𝜕𝛼)𝑉1,2(𝛼)|𝛼=𝛽𝑚(1,2).

The formal solution of the coupled system of Wiener-Hopf equations given by (2.18a)–(2.18h) involves 16 unknown functions, namely, 𝑃1(𝑎,𝛽𝑚(1)),𝑃1(𝑎,𝛽𝑚(2)),𝑃1(𝑎,𝛽𝑚(1)),𝑃1(𝑎,𝛽𝑚(2)),𝑃1(𝑏,𝛽𝑚(1)),𝑃1(𝑏,𝛽𝑚(2)),𝑃1(𝑏,𝛽𝑚(1)),𝑃1(𝑏,𝛽𝑚(2)), 𝑆+1(𝛽𝑚(1)),𝑆+1(𝛽𝑚(2)),𝑆+2(𝛽𝑚(1)),𝑆+2(𝛽𝑚(2)),𝑅1(𝛽𝑚(1)), 𝑅1(𝛽𝑚(2)),𝑅2(𝛽𝑚(1)), and 𝑅2(𝛽𝑚(2)). In order to determine them we substitute 𝛼=𝛽𝑛(1),𝛽𝑛(2),𝑛=1,2, in (2.18a), (2.18b), (2.18g), (2.18h) and 𝛼=𝛽𝑛(1),𝛽𝑛(2),𝑛=1,2, in (2.18c), (2.18d), (2.18e), (2.18f), respectively. So, at a first sight, we get 16 infinite systems of algebraic equations. After some straightforward but tedious algebraic manipulations, one can reduce the number of the unkown sets to 8, namely, 𝑃1(𝑎,±𝛽𝑚(2)),𝑃1(𝑏,±𝛽𝑚(1)),𝑅1(𝛽𝑚(1)),𝑅1(𝛽𝑚(2)),𝑅2(𝛽𝑚(1))and 𝑅2(𝛽𝑚(2)). These coupled systems of algebraic equations will be solved numerically. The approach used in solving the infinite system of algebraic equations is similar to that employed by Rawlins [20]. By using the edge conditions and the asymptotic behavior of 𝛽𝑛(1)and 𝛽𝑛(2), one can show that the convergence of the infinite series appearing in these equations is rapid enough to allow truncation at, say 𝑁. Consequently the infinite systems are replaced by the corresponding finite systems of 8𝑁×8𝑁 algebraic equations and then solved by standard numerical algorithms. The value of 𝑁 was increased until the reflected field amplitude being calculated did not change in a given number of decimal places. A typical result is provided by Figure 3. It can be seen that the reflected field amplitude becomes insensitive to the increase of the truncation number for 𝑁>6.

3. Scattered Field

For 𝜌(𝑎,𝑏) and 𝑧<0, the reflected wave propagating backward is obtained by taking the inverse Fourier transform of 𝐹(𝜌,𝛼), namely, 𝑢11(𝜌,𝛼)=2𝜋𝑏𝑇2(𝜌,𝑏,𝛼)𝑃1(𝑎,𝛼)𝑎𝑇2(𝜌,𝑎,𝛼)𝑃1(𝑏,𝛼)𝑎𝑏𝑇1(𝑎,𝑏,𝛼)𝐹1(𝜌,𝛼)𝑒𝑖𝛼𝑙𝐹+𝑒(𝜌,𝛼)𝑖𝛼𝑧𝑑𝛼.(3.1) The above integral is calculated by closing the contour in the upper half plane and evaluating the residue contributions from the simple poles occurring at the zeros of 𝑇1(𝑎,𝑏,𝛼) lying in the upper 𝛼-half-plane. The reflection coefficient of the fundamental mode is defined as the ratio of reflected and incident waves amplitudes. It is computed from the contribution of the first pole at 𝛼=𝑘. The result is 𝑖=2𝑎𝑏𝑘log(𝑏/𝑎)𝑏𝑃1(𝑎,𝑘)𝑎𝑃1(𝑏,𝑘).(3.2)

Similarly, the transmitted field in the region 𝜌(𝑎,𝑏) and 𝑧>𝑙 is obtained by inverting 𝐹+(𝜌,𝛼): 𝑢11(𝜌,𝛼)=2𝜋𝑏𝑇2(𝜌,𝑏,𝛼)𝑃1(𝑎,𝛼)𝑎𝑇2(𝜌,𝑎,𝛼)𝑃1(𝑏,𝛼)𝑎𝑏𝑇1(𝑎,𝑏,𝛼)𝐹(𝜌,𝛼)𝐹1𝑒(𝜌,𝛼)𝑖𝛼𝑧𝑑𝛼.(3.3) The transmission coefficient 𝒯 of the fundamental mode, which is defined as the ratio of the amplitudes of the transmitted and incident waves, is obtained by evaluating the integral in (3.3) for 𝑧>𝑙. This integral is now computed by closing the contour in the lower half of the complex 𝛼-plane. The pole of interest is at 𝛼=𝑘 whose contribution gives 𝑖2𝑎𝑏𝑘log(𝑏/𝑎)𝑏𝑃1(𝑎,𝑘)𝑎𝑃1𝑒(𝑏,𝑘)𝑖𝑘𝑧𝜌,(3.4a) with 𝑏𝑃1(𝑎,𝑘)𝑎𝑃1=(𝑏,𝑘)2𝑎𝑏𝑘log(𝑏/𝑎)𝑖𝑒𝑖𝑘𝑙𝑉+1(𝑘)2𝑖𝑘𝜂1𝜋𝑚=1𝑉+1𝛽𝑚(1)𝑒𝑖𝛽𝑚(1)𝑙𝑃1𝑏,𝛽𝑚(1)𝒳1𝛽𝑚(1)𝛽𝑚(1)𝑘+𝑏𝑚=1𝑉+1𝛽𝑚(1)𝑒𝑖𝛽𝑚(1)𝑙𝑅1𝛽𝑚(1)𝑉1𝛽𝑚(1)𝛽𝑚(1),𝑘(3.4b) or equivalently 𝑏𝑃1(𝑎,𝑘)𝑎𝑃1=(𝑏,𝑘)2𝑎𝑏𝑘log(𝑏/𝑎)𝑖𝑒𝑖𝑘𝑙𝑉+2(𝑘)2𝑖𝑘𝜂2𝜋𝑚=1𝑉+2𝛽𝑚(2)𝑒𝑖𝛽𝑚(2)𝑙𝑃1𝑎,𝛽𝑚(2)𝒳2𝛽𝑚(2)𝛽𝑚(2)𝑘+𝑎𝑚=1𝑉+2𝛽𝑚(2)𝑒𝑖𝛽𝑚(2)𝑙𝑅2𝛽𝑚(2)𝑉2𝛽𝑚(2)𝛽𝑚(2).𝑘(3.4c) The first term in (3.4a) or in (3.4b) cancels out the incident fundamental mode as expected. The transmission coefficient is then given by 𝑖𝒯=12𝑎𝑏𝑘log(𝑏/𝑎)𝑏𝑃1(𝑎,𝑘)𝑎𝑃1(𝑏,𝑘).(3.5)

4. Computational Results

In order to observe the influence of the different parameters such as the surface impedances (𝜂1 and 𝜂2), the width (𝑙) of the impedance loadings, and the distance 𝑏𝑎 between the two coaxial cylinders on the reflection coefficient, some numerical results are presented in this section. In what follows the impedances are assumed to be purely reactive.

Figures 4(a)4(c) show the amplitude of the reflection coefficient versus the frequency for different values of 𝜂2 while 𝜂1,𝑎,𝑏, and 𝑙 are held fixed. If 𝜂1 and 𝜂2 are both capacitive, the band-stop frequencies of the configuration are shifted to the right for increasing values of |𝜂2| (Figure 4(a)). In the case where 𝜂1 is capacitive while 𝜂2 is inductive, the band stop characteristic of the configuration is observed at lower frequencies for increasing the values of |𝜂2| (Figure 4(b)). Finally, Figure 4(c) displays the case where the both surface impedances are inductive.

Figure 5 depicts the variation of the reflected field amplitude for different values of distance between the coaxial cylinders. It is observed that the amplitude of the reflected field decreases when the separation distance 𝑏𝑎 increases, as expected. Furthermore, for decreasing values of the outer cylinder radius, the band-stop frequencies of the configuration are shifted to the right. This means that total reflections occur at higher frequencies if the radius of the outer cylinder decreases.

The effect of the width for the impedance loadings on the reflection coefficient is shown in Figure 6. The number of resonances corresponding to the variation of the reflected field increases when the value of 𝑙 increases. But the amplitude related to the reflected field is not affected too much by the width of the impedance surfaces. It is also seen that the curves related to reflected field amplitude approaches to the one calculated from (5.2a), when 𝑘𝑙1.

5. Discussion

When we let 𝑙, the Wiener- Hopf equations in (2.10) reduce to 𝑖𝑘𝑎Ψ1(𝑎,𝛼)+𝑉1(𝛼)𝑃1(𝑎,𝛼)=2𝑖𝑘𝜂1𝜋𝑏𝑇1𝑃(𝑎,𝑏,𝛼)1(𝑏,𝛼)+𝑘𝜂1,𝑘+𝛼(5.1a)𝑖𝑘𝑏Ψ2(𝑏,𝛼)𝑉2(𝛼)𝑃1(𝑏,𝛼)=2𝑖𝑘𝜂2𝜋𝑎𝑇1(𝑃𝑎,𝑏,𝛼)1(𝑎,𝛼)+𝑘𝜂2𝑘+𝛼.(5.1b) Then, the reflection coefficient in (3.2) takes the following form: =𝑖2𝑎𝑏𝑘log(𝑏/𝑎)𝑉+1(𝑘)2𝑖𝑘𝜂1𝜋𝑚=1𝑉+1𝛽𝑚(1)𝑃1𝑏,𝛽𝑚(1)𝜒1𝛽𝑚(1)𝛽𝑚(1)+𝑘𝑏𝜂12𝑉+1(𝑘),(5.2a) or =𝑖2𝑎𝑏𝑘log(𝑏/𝑎)𝑉+2(𝑘)2𝑖𝑘𝜂2𝜋𝑚=1𝑉+2𝛽𝑚(2)𝑃1𝑎,𝛽𝑚(2)𝜒2𝛽𝑚(2)𝛽𝑚(2)+𝑘𝑎𝜂22𝑉+2(𝑘),(5.2b) which is nothing but the results related to the junction of perfectly conducting and impedance coaxial waveguides shown in Figure 7.

Equations (5.2a) and (5.2b) are obtained by using the Wiener-Hopf equations in (5.1a) and (5.1b), respectively. It can be checked easily that the two expressions are equivalent. 𝑃1(𝑏,𝛽𝑚(1)) and 𝑃1(𝑎,𝛽𝑚(2)) appearing in the above expressions are to be determined through the solution of the following 2 sets of infinite systems of algebraic equations: 𝑉+1𝛽𝑛(2)𝑃1𝑎,𝛽𝑛(2)+4𝑘2𝜂1𝜂2𝜋2𝑉𝑎𝑏+1𝛽𝑛(2)𝒳1𝛽𝑛(2)𝑉+2𝛽𝑛(2)𝑚=1𝑚𝑛𝑉+2𝛽𝑚(2)𝑃1𝑎,𝛽𝑚(2)𝒳2𝛽𝑚(2)𝛽𝑚(2)𝛽𝑛(2)2𝑖𝑘𝜂1𝜋𝑏𝑚=1𝑉+1𝛽𝑚(1)𝑃1𝑏,𝛽𝑚(1)𝒳1𝛽𝑚(1)𝛽𝑚(1)𝛽𝑛(2)=𝑘𝜂1𝑘+𝛽𝑛(2)𝑉+1(𝑘)2𝑖𝑘2𝜂1𝜂2𝑉+1𝛽𝑛(2)𝑘+𝛽𝑛(2)𝒳1𝛽𝑛(2)𝑉+2𝛽𝑛(2)𝑉+2,(𝑘)𝑉+2𝛽𝑛(1)𝑃1𝑏,𝛽𝑛(1)4𝑘2𝜂1𝜂2𝜋2𝑉𝑎𝑏+2𝛽𝑛(1)𝒳2𝛽𝑛(1)𝑉+1𝛽𝑛(1)𝑚=1𝑚𝑛𝑉+1𝛽𝑚(1)𝑃1𝑏,𝛽𝑚(1)𝒳1𝛽𝑚(1)𝛽𝑚(1)𝛽𝑛(1)+2𝑖𝑘𝜂2𝜋𝑎𝑚=1𝑉+2𝛽𝑚(2)𝑃1𝑎,𝛽𝑚(2)𝒳2𝛽𝑚(2)𝛽𝑚(2)𝛽𝑛(1)=𝑘𝜂2𝑘+𝛽𝑛(1)𝑉+2(𝑘)2𝑖𝑘2𝜂1𝜂2𝑉+2𝛽𝑛(1)𝑘+𝛽𝑛(1)𝒳2𝛽𝑛(1)𝑉+1𝛽𝑛(1)𝑉+1.(𝑘)(5.3)

Finally, for |𝜂2|0, the results obtained in the present work to that previously obtained in [3]. Indeed, from Figure 8 we can see that for decreasing values of |𝜂2|, the curves approach the result related to the finite-length impedance loading in the outer conductor of a coaxial waveguide obtained in [3] (solid line in Figure 8). This can be considered as a check of the analysis made in this paper.

6. Conclusion

In the present work the propagation of TEM wave in a coaxial waveguide with finite-length impedance loading is investigated rigorously through the Wiener-Hopf technique. In order to obtain the explicit expressions of the reflection coefficient, the problem is first reduced into two coupled modified Wiener-Hopf equations and then solved exactly in a formal sense by using the factorization and decomposition procedures. The formal solution involves infinite series with 8 sets of unknown coefficients satisfying 8 infinite sets of algebraic equations which are solved numerically. The advantage of the present method is that the solution obtained here is valid for all frequencies and impedance lengths. Furthermore, it is observed that for certain values of the surface impedances full reflection occurs, showing that this configuration may be used as a band-stop filter.

Finally, it is noteworthy that the Weiner-Hopf solution provided here could be extended to treat the case where the lengths of the impedances on the inner and outer conductors are different. Other future work could lie in the investigation of wave propagation in coaxial waveguides with successive finite-length impedance loadings.