TEM-wave propagation over an impedance-matched periodic RHM to LHM transition in a coaxial waveguide

Balwan Rana; Mariana Dalarsson

doi:10.1364/OME.481637

1. Introduction

During the last two decades, there has been an extraordinary rise in the theoretical and experimental attention given to composite metamaterials, including LHMs. The superior performance of graded RHM-LHM composites in many practical situations is well documented in the literature (see e.g. [1]), despite some occasionally perceived difficulties to experimentally realize LHMs, such as the inherent strong dispersion that comes with negative material parameters. Graded composite structures are built using carefully chosen continuous spatial coordinate functions that describe the permittivity and permeability of the underlying materials. The usage of graded composite structures is widespread, from the manipulation of light beams in optics to the design of vibration-resistant building materials in mechanics [2–5]. The performance of graded composite metamaterials has further led to an interest in the possibility of periodic graded composite metamaterials and their limitations.

Composites with spatially varying permittivities and permeabilities are of interest in several current state-of-the-art research areas. One such central area is transformation optics [1], with unprecedented design flexibility allowing for the creation of novel devices, such as light source collimators, waveguide adapters, and waveguide crossings, useful in integrated photonic chips and compatible with modern fabrication technology. The superior optical performance of the abovementioned devices, along with their efficient integration with other components in an on-chip photonic system, is numerically confirmed in [1]. Transformation optics-based components require spatially-varying dielectric materials only, with no specific magnetic properties. This enables their use in low-loss, broadband, and integrated photonic applications.

Other applications of transformation optics include hyperlenses [6,7], specialized antennas [8,9] as well as subwavelength imaging [10,11]. A further area of interest is electromagnetic cloaking. The principles of transformation optics with metamaterials are considered key enablers for the possibility of manipulating surface waves, in particular, towards the THz and optical regime. In [12], a surface wave cloak, using engineered GRIN materials, was experimentally demonstrated. Another study in [13] demonstrated that broadband electromagnetic cloaking could be achieved using transformation optics in conjunction with elastic deformations.

An area of particular interest for the present study on periodic graded metamaterials concerns their applicability to energy harvesting. In [14], a novel metamaterial rectifying surface (MRS) for electromagnetic energy capture and rectification with high harvesting efficiency is proposed. The metamaterial was fabricated on a layered printed circuit board (PCB) using a periodic arrangement of mirrored split ring resonators, and a a perfect impedance-matching was obtained between each layer of the PCB. In [15], a metamaterial built with superlattice layers of periodically arranged strip-wires and split-ring resonators has been studied experimentally for its transmission and reflection properties of microwave radiation.

In conjunction with energy harvesting and scattering phenomena, periodic metamaterial structures have also shown promise regarding electromagnetic absorption. In [16], a perfect metamaterial absorber (PMA) was fabricated using a water droplet-based metamaterial. In [17], plasmonic and photonic metamaterials with periodic arrangements have shown to also work as efficient electromagnetic narrowband absorbers in the microwave and optical frequency regimes. These structures make use of excitations of plasmonic and photonic resonances and could be useful for electromagnetic absorption and applications therein, such as solar-energy harvesting, photonic detection, broad/multi-band absorption, biosensing, and slowing down light.

In summary, the increased interest in impedance-matched periodic graded metamaterial devices requires new analytical and numerical methods for general description of impedance-matched periodic metamaterial composites. Using the general analytical methods outlined in [18–21], we therefore study a problem of TEM-wave propagation in a periodic impedance-matched metamaterial composite with graded right-handed (RHM) to left-handed (LHM) media transitions in a coaxial waveguide. We derive and study exact analytical solutions for the fields. In addition, we use the COMSOL software to carry out a numerical study of the TEM-wave propagation in a periodic impedance-matched metamaterial composite with graded RHM-LHM transitions in a coaxial waveguide. Thereby, we obtain an excellent agreement between analytical results and the results of numerical simulations. The advantage of the analytical and numerical methods described in the present paper is the ability to model smooth realistic material transitions, and to describe the abrupt transition as a limiting case.

It should be noted here that despite the apparent similarity, there are major differences between the present study and the studies reported in [19] and [21]. In [19], an unbounded periodic RHM-LHM structure in the entire space was considered. Here, we study a more realistic composite bound to a waveguide. On the other hand, in [21] a non-periodic single transition across a one boundary RHM-LHM composite in a coaxial waveguide is studied. The present treatment is very different from that reported in [21], both physically (multiple transitions over many boundaries in a periodic structure instead of a single transition over one boundary) and mathematically (transitions modelled by different mathematical functions that require solutions of very different differential equations). The results presented here cannot be obtained from the results reported in [21], and the physical description is very different. In our work, we have encountered applications (see e.g. [14–17]) that require the analysis of wave propagation through periodic RHM-LHM composites in guided structures. Although from the broad point of view of electromagnetic theory, the description of guided waves is incremental compared to the general EM-wave theory, it is still valuable to have published the present results that are not available elsewhere, despite the apparent similarities to the treatment in [19] and [21].

It is worth mentioning that apart from the above-mentioned practical interest and usefulness of structures with gradient changes in material parameters [1–5], there are also significant mathematical advantages of the present approach, that are applicable even to structures with abrupt material transitions. The analysis of multilayer material structures, like the one described in the present paper, usually requires mode matching over multiple boundaries, and cascading with complex expressions for the fields, which are difficult to put together in closed form. It is therefore common to resort to (semi-)numerical treatment of such problems. On the other hand, in our approach, we provide simple exact analytical solutions in closed form, that are obtained by direct solution of the Maxwell equations in a stratified medium without a need for any boundary conditions. The change of direction of the wave vector when passing between RHM and LHM materials is obtained directly from the solutions, with no need for making any particular assumptions. Our solutions are quite general, and can model both gradient and abrupt transitions as a special case. Thus, even in the case of abrupt transitions, our solutions provide a useful mathematical tool, that is more practical to use compared to the usual mode matching and cascading approach.

Finally, the gradient transitions of material parameters offers the opportunity to fine-tune the fields while retaining their general shape and properties. In most applications, it is not expected that the field patterns should undergo any radical changes when the gradient index designs are used. Thus, the difference in the field distributions is the main consequence of introducing the gradient transitions.

2. Problem formulation and analytical solutions

We consider a coaxial waveguide filled with an impedance-matched periodic metamaterial composite with RHM-LHM transitions, as shown in Fig. 1(a) and Fig. 1(b) for abrupt and linearly graded transitions respectively. The Figs. 1(a) and 1(b) display the behavior of the real and imaginary parts of the stratified relative permittivity $\varepsilon (\omega,z)$ or permeability $\mu (\omega,z)$ along the length $L$ of the waveguide, i.e. between $-L/2$ and $+L/2$. The coaxial waveguide, used in the present study, has a cross-section shown in Fig. 1(c). With the $\exp (\text {j} \omega t)$ time convention, we define the permittivity and permeability in the entire waveguide structure as the following periodic functions

Fig. 1. Waveguide with (a) abrupt and (b) linear RHM-LHM transitions, respectively. The medium inside the hollow waveguide consists of alternating slabs of RHMs and LHMs, in the longitudinal direction of the waveguide. The RHM slabs are denoted by the pink regions while the LHM slabs are denoted as light blue regions. The solid blue curves in each subfigure represent how the real part of the material parameters varies between the different slabs, while the red dashed lines represent the corresponding variation of the imaginary part. The curves also show when positive and negative values are achieved. Subfigure (c) illustrates the coaxial waveguide cross-section.

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(1)$$\varepsilon(z) = \varepsilon_0 \frac{\varepsilon_{I1}+\varepsilon_{I2}}{2 \beta} \left[ f(z) - \text{j} \beta \right] \hspace{3mm} , \hspace{3mm} \mu(z) = \mu_0 \frac{\mu_{I1}+\mu_{I2}}{2 \beta} \left[ f(z) - \text{j} \beta \right]$$

where $f(z)$ is a real-valued periodic function of the spatial coordinate $z$, symmetric about the $z$-axis, as can be inferred from Figs. 1(a) and 1(b). For the sake of brevity, we omit here the explicit $\omega$-dependence of $\varepsilon (z) = \varepsilon (\omega,z)$ and $\mu (z) = \mu (\omega,z)$ as well as in the spatially constant parameters therein ($\varepsilon _R$, $\varepsilon _{I1}$, $\varepsilon _{I2}$, $\mu _R$, $\mu _{I1}$, $\mu _{I2}$ and $\beta$). The impedance-matching condition (see e.g. [20]) reads

(2)$$\beta = \frac{\varepsilon_{I1}+\varepsilon_{I2}}{2 \varepsilon_R - \text{j} (\varepsilon_{I1}-\varepsilon_{I2})} = \frac{\mu_{I1}+\mu_{I2}}{2 \mu_R - \text{j} (\mu_{I1}-\mu_{I2})}$$

where we introduce the attenuation parameter $\beta (\omega )$. Here, the subscripts $R$ and $I$ represent the real and imaginary parts of a given material parameter, respectively. The RHM and LHM are numbered as $1$ and $2$ respectively, for convenience and clarity. Thus, the symbols $\varepsilon _{I1}(\omega )$ and $\varepsilon _{I2}(\omega )$ represent the dimensionless dispersive relative permittivities of the RHM and the LHM. Analogously, the symbols $\mu _{I1}(\omega )$ and $\mu _{I2}(\omega )$ represent the dimensionless dispersive permeabilities of the RHM and the LHM. In the definitions (1) and (2), the relative permittivities and permeabilities of the two materials ($1$ and $2$) are added together or substracted from each other as a part of defining the common stratified model of the entire RHM-LHM composite. These mathematical operations are a part of the mathematical model to describe the composite, and have no other physical significance. It should be noted here that when we use gradient designs, that the impedance-matching in the lossy case must maintain the ratio between the permittivity and permeability unchanged over a graded transition. This is ensured by the form of the graded permittivity and permeability functions (1) and the impedance-matching condition (2). The transverse electric field vector $\boldsymbol {E}(x,y,z)$ (TEM waves have $E_z = 0$), satisfies the following wave equation

(3)$$\nabla^2 \boldsymbol{E} - \frac{1}{\mu} \frac{\text{d} \mu}{\text{d}z} \frac{\partial \boldsymbol{E}}{\partial z} + \omega^2 \varepsilon \mu \hspace{1mm} \boldsymbol{E} = 0 \hspace{3mm} , \hspace{3mm} \boldsymbol{E} = \boldsymbol{E}_T = E_x \hspace{1mm} \boldsymbol{\hat{x}} + E_y \hspace{1mm} \boldsymbol{\hat{y}}$$

When the solution for transverse electric field vector $\boldsymbol {E}(x,y,z)$ is obtained, then we can calculate the transverse magnetic field vector $\boldsymbol {H}$ directly from Maxwell’s equation $\nabla \times \boldsymbol {E} = - \text {j} \omega \mu (z) \boldsymbol {H}$. Thus we do not use the magnetic field vector wave equation. The wave Eq. (3) is solved using variable separation $\boldsymbol {E}(x,y,z) = \boldsymbol {T}(x,y) L(z)$. The longitudinal function $L(z)$ and the transverse vector $\boldsymbol {T}(x,y)$ then satisfy the following differential equations

(4)$$\frac{\text{d}^2 L}{\text{d}z^2} - \frac{1}{\mu} \frac{\text{d} \mu}{\text{d}z} \frac{\text{d}L}{\text{d}z} + k_z^2(z) L = 0 \hspace{3mm} , \hspace{3mm} \nabla_t^2 \boldsymbol{T} = \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \boldsymbol{T} = 0$$

In (4) we denote the square of the spatially dependent wave vector by $k_z^2(z) = \omega ^2 \varepsilon (z) \mu (z)$, such that it can be written in the form

(5)$$k_z^2(z) = \frac{\omega^2}{c^2} \frac{\varepsilon_{I1}+\varepsilon_{I2}}{2 \beta} \frac{\mu_{I1}+\mu_{I2}}{2 \beta} \left[ f(z) - \text{j} \beta \right]^2 = k^2(\omega) \left[ f(z) - \text{j} \beta \right]^2$$

where we introduce a complex-valued spatial constant $k$, not to be confused with the spatially dependent wave number $k_z(z) = k(z)$. In (5), the constant $k = k(\omega )$ is given by

(6)$$k = \frac{\omega}{c} \frac{1}{2 \beta} \sqrt{(\varepsilon_{I1}+\varepsilon_{I2})(\mu_{I1}+\mu_{I2})} = \frac{\omega}{c} \sqrt{(\varepsilon_{R} - \text{j} \frac{\varepsilon_{I1}-\varepsilon_{I2}}{2} ) (\mu_{R} - \text{j} \frac{\mu_{I1}-\mu_{I2}}{2})}$$

It is here important to note that the quantity $\beta k$ is a real-valued spatial constant defined as follows

(7)$$\beta k = \frac{\omega}{2 c} \sqrt{(\varepsilon_{I1}+\varepsilon_{I2})(\mu_{I1}+\mu_{I2})}$$

Let us now consider the cases of abrupt RHM-LHM transitions and linear RHM-LHM transitions within the periodic RHM-LHM composite. These transitions can be described by the following periodic functions with the spatial period of $2a$ and $2(a+d)$ respectively. The abrupt transition can be described as follows:

(8)$$f(z) = \begin{cases} +1 & |z| < a/2 \\ -1 & |z| > a/2 \\ \pm 0 & |z| = a/2 \\ \end{cases}$$

which can be represented by the Fourier series of the form

(9)$$f(z) = \frac{4}{\pi} \sum_{m = 0}^{\infty} \frac{({-}1)^{m}}{2m+1} \cos \left[ \frac{(2m+1)\pi}{a}z \right]$$

The linear transition can be described as:

(10)$$f(z) = \begin{cases} -1 & -a < z <{-}a/2-d/2 \\ (2z+a)/d & -a/2-d/2 < z <{-}a/2+d/2 \\ +1 & -a/2+d/2 < z < a/2-d/2 \\ -(2z-a)/d & a/2-d/2 < z < a/2+d/2 \\ -1 & a/2+d/2 < z < a \\ \end{cases}$$

which can be represented by the Fourier series of the form

(11)$$f(z) = \frac{8a}{\pi^{2}d} \sum_{m = 0}^{\infty} \frac{({-}1)^{m}}{(2m+1)^{2}} \sin \left[ \frac{(2m+1)\pi}{2a}d \right] \cos \left[ \frac{(2m+1)\pi}{a}z \right]$$

where $d$ varies between $(0<d<a)$ and represents the length of the transition region between the RHM and LHM layers. It should be noted here that the solutions obtained in this paper are quite general and can be applied to any periodic function $f(z)$ and the two functions defined in (8) or (10) are just two examples, although important ones.

In order to solve the Eqs. (4), we first observe that the transverse function $\boldsymbol {T}(x,y) = \boldsymbol {T}(\rho )$, which satisfies the waveguide boundary conditions, is not affected by the $z$-dependence of material parameters $\varepsilon (\omega,z)$ and $\mu (\omega,z)$ and does not depend on the angular coordinate $\varphi$. In cylindrical coordinates, it, therefore, assumes the well-known textbook form

(12)$$\boldsymbol{T}(\rho) = U_0 \left( \ln \cfrac{r_2}{r_1} \right)^{{-}1} \frac{1}{\rho} \hspace{1mm} \boldsymbol{\hat{\rho}} \hspace{5mm} , \hspace{5mm} \rho = \sqrt{x^2 + y^2}$$

The voltage applied to the inner conductor of radius $r_1$ is denoted by $U_0$, i.e. the inner conductor has the potential $\Phi = U_0$. The outer conductor of radius $r_2$ is kept at zero potential $\Phi = 0$.

The solution for the electric field in the case of TEM-wave propagation in a coaxial waveguide, filled with a periodic impedance-matched metamaterial composite with graded RHM-LHM transitions, is of the form

(13)$$\boldsymbol{E}(\rho,z) = U_0 \left( \ln \cfrac{r_2}{r_1} \right)^{{-}1} \frac{1}{\rho} \hspace{1mm} L(z) \hspace{1mm} \boldsymbol{\hat{\rho}}$$

We then readily obtain the magnetic field in the form

(14)$$\boldsymbol{H}(\rho,z) = \frac{1}{Z} \hspace{1mm} \boldsymbol{\hat{z}} \times \boldsymbol{E}(\rho,z) = \frac{U_0}{Z} \left( \ln \cfrac{r_2}{r_1} \right)^{{-}1} \frac{1}{\rho} \hspace{1mm} L(z) \hspace{1mm} \boldsymbol{\hat{\varphi}}$$

Here we note that the wave vector direction is $\boldsymbol {\hat {k}} = \boldsymbol {\hat {z}}$, and that $\boldsymbol {\hat {z}} \times \boldsymbol {\hat {\rho }} = \boldsymbol {\hat {\varphi }}$. The real-valued spatially constant wave impedance is given by

(15)$$Z = Z(\omega) = \sqrt{\frac{\mu(\omega,z)}{\varepsilon(\omega,z)}} = \sqrt{\frac{\mu_0}{\varepsilon_0}} \sqrt{\frac{\mu_{I1}+\mu_{I2}}{\varepsilon_{I1}+\varepsilon_{I2}}}$$

In (13–14), $r_1$ and $r_2$ ($r_2 > r_1$) are the dimensions of the coaxial waveguide as shown in Fig. 1(c). The solution for the longitudinal function $L(z)$ is given by

(16)$$L(z) = \exp \left[ - \text{j} \int k_z(z) \ \text{d}z \right] = \exp \left[ - \text{j} \int k(\omega) \left[ f(z)-\text{i}\beta \right] \ dz \right]$$

where $L'(z) = - \text {j} k_z(z) L(z)$ and where we note that for constant $\varepsilon$ and $\mu$, we have $L(z) = \exp \left (- \text {j} k_z z \right )$, as expected. With material parameters given by Eqs. (1), the solution (16) for $L(z)$ can be obtained in the form

(17)$$L(z) = \text{e}^{- \beta k z} \text{e}^{- \text{j} k F(z)} \hspace{3mm} \text{where} \hspace{3mm} F(z) = \int_{}^{z} f(\xi) \ \text{d}\xi$$

Using (17), the exact analytical results for the TEM-fields in a coaxial waveguide become

(18)$$\boldsymbol{E}(\rho,z) = T(\rho) \hspace{1mm} L(z) \hspace{1mm} \boldsymbol{\hat{\rho}} = U_0 \left( \ln \cfrac{r_2}{r_1} \right)^{{-}1} \frac{1}{\rho} \hspace{1mm} \text{e}^{- \beta k z} \text{e}^{- \text{j} k F(z)} \hspace{1mm} \boldsymbol{\hat{\rho}}$$

(19)$$\boldsymbol{H}(\rho,z) = \frac{1}{Z} \hspace{1mm} T(\rho) \hspace{1mm} L(z) \hspace{1mm} \boldsymbol{\hat{\varphi}}= \frac{U_0}{Z} \left( \ln \cfrac{r_2}{r_1} \right)^{{-}1} \frac{1}{\rho} \hspace{1mm} \text{e}^{- \beta k z} \text{e}^{- \text{j} k F(z)} \hspace{1mm} \boldsymbol{\hat{\varphi}}$$

where for $f(z)$ given by (9), we have

(20)$$F(z) = \frac{4a}{\pi^2} \sum_{m = 0}^{\infty} \frac{({-}1)^{m}}{(2m+1)^2} \sin \left[ \frac{(2m+1)\pi}{a}z \right]$$

while for $f(z)$ given by (11), we have

(21)$$F(z) = \frac{8a^{2}}{\pi^{3}d} \sum_{m = 0}^{\infty} \frac{({-}1)^{m}}{(2m+1)^{3}} \sin \left[ \frac{(2m+1)\pi}{2a}d \right] \sin \left[ \frac{(2m+1)\pi}{a}z \right]$$

in the Fourier series description. Here we note that for $f(z)$ given by (8), in the RHM regions ($|z| < a/2$) we have $f(z) = 1$, while in the LHM regions ($|z| > a/2$) we have $f(z) = -1$, such that we obtain the correct directions of the wave propagation. In the RHM, we therefore have

(22)$$\boldsymbol{E}(\rho,|z| < a/2) = U_0 \left( \ln \cfrac{r_2}{r_1} \right)^{{-}1} \frac{1}{\rho} \hspace{1mm} \text{e}^{- \beta k z} \text{e}^{ - \text{j} k z} \hspace{1mm} \boldsymbol{\hat{\rho}}$$

and the wave is propagating in the positive $z$-direction. In the LHM, we have

(23)$$\boldsymbol{E}(\rho,|z| > a/2) = U_0 \left( \ln \cfrac{r_2}{r_1} \right)^{{-}1} \frac{1}{\rho} \hspace{1mm} \text{e}^{- \beta k z} \text{e}^{ - \text{j} ({-}k) z} \hspace{1mm} \boldsymbol{\hat{\rho}}$$

and the wave is propagating backwards in the negative $z$-direction, as required. Here we observe that the wave vector (and refractive index) sign change when propagating from RHM to LHM is obtained by virtue of the graded solutions of Maxwell’s equations alone, and we do not require any boundary conditions or other assumptions. The asymptotic limits of the fields in the linearly graded periodic RHM-LHM composite, described by $f(z)$ given by (10), are equivalent to those of the non-graded periodicity with $f(z)$ given by (9), and will therefore not be repeated. Using (18–19), the complex Poynting vector becomes

(24)$$\boldsymbol{S}(\rho,z) = \frac{1}{2} \boldsymbol{E}(\rho,z) \times \boldsymbol{H}^{{\ast}}(\rho,z) = \frac{|U_0|^2}{2Z} \left( \ln \cfrac{r_2}{r_1} \right)^{{-}2} \frac{\text{e}^{- 2 \beta k z}}{\rho^2} \text{e}^{2 {\mathrm{Im}}{(k)} F(z)} \hspace{1mm} \boldsymbol{\hat{z}} = S(\rho,z) \hspace{1mm} \boldsymbol{\hat{z}}$$

Thus, the power flowing through the cross section of the coaxial waveguide at any given longitudinal position $z$ is given by

(25)$$P(z) = \int_a^b 2 \pi \rho \text{d}\rho \hspace{1mm} {\mathrm{Re}}[S(\rho,z)] = 2 \pi \frac{|U_0|^2}{2Z} \left( \ln \cfrac{r_2}{r_1} \right)^{{-}1} \hspace{1mm} \text{e}^{- 2 \beta k z} \text{e}^{2 {\mathrm{Im}}{(k)} F(z)} \hspace{1mm}$$

expressed in terms of the voltage $U_0$, applied to the inner conductor of radius $r_1$.

3. Numerical solutions and comparison to the analytical results

A lossy impedance-matched periodic RHM-LHM composite media was modeled within a coaxial waveguide with the help of the software COMSOL Multiphysics, which is based on the finite element method. The study involves investigating two different configurations: an abrupt and a linear interface transition within the periodic RHM-LHM media. Table 1 below summarizes the numerical values of the parameters used in the implemented model.

Here we note that the operational frequency was chosen such that only TEM wave propagation is present in the waveguide. The steepness factor $d$ represents the width of the transition layer in the periodic RHM-LHM media, where we see that the composite with abrupt transitions does not have a transition width as opposed to the composite with graded linear transitions. The total waveguide length is occupied by nine periodically alternating slabs of RHM and LHM material sections such that both ends of the waveguide are filled with RHM material. This property is valid for either configuration. In addition, each slab has a width of $a$. The periodicity for the abrupt configuration is $2a$ and for the linear configuration is $2(a+d)$, where the abrupt case only takes into account the slab width whilst the linear case also considers the transition layer width. If the expression for the power distribution within the hollow coaxial waveguide is utilized (25), the induced voltage on the inner conductor becomes $U_0 \approx 2.131$ V and $U_0 \approx 1.445$ V, respectively, for the abrupt and linear configurations. These voltages are equivalent to $1$ W of input power. To model a more realistic RHM-LHM composite for this application, the LHM material has been chosen to have higher losses than the RHM material since LHMs typically have higher losses than RHMs.

Table 1. Summary of the numerical waveguide parameters for both abrupt and linear transitions. Values are rounded to two decimal places.

View Table

The model for the lossy impedance-matched periodic RHM-LHM composite media inside a coaxial waveguide is designed in the same way for metamaterial composites with both abrupt and linear transitions. The above-mentioned design dimensions are used to model the coaxial waveguide using a 2D axial-symmetric geometry. A Perfect Electric Conductor (PEC) boundary condition is applied to the outer conductor.

The inner conductor is set to be made of copper. An inhomogeneous periodic RHM-LHM composite is modeled using the analytical functions (1) with the impedance-matching condition (2) and occupies the hollow volume of the waveguide. The waveguide’s geometric center is at the waveguide’s origin, distributing the inhomogeneous periodic RHM-LHM composite’s behavior following Fig. 1.(c). The receiving port is on the right side and the exciting port is on the left side of the waveguide. An electromagnetic wave can be excited and travel from negative $z$-values (on the left) to positive $z$-values (on the right) due to the ports’ orientation. With the "Slit Condition On Interior Port" active, the Domain-backed slit type is combined with Perfectly Matched Layers (PMLs) for both ports to absorb the exciting mode from a source port and other higher order modes. A lossless section with the same material characteristics as the RHM material is introduced in front of both ports between the PMLs and the inhomogeneous periodic RHM-LHM composite. Internal reflections between the lossy RHM material and the ports and PMLs are prevented by the introduction of these lossless sections. In addition, to absorb internal reflections that occur between the ports and the lossy inhomogeneous periodic RHM-LHM composite, the outward-facing sides of the PMLs were subjected to a scattering boundary condition (SBC). The full implementation has been tested in both 2D and 3D space. With properly chosen mode phases, the results of both simulation types are identical.

In this section, we present the numerical and analytical results of the electric field intensity for the TEM mode in Figs. 2, 3, 4 and 5. These results only represent the electric field distribution inside the inhomogeneous periodic RHM-LHM composite. In Fig. 2, the 1D electric field intensity at a radius of $r = (r_1+r_2)/2 = 0.06$ m is presented. Here, the subfigure pairs (a)-(b) and (c)-(d) represent the analytical and numerical results of the electric field intensity for the abrupt and linear configuration respectively. Figure 3 presents a direct comparison of the analytical and numerical results collected for both configurations shown in Fig. 2.

Fig. 2. Subfigure pairs (a)-(b) and (c)-(d) represent the analytical and numerical results of the 1D electric field intensity $E(x = 0.06,y = 0,z)$ [V/m] for the abrupt and linear transition configuration respectively. The 1D electric field intensity corresponds to the TEM mode at a radius of $r = 0.06$ m of the composite material inside the coaxial waveguide.

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Fig. 3. Subfigure pairs (a)-(b) and (c)-(d) represent a comparison between real and imaginary part of the analytical and numerical results of the 1D electric field intensity $E(x = 0.06,y = 0,z)$ [V/m] for the abrupt and linear transition configuration respectively. The 1D electric field intensity corresponds to the TEM mode at a radius of $r = 0.06$ m of the composite material inside the coaxial waveguide.

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Fig. 4. Subfigure pairs (a)-(b) and (c)-(d) represent the real part of the analytical and numerical results of the 2D electric field intensity $E(x,y = 0,z)$ [V/m] for the abrupt and linear transition configuration respectively. The 2D electric field intensity corresponds to the TEM mode along the $xz$-plane inside the coaxial waveguide.

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Fig. 5. Subfigure pairs (a)-(b) and (c)-(d) represent the analytical and numerical results of the 2D electric field magnitude |$E(x,y = 0,z)$| [V/m] for the abrupt and linear transition configuration respectively. The 2D electric field magnitude corresponds to the TEM mode along the $xz$-plane inside the coaxial waveguide.

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Figure 4 presents the real part of the analytical and numerical results for the 2D electric field intensity $E(x,y = 0,z)$ along the $xz$-plane inside the hollow coaxial waveguide. Here, the subfigure pairs (a)-(b) and (c)-(d) represent the real part of the 2D electric field intensity for the abrupt and linear configuration respectively. In Fig. 5, the subfigure pairs (a)-(b) and (c)-(d) represent the magnitude of the 2D electric field intensity $|E(x,y = 0,z)|$ for the abrupt and linear configuration respectively. Here, we observe that Figs. 4 and 5 are cropped along the $z$-axis and only span half the length of the waveguide. This adjustment along the $z$-axis has been done to visualize the results more clearly, since the lossy RHM-LHM composite media gives rise to heavy attenuation along the waveguide that would otherwise make the 2D electric field intensity and magnitude difficult to observe. An understanding of the attenuation rate could be observed clearly in Fig. 2 and 4. Figures 2, 3 and 4 allow us to observe the phase shift in the real electric field patterns at the interface between RHM and LHM media, where the wave vector reversal occurs.

4. Conclusions

We studied the propagation of TEM-waves inside a hollow coaxial waveguide filled with a lossy impedance-matched periodic RHM-LHM composite. The RHM-LHM transitions within the periodic composite and the TEM-wave propagation were assumed to occur in the direction perpendicular to the boundaries between the two media, chosen to be the $z$-direction. The relative permittivity $\varepsilon (\omega,z)$ and permeability $\mu (\omega,z)$ of the periodic RHM-LHM composite were chosen to vary according to an arbitrary periodic function $f(z)$ along the $z$-direction. In particular, we considered two specific periodic permittivity and permeability profiles: one with abrupt RHM-LHM transitions and one with linear RHM-LHM transitions. The exact analytical solutions to Maxwell’s equations were derived, and the obtained solutions for the field components and wave behavior confirm the expected properties of impedance-matched periodic RHM-LHM structures. In addition, using the COMSOL Multiphysics software, a numerical study of the wave propagation across the impedance-matched graded RHM-LHM composites was performed. The results of the numerical simulations are found to be virtually identical to the analytical results. The current approach includes the abrupt transition as a limiting case and can model smooth, more realistic material transitions.

Funding

Vetenskapsrådet (2018-05001).

Acknowledgements

The work of M. D. was supported by the Swedish Research Council, project number 2018-05001.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Property	Abrupt Transition	Linear Transition
Waveguide Length ( $L$ )	$1.80$ [m]	$2.25$ [m]
Inner Radius ( $r_{1}$ )	$0.03$ [m]
Outer Radius ( $r_{2}$ )	$0.09$ [m]
Operating Frequency ( $f$ )	$1.50$ [GHz]
Operating Wavelength ( $λ$ )	$0.20$ [m]
Steepness Factor ( $d$ )	-	$λ / 4$ [m]
Number Of Slabs ( $N$ )	$9$
Slab Width ( $a$ )	$λ$ [m]
Periodicity	$2 a$ [m]	$2 (a + d)$ [m]
Induced Inner Conductor Voltage ( $U_{0}$ )	$2.13$ [V]	$1.44$ [V]
Relative Permittivity ( $ε_{R}$ )	RHM: $ε_{R} = 2 + 0.04$ j
	LHM: $ε_{R} = - 2 + 0.12$ j
Relative Permeability ( $μ_{R}$ )	RHM: $μ_{R} = 1 + 0.02$ j
	LHM: $μ_{R} = - 1 + 0.06$ j

TEM-wave propagation over an impedance-matched periodic RHM to LHM transition in a coaxial waveguide

Abstract

1. Introduction

2. Problem formulation and analytical solutions

3. Numerical solutions and comparison to the analytical results

4. Conclusions

Funding

Acknowledgements

Disclosures

Data Availability

References

Data Availability

Cited By

Figures (5)

Tables (1)

Equations (25)

Optical Materials Express