TEM-wave propagation in a coaxial waveguide with impedance-matched RHM to LHM transition

Balwan Rana; Brage B. Svendsen; Mariana Dalarsson

doi:10.1364/OE.460924

1. Introduction

Over the last two decades, the superior performance of graded RHM-LHM structures has caused an unprecedented increase in interest in composite metamaterials [1,2]. These graded composite structures are based on selectively chosen continuous functions of spatial coordinates that help characterize the behavior of the material permittivity and permeability [3].

Graded composite structures are used across different disciplines, ranging from metamaterial based antennas and array designs for telecommunication and tumor detection, to metamaterial-inspired matching networks for improving the impedance bandwidth of planar antennas and decoupling mechanisms for SAR and MIMO antenna systems [1,2,4,5]. Graded composite structures could also be applied as metasurfaces for surface wave reduction in SAR and MIMO systems, and for controlling the polarization of scattered electromagnetic waves [6,7].

In optics, there are many published studies on the performance and applicability of graded metamaterials. In [8], an elastic waveguide filled with an array of resonators achieved wave-mode conversion and rainbow trapping, by varying the size of the layered resonators such that their collective material behavior resembled a spatially varying gradient refractive index (GRIN) composite metamaterial. Rainbow trapping concerns the confinement of the wave energy, while wave-mode conversion concerns the reversible conversion of the incident wave-mode oscillation into a different mode of oscillation. In [9], a layered nanostructured waveguide consisting of dielectric silicon nitride (SiN$_{X}$) films squeezed between a glass substrate and air obtained a tunable optical absorption spectrum. This tunable absorption spectrum was manipulated with a varying gradient refractive index profile and the transmittance and reflectance characteristics that vary with the thickness of the dielectric layers and the incidence angle of the light rays. Solar cells with GRIN metamaterials are shown to have a more efficient light absorption and better flexibility with respect to the incidence angle of light rays, compared to traditional solar cells.

In [10], scatterer illusions using transformation optics in combination with anisotropic and inhomogeneous media is demonstrated numerically. The numerical simulation consisted of an enclosed metallic object inside a superscatterer illusion device. The scattering signature from the enclosed metallic object resembled that of an enlarged metallic object with a pre-controlled scaling factor and multiple dielectric objects with pre-designed material properties. GRIN metamaterials were used to enclose the metallic object due to their electromagnetic resonant behavior and unit cell properties. In [11], a novel octave-bandwidth highly-directive half Maxwell fish-eye (HMFE) lens antenna in the super-extended C band is designed using a GRIN metamaterial. The proposed design incorporates a fractal geometry, and the numerical and experimental results show that the lens enables a considerable gain enhancement, without significantly affecting the cross-polarization patterns and impedance matching. Thereby, [11] demonstrates the unique capability of accurately converting quasi-spherical waves to plane waves by the GRIN metamaterial to obtain an unconventionally high gain.

In [12], the possibility of manipulating guided surface magneto-plasmons (SMPs) using a non-reciprocal plasmonic GRIN waveguide system together with an external static magnetic field source, is demonstrated. The waveguide system consisted of a semiconductor-dielectric-metal planar construction, with a static magnetic field applied in the direction perpendicular to the interface normal between material layers. The study has shown that the proposed method could be used for fast modulation of the wave velocity and the spatial field distribution of SMPs, which is beneficial for wave harvesting and selective energy concentration.

In [13], a metamaterial consisting of gate-controlled split-ring resonators is proposed in order to control the magnetic permeability of optical GaInAsP/InP semiconductor waveguides. The split-ring resonators interact magnetically with light in the waveguide, and thereby introduce a variable effective relative permeability at optical frequencies. The variable permeability waveguide devices enable so-called ’permeability engineering’, which aims to facilitate manipulation of light and management of photons, resulting in novel devices with sophisticated functions for photonic integration. In [14], a photon emitter was placed inside and on the surface of a nanostructured hyperbolic metamaterial slab to simulate an optical metamaterial waveguide. The nanostructured metamaterial slab is designed using an array of gold nanorods obtaining a hyperbolic behavior through variation in the design dimensions of the nanorods and their array properties. Nanostructured hyperbolic metamaterials are shown to be useful for designing integrated, fast optical sources for data communications, sensing, or quantum photonic applications.

Graded metamaterials can also be useful for producing efficient optical generators and detectors in the terahertz gap. In [15], an all-metallic epsilon-near-zero (ENZ) GRIN metamaterial lens is experimentally investigated and evaluated. The ENZ GRIN metamaterial is made of an array of narrow hollow rectangular waveguides, operating close to their cutoff frequencies, where the graded-index profile of the metamaterial is obtained by varying the dimensions of the rectangular waveguides and their relation to neighboring waveguide dimensions. An ENZ GRIN device in the terahertz regime can enable full quasioptical processing of terahertz signals.

In [16], a method to improve nanofabrication processes of a nanoscaled transformation-optics wave bender and Luneburg lens by using elasto-optic GRIN metamaterials, is proposed. A GRIN profile with a variable stress-tuneable refractive index was obtained by the combined measures of optics and solid mechanics. In [17], it is shown that a Maxwell fish-eye (MFE) lens could be used as a multiband waveguide crossing medium, by incorporating the waveguide bends into the crossing intersections, and comprising the crossing intersection of a GRIN-based photonic crystal (GPC) MFE lens. In [18], rainbow trapping and releasing using a graphene plasmonic waveguide is demonstrated, where the tunability of light trapping and releasing was achieved by adjusting the chemical potential of the graphene material, giving the waveguide a GRIN profile across the silicon-graded grating structure. Due to the high tunability of the proposed waveguide structure, its applicability could include optical switches, buffers, and storages.

The area of particular interest for the present study is within waveguide applications [19,20]. Particularly interesting are the nanostructured waveguides which are expected to improve the performance of solar cells using a tunable absorption spectrum [21]. Another area of interest concerns light trapping in waveguides [22,23]. In [22] a graded metamaterial waveguide is used to trap and release light in the mid-infrared frequency range. In [23], an improved theoretical structure for trapped storage of light was proposed using a hyperbolic tangent graded metamaterial. The light-trapping characteristics discussed in [22] and [23] propose a graded metamaterial waveguide to be a prime candidate for multi-wavelength absorption, optical modulation, switching, communication, and other light-matter interactions.

Another area of interest concerns the minimization and customization of waveguides [24,25]. In [24] a proposed meta-waveguide structure and a derived transcendental equation were used to reveal the physical mechanisms behind phase control of metamaterial waveguides. The results show that waveguides filled with a graded-index metamaterial allow for customizing phase modulation and absorption in the microwave frequency region. In [25], a semi-analytical method is proposed to calculate the eigenvalues, including the cutoff wavenumbers and dispersion relations, for waveguides filled with GRIN metamaterials. This method is based on modal expansion analysis, and has been shown to highlight the observed below-cutoff backward wave propagation phenomenon in waveguides filled with GRIN metamaterials.

In conclusion, the growing interest in impedance-matched GRIN metamaterials has led to an increased need for general analytical and numerical studies of impedance-matched metamaterial composites. Following the approach employed in [26–28], we now consider a new problem of TEM-wave propagation over an impedance-matched RHM to LHM transition in a coaxial waveguide. The complete exact analytical solutions for the fields are obtained and analyzed. Furthermore, we perform a numerical study of the TEM-wave propagation over an impedance-matched graded RHM-LHM interface in a coaxial waveguide, using the COMSOL software. An excellent agreement between the numerical simulations and analytical results is obtained. It should be mentioned here that the objective of the present paper is to provide a general theory applicable to various waveguide geometries involving metastructures. It is therefore not limited to any specific application. Focus on the design of some real metastructures, for verification and realization, will be a part of our future work.

2. Problem description

The geometry of the problem consists of a hollow waveguide with graded transition from a lossy right-handed material (RHM) filling the left-hand half of the waveguide to the impedance-matched lossy left-handed material (LHM) filling the right-hand half of the waveguide, as shown in Fig. 1(a). In addition, Fig. 1(a) also illustrates the behavior of the permittivity and permeability properties for the RHM-LHM composite, which is shown to vary from a positive real and imaginary part for the RHM to a negative real and imaginary part for the LHM. The cross-section of the coaxial waveguide, used in the present paper, is shown in Fig. 1(b). The red circle represents the inner conductor of the coaxial waveguide structure, and is shown to have a radius of $r = a$ with an arbitrarily applied voltage of $\Phi = U_{0}$. The outer conductor of the coaxial waveguide structure is colored in blue, with a radius of $r = b$, and is grounded ($\Phi = 0$).

Fig. 1. (a) Waveguide with RHM-LHM transition. (b) Coaxial waveguide cross section.

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Using the $\exp (\text {i} \omega t)$ time convention, we define the permittivity and permeability in the entire waveguide structure as continuous functions given by

(1)$$\varepsilon(\omega,z) ={-} \varepsilon_0 \frac{\varepsilon_{I1}+\varepsilon_{I2}}{2 \beta} \left( \tanh \frac{z}{z_0} + \text{i} \beta \right) \hspace{3mm} , \hspace{3mm} \mu(\omega,z) ={-} \mu_0 \frac{\mu_{I1}+\mu_{I2}}{2 \beta} \left( \tanh \frac{z}{z_0} + \text{i} \beta \right)$$

where we recall from e.g. [27] that the impedance-matching condition requires that

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

Asymptotically, for $z \to - \infty$, in the RHM-material we have $\varepsilon _1(\omega,z) = \varepsilon _0 (+\varepsilon _R - \text {i} \varepsilon _{I1})$ and $\mu _1(\omega,z) = \mu _0 (+\mu _R - \text {i} \mu _{I1})$. In the LHM-material asymptotically, for $z \to + \infty$, we have $\varepsilon _2(\omega,z) = \varepsilon _0 (-\varepsilon _R - \text {i} \varepsilon _{I2})$ and $\mu _2(\omega,z) = \mu _0 (-\mu _R - \text {i} \mu _{I2})$, as required by impedance matching in passive materials. The wave equation for the transverse electric field $\boldsymbol {E}(x,y,z)$ (TEM waves have $E_z = 0$) is given by

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

The corresponding wave equation for the transverse magnetic field $\boldsymbol {H}$ is not needed, since the magnetic field can be obtained directly from Maxwell’s equation $\nabla \times \boldsymbol {E} = - \text {i} \omega \mu (z) \boldsymbol {H}$. The approach to solve the wave Eq. (3) is based on variable separation such that $\boldsymbol {E}(x,y,z) = \boldsymbol {T}(x,y) L(z)$, where the transverse vector $\boldsymbol {T}(x,y)$ and the longitudinal function $L(z)$ satisfy following equations

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

where the square of the spatially dependent wave vector is given by $k_z^{2}(z) = \omega ^{2} \varepsilon (z) \mu (z)$, such that we have

(5)$$k_z^{2}(z) = \frac{\omega^{2}}{c^{2}} \frac{\varepsilon_{I1}+\varepsilon_{I2}}{2 \beta} \frac{\mu_{I1}+\mu_{I2}}{2 \beta} \left( \tanh \frac{z}{z_0} + \text{i} \beta \right)^{2} = k^{2}(\omega) \left( \tanh \frac{z}{z_0} + \text{i} \beta \right)^{2}$$

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

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

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

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

3. Solutions for the fields

Here we note that the transverse part of the solution $\boldsymbol {T}(x,y) = \boldsymbol {T}(\rho )$, satisfying the correct boundary conditions, is independent on the angular coordinate $\varphi$, and it is not affected by the $z$-dependence of material parameters $\varepsilon (\omega,z)$ and $\mu (\omega,z)$. Thus, it has a well-known textbook form in cylindrical coordinates

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

where $\Phi = U_0$ is the voltage applied to the inner conductor of radius $a$, while the outer conductor of radius $b$ is kept at zero potential $\Phi = 0$. Thus, we obtain the solution for the electric field in the case of TEM-wave propagation in a coaxial waveguide, with a graded transition from a lossy RHM material filling the left-hand half of the waveguide to the impedance-matched lossy LHM material filling the right-hand half of the waveguide, in the following form

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

The magnetic field is then obtained as follows

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

where the direction of the wave vector is $\hat {\boldsymbol {k}} = \hat {\boldsymbol {z}}$ such that $\hat {\boldsymbol {z}} \times \hat {\boldsymbol {\rho }} = \hat {\boldsymbol {\varphi }}$. The real-valued spatially constant wave impedance is given by

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

In (8–10), $a$ and $b$ ($b > a$) are the coaxial waveguide dimensions shown in Fig. 1(b). The longitudinal function $L(z)$ is obtained in the following form

(12)$$L(z) = \exp \left[ - \text{i} \int k_z(z) \text{d}z \right] = \exp \left[ - \text{i} \int \omega \sqrt{\varepsilon(z) \mu(z)} \text{d}z \right]$$

where we readily see that $L'(z) = - \text {i} k_z(z) L(z)$. Furthermore, we note that for constant $\varepsilon$ and $\mu$, we obtain $L(z) = \exp \left (- \text {i} k_z z \right )$, as required. Introducing the definitions of the material parameters (1), the expression (12) for $L(z)$ can be evaluated as follows

(13)$$L(z) = \text{e}^{- \beta k z} \left( \cosh \frac{z}{z_0} \right)^{\text{i} k z_0}$$

Thus the exact analytical results for the TEM-fields in a coaxial waveguide have the following form

(14)$$\boldsymbol{E}(\rho,z) = U_0 \left( \ln \cfrac{b}{a} \right)^{{-}1} \frac{1}{\rho} \hspace{1mm} \text{e}^{- \beta k z} \left( \cosh \frac{z}{z_0} \right)^{\text{i} k z_0} \hspace{1mm} \hat{\boldsymbol{\rho}}$$

(15)$$\boldsymbol{H}(\rho,z) = \frac{U_0}{Z} \left( \ln \cfrac{b}{a} \right)^{{-}1} \frac{1}{\rho} \hspace{1mm} \text{e}^{- \beta k z} \left( \cosh \frac{z}{z_0} \right)^{\text{i} k z_0} \hspace{1mm} \hat{\boldsymbol{\varphi}}$$

Here we note that in the asymptotic regions ($z \to \mp \infty$) we obtain the correct directions of the wave propagation. Thus in the RHM, far to the left of the interface, we have

(16)$$\boldsymbol{E}(\rho,z \to - \infty) \to U_0 \left( \ln \cfrac{b}{a} \right)^{{-}1} \frac{1}{\rho} \hspace{1mm} \text{e}^{- \beta k z} \text{e}^{ - \text{i} k z} \hspace{1mm} \hat{\boldsymbol{\rho}}$$

such that the wave is propagating in the positive $z$-direction. In the LHM, far to the right of the interface, we have

(17)$$\boldsymbol{E}(\rho,z \to + \infty) \to U_0 \left( \ln \cfrac{b}{a} \right)^{{-}1} \frac{1}{\rho} \hspace{1mm} \text{e}^{- \beta k z} \text{e}^{ - \text{i} ({-}k) z} \hspace{1mm} \hat{\boldsymbol{\rho}}$$

such that the wave is propagating backward in the negative $z$-direction, as required. The sign change when passing from RHM to LHM is achieved automatically by virtue of the graded solutions to Maxwell’s equations, and no boundary conditions or other assumptions are needed. The complex Poynting vector is obtained from (14–15) in the form

(18)$$\boldsymbol{S}(\rho,z) = \frac{1}{2} \boldsymbol{E}(\rho,z) \times \boldsymbol{H}^{{\ast}}(\rho,z) = \frac{|U_0|^{2}}{2Z} \left( \ln \cfrac{b}{a} \right)^{{-}2} \frac{\text{e}^{- 2 \beta k z}}{\rho^{2}} \left( \cosh \frac{z}{z_0} \right)^{- 2 {\textrm{Im}}(k) z_0} \hspace{1mm} \hat{\boldsymbol{z}} = S(\rho,z) \hspace{1mm} \hat{\boldsymbol{z}}$$

The power flowing through the cross section of the coaxial waveguide at an arbitrary longitudinal position $z$ is therefore given by

(19)$$P(z) = \int_a^{b} 2 \pi \rho \text{d}\rho \hspace{1mm} {\textrm{Re}}[S(\rho,z)] = 2 \pi \frac{|U_0|^{2}}{2Z} \left( \ln \cfrac{b}{a} \right)^{{-}1} \hspace{1mm} \text{e}^{- 2 \beta k z} \left( \cosh \frac{z}{z_0} \right)^{- 2 {\textrm{Im}}(k) z_0} \hspace{1mm}$$

and can be expressed in terms of the voltage $U_0$, applied to the inner conductor of radius $a$.

4. Numerical solutions and comparison to the analytical results

The finite element method based software COMSOL Multiphysics is used to model a lossy impedance-matched graded interface between RHM and LHM media inside a coaxial waveguide. A 2D axial-symmetric geometry is used to model the coaxial waveguide with the following design dimensions: an inner radius $a=0.03$ m, an outer radius $b=0.09$ m, and a waveguide length of $L=1$ m. The inner conductor $a$ material is set to copper, while a Perfect Electric Conductor (PEC) boundary condition is applied to the outer conductor $b$. The hollow volume inside the waveguide is occupied by an inhomogeneous RHM-LHM composite modeled according to the analytical functions (1) with the impedance-matching condition (2). The geometrical center of the waveguide is placed at the origin such that the behavior of the inhomogeneous RHM-LHM composite is distributed according to Fig. 1. The exciting port is placed to the left of the waveguide and the receiving port is placed on the right-hand side. This orientation of the ports allows for an electromagnetic wave to be excited and travel from negative $z$-values (the left) to positive $z$-values (the right). It should be noted that the left-hand side of the hollow waveguide volume is filled with the RHM material, while the right-hand side is filled with the LHM material. Consequently, the exciting port is placed to the left of the RHM, and the receiving port is placed to the right of the LHM. Both ports are domain-backed by Perfectly Matched Layers (PMLs) with a scaling factor of 1 and −3 respectively. The negative scaling factor was introduced in order to ensure the proper absorption functionality of the PML when dealing with LHMs. In between the RHM-LHM composite and the PMLs, a lossless section with the same material characteristics as the composite is introduced in front of both ports. These lossless sections are introduced to avoid internal reflections between the lossy composite material and the ports and PMLs. In addition, a scattering boundary condition (SBC) was applied to the outward-facing sides of the PMLs, in order to absorb internal reflections between the ports and the lossy inhomogeneous RHM-LHM composite. The complete implementation was tested in both 2D and 3D domains.

In this section, we present the numerical and analytical results of the electric field intensity in Figs. 2–4 for the TEM mode propagation inside a hollow coaxial waveguide filled with an inhomogeneous impedance-matched RHM-LHM composite. In order to explain how we obtain Figs. 2–4, and how these figures relate to the theory, we note that in Fig. 2, the 1D electric field intensity at a radius of $r = (b+a)/2 = 0.06$ m inside the hollow coaxial waveguide is presented. Here, subfigures (a) and (b) represent the analytical and numerical results for the electric field intensity respectively. In subfigures (c) and (d), a comparison between the real and imaginary parts of the electric field intensity for the analytical and numerical results is shown. Note that the analytical and numerical results in subfigures (c) and (d) are plotted in the same figure for a good comparison.

Fig. 2. Analytical and numerical results of the 1D electric field intensity for the TEM mode at a radius of $r = 0.06$ m of the composite material inside the coaxial waveguide. The subfigures (a) and (b) show the analytical and numerical results respectively. The subfigures (c) and (d) show a comparison between the real and imaginary parts of the analytical and numerical electric field intensities respectively.

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Fig. 3. Analytical and numerical results of the 2D electric field intensity $E(x,y = 0,z)$ [V/m] for the TEM mode along the $xz$-plane inside the coaxial waveguide. The subfigure pairs (a)-(b) and (c)-(d) show the real and imaginary parts of the analytical and numerical electric field intensities respectively.

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Fig. 4. Analytical and numerical results of the 2D electric field magnitude |$E(x,y = 0,z)$| [V/m] for the TEM mode along the $xz$-plane inside the coaxial waveguide. The subfigures (a) and (b) represent the analytical and numerical 2D electric field magnitudes respectively.

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Figure 3 presents 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, respectively. Here, subfigures (a)-(b) and (c)-(d) represent the analytical and numerical results for the 2D electric field intensity along the $xz$-plane respectively. Figure 3 illustrates how the electric field component of the TEM mode varies across the entirety of the hollow coaxial waveguide structure in terms of phase and amplitude.

In order to clarify which component of the electric field is plotted in Figs. 2–4, we recall the Eq. (9) where we see that the electric field only has a radial component. This field can be further decomposed into cartesian transverse components $\boldsymbol {E}_x$ and $\boldsymbol {E}_y$, but it is deemed unnecessary here. Figure 4 represents the analytical and numerical results for the 2D electric field magnitude $|E(x,y = 0,z)|$ along the $xz$-plane inside the hollow coaxial waveguide, respectively. Figure 4 illustrates how the electric field magnitude of the TEM mode varies across the entirety of the hollow coaxial waveguide structure in terms of phase and amplitude. Figure 4 could also be interpreted as illustrating the energy distribution of the component of the TEM mode inside the hollow coaxial waveguide structure.

In all the figures, an excellent agreement between the analytical results and numerical simulations is obtained, with a relative error of less than 0.1%. The relative error is mainly due to a slight phase and amplitude difference between the analytical and numerical results, generated by the wave-PML interactions and internal reflections at the material transitions with the refractive index profile reaching zero at the RHM-LHM transition. Otherwise, the analytical and numerical results are in perfect agreement.

By utilizing the expression for the power distribution inside the coaxial waveguide (19), the voltage induced on the inner conductor which is equivalent to 1 W of input power is $U_{0} \approx 6.11$ V. The gradient transition of the composite’s permittivity and permeability functions has a steepness factor of $z_0=0.01$ m. Furthermore, the RHM has the material properties $\epsilon = 2+0.04$i and $\mu = 1+0.02$i, while the LHM has the material properties $\epsilon = -2+0.2$i and $\mu = -1+0.1$i. For this implementation, we have chosen the LHM section to be more lossy than the RHM section of our composite. This has been done in order to model a realistic RHM-LHM composite, where we know that LHMs typically have larger losses than RHMs. The operational frequency for the simulation is chosen to be 1.5 GHz such that it is only the fundamental TEM mode that is propagating. From Fig. 2, 3 and 4, we can observe the phase shift at the interface between the RHM-LHM media, where the reversal of the wave vector occurs, in the real part of the electric field patterns. The same phase reversal also occurs for the imaginary part of the electric field patterns, although, the effect is not as apparent as for the real part of the electric field patterns. In addition, the traveling electromagnetic wave is observed to obtain an evanescent behavior across the hollow coaxial waveguide. This is expected, given that the propagation media consists of a lossy RHM-LHM composite, where the higher loss factor in the LHM region promotes a higher decay rate for the electromagnetic wave after the phase reversal.

5. Conclusions

We investigated TEM-wave propagation inside a hollow coaxial waveguide filled with a lossy RHM-LHM composite, with a graded transition between an RHM and an impedance-matched LHM. The graded RHM-LHM transition occurs in the $z$-direction, where the relative permittivity $\varepsilon (\omega,z)$ and permeability $\mu (\omega,z)$ vary according to hyperbolic tangent functions. We obtained exact analytical solutions to Maxwell’s equations for lossy media, and showed that the field solutions and wave behavior confirm the expected properties of impedance-matched RHM-LHM structures. Finally, we performed a numerical study of the wave propagation over an impedance-matched graded RHM-LHM interface, using the COMSOL Multiphysics software. An excellent agreement between the analytical results and numerical simulations was obtained, with a relative error of less than 0.1%. This confirms the validity of both the analytical and numerical models employed in the present paper.

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.

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TEM-wave propagation in a coaxial waveguide with impedance-matched RHM to LHM transition

Abstract

1. Introduction

2. Problem description

3. Solutions for the fields

4. Numerical solutions and comparison to the analytical results

5. Conclusions

Funding

Acknowledgements

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Equations (19)

Optics Express