The rigorous electromagnetic analysis

Let us consider a z-directional metallic circular waveguide filled with a metamaterial possesses anisotropic material parameters that are described by tensors diagonalized in a cylindrical coordinate system. Taking the coordinate axis to coincide with the waveguide axis, these take the form $\bar{\bar\varepsilon}=\bar{\bar I}\left( \varepsilon_{rr},\varepsilon_{r \theta},\varepsilon_{rz}\right) \varepsilon_{0}$ and $\bar{\bar\mu}=\bar{\bar I}\left( \mu_{rr},\mu_{r \theta},\mu_{rz}\right) \mu_{0}$ in which $\bar{\bar I}$ is the identity dyadic [Reference Sakli, Yahia, Fathallah, Tao and Aguili1]. In fact $\bar{\bar\varepsilon}$ and $\bar{\bar\mu}$ tensors have zero off-diagonal elements (biaxial material). Substituting electric and magnetic fields E and H describing a wave traveling in the z-direction

(1)\begin{equation} \qquad\qquad\qquad\qquad\qquad\textbf{E}(r,\theta,z)=\left[\textbf{e}_{r}E_{r}(r,\theta)+\textbf{e}_{\theta}E_{\theta}(r,\theta)+\textbf{e}_{z}E_{z}(r,\theta) \right]e^{-jk_{z} z}\;, \end{equation}

(2)\begin{equation} \qquad\qquad\qquad\qquad\qquad\quad \textbf{H}(r,\theta,z)=\left[\textbf{e}_{r}H_{r}(r,\theta)+\textbf{e}_{\theta}H_{\theta}(r,\theta)+\textbf{e}_{z}H_{z}(r,\theta) \right]e^{-jk_{z} z}\;, \end{equation}

into Maxwell curl equations, i.e., $\nabla\times\textbf{E}=-j\omega\bar{\bar\mu}\cdot\textbf{H}$ and $\nabla\times\textbf{H}=j\omega\bar{\bar\varepsilon}\cdot\textbf{E}$, we can determine E_r, E_θ, H_r, and H_θ with respect to partial derivatives of components E_z and H_z, as shown by Eqs. (5)–(8) in [Reference Sakli, Yahia, Fathallah, Tao and Aguili1]. By substituting E_r, E_θ, H_r, and H_θ into z-components of Eqs. (1) and (2) in [Reference Sakli, Yahia, Fathallah, Tao and Aguili1], one may obtain the following coupled equations for components E_z and H_z as

(3)\begin{align} &\qquad\qquad\qquad\qquad\qquad\quad\dfrac{\partial^{2}H_{z}}{\partial r^{2}}+\dfrac{1}{r}\dfrac{\partial H_{z}}{\partial r}+\left( \dfrac{K_{c.\theta}\sqrt{\mu_{r\theta}}}{K_{c.r}\sqrt{\mu_{rr}}}\right)^{2}\dfrac{1}{r^{2}}\dfrac{\partial^{2}H_{z}}{\partial\theta^{2}}\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\quad\quad{}+\left(\dfrac{\sqrt{\mu_{rz}}}{\sqrt{\mu_{rr}}}K_{c.\theta} \right)^{2}H_{z} =\dfrac{k_{z}k_{0}^{2}}{r\omega\mu_{0}\mu_{rr}} \dfrac{\mu_{r\theta}\varepsilon_{rr}-\mu_{rr}\varepsilon_{r\theta}}{K_{c.r}^{2}}\dfrac{\partial^{2}E_{z}}{\partial r\partial\theta}\;, \end{align}

(4)\begin{align} &\dfrac{\partial^{2}E_{z}}{\partial r^{2}}+\dfrac{1}{r}\dfrac{\partial E_{z}}{\partial r}+\left( \dfrac{K_{c.r}\sqrt{\varepsilon_{r\theta}}}{K_{c.\theta}\sqrt{\varepsilon_{rr}}}\right)^{2}\dfrac{1}{r^{2}}\dfrac{\partial^{2}E_{z}}{\partial\theta^{2}}\nonumber\\ &\quad{} +\left(\dfrac{\sqrt{\varepsilon_{rz}}}{\sqrt{\varepsilon_{rr}}}K_{c.r} \right)^{2}E_{z} =\dfrac{k_{z}k_{0}^{2}}{r\omega\varepsilon_{0}\varepsilon_{rr}} \dfrac{\mu_{r\theta}\varepsilon_{rr}-\mu_{rr}\varepsilon_{r\theta}}{K_{c.\theta}^{2}}\dfrac{\partial^{2}H_{z}}{\partial r\partial\theta}\;, \end{align}

where $K_{c.r}^{2}=k_{0}^{2}\varepsilon_{rr}\mu_{r\theta}-k_{z}^{2}$, $K_{c.\theta}^{2}=k_{0}^{2}\varepsilon_{r\theta}\mu_{rr}-k_{z}^{2}$, and $k_{0}^{2}=\omega^{2}\varepsilon_{0}\mu_{0}$, and all other parameters in the equations of the present work were defined in [Reference Sakli, Yahia, Fathallah, Tao and Aguili1].

The TE_z and TM_z modes are defined by assuming $E_{z}=0$ and $H_{z}=0$, respectively. Now, let us explore the possibility of TE_z and TM_z wave propagation in the system. If we assume $E_{z}=0$, then Eqs. (3) and (4) reduce to

(5)\begin{align} \dfrac{\partial^{2}H_{z}}{\partial r^{2}}+\dfrac{1}{r}\dfrac{\partial H_{z}}{\partial r}+\left( \dfrac{K_{c.\theta}\sqrt{\mu_{r\theta}}}{K_{c.r}\sqrt{\mu_{rr}}}\right)^{2}\dfrac{1}{r^{2}}\dfrac{\partial^{2}H_{z}}{\partial\theta^{2}}\nonumber\\ +\left(\dfrac{\sqrt{\mu_{rz}}}{\sqrt{\mu_{rr}}}K_{c.\theta} \right)^{2}H_{z} =0\;, \end{align}

(6)\begin{equation} \dfrac{k_{z}k_{0}^{2}}{r\omega\varepsilon_{0}\varepsilon_{rr}} \dfrac{\mu_{r\theta}\varepsilon_{rr}-\mu_{rr}\varepsilon_{r\theta}}{K_{c.\theta}^{2}}\dfrac{\partial^{2}H_{z}}{\partial r\partial\theta}=0\;. \end{equation}

The TE_z waves must satisfy Eqs. (5) and (6). In Sakli et al. work [Reference Sakli, Yahia, Fathallah, Tao and Aguili1], the authors obtained Eq. (5) for TE_z waves (see Eq. (12) in [Reference Sakli, Yahia, Fathallah, Tao and Aguili1]) but Eq. (6) was neglected. This means that the calculations for TE_z modes in Sakli et al. paper [Reference Sakli, Yahia, Fathallah, Tao and Aguili1] do not satisfy Eq. (6), and therefore there is an obvious and serious error in [Reference Sakli, Yahia, Fathallah, Tao and Aguili1]. In physics, the presence of such a significant error in the investigation of a problem is unacceptable and inevitably leads to incorrect results [Reference Moradi5].

Note that Eqs. (5) and (6) cannot be simultaneously satisfied by the same value of k_z unless in two particular cases. In the first case, material parameters must satisfy the condition

(7)\begin{equation} \mu_{r\theta}\varepsilon_{rr}=\mu_{rr}\varepsilon_{r\theta}\;. \end{equation}

In the second case, decoupling occurs if

(8)\begin{equation} \dfrac{\partial H_{z}}{\partial r}=0,\;\;\;\; \mathrm{or} \;\;\;\;\dfrac{\partial H_{z}}{\partial \theta}=0 \;. \end{equation}

These conditions reduce Eq. (6) to zero. Hence, in general TE_z modes cannot propagate in a metallic circular waveguide filled with anisotropic metamaterial. However, from Eq. (7) it is easy to conclude that TE_z and TM_z modes can propagate separately in a metallic circular waveguide when it is filled with uniaxial media, i.e.,

(9)\begin{equation} \varepsilon_{rr}=\varepsilon_{r\theta},\;\;\;\; \mathrm{and} \;\;\;\;\mu_{r\theta}=\mu_{rr}\;. \end{equation}

In other words, the analysis by Sakli et al. [Reference Sakli, Yahia, Fathallah, Tao and Aguili1] is valid for uniaxial cases. Similarly, if we assume that $H_{z}=0$ (TM_z modes), then Eqs. (3) and (4) reduce to

(10)\begin{align} \dfrac{\partial^{2}E_{z}}{\partial r^{2}}+\dfrac{1}{r}\dfrac{\partial E_{z}}{\partial r}+\left( \dfrac{K_{c.r}\sqrt{\varepsilon_{r\theta}}}{K_{c.\theta}\sqrt{\varepsilon_{rr}}}\right)^{2}\dfrac{1}{r^{2}}\dfrac{\partial^{2}E_{z}}{\partial\theta^{2}}\nonumber\\ +\left(\dfrac{\sqrt{\varepsilon_{rz}}}{\sqrt{\varepsilon_{rr}}}K_{c.r} \right)^{2}E_{z} =0\;, \end{align}

(11)\begin{equation} \dfrac{k_{z}k_{0}^{2}}{r\omega\mu_{0}\mu_{rr}} \dfrac{\mu_{r\theta}\varepsilon_{rr}-\mu_{rr}\varepsilon_{r\theta}}{K_{c.r}^{2}}\dfrac{\partial^{2}E_{z}}{\partial r\partial\theta}=0\;, \end{equation}

Again, Eqs. (10) and (11) cannot be simultaneously satisfied unless material parameters satisfy the condition $\mu_{r\theta}\varepsilon_{rr}=\mu_{rr}\varepsilon_{r\theta}$, or when we have either $\partial E_{z}/\partial r=0, $ or $\partial E_{z}/\partial \theta=0 $. Thus, one may conclude that in the general case, a metallic circular waveguide filled with anisotropic metamaterials cannot support the propagation of TM_z modes. This means that the results for TM_z modes, derived by Sakli et al. (see Eq. (18) in [Reference Sakli, Yahia, Fathallah, Tao and Aguili1]) are also incorrect.