Comment on Sakli, H. Cylindrical Waveguide on Ferrite Substrate Controlled by Externally Applied Magnetic Field. Electronics 2021, 10, 474

Abstract

Recently, Sakli investigated the propagation of electromagnetic waves in metallic cylindrical waveguides filled with longitudinally magnetized ferrite, focusing on TE_z (transverse electric) and TM_z (transverse magnetic) modes relative to the z-axis. This commentary highlights that the proposed system generally cannot support the propagation of the TE_z and TM_z waves, rendering the main results derived by Sakli invalid.

1. Introduction

Recently, Sakli [1] investigated the propagation of TE_z and TM_z modes in metallic cylindrical waveguides filled with lossless, longitudinally magnetized ferrite. He demonstrated how to obtain dispersion diagrams and discussed the impact of anisotropic parameters on dispersion characteristics and cutoff frequencies. Additionally, he presented numerical results for the TE_z and TM_z modes.

However, based on the theory of ferrite-filled cylindrical waveguides obtained in the beginning of the 50s of the last century [2,3], the hybrid wave should be expected for the proposed metallic cylindrical waveguide propagation, and therefore TE_z and TM_z waves are unable to propagate. Additionally, as shown in our recent study [4], a metallic cylindrical waveguide filled with homogeneous anisotropic materials generally supports only hybrid modes. This means that the above-mentioned results for the propagation of TE_z and TM_z modes derived by Sakli [1] are invalid. Let us note that it was also well established that in general cases, the separation of electromagnetic waves into TE and TM modes is not possible in metallic rectangular waveguides [5,6].

We believe that incorrect publications should be corrected to ensure new researchers can build upon accurate prior studies. This motivation drives our commentary, in which we identify the errors in [1].

2. The Rigorous Electromagnetic Analysis

Consider a metallic cylindrical waveguide filled with longitudinally magnetized ferrite as shown in Figure 1 of [1]. We are interested in guided mode solutions in the waveguide propagating in the $z$-direction. Therefore, let us consider the form of the electromagnetic wave propagating in the waveguide as

$$\mathbf{E}\left(r,\theta ,z,t\right)=\mathbf{E}\left(r,\theta \right)e^{j\left(\omega t-k_{z}z\right)},$$

(1)

$$\mathbf{H}\left(r,\theta ,z,t\right)=\mathbf{H}\left(r,\theta \right)e^{j\left(\omega t-k_{z}z\right)},$$

(2)

where $k_z$ is the propagation constant along the $z$-direction. Expanding the Maxwell curl equations (i.e., Equations (1) and (2) in [1]) and using Equation (3) in [1] we obtain

$$\left(\begin{array}{c}{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕E_z}{𝜕\theta }}-{\displaystyle \frac{𝜕E_{\theta }}{𝜕z}}\\ {\displaystyle \frac{𝜕E_r}{𝜕z}}-{\displaystyle \frac{𝜕E_z}{𝜕r}}\\ {\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕\left(r{E}_{\theta }\right)}{𝜕r}}-{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕E_r}{𝜕\theta }}\end{array}\right)=-j\omega {\mu }_0\left(\begin{array}{c}\mu H_r-j\kappa H_{\theta }\\ j\kappa H_r+\mu H_{\theta }\\ {\mu }_{rz}{H}_z\end{array}\right),$$

(3)

$$\left(\begin{array}{c}{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕H_z}{𝜕\theta }}-{\displaystyle \frac{𝜕H_{\theta }}{𝜕z}}\\ {\displaystyle \frac{𝜕H_r}{𝜕z}}-{\displaystyle \frac{𝜕H_z}{𝜕r}}\\ {\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕\left(r{H}_{\theta }\right)}{𝜕r}}-{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕H_r}{𝜕\theta }}\end{array}\right)=j\omega {\epsilon }_0{\epsilon }_{rf}\left(\begin{array}{c}{E}_r\\ E_{\theta }\\ E_z\end{array}\right).$$

(4)

Putting Equations (1) and (2) in Equations (3) and (4), and conducting some manipulation, we obtain

$$E_r={\displaystyle \frac{k_z}{K_c^2}}\left(-j{K}_{c\mu }^2{\displaystyle \frac{𝜕E_z}{𝜕r}}+F{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕E_z}{𝜕\theta }}\right)-{\displaystyle \frac{1}{K_c^2}}\left(A_1{\displaystyle \frac{𝜕H_z}{𝜕r}}+j{A}_2{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕H_z}{𝜕\theta }}\right),$$

(5)

$$E_{\theta }=-{\displaystyle \frac{k_z}{K_c^2}}\left(F{\displaystyle \frac{𝜕E_z}{𝜕r}}+j{K}_{c\mu }^2{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕E_z}{𝜕\theta }}\right)+{\displaystyle \frac{1}{K_c^2}}\left(j{A}_2{\displaystyle \frac{𝜕H_z}{𝜕r}}-A_1{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕H_z}{𝜕\theta }}\right),$$

(6)

$$H_r={\displaystyle \frac{\omega {\epsilon }_0{\epsilon }_{rf}}{K_c^2}}\left(F{\displaystyle \frac{𝜕E_z}{𝜕r}}+j{K}_{c\mu }^2{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕E_z}{𝜕\theta }}\right)-{\displaystyle \frac{1}{K_c^2}}\left(j{k}_z{K}_{c\mu }^2{\displaystyle \frac{𝜕H_z}{𝜕r}}-F{k}_z{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕H_z}{𝜕\theta }}\right),$$

(7)

$$H_{\theta }={\displaystyle \frac{\omega {\epsilon }_0{\epsilon }_{rf}}{K_c^2}}\left(-j{K}_{c\mu }^2{\displaystyle \frac{𝜕E_z}{𝜕r}}+F{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕E_z}{𝜕\theta }}\right)-{\displaystyle \frac{1}{K_c^2}}\left(F{k}_z{\displaystyle \frac{𝜕H_z}{𝜕r}}+j{k}_z{K}_{c\mu }^2{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕H_z}{𝜕\theta }}\right),$$

(8)

where some symbols are introduced for convenience, namely the following:

$$\begin{gathered} K_{c\mu }^2={\epsilon }_{rf}{k}_0^2\mu -k_z^2, \\ F={\epsilon }_{rf}{k}_0^2\kappa , \\ {K}_c^2=K_{c\mu }^4-F^2, \\ {A}_1={\displaystyle \frac{F{k}_z^2}{\omega {\epsilon }_0{\epsilon }_{rf}}}, \\ {A}_2={\displaystyle \frac{K_c^2+k_z^2{K}_{c\mu }^2}{\omega {\epsilon }_0{\epsilon }_{rf}}}. \end{gathered}$$

(9)

We remind the reader that the remaining parameters in the equations used in this work were defined in reference [1]. Note that the longitudinal components satisfy the following coupled equations:

$$\left[{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕}{𝜕r}}\left(r{\displaystyle \frac{𝜕}{𝜕r}}\right)+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{𝜕}^2}{𝜕{\theta }^2}}+{\displaystyle \frac{\omega {\mu }_0{\mu }_{rz}{K}_c^2}{A_2}}\right]H_z=-j{\displaystyle \frac{k_{z}F}{A_2}}\left[{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕}{𝜕r}}\left(r{\displaystyle \frac{𝜕}{𝜕r}}\right)+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{𝜕}^2}{𝜕{\theta }^2}}\right]E_z,$$

(10)

$$\left[{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕}{𝜕r}}\left(r{\displaystyle \frac{𝜕}{𝜕r}}\right)+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{𝜕}^2}{𝜕{\theta }^2}}+{\displaystyle \frac{K_c^2}{K_{c\mu }^2}}\right]E_z={\displaystyle \frac{jF{k}_z}{K_{c\mu }^2\omega {\epsilon }_0{\epsilon }_{rf}}}\left[{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕}{𝜕r}}\left(r{\displaystyle \frac{𝜕}{𝜕r}}\right)+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{𝜕}^2}{𝜕{\theta }^2}}\right]H_z,$$

(11)

so that, in general, a hybrid mode is needed. Note that the right-hand sides of Equations (10) and (11) were neglected in the analysis by Sakli [1]; therefore, his statement that TE_z and TM_z modes can be supported separately in a metallic cylindrical waveguide filled with longitudinally magnetized ferrite is incorrect.

However, the decoupling of $E_z$, from $H_z$ occurs in two particular cases. In the first case, Equations (10) and (11) can be separated into two independent equations of $E_z$ and $H_z$, when the magnetization is equal to zero, i.e., when we have

$$\kappa =0.$$

(12)

In this case, Equations (10) and (11) become

$$\left[{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕}{𝜕r}}\left(r{\displaystyle \frac{𝜕}{𝜕r}}\right)+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{𝜕}^2}{𝜕{\theta }^2}}+{\displaystyle \frac{{\mu }_{rz}}{\mu }}\left({\epsilon }_{rf}{k}_0^2\mu -k_z^2\right)\right]H_z=0,$$

(13)

$$\left[{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕}{𝜕r}}\left(r{\displaystyle \frac{𝜕}{𝜕r}}\right)+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{𝜕}^2}{𝜕{\theta }^2}}+{\epsilon }_{rf}{k}_0^2\mu -k_z^2\right]E_z=0,$$

(14)

and therefore, TE_z and TM_z modes can be supported separately in a metallic cylindrical waveguide filled with longitudinally “unmagnetized” ferrite. In the second case, decoupling occurs if we consider the azimuthal modes, i.e., when we have

$$k_z=0.$$

(15)

Putting the propagation constant equal to zero in Equations (10) and (11), we obtain

$$\left[{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕}{𝜕r}}\left(r{\displaystyle \frac{𝜕}{𝜕r}}\right)+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{𝜕}^2}{𝜕{\theta }^2}}+{\epsilon }_{rf}{\mu }_{rz}{k}_0^2\right]H_z=0,$$

(16)

$$\left[{\displaystyle \frac{1}{r}}{\displaystyle \frac{𝜕}{𝜕r}}\left(r{\displaystyle \frac{𝜕}{𝜕r}}\right)+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{𝜕}^2}{𝜕{\theta }^2}}+{\displaystyle \frac{{\epsilon }_{rf}}{\mu }}\left({\mu }^2-{\kappa }^2\right)k_0^2\right]E_z=0.$$

(17)

In this case, TE_z and TM_z modes with components ($H_z,E_r,E_{\theta }$) and ($E_z,H_r,H_{\theta }$) appear, respectively.

3. Conclusions

In this work, we began by using the general mathematical framework for how electromagnetic waves travel through a metallic cylindrical waveguide that is filled with a ferrite material magnetized along its length (as outlined in references [2] and [3]). Our analysis demonstrated that, in most situations, this type of system does not permit the distinct separation of TE_z and TM_z modes. There are only two special scenarios where this separation might occur. As a result, the main findings presented by Sakli in reference [1] are invalid.