Comments on “Rectangular Waveguide Characterization of Biaxial Material Using TM11 Mode”

Detailed transformations were presented and a substantive error of the theoretical basis of the commented work was shown. For this reason, the considerations presented in the commented work and the resulting conclusions are not justified.


I. INTRODUCTION
I N THE above article [1], the novel measurement technique in the rectangular waveguide was presented to determine the electromagnetic parameters of a biaxial material. The theoretical foundations of this method are presented in [1,Section II]. The authors based on [2] (reference [28]); a correction [3] of [2] exists, assuming the existence of separated TE and TM waves, received (5) and (6), which they then solved. The solution was presented in (7)-(10). This solution is wrong because it does not consider two additional equations (one for TE and one for TM waves) resulting from the boundary problem. In general, solutions (7)-(10) do not satisfy these additional equations. It is explained in Section II.

II. EXPLANATION
Consider a biaxial medium whose principal axes coincide with the axes of the Cartesian coordinate system (x, y, z). The relative permittivity and permeability tensors ε and µ can be represented by diagonal matrices With time dependence exp(+jωt), Maxwell's equations for this medium take the form where ε 0 and µ 0 are the permittivity and permeability of vacuum, respectively. Substituting electric and magnetic fields E and H describing a wave traveling in the z-direction into Maxwell's equations (1) and (2), we obtain The system of Maxwell's equations satisfies the duality principle with respect to substitutions which will be used later in the work. We write (5) and (9) in the form of a system of equations and determine E y and H x with respect to partial derivatives of longitudinal components E z and H z . First, we obtain the principal determinant of this system of equations where β 0 = ω(µ 0 ε 0 ) 1/2 is the wave number in the vacuum. It should be noted that β 0 is an invariant of the principle of duality (β 0 ↔ β 0 ). Following the method of determinants, we obtain a solution to the system of equations in the form of E y and H x relative to the partial derivatives of E z and H z In the same way, from (6) and (8), or by using the duality principle (11), we can obtain E x and H y The transverse field components (12)-(15) have the same form as (3)-(6) in [2].
Differentiating H x with respect to y and H y with respect to x and substituting into (10), we get Moving the terms containing E z to the left-hand side (LHS) and the terms containing H z to the right-hand side (RHS) of the equation, we can obtain Equation (16) has the same form as (7) presented in [2].
Differentiating E x with respect to y and E y with respect to x and substituting into (7) we get Rearranging terms containing H z and E z in the similar way, we can obtain Equation (17) coincides with (8) presented in [2]. Equations (16) and (17) are the system of coupled equations for longitudinal components E z and H z that must be solved. Of course, equations (16) and (17) satisfy the principle of duality, so by finding the first of them it was possible to obtain the second by means of this principle.
In general, hybrid waves propagation should be expected. Instead, let us explore the possibility of TE and TM wave propagation. We substitute E z = 0 into (16) and (17), hence for TE waves Similarly, for TM waves we substitute H z = 0 into (16) and (17), that is, The TE waves must satisfy (18) and (19), while the TM waves must satisfy (20) and (21). It should be strongly emphasized that each of the types of waves must satisfy a different set of two second-order partial differential equations.
As it was stated in [2], the decoupling of E z from H z , which means obtaining separate equations for TM and TE waves, occurs in two particular cases. In the first case, material parameters must satisfy the condition For this case, (18) and (21) are identically equal to zero. This condition is not satisfied for a biaxial medium with any six material parameters. In the second case, decoupling occurs if either It is fulfilled by the TE m0 and TE 0n modes respectively. Only these waves can exist in a biaxial medium. TM waves are unable to propagate. For this reason, the solutions (7)-(10) proposed by the authors of [1] are incorrect.
In the general case, when m ̸ = 0 and n ̸ = 0, this expression is nonzero for all crosssectional points of a rectangular waveguide of dimensions a and b. Equation (18) is not satisfied. Proceeding similarly, it can be shown that solution (9) for the TM mode presented in the above article [1], does not satisfy (21), which ends the proof.

III. CONCLUSION
Those interested in this subject are referred to our work [4], where detailed considerations on the possibility of extracting parameters using TE and TM modes in a rectangular waveguide are presented.