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Relation between the Poynting and Group Velocity Vectors of Electromagnetic Waves in a Bi-Gyrotropic Medium

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Abstract

Analytical formulas are obtained for all components of the high-frequency field, Poynting vector \(\overrightarrow P \), and group velocity vector \(\overrightarrow U \) of electromagnetic waves propagating in an arbitrary direction in an unbounded bi-gyrotropic medium described by the Hermitian permittivity and permeability tensors. It is proven that the corresponding components of vectors \(\overrightarrow P \) and \(\overrightarrow U \) are proportional to each other (therefore, these vectors are parallel) and the ratio between these components is the volume density of the wave energy. The change in the absolute value and orientation of vector \(\overrightarrow U \) and orientation of vector \(\overrightarrow P \) depending on the orientation of the wave vector for different types of electromagnetic waves propagating in a ferromagnetic medium (a particular case of a bi-gyrotropic medium) is calculated. It is found that vectors \(\overrightarrow U \) and \(\overrightarrow P \) are always identically oriented for waves of all types in a ferromagnetic medium.

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Notes

  1. It is often called the Umov‒Poynting vector.

  2. Equation (24) was presented earlier in [9, (4.31)].

  3. To avoid errors when differentiating Eq. (23) with respect to wavenumbers, as well as when differentiating with respect to frequency ω, the designations \(k_{\varphi }^{2},\) Fν, Fg, and Fνg should be replaced by corresponding expressions (20)–(22) containing wavenumbers kx, ky, and kz and quantity k0 by ω/c. After differentiation, these designations can be used again to briefly write expressions (47)–(50).

  4. In terms of stereometry, we can say that the isofrequency dependence is a generatrix for the corresponding figure of revolution (the isofrequency surface).

  5. This is related to the fact that, in the investigated medium, vectors \(\overrightarrow k \), \(\overrightarrow U \), and \(\overrightarrow P \) cannot be always oriented along the z-axis, while they can be directed along the x-axis always and for all types of waves (see the figures below).

  6. The change in P and w values is not presented in this study since they depend on the power of a wave excitation source. Seemingly, they can be normalized to factor \(E_{{z0}}^{2}\); however, such a calculation will lead to an incorrect representation of the change in values P and w as functions of different parameters. For example, \({{P\left( {\theta {\kern 1pt} '} \right)} \mathord{\left/ {\vphantom {{P\left( {\theta {\kern 1pt} '} \right)} {E_{{z0}}^{2}}}} \right. \kern-0em} {E_{{z0}}^{2}}}\) and \({{w\left( {\theta {\kern 1pt} '} \right)} \mathord{\left/ {\vphantom {{w\left( {\theta {\kern 1pt} '} \right)} {E_{{z0}}^{2}}}} \right. \kern-0em} {E_{{z0}}^{2}}}\) dependences are meaningless during propagation of waves along the z-axis, since, in this case, formulas (42)(44) and (51) cannot be used (since they will have sin2θ = 0 in the denominator) and, in addition, Ez0 = Hz0 = 0 (see [9, 12]). Note also that, when waves propagate along the x- and y-axes, formulas (42)(44) and (51) cannot be used either, since the denominator will contain Fg = 0 (substituting kz = 0 into formula (21) and setting \(k_{\varphi }^{2}\) equal to \(k_{0}^{2}{{\varepsilon }_{ \bot }}{{\mu }_{{zz}}}\), according to [9, (4.52)], we obtain Fg = 0). Thus, the investigation of the changes in quantities P and w is a separate problem, which goes beyond this study.

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Funding

This study was carried out within the state assignment, theme no. 0030-2019-0014.

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Correspondence to E. H. Lock.

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Translated by E. Bondareva

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Lock, E.H., Lugovskoi, A.V. & Gerus, S.V. Relation between the Poynting and Group Velocity Vectors of Electromagnetic Waves in a Bi-Gyrotropic Medium. J. Commun. Technol. Electron. 66, 834–843 (2021). https://doi.org/10.1134/S106422692107007X

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  • DOI: https://doi.org/10.1134/S106422692107007X

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