## 2 Results  
### The Hidden Momentum, Refraction, Ray-Wave Tilt, and Bound Charge Waves  
Based on the definitions of the electric and magnetic inductions \(\mathbf{D}=E+4\pi\mathbf{P}\) and \(\mathbf{B}=\mathbf{H}+4\pi\mathbf{M}\), the volumetric density of the total hidden momentum \(\Delta\mathbf{g}\), which is the difference between the [PERSON] momentum \(\mathbf{g}_{Abr}\) and the [PERSON] momentum \(\mathbf{g}_{Min}\), is equal to the sum of the [PERSON] hidden momentum \(\mathbf{g}_{RH}\) and [PERSON] hidden momentum \(\mathbf{g}_{SH}\) (see the scheme in Figure 1):  
\[\Delta\mathbf{g}=\mathbf{g}_{Abr}-\mathbf{g}_{Min}=\frac{1}{4\pi c}\mathbf{D}\times\mathbf{B}- \frac{1}{4\pi c}\mathbf{E}\times\mathbf{H}=\mathbf{g}_{RH}+\mathbf{g}_{SH}=\frac{1}{c}\mathbf{P} \times\mathbf{B}+\frac{1}{c}\mathbf{E}\times\mathbf{M}\]  
Figure 1: Summary of the terminology used here and an outline of some of the findings presented in this paper. We considered a generic linear material with the most general bianisotropic constitutive relations:  
\[ \begin{pmatrix}\mathbf{D}\\ \mathbf{B}\end{pmatrix} =\hat{M} \begin{pmatrix}\mathbf{E}\\ \mathbf{H}\end{pmatrix} = \begin{pmatrix}\hat{\mathbf{e}}&\hat{\mathbf{\chi}}\\ \hat{\mathbf{\gamma}}&\hat{\mu}\end{pmatrix}\begin{pmatrix}\mathbf{E}\\ \mathbf{H}\end{pmatrix} , \tag{4}\]  
A plane wave with wave vector \(\mathbf{k}=(0,0,k_{z})\) propagating through a material described by Equation (4) is carrying fields \(\mathbf{E},\mathbf{H},\mathbf{D},\mathbf{B}\propto e^{i(\mathbf{kr}-\omega t)}\). The time-averaged longitudinal component of the hidden momentum is  
\[\overline{\Delta g_{z}}=\frac{1}{4\pi c}\mathbf{z}\cdot\text{Re}\{\mathbf{D}^{*}\times \mathbf{B}-\mathbf{E}^{*}\times\mathbf{H}\} \tag{5}\]  
We utilized the following identities that follow from [PERSON]'s equations:  
\[\text{Re}\{\mathbf{D}^{*}\times\mathbf{B}\}=\frac{1}{k_{0}^{2}}(\mathbf{k}\cdot\text{Re} \{\mathbf{E}\times\mathbf{H}^{*}\})\ \mathbf{k} \tag{6}\]  
\[\frac{k_{0}}{2}\text{Re}\{\mathbf{E}^{*}\cdot\mathbf{D}+\mathbf{H}^{*}\cdot\mathbf{B}\}=(\bm {k}\cdot\text{Re}\{\mathbf{E}^{*}\times\mathbf{H}\}) \tag{7}\]  
Normalizing the electromagnetic fields such that \(\frac{1}{8\pi}\text{Re}\{\mathbf{E}^{*}\cdot\mathbf{D}+\mathbf{H}^{*}\cdot\mathbf{B}\}=\mathbf{U}\), where \(\mathbf{U}\) is the energy density of electromagnetic field, we arrived at  
\[\frac{c\overline{\Delta g_{z}}}{\mathbf{U}}=nf=n\bigg{(}1-\frac{1}{n^{2}}\bigg{)}, \tag{8}\]  
where the index of refraction is \(n=\mathbf{k}/k_{0}\) and the phase velocity is \(v_{ph}=c/n\). From Equation (8), we see that the index of refraction of an electromagnetic wave is directly related to the longitudinal component of the hidden momentum \(\overline{\Delta g_{z}}\) in the wave and can be expressed as  
\[n=\sqrt{1+\frac{1}{4}\bigg{(}\frac{c\overline{\Delta g_{z}}}{\mathbf{U}}\bigg{)}^ {2}}+\frac{1}{2}\bigg{(}\frac{c\overline{\Delta g_{z}}}{\mathbf{U}}\bigg{)} \tag{9}\]  
This result provides a closed-form direct relationship between the index of refraction of the waves and the amplitudes of the waves \(\mathbf{E},\mathbf{H},\mathbf{D},\mathbf{B}\). Let us turn to the transverse component of the hidden momentum \(\Delta g_{\perp}\). According to the index of refraction operator method [32; 33], the longitudinal fields in the wave can be expressed as  
\[(E_{z},H_{z})^{T}=-\hat{M}_{z}^{-1}\cdot\hat{M}_{z\parallel}\cdot\big{(}E_{x },E_{y},H_{x},H_{y}\big{)}^{T} \tag{10}\]  
where  
\[\hat{M}_{z,\parallel}= \begin{pmatrix}e_{31}&e_{32}&X_{31}&X_{32}\\ Y_{31}&Y_{32}&\mu_{31}&\mu_{32}\end{pmatrix} ,\hat{M}_{z,z}= \begin{pmatrix}e_ {33}&X_{33}\\ Y_{33}&\mu_{33}\end{pmatrix}  \tag{11}\]  
In isotropic media \(\hat{M}_{z,\parallel}=\hat{0}\), all fields are solenoidal with purely transverse amplitudes. The isotropy breaking leads to non-zero \(\hat{M}_{z,\parallel}\) and the appearance of longitudinal components \((\mathbf{k}\cdot\mathbf{E}),(\mathbf{k}\cdot\mathbf{H})\
eq 0\) and divergences \((\
abla\cdot\mathbf{E}),(\
abla\cdot\mathbf{H})\
eq 0\).