I want to calculate the dispersion relation (the relation between $\bf k$ and permittivity and permeability tensors and $\omega$) for a TE and a TM wave with wave vector $\mathbf k=k_x\mathbf {\hat x}+k_y\mathbf {\hat y}$ propagating in an anisotropic medium with general diagonal permeability and permittivity tensors $ \mu=\mu_0 \tilde \mu$ and $\epsilon=\epsilon_0 \tilde \epsilon$ with $\tilde \epsilon$ and $\tilde \mu$ given below.

I write the Maxwell's curl equations for the fields with a space-time dependence of the form $\exp(-ik_0\bf k\cdot r+i\omega t)$ (with $k_0=\omega\sqrt{\epsilon_0\mu_0})$ in the $k$ domain as usual:

$$\bf k\times \bf E=\bf {\tilde \mu} \bf B$$ $$\bf k\times \bf H=-\bf{ \tilde \epsilon} \bf E $$

where $\bf{ \tilde \epsilon}$ and $\bf {\tilde \mu}$ are dimensionless matrices: $$\begin{pmatrix} \epsilon_x & 0 &0 \\ 0 & \epsilon_y & 0\\ 0 &0 &\epsilon_z \end{pmatrix},\begin{pmatrix} \mu_x & 0 &0 \\ 0 & \mu_y & 0\\ 0 &0 &\mu_z \end{pmatrix}$$ writing the vector products $\bf k\times \bf H$ and $\bf k\times \bf E$ as matrix multiplication $\bar k \bf H$ with $$\bar k=\begin{pmatrix} 0 & -k_z &k_y \\ k_z & 0 & -k_x\\ -k_y &k_x &0 \end{pmatrix}$$

I rewrite the curl equations as: $$\cases{ \bar k \bf H=-\tilde \epsilon\bf E\\\bar k \bf E=\tilde \mu \bf H}\to\cases{(\bar k \tilde \mu^{-1}\bar k+\tilde \epsilon)\bf E=0 \\(\bar k \tilde \epsilon^{-1}\bar k+\tilde \mu)\bf H =0 }$$ Now, the determinant of the coefficient matrices $\bar k \tilde \epsilon^{-1}\bar k+\tilde \mu$ and $\bar k \tilde \mu^{-1}\bar k+\tilde \epsilon$ must be zero for the systems to have non-trivial solutions.

My question is, where in this process I should consider the assumption of the waves being TM or TE? I know that the final dispersion relation varies for TE and TM modes.

I want to calculate the dispersion relation (the relation between $\bf k$ and permittivity and permeability tensors and $\omega$) for a TE and a TM wave with wave vector $\mathbf k=k_x\mathbf {\hat x}+k_z\mathbf {\hat z}$ propagating in an anisotropic medium with general diagonal permeability and permittivity tensors $ \mu=\mu_0 \tilde \mu$ and $\epsilon=\epsilon_0 \tilde \epsilon$ with $\tilde \epsilon$ and $\tilde \mu$ given below.

I write the Maxwell's curl equations for the fields with a space-time dependence of the form $\exp(-ik_0\bf k\cdot r+i\omega t)$ (with $k_0=\omega\sqrt{\epsilon_0\mu_0})$ in the $k$ domain as usual:

$$\bf k\times \bf E=\bf {\tilde \mu} \bf B$$ $$\bf k\times \bf H=-\bf{ \tilde \epsilon} \bf E $$

where $\bf{ \tilde \epsilon}$ and $\bf {\tilde \mu}$ are dimensionless matrices: $$\begin{pmatrix} \epsilon_x & 0 &0 \\ 0 & \epsilon_y & 0\\ 0 &0 &\epsilon_z \end{pmatrix},\begin{pmatrix} \mu_x & 0 &0 \\ 0 & \mu_y & 0\\ 0 &0 &\mu_z \end{pmatrix}$$ writing the vector products $\bf k\times \bf H$ and $\bf k\times \bf E$ as matrix multiplication $\bar k \bf H$ with $$\bar k=\begin{pmatrix} 0 & -k_z &k_y \\ k_z & 0 & -k_x\\ -k_y &k_x &0 \end{pmatrix}$$

I rewrite the curl equations as: $$\cases{ \bar k \bf H=-\tilde \epsilon\bf E\\\bar k \bf E=\tilde \mu \bf H}\to\cases{(\bar k \tilde \mu^{-1}\bar k+\tilde \epsilon)\bf E=0 \\(\bar k \tilde \epsilon^{-1}\bar k+\tilde \mu)\bf H =0 }$$ Now, the determinant of the coefficient matrices $\bar k \tilde \epsilon^{-1}\bar k+\tilde \mu$ and $\bar k \tilde \mu^{-1}\bar k+\tilde \epsilon$ must be zero for the systems to have non-trivial solutions.

My question is, where in this process I should consider the assumption of the waves being TM or TE? I know that the final dispersion relation depends on this.(being TE or TM)

I want to calculate the dispersion relation (the relation between $\bf k$ and permittivity and permeability tensors and $\omega$) for a TE and a TM wave with wave vector $\mathbf k=k_x\mathbf {\hat x}+k_y\mathbf {\hat y}$ propagating in an anisotropic medium with general diagonal permeability and permittivity tensors $ \mu=\mu_0 \tilde \mu$ and $\epsilon=\epsilon_0 \tilde \epsilon$ with $\tilde \epsilon$ and $\tilde \mu$ given below.

I write the Maxwell's curl equations for the fields with a space-time dependence of the form $\exp(-ik_0\bf k\cdot r+i\omega t)$ (with $k_0=\omega\sqrt{\epsilon_0\mu_0})$ in the $k$ domain as usual:

$$\bf k\times \bf E=\bf {\tilde \mu} \bf B$$ $$\bf k\times \bf H=-\bf{ \tilde \epsilon} \bf E $$

where $\bf{ \tilde \epsilon}$ and $\bf {\tilde \mu}$ are dimensionless matrices: $$\begin{pmatrix} \epsilon_x & 0 &0 \\ 0 & \epsilon_y & 0\\ 0 &0 &\epsilon_z \end{pmatrix},\begin{pmatrix} \mu_x & 0 &0 \\ 0 & \mu_y & 0\\ 0 &0 &\mu_z \end{pmatrix}$$ writing the vector products $\bf k\times \bf H$ and $\bf k\times \bf E$ as matrix multiplication $\bar k \bf H$ with $$\bar k=\begin{pmatrix} 0 & -k_z &k_y \\ k_z & 0 & -k_x\\ -k_y &k_x &0 \end{pmatrix}$$

I rewrite the curl equations as: $$\cases{ \bar k \bf H=-\tilde \epsilon\bf E\\\bar k \bf E=\tilde \mu \bf H}\to\cases{(\bar k \tilde \mu^{-1}\bar k+\tilde \epsilon)\bf E=0 \\(\bar k \tilde \epsilon^{-1}\bar k+\tilde \mu)\bf H =0 }$$ Now, the determinant of the matrices $\bar k \tilde \epsilon^{-1}\bar k+\tilde \mu$ and $\bar k \tilde \mu^{-1}\bar k+\tilde \epsilon$ must be zero for the systems to have non-trivial solutions.

My question is, where in this process I should consider the assumption of the waves being TM or TE? I know that the final dispersion relation varies for TE and TM modes.

I want to calculate the dispersion relation (the relation between $\bf k$ and permittivity and permeability tensors and $\omega$) for a TE and a TM wave with wave vector $\mathbf k=k_x\mathbf {\hat x}+k_y\mathbf {\hat y}$ propagating in an anisotropic medium with general diagonal permeability and permittivity tensors $ \mu=\mu_0 \tilde \mu$ and $\epsilon=\epsilon_0 \tilde \epsilon$ with $\tilde \epsilon$ and $\tilde \mu$ given below.

I write the Maxwell's curl equations for the fields with a space-time dependence of the form $\exp(-ik_0\bf k\cdot r+i\omega t)$ (with $k_0=\omega\sqrt{\epsilon_0\mu_0})$ in the $k$ domain as usual:

$$\bf k\times \bf E=\bf {\tilde \mu} \bf B$$ $$\bf k\times \bf H=-\bf{ \tilde \epsilon} \bf E $$

where $\bf{ \tilde \epsilon}$ and $\bf {\tilde \mu}$ are dimensionless matrices: $$\begin{pmatrix} \epsilon_x & 0 &0 \\ 0 & \epsilon_y & 0\\ 0 &0 &\epsilon_z \end{pmatrix},\begin{pmatrix} \mu_x & 0 &0 \\ 0 & \mu_y & 0\\ 0 &0 &\mu_z \end{pmatrix}$$ writing the vector products $\bf k\times \bf H$ and $\bf k\times \bf E$ as matrix multiplication $\bar k \bf H$ with $$\bar k=\begin{pmatrix} 0 & -k_z &k_y \\ k_z & 0 & -k_x\\ -k_y &k_x &0 \end{pmatrix}$$

I rewrite the curl equations as: $$\cases{ \bar k \bf H=-\tilde \epsilon\bf E\\\bar k \bf E=\tilde \mu \bf H}\to\cases{(\bar k \tilde \mu^{-1}\bar k+\tilde \epsilon)\bf E=0 \\(\bar k \tilde \epsilon^{-1}\bar k+\tilde \mu)\bf H =0 }$$ Now, the determinant of the coefficient matrices $\bar k \tilde \epsilon^{-1}\bar k+\tilde \mu$ and $\bar k \tilde \mu^{-1}\bar k+\tilde \epsilon$ must be zero for the systems to have non-trivial solutions.

My question is, where in this process I should consider the assumption of the waves being TM or TE? I know that the final dispersion relation varies for TE and TM modes.