Cold collisionless plasma dielectric response (${\epsilon }_r=1-{\omega }_p^2/{\omega }^2$)

In an unmagnetized plasma, an electron displacement driven by $\mathbf{E}{e}^{-i\omega t}$ obeys $-m_e{\omega }^2{\mathbf{x}}_e=-e\mathbf{E}$. Its dipole moment is $-e{\mathbf{x}}_e$, giving electric polarization $\mathbf{P}=-n{e}^2\mathbf{E}/(m_e{\omega }^2)$. With $\mathbf{D}={\epsilon }_0\mathbf{E}+\mathbf{P}$, this yields the displayed relative permittivity and ${\omega }_p^2=n{e}^2/(m_e{\epsilon }_0)$. Ions contribute a further negative term smaller by $Z{m}_e/m_i$ for one ion species of charge $Ze$. Neglecting it requires heavy ions at electron-wave frequencies; collisions, background magnetization and pressure dispersion are excluded. The transverse electromagnetic wave relation is $c^2{k}^2={\omega }^2-{\omega }_p^2$.

For a downward transverse wave in an isotropic plasma, $E_z/E_x=\tan {\theta }_t$ fixes the transmitted direction. Tangential phase matching gives $\sin {\theta }_i=n_p\sin {\theta }_t$. Continuity of tangential electric and magnetic fields then separates incident and reflected amplitudes. A measured transmitted electric amplitude alone cannot be identified with the incident one; mean power for peak phasors includes a factor of one half, and interface-normal flux includes a cosine.

For vacuum incidence on a sharp planar, collisionless plasma interface, the transmitted normal wavenumber obeys $k_z^2=({\omega }^2/c^2)({\cos }^2\theta -{\omega }_p^2/{\omega }^2)$. Negative $k_z^2$ produces an evanescent field and total reflection in this lossless model.