 Solution

For a transverse electromagnetic plane wave in a nonmagnetic medium, Maxwell's equations give $c^2{k}^2={\omega }^2{ϵ}_r(\omega )$. Set damping to zero and define $u=c^2{k}^2/{ϵ}_{\infty }$. Substitution yieldsso ${\omega }^4-({\omega }_{\mathrm{LO}}^2+u){\omega }^2+u{\omega }_{\mathrm{TO}}^2=0$. Therefore the two phonon polariton branches areTake the nonnegative square root for the positive physical frequencies. At $k=0$, ${\omega }_{-}=0$ and ${\omega }_{+}={\omega }_{\mathrm{LO}}$. At small wavevector the lower branch is photon-like with velocity $c/\sqrt{{ϵ}_r(0)}$; at large wavevector it approaches ${\omega }_{\mathrm{TO}}$ from below. The upper branch ultimately becomes photon-like with velocity $c/\sqrt{{ϵ}_{\infty }}$. There is no real propagating transverse branch with frequency strictly between ${\omega }_{\mathrm{TO}}$ and ${\omega }_{\mathrm{LO}}$ in this lossless bulk model. Coupling changes the eigenmodes from a bare photon crossing a flat oscillator into mixed normal modes.