Lyddane-Sachs-Teller relation (${ϵ}_r(0)/{ϵ}_{\infty }={\omega }_{\mathrm{LO}}^2/{\omega }_{\mathrm{TO}}^2$)

For a single lossless polar optical oscillator with frequency-independent electronic background, ${ϵ}_r(\omega )={ϵ}_{\infty }({\omega }_{\mathrm{LO}}^2-{\omega }^2)/({\omega }_{\mathrm{TO}}^2-{\omega }^2)$. Setting frequency to zero yields the displayed static-to-background ratio. Multiple infrared-active branches require the corresponding product relation, rather than selecting an arbitrary pair of modes.