The dispersion relation \( \omega(k) \) of a one-dimensional crystal lattice with a diatomic basis has two branches. These branches result from the solution of two differential equations for this problem of diatomic chains.
The one branch (one solution) \(\omega_-(k) \) is called acoustic because in this case the lattice planes oscillate in phase, as is the case of acoustic waves.
The other branch (the other solution) \(\omega_+(k) \) is called optical because this solution of the respective differential equation gives an opposite-phase oscillation of the lattice planes (with \(m_1\) and \(m_2\)). For example, if the atoms are ions, then they are electrically charged. An antiphase oscillation creates electric dipole moments in the crystal, which affects the optical properties of the crystal.