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.