Why does a difference in refractive index cause a phase shift in an electro-optical crystal?

In an electro-optical crystal, a phase shift occurs because the material exhibits birefringence, which is the difference between the refractive index parallel to the optical axis and the refractive index perpendicular to it.1

Because the refractive index determines the speed at which light travels through a medium, a difference in these indices means that light propagates at different velocities along these two directions. As the light waves travel through the crystal, this velocity disparity causes one component of the light to lag behind the other, resulting in a phase shift. In electro-optical materials, this phase shift can be actively controlled or varied by applying an external voltage, which modulates the refractive index of the crystal. This phenomenon is fundamental to the operation of devices such as optical switches, modulators, and demodulators used in high-speed communications.12