Sharing a common origin between the rotational and linear Doppler effects

The well-known linear Doppler effect arises from the linear motion between source and observer, while the less well-known rotational Doppler effect arises from the rotational motion. Here, we present both theories and experiments illustrating the relationship between the rotational and linear Doppler effects. A spiral phaseplate is used to generate a light beam carrying orbital angular momentum and the frequency shift is measured arising from its rotational and/or linear motion. By considering either the motion-induced time-evolving phase or the momentum and energy conservation in light-matter interactions, we derive the rotational Doppler shift, linear Doppler shift, and overall Doppler shift associated with rotational and linear motions. We demonstrate the relationship between rotational and linear Doppler shifts, either of which can be derived from the other effect, thereby illustrating their shared origin. Moreover, the close relationship between rotational and linear Doppler effects is also deduced for a more general moving rough surface.