Longitudinal patterning of twisted light

Light beams with azimuthal phase dependence [$exp(i \ell\phi)$] carry orbital angular momentum (OAM) which differs fundamentally from spin angular momentum (SAM) associated with polarization. Striking difference between the two momenta is manifested in the allowable values: where SAM is limited to $\hbar k_0$ per photon, the OAM has unbounded value of $\ell\hbar$ per photon ($\ell$ is integer), thus dramatically exceeding the value of SAM \cite{Ref1,Ref2, Ref3}. OAM has thus been utilized in optical trapping \cite{Ref4}, imaging\cite{Ref2}, and material processing \cite{Ref5}. Furthermore, the unbounded degrees-of-freedom in OAM states have been deployed in data communications \cite{Ref6}. Here, we report an \textit{exceptional} behavior for a class of light beams---known as Frozen Waves (FWs)---whose intensity and azimuthal phase profiles can be controlled along the propagation direction, at will. Accordingly, we generate rotating light patterns that can change their sense of rotation and order of phase twist with propagation. Manipulating OAM along the beam axis can open new directions in optical science and its applications.