  To manipulate orbital angular momentum (OAM) carried by light beams, there is
a great interest in designing various optical elements from the
deep-ultraviolet to the microwave. Normally, the OAM variation introduced by
optical elements can be attributed to two terms, namely the dynamic and
geometric phases. Up till now, the dynamic contribution induced by optical
elements has been clearly recognized. However, the contribution of geometric
phase still seems obscure, especially considering the vector vortex beams. In
this work, an analytical formula is derived to fully describe the OAM variation
introduced by the nonabsorbing optical elements, which perform space-variant
polarization-state manipulations. It is found that the geometric contribution
can be further divided into two parts: one is directly related to optical
elements and the other one explicitly relies solely on the vortices before and
after the transformations. Based on this result, the same OAM variation can be
achieved with different combinations of the dynamic and/or geometric
contributions. With numerical simulations, it is shown that transformation of
the optical vortices can be fully and flexibly designed with a family of
optical elements. We believe that these results are helpful to understand the
effect of optical elements and offer a new perspective to design the optical
elements for manipulating the OAM carried by light beams.
