  Recent advances in planar optics with geometric-phase superstructures have
brought a new paradigm in the control of structured light and, in particular,
has substantially enhanced the capabilities of generating and detecting orbital
angular momentum (OAM) states of light and associated spatial modes. However,
the structured modal interface that can reciprocally link OAM states via
adiabatic control and access-associated higher-order geometric phase remains
absent in planar optics. In this work, we propose and experimentally
demonstrate a planar optical astigmatic retarder fabricated with liquid-crystal
(LC) geometric phase. The LC superstructure was designed with the principle of
fractional Fourier transformation and is capable of reciprocal conversion
between all possible OAM states on the same modal sphere. Such a planar device
paves the way towards an easily deployed modal interface of paraxial OAM
states, unlocks the resource of higher-order geometric phase, and has promising
applications in high-dimensional classical/quantum information.
