I would like to understand half wave plate, quarter wave plates and q-plate. (A q-plate is an optical device which can generate light beams with orbital angular momentum of light (OAM)). Quarter-wave plate (QWP), which converts linearly polarized light into circularly polarized light. Half wave plate (HWP), which shifts the polarization direction of linearly polarized light. When HWP at 45 and QWP at 45 then pass to q-plate. What is output mode? Is it radial mode or azimuthal mode?
Perhaps the best way to understand half wave plates and quarter wave plates is to think of their operations on the Poincare sphere. These wave plates rotate the polarization states on the sphere by half a rotation or a quarter of a rotation, respectively. The orientation of the rotation axis is determined by the orientation of the wave plate. The rotation axis always passes through those points on the Poincare sphere that are associated with the normal modes (i.e. those polarization states that are not affected by the wave plate in a given orientation).
While half wave plates and quarter wave plates are homogeneous, q-plate at not homogeneous. A q-plate is basically a half wave plate with an orientation that varies across the surface of the plate. To understand the way that a q-plate works, you first need to understand geometric phase. A q-plate employs geometric phase to modulate an incoming optical field with a given state of polarization to have a helical phase, which carried orbital angular momentum. Depending on the incident state of polarization, the field will obtain either a positive or negative topological charge. Since q-plates produce a helical phase modulation, they only affect the azimuthal degrees of freedom and not the radial degrees of freedom.
Half wave plates and quarter wave plates can be used to prepare the incoming field to have a given state of polarization such that the q-plate would produce the desired topological charge.