Five-dimensional Poincaré sphere system for representing azimuthally varying vector optical fields

Vector optical fields (VOFs) with space-variant polarization on the wave front, have attracted considerable attention and have been applied in many realms from focal engineering to light-matter interaction. Recently, some upgraded Poincaré sphere (PS) models have been presented to describe VOFs, providing new insights in studying and applying structured light. Here, we report a model of five-dimensional (5D) PS system consisting of a series of three-dimensional (3D) spheres (Bloch spheres) located on a two-dimensional (2D) plane. In fact, this model is geometrically similar to the solar system, and the 3D Bloch spheres and the 2D plane are analogous to the “planets” and their orbital plane, respectively. This model is the most generalized model describing the azimuthally varying VOFs we know so far. A reliable and flexible experimental scheme is exploited to generate the VOFs represented by the 5D PS system. Furthermore, the 5D PS system is a complete tool to represent the azimuthally varying VOFs with polarization changing along arbitrary circular path on any 3D PS, and can be implemented for representing and designing the spin and orbital angular momenta. The 5D PS system graphically simplifies the representation of complex structured light and provides a prominent toolkit to the design and application of VOFs as well as the corresponding optical angular momentum.

Figure 1

The schematic of the 5D PS system. The plane is with polar coordinates $(m,\theta )$, and each point on the plane represents an $m\theta$ sphere. The white circles represent the orbits with $m$ = 1–3, respectively. (a)–(d) show the $m\theta$ spheres with coordinates of $(m,\theta )$ = $(1,\pi /2)$, $(1,0)$, $(2,3\pi /2)$, and $(2,7\pi /4)$. The polarization sates of several VOFs in these $m\theta$ spheres are shown with the parameters $(r,2\alpha ,2\phi )$.

Figure 2

Schematic of the experimental setup. SLM: spatial light modulator; L1 and L2: pair of lenses; SF: spatial filter; IC: intensity controller; PC: polarization controller; QWP: quarter-wave plate; HWP: half-wave plate; PBS: polarizing beam splitter; R-Grating: Ronchi phase grating. The coordinates $(m,2\phi )$, $2\alpha$, and $(r,\theta )$ in the 5D PS system are modulated by the SLM, IC, and PC, respectively.

Figure 3

Experimentally measured VOFs in the $m\theta$ spheres with $(m,\theta )$ = $(2,\pi /3)$ in the 5D PS system. The spherical coordinates of the eight VOFs are $(r,2\alpha ,2\phi )$ = $(1,2\pi /5,0)$, $(1,2\pi /5,\pi /2)$, $(0.67,2\pi /5,\pi /2)$, $(0.75,0,\pi /2)$, $(0.5,0,\pi /2)$, $(0.67,0,\pi /3)$, $(0.5,0,0)$, and $(0.67,2\pi /5,0)$, respectively. For each VOF, the four pictures show the total intensity and polarization distributions, Stokes parameters $S_1$, $S_2$, and $S_3$, respectively.

Figure 4

The schematic of deducing the azimuthally varying VOF whose polarization corresponds to arbitrary circular path on the PS. (a1) and (a2) A point with coordinates $(2{\alpha }_1,2{\phi }_1)$ on the PS and the polarization state of the corresponding scalar optical field. (b1) and (b2) The latitude circle on the PS and the corresponding VOF after inserting a vortex retarder with the Jones matrix of ${\mathbf{J}}_1$. (c1) and (c2) The arbitrary circular path on the PS and the corresponding VOF after inserting a phase compensator with Jones matrix of ${\mathbf{J}}_2$.

Figure 5

The OAM and average SAM per photon for the VOFs in the 5D PS system. (a) and (b) show the dependences of the OAM on $m$ and $\alpha$, (c) and (d) show the dependences of the average SAM per photon on $r$ and $\alpha$ when $m$ is a nonzero integer, respectively.

Figure 6

Flow chart of design procedure of the VOF with arbitrary combination of OAM and SAM.