G Sphere: Unified Vector Field Framework

• G Sphere is a geometric model that unifies the description of fully polarized, structured optical fields by incorporating variable ellipticity and orbital angular momentum.
• It employs generalized Stokes parameters to map complex vector fields onto shells in G-space, subsuming traditional Poincaré sphere representations as special cases.
• Experimental setups using interferometric techniques and phase plates allow precise synthesis and control of spin and orbital angular momentum in structured light applications.

The G sphere, or Generalized Poincaré sphere, is a geometric representation designed to systematically unify the description of a wide class of vector optical fields. By extending the conventional Poincaré sphere formalism to incorporate continuously variable ellipticity (spin angular momentum, SAM) and arbitrary orbital angular momentum (OAM), the G sphere serves as a comprehensive framework for characterizing fully polarized azimuthally varying beams. Importantly, the conventional (spin-only) and higher-order (spin–OAM hybrid) Poincaré spheres are recovered as special cases—namely, as specific shells within the G sphere. The associated generalized Stokes parameters (“G parameters”) map arbitrary vector field states onto this geometric structure, enabling direct, unified treatment of geometric phase phenomena and structured light generation (Ren et al., 2014).

1. Motivation and Definition

The conventional Poincaré sphere (S sphere) represents homogeneous polarization states via Stokes parameters, embedding each fully polarized state as a point on the unit sphere. This construction, however, inadequately describes spatially varying or hybrid-cylindrical vector fields. The higher-order Poincaré sphere (H sphere) addresses part of this limitation by adopting basis states composed of circular polarizations carrying opposite OAM (±m), allowing the description of cylindrical vector beams with topological singularities. Yet, it remains restricted to fixed-ellipticity cases.

The G sphere supersedes these by:

• Extending the Jones vector basis at the sphere’s poles to a continuously parametrized family with variable ellipticity (controlling SAM).
• Assigning the radial coordinate not to partial polarization, but to the ellipticity of the pole basis.
• Defining the north and south poles of each shell as orthogonal states with opposite SAM and OAM.

Any fully polarized, azimuthally structured field of OAM index $m$ maps uniquely to a point in three-dimensional G-space, with the radial shell $R \in [0.5,1]$ corresponding to the pole basis ellipticity. The two H spheres of fixed $m$ arise as shells at $R=0.5$ (right-circular) and $R=1$ (left-circular), while $R=0.75$ distinguishes linearly polarized OAM modes (Ren et al., 2014).

2. Construction: Basis States and Generalized Stokes Parameters

For fixed $R$ (ellipticity) and $m$ (OAM index), one constructs an orthonormal pair of Jones vector basis states: $$\begin{align*} |N_R^m\rangle &= \frac{1}{\sqrt{2}} e^{-j m \phi} \left( e^{-jR\pi} \, \hat{e}_x - j e^{+jR\pi} \hat{e}_y \right), \ |S_R^m\rangle &= \frac{1}{\sqrt{2}} e^{+j m \phi} \left( e^{-jR\pi} \, \hat{e}_x + j e^{+jR\pi} \hat{e}_y \right), \end{align*}$$ where $\phi$ is the transverse azimuthal coordinate. The parameter $R\in[0.5,1]$ sets the ellipticity, modulating the SAM content, while $m$ dictates the OAM structure.

General vector fields are represented as normalized superpositions,

$$|\psi^m\rangle = \psi_N^m |N_R^m\rangle + \psi_S^m |S_R^m\rangle, \quad |\psi_N^m|^2 + |\psi_S^m|^2 = 1,$$

with amplitudes parametrized as $\psi_N^m = \sin\beta\,e^{-j\phi_0}$ and $\psi_S^m = \cos\beta\,e^{+j\phi_0}$, $\beta\in[0,\frac{\pi}{2}]$, $\phi_0\in[0,\pi)$.

Generalized Stokes (“G”) parameters, encoding the field’s geometry in G-space, are: $$\begin{align*} G_{0R}^m &= R, \ G_{1R}^m &= R \sin 2\beta \cos 2\phi_0, \ G_{2R}^m &= R \sin 2\beta \sin 2\phi_0, \ G_{3R}^m &= -R \cos 2\beta. \end{align*}$$ Every fully polarized, azimuthally varying vector field of charge $m$ thus corresponds to a point $(G_1, G_2, G_3)$ on the $R$-shell in G-space (Ren et al., 2014).

3. Physical Interpretation of G-Sphere Coordinates

The geometry of the G sphere admits explicit physical interpretation:

• Radial coordinate $R$: Encodes the ellipticity of the pole basis states (thus the intrinsic SAM content). In particular, $\epsilon_N = -\cos(2\pi R)$, $\epsilon_S = +\cos(2\pi R)$.
• Latitude ($2\theta$): Controls the mixing ratio of the north and south pole basis states, determining the local ellipticity of the vector field.
• Longitude ($2\varphi$): Specifies the relative phase between basis components, setting the orientation of polarization ellipses.
• OAM index $m$: Fixed for each shell; sets the azimuthal phase structure through $e^{\pm j m\phi}$.

Together, $(R, 2\theta, 2\varphi)$ coordinatize G-space and fully parametrize the physically accessible structured vector fields (Ren et al., 2014).

4. Relationship to Spheres of Lower Order

The conventional Poincaré sphere (S sphere) and higher-order Poincaré spheres (H spheres) correspond to special cases within the G sphere:

• Standard S sphere: $m=0$, $R=0.5$ (right-circular poles).
• Higher-order H spheres: For each $m$, the shells at $R=0.5$ ($+\hat{z}$ OAM) and $R=1$ ($-\hat{z}$ OAM) are the well-studied H spheres of Milione et al.
• Intermediate shells: $R=0.75$ yields basis states with linear polarization, enabling representation of linearly polarized OAM modes. Thus, the G sphere presents a unifying geometric framework subsuming all previous representations as stratified shells (Ren et al., 2014).

5. Experimental Generation and Field Synthesis

A broadly tunable experimental realization involves a modified Sagnac interferometer:

• An input laser, preconditioned with a polarization beam splitter, is split into counter-propagating orthogonally polarized arms.
• The relative intensity of the two arms (setting $\beta$) is controlled by a half-wave plate.
• Spiral vortex phase plates impose the required OAM phases $\exp(\pm j m\phi)$.
• Geometric phase adjusters (combinatorial quarter- and half-wave plates) set the relative phase $\phi_0$.
• The basis ellipticity parameter ($\tau \equiv R$) is controlled by a rotatable quarter-wave plate.

By jointly tuning these elements, arbitrary G sphere states—hence, any fully polarized, structured vector beam with desired SAM and OAM—can be synthesized (Ren et al., 2014).

6. Geometric Phase and Unification

A salient property of the G sphere representation is its unification of geometric (Pancharatnam–Berry) phases across all cases:

• On the S sphere, the geometric phase is half the solid angle traced by a homogeneously polarized field.
• On H sphere shells, the phase generalizes to fixed-ellipticity, structured (cylindrical vector) beams.
• On the G sphere, arbitrary (spatially varying) polarization trajectories correspond to paths on appropriate shells, and the accumulated Pancharatnam–Berry phase is half the oriented hyper-area enclosed in G-space.

This framework thus unifies the geometric phase analysis of all fully polarized, structured optical fields, with all ordinary and higher-order cases appearing as special shells or loops (Ren et al., 2014).

7. Significance and Applications

The G sphere formalism provides a continuous, geometrically transparent method for representing, generating, and analyzing complex vector fields—including those with spatially varying SAM and OAM. Its capacity to accommodate hybrid and high-order fields positions it as an essential tool for exploring structured light, singular optics, and the associated geometric phase effects. The explicit parametrization by generalized Stokes parameters further facilitates experimental control and theoretical analysis of spin-orbit coupled electromagnetic fields (Ren et al., 2014).