Higher-Order Poincaré Sphere (HOPS)

• HOPS is a geometric framework that extends the Poincaré sphere to describe vector vortex beams with coupled spin and orbital angular momentum.
• It leverages coherent superpositions of circularly polarized Laguerre–Gaussian modes to map spatially inhomogeneous polarization states.
• Experimental implementations using interferometers, metasurfaces, and intra-cavity setups enable high-fidelity synthesis and control of HOPS states.

The Higher-Order Poincaré Sphere (HOPS) is a geometric formalism that generalizes the well-known Poincaré sphere (PS) from homogeneous polarization states (described purely by spin angular momentum, SAM) to vector beams that embody a nontrivial interplay of SAM and orbital angular momentum (OAM). On the HOPS, each point corresponds to a distinct spatially inhomogeneous polarization structure—a "vector vortex beam"—characterized by the coherent superposition of two orthogonal circularly polarized vortex modes of opposite OAM. This generalization underpins the design, synthesis, manipulation, and classification of spatially structured light, with direct relevance to singular optics, high-dimensional classical/quantum communication, and photonic quantum information science.

1. Mathematical Structure and Generalized Stokes Parameters

The HOPS can be rigorously defined as follows: for a given order $\ell$, the basis states are right- and left-circularly polarized Laguerre–Gaussian (LG) modes carrying OAM charges $+\ell$ and $-\ell$: $$|R_\ell\rangle = e^{-i\ell\phi}\frac{x̂ - i ŷ}{\sqrt2}, \quad |L_\ell\rangle = e^{+i\ell\phi}\frac{x̂ + i ŷ}{\sqrt2}$$ An arbitrary pure state on the order-$\ell$ HOPS is

$$|\Psi\rangle = \cos\left(\frac{\theta}{2}\right) e^{-i\frac{\phi}{2}}|R_\ell\rangle + \sin\left(\frac{\theta}{2}\right) e^{+i\frac{\phi}{2}}|L_\ell\rangle$$

where $\theta\in[0, \pi]$ and $\phi\in[0, 2\pi)$ are the polar and azimuthal angles, mapping directly to the latitude and longitude on the HOPS (Chen et al., 2014, Sato et al., 2023, Umar et al., 25 Jun 2025).

The generalized Stokes parameters for the HOPS are

$$S_1^{(\ell)} = 2~\mathrm{Re}(\psi_R\psi_L^*), \quad S_2^{(\ell)} = 2~\mathrm{Im}(\psi_R\psi_L^*), \quad S_3^{(\ell)} = |\psi_R|^2 - |\psi_L|^2$$

where $\psi_R, \psi_L$ are the expansion coefficients in the HOPS basis. The vector $(S_1^{(\ell)}, S_2^{(\ell)}, S_3^{(\ell)})$ lies on the unit sphere $S^2$, reflecting the pure-state constraint (Sato et al., 2023, Fickler et al., 10 Jun 2024).

2. Physical Interpretation: Spin–Orbit Coupling and State Topology

The HOPS encapsulates the set of all nonseparable spin–orbit coupled pure states in the spin ($|R\rangle$, $|L\rangle$) and OAM ($\ell \in \mathbb{Z}$) subspaces. The sphere's north and south poles are $|R_\ell\rangle$ and $|L_\ell\rangle$, while the equator (e.g., $\theta = \pi/2$) contains all cylindrical vector (CV) beams—radially or azimuthally polarized states (depending on $\phi$) (Chen et al., 2014, Ling et al., 2015). Traversing along a meridian (fixed $\phi$) varies the handedness; traversing a parallel (fixed $\theta$) imprints a Pancharatnam–Berry (PB) geometric phase (Yi et al., 2014, Umar et al., 25 Jun 2025). The degree of spatial twisting—i.e., the topological index—is set by $\ell$, and OAM phase winding is manifest in the local polarization field.

3. Experimental Realization and State Manipulation

Several platforms enable dynamic access to arbitrary states on the HOPS:

• Interferometric Generation: A collinear Mach–Zehnder configuration splits an OAM beam, independently modulating each path to select relative amplitude (latitude $\theta$) and phase (longitude $\phi$), before recombination sets the precise HOPS state. Full latitude/longitude scan is controlled via rotating polarizers and waveplates (Chen et al., 2014).
• Sagnac Interferometer: Allows for robust and high-stability generation, with the state vector tuned by waveplates and structured-phase elements for arbitrary HOPS navigation (Ling et al., 2015).
• Intra-cavity Generation: A combination of a quarter-wave plate (QWP) and a spatially inhomogeneous $q$-plate within a solid-state laser cavity breaks the degeneracy between $+\ell$ and $-\ell$ OAM eigenmodes, providing direct emission of arbitrary HOPS states with $>95\%$ modal purity and continuous tunability via the two element angles (1505.02256).
• Metasurface Transformation: Locally structured birefringent metasurfaces (space-variant waveplates) impart OAM-dependent geometrical phases, converting homogeneous input polarization into any desired vector vortex beam on the HOPS (Liu et al., 2014).
• Waveplate SU(2) Gadgets: A universal SU(2) gadget for HOPS utilizes two quarter-wave $q$-plates and one half-wave $q$-plate (all with the same charge $q$). Proper tuning of the retardance and offsets enables arbitrary holonomic evolution on the HOPS (Umar et al., 10 Nov 2025, Umar et al., 13 Sep 2025).

Table: Characteristic Features of Main Generation Platforms

Platform	HOPS State Control	Modal Purity
Mach–Zehnder Interferometer	Lat./long. (polarizer/waveplate)	$>95\%$ (single-beam)
Intra-cavity (QWP + q-plate)	Geometric phase/rotation	$>98\%$ ($\ell=1,10$)
Metasurface	Local axis + retardance	Device-limited
Sagnac Interferometer	Polarizer/SLM phase	High, fast switching
SU(2) Gadget (q-plates)	Offset/retardance angles	Universal holonomic

4. SU(2) Structure, Effective Gadgets, and Holonomy

Every HOPS transformation corresponds to an SU(2) operation in the spin–orbit subspace. The Jones matrix for $q$-plates (with retardance $\delta$ and axis offset $\alpha_0$) acts as

$$M(\delta,\alpha(\phi)) = \begin{pmatrix} \cos\tfrac\delta2 + i\sin\tfrac\delta2 \cos2\alpha(\phi) & i\sin\tfrac\delta2\sin2\alpha(\phi) \ i\sin\tfrac\delta2\sin2\alpha(\phi) & \cos\tfrac\delta2 - i\sin\tfrac\delta2\cos2\alpha(\phi) \end{pmatrix}$$

with $\alpha(\phi) = q\phi + \alpha_0$ (Umar et al., 25 Jun 2025). Rotations on the HOPS are achieved by sequentially applying quarter- and half-wave $q$-plates with fixed offset angles—a direct parallel to the QHQ gadget for the fundamental PS, but with spatially inhomogeneous axes. The condition $q=\eta$ (holonomy) ensures all states remain within the given HOPS. Under this constraint, the three-$q$-plate gadget acts as a single effective waveplate whose combined retardance and fast axis cover all SU(2) rotations on the HOPS (Umar et al., 10 Nov 2025, Umar et al., 13 Sep 2025, Bansal et al., 27 Aug 2025).

5. Extensions: Hybrid-Order Spheres, Generalized and Ellipsoidal Representations

The HOPS formalism has been extended:

• Hybrid-Order Spheres (HyOPS): The hybrid-order Poincaré sphere unifies the familiar Poincaré, orbital, and higher-order spheres by allowing the basis states to differ not only in $\sigma$ (SAM) and $\ell$ (OAM), but also arbitrary total angular momentum (TAM) mismatch. This flexibility supports analysis of spin–orbit interaction and Berry phase in complex media such as $q$-plates and metasurfaces (Yi et al., 2014).
• Generalized (G) Sphere: The G sphere embeds the HOPS as one of its concentric shells, providing a radial coordinate for basis ellipticity (SAM), thus covering all homogeneous and inhomogeneous polarization and spatially varying fields (Ren et al., 2014).
• Ellipsoidal (HOPE) Representation: To distinguish spatial modes with ellipticity (e.g., Ince–Gaussian beams), the HOPS has been generalized to a higher-order Poincaré ellipsoid (HOPE), wherein the Stokes vector is mapped onto a spheroid whose eccentricity reflects the beam’s physical mode ellipticity (Daza-Salgado et al., 26 Feb 2025).

6. Applications and Functional Scope

HOPS-based synthesis and analysis enable:

• Classical and Quantum Communications: Exploitation of the HOPS’s high-dimensional state space allows for mode-division and polarization-division multiplexing, structured-light encoding, robust quantum key distribution in OAM-degenerate Hilbert spaces (Luan et al., 2023, 1505.02256).
• Quantum Information: Preparation, manipulation, and tomography of spin–orbit hybrid qubits and high-dimensional entangled states for advanced protocols in quantum communication, nonlocality tests, and quantum memories (Liu et al., 2 Sep 2025, Ling et al., 2015).
• Singular Optics and Structured Light: Engineering, transformation, and braiding of optical polarization singularities, optical skyrmions, and Möbius strips through controlled paths on the HOPS (Umar et al., 10 Nov 2025, Liu et al., 2 Sep 2025).
• Microscopy and Imaging: Enhanced resolution, focal shaping, and sensitivity through the use of tailored vector vortex beams exploiting the full space of the HOPS (1505.02256, Liu et al., 2014).

7. Limitations, Recent Advances, and Future Perspectives

Traditional approaches to HOPS transformations lacked a universal, minimal gadget analogous to the QHQ device for the PS. Recent works have shown that a three-$q$-plate combination is sufficient for holonomic SU(2) navigation under the holonomy constraint. Further optimizations using metasurfaces, chip-scale photonics, and effective waveplates have realized fast, broadband, scalable access to all HOPS states and their generalizations, with experimental demonstration of high-fidelity transformations and deterministic control (Umar et al., 10 Nov 2025, Luan et al., 2023).

Continuing work involves integration of additional degrees of freedom (frequency, spatio-spectral and temporal modes), complete tomography and control in complex media, and extension of the formalism to encompass multidimensional Poincaré hyperspheres in photonic platforms (Fickler et al., 10 Jun 2024, Daza-Salgado et al., 26 Feb 2025).

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References:

• (Chen et al., 2014) Generation of arbitrary cylindrical vector beams on the higher order Poincare sphere
• (1505.02256) Controlled generation of higher-order Poincare sphere beams from a laser
• (Yi et al., 2014) Hybrid-order Poincaré sphere
• (Umar et al., 25 Jun 2025) SU(2) polarization evolution on higher-order Poincaré sphere by using general $q$-plate
• (Umar et al., 10 Nov 2025) Effective SU(2) gadget: holonomic walk on higher-order Poincaré sphere
• (Ling et al., 2015) Generation of arbitrary full Poincaré beams on the hybrid-order Poincaré sphere
• (Liu et al., 2014) Realization of polarization evolution on higher-order Poincare sphere with metasurface
• (Daza-Salgado et al., 26 Feb 2025) A Higher-Order Poincaré Ellipsoid representation for elliptical vector beams
• (Sato et al., 2023) Higher-order Bloch spheres: A generalized representation of electron spin states with azimuthal phase factor
• (Luan et al., 2023) Fully Tunable On-Chip Meta-Generator for Multidimensional Poincaré Sphere mapping
• (2411.4485) Generalized Poincaré Sphere
• (Bansal et al., 27 Aug 2025) Gadget to realize arbitrary polarization transformation on a higher order Poincaré sphere
• (Fickler et al., 10 Jun 2024) Higher-order Poincaré Spheres and Spatio-Spectral Poincaré Beams
• (Umar et al., 13 Sep 2025) SU(2) gadget for higher-order Poincaré sphere
• (Liu et al., 2 Sep 2025) Synthesis of higher-order Poincaré sphere THz beams