Fractional Orbital Angular Momentum Modes

• Fractional OAM modes are optical states defined by non-integer helical phase structures and branch cuts, allowing continuous-variable quantum protocols.
• They are generated via spatial light modulators and decomposed into superpositions of integer OAM modes, facilitating precise experimental characterization.
• These modes find applications in high-dimensional quantum communications, spectroscopy, and metrology, enhancing secure and efficient light–matter interactions.

Fractional orbital angular momentum (OAM) modes are optical or quantum states in which the helical phase structure associated with the vortex field is characterized by a non-integer (fractional) multiple of $2\pi$ per azimuthal cycle. Unlike conventional OAM states, which have quantized (integer) topological charge, fractional OAM states feature a phase singularity or discontinuity (often referred to as a branch cut or edge dislocation) and are not eigenstates of the standard angular momentum operator. Their experimental realization and characterization enable continuous-variable and high-dimensional quantum information protocols as well as advanced applications in nonlinear optics, spectroscopy, and communications.

1. Mathematical Representation and Physical Origin

Fractional OAM modes generalize the angular dependence of vortex fields from integer to real-valued topological charge. For a scalar paraxial field (e.g., an optical beam or an electron vortex), the basic angular component is:

$\exp(il\phi)$

where $l \in \mathbb{R}$ for fractional OAM and $\phi$ is the azimuthal coordinate. When $l$ is not an integer, this function is multivalued on $[0,2\pi]$, necessitating the introduction of a branch cut (phase discontinuity) characterized by its orientation $\alpha$.

Fractional OAM states are typically constructed by applying a unitary operator $S(l, \alpha)$, corresponding to a non-integer helical phase and branch cut at angle $\alpha$, to a Laguerre–Gaussian (LG) or Bessel beam (Junior et al., 16 Sep 2025). The canonical example is a Laguerre–Gaussian mode:

$$u(r, \phi, z) \propto \left(\frac{\sqrt{2}r}{w(z)}\right)^{|l|} L_p^{|l|}\left(2 r^2 / w(z)^2\right) \exp\left( -\frac{r^2}{w(z)^2} \right) \exp(i l \phi) \ldots$$

with $l$ now real-valued and $w(z)$, $L_p^{|l|}(\cdot)$ as in the standard LG formulation (Shao et al., 2013).

The physical realization of fractional OAM is closely related to the introduction of phase dislocations and to the geometric (Berry) phase, as elucidated by the skyrmionic electron model, which leads to a nonquantized monopole charge and thence to a fractional OAM shift (Bandyopadhyay et al., 2016).

2. Decomposition and Basis-Independent Properties

Fractional OAM states are not angular momentum eigenstates but can be decomposed into a superposition of integer-OAM modes:

$$|\ell, \alpha\rangle = \sum_{m} C_{m'}[M(\alpha)] |m\rangle$$

with expansion coefficients that depend on the fractional part $u = l - \lfloor l \rfloor$, the orientation $\alpha$ of the edge, and the initial starting phase $\theta_0$ (Shao et al., 2013, Junior et al., 16 Sep 2025):

$$C_{m'}[M(a)] = \frac{e^{-ip a}}{i}\, e^{i(l-m')\theta_0} \frac{ e^{i(l-m')a} [1 - e^{i2\pi(l-m')}] }{l-m' }$$

Physical observables for sets of fractional OAM states are most usefully encoded in unitary-invariant two-mode overlaps:

$$r_{ij} = |\langle \ell_i, \alpha_i | \ell_j, \alpha_j \rangle|^2$$

which capture the relational properties between states decoupled from any single basis expansion. These overlaps can be directly measured and are crucial for certifying coherence and dimensionality in ensembles of fractional OAM states (Junior et al., 16 Sep 2025).

3. Experimental Generation and Measurement

Several methodologies enable both the generation and experimental characterization of fractional OAM modes:

• Generation: Fractional OAM beams are commonly created by imprinting a non-integer helical phase onto a spatial light modulator (SLM), often accompanied by an explicit phase discontinuity at a specified orientation angle $\alpha$ (Berger et al., 2018, Junior et al., 16 Sep 2025). In electron vortex beams (EVB), geometric phase manipulation via vortices with nontrivial solid angle yields fractional OAM (Bandyopadhyay et al., 2016).
• Detection and Analysis:
  • Interference-Based Overlap Measurement: A robust technique involves interfering two shifted fractional OAM modes in the focal plane, recording the intensity image, and extracting the overlap from the Fourier transform at a specific spatial frequency (Junior et al., 16 Sep 2025). Only a single image is required per pair, simplifying the experimental setup and bypassing the need for phase locking.
  • OAM Sorting/Transformation: OAM sorters based on coordinate transformation—such as log-polar or spiral coordinate mappings—can convert FOAM to IOAM modes (Mao et al., 19 Apr 2024). The sorted intensity distribution, center-of-mass, and variance can then distinguish among fractional, integer, and mixed states (Berger et al., 2018).
  • Spectral and Modal Probing: Instruments employing rotational Doppler effect or Talbot-based sorting offer resolution for both amplitude and phase, enabling tomography of fractional OAM superpositions (Zhou et al., 2016, Hu et al., 22 Feb 2024).

4. Coherence, Dimensionality, and Certification

Fractional OAM modes enable construction of sets that are provably coherent and high-dimensional. Certification proceeds by evaluating overlap-based witnesses:

• Basis-Independent Coherence: Inequalities (e.g., $r_{AB} + r_{BC} - r_{AC} \leq 1$) borrowed from quantum state discrimination are violated only when the states form coherent quantum superpositions, thus certifying “basis-independent” coherence (Junior et al., 16 Sep 2025).
• Dimensionality: The geometrical configuration of the overlaps imposes boundaries corresponding to the Hilbert space dimension. Demonstrated violations of two-dimensional bounds by triples of FOAM states prove the set spans at least a three-dimensional subspace. More advanced relational witnesses (e.g., higher-order $h_n(r)$ functions) provide dimension estimates for larger sets, though sensitivity decreases with $n$.

This methodology is robust to mode impurity and is applicable independently of the explicit spatial basis, rendering it especially suitable in experimental contexts where purity may be limited.

5. Physical and Practical Implications

The unique structure of fractional OAM states confers several consequences for their role in light–matter interaction and quantum information:

• State Space Expansion: FOAM modes offer a continuous, essentially unbounded state space, substantially enlarging the accessible Hilbert space for communications or quantum computation.
• High-Dimensional Protocols: The basis-independent certification of dimensionality underpins the use of FOAM modes for high-capacity quantum key distribution or entanglement, surpassing qubit paradigms in channel capacity and noise resilience (Junior et al., 16 Sep 2025).
• Mode Sorting and Security: The difficulty in sorting and unambiguously identifying FOAM modes (as opposed to IOAM) provides a natural advantage for secure communications (Mao et al., 19 Apr 2024).
• Spectroscopy and Metrology: FOAM-sensitive technique enables transient and mode-mixed vortex-state discrimination in exciton-polariton microcavities and related semiconductor systems (Berger et al., 2018). The measured OAM moments, including orientation-dependent extrinsic contributions, are especially relevant in studies of vortex dynamics.

6. Theoretical Foundations: Geometric Phase, Stability, and External Fields

The formation and stability of FOAM states are tied to geometric phase effects and system-specific interactions:

• Geometric Phase in Electron Vortex Beams: In EVBs, the geometric phase acquired (proportional to monopole charge $\mu$) results in a shifted expectation value $\langle L_z \rangle = (l - \mu)$, with $\mu = \frac{1}{2}(1 - \cos \theta)$ depending on the tilt of the vortex axis. Renormalization group (RG) flow governs the scale-dependence and thus the propagation stability of the fractional state—instability is suppressed in the presence of an external magnetic field via an effective monopole charge $\mu_{\mathrm{eff}} = \mu + g(2p + |l| + 1)$ chosen by radial index and field sign (Bandyopadhyay et al., 2016).
• Decomposition and Propagation: Fractional OAM beams can be stably propagated and converted in systems with nonlinear interactions and engineered phase matching (e.g., QPM in PPLN) where the conservation of OAM ensures that the output state corresponds to the same fractional content (e.g., SHG: $l' \to 2l'$) as the input superposition, subject to Gouy phase and boundary effects (Shao et al., 2013).

7. Future Directions and Open Problems

Current research is directed toward:

• Extension to Higher Dimensions: Demonstrating coherence and dimensionality in FOAM sets with $d \geq 4$ remains an open problem due to the limitations of existing relational witnesses (Junior et al., 16 Sep 2025).
• Development of Refined Witnesses: Progress is anticipated via higher-order invariants (e.g., Bargmann or out-of-time-order correlators) capturing relational structure beyond pairwise overlaps.
• Integration in Quantum Protocols: Robust, scalable certification methods for FOAM resources are expected to facilitate their adoption in entanglement distribution, high-dimensional teleportation, and quantum-enhanced metrology.
• Control of Stability: The interplay between geometric phase, Gouy phase, and environmental coupling (e.g., magnetic fields, fiber curvature) in determining FOAM state stability is an area of active theoretical and experimental investigation.

A plausible implication is that as mode generation, manipulation, and characterization methods mature, fractional OAM states will become routine resources for high-dimensional photonic quantum technologies and structured light–matter interaction studies.

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This overview synthesizes foundational mathematical structure, experimental methods, relational certification, and practical significance of fractional OAM modes, with all claims grounded in the referenced literature (Shao et al., 2013, Bandyopadhyay et al., 2016, Berger et al., 2018, Mao et al., 19 Apr 2024, Junior et al., 16 Sep 2025).