Transverse Orbital Angular Momentum

• Transverse orbital angular momentum (T-OAM) is the optical angular momentum component perpendicular to the propagation direction, characterized by mixed space–time phase singularities.
• Recent theoretical and experimental studies verify that spatiotemporal optical vortices (STOVs) exhibit strict T-OAM conservation with a half-integer per-photon value.
• Advanced techniques like 4f pulse-shaping and single-shot interferometry facilitate the precise generation, measurement, and manipulation of T-OAM in ultrafast optics.

Transverse orbital angular momentum (T-OAM) denotes the component of optical orbital angular momentum orthogonal to a beam’s mean propagation direction. In contrast to the well-established longitudinal OAM, which arises from azimuthal phase windings in the transverse (x, y) plane and is conserved around the beam axis (z), T-OAM emerges in electromagnetic fields where the phase singularity is embedded in a mixed space–time plane, such as (x, t) or (x, ζ) with ζ = v_g t – z. The spatiotemporal optical vortex (STOV) is the canonical optical object carrying T-OAM, manifesting circulating Poynting flux and energy density in a plane intersecting both space and time. Recent theoretical and experimental advances establish not only the physical definition and conservation laws for T-OAM, but also quantifiable means for generation, measurement, and manipulation on the ultrafast timescale and at high topological charge.

1. Canonical Definition, Density, and Intrinsic/Extrinsic Partition

The rigorous definition of T-OAM in classical fields is set by the instantaneous angular momentum

$$\mathbf{L}(t) = \int d^3\!r\; \mathbf{r} \times \mathbf{P}(\mathbf{r}, t)$$

with the instantaneous momentum density $$\mathbf{P} = \frac{1}{4\pi c} \mathbf{E} \times \mathbf{H}$$ (cgs) or $\varepsilon_0 \mathbf{E} \times \mathbf{B}$ (SI). For pulses, it is natural to decompose the field in comoving spatiotemporal coordinates, so that

$L_y(t) = \int d^3r\, [x P_z - \zeta P_x]$

with ζ = v_g t – z. The total T-OAM can be decomposed into intrinsic and extrinsic components about a defined origin: $$L_y^{\rm (tot)}(t) = \int d^3r\, \left[(\mathbf{r} - \mathbf{R}_{\rm CE}) \times \mathbf{P}(\mathbf{r}, t)\right]_y$$ where $\mathbf{R}_{\rm CE}$ is the pulse’s instantaneous center of energy. The extrinsic term vanishes for a pulse at rest in the lab frame, making the total T-OAM strictly intrinsic under correct measurement conventions (Tripathi et al., 31 Mar 2025).

Correct calculation of global T-OAM, like all extensive field observables, requires integrating momentum densities at a common laboratory time. Summing contributions from different times (as in some "flux-density" approaches) leads to unphysical, frame-dependent results and violates conservation laws.

2. Physical Manifestation in Spatiotemporal Optical Vortices

T-OAM is fundamentally distinct from longitudinal OAM. In a STOV, the phase singularity winds in a mixed spatial and temporal plane, e.g., $A(x, y, \zeta) \propto (\zeta + i x)^l$, yielding

$$\arg A \sim l \tan^{-1}\left(\frac{x}{\zeta}\right)$$

meaning the energy flow instantaneously circulates in a plane containing both propagation and transverse axes. For such a structure, the per-photon T-OAM about the y-axis is generally half-integer in units of ℏ: $$L_y^{\rm (intr)} = \frac{1}{U} \int d^3r\, [x - X_{\rm CE}] P_z - [\zeta - Z_{\rm CE}] P_x$$ yielding exactly $+\tfrac12 \hbar$ per photon for a circularly symmetric $l=1$ STOV (Tripathi et al., 31 Mar 2025). The observable circulation of energy flux in (x, ζ) can be detected as rigorously as the familiar lħ longitudinal OAM in vortex beams.

Table: Distinguishing Longitudinal and Transverse OAM

	Longitudinal OAM	Transverse OAM
Singularities	(x, y) plane, e.g. e^{i m φ}	(x, ζ) or (x, t') plane in pulses
Reference Axis	Propagation axis (z)	Transverse to propagation (e.g., y)
Per-photon OAM	m ℏ	Typically (for STOV) ½ ℏ (

3. Conservation Laws and Operator Structure

For STOVs in free space, T-OAM is strictly conserved under Maxwell’s equations when measured about the center of energy at fixed time. The Hermitian operator governing intrinsic T-OAM for a dispersionless medium is

$$\hat{L}_y = -i\hbar (\zeta \partial_x - x \partial_\zeta)$$

with commutation to the paraxial Hamiltonian ensuring conservation. For each photon, the expectation value of T-OAM is

$$(L_y^{\rm intr})(t) = \frac{\langle E | \hat{L}_y | E \rangle}{\langle E | E \rangle}$$

enforcing both mathematical self-consistency and experimental agreement in torque transfer and spatiotemporal measurements (Tripathi et al., 31 Mar 2025).

4. Experimental Observation, Manipulation, and Measurement

Modern ultrafast optics enables direct generation and detection of T-OAM. Key techniques include:

• 4f pulse-shaper with SLM: Tailored phase masks in the spatial–spectral plane allow realization of arbitrary STOVs with programmable topological charge and orientation (Chen et al., 2021, Liu et al., 29 Jul 2024).
• Single-shot interferometry: Complete reconstruction of the pulse’s spatiotemporal field, and thus of the local OAM density, is accomplished with time-gated spectral interferometry (TG-SSSI) (Hancock et al., 11 Nov 2024).
• Spatiotemporal torque measurements: Amplitude perturbations, such as wire obstructions overlapping a structured field, produce measurable ΔL_y per photon, governed by the overlap integrals of the amplitude mask with the intrinsic tOAM density (Hancock et al., 11 Nov 2024).
• Spectral/amplitude grating analysis: Diffracted STOVs yield a multi-lobe structure with a number of intensity gaps corresponding to the topological charge, providing direct OAM quantization and sign readout (Huang et al., 2022).

Experimental results for l=1 STOVs confirm the theoretically predicted half-integer per-photon T-OAM, with no extrinsic cancellation when evaluated at a fixed time and referenced to the energy centroid.

5. Theoretical and Experimental Controversies

A significant theoretical controversy has involved the magnitude and conservation of T-OAM in STOVs. Competing frameworks have been critically evaluated:

• Integer operator with photon-centroid origin [Bliokh]: Predicts 1 ℏ per photon but fails conservation and experiment (Porras, 2023, Tripathi et al., 31 Mar 2025).
• Flux-density/amplitude-sampled approaches [Porras]: Artificially construct mixed-time profiles, creating an extrinsic T-OAM that exactly cancels the intrinsic contribution and predicts net zero total, violating physical intuition and experimental data (Porras, 2023, Porras, 14 Apr 2024, Tripathi et al., 31 Mar 2025).
• Center-of-energy canonical Maxwell framework [Hancock et al.]: Evaluates OAM density at fixed time, uses energy centroid as the natural reference, and yields rigorously conserved, Hermitian operator–derived T-OAM; in precise agreement with direct measurements and spatiotemporal torque experiments (Tripathi et al., 31 Mar 2025, Hancock et al., 11 Nov 2024).

Resolution requires integrating at a common laboratory time, using the energy-centric reference, and employing the Hermitian canonical operator.

6. High-Order T-OAM, Quantum Structure, and Applications

Control over T-OAM is scalable to extremely high topological charge using inverse-designed phase masks in the spectral domain. Spatiotemporal Bessel optical vortices (STBOVs) have demonstrated robust propagation and pure T-OAM above l > 10², with stability improved at high order via enhanced space-time coupling (Chen et al., 2021).

Quantum theory of T-OAM extends the classical picture, with photon number states or coherent states of arbitrary tilt θ. The quantum OAM operator

$$\hat{\bm L} = -i\hbar\int d^3k \sum_\lambda \hat{a}_{\bm{k},\lambda}^\dagger \left( \bm{k}\times\nabla_k \right) \hat{a}_{\bm{k},\lambda}$$

exhibits polynomial scaling of mean and variance with charge l and tilt θ. The spatial texture of OAM fluctuations presents "smoke-ring" noise patterns, potentially isolable in spatially resolved detection (Das et al., 2 Mar 2024).

These advances open new directions:

• Ultrafast control of T-OAM on picosecond scales for dynamic light–matter interactions (Liu et al., 29 Jul 2024).
• High-dimensional quantum entanglement leveraging orthogonal T-OAM modes, supporting qudit protocols and spatiotemporal-polarization mode-entanglement (Huang et al., 22 Dec 2024).
• Sensing and metrology exploiting the unique spatial structure of T-OAM-induced torque and flight paths, as well as spin–orbit photonic phenomena (Wang, 19 Jun 2024, Hu et al., 2022).

7. Summary of Core Principles and Universal Features

1. Physical observables: Only the intrinsic T-OAM calculated about the center of energy at a fixed laboratory time has fundamental significance and matches experimental outcomes (Tripathi et al., 31 Mar 2025).
2. Operator structure: The Hermitian, conserved operator $\hat{L}_y = -i(\zeta\partial_x - x\partial_\zeta)$ governs intrinsic T-OAM in STOVs.
3. Universal scaling: Circular (|l|=1) STOVs carry per-photon T-OAM of ½ ℏ, with the result preserved under propagation and robust against non-idealities.
4. Critique of alternatives: Approaches ignoring simultaneity and canonical reference lead to erroneous, unphysical, or non-conservative results.

Transverse OAM in spatiotemporal optical fields is therefore a demonstrably real, quantitatively precise, and physically distinct manifestation of photonic angular momentum, with a firm foundation in Maxwell’s theory, validated by direct ultrafast measurements, and presenting rich new avenues in structured light physics, quantum photonics, and spatiotemporal energy flow control (Tripathi et al., 31 Mar 2025, Hancock et al., 11 Nov 2024, Liu et al., 29 Jul 2024, Huang et al., 22 Dec 2024, Chen et al., 2021).