Projected Orbital Momentum in Wave Physics

• Projected orbital momentum is the resolved component of orbital angular momentum along a specified axis, essential for analyzing the spatial structure of fields.
• Its quantification relies on canonical decomposition and Fourier or scalar-potential methods to extract angular momentum in symmetrical systems.
• This concept underpins advancements in optical communications, electron microscopy state reconstruction, and relativistic quantum dynamics with spin–orbit coupling.

Projected orbital momentum is the component of orbital angular momentum (OAM) resolved along a specified axis, typically the propagation axis in systems with cylindrical or paraxial symmetry. In quantum and classical wave physics, projected OAM plays a central role in quantifying the spatial structure of fields, the transfer of mechanical properties in light–matter and electron–matter interactions, and the analysis of optical, electron, and acoustic vortex beams.

1. Canonical Decomposition and Formal Definition

The total time-averaged angular-momentum density $\mathbf{J}(\mathbf{r})$ of an electromagnetic field is given by

$$\mathbf{J}(\mathbf{r}) = \mathbf{r} \times \mathbf{p}(\mathbf{r}) + \mathbf{S}(\mathbf{r})$$

where $\mathbf{p}(\mathbf{r})$ is the linear momentum density, decomposable into orbital (canonical) and spin parts:

$$\mathbf{p}(\mathbf{r}) = \mathbf{p}^o(\mathbf{r}) + \mathbf{p}^s(\mathbf{r})$$

with \begin{align*} \mathbf{p}^o(\mathbf{r}) &= \frac{\varepsilon_0}{4 \omega} \Im \left[ E_i^*(\mathbf{r}) \nabla E_i(\mathbf{r}) + Z_0 H_i^*(\mathbf{r}) \nabla H_i(\mathbf{r}) \right] \ \mathbf{p}^s(\mathbf{r}) &= \tfrac{1}{2} \nabla \times \mathbf{S}(\mathbf{r}), \quad \mathbf{S}(\mathbf{r}) = \frac{\varepsilon_0}{2 \omega} \Im \left[ \mathbf{E}^*(\mathbf{r}) \times \mathbf{E}(\mathbf{r}) + Z_0 \mathbf{H}^*(\mathbf{r}) \times \mathbf{H}(\mathbf{r}) \right] \end{align*} The orbital angular momentum density is then defined as

$$\mathbf{L}(\mathbf{r}) = \mathbf{r} \times \mathbf{p}^o(\mathbf{r})$$

The projection of $\mathbf{L}$ onto an arbitrary unit vector $\mathbf{n}$ (often $\mathbf{n} = \hat{\mathbf{z}}$) is:

$$L_n = \int_V \mathbf{n} \cdot [\mathbf{r} \times \mathbf{p}^o(\mathbf{r})]\, d^3 r$$

This canonical decomposition captures the physical difference between orbital (spatial) and spin (polarization) contributions and underlies the transfer of OAM in both classical and quantum regimes (Jiang et al., 2015).

2. Projected OAM in Optical and Quantum Wavefields

Laguerre–Gaussian (LG) beams illustrate the utility of projected OAM. For an LG mode with field $$E(r, \phi, z) \propto u_{p,\ell}(r, z) e^{i \ell \phi} e^{ikz}$$, the $z$-projection is:

$$L_z = \int (x p^o_y - y p^o_x)\, d^3 r = N\, \ell \hbar$$

where $N$ is the photon number. Thus, each photon in an LG$_{p,\ell}$ mode carries projected OAM $\ell \hbar$ (Jiang et al., 2015). For generic wavefields, projection along $z$ is realized by the operator $\hat{L}_z = -i\hbar \partial/\partial \phi$, with eigenstates $$\Psi_\ell(r,\phi,z) = A(r) e^{i\ell\phi} e^{ik_z z}$$ and eigenvalues $\ell\hbar$ (Clark et al., 2014).

3. Measurement and Experimental Determination

Projected OAM is typically measured by decomposing the field or quantum state into eigenstates of $\hat{L}_z$. In electron microscopy, the multi-pinhole interferometer (MPI) samples the wavefunction at $n$ equally spaced azimuthal angles and reconstructs the OAM spectrum via discrete Fourier analysis:

$$c_\ell = \frac{1}{n} \sum_{j=0}^{n-1} \Psi(a, \phi_j) e^{-i\ell\phi_j}$$

with $|c_\ell|^2$ yielding the distribution in projected OAM $\ell$ (Clark et al., 2014). For light, OAM sorting and tomography employ mode transformations and projective phase elements—see, for example, the parabola-like lens sorting algorithm, which spatially discriminates $(\ell, p_r)$ basis states by mapping them to unique coordinates in the detection plane (Li et al., 2024).

In quantum state characterization, projected OAM distributions can be obtained directly from the quadrature Wigner functions in two orthogonal directions, bypassing the need for explicit angular variables:

$$p(\ell) = \int_0^{2\pi} W(\varphi, \ell) \frac{d\varphi}{2\pi}$$

where $W(\varphi, \ell)$ is the angle–OAM projected Wigner function (Sanchez-Soto et al., 2013).

4. Projected OAM in Relativistic Quantum Theory and Curved Space

For the relativistic electron, projected OAM is constructed by projecting the canonical operator onto the positive-energy Dirac subspace to eliminate zitterbewegung and to properly account for spin–orbit coupling:

$$\hat{\mathbf{L}}_{\mathrm{proj}} = \Lambda_+ \hat{\mathbf{L}} \Lambda_+ = \hat{\mathbf{R}} \times \mathbf{p}$$

with $$\hat{\mathbf{R}} = \mathbf{r} + \frac{\hbar \Sigma \times \mathbf{p}}{2E(E+m)c}$$ and $\Sigma$ the spin operator. The projected operator yields well-defined expectation values and reproduces the orbital–spin exchange encoded in the Dirac–Berry structure (Bliokh et al., 2017).

In locally curved space-time, the extended OAM operator acquires tidal corrections:

$$\mathbf{L} = \mathbf{L}^{(0)} + \mathbf{L}^{(\mathrm{G})}$$

where $\mathbf{L}^{(\mathrm{G})}$ induces half-integer projection quantum numbers $m$ in the presence of curvature, subject to the constraint $l \geq 2$. These half-integer $m$ states vanish as curvature tends to zero, restoring the standard quantum-algebraic structure (Singh et al., 2011).

5. Fourier and Scalar-Potential Approaches to Projected OAM

In wave physics, expectation values or spectral decompositions of projected OAM are most efficiently computed by exploiting the symmetry-adapted basis sets of the Helmholtz equation. For circular symmetry, the Bessel basis diagonalizes $\hat{L}_z$ with coefficients extractable via angular Fourier transforms:

$$\langle L_z \rangle = \sum_{n=-\infty}^{+\infty} n \int_0^\infty k_\perp dk_\perp\, |C_2(n; k_\perp)|^2$$

where

$$C_2(n; k_\perp) = \frac{1}{2\pi} \int_0^{2\pi} d\varphi_k\, e^{-in\varphi_k} \tilde{\Psi}(k_\perp \cos\varphi_k, k_\perp \sin\varphi_k)$$

and $\tilde{\Psi}$ is the transverse Fourier transform (Rodríguez-Lara, 2011).

For propagation-invariant beams, the scalar-potential formalism provides explicit expressions for the local flux density and total projected OAM, including TE/TM mode separation and interference:

$$\langle L_z \rangle = \int_0^{2\pi} d\phi \int_0^{\infty} j_z(r) r dr$$

with $j_z(r)$ dependent on both polarization mode weights and the azimuthal structure of the scalar potential (e.g., Laguerre or Bessel functions) (Rondon et al., 2020).

6. Mechanical and Information-Theoretic Consequences

A rigorous result established for generic monochromatic fields is that only the canonical (orbital) momentum, not the spin momentum, contributes to net mechanical force on matter:

$$\mathbf{F} = \int_{S_\infty} \mathbf{p}^o \cdot \mathbf{n}\, dS$$

Spin momentum yields zero net flux and hence no net force, though both orbital and spin angular momentum transfer contribute to optically induced torque (Jiang et al., 2015).

In holographic and multiplexing applications, projected OAM serves as the encoding/decoding degree of freedom, enabling high-density information channels, 3D volumetric imaging (via helical-phase holography), and robust optical communication bandwidths, leveraging the orthogonality and unboundedness of the $\ell$ index (Shen et al., 2024).

7. Extensions, Generalizations, and Limitations

Projected OAM extends naturally to higher-order projections ($L_n$ for arbitrary $\mathbf{n}$), noncylindrical symmetries (e.g., Mathieu or spheroidal harmonics), and multimodal superpositions, with the expectation value or spectral decomposition always traceable to symmetry-adapted integral transforms. Limitations of direct projection arise in the presence of severe aberrations, spatially varying polarization, or when the underlying topology or curvature breaks the fundamental algebraic structure, as in curved-space extensions. Practical measurement is further limited by modal crosstalk, finite detector aperture, and experimental imperfections in sampling or state preparation (Clark et al., 2014, Li et al., 2024, Singh et al., 2011).

Projected orbital momentum is thus a fundamental and experimentally accessible quantity, central to the mechanical, informational, and structural applications of structured matter waves and electromagnetic fields.