Off-Axis Drift: Dynamics and Applications

• Off-Axis Drift is the displacement measured relative to a system-specific reference axis, highlighting deviations from intended symmetry in optical, plasma, and mechanical settings.
• In optical systems, off-axis drift manifests as vortex displacement and phase tilts that separate topological charge from orbital angular momentum, impacting beam propagation.
• Across plasmas, electron holography, compliant joints, and pulsar emissions, it quantifies transport discrepancies, measurement errors, and parasitic motions with practical design implications.

Searching arXiv for the specified papers and closely related work to ground the article with current citations. arXiv search: "(Liang et al., 4 May 2025) Orbital angular momentum and dynamics of off-axis vortex light" “Off-axis drift” is a relational term for motion or displacement measured with respect to a nominal axis, symmetry direction, field line, fiducial plane, or remote center. Across the literatures considered here, it denotes the free-space motion of a displaced optical phase singularity, the inward displacement of a runaway-electron current channel, the cross-field transport of Solar Energetic Particles in the Parker spiral, frame-to-frame motion in phase-shifting off-axis electron holography, parasitic translation of a compliant Remote Center of Motion joint, and driftband reversals associated with tilted pulsar carousels (Liang et al., 4 May 2025, Hu et al., 2016, Dalla et al., 2013, Lindner et al., 2023, Mariano et al., 30 Mar 2026, Wright et al., 2016). The term therefore spans intrinsic dynamics, guiding-center transport, metrological error, and geometric observational effects.

1. Reference axes, symmetry, and what “off-axis” measures

In each setting, the “axis” is defined differently. For off-axis vortex light it is the geometrical axis of a paraxial Gaussian beam, while the vortex center is displaced by a finite vector $\boldsymbol\rho_0=(x_0,y_0)$. In runaway-electron dynamics it is the magnetic axis or major-radius center $R_0$ of a large-aspect-ratio tokamak. In SEP transport it is the local Parker-spiral field direction $\mathbf{e}_l$, so drift is explicitly perpendicular to the field. In phase-shifting electron holography the relevant reference is the camera-fixed fringe carrier and the aligned hologram stack. In compliant mechanics it is the nominal pivot of a Remote Center of Motion. In pulsar studies it is the fiducial plane containing the rotation and magnetic axes (Liang et al., 4 May 2025, Hu et al., 2016, Dalla et al., 2013, Lindner et al., 2023, Mariano et al., 30 Mar 2026, Wright et al., 2016).

The conserved or controlling quantities are likewise system-specific. Optical vortex propagation is discussed in terms of SO(3) rotational symmetry, topological charge $m$, and the expectation value of $\hat L_z$. Runaway-electron drift is derived from toroidal canonical angular momentum balance. SEP drift follows first-order adiabatic guiding-center theory in a curved and inhomogeneous magnetic field. Electron holography treats drift as a violation of the assumption that each camera pixel samples a fixed cosine law over a phase-shift series. Compliant-joint drift is quantified through compliance matrices, stiffness anisotropy, and parasitic-to-useful rotation. Pulsar bi-drifting is modeled geometrically through a tilted ellipse with angle $\psi$ and line-of-sight impact parameter $\beta$. This suggests that “off-axis drift” is not a single mechanism, but a common descriptor for departures from an intended or symmetry-defined reference geometry.

2. Optical-vortex drift and the separation of topological charge from OAM

For paraxial Gaussian-enveloped off-axis vortex beams, the waist-plane field can be written in transverse complex coordinates as

$$\Psi(\rho,0)=\bigl[\rho-\rho_0\bigr]^m\exp\!\bigl(-|\rho|^2/w_0^2\bigr), \qquad \rho_0=x_0+i y_0,$$

so that $\rho=\rho_0$ is a zero of the field and an $m$-th order phase singularity (Liang et al., 4 May 2025). Under free-space propagation, the displaced singularity follows

$R_0$0

which gives the explicit trajectory

$R_0$1

The transverse velocity components are constant in $R_0$2,

$R_0$3

and the displacement angle evolves as $R_0$4. Near the waist, $R_0$5, the displacement grows linearly with $R_0$6; in the far field, $R_0$7, the drift direction asymptotes to a $R_0$8 rotation of the initial offset (Liang et al., 4 May 2025).

The same analysis is used to distinguish topological charge from orbital angular momentum. The vortex topological charge is defined by

$R_0$9

and remains an exact integer under propagation. By contrast, for a single off-axis vortex the expectation of

$\mathbf{e}_l$0

gives an orbital angular momentum per photon

$\mathbf{e}_l$1

which is generally non-integer for $\mathbf{e}_l$2 and reduces to $\mathbf{e}_l$3 only when $\mathbf{e}_l$4 (Liang et al., 4 May 2025). The paper therefore treats off-axis drift not as a secondary optical imperfection but as the kinematic manifestation of the fact that a displaced vortex beam is no longer an eigenstate of $\mathbf{e}_l$5 alone. Its photon-current streamlines circulate around the displaced vortex and develop a radial component as the beam diffracts, and the drift is attributed to a transverse phase tilt associated with the Gaussian wavefront curvature and Gouy phase (Liang et al., 4 May 2025).

3. Guiding-center and orbit drift in magnetized plasmas and heliospheric fields

In the tokamak problem of runaway-electron plateaus, the central statement is that conservation of toroidal canonical angular momentum couples momentum-space evolution to horizontal orbit displacement. In an axisymmetric torus, $\mathbf{e}_l$6 makes the canonical angular momentum $\mathbf{e}_l$7 an invariant in the absence of non-conservative forces. When radiation drag acts, the balance between mechanical angular-momentum loss and change in the electromagnetic part of $\mathbf{e}_l$8 forces the beam to drift horizontally in configuration space for any given change in momentum space (Hu et al., 2016). In the no-wall limit, the large-aspect-ratio model yields

$\mathbf{e}_l$9

so any deceleration $m$0 produces an inward shift $m$1 (Hu et al., 2016). In the ideal-wall limit, the displacement still grows monotonically inward as $m$2 decreases, but the beam cross-section undergoes a mild “squeeze” rather than pure rigid translation, with $m$3. The effect is explicitly described as nonlinear because the runaway current carries the main poloidal flux, so any shift of the beam center alters both the self-field $m$4 and the eddy-current field $m$5 (Hu et al., 2016).

The time scale is estimated from synchrotron and bremsstrahlung drag through $m$6. For typical parameters, the model gives $m$7 in the no-wall limit and $m$8 in the ideal-wall limit, in good agreement with the $m$9 time for a $\hat L_z$0 inward shift seen on JET or EAST (Hu et al., 2016). The paper also rejects a common simplification: the inward drift is said to be not an $\hat L_z$1 force imbalance of a rigid beam in a fixed field, but the outcome of conserving mechanical plus canonical toroidal angular momentum.

A different off-axis drift appears in the Parker spiral interplanetary magnetic field. Using first-order adiabatic guiding-center theory in local coordinates $\hat L_z$2, the drift velocities have only $\hat L_z$3 and $\hat L_z$4 components, both perpendicular to the field direction $\hat L_z$5 (Dalla et al., 2013). The paper gives explicit forms for the electric drift, gradient-$\hat L_z$6 drift, and curvature drift, with the latter two scaling overall as $\hat L_z$7 in the nonrelativistic limit. In the scatter-free case, curvature drift is present; in the presence of scattering, protons at the high end of the SEP energy range experience significant gradient and curvature drift (Dalla et al., 2013). The magnitude of the drift velocity increases by more than an order of magnitude at high heliographic latitudes compared to near the ecliptic, reaches a maximum at $\hat L_z$8 AU at low heliolatitudes and $\hat L_z$9 AU at high heliolatitudes, and is stronger for partially ionised heavy ions because of the mass-over-charge dependence (Dalla et al., 2013). Quantitatively, near the ecliptic, 100 MeV protons have $\psi$0 or $\psi$1 up to $\psi$2 at $\psi$3 AU, while at high latitudes and $\psi$4 AU the combined drift exceeds $\psi$5 and reaches tens of $\psi$6 (Dalla et al., 2013). In this context, off-axis drift is cross-field transport away from the original Parker-spiral line rather than displacement relative to a fixed geometric center.

4. Drift as a reconstruction error in phase-shifting off-axis electron holography

In phase-shifting off-axis electron holography, off-axis drift is a metrological problem arising from independent motion of the biprism and specimen during acquisition. Biprism drift produces small frame-to-frame changes of the fringe spacing and phase, typically up to $\psi$7 over a 50-image stack, while specimen drift during $\psi$8 exposures causes lateral shifts of the object relative to the carrier fringes. For atomic resolution at $\psi$9, drift must be corrected to a few picometers per frame or better; on the Titan 80–300 kV environmental transmission electron microscope, residual uncorrected specimen drift was $\beta$0–$\beta$1, while after correction the reported error was $\beta$2 (Lindner et al., 2023). Uncorrected drift mixes the phase-shifted series at each pixel, breaks the assumed cosine law for the local intensity, produces “ghost” fringes in the reconstructed phase and amplitude, and degrades both resolution and phase sensitivity.

The mathematical model records a series of holograms $\beta$3 with beam-tilt-induced phase offsets $\beta$4, and fits each pixel to

$\beta$5

Specimen drift is corrected by shifting each raw frame,

$\beta$6

with $\beta$7 determined to sub-pixel precision (Lindner et al., 2023). The workflow is explicit: acquire $\beta$8 reference holograms on vacuum at $\beta$9 biprism and $$\Psi(\rho,0)=\bigl[\rho-\rho_0\bigr]^m\exp\!\bigl(-|\rho|^2/w_0^2\bigr), \qquad \rho_0=x_0+i y_0,$$0 specimen holograms under identical illumination; align the reference stack by phase correlation; average it; extract a vacuum ROI in each specimen hologram; divide by the aligned average reference to suppress Fresnel fringes and camera artifacts; centerband-filter the Fourier transform; use cross-correlation on Bragg-filtered images to estimate specimen drift; apply the shifts to the raw holograms; recompute $$\Psi(\rho,0)=\bigl[\rho-\rho_0\bigr]^m\exp\!\bigl(-|\rho|^2/w_0^2\bigr), \qquad \rho_0=x_0+i y_0,$$1; and finally perform the pixel-wise cosine fit (Lindner et al., 2023).

The performance figures are specific. A biprism voltage of $$\Psi(\rho,0)=\bigl[\rho-\rho_0\bigr]^m\exp\!\bigl(-|\rho|^2/w_0^2\bigr), \qquad \rho_0=x_0+i y_0,$$2 yields fringe spacing of approximately $$\Psi(\rho,0)=\bigl[\rho-\rho_0\bigr]^m\exp\!\bigl(-|\rho|^2/w_0^2\bigr), \qquad \rho_0=x_0+i y_0,$$3 ($$\Psi(\rho,0)=\bigl[\rho-\rho_0\bigr]^m\exp\!\bigl(-|\rho|^2/w_0^2\bigr), \qquad \rho_0=x_0+i y_0,$$4 px at $$\Psi(\rho,0)=\bigl[\rho-\rho_0\bigr]^m\exp\!\bigl(-|\rho|^2/w_0^2\bigr), \qquad \rho_0=x_0+i y_0,$$5); fringe visibility in the reference series is $$\Psi(\rho,0)=\bigl[\rho-\rho_0\bigr]^m\exp\!\bigl(-|\rho|^2/w_0^2\bigr), \qquad \rho_0=x_0+i y_0,$$6 to $$\Psi(\rho,0)=\bigl[\rho-\rho_0\bigr]^m\exp\!\bigl(-|\rho|^2/w_0^2\bigr), \qquad \rho_0=x_0+i y_0,$$7 with mean $$\Psi(\rho,0)=\bigl[\rho-\rho_0\bigr]^m\exp\!\bigl(-|\rho|^2/w_0^2\bigr), \qquad \rho_0=x_0+i y_0,$$8; the information limit reaches the third-order Pt[110] Bragg reflection at $$\Psi(\rho,0)=\bigl[\rho-\rho_0\bigr]^m\exp\!\bigl(-|\rho|^2/w_0^2\bigr), \qquad \rho_0=x_0+i y_0,$$9 ($\rho=\rho_0$0); raw phase sensitivity in vacuum is $\rho=\rho_0$1, or $\rho=\rho_0$2; and after low-pass filtering at $\rho=\rho_0$3 it improves to $\rho=\rho_0$4 (Lindner et al., 2023). Validation against frozen-lattice multislice simulations on a thin Pt sample gives amplitude RMS deviation $\rho=\rho_0$5 and phase RMS deviation $\rho=\rho_0$6 at a best thickness match of $\rho=\rho_0$7 (Lindner et al., 2023). Here, off-axis drift is not a transport phenomenon but a frame-registration error that must be estimated and removed before any physically meaningful phase can be reconstructed.

5. Parasitic off-axis drift in compliant Remote Center of Motion joints

In the mechanics literature considered here, off-axis drift refers to unintended translation of a nominal Remote Center of Motion during end-effector steering. The monolithic compliant joint is modeled by isolating three mobility panels as Euler–Bernoulli beams of length $\rho=\rho_0$8, thickness $\rho=\rho_0$9, width $m$0, and Young’s modulus $m$1, with axial stiffness $m$2 and bending stiffness $m$3 (Mariano et al., 30 Mar 2026). Superposition of the three beams yields a $m$4 global stiffness matrix $m$5, and the translational compliance matrix is $m$6. Under a commanded small angle $m$7, the end-effector tip at distance $m$8 describes an arc $m$9, while the nominal pivot undergoes a smaller parasitic translation $R_0$00. In the small-angle regime,

$R_0$01

and the full expression used later in the paper is

$R_0$02

This formalizes off-axis drift as a parasitic motion normalized by the useful motion (Mariano et al., 30 Mar 2026).

The design objective combines stiffness isotropy with suppression of RCM drift. An anisotropy index $R_0$03 is defined from $R_0$04, and in the 3D-FEM stage the directional stiffness

$R_0$05

is fit by a least-squares ellipse whose principal-axis ratio is

$R_0$06

A five-parameter Ansys sweep uses $R_0$07, $R_0$08, $R_0$09, $R_0$10, and $R_0$11, with radial loads at $R_0$12 to compute isotropy and parasitic-drift metrics $R_0$13 and $R_0$14. A $R_0$15-sample random sweep returns a Pareto set of $R_0$16 candidates, and the selected design minimizes

$R_0$17

(Mariano et al., 30 Mar 2026).

For the chosen configuration, the reported values are $R_0$18 and $R_0$19, i.e. $R_0$20. Under a commanded rotation of $R_0$21, the FEM-predicted parasitic RCM drift lies in the interval

$R_0$22

while the useful end-effector arc displacement is $R_0$23 (Mariano et al., 30 Mar 2026). Benchtop experiments on a PA12 SLS prototype use a $R_0$24 radial load and a 6-DOF Aurora electromagnetic sensor at twelve orientations. The measured stiffness follows the simulated directional trend with local percentage errors from $R_0$25 to $R_0$26, and the global metrics are MAE $R_0$27, RMSE $R_0$28, MAPE $R_0$29, and mean bias $R_0$30 (Mariano et al., 30 Mar 2026). Fatigue analysis with $R_0$31 and $R_0$32 gives workspace limits $R_0$33 and $R_0$34. In this domain, off-axis drift is an error budget component to be minimized while preserving compliance and near-isotropic stiffness.

6. Tilted carousels, observational drift reversals, and recurring distinctions

In pulsar radio-emission modeling, off-axis drift appears through a carousel beam that is not circular but elliptical and tilted by an angle $R_0$35 relative to the fiducial plane $R_0$36. Before tilting, the carousel satisfies $R_0$37; after rotation by $R_0$38, the explicit Cartesian form contains an $R_0$39 term and remains centered on the magnetic axis (Wright et al., 2016). The drift direction of subpulses follows the tangent to this tilted ellipse. For a sightline at constant $R_0$40, the sign of the slope can reverse where the tangent becomes vertical, so the leading and trailing components may drift in opposite directions. This is the geometric origin of pulsar bi-drifting in the model of Wright and Weltevrede (Wright et al., 2016).

The simulations are concrete. For PSR J0815+09, the adopted parameters are $R_0$41, $R_0$42, $R_0$43, $R_0$44, $R_0$45, $R_0$46, $R_0$47, $R_0$48, and $R_0$49, which gives $R_0$50, $R_0$51, anticlockwise circulation, and drift sequence $R_0$52. For PSR B1839–04 in Q-mode, the parameters are $R_0$53, $R_0$54, $R_0$55, $R_0$56, $R_0$57, $R_0$58, $R_0$59, $R_0$60, and $R_0$61, giving $R_0$62, $R_0$63, drift pattern $R_0$64, and a $R_0$65 offset between the profile centroid and the fiducial plane (Wright et al., 2016). The same geometry predicts centroid displacement,

$R_0$66

as well as asymmetric, frequency-dependent component evolution under radius-to-frequency mapping and changes in drift mode if $R_0$67 changes (Wright et al., 2016).

These cases clarify several recurring distinctions. In optical vortices, topological charge is strictly conserved whereas OAM per photon becomes non-integer when the vortex is displaced (Liang et al., 4 May 2025). In tokamaks, inward off-axis motion is tied to canonical-plus-mechanical angular momentum balance rather than a simple $R_0$68 force imbalance (Hu et al., 2016). In SEP transport, drift is not negligible for high-energy particles and high-$R_0$69 ions (Dalla et al., 2013). In electron holography, off-axis drift is not an intrinsic sample property but a registration error that must be corrected before phase retrieval (Lindner et al., 2023). In compliant RCM joints, drift is explicitly normalized against useful rotation through the PRR metric (Mariano et al., 30 Mar 2026). In pulsars, opposing driftbands need not imply circular carousels with anomalous behavior; a tilted elliptical carousel suffices in the cases modeled (Wright et al., 2016). A plausible synthesis is that off-axis drift becomes scientifically informative precisely when it exposes a mismatch between a nominal symmetry description and the actual geometry, transport law, or measurement frame.