Momentum, spin, and orbital angular momentum of electromagnetic, acoustic, and water waves

TL;DR

This work develops a universal Noether-based framework to describe momentum and angular momentum for diverse classical waves (electromagnetic, acoustic, elastic, plasma, and water-surface waves). It defines canonical densities ${\mathbf{P}}$, ${\mathbf{L}}$, and ${\mathbf{S}}$, and their relation ${\mathbf{J}}={\mathbf{L}}+{\mathbf{S}}$, linking them to energy flux via the Belinfante-Rosenfeld relation ${\overline{\bm \Pi}}={\overline{\mathbf{P}}}+\frac{1}{2}{\boldsymbol \nabla}\times{\overline{\mathbf{S}}}$. The framework yields explicit expressions for the densities in each wave type, including Stokes drift for sound, Poynting momentum for EM waves, and spin-orbit features in mixed, structured waves, with special treatment for dispersive media (plasma) and 2D water waves. It further demonstrates that vortex and cylindrical-wave configurations carry well-defined orbital and spin AM, and clarifies longstanding issues such as the Abraham-Minkowski problem in plasma. The results underpin opto-/acousto-mechanical forces and torques on particles and provide a coherent toolkit for analyzing structured waves in complex media and applications to trapping, manipulation, and metamaterials.

Abstract

Waves of various types carry momentum, which is associated with their propagation direction, i.e., the phase gradient. The circulation of the wave momentum density gives rise to orbital angular momentum (AM). Additionally, for waves described by vector fields, local rotation of the wavefield produces spin AM (or simply, spin). These dynamical wave properties become particularly significant in structured (i.e., inhomogeneous) wavefields. Here we provide an introduction and overview of the momentum and AM properties for a variety of classical waves: electromagnetic, sound, elastic, plasma waves, and water surface waves. A unified field-theory approach, based on Noether's theorem, offers a general framework to describe these diverse physical systems, encompassing longitudinal, transverse, and mixed waves with different dispersion characteristics. We also discuss observable manifestations of the wave momentum and AM providing clear physical interpretations of the derived quantities.

Figures (14)

• Figure 1: (a) As Keppler suggested in the 17th century, a comet's tail points away from the sun due to radiation pressure Jones1953. This evidences the electromagnetic wave momentum. (b) The propagation of a water-surface wave induces a drift of water particles in the wave propagation direction, known as the Stokes drift Falkovich_book. This is a manifestation of the wave momentum. Here trajectories of drifting water-surface particles are shown in red. In addition, the circular-like local motion of the particles reveals the presence of transverse spin angular momentum in water waves.
• Figure 2: Mechanical action of optical spin and OAM on matter. (a) The spin of circularly-polarized light induces rotation of an absorbing or birefringent particle around its center (highlighted in yellow). Adapted from Friese1998Nat with permission from Springer Nature. (b) The OAM of optical vortices, characterized by radially-varying intensity (shown in grayscale) and azimuthally-growing phase, generates orbital rotation of an absorbing particle (radially trapped due to the intensity-gradient force) around the vortex center. Adapted from Garces-Chavez2003PRL [Copyright (2003) by the American Physical Society]. The OAM and orbital rotation are produced by the azimuthal momentum density and the corresponding azimuthal radiation-pressure force. The scalebars correspond to $10 \lambda \simeq 10\,{\rm \mu m}$, where $\lambda$ is the wavelength of light.
• Figure 3: Spin, momentum, and OAM in structured water waves visualized through the local circular motion and Stokes drift of water-surface particles (top view) Bliokh2022SA. The experimentally observed particle trajectories are shown in black, their circular motions (spin) are highlighted by red and blue arrows, whereas the Stokes drift is highlighted by green arrows. (a) Interference of two plane water waves with equal amplitudes, frequencies, but different wavevectors ${\bf k}_{1,2}$. The scalebar corresponds to $\lambda/2 \simeq 2.2\,{\rm cm}$, where $\lambda$ is the water-wave wavelength. (b) Interference of two orthogonal standing waves with equal amplitudes and frequencies, phase-shifted by $\pi/2$ with respect to each other, generates alternating wave vortices of opposite signs. The circulating azimuthal Stokes drift produces the OAM in these vortices. The scalebar corresponds to $\lambda/4 \simeq 1.4\,{\rm cm}$. Adapted with permission from Bliokh2022SA. [Copyright (2022) The Authors] CC BY-NC 4.0.
• Figure 4: Types of waves illustrated by the wave-induced instantaneous displacements $\mathbfcal{R}({\bf r},0)$ (magenta arrows) of particles (balck dots) in a medium. (a) Longitudinal plane wave: the displacements are aligned with the wavevector ${\bf k}$ (green arrow). (b) Transverse wave: the displacements are perpendicular to the wavevector. (c) Mixed wave: the displacements have both longitudinal and transverse components. Here a surface wave is shown, with the wave amplitude decaying in the vertical direction. In this case, the wave is mixed with respect to the real part of the wavevector, responsible for the propagation of the wave (see Section \ref{['Water']}).
• Figure 5: Polarization ellipse and spin. A generic monochromatic vector field $\mathbfcal{F}({\bf r},t) = \operatorname{Re}[{\bf F}({\bf r})e^{-i\omega t}]$ traces the polarization ellipse at each point of space. The time-averaged spin angular momentum density $\overline{\bf S} \propto \overline{\mathbfcal{F} \times \dot{\mathbfcal{F}}} = \operatorname{Im}({\bf F}^* \!\times {\bf F})/2$ is directed normally to this ellipse and is proportional to its area.