Optical spin precession

TL;DR

The work presents a gauge-invariant framework for the spin angular momentum of non-monochromatic electromagnetic fields and reveals genuine spin precession, paralleling magnetization dynamics. By deriving a magnetically biased SAM density and a generalized spin continuity equation with a spin-torque source term, it connects photonic spin dynamics to material magnetization via a Landau-Lifshitz–like equation. The paper analyzes concrete polychromatic configurations—rotating observers, circularly polarized waves in static fields, orthogonal beam interference, and near-field dipoles—demonstrating precession frequencies set by beat or drive frequencies and identifying distinct spin and orbital contributions through boundary-sensitive terms. It further proposes experimental approaches, including quasi-monochromatic two-beam interference and split-ring metamaterials, to realize and probe photonic spin precession and photonic spin waves, advancing optical spintronics and magnetization-photonics coupling. $\frac{d \mathbf{S}}{dt} = \omega \mathbf{S} \times \hat{\mathbf{z}}$, $-\mathbf{M} \times \mathbf{B}$, and related relations underpin the core dynamics and back-action mechanisms described.

Abstract

Period-averaged electromagnetic spin angular momentum is a well-established quantity for monochromatic fields, governing phenomena such as light-matter interactions with chiral particles and spin-orbit coupling effects. In contrast, the spin angular momentum of non-monochromatic fields remains unexplored. Here, we extend the concept of optical spin to the domain of non-monochromatic electromagnetic fields. Through this formulation, we uncover the precessional dynamics of electromagnetic spin in specific polychromatic configurations, including the superposition of circularly and linearly polarized plane waves propagating orthogonally at different frequencies, as well as fields generated by a precessing magnetic dipole. We discover that the dynamics of the electromagnetic spin in these cases obeys a Landau-Lifshitz-like equation establishing a profound parallel between dynamics of magnetization and photonic spin.

Figures (4)

• Figure 1: (a) A circularly polarized electromagnetic wave of frequency $\omega$, carrying a static spin angular momentum oriented along the $z$-axis, interacts with a homogeneous static magnetic field $\bm{B}_0$. (b) Their superposition generates a total magnetic field $\bm{B} = \bm{B}_c + \bm{B}_0$ that undergoes precession. (c) The instantaneous spin of the resulting electromagnetic field precesses at the driving frequency $\omega$, remaining perpendicular to the plane defined by the static magnetic field $\bm{B}_0$ and the transverse vector potential $\bm{C}^\perp$.
• Figure 2: (a) A circularly polarized wave at frequency $\omega_1$ propagating along $z$-axis (with static spin along the $z$-axis) interfering with a linearly $z$-polarized wave at frequency $\omega_2$ propagating along the $x$-axis. (b) Total magnetic field trajectory of the combination evolves along quasi-ellipsoidal trajectories gradually rotating at the beating frequency $\delta \omega \equiv \omega_1 - \omega_2$. (b) The instantaneous Belinfante electromagnetic spin experiences a slow precessional motion at the difference frequency $\delta \omega$ with nutations of small amplitude at the sum frequency $\omega_1 + \omega_2$ in the quasi-monochromatic limit $\delta \omega \ll \omega_{1,2}$. Parameters: $\omega_2 = 0.2 \omega_1$, $B_\text{c} = B_\text{l}$.
• Figure 3: (a1-a2) Vector tips trajectories of local near-field electromagnetic spin density on a sphere of unit radius around the time-dependent point magnetic dipole (with time color-coded), for the cases of rotating dipole with $b_0 = 0$ (a1) and precessing dipole with $b_0 = 0.5 m_0$ (a2). (b) The Belinfante electromagnetic total spin experiences a synchronous precessional motion with the same frequency. (c) Possible realization of the precessing electromagnetic dipole as a split-ring meta-atom with individual external current drives for each split-ring, modeling three time-dependent magnetic dipole components.
• Figure 4: (a1-a3) $x$ and $y$ components of spin density of the electromagnetic fields from a precessing magnetic dipole in the horizontal cross-section at $z=\text{const}$ at three time instances inside the precession period: $t=0$, $T/3$, and $2 T/3$. (b1-b3) Local electromagnetic spin density distributions for the same time instances at the unit sphere.