On the origin and properties of dipolar recoil force and torque

TL;DR

The paper investigates the origin and symmetry of the dipolar recoil force and torque, showing they arise from retarded Lorentz forces between charges forming an electric–magnetic dipole. By analyzing a circularly polarized Huygens dipole with retarded interactions, the authors recover the standard recoil term $\langle \mathbf{F}_{rec} \rangle = -\frac{k^4 \eta_0}{12\pi} \operatorname{Re}(\mathbf{p}^* \times \mathbf{m})$ and derive the corresponding recoil torque, highlighting the time-reversal odd nature of recoil. The retardation between leading and trailing charges explains the irreversibility, while Maxwell-field momentum carries the conserved total momentum. The work provides a clear, first-principles bridge between microscopic Lorentz forces and macroscopic recoil phenomena, informing future studies of self-propelling and radiation-driven dipolar systems, and reinforcing the physical interpretation of recoil within light–matter interactions.

Abstract

The recoil optical force and torque acting on an electromagnetic dipole are typically derived by computing the imbalance in radiated linear and angular electromagnetic momentum coming from the source, using Maxwell stress tensor integration. This quantifies the recoil's outcome without revealing its physical origin or the underlying forces that produce it. The recoil force and torque exist even in the absence of external illumination, such that an isolated dipole emitter can experience them. In contrast to the other terms, the recoil terms are odd under time reversal. To clarify their nature and properties, we re-derive the recoil force from first principles using the total Lorentz force on a system of charges that form a simultaneous electric and magnetic dipole. The results agree with the standard momentum-based derivation and reveal their ultimate origin from retardation effects -- arising from the finite speed of light -- in the mutual interactions between charges. This result provides fundamental insight into the recoil mechanism, offering a clearer conceptual foundation for future theoretical and experimental studies of light-induced forces.

Figures (3)

• Figure 1: Huygens dipole constructed from electric, $q_\pm=\pm q_\text{e}$, and magnetic (north/south) charges, $q_\textsc{n/s}=\pm q_\text{m}$, where we assume that $q_\text{e}$, and $q_\text{m}$ are positive.
• Figure 2: Spacetime diagram of four charges constrained to move on a circular trajectory (shown by the blue cylindrical surface). The solid coloured curves represent the world lines of the charges. The past light cone of the red charge is shown in yellow. Its intersections with the other world lines define the corresponding retarded events (opaque points), whose spatial projections on the circular orbit (retarded positions) are shown as transparent markers. The dashed lines indicate the retarded times along each world line.
• Figure 3: Figure illustrating retarded positions of leading and trailing source charges $\vb*{r}_i(t-\upDelta{t}_{ij})$ (denoted by black arrows) as seen from the position of the observer test charge at $\vb*{r}_j(t)$ (green arrow), the separation vector $\vb*{\mathcal{r}}_{ij}$ (blue arrow), separation angle $\upDelta\alpha_{ij}$ (red arc), and finally the angle by which the charge position is retarded $\omega\upDelta t_{ij}$ (purple arc).