Off-Axis Dipole Forces in Optical Tweezers by an Optical Analog of the Magnus Effect

It is shown that a circular dipole can deflect the focused laser beam that induces it and will experience a corresponding transverse force. Quantitative expressions are derived for Gaussian and angular top hat beams, while the effects vanish in the plane wave limit. The phenomena are analogous to the Magnus effect, pushing a spinning ball onto a curved trajectory. The optical case originates in the coupling of spin and orbital angular momentum of the dipole and the light. In optical tweezers the force causes off-axis displacement of the trapping position of an atom by a spin-dependent amount up to $\lambda /2\pi$, set by the direction of a magnetic field. This suggests direct methods to demonstrate and explore these effects, for instance, to induce spin-dependent motion.

Figure 1

Optical analog of the Magnus effect. (a) A linearly polarized ($\mathbf{E}\parallel x$), focused laser induces a circular dipole ($xz$ plane) on a $j=0\to j^{\prime }=1$ ($\mathrm{\Delta }{m}_j=1$) transition, with a magnetic field $\mathbf{B}\parallel y$ setting the quantization axis. The spiral wave scattered by the circular dipole interferes with the incident wave, producing two effects: (b) The incident beam is deflected in the $xz$ plane, with corresponding reaction force on the atom, transverse to the beam. The direction changes sign with the detuning from the atomic resonance. (c) In an optical tweezer (“red” detuning, $\mathrm{\Delta }<0$), the transverse force shifts the trapping position away from the optical axis by an amount $ƛ=\lambda /2\pi$. See Fig. 3 for a far off-resonance case.

Figure 2

Beam deflection: radiant intensities in the plane of the ${\mathbf{u}}_{+}$ dipole, for (a) a Gaussian incident beam with $w_{\theta }=0.6$ and (b) an angular top hat incident beam with $r_{\theta }=0.6$. In both cases, the gray dotted curve shows $J_{\mathrm{in}}(\theta ,\varphi =0)$ of the incident beam, normalized to 1 for $\theta =0$; red solid and blue dashed curves show the outgoing, or total $J(\theta ,0)$, for $\mathrm{\Delta }=-\gamma$ and $+\gamma$, respectively. For clarity, we identify $(\theta ,0)\equiv (-\theta ,\pi )$. Curves remain the same upon switching simultaneously the signs of the detuning and the spin of the dipole.

Figure 3

Optical tweezer operating on a $j=1\to j^{\prime }=0$ transition, leading to $\pm ƛ$ off-axis displacements for the $(m_j{)}_y=\mp 1$ sublevels (upper left). The four panels show, in clockwise order, the effect of a rotation of the quantization axis ($\mathbf{B}$), through a cycle $y\to z\to -y\to -z$. While the $\mathbf{B}$-referenced $(m_j{)}_B$ of an atom is conserved, the space-referenced $(m_j{)}_y$ is not. The locations of the $(m_j{)}_B=\pm 1$ traps move up and down along the $x$ axis, in antiphase. If $\mathbf{B}$ is rotated at the trap frequency, spin-dependent oscillatory motion in the tweezer can be induced.