Fig. 2: Manifestation of gradient and curl torques as the LOT.

a Incident structured light field consisting of two pairs of s- and p − polarized waves. These four waves share an equal amplitude, wave number, and incidence angle θ with respect to the x − y plane. The color scale plot represents the z − component of the momentum-curl ∇ × p for the traveling wave (case I), as well as of the reactive helicity gradient \(\nabla {{{{{\mathcal{H}}}}}}\) for the standing evanescent wave(case II). b, c Field quantities of interest for the traveling (b) and evanescent (c) waves. Insets illustrate the expected LOT due to ∇ × p in (b), and \(\nabla {{{{{\mathcal{H}}}}}}\) in (c). d Calculated Cartesian components of the total torque for λ = 0.58 μm on a Si particle (radius r = 0.1 μm) placed on the x-axis. e, f The LOT as a function of the wavelength and particle radius for cases I (e) and II (f). The torques are normalized to their maxima (7.0 and 14.6 pN ⋅ μm for I and II, respectively).