Research ArticlePHYSICS

Topologically enabled optical nanomotors

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Science Advances  30 Jun 2017:
Vol. 3, no. 6, e1602738
DOI: 10.1126/sciadv.1602738
  • Fig. 1 Optical torque on a Janus particle.

    (A) A linearly polarized plane wave (Embedded Image can exert force and torque on an asymmetric (Janus) particle (silica sphere with a gold half-cap). (B) Particle orientation (cap apex P) is completely determined by angles θ,φ relative to the fixed light source. Absolute value of the torque (C) and the cosine of the angle between the torque and the cap (D), for all possible orientations of the particle. Here, the wavelength of light is λ = 1064 nm. Circles indicate rotationally stable orientations where the torque is zero (gray) or the particle is spinning (colored). (E) Four rotationally stable points corresponding to a spinning particle (as viewed from the direction of the light source).

  • Fig. 2 Rotational dynamics in a linearly polarized plane wave.

    (A) Attractor map for the rotational dynamics of the system (λ = 1064 nm). Background colors indicate attractor basins—regions of initial (θ,φ) in the phase space that equilibrate into one of the eight orientations (attractors) in the damped regime. (B) Examples of orientation trajectories (solid lines) obtained by numerically evolving Eq. 2 for different parameters (cases 1 to 4). (C) Stability of attractors to rotational diffusion and the dependence of particle orientation on the beam intensity. Averaging over many realizations, we find that the probability of the particle to reach and remain at a stable point increases with the beam intensity.

  • Fig. 3 Flow of the torque vector field.

    Streamlines of the N = M × P vector that governs the evolution of the particle orientation P in the damped regime (see main text). Rotational equilibria are the vortex centers of the vector field N. The vortices are indicated with circles and asterisks. Each vortex is also characterized by its topological charge: −1 (asterisks) or +1 (circles). For the current parameters, the poles (θ = 0° and 180°) carry +1 charge and are shown in blue; +1 charges that are also the attractors are colored green. Here, λ = 1064 nm. Notice that the topological invariant Euler characteristic ensures that the sum of all the topological charges equals 2.

  • Fig. 4 Vortex dynamics for the torque vector field.

    Reduced phase space corresponding to θ ∈ [0o, 180o], φ ∈ [0o, 90o]. (A) Attractor map for λ = 532 nm. (B) Increasing the wavelength gives rise to a rich vortex map (B1, B2, and B3), with charge annihilation (B1 → B2, red arrow; see fig. S1) and the emergence of spinning attractors as the result of an avoided crossing of +1 charges (B2 → B3, center, gray arrows). Dashed lines indicate attractor basin boundaries. (C) As the wavelength is further increased, the spinning attractors begin to migrate back to the φ = 0° high-symmetry edge (C1 and C2) and, in an avoided crossing similar to before, finally return to it (C3 and C4, arrows). Another example of charge annihilation (C1, red arrow; see fig. S2) is shown.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/6/e1602738/DC1

    fig. S1. Vortex annihilation and conservation of topological charge.

    fig. S2. Vortex annihilation and conservation of topological charge.

    fig. S3. Attractor map for a larger particle/air as the ambient medium.

  • Supplementary Materials

    This PDF file includes:

    • fig. S1. Vortex annihilation and conservation of topological charge.
    • fig. S2. Vortex annihilation and conservation of topological charge.
    • fig. S3. Attractor map for a larger particle/air as the ambient medium.

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