Nonconservative electric and magnetic optical forces on submicron dielectric particles

We present a study of the total force on a small lossless dielectric particle, which presents both an electric and magnetic response, in a optical vortex wave field. We show that the force is a simple combination of conservative and nonconservative steady forces that can rectify the flow of magnetodielectric particles. In a vortex lattice the electric-magnetic dipolar interaction can spin the particles either in or out of the whirl sites leading to trapping or diffusion. Specifically, we analyze force effects on submicron silicon spheres in the near infrared, proving that the results previously discussed for hypothetical magnetodielectric particles can be observed for these Si particles.

Figure 1

Illustration of the maps of (a) the total force modulus, (b) the force lines and (c) electric field intensity for a dipolar magnetodielectric particle in the intersection region of two standing plane waves oriented along the $x$ and $y$ axes and with the electric field polarized along the $z$ axis.

Figure 2

Nonconservative forces on a Si sphere of radius $a=230$ nm placed at the intersection region of two standing waves with a dephasing $\varphi =\pi /2$ in a medium with $\epsilon =\mu =1$. The light field wavelength, $\lambda =1250$ nm, is tuned to the dipolar electric resonance. Arrows in (a) and (b) point along the total force lines. (a) Contour maps of the modulus of the normalized total force, $|\langle \mathbf{F}\rangle |/F_0$ with $F_0=\left|E_0\right|{}^2/({\mathit{ka}}^3)$. (b) Contour maps of the normalized electric field intensity, $\left|\mathbf{E}\right|{}^2/\left|E_0\right|{}^2$. Near the resonance, the radiation pressure force, $\langle {\mathbf{F}}_{\mathrm{e}}^{\langle S\rangle }\rangle$, overcomes the attractive gradient forces and dominates the total force which presents no stable equilibrium positions. The symbols $\square$ and $◊$ indicate saddle points in the force map.

Figure 3

Normalized force in the $x$ direction, $F_x$ along a line defined by $\mathit{ky}=\pi /2$ for a Si sphere of radius $a=230$ nm in the intersection region of two standing waves with $\varphi =\pi /2$ and for different wavelengths. Notice that, since the force magnitude $F_0$ is inversely proportional to ${\mathit{ka}}^3$, this normalization factor is different for each of the four plots. The symbols sketch the force fields at several positions: $◊$ and $\square$ correspond to saddle points and $▵$ and $◯$ correspond to unstable and stable equilibrium (zero-force) positions, respectively.

Figure 4

Same as Fig. 2 but for a wavelength $\lambda =1300$ nm slightly above (red-shifted) the electric dipolar resonance. The force field, dominated by the scattering force contributions, presents no stable equilibrium position in the system.

Figure 5

Same as Fig. 2 for a wavelength $\lambda =1600$ nm slightly below (blue-shifted) the magnetic dipolar resonance. Equilibrium (zero-force) positions correspond to electric field maxima.

Figure 6

(a) Total force lines, (b) scattering force lines, and (c) conservative force lines for a Si sphere of radius $a=230$ nm placed at the intersection region of two standing waves with a dephasing $\varphi =\pi /2$ in a medium with $\epsilon =\mu =1$. The light field wavelength, $\lambda =1600$ nm, is slightly below (blue-shifted) the magnetic dipolar resonance (same as Fig. 5). The rotation of the Poynting vector around the field nodes [Fig. 6b] induces orbital scattering force lines around these nodes; however, the conservative force [Fig. 6c] pushes the particle out from these nodes and toward stable positions at the corners between adjacent orbiting domains. The interplay between these two forces produces the whirls on the total forces [Fig. 6a].

Figure 7

Same as Fig. 2 for a wavelength $\lambda =1725$ nm slightly above (red-shifted) the magnetic dipolar resonance. Although the force has both contributions, conservative and nonconservative, the conservative magnetic force dominates giving rise to stable equilibrium positions centered where the electric field is less intense. The equilibrium (zero-force) positions, corresponding to magnetic field maxima, are now located at the vortex center.

Figure 8

Same as Fig. 4 when the crossed beams oscillate synchronously ($\varphi =0$). At the dipolar electric resonance the electric forces are zero. The force field in this case is entirely given by magnetic or electric-magnetic forces.