Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows

Based on the Mie theory and on the incident beam model via superposition of two plane waves, we analyze numerically the momentum flux of the field scattered by a spherical, nonmagnetic microparticle placed within the spatially inhomogeneous circularly polarized paraxial light beam. The asymmetry between the forward- and backward-scattered momentum fluxes in the Rayleigh scattering regime appears due to the spin part of the internal energy flow in the incident beam. The transverse ponderomotive forces exerted on dielectric and conducting particles of different sizes are calculated and special features of the mechanical actions produced by the spin and orbital parts of the internal energy flow are recognized. In particular, the transverse orbital flow exerts the transverse force that grows as $a^3$ for conducting and as $a^6$ for dielectric subwavelength particle with radius $a$, in compliance with the dipole mechanism of the field-particle interaction; the force associated with the spin flow behaves as $a^8$ in both cases, which testifies for the nondipole mechanism. The results can be used for experimental identification and separate investigation of the spin and orbital parts of the internal energy flow in light fields.

Figure 1

Geometrical conditions of the light-scattering analysis. The particle is situated in the coordinate origin; incident light comes from the lower hemisphere. Other parameters are explained in the text.

Figure 2

Mechanical forces acting on a probe particle situated in plane $z$ = 0 and illuminated by the superposition of two plane waves with (a) different directions, resulting in the inhomogeneous circularly polarized field described by Eqs. (30, 31, 32, 33, 34, 35) and (b) coinciding directions, resulting in a spatially homogeneous energy distribution but having inclined wave front, according to Eqs. (44), (45). Further explanations are found in the text.

Figure 3

Fourth root of the normalized momentum flux ${\mathbf{P}}_s/P_0$ [see Eqs. (25) and (43)] components of the field scattered by a spherical (a) metallic and (b) dielectric particle, suspended in water, vs the particle-size parameter (37), calculated for the conditions of Eqs. (41), (42). Each curve is marked by the corresponding component notation: subscripts ($x$, $y$) denote the momentum flux Cartesian component; superscripts ($x$, $y$, 45, +) denote the incident field polarization as indicated by Eqs. (31) and (38)–(40). Solid (dashed) lines describe the momentum flux into the forward (backward) hemisphere; the inset shows magnification of the dashed rectangle in panel (b). In both cases (a) and (b), curves $P_{sx}^x(\pm )$ and $P_{sx}^y(\pm )$ coincide with the zero line, and curves $P_{sy}^{45}(\pm )$ and $P_{sy}^{+}(\pm )$ visually merge; however, the small difference between the forward- and backward-scattered contributions can be traced in the inset.

Figure 4

Comparison of the mechanical actions associated with the spin and orbital internal energy flows for (a) metal and (b) dielectric spherical particle suspended in water. Each curve represents the force dependence of the particle-size parameter (37). The SFD-induced forces $F_{sp}^{\pm }$ are calculated as $x$ components of the force experienced by a particle in the field characterized by Eqs. (30, 31, 32, 33, 34, 35, 36) for the conditions of Eqs. (41), (42) [cf. Fig. 2a]; their signs change upon switching the polarization handedness from σ = 1 (solid lines) to $\sigma =-1$ (dashed lines). The OFD-induced contributions $F_{orb}$ are determined as the $y$ components of the force exerted on a particle in the field characterized by Eqs. (44) and (41), (42) [cf. Fig. 2b].

Figure 5

Initial segments of curves, presented by Fig. 4, in double logarithmic scale. Solid lines: metallic particle [Fig. 4a], dashed lines: dielectric particle [Fig. 4b]. Orders of the force growth with the particle radius $a$ are indicated with allowance for the normalization factor $P_0$ (43). For comparison, the behavior of the gradient force $F_y^{grad}$ [Fig. 2a] is included.