Radial dependence of the angular momentum density of a paraxial optical vortex

We obtain general analytic expressions for the linear and angular momentum density of an elliptically polarized paraxial laser beam. As a partial case, we obtain expressions for the linear and angular momentum density of a paraxial elliptically polarized optical vortex. It is shown that for an arbitrary vortex field with rotational symmetry and with circular polarization the effect of the “angular tractor” takes place, which means that the flux of light energy rotates around the optical axis clockwise or counterclockwise at different radii in the beam cross section. It is also shown that the axial component of the angular momentum density of the vortex light field changes its sign at the same radii in the beam cross section. Microparticles trapped in the vortex Bessel beam at different radii are experimentally shown to rotate simultaneously clockwise and counterclockwise.

Figure 1

Intensity distributions of the third-order Bessel modes ($n=3$) with the scaling factor $k_r=k/10$ (i.e., paraxial beam) with right ($\sigma =1$) (a) and left ($\sigma =\text{–}1$) (b) circular polarization. Size of the calculation area is $2R=50\lambda$. Plots (red solid curves) show the longitudinal component of the angular momentum density vector. Dashed arrows show the direction of rotation of the energy flux.

Figure 2

Experimental setup: L is a solid-state laser ($\lambda =532\mathrm{nm}$), $L_1$, $L_2$, $L_3$, $L_4$ are lenses with focal lengths ($f_1=150\mathrm{mm}$, $f_2=350\mathrm{mm}$, $f_3=350\mathrm{mm}$, $f_4=150\mathrm{mm}$), SLM is a spatial light modulator (HOLOEYE, PLUTO-VIS), D is a diaphragm, WP is a quarter-wave plate, ${\mathrm{M}}_1$ and ${\mathrm{M}}_2$ are mirrors, $\mathrm{M}{\mathrm{O}}_1$ is a micro-objective (20×, $\mathrm{NA}=0.4$), $\mathrm{M}{\mathrm{O}}_2$ (16×, $\mathrm{NA}=0.3$), S is a substrate with a solution of 5-μm polystyrene spheres, I is a LED bulb, CMOS is a video camera.

Figure 3

Rotation of 5-μm polystyrene spheres in a third-order Bessel beam with left circular polarization: (a) intensity distribution of the laser beam in the plane of microspheres trapping, (b) stages of motion of the trapped particles (the time interval between the frames is 5 s). Frame size is 60 × 60 μm. Circles with crosses show positions of all the microspheres in each frame, while the arrows show positions of the microspheres moving in opposite directions.