The trapping of Rayleigh metallic particles has been experimentally demonstrated in a conventional optical trap [1] using a microscope objective illuminated with a Gaussian beam. In these experiments, in order to have a significant prevalence of the trapping forces over the scattering forces, given that the scattering forces are proportional to the imaginary part of the metal polarizability, long wavelengths are needed.

To improve the trapping efficiency of metallic particles by minimizing the scattering forces, Zhan [2, 3] proposed the use of radial polarized beams in optical tweezers. These papers have had a large impact in the field of optical forces and trapping. The underlying idea of the abovementioned work is to take advantage of that the radial polarization generates a force field in the focal volume of a high numerical aperture (NA) microscope objective in which the equilibrium location of the conservative forces responsible for the particle trapping is on axis and spatially delocalized respecting the radiation pressure. This is because the radial polarization generates a tight intensity distribution around the focal point whereas the axial component of the radiation pressure, proportional to the time average Poynting vector, surrounds the optical axis. Then, in this picture, a metallic particle close to the focal point would be only significantly affected by the trapping force relaxing to a great extend the need of a small imaginary part of the polarizability.

Unfortunately, the description of the forces acting on a Rayleigh particle in the Zhan’s paper [2] is not complete.

It has been shown recently, that the force exerted by a light field on a small particle can be decomposed in three terms [4],

The first two terms in Eq. (1) follow the spatial distribution of forces mentioned before. The third term is small when the NA is low but when the NA increases it becomes significant altering the trapping potential structure, even in the case of a linear polarized beam [5].

Whereas in the linear uniform polarization case, the third term in Eq. (1) can be considered as a small correction (except for the case of large NA), in the case of a radial polarized beam is fundamental. Must be noticed that here, given that the non-zero electrical field components are radial, and an axial component appears after the refraction by the lens, the spin density is azimuthal and consequently the curl force is axial (parallel to the optical axis) on the focal plane.

Figure 1(a) , shows the normalized spatial distribution at the focal plane of the axial component of the radiation pressure term, (b) the spin density force and (c) the total scattering force (sum of the previous two). At a radius where the radiation pressure is maximum ($\sim 0.45\lambda$), the spin force reaches its minimum value oriented against the light propagation direction, compensating in part the radiation pressure. At the optical axis, the only scattering force is the spin density force. The calculation was done expressing the focal fields in terms of Bessel functions (see reference [6] for the details) for $\text{NA}=n\sin {70}^{\circ }$ ($n=1$). The plot in Fig. 1(d) shows the comparison between the radial profile of the total force of panel (c), solid line, with the total scattering force for a linearly polarized beam along the axis parallel to the polarization direction, dotted line, for the same NA. As can be seen, the scattering force on axis is higher in the radial case than in the linear case for the same NA. Only if the scattering forces are negligible, radial polarization can have an advantage due to the larger gradient force provided by the more compact focal volume intensity distribution the radial polarization provides. Note that the distributions of the scattering force amplitudes, represented in the Fig. 1(a) and (b), are independent of the wavelength given that the radiation pressure and the spin force are both proportional to${\lambda }_0{}^{-3}$ .

 

Fig. 1 (a) Normalized amplitude distribution of the axial radiation pressure component for a radial polarization beam on the transversal focal plane (x,y); (b) axial component of the spin density force; (c), the total axial scattering force; (d) comparison between the total axial scattering force for the radial (solid line) and linear polarization along the beam polarization direction (doted line) for the same NA.

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In conclusion we have shown that when modeling the scattering forces acting on a Rayleigh particle in full, the radial polarization does not provide a spatial distribution of scattering forces in the focal volume of microscope objective advantageous to minimize its effects in optical trapping.