Optical force acting on a particle in the presence of a backward energy flow near the focus of a gradient lens

We show that a 70-nm dielectric nanoparticle placed on the optical axis near the surface (at a distance less than 100 nm) of a high-NA gradient microlens made of silicon, which is illuminated by a laser beam of 1.55 $\mu$m wavelength, is attracted to the lens surface with a piconewton force. The profile of the lens refractive index is described by a hyperbolic secant function. If a cut-out is made in the lens output surface, then the nanoparticle will be pulled into this cut-out, producing a kind of 'optical magnet'. If a reverse energy flow is to be generated on the optical axis near the output surface of such a gradient lens, this will lead to an absorbing dielectric nanoparticle being pulled toward the surface with a greater force than a similar non-absorbing particle. In the absence of a reverse flow, both absorbing and non-absorbing particles will be attracted to the surface with an equal force. The electromagnetic fields involved are calculated using a finite difference time domain (FDTD) method and the acting forces are calculated using a Maxwell stress tensor.