## Computation of forces exerted on a microparticle by a laser beam

##### Abstract

A mathematical description of the electromagnetic fields of non-paraxial laser beams is
derived and used to calculate the trapping forces on spherical particles. The fields are exact
solutions to the wave equation. A set of closed-form expressions for the scalar field of such a
beam is presented first. The solution for the order 00 is equivalent to the wave of a combined
complex-point source and sink. In the far field the two lowest order solutions, 00 and 01,
closely match the energy density produced by a high-numerical aperture lens illuminated by a
paraxial Gaussian beam. At the large beam waist limit these two solutions reduce to the
paraxial beam form. However, it is found that only the 01 order solution is physically
realizable, since the total energy flux through the transverse section of the 00 order beam is
infinite. The scalar solutions of arbitrary order are then used to derive solutions to the vector
wave equation. Next, the electric and magnetic fields that closely fit the far-field boundary
conditions for a focusing lens are constructed from the solutions for the orders 00 and 01.
These fields are in general elliptically polarized at the beam waist. However at the large beam
waist (paraxial) limit and in the far field limit the fields become linearly polarized. The
electromagnetic field due to order 01 is used to calculate the Maxwell stress tensor, and hence
the trapping forces exerted on a dielectric microsphere in a single beam laser tweezers setup. It
is demonstrated that the electromagnetic theory model based on the 5th order Gaussian beam
approximation due to Barton is accurate for almost paraxial beams (numerical aperture
NA<0.25), when compared to the model derived here. However, for strongly focused beams
(NA>l) the 5th order approximation breaks down. Trapping forces on water droplets
suspended in air and on polystyrene spheres suspended in water, exerted by a Gaussian laser
beam focused with lenses of various numerical apertures are calculated. It is established that a
model accurate for a strongly focused beam is vital, since in order to trap a particle effectively
a focusing lens with NA>1 is required.