Theoretical description of optical forces is simple only for very small particles (so-called Rayleigh particles - RP), whose radius fulfills a<<l/20, where l is the trapping wavelength in the medium. Such a small particle behaves as an induced elementary dipole and the optical forces acting on it can be divided into two components - gradient and scattering forces. The gradient force comes from electrostatic interaction of a particle (dielectrics) with an inhomogeneous electric field and the scattering force results from the scattering of the incident beam by the object.
For particles bigger than l/20, a more complex concept of stress tensor of electromagnetic field surrounding the particle must be generally used. This requires the knowledge of the total field outside the confined particle. The original theory based on the plane wave scattering has been gradually modified so that it can be applied for a spherical or spheroidal particle placed into an arbitrary field distribution. It is commonly referred to as the generalized Lorenz-Mie theory (GLMT).
The GLMT uses the scattering procedure presented first by Mie who derived expressions for the field distribution outside a spherical object of arbitrary size placed into a plane wave. This original method was generalized so that it enables an expression of the forces acting on the spherical and oblate objects placed in a general electromagnetic field. For the description of a focused beam, a modification to the fundamental Gaussian beam was presented (so called 5th order corrected Gaussian beam (CGB)). It uses a field expansion in the beam size parameter s=1/(kw0) to the 5th order, achieves better agreement with the wave equation, and, therefore, provides more precise calculation of the optical forces.
Because we consider the standing wave created by the interference of two counter-propagating focused laser beams, we have easily adapted the above mentioned GLMT formalism to this case. Instead of a single CGB we summed field components of two counter-propagating CGB with overlapped beam waists to get the initial field components of the standing wave. Moreover, we assumed that the spherical object was located on the beam axis and so we could employ the radial symmetry of the problem and simplify the calculation. We wrote the modified code ourselves but we do not present a detailed mathematical description, because the method is well described in literature.
We neglected any electrostatic interactions between the surface and the particle as well as multiple scattering of the incident beam. Furthermore we assumed that the beam waist was placed on the surface with reflectivity equal to 100%. The axial positions zs of the sphere center, where we calculated the axial forces, satisfy the inequality a <= zs <= a+l.
Although the adopted simplifications (CGB, absence of spherical aberrations, diffraction, and multiple scattering events) could seem drastic, this model provides at least correct qualitative description of the behavior of dielectric spheres in the GSW and acceptable speed of calculations.
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