Because of the complexity of the LP2i mode in terms of expression, in order to simplify and reduce the complexity of the derivation, we use four superimposed Gaussian beam profile, each light spot is approximated to a Gaussian beam. The four Gaussian beams are then superimposed to obtain the final force acting on cells. First we list the characteristics of Gaussian beams here. The basic properties of the electromagnetic field are described by the Maxwell equations and material equations:
In a non-magnetic, lossless, isotropic homogeneous medium, Maxwell equations and material equations can be simplified to Helmholtz equation under condition of steady-state harmonic electromagnetic wave:
A Gaussian beam is a particular solution to Eq.3.22 for small amplitudes. A fundamental mode Gaussian beam takes a form of:
where w(z) is the width of the Gaussian beam that intersects with the optical axis z; the radius of curvature R(z) is the radius of curvature of the beam wavefront, conforming to iso-phase surface of the beam; waist radius w0 is the radius at the narrowest focus of a Gaussian beam; z0 is the depth of focus, showing the distance from location of w0 to a spot size with V2w0.
Though parameters of a Gaussian beam are correlated, simply use three parameters can describe Gaussian beam characteristics. For example, we can use the waist radius w0, focal depth z0 and waist positions z these three parameters to characterize a particular Gaussian beam. Thus, the entire Gaussian beam was uniquely determined. For the advantage, in the next section a set of four Gaussian beams is to approximate a LP21 mode in calculations, limiting number of beam parameters to the minimum in simulation.