#### Fundamentals

Optical forces | Force fields | Lorentz force distribution in media | Modeling##### Modeling

The Lorentz force on a canonical particle can be directly computed from the known incident and scattered fields. Some illustrations are given below for infinitely long cylinders and various incidences.

Figure 1: Case 1, Example 1, Real(Ez).*Image: T. Grzegorczyk.*

In the case of a single plane wave, the force is constant througout space, independent of the position of the particle. The figures show the total fields (* E_{z}* parallel to the axis of the cylinder in Fig. 1,

*in Fig. 2 and*

**H**_{x}*in Fig. 3) used either in the Maxwell stress tensor method, or in the direct application of the Lorentz force.*

**H**_{y}Figure 4: Case 1, Example 3, a = λ/2,

Real(Ez). *Image: T. Grzegorczyk.*

In the case of 3 plane wave incidences, the interference pattern is hexagonal, creating a periodic array of optical traps (yielding an optical lattice). Fig. 4 shows * E_{z}*, Fig. 5 shows

*, and Fig. 6 shows*

**H**_{x}*. The angles between the plane waves are adjusted such as to cancel the direct forces from each individual plane wave, and only the cross-terms remain. The optical lattice is an array of potential well that attract or repel the particles as function of their properties, giving a direct illustration of the gradient force.*

**H**_{y}Figure 7: Forces on a particle in a Gaussian beam.*Image: T. M. Grzegorczyk.*

In the case of a Gaussian beam (simulating a laser beam), both scattering and gradient forces are present. It is interesting to see how the beams pushes the particle in the direction of its propagation, while also pulling it towards its waist. The equilibrium position is found slightly beyond the waist, where the two forces cancel each other (supposing that the system is not subject to other external forces).