# Optical Binding and Trapping

### MIT Center for Electromagnetic Theory and Applications

#### Fundamentals

Optical forces | Force fields | Lorentz force distribution in media | Modeling Figure 3: Forces on a particle in a Gaussian beam.
Image: J.-M. Fournier.

##### Optical forces

An electromagnetic wave, composed of an electric field and a magnetic field governed by Maxwell's equations, exerts a force when impinging on objects. This force can be either computed via a direct application of the Lorentz force and bound/free current/charges within the volume of the object, or via the Maxwell stress tensor. The advantage of the latter method is its computation efficiency since the electromagnetic fields need to be evaluated only on a surface enclosing the object, while the former method needs the evaluation of the fields within the whole volume. The disadvantage, however, is that the polarizability of the object within its volume is not computed, which is often seen as a more intuitive approach to the force calculation.

The force from the impinging wave is unique, and has to be obtained by calculating the total field surrounding the object, a summation of the incident and the scattered fields. When the particle is small, however, the force can be expressed as a sum of two terms which have been identified as:

• the scattering force: a force that is parallel to the Poynting vector of the propagating wave, pushing or pulling the object in the same direction as the wave propagation [Fig. 1].
• the gradient force: a force due to the gradient of intensity of the electromagnetic radiation. Such gradient is typically obtained by laser beams or in optical lattices. Depending on the property of the object (size, permittivity, permittivity contrast with the background, etc), the gradient force can either be attractive to high intensity regions, or repulsive [Fig. 2]. For small objects and laser beams, the force is attractive and yields an optical tweezer. [See Fig. 3 above.]

A third type of force has been discovered by Burns, Fournier, and Golovchenko [Ref. 1] when multiple particles interact with each other. This force has been called 'binding force', and represents the self-consistent interaction between the multiple particles and the incident wave [Fig. 4].

##### References
1. M. Burns, J-M. Fournier and J. Golovchenko, Optical binding, Phys. Rev. Lett., 1989.
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