Optical forces and torques

The optical force and torque experienced by an illuminated object arises from the energy-momentum tensor $T_\mu^\nu(\vec{r})$ of the electromagnetic field. Among various formulations, the symmetric Maxwell tensor is found to simplify computation of forces and torques (15). In SI units,

$$\mu_0 T_\mu^\nu = - F_{\mu \rho}F^{\rho \nu} + \frac{1}{4} \, F_{\rho \sigma}F^{\rho \sigma} \, \delta_\mu^\nu,$$ (1)

where the momentum flux in the electromagnetic field

$$F_\mu^\nu = \partial_\mu A^\nu - \partial^\nu A_\mu,$$ (2)

emerges from gradients in the light's vector potential, $\vec{A}$ . The force on a particle is then obtained by integration over its surface $\Sigma$ ,

$$F_\mu = \oint_\Sigma T_\mu^\nu \, dS_\nu.$$ (3)

Similarly, the optical torque is obtained as

$$M_\mu = \frac{1}{2} \, \oint_\Sigma \epsilon_{ij\mu} M^{ijk} \, dS_k$$ (4)

where the angular momentum tensor corresponding to the symmetric stress tensor is

$$M_{ijk} = r_i T_{jk} - r_j T_{ik}$$ (5)

and where $\epsilon_{ijk}$ is the Levi-Civita antisymmetric tensor.

In discussing the forces exerted by optical traps, it is conventional to define the trapping efficiency in the direction $\nu$ as (8)

$$Q_\nu = \frac{c}{n_m P} \, F_\nu$$ (6)

where $P$ is the laser power, $n_m$ is the refractive index of the medium, which we assume for simplicity to be homogeneous and isotropic, and $c$ is the speed of light in vacuum. Similarly the torque efficiency for a sphere of radius $a$ rotating about the $\nu$ direction is

$$\tau_\nu = \frac{\omega}{P} \, M_\nu.$$ (7)

This general formulation is common to all theoretical studies of optical trapping. Diverse implementations differ in how they treat the incident and scattered electromagnetic fields surrounding the particle, and in their strategy for performing the surface integrals in Eq. (3) and (4). The formalism described in the following sections computes the superposition of contributions from each pixel in a pixellated DOE to the overall force and torque exerted on an object in a holographic optical trapping system. Although our discussion focuses on optical trapping of isotropic homogeneous spheres, it can be generalized to particles of other shapes and compositions.