Holographic optical trapping (HOT) is an increasingly popular method for applying precisely controlled forces to microscopic objects (1,2). In this technique, a computer-generated hologram is imprinted onto the wavefronts of a laser beam using a diffractive optical element (DOE) such as a spatial light modulator (SLM). The modified beam then is relayed to a high-numerical-aperture lens, which focuses the light into the desired pattern of optical traps. This three-dimensionally structured light field influences the motions of illuminated objects through a combination of induced-dipole forces (3), which arise from local intensity gradients, and radiation pressure (4,5,6), which is directed by local phase gradients (7). The former generally is referred to as the optical gradient force, and the latter as the scattering force. If the combination of these optical forces gives rise either to a mechanical equilibrium or to a dynamical steady state, the illuminated object is said to be trapped, and the structured light field constitutes an optical trap.
The forces and torques exerted by an optical trap can be computed by first calculating the electromagnetic field surrounding an illuminated object and then integrating the Maxwell stress tensor over its surface. This approach has been used (8,9,10,11) to model the forces exerted by conventional optical tweezers (3). More recently, finite-difference time-domain calculations have been used to compute the optical forces and torques on multiple spheres and cylinders arranged in holographically projected arrays of optical tweezers (12). These studies use fully vectorial expansions of the light field, and so capture polarization-dependent effects as well as those due to intensity and phase gradients. Because they are based on particular models for the focused beam, however, they are not easily extended to the more general light fields that can be projected with the HOT technique.
This article presents an efficient method for computing the optical forces and torques exerted by holographic trapping patterns in practical HOT systems. Based on Debye-Wolf theory for light propagation through optical trains and the Lorenz-Mie theory of light scattering by small particles, this approach expresses the incident and scattered fields as solutions of the vector Helmholtz equation and so accurately describes the forces encountered in high-numerical-aperture trapping.
Figure 1 schematically represents the optical train of a typical HOT system. We assume that the DOE consists of an array of discrete pixels, each of which imprints a local phase shift on the wavefronts of a beam of light. An image of the pixel array is projected onto the input pupil of the objective lens by a telescope. This image can be decomposed into a set of plane waves, each of which is brought to a focus by the objective lens. Assuming the objective to be free from aberrations, its focusing properties can be modeled with a well-established angular distribution of plane waves (13). Each pixel in the DOE therefore contributes to the angular distribution of plane waves in the far field of the objective lens. These plane waves, in turn, impinge on the illuminated object, whose scattering pattern can be computed with Lorenz-Mie formulas, T-matrix theory or related methods. Summing up the individual pixels' contributions to the scattered field yields the total electromagnetic field at the particle's surface, and thus the Maxwell stress tensor. This then yields the optical forces and torques experienced by the particle (8,14)