We have combined a Debye-Wolf treatment of light propagation through an optical train with Lorenz-Mie theory for light scattering to develop a vectorial theory for the forces and torques applied by holographic optical traps. This theory of holographic trapping is inherently more accurate than approximations based on scalar diffraction diffraction theory or on the Rayleigh or ray-optics approximations. It not only reproduces previous results obtained for optical tweezers and related single-beam optical traps, but accurately accounts for polarization effects in a realistic model for the optical train of practical holographic trapping systems. Treating the hologram on a pixel-by-pixel basis not only permits detailed analysis of the very general optical force fields that can be projected holographically, but also lends itself to accurate treatment of aberrations, which can be encoded in the DOE.
We have used this vector theory to confirm the behavior of phase-gradient forces in extended traps that was predicted on the basis of scalar diffraction theory (7). The spatially resolved images of the force and torque fields further reveal substantial consequences of polarization rotation in high-numerical-aperture optics. The transformation of the linearly polarized input beam into more general and spatially varying elliptical polarization can give rise to a highly structured torque field whose influence has yet to be observed experimentally. This suggests that polarization engineering can be viewed as an additional channel for control in holographically structured light fields. More generally, the vectorial theory of holographic optical trapping should provide a useful basis for designing and optimizing optical micromanipulation systems for particular applications.
This work was supported in part by the NSF through Grant Number DMR-0606415 and in part by a grant from the Keck Foundation. B.S. acknowledges support from a Kessler Family Foundation Fellowship.