Thermal motion of a holographically trapped SPM-like probe

By holding a complex object in multiple optical traps, it may be harmonically bound with respect to both its position and its orientation. In this way a small probe, or nanotool, can be manipulated in three dimensions and used to measure and apply directed forces, in the manner of a scanning probe microscope. In this paper we evaluate the thermal motion of such a probe held in holographic optical tweezers, by solving the Langevin equation for the general case of a set of spherical vertices linked by cylindrical rods. The concept of a corner frequency, familiar from the case of an optically trapped sphere, is appropriately extended to represent a set of characteristic frequencies given by the eigenvalues of the product of the stiffness matrix and the inverse hydrodynamic resistance matrix of the tool. These eigenvalues may alternatively be interpreted as inverses of a set of characteristic relaxation times for the system. The approach is illustrated by reference to a hypothetical tool consisting of a triangular arrangement of spheres with a lateral probe. The characteristic frequencies and theoretical resolution of the device are derived; variations of these quantities with tool size and orientation and with the optical power distribution, are also considered.