Three-dimensional position detection of optically trapped dielectric particles

Abstract

A theory is presented together with simulation results that describe three-dimensional position detection of a sphere located in a highly focused beam by back-focal plane interferometry. This technique exploits the interference of scattered and unscattered light, which is projected on a quadrant photodiode placed in the back-focal plane of a condenser lens. Due to the Gouy-phase shift inherent in focused beams, it is not only possible to determine the lateral but also the axial position of a spherical particle with nanometer accuracy. In this paper we describe the calculation of arbitrary focused electromagnetic fields, the Gouy phase shift, Mie scattering by focused beams and the resulting position signals using the angular momentum representation. The accuracy and the sensitivity of the detection system are investigated theoretically for various sphere parameters. Both accuracy and sensitivity depend on the incident light distribution as well as on the particle’s properties and position. It is further shown that the maximum capture angle of the detection lens influences the detector’s sensitivity in a nonlinear manner. Additionally, for optical trapping applications the influence of the laser power is taken into account and is considered through a noise analysis. For all investigated trapping conditions the reconstructed position deviates on average <1 nm laterally and <5 nm axially from the actual particle position.