Interference model for back-focal-plane displacement detection in optical tweezers

The lateral position of an optically trapped object in a microscope can be monitored with a quadrant photodiode to within nanometers or better by measurement of intensity shifts in the back focal plane of the lens that is collimating the outgoing laser light. This detection is largely independent of the position of the trap in the field of view. We provide a model for the essential mechanism of this type of detection, giving a simple, closed-form analytic solution with simplifying assumptions. We identify intensity shifts as first-order far-field interference between the outgoing laser beam and scattered light from the trapped particle, where the latter is phase advanced owing to the Gouy phase anomaly. This interference also reflects momentum transfer to the particle, giving the spring constant of the trap. Our response formula is compared with the results of experiments.