  For experiments with ions confined in a Paul trap, minimization of
micromotion is often essential. In order to diagnose and compensate micromotion
we have implemented a method that allows for finding the position of the
radio-frequency (RF) null reliably and efficiently, in principle, without any
variation of direct current (DC) voltages. We apply a trap modulation technique
and focus-scanning imaging to extract 3d ion positions for various RF drive
powers and analyze the power dependence of the equilibrium position of the
trapped ion. In contrast to commonly used methods, the search algorithm
directly makes use of a physical effect as opposed to efficient numerical
minimization in a high-dimensional parameter space. Using this method we
achieve a compensation of the residual electric field that causes excess
micromotion in the radial plane of a linear Paul trap down to 0.09V/m.
Additionally, the precise position determination of a single harmonically
trapped ion employed here can also be utilized for the detection of small
forces. This is demonstrated by determining light pressure forces with a
precision of 135yN. As the method is based on imaging only, it can be applied
to several ions simultaneously and is independent of laser direction and thus
well-suited to be used with, for example, surface-electrode traps.
