The force exerted by an optical trap on a dielectric bead in a fluid is often found by fitting a Lorentzian to the power spectrum of Brownian motion of the bead in the trap. We present explicit functions of the experimental power spectrum that give the values of the parameters fitted, including error bars and correlations, for the best such χ² fit in a given frequency range. We use these functions to determine the information content of various parts of the power spectrum, and find, at odds with lore, much information at relatively high frequencies. Applying the method to real data, we obtain perfect fits and calibrate tweezers with less than 1% error when the trapping force is not too strong. Relatively strong traps have power spectra that cannot be fitted properly with any Lorentzian, we find. This underscores the need for better understanding of the power spectrum than the Lorentzian provides. This is achieved using old and new theory for Brownian motion in an incompressible fluid, and new results for a popular photodetection system. The trap and photodetection system are then calibrated simultaneously in a manner that makes optical tweezers a tool of precision for force spectroscopy, local viscometry, and probably other applications. © 2004 American Institute of Physics.