We explain and demonstrate a new method of force and position calibrations for optical tweezers with back-focal-plane photodetection. The method combines power spectral measurements of thermal motion and the response to a sinusoidal motion of a translation stage. It consequently does not use the drag coefficient of the trapped object as an input. Thus, neither the viscosity, nor the size of the trapped object, nor its distance to nearby surfaces needs to be known. The method requires only a low level of instrumentation and can be applied in situ in all spatial dimensions. It is both accurate and precise: true values are returned, with small error bars. We tested this experimentally, near and far from surfaces in the lateral directions. Both position and force calibrations were accurate to within 3%. To calibrate, we moved the sample with a piezoelectric translation stage, but the laser beam could be moved instead, e.g., by acousto-optic deflectors. Near surfaces, this precision requires an improved formula for the hydrodynamical interaction between an infinite plane and a microsphere in nonconstant motion parallel to it. We give such a formula.
A common approach calibrates far enough above the coverslip surface to ensure that uncertainty about the height does not affect the calibration significantly. This calibration is then used closer to the surface. If an oil-immersion objective is used, this is an error-prone approach: due to the difference in refractive index between oil and water, the trap stiffness decreases rapidly with distance from the surface— from the surface—and the stiffness is typically reduced by more than a factor of 2 for beads with a diameter of (Refs. 12, 20, 21, and 42). Thus, one detrimental effect is swapped for another, and calibration errors can be large.
Equation (34) in Ref. 2 contains a typographical error: The parameter in this equation is , i.e., not the same as the parameter , which occurs in Eq. (32) in Ref. 2. Systematic naming is achieved, if the latter is referred to as , i.e., only its name is changed, not its definition in terms of , where is the mass density of the liquid (Ref. 40). This error is not repeated in the programs described in Refs. 17 and 18.
Note, however, that a function can approximate another function well, even when their derivatives do not approximate each other well. This is the case here: If one considers how Faxén’s result is approached by by expanding Eq. (D6) in powers of , the approach is incorrect to first order.