Calibration of optical tweezers with positional detection in the back focal plane

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—$10μm$ from the surface—and the stiffness is typically reduced by more than a factor of 2 for beads with a diameter of $∼500nm$ (Refs. 12, 20, 21, and 42). Thus, one detrimental effect is swapped for another, and calibration errors can be large.

See Appendix  C for how to square Dirac’s delta-function in Eq. (8).

Equation (34) in Ref. 2 contains a typographical error: The parameter $fm$ in this equation is $fm,0=γ0∕(2πm)$⁠, i.e., not the same as the parameter $fm$⁠, which occurs in Eq. (32) in Ref. 2. Systematic naming is achieved, if the latter is referred to as $fm*,0=γ0∕(2πm*)$⁠, i.e., only its name is changed, not its definition in terms of $m*=m+2πρR3∕3$⁠, where $ρ$ 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 $γ(0,R∕ℓ)$ is approached by $γ(f,R∕ℓ)$ by expanding Eq. (D6) in powers of $f∕fν$⁠, the approach is incorrect to first order.