Trap characterization

The stiffness and viscous drag coefficient can be estimated simultaneously as

$$\frac{\kappa}{k_B T} = \frac{1 - \hat{\phi}^2}{\hat{{\sigma_a^2}}} \frac{\gamma}{k_B T \, \Delta t} = - \frac{1 - \hat{\phi}^2}{\hat{{\sigma_a^2}} \, \ln \hat{\phi}},$$ (41)

with error estimates, $\Delta \kappa$ and $\Delta \gamma$, given by

$$\left(\frac{\Delta \kappa}{\kappa} \right)^2 = \left(\frac{\Delta\hat{{\sigma_a^2}}}{\hat{{\sigma_a^2}}}\right... ... \hat{\phi}^2} \right)^2 \, \left(\frac{\Delta \hat{\phi}}{\hat{\phi}}\right)^2$$ (42)

$$\left(\frac{\Delta \gamma}{\gamma} \right)^2 = \left(\frac{\Delta\hat{{\sigma_a^2}}}{\hat{{\sigma_a^2}}}\right... ... \hat{\phi}} \right)^2 \, \left(\frac{\Delta \hat{\phi}}{\hat{\phi}}\right)^2 .$$ (43)

If the measurement interval, $\Delta t$, is much longer than the viscous relaxation time $\tau = \gamma/\kappa$, then $\phi$ vanishes and the error in the drag coefficient diverges. Conversely, if $\Delta t$ is much smaller than $\tau$, then $\phi$ approaches unity and both errors diverge. Consequently, this approach does not benefit from excessively fast sampling. Rather, it relies on accurate particle tracking to minimize $\Delta \hat{\phi}$ and $\Delta \hat{{\sigma_a^2}}$. For trap-particle combinations with viscous relaxation times exceeding a few milliseconds and typical particle excursions of at least 10 nm, digital video microscopy provides the resolution needed to simultaneously characterize multiple optical traps (29).

In the event that measurement errors can be ignored ( ${\sigma _b^2}\ll {\sigma _a^2}$),

$$\frac{\kappa_0}{k_B T} = \frac{1}{c_0} \, \left[1 \pm \sqrt{ \frac{2}{N} \left(1 + \frac{2c_1^2}{c_0^2 - c_1^2}\right)} \right]$$ and

$$\frac{\gamma_0}{k_B T \, \Delta t} = \frac{1}{c_0 \, \ln\left(\frac{c_0}{c_1}\right)} \, \left( 1 \pm \frac{\Delta \gamma_0}{\gamma_0}\right)$$ (44)

where

$$N \, \left(\frac{\Delta\gamma_0}{\gamma_0}\right)^2 = 2 + \frac{1... ...frac{c_1}{c_0}\right) - c_1^2}{ c_1 \ln\left(\frac{c_1}{c_0}\right)} \right]^2.$$ (45)

These results are not reliable if $c_1 \lesssim {\sigma_b^2}$, which arises when the sampling interval $\Delta t$ is much longer or much shorter than the viscous relaxation time, $\tau$. Accurate estimates for $\kappa$ and $\gamma$ still may be obtained in this case by applying Eq. (40).

As a practical demonstration, we analyzed the thermally driven motions of a single silica sphere of diameter $1.53~\ensuremath{\unit{\mu m}}\xspace$ (Bangs Labs lot number 5328) dispersed in water and trapped in a conventional optical tweezer. With the trajectory resolved to within $6~\unit{nm}$ at 1/30 sec intervals, 1 minute of data suffices to measure both $\kappa$ and $\gamma$ to within 1 percent error using Eqs. (41). The results plotted in Fig. 4(a) indicate trapping efficiencies of $\kappa_x/P = \kappa_y/P = 142 \pm 3~\unit{pN/\ensuremath{\unit{\mu m}}\xspace W}$. Unlike $\kappa$, which depends principally on $c_0$, $\gamma$ also depends on $c_1$, which is accurately measured only for $\tau \gtrsim 1$. Over the range of laser powers for which this condition holds, we obtain the expected $\gamma_x/\gamma_0 = \gamma_y/\gamma_0 = 1.0 \pm 0.1$, as shown in Fig. 4(b). The viscous relaxation time becomes substantially shorter than our sampling time for higher powers, so that estimates for $\gamma$ and its error both become unreliable, as expected.