Adaptive Optimization

Optimal statistical analysis offers insights not only into the traps' properties, but also into the properties of the trapped particles and the surrounding medium. For example, if a spherical probe particle is immersed in a medium of viscosity $\eta$ far from any bounding surfaces, its hydrodynamic radius $a$ can be assessed from the measured drag coefficient using the Stokes result $\gamma = 6 \pi \eta a$. The viscous drag coefficients, moreover, provide insights into the particles' coupling to each other and to their environment. The independently assessed values of the traps' stiffnesses then can serve as a self-calibration in microrheological measurements and in measurements of colloidal many-body hydrodynamic coupling (32). In cases where the traps themselves must be calibrated accurately, knowledge of the probe particles' differing properties gauged from measurements of $\gamma$ can be used to distinguish variations in the traps' intrinsic properties from variations due to differences among the probe particles.

These measurements, moreover, can be performed rapidly enough, even at conventional video sampling rates, to permit real-time adaptive optimization of the traps' properties. Each trap's stiffness is roughly proportional to its brightness. So, if the $m$-th trap in an array is intended to receive a fraction $\left\vert \alpha_m \right\vert ^2$ of the projected light, then instrumental deviations can be corrected by recalculating the CGH with modified amplitudes:

$$\alpha_m \rightarrow \alpha_m \, \sqrt{\frac{\sum_{i=1}^N \kappa_i}{\kappa_m}}.$$ (46)

Analogous results can be derived for optimization on the basis of other performance metrics. A quasiperiodic pattern similar to that in Fig. 2(c) was adaptively optimized for uniform brightness, with a single optimization cycle yielding better than 12 percent variance from the mean. Applying Eqs. (41) to data from images such as Fig. 2(d) allows us to correlate the traps' appearance to their actual performance.

With each trap powered by 3.4 mW, the mean viscous relaxation time is found to be $\tau/\Delta t = 1.14 \pm 0.11$. We expect reliable estimates for the viscous drag coefficient under these conditions, and the result $\gamma/\gamma_0 = 0.95 \pm 0.10$ with an overall measurement error of 0.01, is consistent with the manufacturer's rated 10 percent polydispersity in particle radius. Variations in the measured stiffnesses, $\left< \kappa_x \right> = 0.38 \pm 0.06~\unit{pN/\ensuremath{\unit{\mu m}}\xspace }$ and $\left< \kappa_y \right> = 0.35 \pm 0.10~\unit{pN/\ensuremath{\unit{\mu m}}\xspace }$, can be ascribed to a combination of the particles' polydispersity and the traps' inherent brightness variations. This demonstrates that adaptive optimization based on the traps' measured intensities also optimizes their performance in trapping particles.