# Optimization of Optical Trapping and Laser Interferometry in Biological Cells
## 4 Experimental Results and Discussions
### LI Sensitivity Depending on Radial and Axial Offsets  
Next, in order to optimize the position sensitivity of LI, we investigate the LI signal as a function of the probe position relative to the focus of the probe laser [6]. The black-dotted curve in Figure 6a shows the LI signal as a function of the radial offset (\(x\), \(y\)) of the probe center from the laser focus. In this experiment, the probe particle was kept located at the focus plane (\(z=0\)). LI outputs are proportional to probe displacements when the probe is within the range of less-than \(\sim\)\(\pm\)0.13 \(\upmu\)m from the optical axis. Its gradient shown by the red curve provides the LI sensitivity which is approximately constant in the narrow linear range; the radial displacements of a probe need to be smaller than the range LI experiments. In living cells, however, the probe particle moves more than \(\upmu\)m during the experimental period of time. The feedback-tracking technique is therefore used to stably track a vigorously fluctuating probe in non-equilibrium systems [15]. A piezo stage on which a sample dish is placed is controlled with a feedback in order to maintain a probe in the laser focus. It has been expected that the LI sensitivity could have larger tolerance on the probe offset in the \(z\) direction (axial offset) compared to the radial offsets. However, \(z\) offset also affects the sensitivity when the probe fluctuations are vigorous [37]. Therefore, in this study, we have searched for the best \(z\) position for the sensitivity of LI. Probe particles were dispersed in an elastic hydrogel (0.5 wt% agarose [38]) and LI sensitivity was measured as a function of the probe position in \(z\) direction. The melamine particle with \(2a=1\)\(\upmu\)m diameter was used as a probe. Since the agarose gel is highly elastic at the concentration prepared for our experiments, a weak optical trap does not substantially affect the probe position. Therefore, we measured the LI sensitivity while scanning the axial position (\(z\)) of the probe. As shown in Figure 6b, the best sensitivity was not obtained when the probe is located at the plane of the laser focus (\(z=0\)) [37]. The sensitivity increases when the probe is above the laser focus (\(z>0\)) and reaches its maximum at \(z\sim 0.4\)\(\upmu\)m. Since the probe particle could work as a convex lens and reduce the size of the laser profile at the BFP while the average deflection of the laser is less affected (Figure 6c), the sensitivity takes the maximum there. When the probe is situated beneath the plane of the laser focus (\(z<0\)), on the other hand, the dispersion of sensitivities was larger. Although the mean sensitivity takes its second maximum around \(z\sim-1.2\)\(\upmu\)m, LI experiments are not appropriate at \(z<0\) because of the large errors. Figure 5: (**a**) Normalized LI sensitivity \(A\) in living HeLa cells and their theoretical predictions (green: silica, red: melamine, \(2a=1\)\(\upmu\)m). Symbols (closed squares and circles) are experimental results and error bars indicate standard deviations of the mean (silica: N = 3–5, melamine: N = 5). Solid and broken curves are the same theoretical predictions as those in Figure 4a. (**b**) Amplification of laser refraction at the cell-culture boundary. Scattered laser is refracted by the difference of the refractive index between cells (1.38) and culture medium (1.33). (**c**) \(Af_{\text{R}}(n_{40},n_{\text{p}})/(n_{\text{P}}-n)\) in cells (closed symbols) and aqueous glycerol (open symbols) are shown together with the theoretical prediction\(f_{\text{R}}(n,n_{\text{P}})/(n_{\text{P}}-n)\). Silica probe in living cells shows enormous systematic and random errors from the theoretical prediction. For OT and LI experiments in living cells, since the height of the particle vigorously fluctuates, \(z\) position of the probe particle must be also controlled by the feedback [15]. However, the feedback control in \(z\) direction is not as stable as the other directions, causing once-in-a-while failure of the particle tracking.