# Optimization of Optical Trapping and Laser Interferometry in Biological Cells
## 4 Experimental Results and Discussions
### LI Sensitivity Depending on Radial and Axial Offsets
(**b**) \(\Delta I\) (the difference of pixel intensities at \(\rho=4\) and 22) as a function of the probe height. The definition of \(\Delta I\) is shown by using the vertical solid line and the dotted line in Figure 7a. Blue circles correspond to the data taken in the setup (1) which is the condition used to obtain data shown in (a). Red triangles were obtained in the setup (2) after shifting the focus of the microscopic image by moving the position of the video camera while the radial distances of \(\Delta I\) was fixed (\(\rho=4\) and 22). Upper schematic illustrations show charge-coupled device (CCD) camera setup (1) and (2) together with the microscope images of the probe at \(z=0\). We performed similar experiments in a living cell (Figure 6a). During the measurement, the feedback control was conducted only in radial directions (\(x,y\)), but not in the \(z\) direction since \(z\) positions of the probe must be scanned exceeding the range of heights in which the feedback control is available in the \(z\) direction. Instead, the stage was moved in \(z\) direction faster than the time scale of probe fluctuations in cells. For the same set of images recorded during the scan, \(\Delta I\) was calculated for \(\rho\) = 3 and 17 pix in Figure 8b and \(\rho\) = 7 and 30 pix in Figure 8c. The feedback control of the probe height is accurate and stable when the absolute value of the slope of \(\Delta I\) is sufficiently large (\(z\sim 0.8~{}\upmu\)m in Figure 8b and \(\sim 0.5~{}\upmu\)m in Figure 8c). It is seen that the slope and the linear range of \(\Delta I\) depends on the choice of \(\rho\). The range of heights in which \(\Delta I\) is linearly dependent is largely determined by the relative distance between laser focus and the focus of microscope images. However, it is hard to finely tune the setup before every experiment. Imaging conditions for the microscope such as lighting show subtle changes for every experiment and that could slightly shift the range of \(\Delta I\) and affect feedback stability. Thereby, rather than adjusting the camera position for every experiment, one can optimize the LI sensitivity and the feedback stability by fine tuning the choice of \(\rho\) and setpoint. Finally, we measured the LI sensitivity in living cells as a function of probe offset in \(z\) direction while conducting 3-D feedback tracking. In order to conduct the feedback tracking in wide range of \(z\), the different setup for the camera position was chosen when measuring below and above \(z=1.5~{}\upmu\)m. As shown with closed red circles in Figure 9, experimental results exhibit a similar trend to that observed in the agarose gel (Figure 6b). Note again that the sensitivity is remarkably reduced at \(z=0\), and takes its maximum at \(z\sim 0.5~{}\upmu\)m. Open black diamonds in Figure 9 show the results of numerical calculations based on Lorenz-Mie theory [32]. The simulation results were interpolated to provide  
Figure 8: **(a)**\(I(\rho)\) in living HeLa cells measured while controlling the feedback in \(x\) and \(y\) directions. Height dependence of microscope images were recorded while quickly scanning w/o \(z\) feedback. Results are shown similar to those in Figure 7a. (**b**) \(\Delta I\) as a function of particle height. \(\Delta I\) was determined from \(I(\rho\) = 3 pix) and \(I(\rho\) = 17 pix), as shown by the vertical solid lines in **(a)**. Linear region is \(0.5~{}\upmu\)m \(<z<1~{}\upmu\)m. (**c**) Another height dependence of \(\Delta I\) that was determined by \(I(\rho\) = 7 pix) and \(I(\rho\) = 30 pix), as shown by the broken lines in **(a)**. Linear region is shifted to \(-0.4~{}\upmu\)m \(<z<1~{}\upmu\)m.  
the solid curve as an eye guide. Trend of the measured sensitivity (closed red circles) agrees with the theory in the range 0.0 \(\upmu\)m \(<\)\(z\) \(<\) 0.5 \(\upmu\)m. The numerically-calculated sensitivity takes negative values when the probe particle is located below the laser focus (\(z\) \(<\) 0).