# Optimization of Optical Trapping and Laser Interferometry in Biological Cells
## 4 Experimental Results and Discussions
### LI Sensitivity Depending on Radial and Axial Offsets
We therefore investigate the reason and explore the way to optimize the feedback stability. For the feedback control in \(z\) direction, the probe height was obtained from the microscope images of the particles. The procedure for the image analysis utilizes a similar principle to holographic microscopy [39], though we use it in a more practical manner. First, the pixel intensity \(I\) (\(\rho\)) as a function of radial distance \(\rho\) from the probe center was obtained from the microscope image as shown in Figure 7a. It is seen that the radial intensity profile of the probe image depends on the probe height. We chose two different positions from the probe center (\(\rho\) = 4 and 22 pix) and obtained the difference of the pixel intensities \(\Delta I\) as shown by the solid and broken lines in Figure 7a. By moving the piezo stage in the \(z\) direction, it is observed that \(\Delta I\) changes linearly while the probe height is altered in the rage of \(-0.3\)\(\upmu\)m \(<z<0.2\)\(\upmu\)m, as shown by the blue circles in Figure 7b. It was confirmed that \(\Delta I\) can be used as a measure for probe heights in the linear region. When the feedback control of probe position is conducted in this range in the \(z\) direction, \(\Delta I\) can be used as the \"measure signal\" that should be maintained at a certain assigned value referred to as the \"setpoint\". When a probe accidentally goes out of the proportional range, on the other hand, the feedback tracking fails. Figure 6: **(a)** QPD output voltage (closed circles, left axis) as a function of probe offset (\(x\) or \(y\)) from the laser focus. \(2a\) = 1 \(\upmu\)m melamine particles in 0.5 wt% agarose gel were used. The piezo-sample stage was moved while the QPD signal and the probe position were measured. QPD signals are proportional to the probe offsets at the narrow proximity of the laser focus. LI sensitivity \(S\) was obtained as the gradient of the QPD output as shown by the red curve (right axis). **(b)** Height dependence of the Normalized LI sensitivity measured with \(2a\) = 1 \(\upmu\)m melamine particles in 0.5 wt% agarose gel. Error bars represent the standard deviations of the mean (N = 5). **(c)** Schematic illustrations of a typical laser lay for \(z>0\) (left, solid lines) and \(z=0\) (middle, broken lines), and the BFP image of the laser projected on to the QPD (right). The BFP image of the laser beam becomes smaller for \(z>0\) while the average deflection of the laser is not substantially affected. Consequently, the sensitivity takes its maximum for \(z>0\). In the usual set up of the microscope used for OT and LI, the focus of microscope image and the focus of the trapping laser is overlapped at \(z=0\). In this condition, \(z\) position optimized for LI sensitivity (\(z\sim 0.4~{}\upmu\)m, Figure 6b) is at the edge of the range in which \(\Delta I\) is linearly dependent on the probe height \(z\). Thus, \(z\) position at which the best LI sensitivity is achieved is not ideal for the feedback-tracking stability. However, the focus position for the microscope image can be shifted by inserting TV lens in front of camera and tuning the distance between them. The red triangles in Figure 7b show \(\Delta I\) measured by shifting the microscope focus towards \(z>0\). Note that \(\Delta I\) now changes linearly with probe heights in the range of \(0.4<z<1.1~{}\upmu\)m. The probe height in which the optimized LI sensitivity is achieved then resides in the middle of the range. Figure 7: (**a**) Pixel intensity of particle image \(I(\rho)\) as a function of radial distance \(\rho\) from the center in the range of probe heights between \(z=-1.0~{}\upmu\)m to \(1.0~{}\upmu\)m. \(\rho\) is given in the pixel unit (\(5.1\times 10^{-8}\) m/pix) and the intensities averaged in angular directions are shown. Upper figures show microscope image of probe particles (melamine \(2n=1~{}\upmu\)m) imbedded in agarose gel (0.5 wt%). Colors used for the outline of the microscopic image correspond to the particle height of each curves in the graph.