# Optimization of Optical Trapping and Laser Interferometry in Biological Cells
## 3 Materials and Methods
### LI Sensitivity  
Position sensitivity of LI in a aqueous glycerol was obtained by analyzing the power spectral density of the thermal fluctuation of probe particle [23; 24]. Sensitivities in cells and agarose gels were obtained by sinusoidally oscillating the piezo stage (0.1 Hz, amplitude \(\sim\) 250 nm) while QPD output signals and microscope images were recorded. The sensitivities were then obtained as the ratio of the amplitude of these signals (Figure 3a,b). For measurements in living cells, in order to remove errors caused by the vigorous spontaneous fluctuations, periodical oscillations of the probe particle were  
Figure 2: Experimental setup for OT (optical trapping) and LI (laser interferometry) that is implemented with 3-D feedback-control of piezo stage. Incident beam is tightly focused near the probe particle in a cell by the objective lens. Scattered and unscattered lights are collimated by the condenser lens; the QPD at the BFP of the condenser lens detects the laser refraction that is proportional to the probe offsets in \((x,y)\). QPD signal is applied to the measure input of the PID controller for the feedback control of the piezo stage for horizontal directions \((x,y)\). Microscopic image is captured and analyzed in the personal computer to obtain \(z\) position of the probe. The feedback control signal for \(z\) direction is then created and applied to piezo.  
made under the feedback control of the piezo stage. As shown in Figure 3b, setpoint (target value) for the feedback control was oscillated sinusoidally at the periodicity of 10 s so that QPD output \(V\) (\(\propto\)\(x\)) becomes sinusoidal [15]. Since the feedback delay (\(\tau\) - 10 ms) of the system [15] is much faster than the oscillation period, the feedback control works as intended.  
LI sensitivity and OT strength change owing to the slight alteration of optical settings of the equipment. For instance, separation between the probe particle and the condenser lens (2.80 mm) changes while exchanging samples repeatedly, affecting the optical magnification for the diffraction of the laser. Even the laser emission is slightly altered once the laser is turned off and restarted. These factors affect the sensitivities via \(G\) (\(a\), \(w_{0}\), \(\lambda\)) in Equation (3). As a control, immediately before or after every experiments, sensitivity \(S_{40}\) was measured in a reference media (40 wt% glycerol solution) without changing any other conditions of experimental settings. Further, 40% glycerol solution was chosen as a reference material since its refractive index \(n_{40}\) = 1.384 is close to that of the cell interior [9; 25; 26; 27; 28; 29; 30; 31]. The normalized LI sensitivity \(A\) was then obtained by dividing 5 with \(S_{40}\) as \(A\equiv S/S_{40}=f(n,n_{\mathrm{p}})/f(n_{40},n_{\mathrm{p}})\). Therefore, the theoretical predictions of the normalized sensitivities are given for geometrical and Rayleigh approximations as,  
\[A\equiv\left\{\begin{array}{ll}A_{\mathrm{G}} =n(n_{\mathrm{p}}-n)/n_{40}(n_{\mathrm{p}}-n_{40})&(\mathrm{Geometrical})\\ A_{\mathrm{R}} =n(n_{\mathrm{p}}^{2}-n^{2})(n_{\mathrm{p}}^{2}+2n_{40}^{2})/n_{40}(n_{ \mathrm{p}}^{2}-n_{40}^{2})(n_{\mathrm{p}}^{2}+2n^{2})&(\mathrm{Rayleigh})\\ \end{array}\right. \tag{6}\]