# Optimization of Optical Trapping and Laser Interferometry in Biological Cells
## 2 Laser Interferometry  
Laser interferometry (LI) is a technique to detect the position of a trapped particle with a high spatio-temporal resolution [6]. As shown in Figure 1b, the trapped particle with the displacement \(x\) from the optical axis diffracts the probe laser. After being collimated, the outgoing laser is projected on to the quadrant photodiode that images the back-focal plane of the condenser lens. When the probe particle is smaller than the beam waist (\(w_{0}\)) or the wavelength (\(\lambda\)) of the trapping laser, the scattered and unscattered light creates an interference pattern at the QPD. The technique is therefore conventionally referred to as back-focal-plane laser interferometry (LI) [6]. LI measures the QPD output voltage \(V(t)=M(I_{+}-I_{-})/(I_{+}+I_{-})\) which is proportional to the normalized difference of the laser intensities (\(I_{+}\) and \(I_{-}\)) that arrive at (\(+\)) and (\(-\)) halves of the QPD, respectively. Here, \(M\) indicates the gain of electronic amplifiers. Within the linear range where the QPD output \(V(t)\) is proportional to the probe displacement \(x\) from the optical axis, and LI measures \(x\) with high spatio-temporal resolution. The position sensitivity of LI, which is defined as \(S\equiv V/x=M(I_{+}-I_{-})/\{(I_{+}+I_{-})x\}\), indicates the basic performance of LI once \(M\) is properly normalized. When a Gaussian laser is used for LI in homogeneous media, the sensitivity depends on parameters, e.g., refractive indices of solvent \(n\) and probe particle \(n_{\text{p}}\), radius of the probe \(a\), beam waist \(w_{0}\), and wavelength of the laser \(\lambda\). Scattering of a laser by spherical objects can be predicted by solving the classical electromagnetic theory, which is referred to as Lorenz-Mie theory [19]. The theory supposedly provides for the accurate prediction of the laser diffraction as long as permitted by the power of numerical calculations. However, it is hard to incorporate the actual condition of LI experiments. For instance, various aberrations arise while the laser propagates in the experimental setup and in the sample. Especially, the spherical aberration usually neglected in the calculation appears when the refractive index of the sample media is different from that of immersion oils used for the objective and condenser lenses [20]. When a laser passes through these high-NA lenses, its polarization is compromised. The asphericity of colloids, stray lights from various reflection surfaces, and other misalignments in the optical setup also influence the LI sensitivity, which are hard to take into account in the theoretical predictions. When designing LI experiments, it is worth considering more-drastically simplified approximations in order to obtain a qualitative understanding of how the LI sensitivity depends on various parameters (\(a\), \(n\), \(n_{\text{p}}\), etc.). For a probe particle whose radius is larger than the wavelength of a laser (\(a\gg\lambda\)), the approximation with the geometrical optics is relevant to investigate the propagation of light. That was done in detail by taking multiple scattering at the probe surfaces into account [21]. Here, with more drastic approximation neglecting surface reflections, the position sensitivity of the laser interferometry is approximately given as  
\[\frac{I_{+}-I_{-}}{I_{+}+I_{-}}\cdot\frac{1}{x}\propto\frac{n(n_{\text{p}}-n)}{n _{\text{p}}}\frac{1}{a}.\qquad(\text{Geometrical}) \tag{1}\]  
On the other hand, at the opposite limit (\(a\ll\lambda\)), the effective dipole moment induced at the probe emits the scattered electric field (Rayleigh scattering).