# Holographic Optical Tweezers: Techniques and Biomedical Applications
## 2 Materials and Extended Survey for HOT
### CGH Simulation Methods  
Under the currently used display resolutions, LCD-SLM can encode only a part of the entire CGH information. In HOT applications, a phase-only CGH, which discards the amplitude and retains only the phase information, is typically utilized to regenerate the light field in a 3D field or multiple planes. Therefore, an iterative or non-iterative algorithm [43; 82; 83; 84; 85; 86; 87; 88] is required to optimize and synthesize CGH. Several papers [87; 89; 90] reviewed the computational algorithms used in HOT system and compared their performance. In this context [87; 90], the iterative algorithm exhibits better diffraction efficiency and image uniformity to generate a higher-quality optical trap, whereas the direct-search algorithm exhibits a higher computing speed. To reconstruct the desired amplitude, scalar theory was applied to calculate the phase hologram, \(\phi_{\text{r}}(x_{\text{r}},y_{\text{r}})\), for M traps. The complex amplitude from the \(k\)th pixel, location \((x_{k},y_{k})\) of SLM under  
Figure 1: A typical HOT system. uniform illumination is \(u_{k}=|u|e^{i\phi_{k}}\). After passing through the lens (focal length _f_), the complex amplitude, \(U_{m}\), of the mth trap point \((x_{m},y_{m},z_{m})\) can be written as follows:  
\[U_{m}=\frac{e^{j2\pi(2f+z_{m})/\lambda}}{j}\frac{d}{\lambda f}\sum_{k=1\sim K}| u|e^{i(\phi_{k}-\lambda_{k}^{m})} \tag{1}\]  
where \(d\) is the pixel size and  
\[\Delta_{k}^{m}=\frac{\pi z_{m}}{\lambda f^{2}}\Big{(}x_{k}^{2}+y_{k}^{2}\Big{)} +\frac{2\pi}{\lambda f}(x_{k}x_{m}+y_{k}y_{m}) \tag{2}\]  
Random mask encoding (RM) [88] divides the SLM pixels into subdomains by randomly encoding the information from each CGH. The phase-only hologram of RM can be written as \(\phi_{k,RM}=\Delta_{k}^{m_{k}}\), where the integer \(m_{k}\) is in the range of 1-M for each pixel. The method of superposition of prisms and lens (S) [30; 31] sums up all single trap hologram, \(\phi_{k,S}=arg\Big{[}\sum_{m}e^{j\Delta_{k}^{m}}\Big{]}\), to composite CGH. RM and S perform with very poor efficiency but very fast computation. By further adding a random phase \(\theta_{m}\), \(\phi_{k,SR}=arg\Big{[}\sum_{m}e^{j(\Delta_{k}^{m}+\theta_{m})}\Big{]}\), a random superposition algorithm (SR) [91] has been shown to have the best performance in direct-search algorithms. The iterative algorithm, mainly based on the Gerchberg-Saxton (GS) algorithm, includes a generalized adaptive-additive algorithm (_GAA_) [32], weighted GS (_GSW_) [89], 2D-GSA [34], and 3D-GSA [92]. At the beginning of GS, \(\phi_{k}\) was set as \(\phi_{k,SR}\), and then we computed \(V_{m}=\frac{1}{K}\sum_{k=1\sim K}e^{i(\phi_{k}-\Delta_{k}^{m})}\), and \(\phi_{k}=arg\Big{[}\sum_{m}e^{j\Delta_{k}^{m}}\frac{V_{m}}{|V_{m}|}\Big{]}\) in an iterative procedure until it converged. The diffraction efficiency of GS is high, but the uniformity is not satisfactory. Therefore, _GAA_ adds the bias, \(\xi\), and _GSW_ introduces the weighted factor, \(w_{m}\), to improve the uniformity. The phase information for _GAA_ and _GSW_ during the interaction is modified as \(\phi_{k,GAA}=arg\Big{[}\sum_{m}e^{i\Delta_{k}^{m}}\frac{V_{m}}{|V_{m}|}\Big{(} 1-\xi+\frac{\xi}{|V_{m}|}\Big{)}\Big{]}\), and \(\phi_{k,GSW}=arg\Big{[}\sum_{m}e^{j\Delta_{k}^{m}}\frac{w_{m}V_{m}}{|V_{m}|} \Big{]}\). Both the _GAA_ and _GSW_ perform with high diffraction efficiency and uniformity, but _GSW_ shows remarkable uniformity. HOT often uses FT-based GSA to calculate CGH. Moreover, when 2D-GSA and 3D-GSA use the fast FT, they can generate the CGH to shape the desired amplitude on the multiple planes or in the 3D area. As depicted in Figures 1 and 2, the light with CGH information, \(u_{\text{C}}(p,q)\), is resized by the lenses, \(\text{L}_{1}\) and \(\text{L}_{2}\), and projected onto the front focal plane \((x,y)\) of the objective, i.e., \(u(x,y)=u_{\text{C}}\big{(}\frac{p}{M},\frac{q}{M}\big{)}\). Then the light distribution, \(u(x,y)\), is Fourier transformed (FT) to the trapping area by the objective.