# Holographic Optical Tweezers: Techniques and Biomedical Applications
## 2 Materials and Extended Survey for HOT
### CGH Simulation Methods
Based on Fresnel diffraction, the light distribution on the back focal plane (\(z=f\)) of the objective (real lens) can be expressed as follows:  
\[U(\xi,\eta,z=f)=\frac{1}{j\lambda f}exp\bigg{[}j\frac{\pi}{\lambda f}\bigg{(} 1-\frac{d}{f}\bigg{)}\Big{(}\xi^{2}+\eta^{2}\Big{)}\bigg{]}\iint\limits_{- \infty}^{\infty}u(x,y)exp\bigg{[}-j\frac{2\pi}{\lambda f}(x\xi+y\eta)\bigg{]}dxdy \tag{3}\]  
where \(d\) denotes the distance between the plane (\(x\),\(y\)) and the front focal plane of the real lens. In this HOT system, the (\(x\),\(y\)) plane is on the front focal plane of the objective; hence, the distance, \(d\), is taken to be equal to the objective's focal length, \(f\). Then the light field on the back focal plane of the objective can be expressed as the FT of the resized CGH, i.e., \(U(\xi,\eta,z=f)=\frac{1}{j\lambda f}\Im\{u(x,y)\}\). Furthermore, 2D-GSA is used to calculate the light fields back and forth between the CGH and each regeneration plane, producing a single CGH for multiple regeneration planes. Figure 2 depicts the flowchart of 2D-GSA for M-plane regeneration. On the _n_th recursion, the CGH plane's light field is denoted by \(u_{n}(x,y)\), and the _m_th regeneration plane's light distribution is denoted by \(U_{m,n}(\xi,\eta,z)\). The amplitude of the desired regenerated image is \(T_{m}(\xi,\eta)\), and the distance of the regeneration plane from the focal plane is denoted by \(\Delta_{m}\), where \(m=1\sim M\). To postpone the regeneration plane away from the focal plane, 2D-GSA introduces a virtual lens (focal length \(f_{vm}=-\frac{f^{2}}{\Delta_{m}}\)) into the CGH calculation. The calculation procedure is as follows. Step 1. If \(n=0\), configure the CGH's plane function, \(u_{n}(x,y)=a_{n}e^{j\varphi_{n}(x,y)}\), with random phase, \(\varphi_{0}(x,y)=random(\,0\sim 2\pi)\), and constant amplitude, \(a_{o}\). Step 2. Calculate the light field on the \(m\)th regeneration plane by using FT and then replace the amplitude distribution with the desired image, \(T_{m}\):  
\[U_{mn}(\xi,\eta)=\frac{1}{j\lambda f}\Im\bigg{\{}u_{n}(x,y)e^{j\frac{k}{2f_{ mn}}(x^{2}+y^{2})}\bigg{\}}=A_{mn}(\xi,\eta)e^{j\varphi_{mn}(\xi,\eta)}\to T_{m}( \xi,\eta)e^{j\varphi_{mn}(\xi,\eta)}. \tag{4}\]  
Step 3. Calculate the standard deviation, \(\varepsilon\), between \(A_{m,n}\) and \(T_{m}\). If \(\varepsilon\leq\varepsilon_{0}\), the process is terminated, and \(u_{n}(x,y)\) is output as the final CGH function. Step 4. Take inverse Fourier transform (iFT) to be the regeneration function and pass through a virtual lens, \(-f_{vm}\), back to the CGH field:  
\[\frac{1}{j\lambda f}\Im^{-1}\Big{\{}T_{m}e^{j\varphi_{mn}(\xi,\eta)}\Big{\}}e ^{-j\frac{k}{2f_{vm}}(\xi^{2}+\eta^{2})}=a_{m,n+1}e^{j\varphi_{m,n+1}(x,y)} \tag{5}\]  
Step 5. Sum each phase term corresponding to the same pixel to obtain a single CGH function, \(u_{n+1}(x,y)\), corresponding to the \((n+1)\)th recursion:  
\[\varphi_{n+1}(x,y)=\sum_{m}\sum_{n=1\sim M}\varphi_{m,n+1}(x,y) \tag{6}\]  
\[u_{n+1}(x,y)=a_{0}e^{j\varphi_{n+1}(x,y)} \tag{7}\]  
Furthermore, 2D-GSA with 2D-FT for multiplane regeneration usually converges after a few tens of iterations. However, the uniformity and efficiency of the generated patterns are not very good. Moreover, although the resolution along the \(z\)-axis depends on the plane number, M, overly large values of M increase the calculation time and required resource. [PERSON] et al. [93] generated light fields over a continuous 3D volume, instead of merely on multiple discrete planes, using a 3D-GSA based on iterative 3D-FT between the CGH plane and the regenerated area. Figure 3 depicts the 3D-GSA scheme. During each iteration, the 3D-GSA captures the hemisphere's phase within the CGH area and projects it onto a  
Figure 2: Light field generation away from the real lens’s focal plane by 2D-GSA. single-plane CGH. Based on this, [PERSON] et al. [92] applied the HOT to a continuous space.