# Microparticle Manipulation and Imaging through a Self-Calibrated Liquid Crystal on Silicon Display
## 2 Materials and Methods
### LCoS Display Self-Calibration
#### 2.1.1 Phase-Voltage Calibration  
We use an interferometric method, conducted by self-addressing a reconfigurable two-sectorial split-lens scheme on the LCoS display to be calibrated, to determine its phase-voltage response [15].  
The two-sectorial split-lens configuration applied here represents the classic Billet lens scheme [30], which leads to an interference pattern equivalent to that of the [PERSON]'s double slit experiment. The optical scheme of this method is sketched in Figure 1. The Billet lens scheme consists of a lens cut from its center into two sections and then being separated with a certain distance (Figure 1b). The phase distribution of the digitally generated split-lens scheme, to be addressed to the LCoS display, is depicted in Figure 1a. Under this scenario, the coherent light illuminating the two-sectorial split-lens configuration at the LCoS plane generates two corresponding focal points (F1 and F2) at the focal plane (S plane in Figure 1). These two focal points act as new coherent and punctual light sources. Later, these two divergent beams further propagate and finally encounter at plane B, where a fringes-like interferometric pattern is produced (Figure 1c).  
Figure 1: Two-sectorial split-lens based optical scheme: (**a**) Phase distribution of the split-lens scheme on the LCoS display; (**b**) Billet lens scheme; (**c**) Fringe-like interferometric pattern.  
Once the interferometry pattern is obtained, the phase-voltage relation can be retrieved. To better explain this method, let us review the mathematical expression describing the two split-lens phase distribution shown in Figure 1a. This is given in Equation (1).  
\[U_{N=2}(r,\theta)=U_{1}+U_{2}, \tag{1}\]  
where \(U_{1}\) and \(U_{2}\) are the phase distribution for each one of the two lens sectors and they are written as [15],  
\[U_{1}=\exp[i\frac{\pi}{\lambda f}(r^{2}+{a_{0}}^{2}-2 ra_{0}\cos(\theta_{i}- \theta_{0})+\phi(V))], \tag{2}\]  
\[U_{2}=\exp[i\frac{\pi}{\lambda f}(r^{2}+{a_{1}}^{2}-2 ra_{1}\cos(\theta_{ii}- \theta_{1}))], \tag{3}\]  
where \(f\) refers to the focal length of the split lens, \(r\) to the radial coordinate in the lens plane and \(\lambda\) represents the wavelength for the light source. In addition, \(a_{0}\) is the distance from the first split-lens sector center to the LCoS geometric center, while \(a_{1}\) is the distance from the second split-lens sector center to the same LCoS center. What is more, \(\theta_{0}\) and \(\theta_{1}\) refers to the angular positions of the two split sector centres. Thus, the upper sector angular distribution \(\theta_{i}\) is restricted in the range as \(\pi\) to \(2\pi\), whereas the lower sector angular distribution \(\theta_{ii}\) is restricted in the range as 0 to \(\pi\). Note that these two sectors fully cover the whole angular distribution as 0 to \(2\pi\). Finally, note that the phase distribution for the first split sector (Equation (2)) introduces a constant phase term which depends on the voltage, \(\phi(V)\).  
By setting a term \(\phi(V)\) different to zero in Equation (2) (each possible \(\phi(V)\) is related to a different gray level), a constant phase is added to one of the split sectors (\(U_{1}\) in our case). This produces an extra retardance between wavefronts originated by \(U_{1}\) and \(U_{2}\), which is transferred into a transversal shift of the fringe pattern in a given axial plane (e.g., \(I\) plane). Therefore, by measuring the transversal shifts corresponding to the gray levels, the phase-gray levels relation can be determined, for instance, by using correlation-based calculations [22].