# Microparticle Manipulation and Imaging through a Self-Calibrated Liquid Crystal on Silicon Display
## 2 Materials and Methods
### Microparticle Manipulation through Light Structures Created by a LCoS Display  
We also present a method for microparticle spatial manipulation based on an LCoS display [12]. Before particle manipulation, they are firstly optically trapped into a three-dimensional light structure (optical bottle structure). These light structures are created thanks to the split-lens inspired configuration addressed to the LCoS. The physical mechanism that guarantees the particle trapping is here explained. The particle heated by the optical bottle structure experiences an opto-thermal force (photophoretic force [25]) with its force direction opposite to the gravity. Therefore, once the gravity of the particle is compensated by the photophoretic force, the particle is suspended in the air. To obtain the reliable, easy-implemented light trapping architecture, we send to the LCoS display a digital diffractive \(N\)-sector based spilt-lens pattern [30; 32]. This \(N\)-sector split-lens scheme shares the same principle as the Billet lens but this split-lens is adjusted to more pieces rather than the simple  
Figure 2: Shack–Hartmann micro-lens array scheme: (**a**) Plane wavefront modulated by the diffractive micro-lens array and the spot light configuration demonstrates a uniform distribution; (**b**) Distorted wavefront modulated by the diffractive micro-lens array and the spot light configuration demonstrates the spatial deviation. two pieces. The generated \(N\)-lens sectors are all separated the same distance to the center and share the same focal length. Therefore, we obtain diverse focal points drawing a circle at the focal plane. In particular, if the number of sectors is sufficient enough, a continuous split-lens configuration is created, which leads to a light ring at the focal plane. The configuration of such scheme is depicted in Figure 3.  
In Figure 3, the input light modulated by the continuous split-lens phase distribution is propagated to the focal plane and the complete light ring (green ring) is obtained. Note that the continuous split-lens digital scheme is analogous to a classical lens continuously split and displaced from its center, as depicted in Figure 3a. In addition, a detail of the light cone trapping structure created by using this scheme is depicted in Figure 3b. In particular, light propagation after the light ring creates a hollow light cone that can be used as a basic structure for particle trapping. After the created light cone, interference between beams coming from different positions at the light ring occurs, and a Bessel beam structure is obtained (see the upper right corner in Figure 3). To achieve the above-discussed light cone structure, the particular phase distribution to be sent at the LCoS display is given by Equation (4),  
\[U(r,\theta)=\exp[i\frac{\pi}{\lambda f}(r-a(\theta))^{2}] \tag{4}\]  
where \(\lambda\) is the wavelength of the input light and \(f\) is the focal length shared by all the split-lens. Moreover, we define the phase distribution in the polar coordinate system (\(r\), \(\theta\)). Finally, we use \(a(\theta)\) to describe the distance from the split lens to the LCoS coordinate origin. By using the above-described scheme, we can effectively adjust some light cone characteristics (i.e., the cone length or its spatial position) by simply changing the focal length of the split-lens or the separation distance between the split-lens sectors. In particular, the length of the created light cones can be calculated by using Equation (5), according to geometrical optics relations,  
\[L=a(\theta)f/\phi \tag{5}\]  
where \(\phi\) is the diameter of the generated continuous lens without being spilt (the lens aperture). At this moment, we explain how to trap microparticles inside the light cone.