# Microparticle Manipulation and Imaging through a Self-Calibrated Liquid Crystal on Silicon Display
## 2 Materials and Methods
### LCoS Display Self-Calibration
#### 2.1.2 Surface Profile Calibration  
To conduct the LCoS display surface calibration, a digital micro-lens array pattern (Shack-Hartmann scheme [19; 20]) is self-addressed to the device. The micro-lens array consists of an ordered lattice of micro diffractive convergent lenses integrated at the LCoS plane. Moreover, all these lenses share the same focal length and the distance between any two adjacent lenses is equivalent. In this scenario, by sending the proper S-H phase distribution to the LCoS display, the diffractive micro-lens array is created. Let us assume now that we illuminate the LCoS with a collimated beam perpendicular to the display. In the case of a perfectly flat LCoS display surface (i.e., the screen does not introduce any extra phase distribution), we would obtain a uniformly distributed spot light configuration at the focal plane of the micro-lenses. In other words, each fraction of the input plane wave is focused at the optical axis of its corresponding micro-lens (see Figure 2a). On the contrary, if the LCoS screen presents spatial defects, the plane input wave is spatially distorted, and an extra phase distribution is added to the system, in a situation equivalent to Figure 2b. In such a scenario, corresponding spot lights at the focal plane are not evenly distributed but presents some deviation from the expected centers. Finally, by calculating the deviations of light dots with respect to the expected centers (see Figure 2b), the surface profile can be retrieved by using proper numerical methods.  
We determine the light spot deviations by using centroid calculations thoroughly described in Ref. [20]. The main idea is to use a reference grid at the charge-coupled device (CCD) plane that indicates the theoretical centers. In particular, this reference grid corresponds to a system free of aberrations (light distribution at the focal plane related to a plane wave). Later, this grid is regarded as the criteria and the actual deviations of light dots observed when illuminating the LCoS, are estimated with respect to this reference grid. By determining such deviation, well-known integration methods [31] are used to retrieve the surface of the LCoS.  
Two problems may arise by using this approach. First, we may introduce a reference grid pattern which is shifted from the theoretical grid. This situation leads to a constant shift into the centroid calculation. Nevertheless, this issue is easy to solve as in fact such grid shift can be regarded as a constant value added to the deviation function, and therefore, it only introduces an artificial tilt after using integration methods. Therefore, this situation does not modify the calculated surface profile. Conversely, we may face a second problem. If the size of the grid squares we choose is different from the theoretical one, linear phases are introduced into the centroid calculation and thus, a quadratic function is added to the surface profile after reconstruction during the integration. Therefore, the size of the grid has to be correctly determined as an input parameter to properly retrieve the LCoS surface profile. To solve this situation, we used the grid size calibration method discussed in Ref. [15]. Once the theoretical grid is properly set, experimental light spot deviations are measured with respect to theoretical centers. At this point, we can easily calculate the derivative function (slopes of the actual surface) from shifted light spots and then retrieve the surface profile by simply using the two-dimensional integration. At last, we use cubic splines [31] to interpolate discrete data and obtain a continuous surface profile.