# Microparticle Manipulation and Imaging through a Self-Calibrated Liquid Crystal on Silicon Display
## 2 Materials and Methods
### LCoS Display Based Inline Holographic System
It leads to the following expression Equation (11) [29],  
\[\left|U_{CCD}^{-}\right|^{2}=\frac{1}{4}+\frac{1}{2}\int\limits_{-\infty}^{0 }\!\mathrm{d}\mu\int\limits_{-\infty}^{\infty}\Delta\widetilde{U}_{o}(\mu, \
u)e^{i2\pi(\mu x+\
u y)}\mathrm{d}\
u. \tag{11}\]  
The finally object image, containing the whole spatial frequency distribution of the quasi-transparent object, can be finally written as Equation (12) by combining Equations (10) and (11),  
\[I=\left|U_{CCD}^{-}\right|_{A^{-}}^{2}+\left|U_{CCD}^{+}\right|_{A^{+}}^{2}= \frac{1}{2}+\underbrace{\frac{1}{2}\int\limits_{-\infty}^{0}\!\mathrm{d}\mu \int\limits_{-\infty}^{\infty}\Delta\widetilde{U}_{o}(\mu,\
u)e^{i2\pi(\mu x+ \
u y)}\mathrm{d}\
u}_{\mathrm{A}^{-}}\frac{+\frac{1}{2}\int\limits_{0}^{ \infty}\!\mathrm{d}\mu\int\limits_{-\infty}^{\infty}\Delta\widetilde{U}_{o}( \mu,\
u)e^{i2\pi(\mu x+\
u y)}\mathrm{d}\
u}_{\mathrm{A}^{+}}=\frac{1}{2}U_{0 }(x,y). \tag{12}\]  
In Equation (12), we can clearly distinguish that the conjugate image is eliminated and therefore the object image is obtained without the distortions associated to ghost images. Once the full complex amplitude is assured (i.e., Equation (12)), the holographic wavefront image can be reconstructed at any axial plane by using some diffraction method. We used the Rayleigh-Sommerfeld diffraction method, and the reconstructed wavefront is given by Equation (13) [36; 37],  
\[U(x,y,d)=\iint\widetilde{I}(\mu,\
u)\times\exp\{\frac{i2\pi}{\lambda}d[1- \lambda^{2}(\mu^{2}+\
u^{2})]^{1/2}\}\times\exp[i2\pi(x\mu+\
u\
u)]\mathrm{d} \mu\mathrm{d}\
u, \tag{13}\]  
where \(\tilde{I}(\mu,\
u)\) is the Fourier transfer of Equation (12) and \(d\) is the axial distance from the image input plane to the particular position where we want to reconstruct the wavefront. So far, we have theoretically discussed the mathematical background to implement the IL holographic wavefront imaging through a double-sideband filter. Here, an optical scheme to illustrate how the real implementation of such system can be conducted by using an LCoS display (as the digital double-sideband filter) is presented. The optical sketch is shown in Figure 5, where a polarized collimating light is used as the illumination with its polarization direction rotated \(45^{\circ}\) to Y-axis of the world coordinate system. Afterwards, a convergent lens is inserted after the tested object and therefore, we can determine the Fourier plane at its focal plane \(F\). By now, the LCoS display is precisely implemented at the Fourier plane so that it can provide the frequency filtering. Even though that the spatial light modulator presented in Figure 5 is described as a transmissive element for the sake of clarity, in fact, it is the LCoS display working within the reflective scheme. The actual reflective configuration is conducted by combining the LCoS display with a beam splitter (it is also shown in Figure 5; see red dashed inset image). Into this LCoS display, we set two sectorial phase retardations: half of the display is driven to a phase of \(0^{\circ}\) and the other half to \(180^{\circ}\) (see Figure 5). Under this scenario, the polarization direction of the input light passing through the LCoS is rotated for \(90^{\circ}\)in the upper section (black section, \(\delta\) as 180\({}^{\circ}\)), whereas on the contrary, the polarization direction corresponding to the other section (white section, \(\delta\) as 0\({}^{\circ}\)) is maintained. Later, two linear analyzers (LP1 and LP2) are set before the two CCDs cameras. Their transmissive directions are set perpendicular to each other: LP2 is oriented 45\({}^{\circ}\) to the Y-axis (the same polarization direction as the input light) and LP1 is oriented to 135\({}^{\circ}\) to the Y-axis. Under this scenario, the combination of the LCoS polarization modulation and the analyzers LP1 and LP2 orientations acts as the double sideband filter. Light recorded by CCD1 was previously filtered as half of the object frequencies are blocked. This can be assimilated to Equation (10).