# Microparticle Manipulation and Imaging through a Self-Calibrated Liquid Crystal on Silicon Display
## 2 Materials and Methods
### LCoS Display Based Inline Holographic System  
In this section we introduce an inline (IL) holographic system to monitor particles in arbitrary spatial positions [23; 24]. When performing holographic imaging of objects, the conjugate image (also referred as ghost image) is always added to the final image when dealing with inline holographic architectures, this situation dramatically deteriorating the image quality. To avoid this, off-axis set-ups for holographic imaging have been proposed in literature [33; 34]. However, IL set-ups present some benefits when compared with off-axis set-ups, for instance, inline holographic systems provide a larger spatial bandwidth than off-axis systems [35]. An interesting proposal was given in Ref. [24; 29], where the sideband technique was proposed for holographic imaging of objects, being able to remove ghost images into an IL system. The idea was to use a filter in the Fourier space to remove the contribution of the conjugate image. However, this filtering also removed object content, and the final image was degraded. We present in this section a double-sideband filter IL holographic system from which the undesired conjugate image is eliminated. More importantly, the full content of the object is retrieved, almost at real time, so it can be used for dynamic processes. To achieve the conjugate image removing, we implement the LCoS display as the digital double shutter. To perform the double-sideband filtering, we firstly consider an almost transparent object (i.e., the microparticle with an extremely small diameter) and therefore, the wavefront passing through this object can be written as Equation (6),  
\[U_{o}(x,y)=1+\Delta U_{o}(x,y), \tag{6}\]  
where \(\Delta U_{o}(x,y)\) refers to the small object amplitude variations. Thus, the Fourier transform of the wavefront amplitude distribution can be calculated as Equation (7),  
\[\overset{\sim}{U}_{o}(\mu,\
u)=\delta(\mu,\
u)+\overset{\sim}{\Delta U}_{o}(\mu,\
u), \tag{7}\]  
where \(\delta\) represents the Dirac delta function, \(\mu\) and \(\
u\) refer to the spatial frequencies. The symbol \(\sim\) refers to the Fourier space. Later, by implementing one section of the double-sideband filter at the Fourier plane, which is feasible to block the \(\mu<0\) frequencies, we obtain the filtered wavefront amplitude at the image space (we refer it as CCD plane) as Equation (8) [29],  
\[U_{CCD}^{+}=\frac{1}{2}+\int\limits_{0}^{\infty}\mathrm{d}\mu\int\limits_{- \infty}^{\infty}\Delta\overset{\sim}{U}_{o}(\mu,\
u)e^{i2\pi(\mu x+\
u y)} \mathrm{d}\  
Fourier transform the obtained intensity (i.e., \(I_{CCD}^{+}\) in Equation (9)), and the \(\mu<0\) frequencies are removed. Under this scenario, the complex amplitude is calculated as Equation (10),  
\[\left|U_{CCD}^{+}\right|^{2}\approx\frac{1}{4}+\underbrace{\frac{1}{2}\int \limits_{0}^{\infty}\!\mathrm{d}\mu\int\limits_{-\infty}^{\infty}\Delta \widetilde{U}_{o}(\mu,\
u)e^{i2\pi(\mu x+\
u y)}\mathrm{d}\
u}_{\mathrm{A}^{+}} \tag{10}\]  
Note that by only addressing one sideband filter (which in fact is one section of the double-sideband filter), the undesired conjugate term (i.e., the B\({}^{+}\) term in Equation (9)) is removed. However, the complex amplitude in Equation (10) cannot guarantee a complete wavefront reconstruction as the \(\mu<0\) frequencies are missing. Therefore, a second sideband filter able to only filter the \(\mu>0\) frequencies is addressed. As will be further explained, both sideband filters are simultaneously addressed thanks to the use of the LCoS display. The intensity obtained at the CCD plane by using this second filter can be calculated in a way equivalent to the above-stated filtering process, but now, by removing \(\mu>0\) frequencies.