**1. Introduction**

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Due to the significant capability of Liquid Crystal Displays (LCDs) to spatially manipulate the phase information of an incident light beam, this technology is been widely applied in a large number of optical applications. Nowadays, they are employed as Spatial Light Modulators (SLMs) in many areas, as for instance, in Optical Image Processing (Liu et al., 1985), in Holography Data Storage (Coufal et al., 2000), in Programmable Adaptive Optics (Dou & Giles, 1995), in Medical Optics (Twietmeyer et al., 2008), or in Diffractive Optics (Márquez et al., 2005), among others.

Recently, a new type of reflective LCDs, the Liquid Crystals on Silicon (LCoS) displays, have awakened a great interest due to their specific technical characteristics, which in general, are superior in many aspects to the ones provided by transmissive LCDs (Lee et al., 2004; Wu & Yang, 2005). For instance, as LCoS displays work in reflection, the light impinging such devices performs a double pass through the LC cell, leading to a larger phase modulation than that related to transmissive LCDs with the same thickness. This greater phase modulation capability allows LCoS displays to become very suitable devices for digital holography applications, as for instance, for laser beam shaping (Dickey et al., 2005; Rodrigo et al., 2011) or for optical micro-particle manipulation (Ashkin, 2006).

To maximize the efficiency of digital holograms generated by using LCoS displays, it is required to apply a suitable methodology for optimizing the performance of these devices when generating the specific phase and amplitude distributions. Nowadays, there exist different theoretical models proposed to improve the performance of LCDs (Azzam & Bashara, 1972; Gagnon, 1981; Márquez et. al., 2001). In general, most of these models are based on mathematical formalisms describing fully polarized light, as the Jones formalism (Jones, 1941) or the Berreman formalism (Berreman, 1972). However, some authors have discovered that LCoS displays may introduce non-negligible values of effective depolarization at the reflected beam (Lizana et al., 2008a; Márquez et al., 2008; Wolfe & Chipman, 2006), which are originated by the electrical addressing schemes applied in these devices (Hermerschmidt et al., 2007). This effective depolarization depends on the incident

Study of Liquid Crystal on Silicon Displays for Their Application in Digital Holography 235

Twisted Nematic (TN) LCoS display of 2.46 cm diagonal, in which the twist angle of LC is 45 degrees. The resolution of the display is XGA (1024x768 pixels), with a fill factor of 93%. The pixels are square and they are separated by a distance (center to center) of 19 m. In addition, the device provides digitally controlled gray scales with 256 gray levels (8 bits). To determine the Mueller matrix of the TNLCoS display, the set-up sketched in Fig. 1 is

The TNLCoS display is illuminated by a linear polarized He-Ne laser source with a wavelength equal to 633 nm. In the incident beam, just following the laser source, is set a Half-Waveplate (HWP) that allows us to control the intensity of light transmitted by the linear polarizer LP1. After the HWP element, a Polarization State Generator (PSG), which is formed by a linear polarized (LP1) fixed at 0 degrees of the laboratory vertical and a Quarter-Waveplate (QWP1), is placed. The QWP1 is inserted into a rotating platform that allows us to electronically control the orientation of its fast axis, generating at each different

The angle between the incident and the reflected beams is equal to 4 degrees and so, it can be assumed that the TNLCoS display is operating under quasi-normal incidence. In the light beam reflected by the TNLCoS display is set a Polarization State Detector (PSD), which is formed by Quarter-Waveplate (QWP2) followed by a linear analyzer (LP2). Different

Finally, the radiometric measurements are performed by means of a commercial radiometer (Newport 1830-C) placed at the exit of the PSD. The combination of the PSD and the radiometer results in a complete Stokes polarimeter configuration (Chipman, 1995), which is

The Mueller matrix M of a polarizing sample relates the incident and exiting (reflected, transmitted or scattered) states of polarization, described by the Stokes vectors *S*in and *S*ex

capable of measuring any state of polarization exiting from the TNLCoS display.

used.

respectively:

Fig. 1. Mueller matrix characterization set-up.

orientation a different incident State of Polarization (SoP).

orientations of the QWP2 allow us to analyze different SoPs.

angle (Lizana et al., 2009; Verma et al., 2010), the wavelength (Lizana et al., 2008c) and the state of the polarization of the incident beam (Márquez et al., 2008). In this situation, the application of an optimizing method that includes the unpolarized light contribution observed when working with LCoS display becomes mandatory.

In Ref. (Lizana et al., 2008b), another undesired phenomenon originated by the electrical addressing schemes applied in LCoS displays is reported. We refer to the time-fluctuations of the phase phenomenon, which may notably degrade the efficiency of the digital holograms generated with LCoS displays (Lizana et al., 2008b). Thus, to maximize the efficiency of digital holograms addressed to these devices, different ways to reduce the undesired influence of this damaging phenomenon must be applied.

This Chapter presents a study based on LCoS displays which can be useful as a guideline to optimize the performance of these devices for the generation of digital holograms. In section 2, a characterization and optimization methodology, based on Mueller-Stokes (M-S) formalism is described. This methodology considers the effective depolarization values observed in LCoS displays because the M-S formalism is able to describe fully polarized light, partial polarized light and unpolarized light contributions. In section 3, experimental evidences of the time-fluctuations of the phase phenomenon are presented and its effects on the generation of digital holograms are reviewed. In section 4, a method based on the minimum Euclidean distance principle, devised to reduce the undesired influence of this phenomenon is proposed and experimentally tested by analyzing the efficiency of different optimized digital holograms addressed to an LCoS display. Finally, conclusions are given in section 5.
