**4. Enhancement of the digital hologram efficiency in the presence of timefluctuations of the phase by applying the Minimum Euclidean principle**

In this section, another type of LCoS display, the Parallel Aligned (PA) LCoS display model, is studied for use in the generation of digital holograms. The PALCoS display prototype under analysis is a commercial electrically controlled birefringence LCoS display distributed by HoloEye systems. This prototype is a PLUTO Spatial Light Modulator (SLM) with a diagonal display of 1.8 cm. The pixel pitch is of 8 m with a fill factor of 87% and the display has a resolution of 1920x1080. Besides, this LCoS display model allows us to upload different electrical sequences formats for the digital electrical addressing, which are based on different pulsed-width modulation schemes. The use of different electrical sequences may result in different responses and efficiencies of the PALCoS display (Hermerschmidt et al., 2007; Moore et al., 2008).

A characteristic feature in PALCoS displays is that the LC molecules in such devices are all parallel aligned. Then, by illuminating the display with an incident linear State of Polarization (SoP) parallel to the LC extraordinary axis, the exiting SoP is constant as a function of the time, even in the presence of LC fluctuations. In this situation, the PALCoS display is working in the phase-only regime and the effective depolarization phenomenon can be avoided. However, the damaging effect of the time-fluctuations of the phase phenomenon described in section 3 is still present.

Here, the significance of the time-fluctuations phenomenon in our PALCoS display is experimentally analyzed for the available electrical sequences. Moreover, two different

Study of Liquid Crystal on Silicon Displays for Their Application in Digital Holography 249

In every case, the measurements are done for three different gray levels, which lead to average phase values of 100 degrees, 200 degrees and 300 degrees respectively. In this way,

Figure 10 shows that for all the addressing sequences, periodic time-fluctuations of the phase are obtained. Besides, the amplitude of the phase fluctuations depends on the addressed sequence. In particular, whereas the fluctuations for the sequence #1 (Fig. 10(a)) are very large, they become much smaller for the sequences #2 and #3 (Fig. 10(b) and Fig.

To generate digital holograms with LCoS displays, the required phase distribution has to be implemented. To this aim, the significant parameter is the mean phase. Thus, the mean phase modulation curves for the addressed sequences #1, #2 and #3 are measured as well. The mean phase values have been obtained by calculating the difference between the eigenvalues of the PALCoS display equivalent Jones matrices, which are obtained as

The results are shown in Fig. 11, where the phase modulation for the sequences #1 (in blue) and #2 (in red) are larger than 360 degrees (i.e. 520 degrees and 480 degrees). On the other

Fig. 11. Phase modulation as a function of the gray level for the sequence #1 (in blue), #2 (in

Figure 11 shows how the different electrical sequences available with the PALCoS display provide different phase modulation depths. In general, phase modulations larger than 360

**4.2 Digital hologram efficiency evaluation: Application of the minimum Euclidean** 

hand, the phase depth for the sequence #3 is clearly smaller (about 280 degrees).

we can assume that an estimation of the whole phase domain is conducted.

10(c) respectively).

described in subsection 2.2.

red) and #3 (in green).

**distance principle** 

possible mapping schemes for the ideal phase-only function implementation of digital holograms are considered: a linear phase mismatching scheme and a mapping scheme based on the minimum Euclidean distance principle (Juday, 2001; Moreno et al., 1995). They are experimentally tested in the presence of time-fluctuations of the phase, in order to find the best configuration to maximize the efficiency of digital holograms generated with the PALCoS display (Lizana et al., 2010).

### **4.1 Time-fluctuations of the phase in the PALCoS display**

To operate in the phase-only regime, the PALCoS display under analysis has been sandwiched between two polarizers oriented in the direction of the LC extraordinary axis. By using the diffraction based set-up shown in Fig. 5, we have measured the phase response as a function of the time for three different digital addressing sequences provided by HoloEye (since now labelled as the sequences #1, #2 and #3). For each sequence employed, the phase as a function of the time is measured for different gray levels. In all the cases, the measurements are made by using an unexpanded He-Ne laser beam (633 nm) and at quasinormal incidence (i.e. an incident angle of 2 degrees).

Figure 10 shows the PALCoS phase response as a function of the time for the addressing sequences #1 (Fig. 10(a)), #2 (Fig. 10(b)) and #3 (Fig. 10(c)).

Fig. 10. Phase modulation as a function of time for different gray levels, for the addressing sequences: (a) #1, (b) #2 and (c) #3.

possible mapping schemes for the ideal phase-only function implementation of digital holograms are considered: a linear phase mismatching scheme and a mapping scheme based on the minimum Euclidean distance principle (Juday, 2001; Moreno et al., 1995). They are experimentally tested in the presence of time-fluctuations of the phase, in order to find the best configuration to maximize the efficiency of digital holograms generated with the

To operate in the phase-only regime, the PALCoS display under analysis has been sandwiched between two polarizers oriented in the direction of the LC extraordinary axis. By using the diffraction based set-up shown in Fig. 5, we have measured the phase response as a function of the time for three different digital addressing sequences provided by HoloEye (since now labelled as the sequences #1, #2 and #3). For each sequence employed, the phase as a function of the time is measured for different gray levels. In all the cases, the measurements are made by using an unexpanded He-Ne laser beam (633 nm) and at quasi-

Figure 10 shows the PALCoS phase response as a function of the time for the addressing

Fig. 10. Phase modulation as a function of time for different gray levels, for the addressing

PALCoS display (Lizana et al., 2010).

sequences: (a) #1, (b) #2 and (c) #3.

**4.1 Time-fluctuations of the phase in the PALCoS display** 

normal incidence (i.e. an incident angle of 2 degrees).

sequences #1 (Fig. 10(a)), #2 (Fig. 10(b)) and #3 (Fig. 10(c)).

In every case, the measurements are done for three different gray levels, which lead to average phase values of 100 degrees, 200 degrees and 300 degrees respectively. In this way, we can assume that an estimation of the whole phase domain is conducted.

Figure 10 shows that for all the addressing sequences, periodic time-fluctuations of the phase are obtained. Besides, the amplitude of the phase fluctuations depends on the addressed sequence. In particular, whereas the fluctuations for the sequence #1 (Fig. 10(a)) are very large, they become much smaller for the sequences #2 and #3 (Fig. 10(b) and Fig. 10(c) respectively).

To generate digital holograms with LCoS displays, the required phase distribution has to be implemented. To this aim, the significant parameter is the mean phase. Thus, the mean phase modulation curves for the addressed sequences #1, #2 and #3 are measured as well. The mean phase values have been obtained by calculating the difference between the eigenvalues of the PALCoS display equivalent Jones matrices, which are obtained as described in subsection 2.2.

The results are shown in Fig. 11, where the phase modulation for the sequences #1 (in blue) and #2 (in red) are larger than 360 degrees (i.e. 520 degrees and 480 degrees). On the other hand, the phase depth for the sequence #3 is clearly smaller (about 280 degrees).

Fig. 11. Phase modulation as a function of the gray level for the sequence #1 (in blue), #2 (in red) and #3 (in green).

### **4.2 Digital hologram efficiency evaluation: Application of the minimum Euclidean distance principle**

Figure 11 shows how the different electrical sequences available with the PALCoS display provide different phase modulation depths. In general, phase modulations larger than 360

Study of Liquid Crystal on Silicon Displays for Their Application in Digital Holography 251

implemented phase (denoted as *p*) is represented as a function of the addressed phase (denoted as Note that the diagonal dotted line represents the correct matching between

We assume that the phase values available with the applied LCD are in the range �0, ε�, with ε � ��. First, the red line (model scheme 1) represents a linear phase mismatching. Second, the blue line (model scheme 2) represents a more efficient encoding scheme, which we denote as saturated mismatching encoding. Model 2 represents the perfect phase matching up to the maximum modulation depth φ=ε, while there is a saturation for values φ��. Then, for each phase φ��, the closest available phase value in the modulation domain is taken by following the minimum Euclidean distance principle (see

In this subsection, we compare the efficiency of a basic continuous digital hologram, the blazed grating (Fujita et al., 1982), generated with our PALCoS display when uploading the different electrical sequences (i.e. sequences #1, #2 and #3). For all the sequences, the implementation is conducted by using the linear phase mismatching scheme. In this way, we have limited the phase range between 0 and 360 degrees and we have applied a look-up-table to produce a linear increment for the average phase values. Besides, the phase modulation provided by sequence #3 is lower than 360 degrees (see Fig. 11), and so, the saturated mismatching encoding is applied for this sequence as

In all the cases, the blazed grating is written to the modulator and the corresponding intensity of the zero and of the first diffracted orders is measured as a function of the time by using the diffraction based set-up given in Fig. 5 and by illuminating the PALCoS display with a He-Ne laser (633 nm). The period of the grating is fixed to 16 pixels, being the sufficient number of pixels to neglect the effect of the quantification of the phase

The measurements obtained for the intensity diffracted to the zero (in blue) and to the first (in red) orders are plotted in Fig. 13. In Fig. 13(a) and Fig. 13(b), we have plotted the results obtained when using the sequence #1 and #2 respectively. Next, Fig. 13(c) and Fig. 13(d) show the results obtained when using the sequence #3, the former when applying the linear mismatching and the latter when applying the saturation mismatching

The largest intensity fluctuations are measured when addressing the sequence #1 (Fig. 13(a)) and the best diffraction efficiency is obtained for the sequence #3 with the saturated encoding (Fig. 13(d)). In this way, the sequence #3, even providing a phase modulation lower than 360 degrees (see Fig.11), is the most stable and efficient

The results here provided evidences that to maximize the efficiency of digital holograms generated with LCoS displays, it is important to find a trade-off between the phase modulation depth and the amplitude of the time-fluctuations phenomena in LCoS displays. In this framework, a mathematical model suitable to evaluate the LCoS display response in presence of time-fluctuations on the phase becomes helpful (Lizana et al.,

the designed phase and the displayed phase *p*.

Fig. 12, in blue).

well.

levels.

encoding.

2010).

addressing sequence.

**4.2.2 Experimental results** 

degrees are desired to maximize the efficiency of holograms generated with LCDs. However, as shown in Fig. 10, the electrical sequences applied in LCoS display produce time-fluctuations of the phase, which degrade the efficiency of the generated holograms. Thus, small amplitude of the fluctuations is desired as well.

In this situation, to maximize the efficiency of digital holograms generated with LCoS displays, a trade-off between phase modulation depth (as large as possible) and amplitude of the time-fluctuations of the phase (as small as possible) has to be found.

Next, the suitability of the different electrical sequences to maximize the efficiency of the implemented holograms is experimentally analyzed. Besides, different mapping schemes for phase-only distribution implementation are also reviewed.

### **4.2.1 Linear phase mismatching and saturated mismatching encoding schemes**

To accurately implement digital holograms with LCDs, it is very important to experimentally generate a real phase distribution as close as possible to the designed one. However, as a consequence of diverse non-linearities related to the experimental implementation, the ideal phase distribution is never achieved. To reduce different LCDs degradation sources, some authors have proposed diverse strategies (Davis et al., 1998; Márquez et al., 2001). For instance, if the available dynamic range is less than 360 degrees, the minimum Euclidean distance projection principle (Juday, 2001; Moreno et al., 1995) can be applied. In this way, the modulation diffraction efficiency may be greatly enhanced by selecting an appropriate mapping scheme for the implementation of the phase function onto the restricted phase-only domain (Moreno et al., 1995).

Next, two possible mapping schemes, for the ideal phase-only distribution implementation of holograms, are reviewed. These schemes are represented in Fig. 12, where the truly

Fig. 12. Mapping scheme implementation: In red, the linear mismatching encoding (model 1) and in blue, the saturation mismatching encoding (model 2).

implemented phase (denoted as *p*) is represented as a function of the addressed phase (denoted as Note that the diagonal dotted line represents the correct matching between the designed phase and the displayed phase *p*.

We assume that the phase values available with the applied LCD are in the range �0, ε�, with ε � ��. First, the red line (model scheme 1) represents a linear phase mismatching. Second, the blue line (model scheme 2) represents a more efficient encoding scheme, which we denote as saturated mismatching encoding. Model 2 represents the perfect phase matching up to the maximum modulation depth φ=ε, while there is a saturation for values φ��. Then, for each phase φ��, the closest available phase value in the modulation domain is taken by following the minimum Euclidean distance principle (see Fig. 12, in blue).
