**3.1 Time-fluctuations of the phase: Experimental evidence**

To experimentally measure the time-fluctuations of the phase phenomenon, we have applied the diffraction based set-up sketched in Fig. 5.

We employ an unexpanded beam of a He-Ne laser (633 nm) to illuminate the TNLCoS display, with an angle of incidence equal to 2 degrees. A Polarization State Generator (PSG) is placed in the incident beam, composed of a Linear Polarizer (LP1) and a Quarter Wave Plate (QWP1), and a Half Wave Plate (HWP) is introduced in front of the PSG, allowing us to control the intensity of light incoming to the PSG. In the reflected beam, we have placed a Polarization State Detector (PSD), composed of a Quarter Wave Plate (QWP2) followed by a Linear Polarizer (LP2). We selected the PSG and PSD configuration in Fig. 5, which yields phase-only modulation with constant transmittance (Márquez et al., 2008). Then, we address

around the value 0.65, is achieved. Besides, it is accompanied by a very large phase-shift,

Fig. 4. TNLCoS display phase only response as a function of the gray level.

**3.1 Time-fluctuations of the phase: Experimental evidence** 

applied the diffraction based set-up sketched in Fig. 5.

generated with the TNLCoS display.

**3. Time-fluctuations of the phase: Effects on the digital hologram efficiency**  Experimental evidence of the time-fluctuations of the phase phenomenon, observed when working with LCoS displays (Lizana et al., 2008b; Lizana et al., 2010), are provided in this section. This phenomenon originates from the electrical addressing schemes (Hermerschmidt et al., 2007) applied in LCoS displays, and it may notably degrade the performance of LCoS displays when generating digital holograms (Lizana et al., 2008b). To provide greater insight into this topic, this section also includes a study of the timefluctuations of the phase effects on the efficiency of diverse basic digital holograms

To experimentally measure the time-fluctuations of the phase phenomenon, we have

We employ an unexpanded beam of a He-Ne laser (633 nm) to illuminate the TNLCoS display, with an angle of incidence equal to 2 degrees. A Polarization State Generator (PSG) is placed in the incident beam, composed of a Linear Polarizer (LP1) and a Quarter Wave Plate (QWP1), and a Half Wave Plate (HWP) is introduced in front of the PSG, allowing us to control the intensity of light incoming to the PSG. In the reflected beam, we have placed a Polarization State Detector (PSD), composed of a Quarter Wave Plate (QWP2) followed by a Linear Polarizer (LP2). We selected the PSG and PSD configuration in Fig. 5, which yields phase-only modulation with constant transmittance (Márquez et al., 2008). Then, we address

close to 360 degrees.

a binary grating to the TNLCoS display, and by means of two radiometers (Newport 1830- C) placed in the far diffraction plane, the intensities of the zero and first diffracted orders are captured. The signals detected by the radiometers are synchronized and displayed on an oscilloscope (Tektronix TDS3012B), allowing us to perform intensity measurements of the diffracted orders as a function of the time.

Fig. 5. Diffraction based set-up to measure the phase modulation as a function of the time.

Using the set-up sketched in Fig. 5, the intensity at the zero and at the first diffracted orders was measured as a function of the time, when addressing three different binary gratings. In particular, three binary gratings with a different pair of gray levels are used: (0,120), (0,211) and (0,255). The obtained results are given in Fig. 6. Whereas the intensity at the zero order is plotted in black, the intensity at the first order is plotted in red. The intensity values in Fig. 6 are normalized to the mean value of the intensity measured at the zero order, when a constant image is addressed to the LCoS display with the reference gray level (i.e. zero gray level).

Whereas in Fig. 6 (a) and in Fig. 6 (c), the zero order is in general more intense than the first diffraction order, in Fig. 6(b), the first order intensity is greater than the zero order intensity. This occurs because by changing the addressed gray level pair, the phase difference between the two parts of the binary grating varies. Thus, the quantity of light in each diffracted order in the far diffraction plane varies as well. However, the intensity of the diffraction orders is periodically fluctuating as a function of the time in all cases. This fact points out the existence of time-fluctuations of the phase phenomenon.

A well-known digital hologram is the binary grating with a phase difference of 180 degrees. Theoretically, when addressing such a binary grating, a null zero order has to be obtained. In addition, it has to be accompanied by the 1 diffractive orders, whose intensity values

Study of Liquid Crystal on Silicon Displays for Their Application in Digital Holography 245

From the obtained intensity values in Fig. 6, the phase (t) as a function of the time can be calculated by applying Eq. (14). The results corresponding to the three binary gratings

Fig. 7. Time fluctuations of the phase when addressing a binary grating containing the gray

0 5 10 15 20 25 30 35 40 **Time (ms)**

GL=120 GL=211 GL=255

The results shown in Fig. 7 constitute an experimental evidence for the time-fluctuations of the phase phenomenon, whose fluctuation amplitude depends on the gray levels applied to the binary gratings addressed. Note that the phase fluctuations can reach very high values, as in the case of the grating (0,211), where the mean phase difference is equal to 180 degrees, but it is accompanied with a great fluctuation amplitude of almost 120 degrees (Fig. 7, in red). As proved in this subsection, the phase-shift measured at the TNLCoS reflected beam is far from being constant along the frame period, leading to undesired effects that should

**3.2 Time-fluctuations of the phase: Effects on the digital holograms efficiency** 

In this section, we analyze the effect of time-fluctuations of the phase on the efficiency of two basic digital holograms, generated with the TNLCoS display. Both digital holograms

The TNLCoS display has been configured to work in the phase-only regime by using the configuration related to Fig. 4. Then, by means of the combination of a spatial filter and a convergent lens, the whole TNLCoS display area has been illuminated with a collimated beam. The reflected image is captured with a CCD camera placed in the back focal plane of a second convergent lens set in the reflected beam. In this way, the CCD camera is able to

First, we address a binary phase-only grating with an average phase shift of 180 degrees to the TNLCoS display. The image captured with the CCD camera is shown in Fig. 8, where

Although the intensity related to the 1 diffracted orders is stronger (brighter spots) than the one captured at the zero diffracted order, the later does not completely vanish, as would be

levels: 0-120 (in black), 0-211 (in red) and 0-255 (in blue).


(d)

**Phase modulation (deg)**

be evaluated depending on the application.

have an average phase shift of 180 degrees.

capture the Fourier transform spectrum obtained by diffraction.

expected in a binary grating with a phase difference of 180 degrees.

the typical diffraction orders generated by the binary grating are observed.

applied above (i.e. gray levels (0,120), (0,211) and (0,255)) are given in Fig. 7.

should posses approximately the 40% of the input light. Therefore, the results obtained in Fig. 6(b) correspond in some instants of time to a binary grating with a phase difference equal to 180 degrees. In fact, in some intervals of time, the maximum value of the intensity at the first diffraction order reaches 0.4 and the zero order vanishes.

Fig. 6. Intensity at the zero (in black) and at the first (in red) diffractive orders, when addressing a binary grating containing the gray levels: (a) 0-120, (b) 0-211 and (c) 0-255.

As the TNLCoS display is working in the phase-only regime, the two gray levels applied for the binary gratings addressed to the TNLCoS display present the same values for the amplitude but a phase difference (t) which depends on the time. At this point, an analytical expression for the intensities at the zero and the first diffracted orders can be derived as a function of Zhang et al., 1994. In particular, when the periodic structure of the grating is formed by two levels of the same size (i.e. a duty cycle of 50%), the normalized intensities as a function of the time, in the zero order and in the ±1 orders, can be described as follows:

$$\mathbf{I}\_0(\mathbf{t}) = \frac{1}{2} \left( \mathbf{1} + \cos \Phi \,\mathrm{(t)} \right) \tag{12}$$

$$\mathbf{I}\_{\pm 1}(\mathbf{t}) = \frac{2}{n^2} \left( \mathbf{1} \cdot \cos \Phi \,\mathrm{(t)} \right) \tag{13}$$

From the relations given in Eq. (12) and (13), an expression for the phase modulation as a function of the time can be deduced:

$$\cos\left(\Phi(\mathbf{t})\right) = \frac{4l\_0(t) \cdot \mathbf{u}^2 l\_1(t)}{4l\_0(t) + \mathbf{u}^2 l\_1(t)}\tag{14}$$

should posses approximately the 40% of the input light. Therefore, the results obtained in Fig. 6(b) correspond in some instants of time to a binary grating with a phase difference equal to 180 degrees. In fact, in some intervals of time, the maximum value of the intensity

0.2 0.4 0.6 0.8 1.0

(b)

Zero order First order

**Intensity**

Fig. 6. Intensity at the zero (in black) and at the first (in red) diffractive orders, when addressing a binary grating containing the gray levels: (a) 0-120, (b) 0-211 and (c) 0-255.

as follows:

0.0 0.2 0.4 0.6 0.8 1.0

**Intensity**

I0�t�<sup>=</sup> <sup>1</sup>

I±1�t�<sup>=</sup> <sup>2</sup>

cos�Φ�t�� <sup>=</sup> 4I0�t�-π2I1�t�

function of the time can be deduced:

As the TNLCoS display is working in the phase-only regime, the two gray levels applied for the binary gratings addressed to the TNLCoS display present the same values for the amplitude but a phase difference (t) which depends on the time. At this point, an analytical expression for the intensities at the zero and the first diffracted orders can be derived as a function of Zhang et al., 1994. In particular, when the periodic structure of the grating is formed by two levels of the same size (i.e. a duty cycle of 50%), the normalized intensities as a function of the time, in the zero order and in the ±1 orders, can be described

0 5 10 15 20 25 30 35 40 **Time (ms)**

From the relations given in Eq. (12) and (13), an expression for the phase modulation as a

4I0�t�+π2I1�t�

<sup>2</sup> �1+ cos <sup>Φ</sup> �t�� (12)

0 5 10 15 20 25 30 35 40 **Time (ms)**

Zero order First order

<sup>π</sup><sup>2</sup> �1- cos <sup>Φ</sup> �t�� (13)

(14)

at the first diffraction order reaches 0.4 and the zero order vanishes.

Zero order First order

0 5 10 15 20 25 30 35 40 **Time (ms)**

**Intensity**

(a) 0.0

0.0 0.2 0.4 0.6 0.8 1.0

(c)

From the obtained intensity values in Fig. 6, the phase (t) as a function of the time can be calculated by applying Eq. (14). The results corresponding to the three binary gratings applied above (i.e. gray levels (0,120), (0,211) and (0,255)) are given in Fig. 7.

Fig. 7. Time fluctuations of the phase when addressing a binary grating containing the gray levels: 0-120 (in black), 0-211 (in red) and 0-255 (in blue).

The results shown in Fig. 7 constitute an experimental evidence for the time-fluctuations of the phase phenomenon, whose fluctuation amplitude depends on the gray levels applied to the binary gratings addressed. Note that the phase fluctuations can reach very high values, as in the case of the grating (0,211), where the mean phase difference is equal to 180 degrees, but it is accompanied with a great fluctuation amplitude of almost 120 degrees (Fig. 7, in red). As proved in this subsection, the phase-shift measured at the TNLCoS reflected beam is far from being constant along the frame period, leading to undesired effects that should be evaluated depending on the application.

### **3.2 Time-fluctuations of the phase: Effects on the digital holograms efficiency**

In this section, we analyze the effect of time-fluctuations of the phase on the efficiency of two basic digital holograms, generated with the TNLCoS display. Both digital holograms have an average phase shift of 180 degrees.

The TNLCoS display has been configured to work in the phase-only regime by using the configuration related to Fig. 4. Then, by means of the combination of a spatial filter and a convergent lens, the whole TNLCoS display area has been illuminated with a collimated beam. The reflected image is captured with a CCD camera placed in the back focal plane of a second convergent lens set in the reflected beam. In this way, the CCD camera is able to capture the Fourier transform spectrum obtained by diffraction.

First, we address a binary phase-only grating with an average phase shift of 180 degrees to the TNLCoS display. The image captured with the CCD camera is shown in Fig. 8, where the typical diffraction orders generated by the binary grating are observed.

Although the intensity related to the 1 diffracted orders is stronger (brighter spots) than the one captured at the zero diffracted order, the later does not completely vanish, as would be expected in a binary grating with a phase difference of 180 degrees.

Study of Liquid Crystal on Silicon Displays for Their Application in Digital Holography 247

Fig. 9. Digital hologram designed to reconstruct a butterfly.

al., 2007; Moore et al., 2008).

phenomenon described in section 3 is still present.

**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

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

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

Fig. 8. Diffraction pattern captured with a CCD camera when addressing a binary grating with an averaged phase difference of 180 degrees.

This fact can be understood by comparing the results shown in Fig. 8 with those given in Fig. 6(b). As seen in Fig. 6(b), in those instants of time where the zero diffraction order is exactly equal to zero, the intensity of the first diffracted order is about 0.4. In such instants of time, the results are consistent with a binary phase grating with an instantaneous phase difference equal to 180 degrees. However, the intensity of the diffraction orders detected with the CCD camera (Fig. 8) corresponds to the mean values of the intensity measurements as a function of the time given in Fig. 6 (b). Therefore, as the intensity at the zero and at the 1 diffracted orders varies as a function of the time (because the phase is fluctuating), the mean zero order is higher than 0 and the mean 1 diffracted orders are lower than 0.4 (in particular, 0.059 for the zero order and 0.377 for the 1 orders).

Next, we have displayed a binary phase computer generated hologram designed to reconstruct a butterfly. Again, the averaged phase-shift between gray levels is equal to 180 degrees. The results are shown in Fig. 9. We have added a linear phase along the diagonal direction with the aim of spatially separating the reconstructions of the different orders (i.e. the zero and 1 orders are separated in this way). The butterfly is reconstructed at the 1 orders, with an efficiency equivalent to those obtained in the first diffraction order in Fig. 8. The reconstructed butterflies are accompanied by a zero order diffracted with non-zero intensity that we can see as a bright peak located at the optical axis. As in the binary grating case, the intensity value measured at the zero diffracted order originates from timefluctuations in the phase modulation.

Fig. 8. Diffraction pattern captured with a CCD camera when addressing a binary grating

This fact can be understood by comparing the results shown in Fig. 8 with those given in Fig. 6(b). As seen in Fig. 6(b), in those instants of time where the zero diffraction order is exactly equal to zero, the intensity of the first diffracted order is about 0.4. In such instants of time, the results are consistent with a binary phase grating with an instantaneous phase difference equal to 180 degrees. However, the intensity of the diffraction orders detected with the CCD camera (Fig. 8) corresponds to the mean values of the intensity measurements as a function of the time given in Fig. 6 (b). Therefore, as the intensity at the zero and at the 1 diffracted orders varies as a function of the time (because the phase is fluctuating), the mean zero order is higher than 0 and the mean 1 diffracted orders are lower than 0.4 (in particular, 0.059 for the zero order and 0.377 for

Next, we have displayed a binary phase computer generated hologram designed to reconstruct a butterfly. Again, the averaged phase-shift between gray levels is equal to 180 degrees. The results are shown in Fig. 9. We have added a linear phase along the diagonal direction with the aim of spatially separating the reconstructions of the different orders (i.e. the zero and 1 orders are separated in this way). The butterfly is reconstructed at the 1 orders, with an efficiency equivalent to those obtained in the first diffraction order in Fig. 8. The reconstructed butterflies are accompanied by a zero order diffracted with non-zero intensity that we can see as a bright peak located at the optical axis. As in the binary grating case, the intensity value measured at the zero diffracted order originates from time-

with an averaged phase difference of 180 degrees.

the 1 orders).

fluctuations in the phase modulation.

Fig. 9. Digital hologram designed to reconstruct a butterfly.
