**3. Using a PA-LCoS in a holographic data storage system**

We have a good characterization method and control over the display that will act as a spatial light modulator in our holographic data storage system. We present how we can implement different modulation schemes with a phase-only device as our PA-LCoS.

HDSS enables true three-dimensional (3-D) storage of information and also associative memory retrieval [16]. There are many aspects that need to be addressed for HDSS to be commercially viable [2]. These systems are always taking advantage from the latest technological advances in the various components that form a complete system. In this sense, PA-LCoS microdisplays have replaced previous liquid-crystal display (LCD) technology in most photonics applications [17]; this fact makes PA-LCoS an interesting device to test in HDSS. PA-LCoS are high-resolution reflective devices which enable phase-only operation. They are ideal for binary or multinary phase-only data pages [18, 19]. This leads to DC term cancellation when recording the Fourier transform of the data page, avoiding the premature saturation of the recording material.

#### **3.1. Modulation schemes**

We try to implement the well-known binary intensity modulation (BIM) and the hybrid ternary modulation (HTM) [20, 21]. PA-LCoS devices are designed for displaying phase-only elements without coupled amplitude. This is easily achieved by illuminating them with a linearly polarized light parallel to the director axis. We have investigated that the PA-LCoS can also be used to display the widely applied BIM data pages. We also try to implement the more demanding HTM data pages. HTM had not been studied with PA-LCoS devices. HTM has the advantage that combines the ease of detection of BIM data pages (that only needs to detect intensity) and it reduces the DC term of the Fourier transform. The reduction on the DC term is necessary to avoid saturation of the dynamic range of the recording material [20, 21].

We use the model and characterization technique presented in Section 2.3 that enable us to obtain the average retardance and fluctuation amplitude for every gray level. Before applying the configuration needed, we need to calculate the complex amplitude of the electric field at the exit. To do that, we use the Jones matrix formalism, so we present the basic elements involved,

$$P\_{\chi} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \tag{18}$$

Eq. (18) is the matrix associated to a linear polarizer with its transmission axis along the *X*-axis,

$$\mathcal{W}(\phi) = \begin{pmatrix} \exp\{-j\phi/2\} & 0\\ 0 & \exp\{+j\phi/2\} \end{pmatrix} \tag{19}$$

Eq. (19) is the matrix for a linear retarder of linear retardance *ϕ* with its slow axis along the *X*-axis. When polarization elements are rotated an angle *θ* with respect to the *X*-axis, we need the two-dimensional rotation matrix,

$$R(\theta) = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \tag{20}$$

To produce both BIM and HTM with the PA-LCoS device, we need to insert the display between two rotated linear polarizers at angles *θ*<sup>1</sup> and *θ*<sup>2</sup> , that is, the complex amplitude for the electric field at the output is given by the following equation:

$$\overrightarrow{E}\_{\text{OUT}} = P\_{\text{X}} \cdot \mathcal{R}(\theta\_2) \cdot \mathcal{W}\_{\text{pA}}(\overline{\Gamma}) \cdot \begin{pmatrix} \cos \theta\_1\\ \sin \theta\_1 \end{pmatrix} \tag{21}$$

where ⇀*<sup>E</sup>* OUT corresponds to linearly polarized light at an angle *<sup>θ</sup>*<sup>2</sup> , always with respect to the *X*-axis. *W*PA (¯ <sup>Γ</sup>) is the matrix for the PA-LCoS which is given by Eq. (19), and whose average retardance varies with gray level, and it will be given by our characterization method. Its fluctuation amplitude will not be considered in the calculations: in the calibration we have taken care to select electrical configurations minimizing the existence of this flicker in the retardance [9, 10].

HDSS. PA-LCoS are high-resolution reflective devices which enable phase-only operation. They are ideal for binary or multinary phase-only data pages [18, 19]. This leads to DC term cancellation when recording the Fourier transform of the data page, avoiding the premature

We try to implement the well-known binary intensity modulation (BIM) and the hybrid ternary modulation (HTM) [20, 21]. PA-LCoS devices are designed for displaying phase-only elements without coupled amplitude. This is easily achieved by illuminating them with a linearly polarized light parallel to the director axis. We have investigated that the PA-LCoS can also be used to display the widely applied BIM data pages. We also try to implement the more demanding HTM data pages. HTM had not been studied with PA-LCoS devices. HTM has the advantage that combines the ease of detection of BIM data pages (that only needs to detect intensity) and it reduces the DC term of the Fourier transform. The reduction on the DC term is necessary to avoid saturation of the dynamic range of the recording material [20, 21]. We use the model and characterization technique presented in Section 2.3 that enable us to obtain the average retardance and fluctuation amplitude for every gray level. Before applying the configuration needed, we need to calculate the complex amplitude of the electric field at the exit. To do that, we use the Jones matrix formalism, so we present the basic elements

1 0

Eq. (18) is the matrix associated to a linear polarizer with its transmission axis along the

exp(−*jφ*/2) 0

Eq. (19) is the matrix for a linear retarder of linear retardance *ϕ* with its slow axis along the *X*-axis. When polarization elements are rotated an angle *θ* with respect to the *X*-axis, we need

To produce both BIM and HTM with the PA-LCoS device, we need to insert the display

OUT <sup>=</sup> *PX* <sup>⋅</sup> *<sup>R</sup>*(*θ*2) <sup>⋅</sup> *<sup>W</sup>*PA(¯

and *θ*<sup>2</sup>

Γ) ⋅ ( cos *θ*<sup>1</sup> sin *θ*<sup>1</sup>

<sup>0</sup> <sup>0</sup>) (18)

0 exp(+*jφ*/2)) (19)

cos*<sup>θ</sup>* sin*<sup>θ</sup>* <sup>−</sup>sin*<sup>θ</sup>* cos*θ*) (20)

, that is, the complex amplitude for the

) (21)

saturation of the recording material.

*PX* = (

*<sup>W</sup>*(*φ*) <sup>=</sup> (

the two-dimensional rotation matrix,

*E*

*R*(*θ*) = (

between two rotated linear polarizers at angles *θ*<sup>1</sup>

electric field at the output is given by the following equation:

→

**3.1. Modulation schemes**

144 Holographic Materials and Optical Systems

involved,

*X*-axis,

If we take into account the calculated values for the average retardance, we can calculate the trajectories in the complex plane. In **Figure 9**, we present the configurations used to implement the different modulation schemes.

**Figure 9.** Calculated values of the average retardance and the fluctuation amplitude for *λ* = 532 nm at an angle of incidence of 11.5°, for two different device configurations that enables us to implement BIM (continuous) and HTM (dashed).

The configuration shown in **Figure 9** is obtained at an angle of incidence of 11.5°, which is the incidence angle presented in our setup for holographic data storage. The sequences used are both 5-5, but the voltages applied are changed for the BIM configuration (continuous lines in **Figure 9**). In this case, the voltage has been reduced to *V*bright = 2.02 V and *V*dark = 1.11 V. For BIM, we only need a phase depth of 180°, this is the reason that we can relax the requisites of the device. As shown in **Figure 9**, the fluctuation amplitude has been reduced, in comparison with the configuration used for HTM that is the configuration called "5-5 532 nm 2pi linear" provided by the manufacturer. For more information about the influence of *V*bright and *V*dark, reference [12] can be consulted.

After we have defined the average retardance, from Eq. (21) we can calculate the intensity transmission *I* OUT that is given by the hermitic product of ⇀*<sup>E</sup>* OUT and the phase shift *ϕ*OUTwill be given by the argument. We can optimize in the computer the angles *θ*<sup>1</sup> and *θ*<sup>2</sup> of the linear polarizers to produce the best BIM and HTM regimes.

In the case of BIM, we need to generate the maximum intensity contrast between the on and off values, that is, *I* contrast = *I* oOn/*I* Ooff. We need at least a contrast of 1:20 for achieving an acceptable bit error rate (BER) [22]. Maximum contrast can be obtained with only polarizers, if they are parallel of crossed with each other at 45° with respect to the director axis of the PA-LCoS. Using the data from **Figure 9**, we obtain the intensity transmission and phase-shift curves as a function of the gray level. It is displayed in **Figure 10(a)**, which are also represented as a phasor in the complex plane in **Figure 10(b)**

**Figure 10.** Simulation for BIM. (a) Intensity transmission and phase shift as a function of gray level; (b) phasor evolution in the complex plane. The PA-LCoS is sandwiched between linear polarizers at +45° with respect to the *X*-axis (neutral lines).

In **Figure 10(a)**, the lowest and highest intensity transmission points occur at gray levels 12 and 239, respectively. From the values, we calculate that the theoretical contrast tends to infinity, and the phase-shift values are 270° and 360°. From **Figure 9**, we see that the difference in the retardance value is very close to 180° between these two gray levels. In the experimental measurements, we obtain that the low and high transmission points occur slightly displaced at gray levels 14 and 248, and the contrast we measure is about 1:50. The theoretical contrast value is idealistic since the various degradation effects in the PA-LCoS have a direct impact on the minimum intensity.

The representation in the complex plane (**Figure 10(b)**) is useful to see the complex evolution and the trajectory described with the variation of gray level. We see that the 180° phase jump is produced at the vicinity of the origin (gray level 12). It can be verified that independently of the orientation of the transmission axis of the polarizers, we always obtain a circular trajectory. The same effect is produced when adding wave plates to the system, at the entrance or/and the exit of the PA-LCoS (it will be like adding an offset to the retardance; it will not vary the trajectory). This will be a problem when implementing the HTM scheme.

For HTM scheme, we need three gray level values, two of them with a 180° relative phase shift and an equal and high intensity level (On levels). The third level will be the Off level that has to block as much light as possible to obtain a good contrast. As long as the trajectories in the complex plane are always circular and the 180° phase jump is produced in the origin, we have found that the PA-LCoS device cannot fully meet these requirements. In the case of Twisted-Nematic (TN) LCDs, this is possible. TN LCDs provided a coupled amplitude and phase-shift modulation. This fact enables to produce arbitrary complex amplitude trajectories that can accomplish the requirements of HTM [20, 21].

In the case of BIM, we need to generate the maximum intensity contrast between the on and

bit error rate (BER) [22]. Maximum contrast can be obtained with only polarizers, if they are parallel of crossed with each other at 45° with respect to the director axis of the PA-LCoS. Using the data from **Figure 9**, we obtain the intensity transmission and phase-shift curves as a function of the gray level. It is displayed in **Figure 10(a)**, which are also represented as a

In **Figure 10(a)**, the lowest and highest intensity transmission points occur at gray levels 12 and 239, respectively. From the values, we calculate that the theoretical contrast tends to infinity, and the phase-shift values are 270° and 360°. From **Figure 9**, we see that the difference in the retardance value is very close to 180° between these two gray levels. In the experimental measurements, we obtain that the low and high transmission points occur slightly displaced at gray levels 14 and 248, and the contrast we measure is about 1:50. The theoretical contrast value is idealistic since the various degradation effects in the PA-LCoS have a direct impact

**Figure 10.** Simulation for BIM. (a) Intensity transmission and phase shift as a function of gray level; (b) phasor evolution in the complex plane. The PA-LCoS is sandwiched between linear polarizers at +45° with respect to the *X*-axis

The representation in the complex plane (**Figure 10(b)**) is useful to see the complex evolution and the trajectory described with the variation of gray level. We see that the 180° phase jump is produced at the vicinity of the origin (gray level 12). It can be verified that independently of the orientation of the transmission axis of the polarizers, we always obtain a circular trajectory. The same effect is produced when adding wave plates to the system, at the entrance or/and the exit of the PA-LCoS (it will be like adding an offset to the retardance; it will not vary the trajectory). This will be a problem when implementing

Ooff. We need at least a contrast of 1:20 for achieving an acceptable

off values, that is, *I*

146 Holographic Materials and Optical Systems

on the minimum intensity.

(neutral lines).

the HTM scheme.

contrast = *I*

phasor in the complex plane in **Figure 10(b)**

oOn/*I*

Therefore, we can produce a compromise solution by slightly shifting the circular trajectory. At the cost of leaking some light intensity in the Off level, we can achieve two On levels with an appreciable transmitted intensity and a relative phase shift close to 180°. The closer these On levels are to a phase difference of 180° the lower the DC term is. We have named this modulation scheme as pseudo-HTM modulation (p-HTM). We want to know if this p-HTM scheme is still useful for its application in an HDSS.

In **Figure 11**, we show the complex amplitude for one of the possible p-HTM configurations. It corresponds to input and output polarizers at 55° and −45° with respect to the slow axis. In **Figure 11(a)**, we show the intensity and the phase-shift versus gray level, and in **Figure 11(b)** we plot the phasor evolution in the complex plane where we see that the circular trajectory is slightly displaced from the origin. The two On gray levels considered are 105 and 168 with amplitude transmission values of 0.28 and phase-shift difference of 206°. The Off gray level is 140 with an amplitude transmission of 0.03 and phase shift 170°. The intensity contrast is then 1:10. This is a low value; however, increasing the contrast means almost crossing the two polarizers. This produces that the 180° phase jump is closer to the origin and the difference in phase between On levels is increased rapidly producing a larger DC term, which is what we are trying to reduce [23].

**Figure 11.** Simulation for p-HTM: (a) intensity transmission and phase shift; (b) phasor evolution in the complex plane. input and output polarizers are at +55° and −45° with respect to the slow axis.

#### **3.2. Experimental results**

To test the modulation schemes presented, we use the next experimental setup represented in **Figure 12**.

**Figure 12** shows us the scheme for the experimental HDSS used. We consider a 532-nm laser beam, to which the PVA photopolymer is sensitized. We use PVA/AA because it is a wellstudied photopolymer, and our group had done an intense work to characterize and model it. More information about the photopolymer can be found in references [24, 25]. We have inserted a half wave plate before the shutter and spatial filter to ensure that enough light intensity impinges onto the linear polarizers in the object and reference beam. The intensity ratio between both beams is controlled with one attenuator in reference beam. The linear polarizers LP are used to produce the appropriate SOP for the object and reference beam. In the reference beam, a stop limits the aperture of the beam to about a diameter of 1 cm. Lenses in the reference beam form an afocal system so that the rotating mirror and the recording material are at conjugate planes; this enables angular multiplexing simply by rotation of the mirror. In the object beam, we have combined a divergent and a convergent lens in order to control the curvature of the converging beam onto the PA-LCoS. At the convergence plane, we find the Fourier transform of the data page, in this plane we used a stop that will act as a Nyquist filter. Then we have built a relay system to image the Nyquist filter plane onto the recording plane. We can also introduce some defocusing degree by displacing the recording material plane. This system uses a convergent correlator setup, and finally we read the data saved with the help of a high dynamic CCD camera [25].

**Figure 12.** Schematic representation of our experimental HDSS testing platform.

In the present work, we focus on testing the modulation schemes implemented with our PA-LCoS. We address a data page formed with random data bits. Afterwards, in the reconstruction step, we retrieve the data saved with CCD camera. Compared with the original data, we make a count on the errors detected. This provides us with a measurement or the raw BER. We only apply a simple treatment consisting in looking for the best threshold level that minimizes the errors counted [26].

**3.2. Experimental results**

148 Holographic Materials and Optical Systems

saved with the help of a high dynamic CCD camera [25].

**Figure 12.** Schematic representation of our experimental HDSS testing platform.

in **Figure 12**.

To test the modulation schemes presented, we use the next experimental setup represented

**Figure 12** shows us the scheme for the experimental HDSS used. We consider a 532-nm laser beam, to which the PVA photopolymer is sensitized. We use PVA/AA because it is a wellstudied photopolymer, and our group had done an intense work to characterize and model it. More information about the photopolymer can be found in references [24, 25]. We have inserted a half wave plate before the shutter and spatial filter to ensure that enough light intensity impinges onto the linear polarizers in the object and reference beam. The intensity ratio between both beams is controlled with one attenuator in reference beam. The linear polarizers LP are used to produce the appropriate SOP for the object and reference beam. In the reference beam, a stop limits the aperture of the beam to about a diameter of 1 cm. Lenses in the reference beam form an afocal system so that the rotating mirror and the recording material are at conjugate planes; this enables angular multiplexing simply by rotation of the mirror. In the object beam, we have combined a divergent and a convergent lens in order to control the curvature of the converging beam onto the PA-LCoS. At the convergence plane, we find the Fourier transform of the data page, in this plane we used a stop that will act as a Nyquist filter. Then we have built a relay system to image the Nyquist filter plane onto the recording plane. We can also introduce some defocusing degree by displacing the recording material plane. This system uses a convergent correlator setup, and finally we read the data

The data page is formed by 64 × 64 information bits, and each bit of information is formed by 8 × 8 pixels in the PA-LCoS. This is necessary to avoid interpixel cross-talk effects [16]. We have 4096 information bits of information in each data page.

In **Figure 13**, we present the first result for BIM scheme; in this case, we are not using material to register the hologram. This image enables us to evaluate the optics performance of our experimental setup (**Figure 12**). We see some interesting facts. In the first place, we see how the data page is perfectly reconstructed and the 1's and 0's distribution in gray levels is clearly separated. No errors in the reconstruction have been produced.

**Figure 13.** Experimental results for BIM scheme and no material. (a) Data page; (b) histograms.

In **Figure 14**, we present the results obtained for p-HTM scheme without using material. We see that the histograms are not clearly separated. This implies that some error will be produced in the reconstruction. We detect eight errors, which corresponds to a BER of 2.0 × 10−3. This is due to the lower contrast of the data page, but we still can reconstruct the image with an acceptable BER.

**Figure 15** shows us the results obtained when the data page is stored in a PVA/AA film. The beam intensity ratio used is about 1:400, where the intensity incident onto the recording material is 3.16 mW/cm2 and 8 μW/cm2 , respectively, for the reference and object beam, and for an exposure time of 6 s. As we can see in the histograms, the 0's and 1's show a slight overlap. We detected 52 errors, that is, BER = 1.3 × 10−2. The BER is still in the range that allow to retrieve the information without errors when applying error correction codes [26].

**Figure 14.** Experimental results for p-HTM scheme and no material. (a) Data page; (b) histograms.

**Figure 15.** Experimental results for BIM scheme and PVA/AA. (a) Data page; (b) histograms.

In **Figure 16**, we see the results for a data page stored in PVA/AA using p-HTM scheme. Beam intensity ratio is about 1:800, where the intensity incident onto the recording material is 3.16 mW/cm2 and 4 μW/cm2 , respectively, for the reference and object beam, and for an exposure time of 10 s. As we can see, and due to the lost of contrast, the results are a worse that in BIM case. There are 229 errors detected, that is, BER = 5.6 × 10−2. BER is still in an acceptable range, and it is only five times larger than the BIM case.

We did some simulations to predict the results [23]. Our simulations predicted worse results for p-HTM, when we compare it with BIM, than the ones obtained experimentally. We believe that the better experimental ratio is due to the reduction in DC term, which is to not have into account in our simulations just because we considered a linear material. Further experimental work has to be done with PVA/AA photopolymer to study these saturation effects. In general, we can say that the p-HTM scheme can be used in an HDSS because of its promising results. Maybe, it can show all its potential when the multiplexing capability will be used.

**Figure 16.** Experimental results for p-HTM scheme and PVA/AA. (a) Data page; (b) histograms.
