**2. PA-LCoS characterization**

In this section, we present two characterization methods for PA-LCoS microdisplays. One of them uses any equipment that is present in almost every optics laboratory. The other makes use of a commercial rotating wave plate polarimeter.

#### **2.1. Extending the linear polarimetric method to characterize the PA-LCoS**

Parallel-aligned devices are totally characterized by their linear retardance versus voltage values. For this reason, we can use methods typically used in the characterization of wave plates. All these methods assume that the retardance introduced does not change with time [8, 9]. So, we have to adapt the method for elements that present some levels of fluctuation in the retardance signal.

We use the linear polarimetric method to measure the retardance introduced by the wave plate (our PA-LCoS). We use the next scheme.

In **Figure 1**, we see a wave plate sandwiched between two linear polarizers. For the appropriate angles between wave plate and the linear polarizers, we can calculate the retardance as follows [7]:

In Figure 1, we see a wave plate bandwidth between two linear polarizers. For the appropriate angles between wave plate and the linear polarizers, we can calculate the retardance as follows [7]: 
$$\Gamma = \cos^{-1} \left( \frac{l\_{\rm{OUT}}^{\parallel} - l\_{\rm{OUT}}^{\perp}}{l\_{\rm{OUT}}^{\parallel} + l\_{\rm{OUT}}^{\perp}} \right) \tag{1}$$

where *Γ* is the measured retardance. *I* OUT <sup>∥</sup> and *I* OUT <sup>⊥</sup> are the intensity measured at the exit of the system when the two linear polarizers are oriented at +45° with respect to the slow axis of the wave plate, and when the input linear polarizer is oriented at +45° and the output polarizer is oriented at −45°. These are the so-called "parallel" intensity and "crossed" intensity. With just two measurements, we can calculate the retardance of the wave plate. These intensities are defined as follows:

$$I\_{\rm OUT}^{\prime\prime} = \frac{I\_0}{2} \left[ 1 + \cos \Gamma \right] \tag{2}$$

$$I\_{\rm corr}^{\perp} = \frac{I\_0}{2} \left[ 1 - \cos \Gamma \right] \tag{3}$$

where *I* 0 is the total light intensity introduced in the system.

state of polarization (SOP) converter, phase-only modulator, or amplitude-mostly modulator.

PA-LCoS devices have become widely used in diffractive optics due to its ease of operation and phase-only modulation capabilities, because of its high spatial resolution and high light efficiency [3]. The liquid crystal (LC) technology has achieved a high level of maturity and enables us to have a data entry point in HDS for a high data density recording, and it provides

In order to incorporate this kind of microdisplays into a complete holographic data storage system (HDSS), we have done an intense work to characterize PA-LCoS devices. As a phaseonly device, we need to know the retardance introduced for every gray level. The microdisplay is digitally addressed with a pulsed voltage signal. This fact implies a fluctuation in the phase that will be reflected in the optical response [4–7]. For that reason, for a full characterization, we need to obtain the average retardance and the fluctuation amplitude for every

Even though PA-LCoS are widely used by the photonics community, these phase-only devices have not been intensively applied in holographic data storage applications. These devices could be a key element for a phase multilevel data page coding. If the trend is to use this kind of coding scheme, a right characterization and a profound knowledge will be useful.

In this section, we present two characterization methods for PA-LCoS microdisplays. One of them uses any equipment that is present in almost every optics laboratory. The other makes

Parallel-aligned devices are totally characterized by their linear retardance versus voltage values. For this reason, we can use methods typically used in the characterization of wave plates. All these methods assume that the retardance introduced does not change with time [8, 9]. So, we have to adapt the method for elements that present some levels of fluctuation in

We use the linear polarimetric method to measure the retardance introduced by the wave

In **Figure 1**, we see a wave plate sandwiched between two linear polarizers. For the appropriate angles between wave plate and the linear polarizers, we can calculate the retardance as follows [7]:

( *I* OUT // − *I* OUT \_ ⊥

*I* OUT // + *I* OUT

<sup>⊥</sup> ) (1)

**2.1. Extending the linear polarimetric method to characterize the PA-LCoS**

gray level. This enables us to select the best device configuration for our application.

All these roles are interesting in many research fields [1–3].

134 Holographic Materials and Optical Systems

us with the ability to design different modulation schemes.

**2. PA-LCoS characterization**

the retardance signal.

use of a commercial rotating wave plate polarimeter.

plate (our PA-LCoS). We use the next scheme.

Γ = cos<sup>−</sup><sup>1</sup>

**Figure 1.** Linear polarimeter with the wave plate (WP) to be measured. P1 and P2 are linear polarizers.

As mentioned, the PA-LCoS device introduces a fluctuation due to its digital-addressing scheme. We see how this fact alters the equations described above. As a first approximation, we consider a triangular profile for the periodic variation of retardance with time, *Γ*(*t* ) described by Eq. (4)

described by Eq. (4)

$$
\Gamma(t) = \begin{cases}
\overline{\Gamma} - a + \frac{2a}{T/2}t & 0 \le t < \frac{\gamma}{2} \\
\overline{\Gamma} + 3a - \frac{2a}{T/2}t & \gamma\_2 \le t < T
\end{cases}
\tag{4}
$$

where *Γ*¯ is the average value of the retardance during a period *T*. This function is represented in **Figure 2**.

**Figure 2.** Triangular profile considered for the temporal fluctuation of the linear retardance.

Following the proposed model in Eq. (4), if we calculate the average value for a period of cos(*Γ*(*t*)), we obtain,

$$
\left< \cos(\overline{\Gamma} - a + \frac{2a}{T/2} \, t \right> \right> = \frac{\sin(a)}{a} \cos(\overline{\Gamma} \, t) \tag{5}
$$

We can see how a *sinc* term appears modulating the cosine function. Taking into account Eq. (5), we can rewrite Eqs. (2) and (3) in terms of average intensity as follows:

$$
\langle I\_{\rm OUT}^{\parallel} \rangle = \frac{I\_0}{2} \left[ 1 + \frac{\sin a}{a} \cos \Gamma \right] \tag{6}
$$

$$
\langle I\_{\rm OUT}^{\perp} \rangle = \frac{I\_0}{2} \left[ 1 - \frac{\sin a}{a} \cos \Gamma \right] \tag{7}
$$

We see how the intensity that we really measure is affected by the *sinc(a)* function when we are trying to measure the retardance introduced by a wave plate that presents instabilities. If we combine Eqs. (6) and (7), we obtain the next expression for the average retardance:

$$\frac{\langle l\_{\rm OUT}^{\parallel} \rangle - \langle l\_{\rm OUT}^{\perp} \rangle}{\langle l\_{\rm OUT}^{\parallel} \rangle + \langle l\_{\rm OUT}^{\perp} \rangle} = \frac{\sin a}{a} \cos \Gamma \tag{8}$$

$$
\overline{\Gamma} = \cos^{-1}\left(\frac{\langle \ell\_{\rm int}^{\prime} \rangle \cdot \langle \ell\_{\rm int}^{\prime} \rangle \ell\_{\rm int}^{\prime} \cdot \langle \ell\_{\rm int}^{\prime} \rangle}{\langle \ell^{\rm int} \rangle}\right) \tag{9}
$$

In the case when no fluctuation exist (*a* = 0<sup>o</sup> ), we recover the classical result presented in Eq. (1). In essence, assuming a linear variation with time, as expressed in Eq. (4), this fluctuation is translated into a *sinc* function when averaging the cosine in a period.

To analyze how the fluctuations limit the classical linear polarimeter, we performed the next simulated experiment: using Eqs. (6) and (7), we simulate the intensity values measured in the presence of fluctuations in the retardance. Then, we consider Eq. (1) to obtain the retardance value as if we ignore the existence of these fluctuations.

where *Γ*¯ is the average value of the retardance during a period *T*. This function is represented

Following the proposed model in Eq. (4), if we calculate the average value for a period of cos(*Γ*(*t*)),

We can see how a *sinc* term appears modulating the cosine function. Taking into account Eq.

We see how the intensity that we really measure is affected by the *sinc(a)* function when we are trying to measure the retardance introduced by a wave plate that presents instabilities. If

we combine Eqs. (6) and (7), we obtain the next expression for the average retardance:

( (〈*I* OUT // 〉 − 〈*I* OUT <sup>⊥</sup> 〉 ⁄ 〈*I* OUT // 〉 + 〈*I* OUT <sup>⊥</sup> 〉)

Eq. (1). In essence, assuming a linear variation with time, as expressed in Eq. (4), this fluctua-

*<sup>T</sup>*/2 *<sup>t</sup>* )〉 <sup>=</sup> sin(*<sup>a</sup>* ) \_\_\_\_\_ *<sup>a</sup>* cos(¯

*<sup>a</sup>* cos¯

*<sup>a</sup>* cos¯

*<sup>a</sup>* cos¯

Γ ) (5)

Γ] (6)

Γ] (7)

Γ (8)

\_\_\_\_\_\_\_\_\_\_\_\_ (sin*<sup>a</sup>* <sup>⁄</sup>*a*) ) (9)

), we recover the classical result presented in

Γ − *a*+ \_\_\_ <sup>2</sup>*<sup>a</sup>*

**Figure 2.** Triangular profile considered for the temporal fluctuation of the linear retardance.

(5), we can rewrite Eqs. (2) and (3) in terms of average intensity as follows:

OUT // 〉 <sup>=</sup> *<sup>I</sup>* \_\_0 <sup>2</sup> [ <sup>1</sup> <sup>+</sup> \_\_\_\_ sin*<sup>a</sup>*

OUT <sup>⊥</sup> 〉 <sup>=</sup> *<sup>I</sup>* \_\_0 <sup>2</sup> [ <sup>1</sup> <sup>−</sup> \_\_\_\_ sin*<sup>a</sup>*

OUT // 〉 − 〈*I* OUT <sup>⊥</sup> 〉 \_\_\_\_\_\_\_\_\_\_\_

Γ = cos<sup>−</sup><sup>1</sup>

tion is translated into a *sinc* function when averaging the cosine in a period.

〈*I* OUT // 〉 + 〈*I* OUT <sup>⊥</sup> 〉 <sup>=</sup> \_\_\_\_ sin*<sup>a</sup>*

in **Figure 2**.

136 Holographic Materials and Optical Systems

we obtain,

〈cos(¯

〈*I*

〈*I*

〈*<sup>I</sup>*

¯

In the case when no fluctuation exist (*a* = 0<sup>o</sup>

In **Figure 3**, we show the calculated retardance values as a function of the true retardance ones used in the simulation and for different fluctuation amplitudes, indicated with the curves in the plot.

**Figure 3.** Simulation of the retardance measurement experiment in the presence of fluctuations. The calculated retardance uses the classical expression where fluctuations are not considered. Various fluctuation amplitudes are considered to be compared with the no-fluctuations case.

In **Figure 3**, we see how the fluctuations affect the calculated retardance. The maximum and the minimum measurements are produced at the same places of the theoretical wrapped retardance. This means that they are produced at values multiple of 180°. We also note that the deviation amount of deviation depends on the true retardance value. At true retardance, values multiple of 180° deviation are magnified. Outside of these points, if the fluctuation amplitude is not very large, we also find that the calculated values are very close to the true retardance values.

From the study of Eq. (8), it can be easily deduced that for average retardance values multiple of 180°, the calculated retardance only depends on the fluctuation amplitude *a*. So, in these cases we can uncouple the average retardance measurements from the fluctuation amplitude. These cases are the maximum and minimum points obtained from the measurements. At those points, we can estimate the fluctuation amplitude. The fluctuation difference in these maximum and minimum points can be expressed as

maximum and minimum points can be expressed as 
$$
\Gamma\_{\text{diff}} = \cos^{-1}\left(\frac{\sin a}{a}\right) \tag{10}
$$

Using Eq. (10) and the calculated retardance, for the maximum and minimum points, we obtain Γdiff. Then, we estimate the fluctuation amplitude, and, eventually, we can correct the measurements of the retardance using the exact Eq. (9) instead of the classical one (Eq. (1)) [10].

#### **2.2. Applying the method to a PA-LCoS microdisplay**

To validate and test the method described in the previous section, we use the next experimental setup.

**Figure 4** shows the experimental setup used to measure the average retardance versus the applied voltage (gray level) for a PA-LCoS device. It consists of a light source (He-Ne laser, *λ* = 633 nm), the LCoS, the necessary input and output linear polarizers which are in parallel or crossed configuration, and a radiometer to measure the intensity. For measuring time variations in the retardance, we need to measure the intensity in both cases (parallel and crossed) at the same time, for this reason we have introduced two high-quality nonpolarizing cube beam splitters (model 10BC16NP.4, from Newport): one of them to separate the incident and the reflected beams, and the other to enable synchronized measurement of the parallel and crossed intensities.

**Figure 4.** Experimental setup used to measure the linear retardance as a function of the applied voltage (gray level) for a PA-LCoS. The setup allows to measure both average and instantaneous values.

These simultaneous measurements when obtained with the help of an oscilloscope are used to validate the estimated fluctuation amplitude calculated with the previous method. We demonstrated that the estimated values are in good agreement with the measurements done with the oscilloscope [10]. So, the oscilloscope is no longer necessary. This is the reason why we affirm that the method does not need special equipment (as the mentioned oscilloscope), which maybe is not found in an optical laboratory.

At those points, we can estimate the fluctuation amplitude. The fluctuation difference in these

Using Eq. (10) and the calculated retardance, for the maximum and minimum points, we obtain Γdiff. Then, we estimate the fluctuation amplitude, and, eventually, we can correct the measurements of the retardance using the exact Eq. (9) instead of the classical one (Eq. (1)) [10].

To validate and test the method described in the previous section, we use the next experimen-

**Figure 4** shows the experimental setup used to measure the average retardance versus the applied voltage (gray level) for a PA-LCoS device. It consists of a light source (He-Ne laser, *λ* = 633 nm), the LCoS, the necessary input and output linear polarizers which are in parallel or crossed configuration, and a radiometer to measure the intensity. For measuring time variations in the retardance, we need to measure the intensity in both cases (parallel and crossed) at the same time, for this reason we have introduced two high-quality nonpolarizing cube beam splitters (model 10BC16NP.4, from Newport): one of them to separate the incident and the reflected beams, and the other to enable synchronized measurement of the parallel and crossed intensities.

These simultaneous measurements when obtained with the help of an oscilloscope are used to validate the estimated fluctuation amplitude calculated with the previous method.

**Figure 4.** Experimental setup used to measure the linear retardance as a function of the applied voltage (gray level) for

a PA-LCoS. The setup allows to measure both average and instantaneous values.

( \_sin*a*

*<sup>a</sup>* ) (10)

maximum and minimum points can be expressed as

**2.2. Applying the method to a PA-LCoS microdisplay**

tal setup.

Γdiff = cos<sup>−</sup><sup>1</sup>

138 Holographic Materials and Optical Systems

In our experiments, we have used and analyzed an LCoS display distributed by the company HOLOEYE. It is an active matrix reflective mode device with 1920 × 1080 pixels and 0.7'' diagonal named PLUTO spatial light modulator. The pixel pitch is of 8.0 μm and the display has a fill factor of 87%. The signal is addressed via a standard digital visual interface (DVI) signal. Using an RS-232 interface and its provided software, we can perform gamma control to configure the modulator for different applications and wavelengths. The manufacturer provides some configuration files for the equipment. These configurations are designed for different applications. The configuration files are labeled as "18-6 default," "18-6 2pi linear 633 nm," and "5-5 2pi linear 633 nm." From the name, we can infer that the ones with "2pi linear 633 nm" are designed for a linear response in retardance for 633-nm wavelength. The labeled as "default" means that no gamma correction has been applied and the retardance response will be nonlinear. The other part of the label (18-6 and 5-5) makes reference to the configuration of the digital addressed signal. This part reflects the number of bits used in the signal, so the signal has different lengths depending on the configuration file. In principle, a shorter length presents less flicker. You can find more information about the digital signal and configuration in Refs. [10, 11].

The method presented in Section 2.1 enables us to correct the measures done for characterizing the PA-LCoS in order to know the exact average retardance introduced by each gray level. To do that, we extrapolate the fluctuation amplitude value calculated from the extremals (180° multi ples) to a wider gray level range: the fluctuation amplitude value from the first extremal is consi dered to be roughly valid until half of the gray level distance between itself and the next extremal.

In **Figure 5**, we show the directly taken measurements ("Uncorrected" ones) that we obtain by applying Eq. (1) to the parallel and crossed intensities measured. We know that we are not taking into account the presence of fluctuations. From the extremal points, which are multiples of 180° in the retardance, we can calculate the fluctuation amplitude using Eq. (11). In this way, we obtain a fluctuation amplitude that we apply by intervals between the different extremals. Using Eq. (9), and knowing the value of the fluctuation, we recalculate the average retardance correcting the previous obtained curves ("Corrected" ones).

**Figure 5.** Average retardance versus gray level considering the existence of the fluctuation amplitude in the retardance, "corrected" curve, and considering the ideal *a* = 0°, "uncorrected" curve. (a) "18-6 default"; (b) "18-6 2pi linear 633 nm"; (c) "5-5 2pi linear 633 nm".

To sum up: we are able of obtain the corrected curves. With this information, we can change the gamma curve configuration to obtain a selected behavior. And we can just evaluate Γdiff in the extremals to select the configuration that has less fluctuation amplitude [10, 12].

#### **2.3. Characterization using averaged stokes polarimetry**

The method showed in the previous section enables us to configure the device but it does not provide a full characterization. We only have information about the fluctuation amplitude in some points. In this section, we present a method that provides full information about the average retardance and the fluctuation amplitude for all gray levels.

We will use the Mueller-Stokes formalism [8], which enables to deal both with polarized and with unpolarized light. In this formalism, the retardance wave plate matrix is

$$M\_k(\Gamma) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos \Gamma & \sin \Gamma \\ 0 & 0 & -\sin \Gamma & \cos \Gamma \end{pmatrix} \tag{11}$$

Eq. (11) describes a wave plate with the fast axis along the *X*-axis and a retardance *Γ*. Let us consider a unit intensity Stokes vector corresponding to a state of polarization linearly polarized at +45° with respect to the *X*-axis impinging perpendicular onto the linear retarder. Then, the SOP at the exit is calculated as follows:

$$
\begin{pmatrix} 1 \\ 0 \\ \cos \Gamma \\ -\sin \Gamma \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos \Gamma & \sin \Gamma \\ 0 & 0 & -\sin \Gamma & \cos \Gamma \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix} \tag{12}
$$

In Eq. (12), we see how the second element of the resultant Stokes vector is zero independently of the retardance introduced by the wave plate, and it does not depend on the possible temporal retardance fluctuations. If we introduce the signal variation model described in Eq. (4), we calculate the average value for *S*<sup>2</sup> and *S*<sup>3</sup> components as follows:

$$\langle S\_2 \rangle = \frac{1}{T/2} \int\_0^{\epsilon\_i} \cos \left( \Gamma - a + \frac{2a}{T/2} t \right) dt = \frac{\sin(a)}{a} \cos(\Gamma) \tag{13}$$

$$\langle S\_3 \rangle = \frac{-1}{T/2} \int\_0^{\langle \cdot \rangle} \sin \left( \Gamma - a + \frac{2a}{T/2} \, t \right) dt = \frac{-\sin(a)}{a} \sin(\Gamma) \tag{14}$$

Therefore, the average exit vector *S*out reflected by the PA-LCoS when a linear polarized beam impinges, and the angle between the slow axis and the polarizer is 45° is given by

$$\langle S\_{\rm out} \rangle = < \begin{pmatrix} 1 \\ 0 \\ \cos \Gamma(t) \\ -\sin \Gamma(t) \end{pmatrix} > = \begin{pmatrix} 1 \\ 0 \\ \frac{\sin(a)}{a} \cos(\Gamma) \\ -\frac{\sin(a)}{a} \sin(\Gamma) \end{pmatrix} \tag{15}$$

From Eq. (15), we see how, when we calculate the degree of polarization (DoP), it depends only on the fluctuation amplitude

only on the fluctuation amplitude

$$\text{DoP} = \frac{\sqrt{(S\_1)^2 + (S\_2)^2 + (S\_3)^2}}{S\_0} = \frac{\sin(a)}{a} \tag{16}$$

Eq. (16) shows that the DoP is produced by the fluctuation in the retardance and provides as a way to calculate the fluctuation amplitude for every gray level by measuring the DoP.

From Eq. (15), it is clear that the average retardance can be obtained by calculating the ration between third and fourth Stoke vector components,

$$-\langle S\_{\mathfrak{z}}\rangle / \langle S\_{\mathfrak{z}}\rangle = \text{tg}(\overline{\Gamma}) \tag{17}$$

We have presented a method that can be applied to any electro-optic element acting as a wave plate and presenting fluctuations in its optical response [10]. It is not only applicable to PA-LCoS devices. We have applied them to the mentioned PLUTO device obtaining a complete characterization [13].

#### *2.3.1. Experimental setup for characterization*

To sum up: we are able of obtain the corrected curves. With this information, we can change the gamma curve configuration to obtain a selected behavior. And we can just evaluate Γdiff in

The method showed in the previous section enables us to configure the device but it does not provide a full characterization. We only have information about the fluctuation amplitude in some points. In this section, we present a method that provides full information about the

We will use the Mueller-Stokes formalism [8], which enables to deal both with polarized and

Eq. (11) describes a wave plate with the fast axis along the *X*-axis and a retardance *Γ*. Let us consider a unit intensity Stokes vector corresponding to a state of polarization linearly polarized at +45° with respect to the *X*-axis impinging perpendicular onto the linear retarder.

1 0 0 0 0 <sup>1</sup> <sup>0</sup> <sup>0</sup> 0 0 cosΓ sinΓ

0 0 −sinΓ cosΓ

1 0 0 0 0 <sup>1</sup> <sup>0</sup> <sup>0</sup> 0 0 cosΓ sinΓ

0 0 −sinΓ cosΓ

In Eq. (12), we see how the second element of the resultant Stokes vector is zero independently of the retardance introduced by the wave plate, and it does not depend on the possible temporal retardance fluctuations. If we introduce the signal variation model described in Eq.

and *S*<sup>3</sup>

<sup>Γ</sup> <sup>−</sup> *<sup>a</sup>*<sup>+</sup> \_2*<sup>a</sup>*

<sup>Γ</sup> <sup>−</sup> *<sup>a</sup>*<sup>+</sup> \_2*<sup>a</sup>*

Therefore, the average exit vector *S*out reflected by the PA-LCoS when a linear polarized beam

⎞ ⎟ ⎠

> =

⎛

⎜ ⎝

−\_ sin(*a* ) *<sup>a</sup>* sin(¯ Γ )

impinges, and the angle between the slow axis and the polarizer is 45° is given by

1 0 cosΓ(*t*) −sinΓ(*t*)

⎛ ⎜ ⎝ (11)

(12)

(15)

Γ ) (13)

Γ ) (14)

⎞ ⎟ ⎠

⎛ ⎜ ⎝

⎞ ⎟ ⎠

⎞ ⎟ ⎠

components as follows:

*<sup>T</sup>*/2 *<sup>t</sup>*)*dt* <sup>=</sup> sin(*<sup>a</sup>* ) \_\_\_\_\_ *<sup>a</sup>* cos(¯

*<sup>T</sup>*/2 *<sup>t</sup>*)*dt* <sup>=</sup> <sup>−</sup>sin(*<sup>a</sup>* ) \_\_\_\_\_\_ *<sup>a</sup>* sin(¯

1 0 \_ sin(*a* ) *<sup>a</sup>* cos(¯ Γ ) 

⎞

⎟ ⎠

the extremals to select the configuration that has less fluctuation amplitude [10, 12].

**2.3. Characterization using averaged stokes polarimetry**

*MR*

140 Holographic Materials and Optical Systems

Then, the SOP at the exit is calculated as follows:

(4), we calculate the average value for *S*<sup>2</sup>

〈*S*<sup>2</sup>

〈*S*<sup>3</sup>

〈*S*out〉 = <

⎛ ⎜ ⎝

〉 <sup>=</sup> \_\_\_1 *<sup>T</sup>*/2 ∫ 0 *<sup>T</sup>*⁄<sup>2</sup> cos(¯

〉 <sup>=</sup> \_\_\_ <sup>−</sup><sup>1</sup> *<sup>T</sup>*/2 ∫ 0 *<sup>T</sup>*⁄<sup>2</sup> sin(¯

1 0 cosΓ −sinΓ

⎞ ⎟ ⎠ = ⎛ ⎜ ⎝

average retardance and the fluctuation amplitude for all gray levels.

with unpolarized light. In this formalism, the retardance wave plate matrix is

⎛ ⎜ ⎝

(Γ) =

To use the method described above, we just need a polarimeter. In our case, we have used a commercial PAX5710VIS-T model from THORLABS. This is a rotating wave plate-based polarimeter, which belongs to time-division mode polarimeters. They are not able to provide instantaneous values if the state of polarization changes more rapidly than its measurement time interval. In our case, the THORLABS polarimeter has a measurement time interval of 3 ms (maximum rotation frequency is 333 Hz). But as long as we need an average value, and the software allows to enlarge the measurement time interval, we just select an appropriate time interval larger than the fluctuation time period of the PLUTO device, which is about 8.66 ms (120 Hz).

In **Figure 6**, we show the experimental setup used for characterizing our PA-LCoS, the first wave plate shown is used just to guarantee enough light intensity regardless of the polarizer angle, since the output light from the laser is linearly polarized. A quarter wave plate, as in the figure, with its neutral lines properly oriented can do the work, even though it is not the only possibility. As we show in the method the input polarizer has to be at 45° from the neutral lines of our PA-LCoS, in our case it is the *x*-*y*-axis.

We test the method with the "18-6 633 nm 2pi linear" and "5-5 633 nm 2pi linear" configurations, already introduced. We measure the Stokes vector and degree of polarization of each gray level. We have obtained the next results.

**Figure 7** shows the data obtained for the two different configurations tested when linearly polarized light impinges onto the device at 45°. In dashed lines, we show the measurements for "5-5 633 nm 2pi linear" configuration. In continuous lines, we show the measurements for "18-6 633 nm 2pi linear." The first thing that we have to note is that parameter *S1* is close to zero for all gray levels, in clear confirmation of the result obtained in Eq. (15). To obtain the retardance value and the fluctuation amplitude, we have to apply Eqs. (17) and (16), respectively.

**Figure 6.** Experimental setup for characterizing a PA-LCoS.

**Figure 7.** Measurements obtained from the PLUTO for "5-5 633 nm 2pi linear" (dashed line) and "18-6 633 nm 2pi linear" (continuous line).

In **Figure 8**, we have calculated the fluctuation amplitude and the average retardance. We have obtained a full characterization of the device. We observe that the retardance range is about 360° for both sequences with very good linearity. The fluctuation amplitude is clearly smaller for the 5-5 sequence.

for "18-6 633 nm 2pi linear." The first thing that we have to note is that parameter *S1*

respectively.

142 Holographic Materials and Optical Systems

**Figure 6.** Experimental setup for characterizing a PA-LCoS.

(continuous line).

to zero for all gray levels, in clear confirmation of the result obtained in Eq. (15). To obtain the retardance value and the fluctuation amplitude, we have to apply Eqs. (17) and (16),

**Figure 7.** Measurements obtained from the PLUTO for "5-5 633 nm 2pi linear" (dashed line) and "18-6 633 nm 2pi linear"

is close

**Figure 8.** Calculated values for the average retardance and the fluctuation amplitude for *λ* = 633 nm, and for sequences "5-5 633 nm 2pi linear" (dashed) and "18-6 633 nm 2pi linear" (continuous).

This method has been validated with the direct measurements of the retardance fluctuations and with the predictive capability of the SOP at the exit regardless of the SOP at the input [13]. The direct measurements of the fluctuation amplitude have been done with the help of an oscilloscope [14], and we have also tested the predictive capability of the model when applied to a more complex diffractive optical element (DOE) as a blazed grating [15].
