**2.1 LCoS display Mueller matrix characterization: Intensity prediction**

The LCoS display under analysis is a commercial Philips LCoS model X97c3A0, sold as the kit LC-R2500 by HoloEye Systems. This prototype presents a monochrome reflective

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

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

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

**2. Liquid Crystal on Silicon displays characterization and optimization** 

experimentally obtained Mueller matrices is carried on.

In this section, we propose a characterization-optimization method, based on a combination of the Mueller-Stokes (M-S) formalism (Parke, 1949) and the Jones formalism (Jones, 1941), which is able to predict and optimize the intensity and the phase response of LCoS displays. The characterization of the LCoS display is conducted as a function of the addressed gray level (which is related to the applied voltages), by experimentally calibrating its corresponding Mueller matrices, which are able to describe depolarizing samples. In this situation, the intensity response of the LCoS display can be predicted but since Mueller matrices are calculated from intensity measurements, no information about the phase modulation response is obtained. As Jones matrices contain the required phase information, a conversion from Mueller matrices to Jones matrices is applied. To perform a proper conversion, a polar decomposition (Lu & Chipman, 1996; Foldyna et al., 2009) of the

Once the intensity and phase response of the LCoS display can be predicted, an optimization procedure is applied, in order to get the desired LCoS display behaviour, i.e. to work as amplitude-only modulator or as phase-only modulator. In next subsections, the characterization and optimization methodology is described in detail and some results are

The LCoS display under analysis is a commercial Philips LCoS model X97c3A0, sold as the kit LC-R2500 by HoloEye Systems. This prototype presents a monochrome reflective

**2.1 LCoS display Mueller matrix characterization: Intensity prediction** 

observed when working with LCoS display becomes mandatory.

undesired influence of this damaging phenomenon must be applied.

section 5.

provided.

**methodology** 

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 used.

Fig. 1. Mueller matrix characterization set-up.

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 orientation a different incident State of Polarization (SoP).

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 orientations of the QWP2 allow us to analyze different SoPs.

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

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 respectively:

Study of Liquid Crystal on Silicon Displays for Their Application in Digital Holography 237

described. The PSG in Fig. 1 is formed by the linear polarizer LP1, fixed at 0 degrees to the laboratory vertical, followed by the quarter-waveplate QWP1. The general expression for the states of polarization exiting from the PSG can be obtained by multiplying the Stokes vector of linearly polarized light in the horizontal direction (i.e. SP1 = (1, 1, 0, 0)T) by the Mueller matrix of a linear retarder with a retardance equal to degrees and whose orientation

where different incident States of Polarization (SoPs) can be generated by varying the orientation angle 1 of the QWP1. By replacing the Stokes vector given in Eq. (4) into Eq. (1),

<sup>2</sup> cos 4θ<sup>1</sup> <sup>+</sup> mk2

where k (k = 0, 1, 2, 3) indicates the different Stokes parameter and mk,j (j = 0, 1, 2, 3) are the

Equation (5) comprises different sinusoidal functions whose arguments are entire multiples of θ1, and thus, the method of synchronous detection can be applied again. In particular, by performing a summation of different reflected SoPs corresponding to N different equally

<sup>2</sup> cos 4θ1,r

where N is the number of SoPs, r (r = 1,2,…,N) indicates the specific term in the summation

By performing a summation of different reflected SoPs related to N different equally spaced values of θ1 on a complete rotation of 360 degrees, some terms of Eq. (6) vanish because of the orthogonal properties of the sinusoidal sampled functions (Lizana et al., 2009). Besides, by performing a summation of different reflected SoPs related to N different equally spaced values of θ1, but now, multiplying these SoPs by the sine or by the cosine of the corresponding angle θ1, different terms in Eq. (6) vanish. By doing this computation, an

NN N N N

r1 r1 r 1 r 1 r 1 NN N N N

r1 r1 r 1 r 1 r 1

NN N N N

r1 r1 r 1 r 1 r 1

0,r 0,r 1,r 0,r 1,r 0,r 1,r 0,r 1,r

S 2 S cos4 4 S cos4 4 S sin4 2 S sin2

S 2 S cos4 4 S cos4 4 S sin4 2 S sin2

1,r 1,r 1,r 1,r 1,r 1,r 1,r 1,r 1,r

3,r 3,r 1,r 3,r 1,r 3,r 1,r 3,r 1,r

S 2 S cos4 4 S cos4 4 S sin4 2 S sin2

N NN

2,r 1,r 2,r 1,r 2,r 1,r r 1 r 1

S cos4 4 S sin4 2 S sin2

r=1 <sup>+</sup> <sup>∑</sup> mk2

N

� � · � 1 1 0 0

� = �

<sup>2</sup> sin 4θ1,r

1 cos<sup>2</sup> 2θ<sup>1</sup> sin 2θ<sup>1</sup> cos 2θ<sup>1</sup> sin 2θ<sup>1</sup>

<sup>2</sup> sin 4θ<sup>1</sup> +mk3 sin 2θ1 (5)

r=1 + ∑ mk3 sin 2θ1,r N

r=1 (6)

(7)

� (4)

depends on the angle 1 (Goldstein, 2003). We obtain the following Stokes vector:

10 0 0 0 cos<sup>2</sup> 2θ<sup>1</sup> sin 2θ<sup>1</sup> cos 2θ<sup>1</sup> - sin 2θ<sup>1</sup> 0 sin 2θ<sup>1</sup> cos 2θ<sup>1</sup> sin2 2θ<sup>1</sup> cos 2θ<sup>1</sup> 0 sin 2θ<sup>1</sup> - cos 2θ<sup>1</sup> <sup>0</sup> �

a general expression for the SoP exiting from the LCoS display is obtained:

<sup>2</sup> � <sup>+</sup> mk1

N

expression for the Mueller matrix of the LCoS display is obtained:

SPSG=

�

∑ Sk�θ1,r� <sup>N</sup>

and with θ1,r= <sup>2</sup>π(r-1)

<sup>1</sup> <sup>M</sup> N

Sk�θ1�<sup>=</sup> �mk0+ mk1

different coefficients of the Mueller matrix.

r=1 <sup>=</sup> <sup>∑</sup> �mk0+ mk1

<sup>N</sup> .

N N

2,r 2,r 1,r r1 r1 r 1

S 2 S cos4 4

spaced values of θ1 we obtain the following expression:

<sup>2</sup> � <sup>N</sup> r=1 <sup>+</sup> <sup>∑</sup> mk1

� �

$$\mathbf{S}\_{\mathbf{ex}} = \mathbf{M} \cdot \mathbf{S}\_{\mathbf{ex}} = \begin{pmatrix} \mathbf{m}\_{00} & \mathbf{m}\_{01} & \mathbf{m}\_{02} & \mathbf{m}\_{03} \\ \mathbf{m}\_{10} & \mathbf{m}\_{11} & \mathbf{m}\_{12} & \mathbf{m}\_{13} \\ \mathbf{m}\_{20} & \mathbf{m}\_{21} & \mathbf{m}\_{22} & \mathbf{m}\_{23} \\ \mathbf{m}\_{03} & \mathbf{m}\_{31} & \mathbf{m}\_{32} & \mathbf{m}\_{33} \end{pmatrix} \mathbf{S}\_{\mathbf{in}} \tag{1}$$

where mj,k (j = 0, 1, 2, 3 and k = 0, 1, 2, 3) are the coefficients of the Mueller matrix.

Due to the linear relation shown in Eq. (1), the Mueller matrix M of a polarizing sample can be completely characterized by implementing a convenient independent linear equation system. In this way, the experimental LCoS display Mueller matrix is obtained by generating different incident SoPs (by means of the PSG in Fig. 1) and measuring their respective reflected ones by using different SoPs analyzers (implemented by means of the PSD in Fig. 1).

To completely characterize the 4x4 coefficients of the Mueller matrix describing the LCoS display, at least 4 independent incident SoPs and 4 independent analyzer SoPs must be used. Different sets of incident and analyzer SoPs are valid to this aim but the quality of the measurements will depend on different factors, such as the condition number (Taylor, 1974) of the matrices composed by the incident SoPs and the analyzer SoPs (De Martino et al., 2007; Peinado et al., 2010) or the number of SoPs selected (Peinado et al., 2010). Here, the calculation of the Mueller matrix is conducted as follows:

The intensity detected behind the PSD in Fig. 1 depends on the Stokes parameters of the light reflected by the LCoS display and on the orientation θ2 of the quarter-waveplate QWP2. It can be written as follows (Lizana et al., 2009):

$$\mathbf{I}(\boldsymbol{\theta}\_{2}) = \frac{1}{2} \left[ \mathbf{S}\_{0} + \mathbf{S}\_{1} \cos^{2}(2\boldsymbol{\theta}\_{2}) + \mathbf{S}\_{2} \sin(2\boldsymbol{\theta}\_{2}) \cos(2\boldsymbol{\theta}\_{2}) \mathbf{S}\_{3} \sin(2\boldsymbol{\theta}\_{2}) \right] \tag{2}$$

where S0, S1, S2 and S3 are the Stokes parameters of the light reflected from the LCoS display. As the intensity in Eq. (2) is a periodical signal with respect to the angle θ2, because it consists of different sinusoidal functions whose arguments are entire multiples of θ2, the method of synchronous detection (Goldstein, 2003) is applied. The synchronous detection represents an estimation of the coefficients of the Fourier series of this function.

By performing a summation of intensities corresponding to N different equally spaced values of θ2, completing a rotation of 360 degrees, some terms of Eq.(2) vanish due to the orthogonal properties of the sinusoidal sampled functions (Lizana et al., 2009). Besides, by performing another summation of different intensities, but now, multiplying the intensities by the sine or by the cosine of the corresponding angle θ2, different terms in Eq. (2) vanish. In this way, the following mathematical expression for the Stokes parameters is obtained (Lizana et al., 2009):

$$
\begin{pmatrix} \mathbb{S}\_{0} \\ \mathbb{S}\_{1} \\ \mathbb{S}\_{2} \\ \mathbb{S}\_{3} \end{pmatrix} = \begin{pmatrix} 2 \cdot \sum\_{r=1}^{\mathrm{N}} \mathrm{I}\left(\frac{n}{2}, \theta\_{2,r}\right) \cdot 4 \cdot \sum\_{r=1}^{\mathrm{N}} \mathrm{I}\left(\frac{n}{2}, \theta\_{2,r}\right) \cos\{4\theta\_{2,r}\} \\\ 8 \cdot \sum\_{r=1}^{\mathrm{N}} \mathrm{I}\left(\frac{n}{2}, \theta\_{2,r}\right) \cos\{4\theta\_{2,r}\} \\\ 8 \cdot \sum\_{r=1}^{\mathrm{N}} \mathrm{I}\left(\frac{n}{2}, \theta\_{2,r}\right) \sin\{4\theta\_{2,r}\} \\\ \cdots \cdot \sum\_{r=1}^{\mathrm{N}} \mathrm{I}\left(\frac{n}{2}, \theta\_{2,r}\right) \sin\{2\theta\_{2,r}\} \end{pmatrix} \tag{3}
$$

where N is the number of selected angles and θ2,r= <sup>2</sup>π(r-1) <sup>N</sup> with r = 1, 2,…, N.

Once the SoP reflected by the LCoS display can be measured as described above, the next step is to fully characterize the LCoS display Mueller matrix. Next, the applied technique is

Due to the linear relation shown in Eq. (1), the Mueller matrix M of a polarizing sample can be completely characterized by implementing a convenient independent linear equation system. In this way, the experimental LCoS display Mueller matrix is obtained by generating different incident SoPs (by means of the PSG in Fig. 1) and measuring their respective reflected ones by using different SoPs analyzers (implemented by means of the

To completely characterize the 4x4 coefficients of the Mueller matrix describing the LCoS display, at least 4 independent incident SoPs and 4 independent analyzer SoPs must be used. Different sets of incident and analyzer SoPs are valid to this aim but the quality of the measurements will depend on different factors, such as the condition number (Taylor, 1974) of the matrices composed by the incident SoPs and the analyzer SoPs (De Martino et al., 2007; Peinado et al., 2010) or the number of SoPs selected (Peinado et al., 2010). Here, the

The intensity detected behind the PSD in Fig. 1 depends on the Stokes parameters of the light reflected by the LCoS display and on the orientation θ2 of the quarter-waveplate QWP2.

where S0, S1, S2 and S3 are the Stokes parameters of the light reflected from the LCoS display. As the intensity in Eq. (2) is a periodical signal with respect to the angle θ2, because it consists of different sinusoidal functions whose arguments are entire multiples of θ2, the method of synchronous detection (Goldstein, 2003) is applied. The synchronous detection

By performing a summation of intensities corresponding to N different equally spaced values of θ2, completing a rotation of 360 degrees, some terms of Eq.(2) vanish due to the orthogonal properties of the sinusoidal sampled functions (Lizana et al., 2009). Besides, by performing another summation of different intensities, but now, multiplying the intensities by the sine or by the cosine of the corresponding angle θ2, different terms in Eq. (2) vanish. In this way, the following mathematical expression for the Stokes parameters is obtained

<sup>2</sup> ,θ2,r� -4· <sup>∑</sup> <sup>I</sup> �

π

π

π

Once the SoP reflected by the LCoS display can be measured as described above, the next step is to fully characterize the LCoS display Mueller matrix. Next, the applied technique is

r=1

<sup>2</sup> ,θ2,r� ·cos�4θ2,r� <sup>N</sup>

<sup>2</sup> ,θ2,r� ·sin�4θ2,r� <sup>N</sup>

<sup>2</sup> ,θ2,r� ·sin�2θ2,r� <sup>N</sup>

r=1 �

π

<sup>2</sup> ,θ2,r� ·cos�4θ2,r� <sup>N</sup>

� � � �

<sup>N</sup> with r = 1, 2,…, N.

(3)

represents an estimation of the coefficients of the Fourier series of this function.

<sup>2</sup> �S0+S1cos<sup>2</sup>�2θ2�+S2sin�2θ2�cos�2θ2�-S3sin�2θ2�� (2)

where mj,k (j = 0, 1, 2, 3 and k = 0, 1, 2, 3) are the coefficients of the Mueller matrix.

m00 m01 m02 m03 m10 m11 m12 m13 m20 m21 m22 m23 m03 m31 m32 m33

� ·Sin (1)

Sex=M·Sex= �

calculation of the Mueller matrix is conducted as follows:

It can be written as follows (Lizana et al., 2009):

<sup>I</sup>�θ2�<sup>=</sup> <sup>1</sup>

(Lizana et al., 2009):

�

S0 S1 S2 S3

� =

�

where N is the number of selected angles and θ2,r= <sup>2</sup>π(r-1)

� � � �

2· ∑ I �

N r=1 π

8· ∑ I �

r=1 8· ∑ I �

r=1 -4· ∑ I �

PSD in Fig. 1).

described. The PSG in Fig. 1 is formed by the linear polarizer LP1, fixed at 0 degrees to the laboratory vertical, followed by the quarter-waveplate QWP1. The general expression for the states of polarization exiting from the PSG can be obtained by multiplying the Stokes vector of linearly polarized light in the horizontal direction (i.e. SP1 = (1, 1, 0, 0)T) by the Mueller matrix of a linear retarder with a retardance equal to degrees and whose orientation depends on the angle 1 (Goldstein, 2003). We obtain the following Stokes vector:

$$\mathbf{S\_{PSG}} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos^2 2\theta\_1 & \sin 2\theta\_1 \cos 2\theta\_1 & -\sin 2\theta\_1 \\ 0 & \sin 2\theta\_1 \cos 2\theta\_1 & \sin^2 2\theta\_1 & \cos 2\theta\_1 \\ 0 & \sin 2\theta\_1 & -\cos 2\theta\_1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ \cos^2 2\theta\_1 \\ \sin 2\theta\_1 \cos 2\theta\_1 \\ \sin 2\theta\_1 \end{pmatrix} \tag{4}$$

where different incident States of Polarization (SoPs) can be generated by varying the orientation angle 1 of the QWP1. By replacing the Stokes vector given in Eq. (4) into Eq. (1), a general expression for the SoP exiting from the LCoS display is obtained:

$$\mathbf{S}\_{\mathbf{k}}(\theta\_1) = \left(\mathbf{m}\_{\mathbf{k}0} + \frac{\mathbf{m}\_{\mathbf{k}1}}{2}\right) + \frac{\mathbf{m}\_{\mathbf{k}1}}{2}\cos 4\theta\_1 + \frac{\mathbf{m}\_{\mathbf{k}2}}{2}\sin 4\theta\_1 + \mathbf{m}\_{\mathbf{k}3}\sin 2\theta\_1 \tag{5}$$

where k (k = 0, 1, 2, 3) indicates the different Stokes parameter and mk,j (j = 0, 1, 2, 3) are the different coefficients of the Mueller matrix.

Equation (5) comprises different sinusoidal functions whose arguments are entire multiples of θ1, and thus, the method of synchronous detection can be applied again. In particular, by performing a summation of different reflected SoPs corresponding to N different equally spaced values of θ1 we obtain the following expression:

$$\sum\_{\mathbf{r}=1}^{N} \sum\_{\mathbf{k}} \mathbf{S}\_{\mathbf{k}} \left( \boldsymbol{\theta}\_{1,\mathbf{r}} \right) = \sum\_{\mathbf{r}=1}^{N} \left( \mathbf{m}\_{\mathbf{k}0} + \frac{\mathbf{m}\_{\mathbf{k}1}}{2} \right) + \sum\_{\mathbf{r}=1}^{N} \frac{\mathbf{m}\_{\mathbf{k}1}}{2} \cos 4\boldsymbol{\theta}\_{1,\mathbf{r}} + \sum\_{\mathbf{r}=1}^{N} \frac{\mathbf{m}\_{\mathbf{k}2}}{2} \sin 4\boldsymbol{\theta}\_{1,\mathbf{r}} + \sum\_{\mathbf{r}=1}^{N} \mathbf{m}\_{\mathbf{k}3} \sin 2\boldsymbol{\theta}\_{1,\mathbf{r}} \tag{6}$$

where N is the number of SoPs, r (r = 1,2,…,N) indicates the specific term in the summation and with θ1,r= <sup>2</sup>π(r-1) <sup>N</sup> .

By performing a summation of different reflected SoPs related to N different equally spaced values of θ1 on a complete rotation of 360 degrees, some terms of Eq. (6) vanish because of the orthogonal properties of the sinusoidal sampled functions (Lizana et al., 2009). Besides, by performing a summation of different reflected SoPs related to N different equally spaced values of θ1, but now, multiplying these SoPs by the sine or by the cosine of the corresponding angle θ1, different terms in Eq. (6) vanish. By doing this computation, an expression for the Mueller matrix of the LCoS display is obtained:

NN N N N 0,r 0,r 1,r 0,r 1,r 0,r 1,r 0,r 1,r r1 r1 r 1 r 1 r 1 NN N N N 1,r 1,r 1,r 1,r 1,r 1,r 1,r 1,r 1,r r1 r1 r 1 r 1 r 1 N N 2,r 2,r 1,r r1 r1 r 1 S 2 S cos4 4 S cos4 4 S sin4 2 S sin2 S 2 S cos4 4 S cos4 4 S sin4 2 S sin2 <sup>1</sup> <sup>M</sup> N S 2 S cos4 4 N NN 2,r 1,r 2,r 1,r 2,r 1,r r 1 r 1 NN N N N 3,r 3,r 1,r 3,r 1,r 3,r 1,r 3,r 1,r r1 r1 r 1 r 1 r 1 S cos4 4 S sin4 2 S sin2 S 2 S cos4 4 S cos4 4 S sin4 2 S sin2 (7)

Study of Liquid Crystal on Silicon Displays for Their Application in Digital Holography 239

conversion of the experimentally obtained LCoS display Mueller matrices to Jones matrices

As stated above, LCoS displays introduce a certain amount of unpolarized light in the reflected beam (Lizana et al., 2008a; Márquez et al., 2008; Wolfe & Chipman, 2006). Since Jones matrices do not describe unpolarized samples, a direct conversion from the experimental Mueller matrices to the Jones matrices can not be directly performed. In this situation, a decomposition of the LCoS display Mueller matrices into different basic Mueller matrices becomes very helpful. There exist different ways to decompose depolarizing Mueller matrices (Ossikovski et al., 2008). All these different decompositions provide significant polarimetric information and they become more or less practical as a function of

Here, the Lu-Chipman decomposition (Goldstein, 2003), which is a natural generalization of the polar decomposition (Lu & Chipman, 1996) to the depolarizing case, is applied. The Lu-Chipman decomposition states that a general Mueller matrix can be decomposed as the product of three basic Mueller matrices: the Mueller matrix of a pure diattenuator MD (an optical element that changes the orthogonal amplitudes unequally), the Mueller matrix of a pure retarder MR (an optical element that introduces a phase-shift between the orthogonal components), and the Mueller matrix of a depolarizer Man optical element that introduces certain amount of unpolarized light. In the case of LCoS displays, diverse polarimetric studies have proved that they are non-diattenuating devices (Lizana et. al, 2008c; Lizana et al., 2009; Márquez et al., 2008) and the corresponding MD matrices can be approximated to the identity matrix. Therefore, when applying the Lu-Chipman polar decomposition to the

T

���1 0��

� refers to the polarizance vector (Goldstein, 2003), and the matrices

T

� (8)

<sup>A</sup>\*� (9)

0�� mR

P��� <sup>Δ</sup> m<sup>Δ</sup>

denoted as m and mR are 3x3 submatrices whose coefficients can be calculated by following

The retarder component MR of the Mueller matrix given in Eq. (8) is a non-absorbing, unitary and fully polarized Mueller matrix. Under these conditions, the equivalent Jones matrix of the retarder can be obtained from the MR component (Moreno et al., 2008). It can be done by following the relationships between the Mueller matrix coefficients and the Jones

Once the equivalent Jones matrices of the TNLCoS are calculated, a methodology for the evaluation of the complex phase modulation is applied. We follow the technique proposed in Ref. (Moreno et al., 2003), being suitable to predict the phase response in LCDs. This technique considers that any non-absorbing reciprocal polarization device can be described by an unimodular unitary Jones matrix (Fernández-Pousa et al., 2000), the TNLCoS displays being a particular case. Then, such devices can be described by the following relation:

where A=ARiAI and B=BRiBI are complex parameters (subscripts R and I indicate the real and imaginary parts) which depend on the applied voltage V and fulfill the condition

A B -B\*

the specific application for which they are required.

MLCoS=MΔMR= � 1 0��

the procedure described in (Goldstein, 2003).

matrix coefficients given in Ref. (Goldstein, 2003).

JLCoS=e-i<sup>β</sup> �

where 0��� = �0, 0,0,0�, P��


� <sup>R</sup> 

LCoS display case, the following relation holds (Moreno et al., 2008):

is applied.

where N is the number of SoPs present in the summation, r (r = 1, 2,…, N) indicate the specific term of the summation and with θ1,r= <sup>2</sup>π(r-1) <sup>N</sup> .

By experimentally measuring the quantities given in Eq. (7), the Mueller matrices of the Twisted Nematic LCoS display for different gray levels (from 0 to 255 in steps of 20 gray levels) have been obtained. The validity of the obtained results is tested by analyzing their capability to predict the SoP of the light reflected by the LCoS display.

As an example, in Fig. 2 we have plotted the Stokes parameters of the reflected SoP as a function of the gray level, when using an incident linear SoP at 135 degrees to the laboratory vertical. The spots represent the experimental data, which is measured by applying the SoP measuring method described above (see Eq. (2)). The continuous lines represent the theoretical values, which are calculated by multiplying the Stokes vector of a linear polarized light at 135 degrees (i.e. S135 = (1, 0, -1, 0)T) by the measured Mueller matrices of the TNLCoS display (see Eq. (1)). In all the cases, there is an excellent agreement between the predicted and the experimentally obtained Stokes parameters. Thus, the results given in Fig. 2 become an evidence of the effectiveness of the measured TNLCoS display Mueller matrices and validates the characterization method proposed.

It should be pointed out that depending on the specific application of the LCoS display, different physical parameters, such as the incident angle or the wavelength, can be required. In such situations, the LCoS display response can be notably different. However, the characterizing method described in this section can be extended to different experimental conditions, such as the incident angle (Lizana et al., 2009) or the wavelength (Lizana et al., 2008c).

Fig. 2. Stokes parameters as a function of the gray level for an incident SoP linear at 135 degrees of the laboratory vertical. Whereas the spots represent the experimental data, the continuous lines represent the theoretical values.

### **2.2 Mueller to Jones conversion: Phase prediction**

In this subsection, a methodology useful to predict the phase response of LCoS displays is proposed. To this aim, since Jones matrices contains the required phase information, a

where N is the number of SoPs present in the summation, r (r = 1, 2,…, N) indicate the

As an example, in Fig. 2 we have plotted the Stokes parameters of the reflected SoP as a function of the gray level, when using an incident linear SoP at 135 degrees to the laboratory vertical. The spots represent the experimental data, which is measured by applying the SoP measuring method described above (see Eq. (2)). The continuous lines represent the theoretical values, which are calculated by multiplying the Stokes vector of a linear polarized light at 135 degrees (i.e. S135 = (1, 0, -1, 0)T) by the measured Mueller matrices of the TNLCoS display (see Eq. (1)). In all the cases, there is an excellent agreement between the predicted and the experimentally obtained Stokes parameters. Thus, the results given in Fig. 2 become an evidence of the effectiveness of the measured TNLCoS display Mueller

It should be pointed out that depending on the specific application of the LCoS display, different physical parameters, such as the incident angle or the wavelength, can be required. In such situations, the LCoS display response can be notably different. However, the characterizing method described in this section can be extended to different experimental conditions, such as the incident angle (Lizana et al., 2009) or the wavelength (Lizana et al.,

Fig. 2. Stokes parameters as a function of the gray level for an incident SoP linear at 135 degrees of the laboratory vertical. Whereas the spots represent the experimental data, the

In this subsection, a methodology useful to predict the phase response of LCoS displays is proposed. To this aim, since Jones matrices contains the required phase information, a

continuous lines represent the theoretical values.

**2.2 Mueller to Jones conversion: Phase prediction** 

<sup>N</sup> . By experimentally measuring the quantities given in Eq. (7), the Mueller matrices of the Twisted Nematic LCoS display for different gray levels (from 0 to 255 in steps of 20 gray levels) have been obtained. The validity of the obtained results is tested by analyzing their

specific term of the summation and with θ1,r= <sup>2</sup>π(r-1)

capability to predict the SoP of the light reflected by the LCoS display.

matrices and validates the characterization method proposed.

2008c).

conversion of the experimentally obtained LCoS display Mueller matrices to Jones matrices is applied.

As stated above, LCoS displays introduce a certain amount of unpolarized light in the reflected beam (Lizana et al., 2008a; Márquez et al., 2008; Wolfe & Chipman, 2006). Since Jones matrices do not describe unpolarized samples, a direct conversion from the experimental Mueller matrices to the Jones matrices can not be directly performed. In this situation, a decomposition of the LCoS display Mueller matrices into different basic Mueller matrices becomes very helpful. There exist different ways to decompose depolarizing Mueller matrices (Ossikovski et al., 2008). All these different decompositions provide significant polarimetric information and they become more or less practical as a function of the specific application for which they are required.

Here, the Lu-Chipman decomposition (Goldstein, 2003), which is a natural generalization of the polar decomposition (Lu & Chipman, 1996) to the depolarizing case, is applied. The Lu-Chipman decomposition states that a general Mueller matrix can be decomposed as the product of three basic Mueller matrices: the Mueller matrix of a pure diattenuator MD (an optical element that changes the orthogonal amplitudes unequally), the Mueller matrix of a pure retarder MR (an optical element that introduces a phase-shift between the orthogonal components), and the Mueller matrix of a depolarizer Man optical element that introduces certain amount of unpolarized light. In the case of LCoS displays, diverse polarimetric studies have proved that they are non-diattenuating devices (Lizana et. al, 2008c; Lizana et al., 2009; Márquez et al., 2008) and the corresponding MD matrices can be approximated to the identity matrix. Therefore, when applying the Lu-Chipman polar decomposition to the LCoS display case, the following relation holds (Moreno et al., 2008):

$$\mathbf{M}\_{\rm LCsS} = \mathbf{M}\_{\Delta} \mathbf{M}\_{\rm R} = \begin{pmatrix} 1 & \vec{0} \\ \vec{\mathbf{P}}\_{\Delta} & \mathbf{m}\_{\Delta} \end{pmatrix} \cdot \begin{pmatrix} 1 & \vec{0} \\ \vec{0} & \mathbf{m}\_{\rm R} \end{pmatrix} \tag{8}$$

where 0��� = �0, 0,0,0�, P�� � refers to the polarizance vector (Goldstein, 2003), and the matrices denoted as m and mR are 3x3 submatrices whose coefficients can be calculated by following the procedure described in (Goldstein, 2003).

The retarder component MR of the Mueller matrix given in Eq. (8) is a non-absorbing, unitary and fully polarized Mueller matrix. Under these conditions, the equivalent Jones matrix of the retarder can be obtained from the MR component (Moreno et al., 2008). It can be done by following the relationships between the Mueller matrix coefficients and the Jones matrix coefficients given in Ref. (Goldstein, 2003).

Once the equivalent Jones matrices of the TNLCoS are calculated, a methodology for the evaluation of the complex phase modulation is applied. We follow the technique proposed in Ref. (Moreno et al., 2003), being suitable to predict the phase response in LCDs. This technique considers that any non-absorbing reciprocal polarization device can be described by an unimodular unitary Jones matrix (Fernández-Pousa et al., 2000), the TNLCoS displays being a particular case. Then, such devices can be described by the following relation:

$$\mathbf{J}\_{\rm LCoS} \triangleq \mathbf{e}^{i\boldsymbol{\beta}} \begin{pmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{-B} & \mathbf{A}^\* \end{pmatrix} \tag{9}$$

where A=ARiAI and B=BRiBI are complex parameters (subscripts R and I indicate the real and imaginary parts) which depend on the applied voltage V and fulfill the condition || || � <sup>R</sup> 

Study of Liquid Crystal on Silicon Displays for Their Application in Digital Holography 241

coefficients enabling us to vary the specific weight of the three parameters involved into the

By maximizing the figure of merit given in Eq. (11), different TNLCoS display configurations have been obtained, providing excellent results for phase-only response

In Fig. 3, the intensity values (left axis) and the phase values (right axis) are represented as a function of the gray level, the former being plotted in pink and the latter plotted in brown. Whereas the predicted values are represented by curves, the measured experimental data are represented by spots. For the measurement of the phase (brown squares), the interferometer based set-up given in Ref. (Lizana et al., 2008b) has been applied. The experimental orientations of external waveplates (WP) and linear polarizers (LP) used to achieve the incident and analyzer SoPs for this optimized configuration are: incident

We see an excellent agreement between experimental and theoretical values, validating the suitability of the employed optimization methodology. Besides, good results for phase-only modulation are achieved. In fact, an almost constant intensity curve as a function of the gray

(LP1=45 degrees, WP1=160 degrees); analyzer (LP2= -41 degrees, WP2=18 degrees).

(Márquez et al., 2008), as for instance, those shown in Fig. 3.

level, with a phase-shift approximately of 300 degrees is obtained.

Fig. 3. TNLCoS display phase only response as a function of the gray level.

Finally, we have allowed the numerical search procedure to decrease the minimum value of transmittance Tm, in order to reach a higher phase-shift. The obtained result is shown in Fig. 4. The experimental orientations of external waveplates (WP) and linear polarizers (LP) are the following: incident beam (LP1=88 degrees, WP1=7 degrees); analyzer (LP2= 90 degrees,

In Fig. 4, we see an excellent result in terms of phase-only regime. In particular, a continuous intensity response as a function of the gray level, which keeps almost constant

optimization process.

WP2= -15 degrees).

From the calculated matrices MR and the transformations given in (Goldstein, 2003), the complex parameters A and B are fully calibrated as a function of the gray level. However, no information about the external phase in Eq. (9) is obtained. The knowledge of the external phase is critical to accurately predict the phase modulation of LCoS displays and so, the application of a technique to determine its value becomes mandatory.

A general expression for the total phase of a LCD inserted between two polarizers (whose orientations are 1 and 2 respectively) is provided in Ref. (Moreno et al., 2003):

$$\delta = \emptyset + \delta\_{\mathsf{M}} = \emptyset + \operatorname{atan}\left(\frac{\mathsf{A}\_{\mathsf{I}}\cos(\theta\_{\mathsf{I}} + \theta\_{\mathsf{Z}}) + \mathsf{B}\_{\mathsf{I}}\sin(\theta\_{\mathsf{I}} + \theta\_{\mathsf{Z}})}{\mathsf{A}\_{\mathsf{K}}\cos(\theta\_{\mathsf{I}} \cdot \theta\_{\mathsf{Z}}) + \mathsf{B}\_{\mathsf{R}}\sin(\theta\_{\mathsf{I}} \cdot \theta\_{\mathsf{Z}})}\right) \tag{10}$$

where M is a phase contribution that depends on the polarizer orientations 1 and 2.

As the coefficients AR, AI, BR and BI are known, the phase M in Eq. (10) can be calculated for a given pair of orientations 1 and 2. Besides, the total phase as a function of the gray level can be experimentally determined, for instance, by using the interferometric based set-up given in Ref. (Lizana et al., 2008b). In this situation, the external phase is readily obtained by isolating it from Eq. (10).

Since the external phase does not depend on the incident and analyzer SoPs, once it is measured, the phase response of LCoS display as a function of the gray level can be completely predicted.

### **2.3 LCoS display optimizing procedure**

In digital holography applications, the diffraction efficiency is maximized when operating in the phase-only modulation regime (Moreno et al., 2004). Therefore, in such applications is very desirable to operate with an LCoS display response achieving a linear phase modulation up to 360 degrees, without coupled depolarization or coupled amplitude modulation.

By following the methods given in subsections 2.1 and 2.2, we are able to predict the intensity and the phase response of LCoS display. By taking advantage of this prediction capability, an optimization procedure can be devised to maximize the efficiency of digital holograms generated with LCoS displays.

Here, we apply an optimizing method, based on a numerical search algorithm, which allows us to achieve pairs of incident and analyzer SoPs that provide an LCoS display working in the phase-only regime. By starting from an initial pair of incident and analyzer SoPs, the numerical search algorithm varies these parameters in order to maximize the figure of merit QPO shown below. The SoP searching process enables to use any type of fully polarized light (i.e. linearly polarized light, circularly polarized light and elliptically polarized light). In this way, better results than those obtained when using only linear polarized light are achieved (Márquez et al., 2008).

The figure of merit QPO is based on a specific criterion to evaluate the quality of the response in the phase-only regime. In particular, it is based on three conditions: minimum transmittance value as high as possible, maximum phase modulation and minimum transmittance contrast.

$$\mathbf{Q\_{PO}} = \frac{1}{\lambda\_1 + \lambda\_2 + \lambda\_3} \left[ \lambda\_1 \frac{\Delta \mathbf{r}}{2\mathbf{n}} + \lambda\_2 \{1 \cdot \Delta \mathbf{T}\} \star \lambda\_3 \mathbf{T\_m} \right] \tag{11}$$

where is the maximum phase difference, T is the maximum transmittance difference and Tm denotes the minimum transmission value. Moreover, the parameters 1, 2 and 3 are

From the calculated matrices MR and the transformations given in (Goldstein, 2003), the complex parameters A and B are fully calibrated as a function of the gray level. However, no information about the external phase in Eq. (9) is obtained. The knowledge of the external phase is critical to accurately predict the phase modulation of LCoS displays and so, the

A general expression for the total phase of a LCD inserted between two polarizers (whose

As the coefficients AR, AI, BR and BI are known, the phase M in Eq. (10) can be calculated for a given pair of orientations 1 and 2. Besides, the total phase as a function of the gray level can be experimentally determined, for instance, by using the interferometric based set-up given in Ref. (Lizana et al., 2008b). In this situation, the external phase is readily obtained

Since the external phase does not depend on the incident and analyzer SoPs, once it is measured, the phase response of LCoS display as a function of the gray level can be

In digital holography applications, the diffraction efficiency is maximized when operating in the phase-only modulation regime (Moreno et al., 2004). Therefore, in such applications is very desirable to operate with an LCoS display response achieving a linear phase modulation up to 360 degrees, without coupled depolarization or coupled amplitude

By following the methods given in subsections 2.1 and 2.2, we are able to predict the intensity and the phase response of LCoS display. By taking advantage of this prediction capability, an optimization procedure can be devised to maximize the efficiency of digital

Here, we apply an optimizing method, based on a numerical search algorithm, which allows us to achieve pairs of incident and analyzer SoPs that provide an LCoS display working in the phase-only regime. By starting from an initial pair of incident and analyzer SoPs, the numerical search algorithm varies these parameters in order to maximize the figure of merit QPO shown below. The SoP searching process enables to use any type of fully polarized light (i.e. linearly polarized light, circularly polarized light and elliptically polarized light). In this way, better results than those obtained when using only linear polarized light are achieved

The figure of merit QPO is based on a specific criterion to evaluate the quality of the response in the phase-only regime. In particular, it is based on three conditions: minimum transmittance value as high as possible, maximum phase modulation and minimum

where is the maximum phase difference, T is the maximum transmittance difference and Tm denotes the minimum transmission value. Moreover, the parameters 1, 2 and 3 are

<sup>2</sup><sup>π</sup> <sup>+</sup>λ2൫1-ΔT൯+λ3Tmቃ (11)

λ1+λ2+λ<sup>3</sup>

ቂλ<sup>1</sup> Δτ

AI cosሺθ1+θ2ሻ+BI sinሺθ1+θ2ሻ AR cos൫θ1-θ2൯+BR sin൫θ1-θ2൯

൰ (10)

application of a technique to determine its value becomes mandatory.

δ=β+δM=β+ atan ൬

by isolating it from Eq. (10).

**2.3 LCoS display optimizing procedure** 

holograms generated with LCoS displays.

QPO= <sup>1</sup>

completely predicted.

(Márquez et al., 2008).

transmittance contrast.

modulation.

orientations are 1 and 2 respectively) is provided in Ref. (Moreno et al., 2003):

where M is a phase contribution that depends on the polarizer orientations 1 and 2.

coefficients enabling us to vary the specific weight of the three parameters involved into the optimization process.

By maximizing the figure of merit given in Eq. (11), different TNLCoS display configurations have been obtained, providing excellent results for phase-only response (Márquez et al., 2008), as for instance, those shown in Fig. 3.

In Fig. 3, the intensity values (left axis) and the phase values (right axis) are represented as a function of the gray level, the former being plotted in pink and the latter plotted in brown. Whereas the predicted values are represented by curves, the measured experimental data are represented by spots. For the measurement of the phase (brown squares), the interferometer based set-up given in Ref. (Lizana et al., 2008b) has been applied. The experimental orientations of external waveplates (WP) and linear polarizers (LP) used to achieve the incident and analyzer SoPs for this optimized configuration are: incident (LP1=45 degrees, WP1=160 degrees); analyzer (LP2= -41 degrees, WP2=18 degrees).

We see an excellent agreement between experimental and theoretical values, validating the suitability of the employed optimization methodology. Besides, good results for phase-only modulation are achieved. In fact, an almost constant intensity curve as a function of the gray level, with a phase-shift approximately of 300 degrees is obtained.

Fig. 3. TNLCoS display phase only response as a function of the gray level.

Finally, we have allowed the numerical search procedure to decrease the minimum value of transmittance Tm, in order to reach a higher phase-shift. The obtained result is shown in Fig. 4. The experimental orientations of external waveplates (WP) and linear polarizers (LP) are the following: incident beam (LP1=88 degrees, WP1=7 degrees); analyzer (LP2= 90 degrees, WP2= -15 degrees).

In Fig. 4, we see an excellent result in terms of phase-only regime. In particular, a continuous intensity response as a function of the gray level, which keeps almost constant

Study of Liquid Crystal on Silicon Displays for Their Application in Digital Holography 243

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

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

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

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

existence of time-fluctuations of the phase phenomenon.

diffracted orders as a function of the time.

level).

around the value 0.65, is achieved. Besides, it is accompanied by a very large phase-shift, close to 360 degrees.

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