## Abstract

A polarimetry based method able to characterize optical properties of linear Liquid Crystal Displays (LCDs), even in presence of time-fluctuations of the phase, is proposed in this work. In particular, mean linear retardance, Liquid Crystal (LC) fast axis orientation and phase fluctuation amplitude of LCDs can be obtained with the proposed alternative technique. This technique enables to achieve these important features of LCDs with a set-up significantly less complicated to build up and with faster measurements than previously proposed techniques, which are based on diffraction or interferometry experiments. The validity of the technique is tested by measuring two different LCDs: one monopixel PA-LC panel working in transmission and a reflective PA-LCoS display. The technique provides similar results than those obtained by using previously proposed methods, confirming the validity of our alternative technique.

© 2013 OSA

## 1. Introduction

Liquid crystals Displays (LCDs) are commonly used in diffractive optics applications due to their great capability to act as Spatial Light Modulators (SLMs) [1,2]. To optimize the performance of LCDs working as SLM, it is crucial to accurately calibrate the phase modulation provided by the device as a function of the gray level addressed to it. To this aim, methods based on interferometric measurements or on diffractive measurements are typically used [3,4]. A different approximation is applied in Ref [5], where a measuring technique based on the well-known linear polarimeter is applied for the measurement of the linear retardance provided by a ferroelectric LCD. In spite of this technique is limited to the measurement of linear retardance in LCDs (e.g. elliptical retardance in twisted nematic LCDs cannot be measured), it can be used to determine important physical parameters of linear LCDs, like the orientation of the liquid crystal director or the phase-shift. Last but not least, this polarimetric method results faster than those based on diffractive or interferometric set-ups (due to the time of acquisition and the data post processing that those techniques require to obtain phase modulation measurements) and its experimental implementation is significantly less complicated to build up.

Recently, Liquid Crystal on Silicon (LCoS) displays, a type of reflective LCDs, are being used in diffractive optics applications due to their high spatial resolution, high light efficiency and very large phase modulation [6,7]. In fact, because of the double pass that light beams produce into LCoS displays, they lead to an increase of the phase modulation when compared to transmissive LCDs of the same thickness. This phase modulation increase is very desirable in diffractive optics applications, where a phase-shift equal or larger than 2π becomes very important to achieve high efficiency for the diffractive optical elements addressed [8].

In spite of this and other appealing properties exhibited by LCoS displays, some authors have shown that due to the electrical addressing schemes applied in these devices, the liquid crystal molecules cannot keep still in a frame period, but fluctuate as a function of time. This fact originates different non-desired phenomena, as an effective depolarization [9,10] or time-fluctuations of phase [11], which depend of the gray level addressed to the modulator and decrease the efficiency of the addressed diffractive optical elements. The effective depolarization phenomena can be avoided by using a particular type of LCoS displays, the Parallel Aligned (PA) LCoS displays, which allow a phase-only modulation without coupled amplitude modulation. However, time-fluctuations of phase are still present on (PA) LCoS displays [12]. To overcome this problem, new LCoS displays optimizing techniques have been proposed, reducing to certain extent the influence of the time-fluctuations of phase on the device efficiency [12,13].

In this framework, to properly evaluate the optical response of LCoS displays, it is essential to use an efficient characterizing technique accounting for phase fluctuations. In fact, the parameter of interest in presence of phase fluctuations is the mean phase given by the SLM for each gray level addressed [11]. Note that this value cannot be measured by the characterizing methods previously stated as they assume a constant phase in time.

A procedure valid to experimentally determine the mean phase modulation of LCoS displays in presence of phase fluctuations is provided in Ref [11]. Two different characterizing set-ups are proposed: a diffraction based set-up and an interferometry based set-up. In both cases, the set-ups are difficult to built and require a large time of acquisition. To find an alternative, faster and easier to assemble in laboratories, the applicability of the linear polarimeter to the measurement of the linear retardance in presence of phase flicker is examined in Ref [14]. It is shown that by using the linear polarimeter for the measure of the linear retardance of (PA) LCoS displays, large errors in the mean phase determination may be obtained. In particular, the larger the amplitude of fluctuation, the higher the error for the mean phase obtained. However, they demonstrate as discrepancies between mean phase values calculated in absence of phase fluctuations and true measured values can be used to accurately estimate the amplitude of the (PA) LCoS flicker. Moreover, they show as once the flicker values are determined, they can be used to determine the correct values for the mean phases.

In this work we go a step further. We propose a polarimetric method, based on linear polarimeter, which is able to obtain the mean linear retardance, the liquid crystal fast axis orientation and the phase fluctuation amplitude directly from measurements, without the necessity of performing any subsequent processing of data. With this aim, we have modified the measuring procedure given in [5] by including as well intensity measurements obtained by illuminating the SLM with circular polarized light. By doing this, we add extra information to the system, allowing to uncouple mean phase contribution and amplitude of the phase fluctuation contributions from intensity measurements.

The technique proposed in this paper can be useful for a large number of researchers because it is based on a simple set-up, which results easier to build up than more complex existing set-ups, as those based in diffractive or interferometric set-ups [11]. In addition, although our proposed technique is not devised to compete in velocity with snapshot characterizing techniques already existing [15,16], it provides different information than such polarimetric approaches, as the amplitude for the time-fluctuations of the phase or the mean phase modulation, in a reasonable short period of time (acquisition time may be decreased by controlling the number of optical configurations used, i.e. by reducing redundant data). Moreover, for the particular case where the orientation of the LCDs director is known (for instance, for commercial LCDs, where this parameter is usually given by manufacturers), the required number of measurements is reduced to four (i.e. four different configurations of the optical elements), and so, the obtaining of the optical properties of LCDs may be achieved even faster.

The outline of this work is as follow. In section 2, we give the mathematical background of the polarimetric method proposed. In section 3, the method is experimentally tested by measuring the mean phase, the flicker amplitude and the liquid crystal director orientation for two different LCDs: one monopixel parallel aligned liquid crystal panel working in transmission and one parallel aligned LCoS display. Finally, the main conclusions of the present work are given in section 4.

## 2. Theory

As said at the introduction, the polarimetric method proposed in this paper is valid to determine the mean phase modulation, the liquid crystal molecules orientation and the amplitude of the phase fluctuations of a linear LCD. In the following, the mathematical description of such polarimetric method is described.

Let us assume a linear waveplate set at a given angle *α* with respect to the vertical of the laboratory and sandwiched between two linear polarizers (labeled as LP_{1} and LP_{2}) with their transmission angles parallel one to each other (i.e. *θ _{1}* =

*θ*=

_{2}*θ*). In such case, if we illuminate the system with a monochromatic light source with a linear state of polarization, the light exiting from the LP

_{2}can be written, in the Mueller-Stokes mathematical formalism [17], as follows:

*M*

_{LP2}is the Mueller matrix of the polarizer LP

_{2}and

*M*is the Mueller matrix of the waveplate set an angle

_{WP}*α*and with a retardance of Δ.

The first quantity of a Stokes vector (S_{0} parameter) describes the total intensity of the light beam. Thus, by solving Eq. (1) we obtain the intensity at the exit of the system, which is written as:

Similarly, if the two linear polarizers are crossed (i.e. *θ*_{2} = *θ* + 90°), the intensity exiting from the second linear polarizer is:

Afterwards, let us consider the same system, but now, instead of illuminate it with a linear state of polarization, we use a circular state of polarization (i.e. S_{1} = S_{2} = 0). In such case, the light exiting from the LP_{2} can be written as:

Therefore, the intensity exiting from the system will be, for an incident right-handed circular polarization (i.e. S_{3} = 1) and an incident left-handed circular polarization (S_{3} = −1) respectively:

The Eqs. (2), (3), (5) and (6) are valid for a constant value of the waveplate retardance, Δ. However, the instantaneous phase value *δ* of the LCD may fluctuate as a function of time. In such case, these relations must take into account the time dependence of *δ*, and are rewritten as:

*δ*which fluctuates as a function of the time by following a triangle function of period T, around a mean value $\overline{\Delta}$and with a difference between maximum and minimum values equal to 2a (see Fig. 1 ).

In this situation, the time averaging of the cosine is:

*β*= sin(

*a*)/

*a*.

Similarly,

By replacing Eqs. (12) and (13) into Eq. (7), (8), (9) and (10), we obtain:

Let us assume that the linear polarizers are ideal, therefore we make the constants *A* = *B* = ½. Finally, we obtain two relations for the normalized intensities:

From the above relations, we observe that the LCDs can be directly characterized from intensity measurements at the exit of the LP_{2}, obtained by using different configurations of the system (i.e. different values of *θ*) and two different illuminations (i.e. linear and circular polarized light). In particular, the mean phase Δ, the fluctuation ratio *β*, and the orientation of the liquid crystal fast axis *α* are obtained by using the least square minimization in the following form:

*m*different orientations

*θ*of the polarizer, error

_{L}and error

_{C}are the figures of merit for the linear and circular polarization based measurements. Note that by increasing the number of equally spaced orientations, the error associated to the determination of the parameters may be decreased. Finally, by minimizing the addition of error

_{L}and error

_{C}figures of merit, we can obtain estimations for the values of

*β*, Δ and

*α*. We have used the minimization functions of Matlab software to obtain the best fitting parameters.

If the orientation of the liquid crystal fast axis *α* is known (for instance, by means of a previous calibration process, or by the data provided by the manufacturer), further characterizations of the LCDs can be obtained by significantly reducing the number of intensity measurements. In particular, by knowing the value of *α* we can choose the orientation of the polarizer leading to *θ* −*α* = 45°. In this way, relations given above (Eq. (20) and (21)) are rewritten as:

By combining Eq. (22) and (23) we obtain:

Consequently, the mean phase Δ and the fluctuation ratio *β* are fully characterized just by performing four intensity measurements related to the polarizer configuration *θ* = 45° + *α* (i.e. two intensity measurements corresponding to the linear polarization and to the circular polarization illuminations, and two extra intensity measurements for the required intensity normalizations).

## 3. Experimental results

In this section, we experimentally test the polarimetric technique explained in section 2. With this aim, we have applied the proposed technique to the measurement of the optical response of two different LCDs. The first one is a monopixel Parallel Aligned (PA) liquid crystal panel working in transmission and which does not show significant phase flickering. The second one is a reflective parallel aligned LCoS display in which time-fluctuations of the phase are present.

#### 3.1. Monopixel (PA) liquid crystal panel

We have implemented an experimental set-up able to ensure the physical conditions detailed in section 2. For the measurement of the transmissive Parallel Aligned (PA) liquid crystal panel, the set-up used is sketched in Fig. 2
. In Fig. 2(a), we sketch the set-up, based on the linear polarimeter, which has been used to register intensity measurements at the exit of the (PA) liquid crystal panel when illuminated with linear polarized light. As a light source, a He-Ne laser (*λ* = 632.8 nm) with a maximum power of 30 mW and distributed by Melles Griot (05-LHP-991 model) is used. Because of the light exiting from the laser is linearly polarized, we have placed a quarter waveplate after it. By setting the proper orientation of the quarter waveplate, circular polarized light is illuminating the Linear Polarizer 1 (LP_{1}), ensuring an almost constant intensity at the exit of the LP_{1}, independently of its orientation. Although an ideal circular polarization cannot be experimentally implemented, small intensity differences at the light exiting from LP_{1} are corrected by the subsequent intensity normalization.

The (PA) liquid crystal panel is sandwiched between two linear polarizers (labeled as LP_{1} and LP_{2} respectively). The orientation of both linear polarizers is electronically controlled by means of two mechanical motors distributed by Thorlabs (PRM1Z8 model). The PA liquid crystal panel was built by using a liquid crystal cell with an active area of 3x3 cm^{2}. The cell is a flat structure sandwiched between two flat glasses with a thickness of 0.7 mm. The glasses are coated in their internal face with a transparent semiconductor (Indium Tin Oxide, ITO) which acts as electrode. The liquid crystal molecules are parallel aligned. When a voltage is applied between electrodes, a tilt in the liquid crystal director is induced but its orientation remains constant. This induced tilt tends to align liquid crystal director parallel to direction of propagation of the electric field, i.e. perpendicular to the glass surfaces. Different voltages provide different tilts of the liquid crystal molecules, and so, different linear retardances can be obtained. Finally, a radiometer (Optical Power Meter model 1830-C, distributed by Newport) is placed at the exit of the LP_{2} to obtain the required intensity measurements.

By using the set-up given in Fig. 2(a), we take *n* intensity measurements corresponding to different and equally spaced orientations *θ _{n}* of the transmission axes of the LP

_{1}and LP

_{2}polarizers, when they are parallel aligned. To normalize intensity data, we also perform intensity measurements for different orientations

*θ*of the LP

_{n}_{1}and LP

_{2}transmission axes, when they are crossed one to each other (i.e. Eq. (18) is obtained).

To illuminate the (PA) liquid crystal panel with circular polarized light, a quarter waveplate is inserted on the previous set-up, between the parallel aligned liquid crystal panel and the LP_{1} (see Fig. 2(b)). By orienting the fast axis at 45° (135°) of the LP_{1} transmission axis, right-handed (left-handed) circular polarized light is generated. Then, intensity measurements for different orientations *θ _{n}* of the of the LP

_{2}transmission axis, when the (PA) liquid crystal panel is illuminated by using circular right-handed polarized light are obtained. Finally, to normalize the obtained data, intensity measurements for the same orientations of the polarizer LP

_{2}, but now by illuminating the device with left-handed circular polarized light are recorded as well, then the normalized intensities related to Eq. (19) are obtained. Once the series of intensity measurement are collected, they are used in the least square minimization procedure based on relations Eqs. (20) and (21). In this way, the characterization of the (PA) liquid crystal panel is achieved.

Nevertheless, as said in section 2, for the particular case where the liquid crystal orientation axis is known, the number of required measurements is reduced to four.

The results obtained for the mean phase Δ and the fluctuation ratio *β*, as a function of the voltage, are given in Figs. 3
and 4
respectively. In addition, the value obtained for the orientation of the liquid crystal fast axis is of $<\alpha >=-5.40$degrees, which corresponds to the average of the *α _{n}* values obtained for the different voltages addressed.

From Fig. 3 we see that the monopixel parallel aligned liquid crystal panel analyzed provides a phase-shift of about 400 degrees, being a very large modulation for the applied wavelength (632.8 nm). Moreover, there is a very good agreement between the experimental data and the fitted data. From Fig. 4, we see as the contrast ratio *β* is almost one for all the voltages addressed, showing that the time-fluctuations of the phase phenomena is not significant for this transmissive device.

#### 3.2. Reflective PA-LCoS display

In this section, the same experimental characterization procedure applied in section 3.1. is used to obtain the optical response of a reflective LCD. The LCD under analysis is a parallel aligned LCoS display distributed by HoloEye, with an active matrix reflective mode phase-only LCD with 1920x1080 resolution and 0.7” diagonal named the PLUTO Spatial Light Modulator (SLM). The pixel pitch is of 8.0 μm and the display has a fill factor of 87%. The signal is addressed via a standard DVI (Digital Visual Interface) signal. By means of the RS-232 interface and its corresponding provided software, we can perform gamma control to configure the modulator for different applications and wavelengths. Besides, different pulsed width modulation (PWM) addressing schemes (addressing sequences) can be generated by the driver electronics [18]. In particular, we use an addressing sequence provided by HoloEye and labeled as “5-5 2π linear 633 nm”. Whereas the first five denotes the quantity of “equally weighted” bit-planes, the second five indicates de quantity of “binary” bit-planes [18]. This addressing sequence is optimized to 633 nm and so, a phase modulation of about 2π for this wavelength is expected.

Since Parallel Aligned (PA) LCoS displays work in reflection, the set-up shown in Fig. 2. is adapted to perform measurements in reflection. Such modified set-up is sketched in Fig. 5
. In particular, a beam-splitter cube is inserted between the LP_{1} and the (PA) LCoS display, sending the light reflected by the (PA) LCoS to 90 degrees of the incident beam. Even beam splitters may introduce some small polarimetric variation in the optical response of the LCoS display analyzed, this configuration is preferred than oblique incidence, as the global phase retardation of LCoS displays decreases as the incident angle increases [19]. In this situation, the optical parameters given by our experimental technique will be related to the optical system formed by the (PA) LCoS display plus beam splitter.

The results obtained for the sequence “5-5 2π linear 633 nm” in terms of mean phase Δ and fluctuation ratio *β*, as a function of the gray level, are given in Fig. 6
and 7
respectively. Figure 6 shows as our (PA) LCoS display provides a phase modulation about 350 degrees. This result was expected, as sequence “5-5 2π linear 633 nm” was designed to provide a phase modulation of 2π with a linear relation between phase and gray level addressed to the LCoS display.

Figure 7 gives the results obtained for the fluctuation ratio *β* showing as our (PA) LCoS display introduces non negligible values of this parameter. Whereas for low and high values of the gray level addressed to the (PA) LCoS display the fluctuation ratio is close to one (i.e. very small amplitude for the phase fluctuations are present), a significant decrease of the fluctuation ratio is obtained around the 175 gray level, with a minimum value close to 0.6. Finally, the mean orientation of the fast axis with respect to the laboratory vertical obtained for this device is of $<\alpha >=-8.61$degrees.

We want to emphasize that results obtained for our parallel aligned LCoS display operating with the “5-5 2pi linear 633 nm” addressing sequence, are similar to those presented in Ref [12,14]. In this way, the capability of our alternative technique to characterize optical parameters of LCoS displays in presence of phase fluctuations is provided.

## 4. Conclusions

In this paper, we propose a new polarimetric method, based on linear polarimeter, which is able to obtain important optical features of linear LCDs in presence of time-fluctuations of the phase. In particular, the mean phase modulation, the liquid crystal director orientation and the amplitude of the phase fluctuation present in the device can be determined by applying our technique.

Unlike other existing polarimetric techniques, our method is able to fully characterize all these significant LCD parameters directly from intensity measurements, without the necessity of performing any subsequent processing of data.

The technique is tested by experimentally determining the mean phase-modulation, the fast axis orientation and the fluctuation ratio of two different LCDs: one monopixel parallel aligned liquid crystal panel working in transmission and which does not show significant phase flickering and a reflective parallel aligned LCoS display, in which time-fluctuations of the phase phenomena is significant.

We have obtained similar results than those obtained by using existing techniques, confirming the validity of our proposed alternative technique.

## Acknowledgments

We acknowledge financial support from the Spanish Ministerio de Educación y Ciencia and funds from FEDER (FIS 2009-13955-C02-01).

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