## Abstract

In this paper we experimentally analyze the performance of a twisted nematic liquid crystal on silicon (LCoS) display as a function of the angle of incidence of the incoming beam. These are reflective displays that can be configured to produce amplitude or phase modulation by properly aligning external polarization elements. But we demonstrate that the incident angle plays an important role in the selection of the polarization configuration. We performed a Mueller matrix polarimetric analysis of the display that demonstrates that the recently reported depolarization effect observed in this type of displays is also dependant on the incident angle.

©2009 Optical Society of America

## 1. Introduction

The capability of the liquid crystal displays (LCD) to work as spatial light modulators (SLM) have caused a widespread use of these devices in optical applications such as diffractive optics [1], holographic data storage [2], optical metrology [3], or in programmable adaptive optics [4]. As a consequence, optimizing the LCDs response has become a challenge to many authors, and ways to obtain a desired intensity and phase response have been thoroughly studied [5,6]. A type of LCD used in numerous optical applications is the Liquid Crystal on Silicon (LCoS) display. These devices are LCDs that work in reflection, giving high phase modulation. However, a certain amount of unpolarized light has been detected at the LCoS displays reflected beam [7-9]. The origin of this depolarization effect was investigated in [9], showing that it is related to temporal fluctuations of the liquid crystal orientation caused by the electrical signal addressed to the display. This depolarization effect can adversely affect applications, as for instance in diffractive optics where it was demonstrated to reduce the diffraction efficiency [10].

Because of this depolarization effect, the Mueller-Stokes (M-S) formalism has been adopted for LCoS displays, and a protocol to optimize the intensity [7] and phase [11] modulation responses has been developed. By extension, this protocol is valid to characterize any polarizing or depolarizing optical element. In [12] we showed that the intensity, phase and degree of polarization of the light beam modulated at the LCoS display have a strong dependence with the wavelength. In this work we study the modulation performance as a function of another parameter: the angle of incidence. For that purpose we have performed a complete polarimetric characterization of the LCoS display for different angles of incidence. We show how the angle of incidence plays an important role and strongly affects the modulation response. Then, we show that optimized phase modulation can be obtained for the different angles, but the polarization configuration must be readjusted. This study can be especially relevant for LCoS displays applications involving high numerical apertures, where a wide range of incident angles are used, as for instance in optical trappings set-ups [13].

The outline of the paper is as follows. In Section 2, the experimental method and the setup used to characterize the LCoS display are described. In Section 3, the results of the polarimetric analysis of the display are presented. In particular, the degree of polarization, diattenuation, polarizance and retardance parameters are thoroughly studied as a function of the addressed gray level and as a function of the incident angle. In Section 4, the phase and intensity modulation provided by the LCoS display is analyzed as a function of the incident angle, and optimized configurations are demonstrated for the different angular positions. The results are compared with those obtained with normal incidence and employing a beam splitter. Finally, the conclusions are presented in Section 5.

## 2. Experimental characterization based on a synchronous method

In this work we have characterized the Mueller matrix of an LCoS display as a function of the addressed gray level for five different incident angles: α=2°, 12.5°, 23°, 34° and 45°. The characterization of the LCoS display has been performed by using a modification of the methodology described at [7]. The proposed procedure is based on the method of synchronous detection [14] and it is valid to characterize the Mueller matrix of any polarizing or depolarizing optical element, being the LCoS display a particular case.

The experimental set-up used to implement this method is shown in Fig. 1. We use a 633 nm He-Ne laser and the LCoS display under analysis is a Philips model X97c3A0, sold as the kit LC-R2500 by Holoeye. The LC-R2500 is a 2.46 cm diagonal reflective LCoS display of the 45° twisted nematic type, with XGA resolution (1024 × 768 pixels), with digital data input and digitally controlled gray scales with 256 gray levels. The pixels are square with a center to center separation of 19 μm and an excellent fill factor of 93%. The LCoS is placed on the top of a rotating platform that enables choosing the incident angle with a precision of 1°. We have set a polarization state generator (PSG) at the incident beam and a polarization state detector (PSD) at the reflected beam. The PSG is composed by a polarizer and an achromatic quarter wave plate and the PSD is composed by an achromatic quarter wave plate and an analyzer. Both polarizer and analyzer are fixed at 0°, considered parallel to the laboratory vertical direction, and both wave plates can be electronically rotated by 360°.

It is well known that the Mueller matrix (M) of an optical polarizing element relates the input and output states of polarization (SoPs), described by the Stokes vectors S_{input} and S_{output}. By generating different input SoPs and measuring its corresponding output SoPs (using simply intensity measurements), it is possible to construct an independent equations system from which the whole Mueller matrix can be derived, as it was done in [7]. In this work, we alternatively measure the SoPs reflected from the LCoS display by using the method of synchronous detection [14]. The analyzer LP_{2} is fixed at 0°. Then, the intensity behind the PSD is function of the Stokes parameters of the reflected beam, and of the phase-shift (ϕ) and orientation (θ_{2}) of the waveplate WP_{2}. As a particular case, when using a quarter wave plate (ϕ=π/2), the intensity as a function of the angle θ_{2} can be written as follows:

where *S*
_{0}, *S*
_{1}, *S*
_{2} and *S*
_{3} are the Stokes parameters of the light reflected from the LCoS display.

The intensity in Eq. (1) is a periodical signal with respect to the angle θ_{2} since it contains several sinusoidal functions whose arguments are entire multiples of θ_{2}. The synchronous detection consists on the evaluation of the coefficients of the Fourier series of this function. By performing a summation of intensities corresponding to *N* different equidistant values of θ_{2}, completing a rotation of 360°, some terms of Eq. (1) vanish due to the orthogonal properties of the sinusoidal sampled functions. In particular the following relations hold:

where *N* is the number of samples and *δ _{ij}* the Kronecker delta. Therefore, using these relations it is possible to describe the reflected SoP as a function of summations of intensity measurements obtained for the different equidistant analyzer angles θ

_{2}as:

where *N* is the number of selected angles θ_{2} and θ_{2,r}=*r*2π/*N*. On the other hand, keeping the PSG polarizer fixed at 0°, the SoPs impinging the LCoS display only depend on the WP_{1} rotation angle θ_{1}. The incident SoP can be expressed as:

Then, Stokes parameters of the corresponding reflected beam can be written as:

where *k* = 0, 1, 2, 3, and *m _{k,j}* (

*j*= 0,1,2,3) are the different elements of the Mueller matrix. Next, taking into account the orthogonal properties of the sinusoidal sampled functions (Eqs. (2)), and performing a summation for different output SoPs (corresponding to equidistant angles θ

_{1}from a complete rotation of WP1) in Eq. (5), it is possible to obtain all the LCoS Mueller matrix coefficients as a function of output SoPs summations as:

where *N* is the number of selected equidistant angles θ_{1}. For every value θ_{1,r} the corresponding Stokes parameters *S ^{r}_{k}* are measured according to Eq. (3).

The advantage of this characterization procedure with respect to that used in [7] is that here some redundant information is employed, which result in a reduction of the experimental measurement error. In this work, these polarimetric measurements were acquired with steps of 51.4° (on both angles θ_{1} and θ_{2}), and the whole system was automated by means of rotation motorized devices with a precision of 0.1°.

## 3. LCoS display as a function of the incident angle: polarimetric analysis

In [7] we presented a rigorous study of the polarimetric properties of a twisted nematic LCoS display, illuminated with a 633 nm laser beam at quasi-normal incidence. Here, we extend that study to different incident angles in order to analyze its influence on the LCoS display performance. In particular, we have analyzed the degree of polarization, diattenuation, polarizance and retardance dependence with respect to the incident angle.

#### 3.1 Degree of polarization as a function of the incident angle

As mentioned previously, the LCoS display has been shown to produce a reduction in the degree of polarization (DOP) that depends on the gray level and on the input SoP [7-9,12]. Figure 2 shows the measured DOP as a function of the gray level for various angles of incidence (α=2°, 12.5°, 23°, 34° and 45°), calculated from the experimentally measured Stokes parameters as $\mathrm{DOP}=\sqrt{{S}_{1}^{2}+{S}_{2}^{2}+{S}_{3}^{2}}/{S}_{0}$ [14]. By definition, the DOP takes values from 0 to 1 but in Fig. 2, the y axis has been zoomed in order to show the results more clearly. The results correspond to three input SoPs: linear polarized light at 0°, linear polarized light at 135° and left-handed circular polarized light. Figure 2 shows some relevant information about the DOP dependence with the incident angle. Note that some DOP values are slightly higher than 1, as a consequence of the instrumental error associated to the intensity measurements in Eq. (3) and its corresponding error propagation. From Fig. 2, we see that for all the selected incident angles, the DOP depends on the input SoP. For a fixed input SoP, there is a quite relevant difference in the DOP evolution as a function of the gray level. For quasi-normal incidence (Fig. 2(a)), the reflected light remains fully polarized (DOP close to one) for gray level ranges below 100 or above 240. However, important depolarization effects are detected for gray levels in between 100 and 240, reaching depolarization values higher than 10%. This depolarization effect is related to SoP time fluctuations originated from the electrical signal addressing of the LCoS display [9]. For low gray levels, the DOP remains close to one (Fig. 2(a), gray levels below 100) because the liquid crystal molecules are oriented basically parallel to the glass windows and their orientation is not so sensitive to fluctuations in the electrical signal. On the contrary, for gray levels above 240 (Fig. 2(a)), the LC molecules are almost completely tilted parallel to the electric field direction, despite the fluctuations in the electrical signal, and the reflected beam also remains fully polarized. However, for gray levels in between 100 and 240, the liquid crystal molecules tilt has an intermediate value, and the optical modulation is very sensitive to the fluctuations of the electrical signal, resulting in the highest depolarization effect for gray level 180.

When increasing the incident angle (Figs. 2(b)-2(e)), we detect unpolarized light in the gray level range around gray level 180 (as in the quasi-normal incidence case shown in Fig. 2(a)), but also for higher or lower gray level ranges, where the depolarization increases as the incident angle increases. For instance, for incident angles α=12.5° and α=23° (Figs. 2(b) and 2(c)), depolarization overpass 5%, while it is greater than 10% for incident angles of α=34° and α=45° (Figs. 2(d) and 2(e)). For high incident angles and for some input SoPs, depolarization reaches approximately a 10% along the whole gray level range (black triangles at Fig 2(d) and Fig 2(e)).

We want to emphasize that part of the depolarized light detected around the 180 gray level and at oblique incidence can be attributed to the signal fluctuations discussed before. However, there is another depolarization source when the LCoS display is used at high oblique incidences. The amount of depolarization is higher when increasing the incident angle and it is not only caused by the fluctuations phenomena. In order to prove this last statement, we have measured the DOP corresponding to different input SoPs and with the LCoS display switched off (no voltage addressed). This measurement has been performed with the incident angles of to 2° and 45°. The results are shown in Fig. 3.

When no voltage is addressed to the LCoS display, the light reflected by the device is almost fully polarized, for an incident angle equal to 2° and for all the tested incident SoPs (Fig. 3(a)). Unlikely, for an incident angle equal to 45°, the DOP strongly depends on the incident SoP although no voltage is addressed to the LCoS display (Fig. 3(b)). In fact, using linear polarized light at 0° of the lab vertical the DOP is almost one but when using an input linear polarized light at 135° or left-handed circular light, we reach values close to 10% of depolarization. Therefore, Fig. 3 proves that we identified a new depolarization source which is not originated by the fluctuations in the electrical signal addressed to the LCoS display (which are the cause of the effective depolarization effect previously reported [9]). Moreover, this new depolarization source is not simply a constant offset equally added to the effective depolarization effect. A constant offset would mean that the DOP should be, along the whole gray level range, equal or lower than the DOP measured with the LCoS switched off. We see for example in the case of left-handed circular light and α=45° incident angle (squares in Fig. 2(e)) that the DOP is bigger than the value 0.9, measured in the off-state (Fig. 3(b)), for most of the gray level range. Therefore, we conclude that the new depolarization probably depends on the optical director distribution in the LC layer (which changes with the addressed voltage). It would be necessary further experiments to get a tighter grip on which is the origin of the new depolarization source detected. A list of possible depolarization sources are described in [8,15].

#### 3.2. Diattenuation, polarizance and retardation dependence with the incident angle

Using the synchronous method described in Section 2, we have measured the experimental Mueller matrix of the LCoS display for the incident angles of α=2°, 12.5°, 23°, 34° and 45°. The experimentally measured Mueller matrices provide the polarimetric information of the analyzed LCoS display. On one hand, the first row of the Mueller matrix is related to the diattenuation vector, which gives the dependence of the transmittance with the incident SoP [16]. On the other hand, the first column of the Mueller matrix is related to the polarizance vector, which indicates the capability of the polarization element to polarize a fully unpolarized beam [16]. Finally, the bottom right 3×3 submatrix gives the information about the retardance and depolarization of the optical polarization element.

Figure 4 shows the first row (Fig. 4(a)) and the first column (Fig. 4(b)) coefficients of the obtained experimental Mueller matrices, as a function of the gray level and for an incident angle α=2°. The corresponding equivalent results are plotted in Figs. 4(c) and 4(d) for an incident angle α=45°.

All these Mueller matrix coefficients have values very close to zero (except m_{00} which is equal to one). Similar results are obtained for the other measured angles of incidence. As these coefficients remain null as a function of the gray level, the LCoS display can be regarded as a non-diattenuating and non-polarizing polarization device independently of the chosen incident angle.

Next, we have analyzed the coefficients of the bottom right 3×3 submatrix, which provide the retardance and depolarization information. As an example, Fig. 5 shows a comparison, for the different incident angles, of the evolution with gray level of the m_{21}, m_{22} and m_{23} coefficients of the experimental Mueller matrices. The large coefficients modulation shown in Fig. 5 has been observed also for all the 3×3 submatrix coefficients. Note that the coefficient evolution as a function of the gray level, shown at Fig. 5, varies gradually when increasing the incident angle, finally leading to large variations between the results for quasi-normal incidence (Fig. 5(a)) in comparison with the incidence at α=45° (Fig. 5(e)). This result indicates that the retardance and depolarization effects will have a relevant dependence on the operating incident angle.

In order to extract more information from the Mueller matrices, we have used the combined method exposed in [11], where the Lu-Chipman polar decomposition [14] (based on the polar decomposition theorem [17]) is applied to the LCoS Mueller matrix. In this sense, the Mueller matrix of any depolarizing element can be expressed as the product of three Mueller matrices: the depolarizer, the retarder and the diattenuator matrices. By taking into account the results shown in Fig. 4, the diattenuation matrix can be approximated to the identity in all cases. Therefore, it is possible to write the Mueller matrix of the LCoS display just as the product of the depolarizer and the retarder matrices. Finally, from the retarder matrix, we are able to find the Jones matrix of the equivalent retarder [11]. Next, we focus on the analysis of the Jones matrix for the equivalent retarder.

Non-absorbing reciprocal polarization devices in reflection are theoretically equivalent to a linear retarder [18]. Then, if we concentrate on the equivalent retarder Jones matrix for the LCoS we may consider that under normal incidence (forward and backward path after reflection are the same) the LCoS can be expected to act as linear retarder, whose neutral lines orientation and retardance depend on the addressed voltage. However, when increasing the incident angle, the LCoS may act as an elliptical retarder since the forward and backward paths in the LC layer are no longer coincident. In order to evaluate this effect, we have calculated the eigenvectors and eigenvalues of the equivalent retarder Jones matrix, and the eigenvectors orientation and ellipticity is derived as a function of the gray level. Note that the eigenvectors indicate the neutral lines in a linear retarder and the phase-shift between eigenvalues gives the retardance. Figure 6 shows the retardance as a function of the gray level for all the incident angles used along the experience. The minimum phase value corresponds to the gray level 0 and an incident angle α=2° (rhombus spots), whereas the maximum phase is obtained for the gray level 240 and α=45° (circular spots). These results show that small incident angles (2°-12.5°) show a higher phase-shift dynamic range than high incident angles (34°-45°). This fact should be taken into account when searching configurations of maximum phase modulation, as we show in Section 4.

Next, the equivalent retarder eigenvectors are represented in Fig. 7, as a function of the gray level, and for quasi-normal incidence (α=2°). They remain almost linearly polarized in the whole range of gray levels, and their orientation rotates counter-clockwise as the gray level increases. Therefore, we can consider the LCoS display at quasi-normal incidence as a linear retarder whose retardation (Fig. 6) and neutral lines orientation (Fig. 7) change with gray level, in agreement with [7]. Moreover, as expected in retarders, the eigenvectors are orthogonal to each other at every gray level, pointing out that the LCoS display is a homogeneous element [19].

Finally, in Fig. 8, we represent the eigenvectors as a function of the gray level for the other incident angles α=12.5°, 23°, 34° and 45°. Now their ellipticity increases as gray level increases, being this effect stronger as the incident angle increases. Thus these result show that, by increasing the incident angle and the gray level, the LCoS displays becomes an elliptical retarder [20] or even a circular retarder for the particular case of gray level 240 and incident angle α=45°. Let us note that transmissive twisted nematic liquid crystal displays have, in general, two eigenvectors that are elliptically polarized [6], while reflective twisted nematic displays operating at perfectly normal incidence act as an equivalent linear retarder, thus having linear eigenvectors [21]. The result in Fig. 7 for quasi-normal incidence verifies this situation, but the results in Fig. 8 evidence that the equivalent linear retarder behavior is lost when the angle of incidence increases.

## 4. LCoS response optimization results: comparison between different incident angles

The performance of LCoS displays in optical applications require the adjustment of optimal PSG and PSD configurations, providing specific intensity and phase modulation regime. Typically the two desired modulation regimes are: maximum intensity contrast modulation with constant phase, or constant amplitude modulation with maximum phase modulation. These intensity or phase regimes can be achieved by using elliptically polarization PSG and PSD configurations [22]. In our previous works we have demonstrated a useful methodology suitable to optimize the intensity response [7] or the phase response [11] of a LCoS display. In fact, in [11] a PSG and PSD configuration giving high phase modulation and also a constant intensity response was obtained for a 633 nm wavelength, and for the incident angle of α=2°.

Here we study how the optimized modulation results are affected by increasing the incident angle. We have measured the intensity and phase of the LCoS response when setting the optimized configuration for phase modulation used in [11]. The experimental modulation results are shown in Fig. 9, where an optimized configuration designed for phase-only modulation regime at the incident angle α=2° is then tested at higher angles α=12.5° and α=45°. The lines represent the theoretical simulations calculated using the combined formalism described in [11], being the continuous line the simulated intensity (left axis) and the dotted line the simulated phase (right axis). The symbols represent the experimental values and they have been measured following the techniques described in [7]. The black circles represent the experimental intensity and the squares the experimental phase.

Figure 9 shows, in all cases, a great agreement between simulated and experimental values. Figure 9(a) evidence a phase-only modulation response of the LCoS display, where a phase modulation up to almost 360° is accompanied with a constant intensity modulation. Figure 9(b) shows that the phase-only modulation response is only slightly modified at low incident angles (α=12.5°), but Fig. 9(c) shows that the phase-only performance is lost for α=45°. Figure 9(c) shows significantly lower values of phase-shift and noticeable coupled amplitude. This result indicates that the optimization performed for a given angle of incidence works well within a reduced incident angle range. Out of this range the response is rather different and good modulation results require applying the optimization technique to obtain the specific optimal PSG and PSD configurations. As an example, Fig. 10 shows the modulation results for two phase modulation configurations and for two incident angles (α=12.5° and α=45°). The PSG and the PSD optimized values are P_{1}=95°, WP_{1}=85°, P_{2}=61° and WP_{2}=64° for the α=12.5° incident angle and P_{1}=91°, WP_{1}=104°, P_{2}=54° and WP_{2}=53° for the α=45° incident angle.

On one hand, Fig. 10(a) shows a very constant intensity response (black line and circles) as a function of the gray level. Moreover, we obtain almost 2π phase-shift (dotted line and squares). Therefore, the modulation response optimization with a=12.5° provides similar results to those obtained with α=2° (Fig. 9(a)). Thus, for a small range of incident angles (around 10°) a single optimization is enough. On the other hand, Fig. 10(b) (α=45°) gives also a constant intensity response but the phase-shift is significantly shorter (only slightly over 240°) than the obtained at quasi-normal incidence (Fig. 9(a)). Then, even by optimizing the LCoS display phase modulation response at high incident angles, the results remain worse than the obtained at low incident angles (Fig. 9(a) and Fig. 10(a)). This result is in agreement with Fig. 6, where the phase-shift between the equivalent retarder eigenvectors is higher for quasi-normal incidence than for oblique incidence.

As we have shown above, by using high incident angles the obtained phase modulation response is remarkably lower than by using quasi-normal incidence. However, there are some optical applications where a right angle between the incident and reflected beam is required, and a high phase-shift is also desired to achieve good diffraction efficiency [13, 23]. Normal incidence may be achieved in this case using a beam splitter, in a set-up like shown in Fig. 11(a), where the PSG and the PSD are located before and behind the beam splitter. While the beam splitter permits to build this compact setup, it presents the disadvantage of loosing light power (half power is lost on every pass) and eventually may introduce additional polarization effects that must be taken into account. By following the procedure previously discussed, the whole system composed of the beam splitter and the LCoS display has been characterized as a polarization device, and its phase modulation response has been optimized. The results are shown at Fig. 11(b). It shows modulation results very similar to those obtained with quasi-normal incidence (Fig. 9(a)), with a very small reduction in the phase-shift, caused by the retardance introduced by the beam splitter. However, in order to work at 90° between the incident and the reflected beams, while maintaining a high phase-shift response, the beam splitter option is recommended.

## 5. Conclusion

Summarizing, in this work we provide a study of the performance of an LCoS display as a function of the incident angle. We analyzed how optimized phase modulation configurations employing elliptically polarized light respond when changing the incidence. Here we presented results on the polarimetric properties of an LCoS display, illuminated by 633 nm wavelength laser. The experimental measurements presented evidence that a previously reported effective depolarization effect, which shows dependence on the addressed gray level and on the input SoP, also presents an important dependence on the incident angle. Moreover, we detected an additional source of depolarization, not related to the fluctuations of the electrical signal, and which is more significant at high incident angles. In addition, the polarimetric study revealed that LCoS display acts as a non diattenuating and non polarizing element for every tested incident angle. On the contrary, we have observed a strong relation between the LCoS display retardation and the incident angle. Moreover, the experimental measurements show that LCoS display becomes an elliptical retarder when increasing the incident angle. The retardance dependence with the incident angle has an important effect at the LCoS display phase modulation response. We proved that incident angle deviations less than 10° do not modify substantially the modulation properties. However, we showed that a fixed configuration of polarizers and waveplates giving very good phase response at normal incidence shows a degraded phase-only modulation response (reduced phase-shift and couple amplitude modulation) when increasing the incident angle.

We have also proved that optimized phase-only modulation configurations can be achieved for every incident angle, although the optimization procedure must be applied in each case. However, we have obtained less phase modulation depth in the phase-only modulation configurations as the incident angle increases. Finally, in order to retain a good phase modulation depth in a setup with perpendicular incident and reflected beams, we included a beam splitter. The system composed of the beam splitter and LCoS display has been characterized as a single polarization modulator, and the optimization process led to a phase-only configuration providing results almost equivalent to those obtained with quasi-normal incidence, in spite of the retardance introduced by the beam splitter. Therefore, when good phase modulation is required, simultaneously with perpendicular incident and reflected beams, the use of a beam splitter is recommended.

All these results are relevant since LCoS displays are becoming a device useful for a number of optical applications, and care must be taken when selecting the incident angle. In addition, these effects may be relevant when employing the device illuminated with a wide range of incident angles, as it is the case for instance in optical trapping systems.

## Acknowledgments

We acknowledge financial support from Spanish Ministerio de Educación y Ciencia (FIS2006-13037-C02-01 and 02) and Generalitat de Catalunya (2006PIV00011). C. Iemmi acknowledges support from Univ. Buenos Aires and CONICET (Argentina).

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