Analysis of multiple internal reflections in a parallel aligned liquid crystal on silicon SLM

Multiple internal reflection effects on the optical modulation of a commercial reflective parallel-aligned liquid-crystal on silicon (PAL-LCoS) spatial light modulator (SLM) are analyzed. The display is illuminated with different wavelengths and different angles of incidence. Non-negligible Fabry-Perot (FP) effect is observed due to the sandwiched LC layer structure. A simplified physical model that quantitatively accounts for the observed phenomena is proposed. It is shown how the expected pure phase modulation response is substantially modified in the following aspects: 1) a coupled amplitude modulation, 2) a non-linear behavior of the phase modulation, 3) some amount of unmodulated light, and 4) a reduction of the effective phase modulation as the angle of incidence increases. Finally, it is shown that multiple reflections can be useful since the effect of a displayed diffraction grating is doubled on a beam that is reflected twice through the LC layer, thus rendering gratings with doubled phase modulation depth. ©2014 Optical Society of America OCIS codes: (120.2040) Displays; (060.5060) Phase modulation; (120.2230) Fabry-Perot. References and links 1. G. Lazarev, A. Hermerschmidt, S. Krüger, and S. Osten, “LCOS spatial light modulators: Trends and applications,” in W. Osten, Optical Imaging and Metrology (Edt. Wiley-VCH, 2012), Chap. 1. 2. R. Carey, P. A. Gago-Sandoval, D. M. Newman, and B. W. J. Thomas, “Programmable liquid crystal waveplates in ellipsometric measurements,” Meas. Sci. Technol. 7(4), 505–510 (1996). 3. A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable femtosecond pulse shaping by use of a multielement liquid-crystal phase modulator,” Opt. Lett. 15(6), 326–328 (1990). 4. G. Lester, A. Strudwick, and S. Coulston, “Electronically switchable diffractive optical elements,” OptoElectron. Rev. 12, 313–316 (2004). 5. L. Lobato, A. Márquez, A. Lizana, I. Moreno, C. Iemmi, and J. Campos, “Characterization of a parallel aligned liquid cristal on silicon and its application on a Shack-Hartmann sensor,” Proc. SPIE 7797, 77970Q (2010). 6. J. A. Davis, J. B. Chambers, B. A. Slovick, and I. Moreno, “Wavelength-dependent diffraction patterns from a liquid crystal display,” Appl. Opt. 47(24), 4375–4380 (2008). 7. S. Reichelt, “Spatially resolved phase-response calibration of liquid-crystal-based spatial light modulators,” Appl. Opt. 52(12), 2610–2618 (2013). 8. D. Engström, M. Persson, J. Bengtsson, and M. Goksör, “Calibration of spatial light modulators suffering from spatially varying phase response,” Opt. Express 21(13), 16086–16103 (2013). 9. C. Lingel, T. Haist, and W. Osten, “Optimizing the diffraction efficiency of SLM-based holography with respect to the fringing field effect,” Appl. Opt. 52(28), 6877–6883 (2013). 10. I. Moreno, A. Lizana, A. Márquez, C. Iemmi, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express 16(21), 16711–16722 (2008). 11. Y. Bao, A. Sneh, K. Hsu, K. M. Johnson, J.-Y. Liu, C. M. Miller, Y. Monta, and M. McLain, “High-speed liquid crystal fiber Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(9), 1190–1192 (1996). 12. Y.-K. Jang and P. Bos, “Analysis of the multireflection effects in compensated liquid crystal devices,” J. Appl. Phys. 101(3), 033131 (2007). #217087 $15.00 USD Received 15 Jul 2014; revised 16 Sep 2014; accepted 17 Sep 2014; published 14 Oct 2014 (C) 2014 OSA 20 October 2014 | Vol. 22, No. 21 | DOI:10.1364/OE.22.025866 | OPTICS EXPRESS 25866 13. A. Vargas, M. M. Sánchez-López, P. García-Martínez, J. Arias, and I. Moreno, “Highly accurate spectral retardance characterization of a liquid crystal retarder including Fabry-Perot interference effects,” J. Appl. Phys. 115(3), 033101 (2014). 14. J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38(6), 1051–1057 (1999). 15. F. Gires and P. Tournois, “Interféromètre utilisable pour la compression d’impulsions lumineuses modulées en fréquence,” C. R. Acad. Sci. Paris 258, 6112–6115 (1964). 16. A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Broadband suppression of the zero diffraction order of an SLM using its extended phase modulation range,” Opt. Express 22(14), 17590–17599 (2014). 17. V. Calero, P. García-Martínez, J. Albero, M. M. Sánchez-López, and I. Moreno, “Liquid crystal spatial light modulator with very large phase modulation operating in high harmonic orders,” Opt. Lett. 38(22), 4663–4666 (2013). 18. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley, 1999). 19. K. Lu and B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29(3), 240–246 (1990). 20. A. Márquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays,” Opt. Eng. 40(11), 2558–2564 (2001). 21. M. L. Álvarez, A. Márquez, L. A. Puerta, R. Estévez, E. Fernández, A. Beléndez, and I. Pascual, “Characterization of a low resolution liquid crystal display to be used as an optical modulator,” J. Opt. A.: Pure Appl. Opt. 40, 95–103 (2007). 22. H. Fujiwara, Espectroscopic Ellipsometry, (John Wiley, 2007). 23. A. Lizana, N. Martín, M. Estapé, E. Fernández, I. Moreno, A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Influence of the incident angle in the performance of Liquid Crystal on Silicon displays,” Opt. Express 17(10), 8491–8505 (2009). 24. F. J. Martínez, A. Márquez, S. Gallego, M. Ortuño, J. Francés, A. Beléndez, and I. Pascual, “Averaged Stokes polarimetry applied to evaluate retardance and flicker in PA-LCoS devices,” Opt. Express 22(12), 15064–15074 (2014). 25. I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. J. Yzuel, “Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display,” Appl. Opt. 43(34), 6278–6284 (2004). 26. J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51(2), 111–115 (2013).


Introduction
Parallel-aligned liquid-crystal spatial light modulators (PAL-LC-SLM) may be ideally regarded as pixelated linear retarders with tunable retardance, useful for applications where pure phase modulation is a key feature [1].These include phase or amplitude control in ellipsometric systems [2], shaping of femto-second pulses [3], generation of diffractive optical elements [4] and wavefront sensors [5], among others.PAL-SLMs ideally offer pure phase modulation when illuminated with light that is linearly polarized parallel to the LC director.They are usually designed to produce at least 2π phase modulation at the operating wavelength.Their phase response versus addressed gray level is sometimes linearized at the electronic driver level.In addition, either normal or small oblique angles of incidence are usually recommended by manufacturers for reflective liquid crystal on silicon (LCoS) devices.
In practice, however, the response of PAL-SLMs as ideal programmable retarders is deteriorated by several secondary effects.First, the pixelated structure introduces diffraction losses [6].In LCoS devices these losses have been reduced thanks to their better fill factor, since pixelated electrodes are required only on one side of the panel.However, LCoS-SLMs are affected by other drawbacks like non-uniformities in the backplane [7,8], and fringing field effects, which become important due to their increased spatial resolution [9].Temporal phase fluctuations or flickering due to digital addressing schemes might also be present in many LCoS devices [10].All these effects deteriorate their effective modulation, and they must be considered if optimal diffraction efficiency and accurate measurements are required.
In this work we analyze another effect that might substantially modify the optical modulation: Fabry-Perot (FP) multiple-beam interference.Although FP effects have found uses in LC devices to build tunable filters [11], they are usually secondary effects that degrade their performance [12].When the LC-SLM is illuminated with light of a broadband spectrum, FP effects can be easily noticed as rapid oscillations in the transmitted spectrum [13].Using monochromatic light, FP effects become also apparent as intensity oscilations versus applied voltage when the input beam is linearly polarized along the LC director [14].
In all the aforementioned works, the devices were transmissive PAL-LC modulators illuminated under normal incidence.However, LCoS devices operate in reflection, and therefore the FP interference effect adopts the Gires-Tournois interferometer configuration [15], with a first low reflective layer, and a second highly reflective layer.
LCoS devices are sometimes illuminated under angles of incidence other than normal incidence.In this situation the FP interference condition is substantially affected.In addition, achieving phase modulation regimes substantially higher than 2π is interesting to produce unusual chromatic applications [16].Since SLMs with such a very large phase modulation are not available, we recently proposed the use of visible light with a PAL-LCoS designed to be operated with infrared light (IR) [17].In such cases, operating far from the design conditions, multiple reflection interference becomes even more relevant.
Therefore, the goal of this paper is to analyze FP interference effects in LCoS devices, and their implications in their phase-only modulation, including changing the angle of incidence.The observed effects are: 1) undesired coupled amplitude modulation, 2) a non-linear phase modulation, 3) a fraction of unmodulated light, and 4) a phase depth reduction as the angle of incidence increases.We propose a simple physical model that quantitatively accounts for all the observed effects with a reduced set of optical parameters.
In addition, the diffraction efficiency losses due to FP effects are analyzed.We will check that, for small diameter beams illuminating the device under a large-enough angle of incidence, multiple reflected beams can be spatially separated.In this situation, it will be shown how the effect of a displayed diffraction grating is doubled on a beam that has been reflected twice through the LC layer.We will show that this effect can render doubled phase modulation diffractive gratings.
The paper structure is the following: In section 2 the LCoS device is presented and interference effects are experimentally characterized for different wavelengths and angles of incidence.In section 3, the FP simplified physical model is presented.In section 4 the theoretical predictions are compared to the experimental modulation curves in order to obtain the parameters that fit the model.Section 5 includes an analysis of the phase-modulation characteristics, and section 6 includes experimental diffraction results with binary and blazed gratings are displayed on the device.The consequences of FP effects on the diffraction efficiency are discussed.The provided experimental results show that the beam reflected twice by the LC layer presents a double diffraction grating effect in comparison to the main first reflection.Finally, in section 7 conclusions of the work are presented.

Experimental evidence of multiple reflections
A commercial LCOS device was used in this work.It is a PAL-LCoS-SLM model X10468-08 from Hamamatsu, with 792 × 600 pixels, 20 × 20 μm 2 pixel size and video-rate operation (60 Hz).Although it was designed to work in the 1000 nm to 1500 nm range, we use visible light in order to achieve a very large phase modulation.Such large phase modulation is of interest to produce diffractive elements with wavelength compensated efficiency [16] or wavelength compensated dispersion [17].Nevertheless, similar FP phenomena as presented here have been observed in other devices from the same series designed to operate with visible light.
Figure 1 shows the layer structure of the LCoS device described by the manufacturer.It consists on the following series of dielectric and conductor layers:   Figure 2(a) depicts the illumination scheme suggested by the manufacturer.The device is illuminated by a horizontally linearly polarized TM beam with angle of incidence θ 0 .The linear polarization plane coincides with the LC director axis.Thus, extraordinary waves are excited inside the LC medium.At normal incidence, multiple reflected beams overlap.However, if the beam width is narrow enough and the angle of incidence is large enough, multiple reflected beams can be spatially separated.Figure 2(b) shows evidence of these multiple reflections.Here a non-expanded laser beam illuminates the LCoS screen with θ 0 = 45°.The reflected beam is captured with a CCD camera.Three main reflections, denoted as I 1 , I 2 and I 3 , are clearly observed (the CCD image is shown in the inset of Fig. 2(b)).The first reflection (I 1 ) is generated on the front surface of the protective glass.The most intense central reflection (I 2 ) is the main beam, which is reflected at the back surface, thus traversing the LC layer.Finally, the third reflected beam (I 3 ) is generated after a reflection inside the protective glass, and another reflection on the LC layer.Lines indicating these beams are drawn on the photograph in Fig. 2(b) to illustrate them.
The intensity of these reflected beams was measured versus the addressed gray level.Three wavelengths from an Ar-Kr laser were used: λ = 488 nm, λ = 568 nm, and λ = 647 nm.Two different angles of incidence were selected.In Figs.3(a), 3(b), and 3(c) the angle was small, approximately θ 0 = 15°.Here, although the three reflected beams had a slight lateral separation, it was difficult to spatially filter them.In consequence, they were measured together and the measurements represent the addition of the intensities of the three reflected beams.On the contrary, Figs.3(d), 3(e) and 3(f) correspond to θ 0 = 45°, where the three beams have enough lateral separation to be spatially filtered, and be measured independently.Reflected Intensity (a.u.)These figures highlight a second multiple reflection effect originated at the LC layer.The observed characteristic oscillation in the output intensity versus the gray level denotes the presence of FP interference, similar to that observed in [14] for a transmission display under normal incidence.Since the LC layer is very narrow, multiple reflected beams in this layer overlap and interference cannot be avoided.

Gray level
Two interesting effects are noticeable in Fig. 3. First, the number of oscillations decreases as the wavelength increases.For θ 0 = 15°, more than 4.5 oscillations are observed for 488 nm (Fig. 3(a)), 3.5 oscillations for 568 nm (Fig. 3(b)), and 3 oscillations for 647 nm (Fig. 3(c)).This is in agreement with the smaller phase obtained for larger wavelengths.Second, for θ 0 = 45°, the number of oscillations for the I 2 main reflected beam is reduced for the three wavelengths.Now only 3 oscillations are observed for 488 nm (Fig. 3(d)), 2.5 oscillations for 568 nm (Fig. 3(e)), and 2 oscillations for 647 nm (Fig. 3(f)).The first reflected beam (I 1 ) shows a constant intensity insensitive to changes in the gray level, thus confirming that it is generated at the most external air-glass reflection.The third reflection (I 3 ) is less intense than the second one (I 2 ), but oscillates in the same manner.
In order to explain this modulation behaviour, a simple physical model that considers a three layer reflective structure is proposed.We will show that such a model quantitatively describes the observed phenomena, with a reduced set of optical parameters.The solid lines in Fig. 3 show the predictions of the proposed model, which will be explained next.

Simplified model of the LC-device layer structure
As expected, the main FP effect is produced at the birefringent LC-layer, where the optical path lengths are modulated by the driving voltage.Therefore, intermediate electrode and substrate layers will be discarded in the simple model proposed here.Therefore, we will consider the LCoS device as a three layer medium with the following layers: 1.A thick layer of protective glass (with refractive index n 1 ), 2. A thin layer of liquid crystal (with refractive index n 2 ), 3. A highly reflective aluminum layer (with refractive index n 3 ), while the external medium (referred to as 0) is air.
Figure 4(a) depicts the outline of the principal reflected beams (I 1 , I 2 and I 3 ).The incident beam, with intensity I i , impinges onto the LCoS device at an angle of incidence θ 0 .The first beam is reflected on the air-glass interface (point P 1 in Fig. 4(a)), with intensity I 1 .The transmitted beam, with a refraction angle θ 1 , propagates through the glass and reaches the LC layer (point Q 1 ).Here a beam is reflected after traversing the thin LC layer.This reflection is affected by FP interference, which depends on the addressed gray level.The reflected beam reaches the glass-air interface (point P 2 ), and produces the second beam (main modulated beam), of intensity I 2 .But another beam is reflected back at point P 2 in this interface and, after a second reflection through the LC layer (point Q 2 ), ends up producing the third reflected beam (point P 3 ), of intensity I 3 .Note that, from Fig. 4(a), the lateral displacement, 2Δx, between consecutive reflected beams can be calculated through geometrical considerations to be: where d g denotes the width of the glass layer.Therefore, the condition to avoid overlapping of these multiple reflected beams (I 1 , I 2 and I 3 ) is: that being w the waist of the laser beam.Thus, the following limit to w can be derived to obtain spatially separated beams can be retrieved: For instance, for θ 0 = 45°, and assuming d g = 1 mm and n 1 = 1.5, Eq. ( 3) indicates that the laser beam on the LCoS screen must have a width w < 0.7 mm, which is in good agreement with the observed experiment.Figure 4(b) illustrates in more detail the interference that occurs at the LC layer (i.e. the reflection at point Q 1 in Fig. 4(a)).Optical modulation is produced here by realignment of the LC director.Let us point out that since the LC layer is an anisotropic medium, the observed FP effects are noticeable only for the polarization component parallel to the LC director, which excites the extraordinary wave inside such medium.Therefore, refraction inside the LC layer does not follow Snell's law, and formulations for analyzing reflection at anisotropic layers [18] are required.However, as we shall show next, a good approximation of the observed FP effect can be obtained by considering the LC media as isotropic with a refractive index n 2 that changes under the action of the applied voltage (it corresponds to the effective extraordinary index).Since the LC layer is so thin, deviations from Snell's law are expected to have relatively small impact.Note that such an approximation has been already successfully applied in many other physical models of LC devices [19][20][21].
The overall reflection coefficient of the LC three layer structure (i.e. the effective reflection coefficient occurring at Q 1 in Fig. 4(a)) reads [22]: ) where ρ 12 is the reflection coefficient from glass to LC, ρ 23 is the reflection coefficient from LC to the reflective layer, and ΔL is the optical path difference between trajectories ABC and EC in Fig. 4(b).The related phase shift k 0 ΔL is given by: where k 0 is the wave number, λ 0 is the vacuum wavelength, n 2 is the LC layer refractive index and d is its thickness.If the LC layer were isotropic, from the geometry of Fig. 4(b), the optical path difference would be ΔL = n 2 ABC−n 1 EC = 2dn 2 cosθ t , being θ t , the transmitted angle inside the LC layer.However, since it is an anisotropic layer, a parameter t≤1 is used instead, with no direct relation to the angle θ t .This parameter will be fitted to the experimental data.Nevertheless, note that a relevant dependence of the phase shift in Eq. ( 5) on the angle of incidence θ 0 arises through this parameter.For normal incidence t can be assumed to be equal to one, and to significantly reduce its value as θ 0 increases.

Modulation characterization and fitting results
In order to characterize the observed FP effects, and obtain the model parameters, we evaluate the data shown in Figs.3(d)-3(f), corresponding to θ 0 = 45°.These curves correspond to the three following paths, shown in Fig. 4(a): Beam 1: First reflection at the air-glass interface whose intensity is expected to be a constant value given by: where ρ 01 is the reflection coefficient at the air-glass interface, which we assume to be a realvalued number.Beam 2: Transmission at the air-glass interface, reflection at the LC layer and transmission from glass to air.The beam intensity is given by: where the overall LC layer reflection coefficient ρ FP was given in Eq. ( 4).Beam 3: Transmission at the air-glass interface, reflection at the LC layer, reflection at the glass-air interface, a second reflection at the LC layer, and transmission through interface glass-air.Its corresponding intensity is expected to be Note that the above relations imply no interference between the first, second and third reflected beams.This can be assumed in our system as long as the laser beam is maintained with a small diameter and the incidence is not selected normal.
Using Eqs. ( 3)-( 5), 2 01 ρ and 2 ρ FP are calculated for each addressed gray level as follows: and ( ) Let us remark that with this procedure the determination of ρ 01 and ρ FP is independent of the incident beam intensity I i and relies entirely on the intensity measurement of the three consecutive reflected beams (I 1 , I 2 and I 3 ).Moreover, isolating the FP interference term at the LC layer from undesired terms avoids changing the measurement conditions.The results derived from the experimental data in Figs.  4) and (5).
The model parameters are: a) Reflection coefficients ρ 12 and ρ 23 , which are assumed to be independent of θ 0 , but dependent on the wavelength.As initial search values in the fitting procedure, values close to 0.1 and 0.9 were selected for ρ 12 , and ρ 23 , respectively.b) Parameter t, which is assumed to be wavelength independent, but depends on θ 0 ; t≅1 is expected for small θ 0 , and smaller values are expected as θ 0 increases.
c) The variation with the addressed gray level of the phase shift k 0 ΔL given in Eq. ( 5).For convenience, this phase shift will be expressed as: ( ) where g∈[0,255] denotes the addressed gray level, β 0 is the phase shift for g = 0, and Δβ(g) represents the birefringence variation with g.In general, LC displays provide a phase modulation with a non linear relation with g.This reduces the efficiency of displayed diffractive elements, and compensation at a software level is required to linearize the phase levels.However, some manufacturers, as it is here the case, provide compensation at the hardware level, so the relation between phase levels and addressed gray levels is linear.This compensation is valid for normal incidence.Therefore, Δβ(g) in Eq. ( 11) can be assumed as:  ( ) where Δβ max is the maximum phase change due to the LC birefringence variation.The maximum phase modulation the LCoS device can provide is regarded to be 2Δβ max .Therefore, the remaining two parameters required to fit the physical model are β 0 and Δβ max , which both depend on the wavelength and take larger values for shorter wavelengths.It is very important to note that, while in an isotropic medium β 0 and Δβ max would not depend on the angle of incidence θ 0 , they do depend on it in an anisotropic medium since they are directly related to the effective extraordinary index (which is not constant with the angle of incidence).Thus, we expect to obtain close (but not the same) values for these parameters at different angles of incidence.For the sake of simplicity, we will keep Δβ max independent of the angle of incidence, but small variations on β 0 will be allowed.
Therefore, the experimental curves in Figs.5(a), 5(b) and 5(c) corresponding to θ 0 = 45°, were fitted to numerical calculations based on Eqs. ( 4), ( 5), (11), and (12), by adjusting parameters ρ 12 , ρ 23 , t, β 0 and Δβ max with the above mentioned constrains.The results of this fitting process are shown in Fig. 5 as solid curves that overlap the experimental data.Excellent agreement is found in all cases.Table 1 shows the fitting parameters retrieved for each wavelength.As expected, values for ρ 12 are much lower than ρ 23 in all cases.The birefringence Δβ max diminishes with increasing λ and t = 0.683.
Next, the same set of parameters are used in Eqs. ( 3), (4), and (5) to match the intensity of the three reflected beams measured at θ 0 = 45° which are shown in Figs.3(d), 3(e), and 3(f).The fitting results are depicted as solid lines in these figures which, again, keep an excellent agreement with the experimental data.Finally, the same values are applied to predict the modulation for θ 0 = 15° observed in Figs.3(a), 3(b) and 3(c).In these predictions, t = 1 was selected and simply β 0 was readjusted to provide the best fit.The newly retrieved β 0 values, also displayed in Table 1, are slightly different from the values at θ 0 = 45°.The numerical simulation, included in Figs.3(a), 3(b) and 3(c) as solid lines, accounts excellently for the measured data, thus confirming the predictions of the proposed physical model.A quantitative estimation of this agreement is given in Table 2.The mean square error is calculated for the of incidence of 15° and 45°, between each experimental data and its corresponding simulated data, and for all three measured wavelengths.For θ 0 = 15°, the error is calculared for the curves in Figs.3(a), 3(b) and 3(c), while for θ 0 = 45°, it is calculated only for the I 2 curve, since it is the only one that shows the oscillatory behavior.As expected the error is very low for the incidence of 45°, since it is being used to fit the model parameters, and it is greater for incidence of 15°, although providing rather good agreement.

Phase modulation characteristics
Once the physical model has been set, it can be used to predict the phase modulation characteristics of the device.Note that the phase modulation Δφ(g) is given by the phase difference between the complex coefficient ρ FP in Eq. ( 1) for different gray levels, i.e., ( ) At normal incidence, and ignoring FP interference effects, the maximum phase modulation can be approached to be equal to the maximum birefringence variation, i.e., Δφ(g) ≅2Δβ(g).However, phase modulation significantly reduces when the angle of incidence increases up to 45°.
These results confirm that the optical path difference ΔL in Eq. ( 2) depends on parameter t.The reduction in parameter t from t = 1 at normal and small incidence (θ 0 = 15°), to t = 0.683 at θ 0 = 45° explains phase modulation depth reduction that is observed for greater incidence angles.This is a relevant conclusion of our work.This phase modulation reduction with θ 0 was already noticed in [23] for a twisted nematic LCoS display, and recently in [24] for a PAL-LCoS display.The model here developed provides a physical explanation.
To better understand the physical insights of this phenomena, we can assume that ρ 12 <<ρ 23 and ρ 12 ρ 23 <<1 to approximate the complex modulation in Eq. (4) as: Since t is related to θ 0 , it can be understood that phase modulation should directly depend on θ 0 .If the LC layer were isotropic, then t = cosθ t , and the transmitted (θ t ) and incident (θ 0 ) angles would be directly linked through Snell's law.Therefore they would increase or decrease together.The maximum phase modulation 2Δβ max t should get its maximum value, 2Δβ max , at normal incidence where θ t = θ 0 = 0° and t = 1, and it should decrease with larger angles of incidence.Since the LC layer is anisotropic, an exact description of the t factor still needs to be performed.Nevertheless, the experimental results shown here indicate that maximum phase modulation is produced at normal incidence whereas it is reduced by a factor t<1 if greater angles of incidence are used.

Effects on phase-only diffraction gratings
In this final section, we incorporate the above described multiple reflection effects in the modulation diffraction efficiency of the display, and analyze how they affect encoded phaseonly diffraction gratings.Results with binary-phase gratings and with continuous blazedphase gratings are discussed.For these results, angle of incidence θ 0 = 45°, wavelength λ = 568 nm and the main modulated reflected beam (I 2 ) are only considered.Experiments have been conducted with other wavelengths and normal incidence in [17].
As it has been shown, interference effects produce non-desired behaviour as coupled amplitude modulation and phase non-linearities, which reduce the diffraction efficiency.After [25], the following general expression can be derived for the amplitude of the m th diffraction order generated by a displayed linear phase-only diffraction grating: where P is the period of the grating, N is the number of steps in each period, w is the width of each step, so Nw = P, and ρ FP (g n ) is the constant reflection coefficient defined in Eq. ( 4), at level n, which is addressed by gray level g n .The diffraction efficiency at each diffraction order m can be calculated as η m = |c m | 2 Special situations can be obtained with binary-phase gratings (N = 2) and blazed gratings, which allow to easily identify the maximum phase level achieved in phase regimes [26].A binary phase grating produces a diffraction pattern where the zero (DC) order is cancelled whenever the phase difference between the two phase levels is π, 3π, 5π, ... On the contrary, if the phase difference is 0, 2π, 4π,..., there is not an effective phase difference, and no diffraction orders arise except the zero order.
For blazed gratings, light gets progressively diffracted to higher diffraction orders as the maximum phase modulation (Δφ max ) increases.If Δφ max = 2π, all the energy is diffracted to the first diffraction order; iIf Δφ max = 4π, then all light is diffracted onto the second diffraction order, etc.On the contrary, if Δφ max = π, most of the energy is split between the zero and first diffracted orders, which have equal intensity.For Δφ max = 3π, most of the energy is now split between the first and second diffraction orders again with the same intensity [26].
In Fig. 7 the diffraction efficiency of the second reflected beam (I 2 ) is simulated for λ = 568 nm and θ 0 = 45°, taking into account the coupled amplitude modulation (I 2 curve in Figs.3(e)) and the effective phase modulation Δφ (Figs. 6(b), solid curve).In Fig. 7(a), the efficiencies at the zero and first diffracted orders, η 0 = |c 0 | 2 and η 1 = |c 1 | 2 , are calculated for a binary phase grating as one gray level is increased while the other is set to g = 0.An effective reduction of the overall maximum efficiency due to losses induced by the ρ FP magnitude is observed.Nevertheless, whenever Δφ reaches an odd integer value of π, the zero order is cancelled and an approximate 40.5% efficiency for η 1 relative to the maximum value η 0 is reached.Note the non-symmetric features of the efficiency curves, caused by the nonlinearities in the phase modulation.
Figure 7(b) depicts the 0 th , 1 st and 2 nd diffraction order intensities for a blazed grating, with linear gray levels increasing from zero to a maximum value g, and diffraction efficiencies are evaluated as a function of g.Therefore, the maximum phase increases from zero up to a maximum value Δφ(g).In this case, the intensity maxima do not reach 100% and the minima do not extinguish for phase values integer multiples of 2π, as it is expected for an ideal blazed grating.This is a consequence of the amplitude modulation and the phase nonlinearities introduced by the interference effect.Nevertheless, it can be appreciated that most of the energy is diffracted onto the first diffraction order when the maximum phase reaches 2π (red curves), and onto the second diffracted order where it reaches 4π (green curves).Figure 8 shows CCD images of the binary grating experimental diffracted orders for the three main reflected beams, corresponding to λ = 568 nm.The period was chosen to 50 pixels and the orientation of the grating was selected to diffract in a direction perpendicular to the multiple reflected beams.This rather large period helps to avoid other secondary effects that affect the diffraction efficiency, such as fringing effect [9].
Figure 8(a) displays the three reflected beams when a uniform screen is addressed to the SLM.The position of the beams (I 1 , I 2 and I 3 ) is labeled on the picture.The rest of the figures show images for selected phase values every π/2 radians.Note that the first reflected beam (I 1 ) is completely unmodulated.Since it is generated by reflection at the outer surface, it is not modulated by the liquid crystal layer, and no diffraction grating is acting on it.The second beam (I 2 ) is the main modulated beam, and follows the diffraction efficiency shown in Fig. 7(a).Finally, it is very interesting to compare the diffraction results of the second reflected   beam (I 2 ) with the third reflected beam (I 3 ).Since the latter travels twice through the LC layer,the addressed grating acts twice on the beam, and a double effective phase modulation is observed.For instance, in Fig. 8(c), beam I 2 undergoes a complete cancellation of the zero order, indicating a π phase modulation, while beam I 3 shows one single zero diffraction order, indicating a 2π phase modulation.Similarly, in Fig. 8(e) I 2 shows a single zero diffraction order, signaling a 2π phase modulation, as it is the case for I 3 , which is now affected by a 4π phase binary grating.The same situation is presented in all cases in Fig. 8, and the effective phase modulations Δφ 2 and Δφ 3 acting on beams I 2 and I 3 are indicated on top of the image, showing that Δφ 3 = 2Δφ 2 in all cases.
Figure 9 shows the equivalent results with blazed gratings of the same period.In this case, the zero order does not vanish and other diffracted orders appear, as a direct consequence of the diffraction efficiency degradation provoked by non-perfect phase-only modulation [20].Nevertheless, the main characteristics of the blazed gratings are observed on the main reflected beam (I 2 ).When the maximum phase modulation reaches Δφ 2 = 2π, most of the energy is diffracted onto the first (m = 1) diffraction order (Fig. 9(e)), and when it reaches 4π, most of the energy is diffracted onto order m = 2 (Fig. 9(i)).
These results also confirm that the third reflected beam (I 3 ) is affected by a blazed grating with twice the maximum phase modulation than the blazed grating affecting the second reflected beam (I 2 ), i.e.Δφ 3 = 2Δφ 2 .For instance Figs.9(e) and 9(i) show that beam I 3 is mainly diffracted to orders m = 2 and m = 4, indicating that the effective phase modulation in this reflection is Δφ 3 = 4π and Δφ 3 = 8π respectively.Similarly, Fig. 9(g) shows that beam I 2 splits mainly to orders m = 1 and m = 2 indicating a maximum phase shift of Δφ 2 = 3π radians in this reflected beam, while beam I 3 diffracts mostly to order m = 3, as it corresponds to a maximum phase modulation of Δφ 3 = 6π.Again, the maximum phase modulation values are indicated on top of each image, and they all agree well with the diffraction results.

Conclusions
In summary, we have estimated the complex response of a PAL-LCoS-SLM rendered as a consequence of multiple reflections generated by the intrinsic layer structure of the device.We have shown that maximum phase modulation is achieved at normal incidence whereas increasing the angle of incidence leads to a reduction of the phase modulation range.We have also shown that multiple reflections at the LC layer generate FP interferences that cause a coupled amplitude modulation.
A simplified physical model that accounts for the observed phenomena was developed.By fitting experimental curves for different wavelengths with a reduced set of physical parameters, related to reflection coefficients and LC birefringence, it is possible to quantitatively describe with very good accuracy all the measured modulation curves.
Finally, it was shown that by operating with a large angle of incidence θ 0 = 45° and using a beam with small diameter, it is possible to spatially isolate different reflected beams and generate interesting effects when addressing phase gratings.A second reflection through the LC layer makes double the effect of the diffraction grating, and doubles the effective maximum phase modulation.This can be a useful effect since it can compensate for the phase modulation reduction that occurs at large angles of incidence.
i.A protective glass (thick layer), ii.A transparent electrode, iii.An alignment film, iv.The liquid crystal layer, v.An alignment film, vi.An aluminum mirror (or dielectric mirror in other models), vii.The silicon substrate.

Fig. 2 .
Fig. 2. (a) Illumination scheme, where incident polarization is selected parallel to LC director.(b) Picture of the LCoS-SLM illuminated with θ 0 = 45° and λ = 568 nm, where light scattered at the reflecting surfaces can be noticed.Incoming and emerging light are outlined by dashed lines.The reflected beams captured by a CCD camera are shown in the upper left inset box.

Fig. 4 .
Fig. 4. (a) Simplified layer structure and main transmitted/reflected beams.(b) Schematic of the interference condition at the LC layer.Voltage dependent LC director realignment in this layer is indicated 3(d)-3(f) are shown in Figs.5(a)-5(c).|ρ 01 | 2 takes a constant value independent of the gray level.On the contrary, |ρ FP | 2 exhibits interference oscillations.These experimental values are used to obtain the physical parameters of the model through Eqs. (

Fig. 7 .
Fig. 7. Diffraction efficiency (η m ) at the m-th order diffraction efficiency versus gray level for (a) binary grating, and (b) blazed grating corresponding to the reflectance obtained for 568 nm.