Influence of the incident angle in the performance of Liquid Crystal on Silicon displays A. Lizana1, N. Martin1, M. Estapé1, E. Fernández2, I. Moreno3, A. Márquez2, C. Iemmi4, J. Campos1 and M. J. Yzuel1 1 Departamento de Física, Universidad Autónoma de Barcelona, 08193 Bellaterra, Spain Dept. de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Ap. 99, 03080 Alicante, Spain 3 Dept. de Ciencia de Materiales, Óptica y Tecnología Electrónica, Universidad Miguel Hernández, Elche, Spain 4 Dept. de Física, Fac. de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina *Corresponding author: [email protected] 2 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 OCIS codes: (120.2040) Displays; (120.5410) Polarimetry; (230.3720) Liquid-crystal devices; (230.6120) Spatial light modulators. References and links P. J. Turunen and F. Wyrowski eds., Diffractive Optics for Industrial and Commercial Applications, (Akademie Verlag, Berlin, 1997). 2. H. J. Coufal, D. Psaltis and B. T. Sincerbox, eds., Holographic Data Storage, (Springer-Verlag, Berlin, 2000). 3. W. Osten, C. Kohler, and J. Liesener, “Evaluation and application of spatial light modulators for optical metrology,” Opt. Pura Apl. 38, 71-81 (2005). 4. R. Dou and M. K. Giles, “Closed-loop adaptative optics system with a liquid crystal television as a phase retarder,” Opt. Lett. 20, 1583-1585 (1995). 5. J. Campos, A. Márquez, J. Nicolas, I. Moreno, C. Iemmi, J. C. Escalera, J. A. Davis and J. M. Yzuel, “Optimization of liquid crystals displays behaviour in optical image processing and in diffractive optics”, in Optoelectronic information processing: Optics for information systems, (SPIE Press, Critical Reviews) Vol. CR81, 335-364 (2001). 6. J. A. Davis, I. Moreno and P. Tsai, “Polarization eigenstates for twisted-nematic liquid-crystal displays,” Appl. Opt. 37, 937-945 (1998). 7. A. Márquez, I. Moreno, C. Iemmi, A. Lizana, J. Campos and M. J. Yzuel, “Mueller-Stokes characterization and optimization of a liquid crystal on silicon display showing depolarization,” Opt. Express 16, 1669-1685 (2008). 8. J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt. 45, 1688-1703 (2006). 9. A. Lizana, I. Moreno, C. Iemmi, A. Márquez, J. Campos and M. J. Yzuel, “Time-resolved Mueller matrix analysis of a liquid crystal on silicon display,” Appl Opt. 47, 4267-4274 (2008). 10. A. Lizana, I. Moreno, 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, 16711-16722 (2008). 11. I. Moreno, A. Lizana, J. Campos, A. Márquez, C. Iemmi, and M. J. Yzuel, “Combined Mueller and Jones matrix method for the evaluation of the complex modulation in a liquid-crystal-on-silicon display,” Opt. Lett. 33, 627-629 (2008). 12. A. Lizana, A. Márquez, I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Wavelength dependence of polarimetric and phase-shift characterization of a liquid crystal on silicon display,” J. Eur. Opt. Soc. – Rapid Pub. 3, 08011 1-6 (2008). 1. #108308 - $15.00 USD (C) 2009 OSA Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8491 13. E. Martín-Badosa, M. Montes-Usategui, A. Carnicer, J. Andilla, E. Pleguezuelos, and I. Juvells, “Design strategies for optimizing holographic optical tweezers set-ups,” J. Opt. A - Pure Appl. Op. 9, S267-S277 (2007). 14. D. Goldstein, Polarized Light, (Marcel Dekker, NY, 2003). 15. J. L. Pezzaniti, S. C. McClain, R. A. Chipman and S.-Y. Lu, “Depolarization in liquid-crystal televisions,” Opt. Lett. 18, 2071-2073 (1993). 16. S. Y. Lu and R. A.Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106-1113 (1996). 17. P. Lancaster and M Tismenetsky, The Theory of Matrices, 2nd Ed. (Academic, San Diego, 1985). 18. A. Marquez, I. Moreno, J. Campos, and M. J.Yzuel, “Analisis of Fabry-Perot interference effects on the modulation properties of liquid crystal displays,” Opt. Commun. 265, 84-94 (2006). 19. S. Y. Lu and R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766-773 (1994). 20. S. Huard, Polarisation de la lumière, (Masson, Paris, 1993), pg. 109. 21. S. Stallinga, “Equivalent retarder approach to reflective liquid crystal displays,” J. Appl. Phys. 86, 47564766 (1999). 22. J. Nicolas, J. Campos and M. J. Yzuel, “Phase and amplitude modulation of elliptic polarization states by nonabsorbing anisotropic elements: application to liquid-crystal devices,” J. Opt. Soc. Am. A 19, 1013-1020 (2002). 23. C. Glasenapp, W. Mönch, H. Krause and H. Zappe, “Biochip reader with dynamic holographic excitation and hyperspectral fluorescence detection,” J. Biomed. Opt. 2, 014038 (2007). 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 #108308 - $15.00 USD (C) 2009 OSA Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8492 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=2º, 12.5º, 23º, 34º and 45º. 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. Fig. 1. Set up used to obtain the experimental LCoS Mueller matrix. 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 45º twisted nematic type, with XGA resolution (1024 x 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 1º. 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 0º, considered parallel to the laboratory vertical direction, and both wave plates can be electronically rotated by 360º. 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 Sinput and Soutput. 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 LP2 is fixed at 0º. Then, the intensity behind the PSD is function of the Stokes parameters of the reflected beam, and of the phase-shift (φ) and orientation (θ2) of the waveplate WP2. As a particular case, when using a quarter wave plate (φ=π/2), the intensity as a function of the angle θ2 can be written as follows: #108308 - $15.00 USD (C) 2009 OSA Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8493 S S S 1 π I , θ 2 = S 0 + 1 + 1 cos (4θ 2 ) + 2 sin (4θ 2 ) − S 3 sin (2θ 2 ) , 2 2 2 2 2 (1) where S0, S1, S2 and S3 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 θ2 since it contains several sinusoidal functions whose arguments are entire multiples of θ2. 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 θ2, completing a rotation of 360º, some terms of Eq. (1) vanish due to the orthogonal properties of the sinusoidal sampled functions. In particular the following relations hold: 2 N N ∑ r =1 2πri 2πrj 2 sin = ·sin N N N N ∑ sin r =1 N ∑ r =1 N ∑ cos r =1 2πri 2πrj = δij , ·cos N N 2πri 2πrj = 0, ·cos N N 2πri sin = N N ∑ cos r =1 2πri = 0. N (2a) (2b) (2c) where N is the number of samples and δij 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 θ2 as: N N π π 2· I , θ 2 ,r − 4 I , θ 2,r ·cos(4θ 2,r ) 2 2 r =1 r =1 N S0 π 8 I , θ 2,r ·cos(4θ 2,r ) 2 S1 1 r =1 , S = N N π 2 8 I , θ 2,r ·sin (4θ 2,r ) S 2 3 r =1 N π −4 I , θ 2,r ·sin (2θ 2,r ) 2 r =1 ∑ ∑ ∑ ∑ (3) ∑ where N is the number of selected angles θ2 and θ2,r=r2π/N. On the other hand, keeping the PSG polarizer fixed at 0º, the SoPs impinging the LCoS display only depend on the WP1 rotation angle θ1. The incident SoP can be expressed as: S input 1 2 cos (2θ1 ) =1 . sin (4θ1 ) 2 sin(2θ1 ) (4) Then, Stokes parameters of the corresponding reflected beam can be written as: #108308 - $15.00 USD (C) 2009 OSA Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8494 S kr (θ1 ) = mk 0 + output mk 1 mk1 m + cos(4θ1 ) + k 2 sin (4θ1 ) + mk 3 sin(2θ1 ) , 2 2 2 (5) where k = 0, 1, 2, 3, and mk,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 θ1 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: N S 0r rN=1 S1r 1 r =1 M= N N r S2 r =1 N S 3r r =1 N ∑ −2 ∑ ∑ −2 ∑ −2 ∑ −2 r =1 N S 0r cos(4θ1,r ) 4 ∑ ∑S 1 ∑S 2 ∑S 3 r =1 N r =1 N r =1 N r r r cos(4θ1,r ) 4 r =1 N S 0r cos(4θ1, r ) 4 ∑ ∑S cos(4θ1,r ) 4 r =1 N ∑S cos(4θ1,r ) 4 r =1 N ∑S r =1 N r 1 r 2 r 3 cos(4θ1, r ) 4 r =1 N ∑S cos(4θ1, r ) 4 r =1 N ∑S cos(4θ1, r ) 4 r =1 N ∑S r =1 ( ) θ sin 2 r 1 , 0 r =1 N r 2 S 1 sin (2θ1, r ) r =1 ,(6) N 2 S 2r sin (2θ1, r ) r =1 N 2 S 3r sin (2θ1, r ) r =1 S 0r sin (4θ1,r ) 2 N ∑S r 1 r 2 r 3 sin (4θ1,r ) sin (4θ1,r ) sin (4θ1,r ) r ∑ ∑ ∑ where N is the number of selected equidistant angles θ1. For every value θ1,r the corresponding Stokes parameters Skr 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 51.4º (on both angles θ1 and θ2), and the whole system was automated by means of rotation motorized devices with a precision of 0.1º. 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=2º, 12.5º, 23º, 34º and 45º), calculated from the experimentally measured Stokes parameters as DOP = S12 + S 22 + S 32 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 0º, linear polarized light at 135º 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 #108308 - $15.00 USD (C) 2009 OSA Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8495 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. Fig. 2. Degree of polarization as a function of the gray level and for an angle of incidence equal to: a) α=2º, b) α=12.5º, c) α=23º, d) α=34º and e) α=45º. 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=12.5º and α=23º (Figs. 2(b) and 2(c)), depolarization overpass 5%, while it is greater than 10% for incident angles of α=34º and α=45º (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. #108308 - $15.00 USD (C) 2009 OSA Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8496 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 2º and 45º. The results are shown in Fig. 3. Fig. 3. DOP as a function of different incident SoPs and with an incident angle equal to: a) 2º; b) 45º. The LCoS display is switched off. 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 2º and for all the tested incident SoPs (Fig. 3(a)). Unlikely, for an incident angle equal to 45º, 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 0º of the lab vertical the DOP is almost one but when using an input linear polarized light at 135º 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=45º 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=2º, 12.5º, 23º, 34º and 45º. 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 3x3 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 #108308 - $15.00 USD (C) 2009 OSA Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8497 angle α=2º. The corresponding equivalent results are plotted in Figs. 4(c) and 4(d) for an incident angle α=45º. Fig. 4. First row and column Mueller coefficients as a function of the gray level for an incident angle of: a, b) α=2º; c, d) α=45º. All these Mueller matrix coefficients have values very close to zero (except m00 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 3x3 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 m21, m22 and m23 coefficients of the experimental Mueller matrices. The large coefficients modulation shown in Fig. 5 has been observed also for all the 3x3 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=45º (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 #108308 - $15.00 USD (C) 2009 OSA Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8498 Fig. 5. Mueller matrix third row coefficients as a function of the gray level and an incident angle equal to: a) α=2º; b) α=12.5º; c) α=23º; d) α=34º and e) α=45º. 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=2º (rhombus spots), whereas the maximum phase is obtained for the gray level 240 and α=45º (circular spots). These results show that small incident angles (2º-12.5º) show a higher phase-shift dynamic range than high incident angles (34º-45º). This fact should be taken into account when searching configurations of maximum phase modulation, as we show in Section 4. #108308 - $15.00 USD (C) 2009 OSA Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8499 Retardance 270 250 230 210 190 170 150 130 110 90 0 20 40 60 80 100 120 140 160 180 200 220 240 2º 12.5º 23º 34º 45º Gray level Fig. 6. Retardance as a function of the gray level and different incident angles. Fig. 7. Equivalent retarder eigenvectors as a function of the gray level for the incident angle α=2º. Next, the equivalent retarder eigenvectors are represented in Fig. 7, as a function of the gray level, and for quasi-normal incidence (α=2º). 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 #108308 - $15.00 USD (C) 2009 OSA Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8500 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=12.5º, 23º, 34º and 45º. 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=45º. 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=2º. 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=2º is then tested at higher angles α=12.5º and α=45º. 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 360º 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=12.5º), but Fig. 9(c) shows that the phase-only performance is lost for α=45º. 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=12.5º and α=45º). The PSG and the PSD optimized values are P1=95º, WP1=85º, P2=61º and WP2=64º for the α=12.5º incident angle and P1=91º, WP1=104º, P2=54º and WP2=53º for the α=45º incident angle. #108308 - $15.00 USD (C) 2009 OSA Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8501 Fig. 8. Equivalent retarder eigenvectors as a function of the gray level for the incident angles α=12.5º, α=23º, α=34º and α=45º. #108308 - $15.00 USD (C) 2009 OSA Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8502 Fig. 9. Theoretical (lines) and experimental (spots) intensity and phase values, when using an incident angle equal to: a) 2º; b) 12.5º; c) 45º. The rotation angle values of polarizers and waveplates used at the PSG and PSD systems are: P1=88º and WP1=7º; P2=90º and WP2=-15º. 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 α=12.5º provides similar results to those obtained with α=2º (Fig. 9(a)). Thus, for a small range of incident angles (around 10º) a single optimization is enough. On the other hand, Fig. 10(b) (α=45º) gives also a constant intensity response but the phase-shift is significantly shorter (only slightly over 240º) 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. Fig. 10. Phase modulation optimization when using an incident angle equal to: a) α=12.5º; b) α=45º. 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 #108308 - $15.00 USD (C) 2009 OSA Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8503 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 quasinormal 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 90º between the incident and the reflected beams, while maintaining a high phase-shift response, the beam splitter option is recommended. Fig. 11. (a). Experimental set-up. (b). Optimized phase modulation response obtained when using the beam splitter set-up. On one hand, the intensity values are represented in continuous line (simulation) and black circles (experimental values). On the other hand, the phase values are represented with a dotted line (simulation) and squares (experimental values). The rotation angle values of polarizers and waveplates used at the PSG and PSD systems are: P1=105º and WP1=94º; P2=105º and WP2=82º. 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 10º 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. #108308 - $15.00 USD (C) 2009 OSA Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8504 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 quasinormal 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). #108308 - $15.00 USD (C) 2009 OSA Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009 11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8505

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