Influence of the incident angle in the performance of Liquid Crystal on Silicon displays

Transcription

1 Influence of the incident angle in the pefomance of Liquid Cystal on ilicon displays A. Lizana,. Matin, M. Estapé, E. Fenández, I. Moeno 3, A. Máquez, C. Iemmi 4, J. Campos and M. J. Yzuel Depatamento de Física, Univesidad Autónoma de Bacelona, 0893 Bellatea, pain Dept. de Física, Ingenieía de istemas y Teoía de la eñal, Univesidad de Alicante, Ap. 99, Alicante, pain 3 Dept. de Ciencia de Mateiales, Óptica y Tecnología Electónica, Univesidad Miguel Henández, Elche, pain 4 Dept. de Física, Fac. de Ciencias Exactas y atuales, Univesidad de Buenos Aies, 48 Buenos Aies, Agentina *Coesponding autho: angel.lizana@uab.es Abstact: In this pape we expeimentally analyze the pefomance of a twisted nematic liquid cystal on silicon (LCo display as a function of the angle of incidence of the incoming beam. These ae eflective displays that can be configued to poduce amplitude o phase modulation by popely aligning extenal polaization elements. But we demonstate that the incident angle plays an impotant ole in the selection of the polaization configuation. We pefomed a Muelle matix polaimetic analysis of the display that demonstates that the ecently epoted depolaization effect obseved in this type of displays is also dependant on the incident angle. 009 Optical ociety of Ameica OCI codes: (0.040 Displays; (0.540 Polaimety; ( Liquid-cystal devices; (30.60 patial light modulatos. Refeences and links. P. J. Tuunen and F. Wyowski eds., Diffactive Optics fo Industial and Commecial Applications, (Akademie Velag, Belin, H. J. Coufal, D. Psaltis and B. T. incebox, eds., Hologaphic Data toage, (pinge-velag, Belin, W. Osten, C. Kohle, and J. Liesene, Evaluation and application of spatial light modulatos fo optical metology, Opt. Pua Apl. 38, 7-8 ( R. Dou and M. K. Giles, Closed-loop adaptative optics system with a liquid cystal television as a phase etade, Opt. Lett. 0, ( J. Campos, A. Máquez, J. icolas, I. Moeno, C. Iemmi, J. C. Escalea, J. A. Davis and J. M. Yzuel, Optimization of liquid cystals displays behaviou in optical image pocessing and in diffactive optics, in Optoelectonic infomation pocessing: Optics fo infomation systems, (PIE Pess, Citical Reviews Vol. CR8, ( J. A. Davis, I. Moeno and P. Tsai, Polaization eigenstates fo twisted-nematic liquid-cystal displays, Appl. Opt. 37, ( A. Máquez, I. Moeno, C. Iemmi, A. Lizana, J. Campos and M. J. Yzuel, Muelle-tokes chaacteization and optimization of a liquid cystal on silicon display showing depolaization, Opt. Expess 6, ( J. E. Wolfe and R. A. Chipman, Polaimetic chaacteization of liquid-cystal-on-silicon panels, Appl. Opt. 45, ( A. Lizana, I. Moeno, C. Iemmi, A. Máquez, J. Campos and M. J. Yzuel, Time-esolved Muelle matix analysis of a liquid cystal on silicon display, Appl Opt. 47, ( A. Lizana, I. Moeno, A. Máquez, C. Iemmi, E. Fenández, J. Campos and M. J. Yzuel, Time fluctuations of the phase modulation in a liquid cystal on silicon display: chaacteization and effects in diffactive optics, Opt. Expess 6, (008.. I. Moeno, A. Lizana, J. Campos, A. Máquez, C. Iemmi, and M. J. Yzuel, Combined Muelle and Jones matix method fo the evaluation of the complex modulation in a liquid-cystal-on-silicon display, Opt. Lett. 33, (008.. A. Lizana, A. Máquez, I. Moeno, C. Iemmi, J. Campos, and M. J. Yzuel, Wavelength dependence of polaimetic and phase-shift chaacteization of a liquid cystal on silicon display, J. Eu. Opt. oc. Rapid Pub. 3, (008. # $5.00 UD Received 3 Ma 009; evised 3 Ap 009; accepted 6 Ap 009; published 5 May 009 (C 009 OA May 009 / Vol. 7, o. 0 / OPTIC EXPRE 849

2 3. E. Matín-Badosa, M. Montes-Usategui, A. Canice, J. Andilla, E. Pleguezuelos, and I. Juvells, Design stategies fo optimizing hologaphic optical tweezes set-ups, J. Opt. A - Pue Appl. Op. 9, ( D. Goldstein, Polaized Light, (Macel Dekke, Y, J. L. Pezzaniti,. C. McClain, R. A. Chipman and.-y. Lu, Depolaization in liquid-cystal televisions, Opt. Lett. 8, ( Y. Lu and R. A.Chipman, Intepetation of Muelle matices based on pola decomposition, J. Opt. oc. Am. A 3, 06-3 ( P. Lancaste and M Tismenetsky, The Theoy of Matices, nd Ed. (Academic, an Diego, A. Maquez, I. Moeno, J. Campos, and M. J.Yzuel, Analisis of Faby-Peot intefeence effects on the modulation popeties of liquid cystal displays, Opt. Commun. 65, ( Y. Lu and R. A. Chipman, Homogeneous and inhomogeneous Jones matices, J. Opt. oc. Am. A, ( Huad, Polaisation de la lumièe, (Masson, Pais, 993, pg tallinga, Equivalent etade appoach to eflective liquid cystal displays, J. Appl. Phys. 86, (999.. J. icolas, J. Campos and M. J. Yzuel, Phase and amplitude modulation of elliptic polaization states by nonabsobing anisotopic elements: application to liquid-cystal devices, J. Opt. oc. Am. A 9, ( C. Glasenapp, W. Mönch, H. Kause and H. Zappe, Biochip eade with dynamic hologaphic excitation and hypespectal fluoescence detection, J. Biomed. Opt., (007.. Intoduction The capability of the liquid cystal displays (LCD to wok as spatial light modulatos (LM have caused a widespead use of these devices in optical applications such as diffactive optics [], hologaphic data stoage [], optical metology [3], o in pogammable adaptive optics [4]. As a consequence, optimizing the LCDs esponse has become a challenge to many authos, and ways to obtain a desied intensity and phase esponse have been thooughly studied [5,6]. A type of LCD used in numeous optical applications is the Liquid Cystal on ilicon (LCo display. These devices ae LCDs that wok in eflection, giving high phase modulation. Howeve, a cetain amount of unpolaized light has been detected at the LCo displays eflected beam [7-9]. The oigin of this depolaization effect was investigated in [9], showing that it is elated to tempoal fluctuations of the liquid cystal oientation caused by the electical signal addessed to the display. This depolaization effect can advesely affect applications, as fo instance in diffactive optics whee it was demonstated to educe the diffaction efficiency [0]. Because of this depolaization effect, the Muelle-tokes (M- fomalism has been adopted fo LCo displays, and a potocol to optimize the intensity [7] and phase [] modulation esponses has been developed. By extension, this potocol is valid to chaacteize any polaizing o depolaizing optical element. In [] we showed that the intensity, phase and degee of polaization of the light beam modulated at the LCo display have a stong dependence with the wavelength. In this wok we study the modulation pefomance as a function of anothe paamete: the angle of incidence. Fo that pupose we have pefomed a complete polaimetic chaacteization of the LCo display fo diffeent angles of incidence. We show how the angle of incidence plays an impotant ole and stongly affects the modulation esponse. Then, we show that optimized phase modulation can be obtained fo the diffeent angles, but the polaization configuation must be eadjusted. This study can be especially elevant fo LCo displays applications involving high numeical apetues, whee a wide ange of incident angles ae used, as fo instance in optical tappings set-ups [3]. The outline of the pape is as follows. In ection, the expeimental method and the setup used to chaacteize the LCo display ae descibed. In ection 3, the esults of the polaimetic analysis of the display ae pesented. In paticula, the degee of polaization, diattenuation, polaizance and etadance paametes ae thooughly studied as a function of the addessed gay level and as a function of the incident angle. In ection 4, the phase and intensity modulation povided by the LCo display is analyzed as a function of the incident angle, and optimized configuations ae demonstated fo the diffeent angula positions. The # $5.00 UD Received 3 Ma 009; evised 3 Ap 009; accepted 6 Ap 009; published 5 May 009 (C 009 OA May 009 / Vol. 7, o. 0 / OPTIC EXPRE 849

3 esults ae compaed with those obtained with nomal incidence and employing a beam splitte. Finally, the conclusions ae pesented in ection 5.. Expeimental chaacteization based on a synchonous method In this wok we have chaacteized the Muelle matix of an LCo display as a function of the addessed gay level fo five diffeent incident angles: α=º,.5º, 3º, 34º and 45º. The chaacteization of the LCo display has been pefomed by using a modification of the methodology descibed at [7]. The poposed pocedue is based on the method of synchonous detection [4] and it is valid to chaacteize the Muelle matix of any polaizing o depolaizing optical element, being the LCo display a paticula case. Fig.. et up used to obtain the expeimental LCo Muelle matix. The expeimental set-up used to implement this method is shown in Fig.. We use a 633 nm He-e lase and the LCo display unde analysis is a Philips model X97c3A0, sold as the kit LC-R500 by Holoeye. The LC-R500 is a.46 cm diagonal eflective LCo display of the 45º twisted nematic type, with XGA esolution (04 x 768 pixels, with digital data input and digitally contolled gay scales with 56 gay levels. The pixels ae squae with a cente to cente sepaation of 9 µm and an excellent fill facto of 93%. The LCo is placed on the top of a otating platfom that enables choosing the incident angle with a pecision of º. We have set a polaization state geneato (PG at the incident beam and a polaization state detecto (PD at the eflected beam. The PG is composed by a polaize and an achomatic quate wave plate and the PD is composed by an achomatic quate wave plate and an analyze. Both polaize and analyze ae fixed at 0º, consideed paallel to the laboatoy vetical diection, and both wave plates can be electonically otated by 360º. It is well known that the Muelle matix (M of an optical polaizing element elates the input and output states of polaization (ops, descibed by the tokes vectos input and output. By geneating diffeent input ops and measuing its coesponding output ops (using simply intensity measuements, it is possible to constuct an independent equations system fom which the whole Muelle matix can be deived, as it was done in [7]. In this wok, we altenatively measue the ops eflected fom the LCo display by using the method of synchonous detection [4]. The analyze LP is fixed at 0º. Then, the intensity behind the PD is function of the tokes paametes of the eflected beam, and of the phase-shift (φ and oientation (θ of the waveplate WP. As a paticula case, when using a quate wave plate (φ=π/, the intensity as a function of the angle θ can be witten as follows: # $5.00 UD Received 3 Ma 009; evised 3 Ap 009; accepted 6 Ap 009; published 5 May 009 (C 009 OA May 009 / Vol. 7, o. 0 / OPTIC EXPRE 8493

4 π I, θ = cos( 4θ + sin( 4θ 3 sin( θ, ( whee 0,, and 3 ae the tokes paametes of the light eflected fom the LCo display. The intensity in Eq. ( is a peiodical signal with espect to the angle θ since it contains seveal sinusoidal functions whose aguments ae entie multiples of θ. The synchonous detection consists on the evaluation of the coefficients of the Fouie seies of this function. By pefoming a summation of intensities coesponding to diffeent equidistant values of θ, completing a otation of 360º, some tems of Eq. ( vanish due to the othogonal popeties of the sinusoidal sampled functions. In paticula the following elations hold: πi πj sin sin = πi πj cos cos = δ ij, (a πi πj sin cos = 0, (b = πi = = πi sin cos = 0. (c whee is the numbe of samples and δ ij the Konecke delta. Theefoe, using these elations it is possible to descibe the eflected op as a function of summations of intensity measuements obtained fo the diffeent equidistant analyze angles θ as: 0 3 = π I, θ 8 8 4, 4 π I, θ π I, θ,, π I, θ π I, θ cos sin, sin, cos ( 4θ, ( 4θ, ( 4θ ( θ,,, (3 whee is the numbe of selected angles θ and θ, =π/. On the othe hand, keeping the PG polaize fixed at 0º, the ops impinging the LCo display only depend on the WP otation angle θ. The incident op can be expessed as: cos ( θ =. (4 input sin( 4θ sin( θ Then, tokes paametes of the coesponding eflected beam can be witten as: # $5.00 UD Received 3 Ma 009; evised 3 Ap 009; accepted 6 Ap 009; published 5 May 009 (C 009 OA May 009 / Vol. 7, o. 0 / OPTIC EXPRE 8494

5 mk mk mk k ( θ = mk cos( 4θ + sin( 4θ + mk3 sin( θ, (5 output whee k = 0,,, 3, and m k,j (j = 0,,,3 ae the diffeent elements of the Muelle matix. ext, taking into account the othogonal popeties of the sinusoidal sampled functions (Eqs. (, and pefoming a summation fo diffeent output ops (coesponding to equidistant angles θ fom a complete otation of WP in Eq. (5, it is possible to obtain all the LCo Muelle matix coefficients as a function of output ops summations as: M = cos cos cos cos ( 4θ 4 cos( 4θ 4 sin( 4θ sin( θ, ( 4θ 4 cos( 4θ 4 sin( 4θ sin( θ, ( 4θ 4 cos( 4θ 4 sin( 4θ sin( θ, ( ( ( ( 4θ 4 cos 4θ 4 sin 4θ sin θ, 0 3,,,, 0 3,,,, 0 3,,,,,(6 whee is the numbe of selected equidistant angles θ. Fo evey value θ, the coesponding tokes paametes k ae measued accoding to Eq. (3. The advantage of this chaacteization pocedue with espect to that used in [7] is that hee some edundant infomation is employed, which esult in a eduction of the expeimental measuement eo. In this wok, these polaimetic measuements wee acquied with steps of 5.4º (on both angles θ and θ, and the whole system was automated by means of otation motoized devices with a pecision of 0.º. 3. LCo display as a function of the incident angle: polaimetic analysis In [7] we pesented a igoous study of the polaimetic popeties of a twisted nematic LCo display, illuminated with a 633 nm lase beam at quasi-nomal incidence. Hee, we extend that study to diffeent incident angles in ode to analyze its influence on the LCo display pefomance. In paticula, we have analyzed the degee of polaization, diattenuation, polaizance and etadance dependence with espect to the incident angle. 3. Degee of polaization as a function of the incident angle As mentioned peviously, the LCo display has been shown to poduce a eduction in the degee of polaization (DOP that depends on the gay level and on the input op [7-9,]. Figue shows the measued DOP as a function of the gay level fo vaious angles of incidence (α=º,.5º, 3º, 34º and 45º, calculated fom the expeimentally measued tokes paametes as DOP = [4]. By definition, the DOP takes values fom 0 to but in Fig., the y axis has been zoomed in ode to show the esults moe clealy. The esults coespond to thee input ops: linea polaized light at 0º, linea polaized light at 35º and left-handed cicula polaized light. Figue shows some elevant infomation about the DOP dependence with the incident angle. ote that some DOP values ae slightly highe than, as a consequence of the instumental eo associated to the intensity measuements in Eq. (3 and its coesponding eo popagation. Fom Fig., we see that fo all the selected incident angles, the DOP depends on the input op. Fo a fixed input op, thee is a quite elevant diffeence in the DOP evolution as a function of the gay level. Fo quasi-nomal # $5.00 UD Received 3 Ma 009; evised 3 Ap 009; accepted 6 Ap 009; published 5 May 009 (C 009 OA May 009 / Vol. 7, o. 0 / OPTIC EXPRE 8495

6 incidence (Fig. (a, the eflected light emains fully polaized (DOP close to one fo gay level anges below 00 o above 40. Howeve, impotant depolaization effects ae detected fo gay levels in between 00 and 40, eaching depolaization values highe than 0%. This depolaization effect is elated to op time fluctuations oiginated fom the electical signal addessing of the LCo display [9]. Fo low gay levels, the DOP emains close to one (Fig. (a, gay levels below 00 because the liquid cystal molecules ae oiented basically paallel to the glass windows and thei oientation is not so sensitive to fluctuations in the electical signal. On the contay, fo gay levels above 40 (Fig. (a, the LC molecules ae almost completely tilted paallel to the electic field diection, despite the fluctuations in the electical signal, and the eflected beam also emains fully polaized. Howeve, fo gay levels in between 00 and 40, the liquid cystal molecules tilt has an intemediate value, and the optical modulation is vey sensitive to the fluctuations of the electical signal, esulting in the highest depolaization effect fo gay level 80. Fig.. Degee of polaization as a function of the gay level and fo an angle of incidence equal to: a α=º, b α=.5º, c α=3º, d α=34º and e α=45º. When inceasing the incident angle (Figs. (b-(e, we detect unpolaized light in the gay level ange aound gay level 80 (as in the quasi-nomal incidence case shown in Fig. (a, but also fo highe o lowe gay level anges, whee the depolaization inceases as the incident angle inceases. Fo instance, fo incident angles α=.5º and α=3º (Figs. (b and (c, depolaization ovepass 5%, while it is geate than 0% fo incident angles of α=34º and α=45º (Figs. (d and (e. Fo high incident angles and fo some input ops, depolaization eaches appoximately a 0% along the whole gay level ange (black tiangles at Fig (d and Fig (e. We want to emphasize that pat of the depolaized light detected aound the 80 gay level and at oblique incidence can be attibuted to the signal fluctuations discussed befoe. # $5.00 UD Received 3 Ma 009; evised 3 Ap 009; accepted 6 Ap 009; published 5 May 009 (C 009 OA May 009 / Vol. 7, o. 0 / OPTIC EXPRE 8496

7 Howeve, thee is anothe depolaization souce when the LCo display is used at high oblique incidences. The amount of depolaization is highe when inceasing the incident angle and it is not only caused by the fluctuations phenomena. In ode to pove this last statement, we have measued the DOP coesponding to diffeent input ops and with the LCo display switched off (no voltage addessed. This measuement has been pefomed with the incident angles of to º and 45º. The esults ae shown in Fig. 3. Fig. 3. DOP as a function of diffeent incident ops and with an incident angle equal to: a º; b 45º. The LCo display is switched off. When no voltage is addessed to the LCo display, the light eflected by the device is almost fully polaized, fo an incident angle equal to º and fo all the tested incident ops (Fig. 3(a. Unlikely, fo an incident angle equal to 45º, the DOP stongly depends on the incident op although no voltage is addessed to the LCo display (Fig. 3(b. In fact, using linea polaized light at 0º of the lab vetical the DOP is almost one but when using an input linea polaized light at 35º o left-handed cicula light, we each values close to 0% of depolaization. Theefoe, Fig. 3 poves that we identified a new depolaization souce which is not oiginated by the fluctuations in the electical signal addessed to the LCo display (which ae the cause of the effective depolaization effect peviously epoted [9]. Moeove, this new depolaization souce is not simply a constant offset equally added to the effective depolaization effect. A constant offset would mean that the DOP should be, along the whole gay level ange, equal o lowe than the DOP measued with the LCo switched off. We see fo example in the case of left-handed cicula light and α=45º incident angle (squaes in Fig. (e that the DOP is bigge than the value 0.9, measued in the off-state (Fig. 3(b, fo most of the gay level ange. Theefoe, we conclude that the new depolaization pobably depends on the optical diecto distibution in the LC laye (which changes with the addessed voltage. It would be necessay futhe expeiments to get a tighte gip on which is the oigin of the new depolaization souce detected. A list of possible depolaization souces ae descibed in [8,5]. 3.. Diattenuation, polaizance and etadation dependence with the incident angle Using the synchonous method descibed in ection, we have measued the expeimental Muelle matix of the LCo display fo the incident angles of α=º,.5º, 3º, 34º and 45º. The expeimentally measued Muelle matices povide the polaimetic infomation of the analyzed LCo display. On one hand, the fist ow of the Muelle matix is elated to the diattenuation vecto, which gives the dependence of the tansmittance with the incident op [6]. On the othe hand, the fist column of the Muelle matix is elated to the polaizance vecto, which indicates the capability of the polaization element to polaize a fully unpolaized beam [6]. Finally, the bottom ight 3x3 submatix gives the infomation about the etadance and depolaization of the optical polaization element. Figue 4 shows the fist ow (Fig. 4(a and the fist column (Fig. 4(b coefficients of the obtained expeimental Muelle matices, as a function of the gay level and fo an incident # $5.00 UD Received 3 Ma 009; evised 3 Ap 009; accepted 6 Ap 009; published 5 May 009 (C 009 OA May 009 / Vol. 7, o. 0 / OPTIC EXPRE 8497

8 angle α=º. The coesponding equivalent esults ae plotted in Figs. 4(c and 4(d fo an incident angle α=45º. Fig. 4. Fist ow and column Muelle coefficients as a function of the gay level fo an incident angle of: a, b α=º; c, d α=45º. All these Muelle matix coefficients have values vey close to zeo (except m 00 which is equal to one. imila esults ae obtained fo the othe measued angles of incidence. As these coefficients emain null as a function of the gay level, the LCo display can be egaded as a non-diattenuating and non-polaizing polaization device independently of the chosen incident angle. ext, we have analyzed the coefficients of the bottom ight 3x3 submatix, which povide the etadance and depolaization infomation. As an example, Fig. 5 shows a compaison, fo the diffeent incident angles, of the evolution with gay level of the m, m and m 3 coefficients of the expeimental Muelle matices. The lage coefficients modulation shown in Fig. 5 has been obseved also fo all the 3x3 submatix coefficients. ote that the coefficient evolution as a function of the gay level, shown at Fig. 5, vaies gadually when inceasing the incident angle, finally leading to lage vaiations between the esults fo quasi-nomal incidence (Fig. 5(a in compaison with the incidence at α=45º (Fig. 5(e. This esult indicates that the etadance and depolaization effects will have a elevant dependence on the opeating incident angle. In ode to extact moe infomation fom the Muelle matices, we have used the combined method exposed in [], whee the Lu-Chipman pola decomposition [4] (based on the pola decomposition theoem [7] is applied to the LCo Muelle matix. In this sense, the Muelle matix of any depolaizing element can be expessed as the poduct of thee Muelle matices: the depolaize, the etade and the diattenuato matices. By taking into account the esults shown in Fig. 4, the diattenuation matix can be appoximated to the # $5.00 UD Received 3 Ma 009; evised 3 Ap 009; accepted 6 Ap 009; published 5 May 009 (C 009 OA May 009 / Vol. 7, o. 0 / OPTIC EXPRE 8498

9 Fig. 5. Muelle matix thid ow coefficients as a function of the gay level and an incident angle equal to: a α=º; b α=.5º; c α=3º; d α=34º and e α=45º. identity in all cases. Theefoe, it is possible to wite the Muelle matix of the LCo display just as the poduct of the depolaize and the etade matices. Finally, fom the etade matix, we ae able to find the Jones matix of the equivalent etade []. ext, we focus on the analysis of the Jones matix fo the equivalent etade. on-absobing ecipocal polaization devices in eflection ae theoetically equivalent to a linea etade [8]. Then, if we concentate on the equivalent etade Jones matix fo the LCo we may conside that unde nomal incidence (fowad and backwad path afte eflection ae the same the LCo can be expected to act as linea etade, whose neutal lines oientation and etadance depend on the addessed voltage. Howeve, when inceasing the incident angle, the LCo may act as an elliptical etade since the fowad and backwad paths in the LC laye ae no longe coincident. In ode to evaluate this effect, we have calculated the eigenvectos and eigenvalues of the equivalent etade Jones matix, and the eigenvectos oientation and ellipticity is deived as a function of the gay level. ote that the eigenvectos indicate the neutal lines in a linea etade and the phase-shift between eigenvalues gives the etadance. Figue 6 shows the etadance as a function of the gay level fo all the incident angles used along the expeience. The minimum phase value coesponds to the gay level 0 and an incident angle α=º (hombus spots, wheeas the maximum phase is obtained fo the gay level 40 and α=45º (cicula spots. These esults show that small incident angles (º-.5º show a highe phase-shift dynamic ange than high incident angles (34º-45º. This fact should be taken into account when seaching configuations of maximum phase modulation, as we show in ection 4. # $5.00 UD Received 3 Ma 009; evised 3 Ap 009; accepted 6 Ap 009; published 5 May 009 (C 009 OA May 009 / Vol. 7, o. 0 / OPTIC EXPRE 8499

10 Retadance º.5º 3º 34º 45º Gay level Fig. 6. Retadance as a function of the gay level and diffeent incident angles. Fig. 7. Equivalent etade eigenvectos as a function of the gay level fo the incident angle α=º. ext, the equivalent etade eigenvectos ae epesented in Fig. 7, as a function of the gay level, and fo quasi-nomal incidence (α=º. They emain almost linealy polaized in the whole ange of gay levels, and thei oientation otates counte-clockwise as the gay level inceases. Theefoe, we can conside the LCo display at quasi-nomal incidence as a linea etade whose etadation (Fig. 6 and neutal lines oientation (Fig. 7 change with # $5.00 UD Received 3 Ma 009; evised 3 Ap 009; accepted 6 Ap 009; published 5 May 009 (C 009 OA May 009 / Vol. 7, o. 0 / OPTIC EXPRE 8500

11 gay level, in ageement with [7]. Moeove, as expected in etades, the eigenvectos ae othogonal to each othe at evey gay level, pointing out that the LCo display is a homogeneous element [9]. Finally, in Fig. 8, we epesent the eigenvectos as a function of the gay level fo the othe incident angles α=.5º, 3º, 34º and 45º. ow thei ellipticity inceases as gay level inceases, being this effect stonge as the incident angle inceases. Thus these esult show that, by inceasing the incident angle and the gay level, the LCo displays becomes an elliptical etade [0] o even a cicula etade fo the paticula case of gay level 40 and incident angle α=45º. Let us note that tansmissive twisted nematic liquid cystal displays have, in geneal, two eigenvectos that ae elliptically polaized [6], while eflective twisted nematic displays opeating at pefectly nomal incidence act as an equivalent linea etade, thus having linea eigenvectos []. The esult in Fig. 7 fo quasi-nomal incidence veifies this situation, but the esults in Fig. 8 evidence that the equivalent linea etade behavio is lost when the angle of incidence inceases. 4. LCo esponse optimization esults: compaison between diffeent incident angles The pefomance of LCo displays in optical applications equie the adjustment of optimal PG and PD configuations, poviding specific intensity and phase modulation egime. Typically the two desied modulation egimes ae: maximum intensity contast modulation with constant phase, o constant amplitude modulation with maximum phase modulation. These intensity o phase egimes can be achieved by using elliptically polaization PG and PD configuations []. In ou pevious woks we have demonstated a useful methodology suitable to optimize the intensity esponse [7] o the phase esponse [] of a LCo display. In fact, in [] a PG and PD configuation giving high phase modulation and also a constant intensity esponse was obtained fo a 633 nm wavelength, and fo the incident angle of α=º. Hee we study how the optimized modulation esults ae affected by inceasing the incident angle. We have measued the intensity and phase of the LCo esponse when setting the optimized configuation fo phase modulation used in []. The expeimental modulation esults ae shown in Fig. 9, whee an optimized configuation designed fo phase-only modulation egime at the incident angle α=º is then tested at highe angles α=.5º and α=45º. The lines epesent the theoetical simulations calculated using the combined fomalism descibed in [], being the continuous line the simulated intensity (left axis and the dotted line the simulated phase (ight axis. The symbols epesent the expeimental values and they have been measued following the techniques descibed in [7]. The black cicles epesent the expeimental intensity and the squaes the expeimental phase. Figue 9 shows, in all cases, a geat ageement between simulated and expeimental values. Figue 9(a evidence a phase-only modulation esponse of the LCo display, whee a phase modulation up to almost 360º is accompanied with a constant intensity modulation. Figue 9(b shows that the phase-only modulation esponse is only slightly modified at low incident angles (α=.5º, but Fig. 9(c shows that the phase-only pefomance is lost fo α=45º. Figue 9(c shows significantly lowe values of phase-shift and noticeable coupled amplitude. This esult indicates that the optimization pefomed fo a given angle of incidence woks well within a educed incident angle ange. Out of this ange the esponse is athe diffeent and good modulation esults equie applying the optimization technique to obtain the specific optimal PG and PD configuations. As an example, Fig. 0 shows the modulation esults fo two phase modulation configuations and fo two incident angles (α=.5º and α=45º. The PG and the PD optimized values ae P =95º, WP =85º, P =6º and WP =64º fo the α=.5º incident angle and P =9º, WP =04º, P =54º and WP =53º fo the α=45º incident angle. # $5.00 UD Received 3 Ma 009; evised 3 Ap 009; accepted 6 Ap 009; published 5 May 009 (C 009 OA May 009 / Vol. 7, o. 0 / OPTIC EXPRE 850

12 Fig. 8. Equivalent etade eigenvectos as a function of the gay level fo the incident angles α=.5º, α=3º, α=34º and α=45º. # $5.00 UD Received 3 Ma 009; evised 3 Ap 009; accepted 6 Ap 009; published 5 May 009 (C 009 OA May 009 / Vol. 7, o. 0 / OPTIC EXPRE 850

13 Fig. 9. Theoetical (lines and expeimental (spots intensity and phase values, when using an incident angle equal to: a º; b.5º; c 45º. The otation angle values of polaizes and waveplates used at the PG and PD systems ae: P =88º and WP =7º; P =90º and WP =-5º. On one hand, Fig. 0(a shows a vey constant intensity esponse (black line and cicles as a function of the gay level. Moeove, we obtain almost π phase-shift (dotted line and squaes. Theefoe, the modulation esponse optimization with α=.5º povides simila esults to those obtained with α=º (Fig. 9(a. Thus, fo a small ange of incident angles (aound 0º a single optimization is enough. On the othe hand, Fig. 0(b (α=45º gives also a constant intensity esponse but the phase-shift is significantly shote (only slightly ove 40º than the obtained at quasi-nomal incidence (Fig. 9(a. Then, even by optimizing the LCo display phase modulation esponse at high incident angles, the esults emain wose than the obtained at low incident angles (Fig. 9(a and Fig. 0(a. This esult is in ageement with Fig. 6, whee the phase-shift between the equivalent etade eigenvectos is highe fo quasi-nomal incidence than fo oblique incidence. Fig. 0. Phase modulation optimization when using an incident angle equal to: a α=.5º; b α=45º. As we have shown above, by using high incident angles the obtained phase modulation esponse is emakably lowe than by using quasi-nomal incidence. Howeve, thee ae some optical applications whee a ight angle between the incident and eflected beam is equied, and a high phase-shift is also desied to achieve good diffaction efficiency [3, 3]. omal # $5.00 UD Received 3 Ma 009; evised 3 Ap 009; accepted 6 Ap 009; published 5 May 009 (C 009 OA May 009 / Vol. 7, o. 0 / OPTIC EXPRE 8503

14 incidence may be achieved in this case using a beam splitte, in a set-up like shown in Fig. (a, whee the PG and the PD ae located befoe and behind the beam splitte. While the beam splitte pemits to build this compact setup, it pesents the disadvantage of loosing light powe (half powe is lost on evey pass and eventually may intoduce additional polaization effects that must be taken into account. By following the pocedue peviously discussed, the whole system composed of the beam splitte and the LCo display has been chaacteized as a polaization device, and its phase modulation esponse has been optimized. The esults ae shown at Fig. (b. It shows modulation esults vey simila to those obtained with quasinomal incidence (Fig. 9(a, with a vey small eduction in the phase-shift, caused by the etadance intoduced by the beam splitte. Howeve, in ode to wok at 90º between the incident and the eflected beams, while maintaining a high phase-shift esponse, the beam splitte option is ecommended. 5. Conclusion Fig.. (a. Expeimental set-up. (b. Optimized phase modulation esponse obtained when using the beam splitte set-up. On one hand, the intensity values ae epesented in continuous line (simulation and black cicles (expeimental values. On the othe hand, the phase values ae epesented with a dotted line (simulation and squaes (expeimental values. The otation angle values of polaizes and waveplates used at the PG and PD systems ae: P =05º and WP =94º; P =05º and WP =8º. ummaizing, in this wok we povide a study of the pefomance of an LCo display as a function of the incident angle. We analyzed how optimized phase modulation configuations employing elliptically polaized light espond when changing the incidence. Hee we pesented esults on the polaimetic popeties of an LCo display, illuminated by 633 nm wavelength lase. The expeimental measuements pesented evidence that a peviously epoted effective depolaization effect, which shows dependence on the addessed gay level and on the input op, also pesents an impotant dependence on the incident angle. Moeove, we detected an additional souce of depolaization, not elated to the fluctuations of the electical signal, and which is moe significant at high incident angles. In addition, the polaimetic study evealed that LCo display acts as a non diattenuating and non polaizing element fo evey tested incident angle. On the contay, we have obseved a stong elation between the LCo display etadation and the incident angle. Moeove, the expeimental measuements show that LCo display becomes an elliptical etade when inceasing the incident angle. The etadance dependence with the incident angle has an impotant effect at the LCo display phase modulation esponse. We poved that incident angle deviations less than 0º do not modify substantially the modulation popeties. Howeve, we showed that a fixed configuation of polaizes and waveplates giving vey good phase esponse at nomal incidence shows a degaded phase-only modulation esponse (educed phase-shift and couple amplitude modulation when inceasing the incident angle. # $5.00 UD Received 3 Ma 009; evised 3 Ap 009; accepted 6 Ap 009; published 5 May 009 (C 009 OA May 009 / Vol. 7, o. 0 / OPTIC EXPRE 8504

15 We have also poved that optimized phase-only modulation configuations can be achieved fo evey incident angle, although the optimization pocedue must be applied in each case. Howeve, we have obtained less phase modulation depth in the phase-only modulation configuations as the incident angle inceases. Finally, in ode to etain a good phase modulation depth in a setup with pependicula incident and eflected beams, we included a beam splitte. The system composed of the beam splitte and LCo display has been chaacteized as a single polaization modulato, and the optimization pocess led to a phase-only configuation poviding esults almost equivalent to those obtained with quasinomal incidence, in spite of the etadance intoduced by the beam splitte. Theefoe, when good phase modulation is equied, simultaneously with pependicula incident and eflected beams, the use of a beam splitte is ecommended. All these esults ae elevant since LCo displays ae becoming a device useful fo a numbe of optical applications, and cae must be taken when selecting the incident angle. In addition, these effects may be elevant when employing the device illuminated with a wide ange of incident angles, as it is the case fo instance in optical tapping systems. Acknowledgments We acknowledge financial suppot fom panish Ministeio de Educación y Ciencia (FI C0-0 and 0 and Genealitat de Catalunya (006PIV000. C. Iemmi acknowledges suppot fom Univ. Buenos Aies and COICET (Agentina. # $5.00 UD Received 3 Ma 009; evised 3 Ap 009; accepted 6 Ap 009; published 5 May 009 (C 009 OA May 009 / Vol. 7, o. 0 / OPTIC EXPRE 8505