Optimal period for diffraction gratings recorded in polymer dispersed liquid crystals

OPTO-ELECTRONICS REIEW 15(1) 66 70 DOI: 10.478/s1177-006-0058-1 Optimal period for diffraction gratings recorded in polymer dispersed liquid crystals A. ASLANYAN and A. GALSTYAN Department of Physics Yerevan State University 1 Manougyan Str. 375049 Yerevan Armenia New diffusion model of recording diffraction gratings in the media of PDLC is described in which besides diffusion of monomer molecules also diffusion of polymer molecules and non-locality of diffusion coefficient are taken into account. It lets us to explain why diffraction efficiency is low for low and high values of intensities of grating recording beam. With the considered model we have theoretically got optimal period for grating recording. Keywords: liquid crystal polymer grating diffraction. 1. Introduction In the recent years photo polymeric materials have attracted researchers attention. They have sequence of advantages against other types of recording materials such as photo resistors gelatins and so on. Polymer dispersed liquid crystal (PDLC) medium is convenient for information recording and storage [1 4]. The main advantage of such media is that we can follow up the appearing of formations in the real time [5]. In this composite material polymer serves as a matrix where the molecules of liquid crystals are dispersed as drop domains. Optical properties of the obtained PDLC materials basically depend on the spatial distribution and morphology of drop-shaped domains arrangement of LC director in the drops dimensions of the liquid crystalline formations and so on. The arising of the forms is caused by modulation of a concentration of the drops and polymer matrix holes. PDLC media have mechanical properties of polymer matrix and optical anisotropy of LC dispersed in the matrix [6]. That anisotropy in the case of weak anchoring energy at domains walls could be controlled by external fields. Mechanical strength and easy controllable anisotropy of such media provide possibility of creation of plane LC monitor microlenses with controllable focal distances [7] and so on. The simplest forms with the discussed properties are the diffraction gratings with periodical distribution of concentration and refractive index which are produced with the help of interference pattern of two plain waves [89]. In Refs. 10 11 and 1 the recording dynamics in photopolymers dispersed liquid crystals media of gratings and choosing of optimal parameters for receiving effective gratings are considered. The forms appear with the following scheme photomonomer transforms into a polymer because of chain reaction *e-mail: 66 ar_aslanyan@yahoo.com which arises from light affection. At that if the photomonomer is being illuminated with the interference pattern then in the regions of space where the intensity of light is higher the process passes faster [13]. When monomerpolymer transformation takes place spatial redistribution of monomer concentration arises. Molecules of the monomer diffuse from the dark regions to the illuminated regions because the decrease in monomer molecules takes place in those regions. Correspondingly in the illuminated regions the amount of polymerised monomers increases and LC is being pushed out from those places. So modulation of refractive index arises and consequently diffraction grating arises too. Monomer and LC in PDLC media is being chosen in such a way that they dissolve in each other. Situation is different in the case of polymer and liquid crystal. There is a phase division between them and if there is a polymer in some volume liquid crystal is being forced out from that volume. In fact in the dark regions concentration of a liquid crystal is higher than in light regions what leads to anisotropic diffraction grating formation which has controllable modulation of refractive index. There are sequences of models describing these processes [1314]. Nevertheless in the case of low intensities of grating recording laser the process models give incorrect results because in that case modulation depth of the recorded diffraction grating is maximal [9]. As we will discuss below this does not coincide with reality. As it can be seen from qualitative experiments for very high and very low intensities modulation of polymer concentration and efficiency of recorded diffraction gratings are small. A new model is described where besides diffusion of monomer molecules diffusion of polymer molecules and non-locality of diffusion coefficient are taken into account. This gives an opportunity to explain entirely the recording dynamics of diffraction gratings in PDLC media. In the Opto-Electron. Rev. 15 no. 1 007

frame of the considered model it is theoretically shown that there is a spatial period at which diffraction efficiency is maximal.. Diffusion model with consideration of polymer diffusion Photopolymeric systems basically consist of monomer polymeric cross linking agent and photoinitiator. Monomer is being polymerized under light influence and percentage of polymerized molecules increases proportionally to the illuminating time. Concurrent with the process of monomer-polymer transformation diffusive processes are being slowed down and finally broken off what means that coefficient of monomer diffusion decreases during exposition. It can be explained in the following manner increasing polymer concentration makes substance more opaque for monomer molecules. Dynamics of diffraction gratings recording in PDLC media is being widely investigated and is described with so-called diffusion equations of polymerization. There are two basic models for polymerization local [58] and non local one [91314]. In the first one it is considered that coefficient of polymerization in each point depends on illumination value at that point and in the second one it is considered that coefficient of polymerization also depends on the illumination value at the neighbouring points. In the following discussion we will concentrate only on the case of non-local response as its generality. Let us consider dynamics of diffraction gratings recording in PDLC media which arises from the result of two plane waves interference. Spatial distribution of two plane waves can be presented in the following form [4] I (x t ) = I 0 (1 + cos(kx )) (1) here I 0 is the average intensity is the visibility of an interference pattern k = p L and L is the period of interference pattern. Schematic representation of diffraction grating recording is shown in Fig. 1. But in the case of low intensities of the recording laser the results obtained with this model differ from experimental results. In this case diffraction efficiency of a grating achieves its maximal value. Such a result is incorrect. Experiments show that in the cases of very high and very low intensities diffraction efficiencies of the recorded gratings are low [15]. To develop a model which will explain experimental results in the proper way let us take into account the following fact. In the lightened regions monomer is being polymerized and naturally in that region concentration of monomer decreases. Diffusive processes of the monomer tend to fill up the lack of concentration in the lightened regions i.e. monomer diffusion from dark regions to the lightened ones takes place. On the other hand concurrent with polymerization the velocity of diffusive processes in the medium decreases. Consequently if the characterization times of the polymerization be incomparably smaller than monomer diffusive process ones then it would not be modulation of concentration. Otherwise if monomer has time to diffuse from dark regions to light regions completely and concurrent with polymerization the decrease in diffusion would not impede monomer diffusion then maximal modulation of concentration will be obtained. For low intensities only this case occurs because of a very slow process of polymerization the diffusive processes cannot be stopped and maximal modulation is obtained. According to the presented discussions in the case of low intensities neglecting of polymer diffusion is rough approximation. Naturally the processes of polymer diffusion are much slower than monomer s ones but when the times of polymerization are very long even slower diffusive processes would have time definitely average polymer concentration. That is in the case of slow polymerization polymer diffusion from the dark to the lightened regions must not be neglected. So diffusive equations of polymerization where both polymer and monomer diffusion processes are taken into account will be written in the following form U(x t ) æ U(x t ) N (x t ) = ç D (x t ) t x U x t N (x t ) æ N (x t ) = ç D (x t ) t x N x + + () (3) ò R(x x ) F(x )U(x t)dx - here U(x t ) is the monomer concentration F(x t ) is the local coefficient of polymerization [710] and DN is the polymer diffusion coefficient. The function R(x x ) describes spatial non-locality. Unlike previous models first item in the right-hand side of Eq. (3) describes diffusion of polymer molecules. A member considering spatial non-locality is represented in the form of Gaussian function [13] Fig. 1. Schematic representation of diffraction grating recording. Opto-Electron. Rev. 15 no. 1 007 R(x - x ) = æ (x - x ) expç s ps (4) 67 A. Aslanyan

where s describes the length of response non-locality i.e. spatial scale where influence of neighbouring points is still essential. In the future calculations we will admit that the coefficient of polymerization is proportional to the radiation intensity [5] F(x t ) = F0 (1 + cos(kx )) (5) where F0 = KI 0 K is the coefficient of proportionality. Expanding Eqs. () and (3) in the Fourier series and keeping first three spatial harmonics we obtain the following system of coupled differential equations du 0 = - F0U 0 (t ) - F0 U1 (t ) (6) æ k s du1 U (t ) = - F0 expç 0 æ æ k s + DU 0 (t )k U1 (t ) - çç F0 expç (7) æ æ k s + DN1 (t )k U (t ) - çç F0 expç DN 0 (t ) and DN1 (t ) are the first two components of Fourier series of a diffusion coefficient of polymer DU0 (t ) and DU1 (t ) are the first two components of Fourier series of diffusion coefficient of monomer a U and a N are the decreasing velocities of diffusion coefficient for monomer and polymer respectively concurrent with the polymerization process. Consideration of diffusion coefficient non-locality becomes more significant in the case of the higher magnitude s of and neglecting of non-locality may bring to non-physical result. As we know for increasing s parameter the modulations of concentration of monomer and polymer decrease because polymerization excited with one maximum is being overlapped with the neighbouring maximum. Particularly if the radius of non-locality is bigger than the period of light modulation then polymerization in the dark and light regions will run with the same speed. It means that diffusion coefficient of monomer is not modulated that is DU1 (t ) = 0. This condition will not be satisfied if in the argument of hyperbolic functions of Eqs. (1) (15) the member exp( -k s ) is absent. Initial conditions for this system are U 0 (0) = U 0 + DN 0 (t )k N1 + DN1 (t )k N U1 (0) = U (0) = N 0 (0) = N1 (0) = N (0) = 0 du æ = -ç F0 exp( -k s ) + DU1 (t )k U1 (t ) - ( F0 exp( -k s ) + 4 DU 0 (t )k )U (t ) (8) + DN1 (t )k N1 + 4 DN 0 (t )k N dn 0 = F0U 0 (t ) + F0 U1 (t ) æ k s æ k s dn1 U 0 (t ) + F0 expç U (t ) = F0 expç 1 (9) (10) æ k s U (t ) -DN 0 (t )k N1 -DN1 (t )k N + F0 expç dn = F0 exp( -k s )U1 (t ) + F0 exp( -k s )U (t ). (11) that is in the initial solution concentration of polymer is 0 and the initial concentration of monomer is not modulated. As LC does not participate in a polymerization process and in a very good approximation does not affect the final distribution of molecules so its concentration is not included in the diffusion equation. Let us denote with L(x t ) concentration of a liquid crystal c = L 0 N 0 is a ratio between average concentrations of liquid crystal and polymer. Concentration of liquid crystal in the definite time and definite place can be found from the normalization condition U(x t ) + N (x t ) + L(x t ) = 1. Hence concentration of a liquid crystal at the time moment and in the space point can be represented in the following form L(x t ) = L 0 (t ) + L1 (t ) cos(kx ) - DN1 (t ) N1-4 DN 0 (t )k N Here = cn1 (t ) - N (t ) cos(kx ) DU 0 (t ) = DUa exp( -a u F0 (t )t )ch(a U F0 (t ) (1) exp( -k s )t ) DU1 (t ) = - DUa exp( -a u F0 (t )t )sh(a U F0 (t ) exp( -k s )t ) DN 0 (t ) = DNa exp( -a N F0 (t )t )ch(a N F0 (t ) exp( -k s )t ) DN1 (t ) = - DNa exp( -a N F0 (t )t )sh(a N F0 (t ) exp( -k s )t ) 68 (13) (14) (15) (16) sign in front of modulation term in Eq. (16) indicates the concentration maximums of polymer and LC are shifted with the half of a period. Solving Eqs. (6) (11) and using normalization condition we will obtain spatial distributions of monomer polymer and LC at the arbitrary moment of time. As an example in Fig. (a) and (b) there are shown time dependences of the first two harmonics of monomer and polymer which were obtained with a numerical solution of Eqs. (6) (11) system. It can be seen in Fig. that the concentration of monomer decreases concurrently with increasing polymer concentration. Opto-Electron. Rev. 15 no. 1 007 007 SEP Warsaw

Fig. 3. Diffraction efficiency dependence on average intensity of recording laser. Dashed line the case of the absence of polymer diffusion solid line the case of presence of polymer diffusion. Fig.. Time dependences of monomer concentration and concentration modulation (a) time dependences of polymer concentration and concentration modulation (b). 3. Diffraction efficiency dependence on writing beam average intensity and gratings periods Figure 3 shows dependence of h diffraction efficiency on the average I 0 intensity of the recording laser radiation. In the case of the absence of polymer diffusion h dependence on the average I 0 intensity of the recording laser radiation is shown with the dashed line and the same dependence in the case of presence of polymer diffusion is shown with the solid line using the following numerical values s = 01. mm = 0.5 a U =.5 a N = 10 K = 0.003cm mj L = 1 mm k = p L DUa = 0. mm s DNa = 017. mm s and the following initial conditions U 0 = 07. L0 = 0.3 and N 0 = 0. As it can be seen in Fig. 3 in the absence of polymer molecules diffusion diffraction efficiency of the grating rises concurrently with intensity decrease. However when the diffusion of polymer is considered diffraction efficiency of the grating in the case of low intensities is small because during the process of polymer molecules diffusion averaging of polymer concentration takes place. CalOpto-Electron. Rev. 15 no. 1 007 culations show that an optimal intensity for hologram recording exists which for maximal numerical values amounts» 15mW cm and it is in good agreement with qualitative experimental results. Diffraction features of PDLC gratings depend not only on average intensity of writing laser but also on a sequence of other parameters such as properties of PDLC components and period of grating as well. In this work we have investigated influence of period of gratings which are written in PDLC on their diffraction properties. Dependence of diffraction efficiency on grating periods represents special interest because in the process of grating writing a period is easily controllable magnitude. Diffraction efficiency for the -polarized laser beam is being counted with well known formula [16] h = sin (pn1 d l cos q 0 ) where h is the diffraction efficiency of the holographic grating n1 is the spatial modulation of the refractive index d is the thickness of the grating l is the wavelength of the incidence light and q 0 is the Bragg angle. Solving the system of Eqs. (6) (11) for each L and after obtaining n1 modulation by the method presented in Ref. 5 we can plot h versus L. In Fig. 4 curve 1 shows dependence of diffraction efficiency on the recorded gratings periods without diffusion of polymer molecules. As it can be seen from the figure for the periods smaller than L» 0.5mm the diffraction efficiency of the gratings is so small that it can be neglected. It is explained in following way. For small periods of the grating non locality of polymerization process becomes essential which impedes arising of concentration modulation. Calculations show that also exists an optimal value of grating period (L» mm) for which the maximal diffraction efficiency is obtained (95%). The reason for this is that for a big period the monomer s diffusion does not keep pace with diffusion of illuminated regions and modulation of concentration does not arise. 69 A. Aslanyan

References Fig. 4. Dependence of diffraction efficiency on gratings periods 1 without polymer diffusion and with polymer diffusion. Consideration of polymer molecules diffusion leads to definite changes. As it can be seen from Fig. 4 (curve ) the gratings arise for the value higher than 0.6 µm. Maximal value of diffraction efficiency decreases (75%). The period of the grating with the maximal efficiency becomes.5 µm. Decrease in maximal value of diffraction efficiency is caused with the fact that modulation of concentration decreases due to polymer molecules diffusion. Rising of minimal and optimal periods is caused with the fact that modulation of concentration besides distorting with non-locality is also being distorted due to polymer molecules diffusion. 4. Conclusions New diffusion model of recording diffraction gratings in the media of PDLC is considered where besides diffusion of monomer molecules the diffusion of polymer molecules and non locality of diffusion coefficient is taken into account. It describes the features of gratings recorded in PDLC most completely. Particularly consideration of polymer molecules diffusion allows us correctly explain behaviour of gratings diffraction efficiency during gratings recording with low-intensity laser. It is shown that for a fixed intensity minimal period for diffraction grating recording exists. This is a sequence of non locality of a polymerization process. An optimal period of the grating recording is observed when maximal diffraction efficiency is obtained. Consideration of polymer molecules diffusion leads to minimal and optimal periods rising. Calculations show that in this case maximal value of diffraction efficiency decreases. Acknowledgements The authors thank Prof. R. Hakobyan and L. Aslanyan for stimulating discussions. 70 1. P. Pilot Y.B. Boiko and T.. Galstian Near-IR (800 855 nm) sensitive holographic photopolymer dispersed liquid crystal materials Proc. SPIE 3635 143 150 (1999).. P. Nagtegaele and T.. Galstian Holographic characterization of near infrared photopolymerizable materials Synthetic Metals 17 85 87 (00). 3. R. Kaputo A. Sukhov C. Umeton and R. Ushakov Formation of a grating of submicron nematic layers by photopolymerization of nematic-containing mixtures JETP 118 1374 1383 (000). 4. R.L. Sutherland.P. Tondiglia L.. Natarajan T.J. Bunning and W.W. Adams olume holographic image storage and electro-optical readout in a polymer-dispersed liquid-crystal film Opt. Lett. 0 135 (1995). 5. I. Aubrecht M. Miler and I. Koudela Recording of holographic diffraction gratings in photopolymers: theoretical modelling and real-time monitoring of grating growth J. Mod. Opt. 45 1465 1477 (1998). 6. F. Roussel and B. Fung Anchoring behaviour orientational order and reorientation dynamics of nematic liquid crystal droplets dispersed in cross-linked polymer networks droplets dispersed in cross-linked polymer networks Phys. Rev. E67 041709.1 041709.4 (003). 7. Y.H. Fan H. Ren and S.T. Wu Switchable fresnel lens using polymer-stabilised liquid crystals Optics Express 11 3080 3086 (003). 8. G. Zhao and P. Mouroulis Extension of a diffusion model for holographic photopolymers J. Mod. Opt. 41 199 573 (1994). 9. J.T. Sheridan and J.R. Lawrence Nonlocal-response diffusion model of holographic recording in photopolymer J. Opt. Soc. Am. 17 1108 (000). 10. D. Duca A.. Sukhov and C. Umeton Detailed experimental investigation on recording of switchable diffraction gratings in polymer dispersed liquid crystal films by U laser curing Liquid Crystals 6 931 937 (1999). 11. J.T. Sheridan T.O. Neill and J.. Kelly Holographic data storage: optimized scheduling using the nonlocal polymerization-driven diffusion model J. Opt. Soc. Am. B1 1443 1451 (004). 1. S. Gallego M. Ortuno C. Neipp I. Pascual J.. Kelly and J.T. Sheridan 3 Dimensional analysis of holographic photopolymers based memories Optics Express 13 3543 3557 (005). 13. J.R. Lawrence F.T. O Neill and J.T. Sheridan Adjusted intensity nonlocal diffusion model of photopolymer grating formation J. Opt. Soc. Am. B19 61 69 (00). 14. S.D. Wu and E.N. Glytsis Holographic grating formation in photopolymers: analysis an experimental results based on a nonlocal diffusion model and rigorous coupled-wave analysis J. Opt. Soc. Am. 0 1177 1187 (003). 15. R.S. Akopyan A.L. Aslanyan and A.. Galstyan About not monotonous hologram diffraction efficiency dependence in photopolymeric materials on average intensity of the writing laser J. Contemp. Phys. 39 37 330 (004). 16. H. Kogelnik Coupled wave theory for thick holographic gratings Bell Syst. Tech. J. 48 909 947 (1969). Opto-Electron. Rev. 15 no. 1 007 007 SEP Warsaw