In this paper we present results of experiments designed to increase our understanding of the photorefractive effect occurring during processes of dynamic hologram generation in Hybrid Photorefractive Liquid Crystal Structures (HPLCS). We also propose equivalent mathematical model which can be used to optimize those structures in order to obtain the highest diffraction efficiency in possibly shortest time.
© 2011 OSA
Dynamic holography also known as a real time holography is a technique that allows us to record and read information in the same time. It has found numerous photonic applications [1–3] still, the biggest problem facing scientists working in this field is to find new materials implementing general development trends of information technology e.g.: larger amounts of information, processed faster and at lower energy cost.
The materials used in dynamic holography technique must show an ability for rapid and reversible change of their refractive index accordingly to the incident light intensity pattern, which narrows the field of exploration to photorefractive and photochromic materials . Our studies presented in this manuscript are focused on hybrid organic structures based on liquid crystals, which attracted significant attention due to their strong optical nonlinearities at relatively low voltages and light intensities [5,6]. The mechanisms of photoinduced modulation of refractive index in liquid crystal structures was investigated by many research groups since 1994. Proposed models assumed that the photorefractive effect could be induced by photogeneration of electrical charges in the liquid crystal volume [7–10], connected this effect with generation and mobility of charges in the photosensitive layer, adjacent to the liquid crystal volume [11,12] or explained modulation of external electric field affecting the LC bulk orientation with photoinduced surface effects [13–15].
In contrast to the models proposed by other research groups the model presented in this paper describes the kinetics of the diffraction gratings generation processes in the form of mathematical formulas, which are versatile and could be applied to various photorefractive systems comprising photoconducting layers. It allows us to conduct experimental data approximation which provides information about the speed and the amplitudes of processes responsible for the reorientation of liquid crystal molecules.
First condition that must be met to record optical information as a hologram in the LC structure is preparation of the thin layer of the holographic material, which can be achieved by sandwiching liquid crystalline mixture between two glass plates separated with spacers . Such structure, known as liquid crystal panel is schematically shown in Fig. 1.a. The second requirement is the preliminary ordering of the LC molecules which can be performed by connecting the LC panel to the external electric field, resulting in the preliminary reorientation of the liquid crystal molecules along the field force lines . For samples preparation we have chosen nematic liquid crystal 1658 characterized at room temperature by birefringence Δn = 0.380 at 633 nm, positive dielectric anisotropy Δε = 15.6, crystallization temperature Tc = –20°C and isotropisation temperature Ti = 114°C . LC mixture was inserted into three identical pre–prepared cells made of two indium tin oxide (ITO) glass plates separated by 5μm thick polypropylene spacers, with one of the glass plates covered with a photoconductive layer of polyvinylcarbazole polymer doped with trinitrofluorenone (PVK + TNF).
Experimental setup shown in Fig. 1.b. was a typical two wave mixing system equipped with an additional laser used to read generated holograms. Measured variable was the time dependence of the reading beam intensity in the plus first diffraction order which was later converted into the diffraction efficiency of the system.
The diffraction efficiency was measured as a function of experimental setup parameters such as: applied voltage U, angle 2θ between writing beams and polarization of the electrodes. Polarization (+/–) translated to the sample setting, hence the location of the PVK + TNF photoconductive layer. Angle between writing beams 2θ had direct impact on the period values of recorded gratings and was experimentally set to 0.39, 0.26 and 0.11 rad which resulted in Λ = 0.98, 1.45 and 3.33 μm respectively. Measurements were taken for two different voltages: ±15 and ± 20V. Collected data have been approximated with the mathematical model described in section 3.
Electric charge carriers generated during irradiation of the Photorefractive Structures with Photoconducting Layers (PSPL) can move in a certain, pre-defined way, therefore we can predict their behavior. In the photoconductive layer, both holes and electrons can move along the lines of the electric field and/or diffuse. Consequently we have three possible processes that may occur in this layer and result in diffraction grating formation. The first one can be described as a flow of charges across the layer in the photorefractive medium direction and their removal, second one as the diffusion of these charges accordingly to the concentration gradient and the third one as a flow of charges to the electrode and their removal.
In thin photorefractive medium layer diffusion is rendered impossible due to the high electric field intensity, which means that the charges can only move towards the electrodes and be further removed. This limits the number of possible processes leading to grating generation to two.
Since each process corresponds to the diffraction grating formation, the behavior of the PSPL during hologram recording can be expressed as a reversible three–stage system. Stage LC0 is the state of LC in which LC molecules are under the influence of an external electric field and align themselves along the field force lines. In this state there are no diffraction gratings, but the orientation of LC molecules can be modified through the irradiation of the sample, leading to diffraction gratings generation, i.e. pushing the system to the state LC1 and LC2. Stage LC1 can be understood as a certain rearrangement of LC molecules caused by their reorientation on the direction of the evanescent electric field connected with the photoconductive layer, which is equivalent to the presence of the first diffraction grating. Stage LC2 can be understood as the reordering of LC molecules caused by the flow of electrical charges through the LC volume, which is equivalent to the presence of the second diffraction grating. Since each diffraction grating is formed due to the reorientation of certain amount of LC molecules in each state, we can write down the equations describing dynamics between LC molecules participating (LCi) and not participating (LC0i) in i-th grating formation:
The processes of gratings growth and decay are characterized by five rate constants k1, k'1, k”1, k2 and k'2 where ki = 1/τi. Those rate constants represent the transportation processes of certain electrical charges inside the PSPL which were described previously (three types of movement in the photoconducting layer and two types of movement in photorefractive medium). The rates at which LC molecules quantities change would be given as follows, where ci is the concentration of LC molecules responsible for i-th grating formation and Ai is the amplitude which determines the quantitative contribution of i-th grating.
The modulation of refractive index corresponding to the presence of each diffraction grating is proportional to the concentration of the molecules participating in gratings formation: Δni ~ci. Proposed model is dynamic, which means that it takes into account not only the steady state of the system, but also the state prior to its equilibrium, which is essential in description of such complex systems as photorefractives. It is in agreement with the diffraction gratings coupling theory , therefore modulation of the refractive index Δn can be expressed as the sum of coupling gratings vectors, where Χ is a cancelling function and φ is a coupling angle between the gratings:
The angle is time dependent and determines relative position of the gratings. It shifts from 0 to π with constant rate kφ and can be given as φ = π [1–exp(–kφ t)].
Cancelling function represents an additional, unknown process responsible for the diffraction efficiency decrease, which has the form X = exp(–kX ∙ t)]. It could be e.g. a consequence of reaching equilibrium state, when electrical charge carriers are generated and simultaneously removed from the system.
If the thickness of the holographic material is comparable to the recorded grating period, PSPL behavior has to be considered in the Raman–Nath regime , which connects diffraction efficiency η with modulation of refractive index Δn through the Bessel function J1, where d is the thickness of the holographic material, λ – the wavelength of the reading beam, Δn – modulation of the refractive index and β – the angle of the hologram readout.
Based on presented model, we proposed the following explanation of the phenomenon occurring during hologram recording and erasing processes in optically addressed Hybrid Photorefractive Liquid Crystal Structures. The first stage understood as first grating growth, starts right after the irradiation of the sample and for negative polarization can be divided into two following processes: generation of electrons and holes in the bright regions of PVK:TNF layer and holes removal on the electrode (–) which leads to the rapid growth of diffraction efficiency (Fig. 2.a). Simultaneously, the first grating is being erased by two parallel processes. First can be explained as diffusion of electrons into the dark regions of PVK:TNF layer accordingly to the gradient of concentration and their movement along the lines of the electric field in the direction of the liquid crystal layer (Fig. 2.b); second as the outflow of electrons from the PVK + TNF layer due to their injection into the LC volume (Fig. 2.c), which causes flow of negative ions through the sample volume towards the positive electrode. In this case this process is equivalent to the second diffraction grating growth, which is later erased by the outflow of charges on the positive electrode. (Fig. 2.d).
Charges flow leads to the generation of internal electric field, which in superposition with an applied voltage causes reorientation of liquid crystal molecules in the sample volume, and modulates refractive index of the LC layer.
Experimental results obtained for the sample with highest diffraction efficiency are presented together with theoretical calculations in Fig. 3 . Presented charts show, that we’ve got very good approximation results for different setup conditions. Approximated parameters were summarized in Tab.1 .
Approximation results show that for both polarizations the speed of the second erasing process associated with polymeric layer is independent on grating period and equals the speed of the first process associated with the LC layer (τ”1(+/–) = τ2(+/–), where superscript refers to polarization type). It confirms the model assumptions that in fact, it is the same process and that the second grating is being built at the expense of the first. One can also notice, that the speed of first diffraction grating built–up process for negative polarization equals to the speed of the second grating erasure process for the opposite polarization (τ1(–) = τ'2(+) and τ1(+) = τ'2(–)),which means that both processes are related to the certain charge type mobility. To determine which polarization concerns which charge type we can compare the speed of the individual processes. We can see that the speed of the τ1(–) = τ'2(+) is higher than speed of the τ1(+) = τ'2(–), consequently it will concern the flow of holes towards the electrodes and their removal from the system (in the photoconductive polymers such as PVK, the hole mobility is much higher compared to the electrons ). On the contrary, the outflow of electrons will be the driving force of the first diffraction grating built–up process for positive polarization, and the second grating erasure process for negative polarization (τ1(+) and τ'2(–)).
What’s more, the speed of first diffraction grating built–up process τ1(–) does not depend on the applied voltage and period of the grating, which means that the process of holes movement along the electric field lines is faster than the process of reorientation of LC molecules. We can also observe that the amplitude of each grating grows with the period as well as time constant of angle between gratings during the hologram recording. These dependencies allow us to control and adjust certain experimental set-up parameters which could be optimized for the best combination of high diffraction efficiency and quick material response. Results show that highest diffraction efficiency is observed for higher periods and lower voltages but at the expense of longer time required for hologram generation and erasure. Optimization must therefore be carried out on a compromise between both parameters.
Unfortunately we couldn’t find any explanation as to why time constants of external process cancelling both gratings calculated for –15V don’t follow any order in contrast to results obtained for –20V or why the external cancelling is not observed for + 20V at all.
Examined liquid crystalline systems can work as active optical components, hence they can be used in many optical devices based on parallel optical processes, i.e., devices using dynamic holography technique. We proposed mathematical model enabling the optimization of the parameters of these devices such as the individual set–up elements geometry or applied voltage. Presented mathematical model based on generation of electrical charges, their removal and mobility is versatile and can be used to describe any photorefractive system comprising photoconducting layers. It works best with liquid crystal structures as it was confirmed by obtaining strong convergence of theoretical function values and experimental data. As for polymer and other photorefractives research is yet to be done.
The authors wish to thank Wroclaw University of Technology (Fellowship co-financed by European Union within European Social Fund POKL ‘Mloda Kadra’) and the Polish Ministry of Science and Higher Education for financial support (Grants nos. NN 507 475237, IP2010 0277 70). J.M. and S.B. acknowledge the financial support by University of Angers and Region Pays de la Loire.
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