1. Introduction

Photorefractive (PR) materials have been extensively studied because of their large optical nonlinearity and wide range of potential applications, such as holographic recording, optical image processing, phase conjugation, spatial filtering, beam amplification, and others [1, 2]. Research on PR effects has been focused exclusively on the inorganic photorefractive crystals such as LiNbO₃, BaTiO₃, InP, GaAs and SBN. However, since Rudenko and Sukhov proposed and demonstrated the PR effect in dye-doped nematic liquid crystals (NLCs) [3], considerable progress has been achieved in the PR performance of these materials [4-7]. In particular, Khoo et al. observed a director axis reorientation effect induced by the space charge field in dye-doped NLCs [8, 9]. They discussed all contributing processes of space charge field formation, the resulting torques, the director axis reorientation and optical wave mixing effects. The electro-optic responses in inorganic materials originate from the Pockels effect or linear electro-optic effect, while in dye-doped NLCs the electro-optic responses come from the quadratic electro-optic effect due to the director axis reorientation of the LCs, so called ‘orientational photorefractive (OPR) effects’ [9]. It is known that pure PR (PPR) effects are attributed to the fast electronic and/or ionic processes, whereas the OPR effects are due to the slow molecular reorientational motions. Janossy et al. also presented that the optical torque increases significantly when small amounts of appropriate absorbing dyes are added to NLCs [10]. Several dopant dyes such as Methyl red, C₆₀ and carbon nanotubes have been known effective to increase the OPR effect [11-13].

The purpose of this work is to derive the transient behaviors of the OPR gratings via director axis torque of NLCs, which is caused by fast pure PR gratings in conjunction with applied electric field and to compare with the experiments. Dependences of transient OPR holographic gratings on the applied dc field for various grating periods and intensities of the writing beams are investigated in porphyrin:Zn-doped NLC cells.

2. Theory

2.1 Kinetics of space charge field gratings in nematic liquid crystals

The material equations for NLCs are given by [3]

where n ^± are the positive and negative charge carrier densities, γ_R is the recombination rate, J ^± are the current densities, µ ^± are the mobilities, α is the charge generation rate, e is the elementary charge, ε is the relative dielectric constant, ε ₀ is the dielectric constant in the vacuum, k_B is the Boltzmann’s constant, T is the absolute temperature, I is the light intensity, and E is the total electric field, consisting of the applied electric field E ₀ and the induced space charge field E ₁. Eq. (1a) represents the rate equations for the positive and negative charge carrier densities, Eq. (1b) is the total current density equations, consisting of contributions from the drift of charge carriers due to the electric field and from the diffusion due to the gradient of carrier density and Eq. (1c) is the Poisson equation. Considering the two coherent writing beams incident onto dye-doped NLCs, as shown in Fig. 1, the light intensity distribution for a grating formation is then given by

where $m=\frac{2\sqrt{I_a{I}_b}}{\left(I_a+I_b\right)}$ is the modulation depth, I _a and I _b are the incident intensities of the writing beams, I ₁(t)= mI ₀(t), I ₀=I_a+I_b is the total input intensity, q is the grating wave vector, q=|q|=2π/Λ_g and Λ_g is the grating period, and c.c. is the complex conjugate.

 

Fig. 1. Geometry for writing orientational photorefractive hologram in porphyrin:Zn-doped NLCs sample. I_a and I_b are intensities of writing beams, θ _inc is the incident half-angle between two incident beams, β is the tilt angle, q is the grating vector, and E ₀ is the applied electric field, parallel to the z-axis.

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We assume that the physical variables used in Eqs. (1) take the same periodic function with the intensity distribution I(r,t) as:

where Y_i(i=0, 1) stands for the variables n ^±, J ^± and E, and Y₁(t)=mY ₀(t). Substituting Eqs. (2) and (3) into Eq. (1) and separating the variables with subscripts 0 and 1, yields the following equations for the subscript 0:

and for the subscript 1:

where n ₀ is the average (spatially uniform) value for the positive and negative charge carrier densities, Δn=n ⁺ ₁-n ^- ₁ and D ^±=k_BTµ ^±/e is the diffusion coefficient. It is noted that in deriving Eqs. (5) we neglect the cross products of two quantities with subscript 1, which are the nonlinear driving sources of some interesting phenomena such as spatial subharmonic instability observed in inorganic PR crystals like BSO [14, 15]. Assuming the recombination rate γ_R between opposite ions, which is inversely proportional to the photo-ion lifetime is very large, Eq. (4a) with Eq. (4b) reduces to $n_0=\sqrt{\frac{\alpha I_0}{{\gamma }_R}}$ _R for the steady state. Eliminating n ⁺ ₁(t) and n ^- ₁(t) from Eqs. (4) and (5), after some lengthy calculations, we obtain the following differential equation for the space charge field E ₁.

where a=γ_Rn₀+iq_zµ ⁺ E ₀+q ² D ⁺, b=γ_Rn₀-iq_z µ ^- E ₀+q ² D ^-, c=γ_Rn ₀, q_z=qsinβ and β is the tilt angle. The applied dc field E ₀ and q are related by

and the space charge field E ₁ has the same direction with q, where q̂‖ and q̂ _⊥ are the unit vectors parallel and perpendicular to the grating wave vector q, respectively, and β is the tilt angle. From Eqs. (6) and (7), we readily obtain the following scalar second-order differential equation for E ₁.

with $A=\frac{1}{{\tau }_d}\left(1+\frac{2{\tau }_d}{\tau }+\frac{E_D}{E_q}+i\frac{E_0\sin \beta }{E_q}\nu \right)$,

$B=\frac{2}{{\tau }_d\tau }\left(1+\frac{E_D}{E_M}+\frac{E_D}{2E_q}+\frac{{E_0}^2{\sin }^2\beta }{2E_q{E}_M}+\frac{{E_D}^2}{2E_q{E}_M}+i\frac{E_0\sin \beta }{2E_q}\nu \right)$,

$C=\frac{1}{{\tau }_d\tau }\left(i{E}_D\nu -E_0\sin \beta \right)$,

where µ=µ ⁺ µ ^-/(µ ⁺+µ ^-), ν=(µ ⁺-µ ^-)/(µ ⁺+µ ^-)=(D ⁺-D ^-)/(D ⁺+D ^-), τ=1/(γ_Rn ₀) is the photoion lifetime, τ_d=εε ₀/[en ₀(µ ⁺+µ ^-)] is the Maxwell relaxation time, E_D=k_BT_q/e is the diffusion field, E_M=γ_Rn ₀(µ ⁺+µ ^-)/(qµ ⁺ µ ^-)=γ_Rn ₀/(qµ) is the drift field, and E_q=en ₀/qεε ₀ is the saturating field. If we take the slowly-varying amplitude approximation of E ₁ in time, the second order derivative, d ² E ₁/dt ², can be neglected and then Eq. (8) becomes

with $g=\frac{B}{A}=\frac{2}{\tau }\frac{\left(1+\frac{E_D}{E_M}+\frac{E_D}{2E_q}+\frac{{E_0}^2{\sin }^2\beta }{2E_q{E}_M}+\frac{{E_D}^2}{E_q{E}_M}+i\frac{E_0\sin \beta }{2E_q}\nu \right)}{\left(1+\frac{2{\tau }_d}{\tau }+\frac{E_D}{E_q}+i\frac{E_0\sin \beta }{E_q\nu }\right)}$,

$h=\frac{C}{A}=\frac{1}{\tau }\frac{\left(i{E}_D\nu -E_0\sin \beta \right)}{\left(1+\frac{2{\tau }_d}{\tau }+\frac{E_D}{\mathit{Eq}}+i\frac{E_0\sin \mathrm{}\beta }{E_q}\nu \right)}$.

The transient solution for E ₁ can then be written as

The steady state space-charge field E ₁(∞) is given by

where X=1+E_D/E_M+E_D/(2E_q)+E ² ₀sin² β/(2E_qE_M)+E ² _D/(2E_qE_M) and Y=E₀νsinβ/(2E_q). Using Eq. (11), we obtain the magnitude of the steady state space-charge field |E ₁(∞)| and the phase shift ϕ between the space-charge field grating and the intensity grating as follows

 

Fig. 2. (a) Complex representation of the steady state space charge field E ₁(∞) as positively (E _o) and negatively (-E _o) increasing the applied dc field and (b) the phase shift variation ϕ against the applied dc field for various grating periods at I ₀=220mW/cm².

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The applied dc field E ₀ not only changes the magnitude of the space-charge field |E ₁(∞)|, but also alters the spatial phase ϕ. Figure 2(a) shows the complex representation of the steady state space-charge field. As increasing the applied dc field, the magnitude of the space charge field |E ₁(∞)| gradually increases to a maximum value and then rapidly diminishes, irrespective of the direction of the dc field E ₀. Figure 2(b) represents the phase shift ϕ against E ₀ for various grating periods in Bragg region. The spatial phase shift plays a key role in the energy transfer in the two beam coupling. For the case of E ₀=0 in Eq. (13), the space charge gratings are spatially shifted by ϕ=90° relative to the intensity gratings. As positively (negatively) increasing the applied dc field, however, the phase shift ϕ steeply approaches to 180° (0°) as shown in Fig. 2(b), and then the PR effect disappears. Similarly, Fig. 3(a) and 3(b) show the complex representation of space-charge field and the phase shift against E ₀ for various total input beam intensities, respectively. As the total beam intensity I ₀ increases, the charge carrier densities n ₀ and |E ₁(∞)| also increase and consequently PR effect is enhanced. In plotting Fig. 2 and Fig. 3, we have used the following parameters: E_d=0.106 V/µm, E_c=1.44 V/µm, E_m=0.270 V/µm and E_q=0.295V/µm when Λ_g=1.48µm and I ₀=250mW/cm².

 

Fig. 3. (a) Complex representation of the steady state space charge field E ₁(∞) as positively (E _o) and negatively (-E _o) increasing the applied dc field and (b) the phase shift variation ϕ against the applied dc field for various input intensities at Λg=1.48µm.

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2.2 Orientational photorefractive gratings induced by director axis reorientation of NLCs

In this section, we will derive the kinetics of the OPR gratings via director axis torque of NLCs, which is caused by fast pure PR (PPR) gratings in conjunction with applied electric field, as will be seen below. The underlying physical origins of PPR gratings are attributed to the fast electronic and/or ionic processes, whereas the OPR gratings are due to the slow molecular reorientational motions. Therefore, it is quite natural to assume that the response time (or the grating formation time) of the PPR grating is much faster than that of the OPR grating. Keeping this in mind, we only consider the steady state value of the PPR gratings. As in [9], we define an angle θ as a director axis reorientation angle, where θ is the angle between the direction of the applied dc field (i.e., z -direction in Fig. 1) and the reoriented director axis of NLCs, being a spatially and temporally varying. Using the small reorientation angle approximation (θ≪1) with the one elastic constant K, the torque balance equation is given by [16]

γ_vis is the Leslie viscosity coefficient, |Γ _E|=Δεε ₀|n′·E(n′×E) is the magnitude of the director axis torque, which is induced by the applied dc electric field E ₀ and the steady state space charge field E ₁(∞), and n′ is a unit vector parallel to the reoriented director axis of NLCs. n′ and E are written as

For small reorientation angle (i.e., θ≪1), |Γ _E| is approximately given by

where Δε=ε‖-ε _⊥ is the dielectric anisotropy. We take a trial solution of the Eq. (14) as

It should be emphasized that since no surface treatments are made to NLCs sample, the director axis orientations of our sample are random before they are subject to any illumination or applied electric field, so that we can not impose the hard boundary condition unlike used in the literatures [8, 9, 16]. Substituting Eq. (17) with Eqs. (15) and (16) into Eq. (14)) and equating the coefficient of cos(q·r+ϕ) to zero we obtain the following first-order differential equation for θ ₁(t)

with $\frac{1}{{\tau }_g}=a\left({E_C}^2+{E_0}^2\right)\mathrm{and}{\theta }_1(\infty )=\frac{E_0\mid E_1(\infty )\mid }{E_C^2+E_0^2}\cos \beta$,

where a=Δεε ₀/γ_vis and $E_C=\sqrt{\frac{K{q}^2}{\left(\mathrm{\Delta }\epsilon {\epsilon }_0\right)}}$ is a critical field, which is analogous to the Freedericksz transition field, and τ _g is a characteristic time constant, which is related to the OPR grating formation and erasing processes. It should be noted that since |E ₁(∞)| is proportional to the modulation depth, $m=\frac{2\sqrt{I_a{I}_b}}{\left(I_a+I_b\right)}$, of the two writing beams as in Eq. (11), the right hand side of Eq.(18), that is a source term giving rise to OPR gratings, also depends on m as well as E ₀. In case when two writing beams are turned on (i.e., m≠0), which corresponds to the OPR grating formation process, θ ₁(t) is then given by

After reaching to a steady state value of θ ₁(∞), if one of the two writing beams is turned off (i.e., m = 0), which corresponds to the OPR grating erasing process, using Eq.(18) θ ₁(t) can be written as

It is interesting to point out that according to Eqs. (19) and (20) the grating formation time is equal to the grating erasing time as long as the same magnitude of applied dc field maintains for these two processes. Since the NLC molecules have a uniaxial symmetry, the refractive index for the extraordinary wave is given by

where n _‖ and n _⊥ are the refractive index for field parallel and perpendicular to the director axis, respectively. As a result of the orientational birefringence of the director axis, the induced extraordinary wave phase grating (i.e., the OPR grating) is defined as Δn(t)=n_e(β+θ)-n_e(β) [9] and can be calculated by using small reorientation angle approximation (i.e., θ≪β) as

where δn ₁(t) is the amplitude of the OPR grating and is given by

Here, the steady state value of the amplitude of the OPR grating is given by

It is seen from Eq. (24) that the OPR gratings are rather quadratic electro-optic effect, which is associated with the space charge field and the externally applied field as well the critical field. The diffraction efficiency η for the OPR gratings can be written as [2]

where d is the sample thickness, λ_r is the wavelength of the reading beam and θ_B is the Bragg angle of the reading beam. Since δn ₁(∞) depends on the direction of E ₀ and β in Eqs. (23) and (24), it is clear that δn ₁(-E ₀)=-δn ₁(E ₀) and δn ₁(-β)=-δn ₁(β), but the diffraction efficiency η does not influenced by the reverse directions of E ₀ and β because of square of the sine function.

3. Experimental results and analysis

We fabricated porphyrin:Zn-doped NLC cells filled by capillary phenomenon between two indium tin oxide (ITO) coated glass substrates with _20µm-thick beads as a spacer. The NLCs, E7, were purchased from Merck Korea, which have the dielectric anisotropy Δε=13.8, the elastic constant K ₁₁=1.11×10^-11N and K ₃₃=1.71×10-¹¹N at room temperature, and wavelength λ=589 nm. We made no surface treatments to NLCs sample, so the director axis orientations of our sample are random before they are subject to any illumination or applied electric field. Zn-doped porphyrin dye [5, 10, 15, 20-tetraphenylporphyrinatozinc (ZnTPP)] was supplied by Busan National University and is photosensitive to blue-green wavelength region. The Zn-doped porphyrin dye was added to enhance the OPR effects, so-called dye effect and the concentration of the dopant dye in NLCs was _{0.5 wt%}. Experimental setup for measuring the diffraction efficiency and two beam coupling (TBC) gain is schematically shown in Fig. 4. We used two coherent and p -polarized Ar-ion laser beams (λ_w=514 nm) as two writing beams and the intensity beam ratio was kept to be unity (i.e., m=1). An incoherent He-Ne laser beam (λ _r=633 nm) was used for measuring the real-time diffraction efficiency and the intensity of the reading beam was 4.2mW/cm². The polarization angle of the reading beam was controlled by a λ/4-plate and a polarizer. A tilt angle β was +35° and a dc field, ranging from 0 to 2.0 V/µm, was applied across to the sample. The Bragg angle of the reading beam θ_B and the grating period Λ_g were determined by the incident half angle θ_inc of the writing beams.

 

Fig. 4. Experimental setup for measuring real-time diffraction efficiency and two beam coupling gain of NLC sample (BS : beam splitter, M1~M4 : mirrors, D1~D4 : detectors).

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Fig. 5. Typical experimental data for two beam coupling experiment at E ₀=1.2V/µm.

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In order to check out whether our sample exhibits photorefractivity or not, we performed a two beam coupling experiment in the Bragg diffraction regime. Figure 5 represents a typical experimental result for TBC. The asymmetric energy transfer is clearly seen by the decrease in the intensity I_a beam and increase in that of I_b beam, revealing the photorefractive nature of our sample. Figure 6 shows real-time diffraction efficiencies for OPR holographic gratings at the grating period of Λ_g=1.0µm. The solid lines are theoretical curves of Eq. (25) with Eqs. (23) and (24), at which the arrows (↓) represent the moment one of the two writing beams turned off. As theoretically predicted, the grating formation time is equal to the grating decay time during the same magnitude of applied dc field maintains. As the applied dc electric field increases, the diffraction efficiency and the inverse of the grating characteristic time also increase. To investigate the dependence of the diffraction efficiency and the grating characteristic time on the grating periods, we also measured the real-time diffraction efficiencies at different grating periods of Λ_g=1.24µm and Λ_g=1.48µm. It is shown that the diffraction efficiency is greatly enhanced up to about 20% at Λ_g=1.48µm and E ₀=1.5V/µm, as shown in Fig. 7.

 

Fig. 6. Real-time diffraction efficiencies for a grating period of Λ_g=1.0µm. The solid lines are the theoretical curves. The arrows (↓) represent the moment one of writing beams turned off.

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Fig. 7. Real-time diffraction efficiencies for a grating period of Λ_g=1.48µm. The solid lines are the theoretical curves. The arrows (↓) represent the moment one of writing beams turned off.

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The experimental data with the theory support out assumption that the transient behaviors of diffraction efficiencies do not come from fast PPR gratings but from slow OPR gratings. The dependence of 1/τ _g on the applied dc field for various grating periods is shown in Fig. 8 and the solid lines are theoretical predictions of Eq.(18), in which 1/τ _g is proportional to E ² ₀ and is inversely proportional to Λ² _g. From the best curve fittings with the experimental data, we get a critical field of E_C≅1.43 V/µm for Λ_g=1.0µm and the constant of a=Δεε ₀/γ_vis=98±2µm²/(V²·s), yielding the viscosity coefficient of γ_vis=1.24±0.02Pa·s. This value of γ_vis in porphyrin:Zn doped NLCs is four times larger than that of undoped NLCs (E7), which has γ_vis=0.3Pa·s at 18°C [17].

 

Fig. 8. Dependence of 1/τ _g on applied dc electric field at various grating periods. The solid lines are the theoretical predictions of Eq.(18).

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Figure 9 represents the steady state values of the diffraction efficiencies against the total writing beam intensity I ₀ at E ₀=1.4 V/µm and Λ_g=1.48µm. We have a maximum diffraction efficiency of η=42.2% at I0=700mW/cm² and 0.5 wt% of the dye concentration. For I ₀≈1000mW/cm², the diffraction efficiency reaches η≈85% theoretically, as shown in Fig. 9.

 

Fig. 9. Semilog plot of diffraction efficiencies as a function of total writing beam intensity at E ₀=1.4V/µm and Λ_g=1.48µm. The solid line is the theoretical curve.

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Figure 10 depicts the diffraction efficiencies against the applied dc field for several total writing beam intensities at Λ_g=1.48µm with the theoretical predictions, showing good agreements. From the theoretical curves with the experimental data, we get an optimum electric field, E _{0, opt}, which is defined by the applied dc field to obtain maximum diffraction efficiency. When the total writing beam intensities increase, say, 178mW/cm², 252mW/cm², and 414mW/cm², the optimum electric fields also slowly increase and are given by 1.06V/µm, 1.14 V/µm, and 1.25 V/µm, respectively.

 

Fig. 10. Diffraction efficiencies against applied dc field for several writing beam intensities at Λ_g=1.48µm. The solid lines are the theoretical curves of Eq.(25) with Eq.(24).

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The dependences of the diffraction efficiency on the grating period Λ_g are also shown as Fig 11. We also obtained the optimum electric field, E _{0, opt} for various grating periods. When the grating periods Λ_g increase, say, 1.00µm, 1.24µm, and 1.48µm, the optimum electric fields rather decrease unlike Fig. 10 and are given by 1.31V/µm, 1.19 V/µm, and 1.14 V/µm, respectively.

 

Fig. 11. Diffraction efficiencies against applied dc field for several grating periods at I ₀=224mW/cm². The solid lines are the theoretical curves of Eq.(25) with Eq.(24).

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We have used the following physical parameters from the best curve fittings for all experimental results; E_d=0.106 V/µm, E_c=1.44 V/µm, E_m=0.270V/µm and E_q=0.295 V/µm when Λ_g=1.48µm and I ₀=250mW/cm². The amplitude of the OPR grating is estimated as δn ₁(∞)≈7×10^-3 and the nonlinear index coefficient n ₂, defined by δn ₁(∞)=n ₂ I ₀, is n ₂≈10^-2 cm²/W when E ₀=1.4 V/µm, Λ_g=1.48µm and I ₀=700mW/cm².

In summary, we found that a porphyrin:Zn-doped nematic liquid crystal sample reveals a photorefractivity owing to asymmetric two beam energy couplings and also observed a high diffraction efficiency of up to η=42.2% for values of the grating period of around Λ_g=1 µm, much smaller than the sample thickness (d=20µm). This implies that our orientational photorefractive gratings correspond to the Bragg diffraction or thick grating regime. The Q parameter, defined as Q=2πλd/(n _oΛ² _g), where d is the sample thickness, λ is the wavelength of light in vacuum, n _o is the linear refractive index and Λ_g is the grating period, has been used as a criterion for the Bragg and Raman-Nath regimes [2]. Values of Q<1 are believed to be the Raman-Nath or thin grating regime, while large values of Q(Q>10) to be the Bragg or thick grating regime. We roughly estimate Q≈40 for the experimental conditions, which confirms the Bragg diffraction regime. Finally, it should be mentioned that the diffraction efficiency of the OPR gratings has been known to have a maximum for a grating period about twice the sample thickness (i.e., Λ_g=2d) and to decrease dramatically for grating periods below the sample thickness, as described in [9] and others, which is obviously contrary to our results. In addition, for this situation, the diffraction grating is apt to be the Raman-Nath regime with multiple higher order diffractions, deteriorating the diffraction efficiencies. We believe that it is owing to the fact that the director axes of the NLCs anchor to the surfaces of the sample via a sample treatment like a rubbing, so the hard boundary condition (i.e., θ=0 at z=0 and z=d) has to be imposed. One the other hand, we made no surface treatments to NLCs sample, being random director axis orientations before any illumination or applied electric field, which is connected to the real nature of the OPR grating and has relevant implications for possible applications, since it basically would mean that the resolution of photorefractive gratings obtained using liquid crystals can be much higher.

4. Conclusions

We fabricated a porphyrin:Zn-doped nematic liquid crystal sample and investigated holographic diffraction properties by measuring the time dependent diffraction efficiency of the OPR gratings for various grating periods and total beam intensities under the influence of applied dc field. Based on the material equations and the torque balance equation of the director axis reorientation of NLCs, we have also theoretically derived the expressions for the diffraction efficiency and compared with the experimental data, showing excellent agreements.