1. Introduction

Photorefractive materials are among the most promising materials being used in optical signal processing. Potential applications of photorefractive materials include real-time holography, image amplification, laser beam steering, optical interconnections, and holographic memory, to name a few. To gain better understanding of both the physical mechanism of the holographic grating formations and the intrinsic material parameters of photorefractive materials, it is of particular important to investigate and learn about the mixed gratings (phase and absorption) and phase shifts with respect to the intensity grating.

As a means of measuring photorefractive grating in photorefractive materials, many researchers proposed several experimental techniques based on two wave-mixing geometries [1–6]. Sutter and Günter [2] presented a relatively simple method based on two-beam coupling experiment, often called grating translation technique, which relies on the fact that the sample should be translated in a time less than the grating growth or decay times along the grating wave vector. The grating translation technique has two distinctive features: it allows us to easily determine (i) the amplitude of both photorefractive and photochromic gratings and (ii) their phase shifts with respect to the intensity grating. Kwak and Lee [6] also presented an approximated solution to determine the amplitudes and phases of photorefractive and photochromic gratings by adopting the grating translation technique. The grating translation technique, however, was adopted within a very short translation time, and the grating phase shift ϕ_g against the intensity grating was just replaced by ϕ_g+Ωt, where Ω=K_gv is the frequency of the moving beam fringe pattern, K_g is the magnitude of the grating wave vector and v is the moving velocity of the sample by the grating translation. In this paper, we have experimentally found that when the grating translation time is comparable to the material characteristic times such as grating growth or decay times, or in case of very long translation time, transient two beam coupling gains behave as transient damped harmonic oscillations. In this situation, simply replacing the grating phase shift ϕ_g by ϕ_g+Ωt can not be true and that the theory for the grating translation technique developed in [2, 6] breaks down.

The purpose of this study was twofold: first, to derive an analytic expression of the space-charge field for the grating translation technique that is valid for all time regions whether or not the grating translation time is less than the material characteristic times; and second, to compare our findings with the experimental results. In section 2, we will briefly review the two wave mixing for moving gratings in photorefractive crystal. In section 3, from the Kukhtarev material equations [7] we will derive the analytic expression of the transient space charge field for the moving grating under the influence of the external applied field and represent the kinetics of the space charge field in a complex plane. In section 4, we will analyze the experimentally measured transient gains based on our theory, and in section 5, we will summarize the results.

2. Two wave mixing for moving grating or grating translation in photorefractive crystal

We consider the grating translation technique [2] or the moving grating technique [8, 9] by means of two wave mixing in a photorefractive crystal. For a moving interference pattern, which is caused by translating the crystal parallel to the grating wave vector K_g using a PZT transducer with a velocity v or by a frequency detuning one of the two input beams using a PZT mirror by an amount Ω, the intensity grating can be expressed as

where I ₀=I_P+I_S is the total incident beam intensity, I_P and I_S are the pump and the signal beam intensities, respectively, I ₁=mI ₀, $m=2\frac{\sqrt{I_P{I}_S}}{\left(I_P+I_S\right)}$ is the modulation depth between the two writing beams, Ω=K_gv, K_g is the grating wave vector, and c.c. stand for the complex conjugate. The time-dependent refractive index change induced by the moving intensity grating can then be written as [10]

where n ₀ and n₁ (t) are the linear refractive index and the amplitude of nonlinear refractive index change, respectively, ϕ_g is the initial photorefractive grating phase shift relative to the intensity grating, Ψ(t) is the additional phase shift induced by the moving grating or the grating translation periods. For simplicity, we neglect the static phase difference between the pump and signal beam amplitudes in Eq. (2). The nonlinear refractive index change n ₁ (t) is related to the space charge field E₁(t) as [7, 10, 12]

where n_b is the background refractive index of the crystal, r_eff is the effective electro-optic coefficient depending on the incident beam polarization and crystal orientation, and |E ₁(t)|is the absolute value of the time-dependent space charge field induced by the intensity grating. In general, the photorefractive and the photochromic gratings are simultaneously generated in a photorefractive crystal [6]. In a low absorption medium, however, the influence of photorefractive grating is dominant to the photochromic grating in two beam energy coupling. For this case, the time-dependent pump beam and the signal beam intensities for two wave mixing are well known as [7, 8, 10, 14]

where α is the linear absorption coefficient, θ is the half-angle between the two incident beams inside the crystal, I_P(z=0) and I_S(z=0) are the pump and signal beam intensities at entrance face of the crystal, respectively, β=I_P(0)/I_S(0) is the incident beam intensity ratio, and z=L is the medium thickness. Here, the time-dependent gain coefficient Γ(t) is given by [7, 8, 10]

where λ is the wavelength of the light. From Eq. (4c), the gain coefficient is directly proportional to the imaginary part of space charge field. It should be noted that Eqs. (4a) through (4c) do not take into account all effects of anisotropies of the material and are valid only in a symmetric situation.

3. Derivation of the transient space charge field for the grating translation

In this section, we will derive an analytic expression for describing the transient behaviors of photorefractive moving grating with an external electric field. Kukhtarev’s band transport model [7] of the inorganic photorefractive crystal is known as the standard model for the photorefractive effect. In order to elucidate the physical essence of the transient space charge field, we only consider a single charge carrier, say electron, model and neglect the effect of electron-hole competition of the material, if any. A set of the diffusion dominant material equations may be written as [7]

where N ⁺ _D and N_D are the ionized donor density and the donor density, respectively, N_A is the acceptor density, N is the electron density, J is the current density, E is the electric field, s is the cross section of photoionization, γ_R is the recombination constant, ε ₀ is the free space permittivity, ε is the dielectric constant, µ is the mobility, k_B is the Boltzmann constant, T is the absolute temperature, and e is the electron charge. Equations (5a) and (5b) describe the time dependent charge density equation and the continuity equation, respectively, while Eqs. (5c) and (5d) represent the Poisson equation and the current density equation, respectively. In Eq. (5d), we neglect the photovoltaic current density, for simplicity. We assume that all the physical variables used in Eqs. (5a) to (5d) are of the same periodic function of space and time as the intensity grating in Eq. (1) and take the following form

where F_i(t) (i=0, 1) stands for the physical variables such as N ⁺ _D, N, J and E. Following the same procedure of [8, 11] and substituting Eq. (6) into Eqs. (5) and eliminating all the variables except for the space charge field E ₁(t), we can get the differential equation for the space charge field of the moving grating of the form

where τ=1/γ_RN_A is the photoelectron lifetime, τ_d=ε₀ε/eµN ₀ is the Maxwell relaxation time, E_D=k_BTK_g/e is the diffusion field, E_M=γ_RN_A/µK_g is the drift field, and E_q=eN_A/εε₀K_g/e is the limiting space charge field. In obtaining Eqs. (7), the average electron density N ₀ is assumed to be a constant with time and is given by N ₀=sI₀N_D/γ_RN_A and the second-order time derivative term for E ₁ (t) is neglected by using the slowly varying amplitude approximation. Kwak et al. [11] derived a similar form of Eqs. (7) that contains the terms of time derivatives of applied alternating electric field, ∂E₀/∂t. Equations (7), however, contains new terms not only the time derivatives of the external applied electric field but the moving frequency Ω of the incident beam fringe pattern.

For simplicity, in what follows we will restrict our attention to the case of applied DC field (i.e. ∂E₀/∂t=0) and solve Eqs. (7) to obtain the analytic solution of the transient space charge field. The problem is divided into two stages: the first one is that the photorefractive grating is formed without the grating translation (i.e., Ω=0) and the other is that once the grating reaches to its steady state the grating translation starts. Before proceeding further, it is convenient to define the new parameters as g(Ω=0)≡g _o=1/τ_g+jω_o and g(Ω≠0)≡g=g_o-jΩ, where the characteristic response time τ_g and the characteristic frequency ω_o of the photorefractive material are given by

Note that ω_o=Im[g_o] becomes zero when applied field is absent (i.e. E_o=0).

3.1(i) First stage: Grating formations with no grating translation (i.e., Ω=0)

At first stage, the transient solution to Eqs. (7) is simply given by

where E ₁₀=h/g ₀ is the steady state value of the space charge field and is given by

The amplitude |E ₁₀| and the grating phase shift ϕ_g with respect to the intensity grating are given by, respectively,

which are equivalent to the well known expressions [12]. In this situation the transient gain coefficient becomes

, which reveals an overdamped oscillatory behavior with a response time τ_g and a frequency ω_o. The gain coefficient evolves no oscillatory behavior with time when an applied field is absent (i.e. ω_o=0).

3.2 (ii) Second stage: Grating translations (i.e., Ω≠0) after first stage

The general solution to the second stage (i.e., Ω≠0), after reaching a steady state value of the space charge field without a grating translation, can then be calculated by using Eq. (7) with an initial condition of E ₁(0)=mE ₁₀ at time t=0 and is given by

where Ψ(t) is an additional phase shift induced by during a grating translation and g* is complex conjugation of g=g_o-jΩ. Equation (11) are the time-dependent solution of the space charge field and phase shift with moving fringes and represent the damped oscillatory behaviors during the grating translation. In order to show the validity of our theory, we find out the steady state value of Eqs.(11). For the final steady state (i.e. t→∞), the moving space charge field, after some calculations, becomes

or, more explicitly

where b=Ωτ_g is the detuning or the dimensionless moving frequency. It should be noted that Eqs. (12) recover the same expression of the well known standard theory of moving gratings for the steady state [8, 9].

We are now in a position to discuss the transient behaviors of the moving grating Eqs. (11). The effect of the moving frequency Ω of the grating translation on the transient space charge field can be easily understood in the complex plane. It is convenient to define the normalized space charge field Z(t,Ω)=X(t,Ω)+jY(t,Ω) where X(t,Ω)≡Re[E ₁(t,Ω)/(m|E ₁₀|)] and Y(t,Ω)≡Im[E ₁(t,Ω)/(m|E ₁₀|)]for graphical representation in a complex plane. For the steady state, we obtain the following parametric equation by using Eqs. (11) or Eqs. (12) as

where

The resonance frequency Ω_RES=ω₀ is defined as the frequency where the absolute value of the space charge field at the steady state, |Z(∞,Ω)|≡|E ₁(∞,Ω)|/(m|E ₁₀|), has a maximum. Equations (13) represent a circle centered at (X _RES/2, Y _RES/2) with a radius of |Z _RES|/2 in complex plane, which implies that the transient space charge field eventually resides in a point on the circle at the final steady state time.

 

Fig. 1. Representations of the complex space charge field during the grating translation when an external DC field applies (i.e., E ₀=1 kV cm). (a) Complex representation of the space charge field. The circle exhibits the steady state representation and the spirals depict the transient behaviors for each moving frequency, (b) the imaginary part of the space charge field Y(t,Ω), (c) the amplitude of the space charge field |Z(t,Ω)and (d) the phase shift Ψ(t,Ω) with an initial grating phase ϕ_g against dimensionless time t/τ_g for various r=Ω/Ω_OPT, where the moving frequency Ω is divided by the optimum frequency Ω_OPT, where Ω_OPT=4 Hz and τg=4.3×10^-2 s are used for calculations.

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Figure 1(a) depicts a circle and some spiral curves in the complex plane for the case of external field present (i.e., E ₀≠0). The spiral curves depict the transient behaviors of the space charge field in the complex plane during the grating translation stage for each dimensionless frequency r=Ω/Ω_OPT, which is defined as the moving frequency Ω divided by the optimum frequency Ω_OPT. The optimum frequency is defined as the frequency where the magnitude of the imaginary part of space charge field, Y(∞,Ω)=Im[E ₁(∞,Ω)/(m|E ₁₀|)], has a maximum value. The optimum frequency Ω_OPT can be readily derived from the relation of X(∞,Ω_OPT)=X _RES/2 in Eq. (13a) and is given by

where ϕ_g is an initial grating phase shift relative to the intensity grating before launching the grating translation and is given by Eq. (9d). The maximum value of the imaginary part of space charge field is also given byY(∞,Ω_OPT)=(Y _RES+|Z _RES|)/2. Figure 1(b) in conjunction with Fig. 1(d) describes the transient behaviors of two wave mixing gain, whereas Fig. 1(c) exhibits the temporal evolutions of the diffraction efficiency during the grating translation. It is well known that there are several kinds of method that could enhance the two wave mixing gain in photorefractive crystal. One of the most effective methods is the applied DC field accompanying with the moving grating [8, 9]. Figure 1(b) also shows the enhancement of the imaginary part of the space charge field when the moving frequency approaches to the optimum value (i.e., r→1), in which the parameters used in calculations are not optimized to have a maximum gain enhancement.

 

Fig. 2. Representations of the complex space charge field during the grating translation when no external field applies (i.e., E ₀=0kV/cm). (a) Complex representation of the space charge field. The circle exhibits the steady state representation and the spirals depict the transient behaviors for each dimensionless moving frequency b=Ωτ_g, (b) the imaginary part of the space charge field Y(t,Ω), (c) the amplitude of the space charge field |Z(t,Ω) and (d) the phase shift Ψ(t,Ω) with an initial phase ϕ_g against dimensionless time t/τ_g for various b values. In the case of no external field, the optimum frequency Ω_OPT becomes zero.

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Figure 2(a) depicts a circle for the case of no external applied field (i.e., E ₀=0). The spiral curves depict the transient behaviors of the space charge field in the complex plane during the grating translation stage for each dimensionless moving frequency b=Ωτ_g. The transient behaviors of Y(t,Ω), Z(t,Ω) and Ψ(t,Ω) are also shown in Figs. 2(b), 2(c) and (2d), respectively, for various b=Ωτ_g. It should be emphasized that the initial grating phase shift ϕ_g relative to the intensity grating can not be replaced by ϕ_g+Ωt as shown in Fig. 1(d) and Fig. 2(d) during the grating translation, except that a grating translation time is much less than the material characteristic time τ_g.

We also try to compare our theory with the fast moving grating technique proposed by Sutter and Günter [2]. When the grating translation speed is very fast (i.e. b=Ωτ_g≫1), retaining only the (Ωτ_g)² terms in Eqs. (11) yields

and that when the grating translation is performed within a very fast time less than the grating characteristic time τ_g, (i.e., t≪τ_g), Eqs. (15) reduces to the usual undamped expression of E ₁(t)=m|E ₁₀|exp[j(ϕ_g+Ω-ω_o)t)], so that the gain coefficient is Γ(t)∝|E ₁₀|sin[ϕ_g+Ω-ω_o)t]. Note that in case of no external electric field, the characteristic frequency ω_o=0, so the gain coefficient as well as the phase shift Ψ(t)=Ωt reduce to the result of the fast moving grating technique [2]. (Refer to Figs. 3(b) and 3(d)).

 

Fig. 3. Representations of the complex space charge field for a fast grating translation (b=100) and no external field applies (i.e., E ₀=0kV/cm). (a) Complex representation of the space charge field. The circle exhibits the steady state representation and the spirals behaves like a squirrel cage with gradually decreasing the radii, (b) the transient behavior of Y(t,Ω)represent nearly sinusoidal oscillation, (c) |Z(t,Ω)|decays very slowly with time and (d) the phase shift Ψ(t,Ω) is linearly proportional to the grating translation time.

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Figure 3 describe a typical fast grating translation, which is an extreme case of our theory. For a large b=100 and within a very fast time t≤0.2τ_g, Figs. 3 depict well the conventional fast grating translation situations, as mentioned above with Eqs. (15). However, it is noted that even though the grating translation is performed within a very fast time less than the grating characteristic time τ_g, it is inevitable to gradual exponential decay of the envelope of the amplitude of the complex space charge field, as described in Figs. 3(a) through 3(c), whereas the phase shift Ψ(t) is almost linearly proportional to the grating translation time.

4. Experiment and analysis

To prove our theoretical results for kinetics of two wave mixing in the photorefractive crystal, we simultaneously measured the pump and signal beam intensities in real time during two beam energy couplings and sequentially conducted a grating translation experiment in a BaTiO₃ crystal (5×5.5×10 mm³, thickness L=5mm). The crystal c-axis is parallel to the side of the crystal along to 10 mm length. Two coherent Ar-ion laser beams (wavelength of 514 nm) were used to record dynamic photorefractive gratings and the output beam powers were measured in real time. We also used extraordinary polarized beams throughout the experiments. In order to show an alternative transient energy transfer between two input beams, the input intensity ratio was taken to be unity during the grating translation period. The input intensity of each writing beam, taking into account the Fresnel loss at the surface of the crystal, was 2 I ₀=200mW/cm ² and the beam diameter at the entrance face of the crystal was 2.5mm. The incident half-angle between the two incident beams was θ_air=10° in air (θ≅4.1° in the crystal). The measured linear absorption coefficient was α=0.46 cm^-1.

 

Fig. 4. Experimental setup for two wave mixing with grating translational method.

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Figure 4 depicts a schematic geometry for the grating translation technique by means of two wave mixing. For grating translation experiments, the crystal was mounted on a sample holder attached to a piezoelectric transducer that could be translated parallel to the grating wave vector K_g. The incident angles of two input beams with respect to the z-axis were symmetric, so the grating wave vector is parallel to the crystal c-axis as shown in Fig. 4. This unslanted grating geometry renders a translation of the sample accurately along the direction of the grating vector as well as crystal c-axis, which is not an optimum experimental condition to maximize the two wave mixing gain. No external electric field is applied to the crystal during the experiments (i.e., E ₀=0). At first stage, the photorefractive grating is formed without the grating translation. In this case (i.e., E ₀=Ω=ω_o=0), the space charge field evolution is simply given by using Eq.(9a) as

After the grating formation reaches to its steady state, the grating translation launches. For this grating translation stage, Eqs. (11a) and (11b) then reduce to

Figure 5 shows typical temporal evolutions of the pump beam and signal beam powers during the grating formation and the grating translation periods. It is shown that the transient behaviors of measured powers during the grating translation period look like damped harmonic oscillations. The origin of the time is shifted to the moment when the grating translation starts. In case of no applied DC field, the optimum and resonance frequencies disappear (i.e., Ω_OPT=0 and Im[g₀]=ω ₀=0), which means that no physical mechanisms remain to enhance the two wave mixing gain. So, the pump and signal beam powers start to decay and behave as damped harmonic oscillations once the grating formed at first stage is translated parallel to the grating wave vector, as seen in Fig. 5.

 

Fig. 5. Typical experimental data for the transient behaviors of the pump beam and signal beam powers during the grating formation and the grating translation periods. No external electric field was applied to a BaTiO₃ photorefractive crystal.

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Figure 6 represents the comparison of the transient kinetics of the measured two wave mixing gain with the theory during the grating translation for various dimensionless moving frequencies b=Ωτ_g. The experimental gain G(t) is defined as [7, 8, 10]

and the theoretical gain G(t) is given by, using Eq. (4b)

from which the transient gain coefficient Γ(t) is determined by the following formula with the experimental gain G(t)

 

Fig. 6. Transient behaviors of two wave mixing gains for various dimensionless moving frequencies, b=Ωτ_g. Theoretical gain curves are from Eq. (18b) with Eq. (19).

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On the one hand, the theoretical gain coefficient is obtained from Eqs. (17) with Eq. (4c)

where Γ(0)=2πn³_br_eff|E ₁₀|/(λcosθ) and the additional phase shift Ψ(t) is from Eq. (17b). The gains approach to the steady state values as time goes by much larger than the characteristic time and these phenomena also could be explained with the previous theoretical results, Eqs. (18) and (19). The theory seems to be excellent predictions for the entire time range, as shown in Fig. 6. From the best curve fittings we have the following physical parameters as: the characteristic time or the grating formation time τ_g=2.25 sec, the gain coefficient Γ(0)=2.57cm^-1, the nonlinear refractive index change n ₁(0)=2.1×10^-5, and the initial photorefractive grating phase ϕ_g=90°±10°. Note that the measured value of Γ(0)=2.57cm^-1 is relatively lower than Γ(0)=12.1 cm^-1 of previously published result [11], because our experimental conditions, especially for incident beam angles relative to the c -axis, grating spacing, beam intensity ratio and input beam powers, were not adjusted to the optimum conditions to obtain a large two wave mixing gain. In addition, the modulation depth m was kept to be unity during our experiments. It should be emphasized that the measured gain decreases significantly at large modulation depth m [8, 13], which means that the calculated gain coefficient, Eq. (19) is no more constant on m. Refregier et al. [8] and Kwak et al. [13] developed an empirical correction function f(m) to replace the modulation depth m in the space charge field or gain coefficient expressions in order to describe the nonlinear dependence of the gain on m : Γ(0)→Γ(0)f(m)/m where f(m)=[1-exp(-am)]/a [8] or f(m)=m/(1+cm) [13], (a and c are fitting parameters depending on the experimental conditions. Note that form ≪1, f(m)→m(i.e. linear modulation theory).

 

Fig. 7. Normalized gain coefficient as a function of dimensionless moving frequency b. The lines are theoretical curves of Eq.(20).

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As a final remark, it is immediately shown from Eq.(19) that the steady state value of the gain coefficient is given by

which is exactly the same expression [14, 15]. For ϕ_g=90°, the gain coefficient becomes Γ(∞)=Γ(0)/(1+ b ²) and reveals symmetric function of b, while for ϕ_g≠90°, Γ(∞) is an asymmetric function. Figure 7 shows measured steady state normalized gain coefficients, Γ(∞)divided by Γ(0), against dimensionless moving frequency b with theoretical curves for different grating phase shift ϕ_g. From the best curve fittings with several grating phase shifts ϕ_g, we conclude that the grating phase shift of our BaTiO₃ crystal is in-between ϕ_g=90°±10°, implying a possibility to take into account the photovoltaic effect to the grating phase shift [15]. More careful measurements would be required to accurately determine the magnitude of the phase shift of the photorefractive grating relative to the intensity grating.

5. Conclusion

In conclusion, we experimentally found that when the grating translation time is comparable to the material characteristic times such as grating growth or decay times, or in case of very long translation time, transient two-beam coupling gains look like transient damped harmonic oscillations. In this situation, simply replacing the grating phase shift ϕ_g to ϕ_g+Ωt can not be true and the theory for the grating translation technique developed in literatures [2, 6] breaks down. We presented an analytic expression of the transient space-charge field under the external electric field accompanying with the grating translation technique that is valid for all time regions, whether or not the grating translation time is less than the material characteristic time. We measured a transient two-wave mixing gain in photorefractive BaTiO₃ crystal by employing the grating translation technique and compared with the theory, which revealed excellent agreements for the entire time regions. We also found that when the grating translation time is comparable to the material characteristic times, such as grating growth or decay times, or in the case of very long translation time, the transient two-beam coupling gains behave as transient damped harmonic oscillations. We expect that the theory developed in this work would be applicable to the organic photorefractive materials like dye-doped nematic liquid crystals [16, 17], whose gain coefficient appears much larger than the conventional inorganic photorefractive crystals.