## Kinetics of two wave mixing gain for moving grating technique in photorefractive BaTiO_{3} crystal

Optics Express, Vol. 16, Issue 24, pp. 19615-19628 (2008)

http://dx.doi.org/10.1364/OE.16.019615

Acrobat PDF (291 KB)

### Abstract

We present analytical expressions for time-dependent space charge fields and two wave-mixing gains under the external applied fields accompanying with the grating translation. We analyzed the variations of complex space charge fields in a complex plane, and also obtained the explicit expressions for the resonance and optimum frequencies (or moving grating velocities), which maximize the magnitude and imaginary part of space charge fields. We also conducted two wave-mixing experiments with the grating translation technique without an external applied field in a BaTiO3 crystal. The transient behaviors of measured gains look like damped harmonic oscillations, showing excellent agreement with the theory for the entire time range.

© 2008 Optical Society of America

## 1. Introduction

1. M. Z. Zha, P. Amrhein, and P. Günter, “Measurement of phase shift of photorefractive gratings by a novel method,” IEEE Quantum Electron. **26**, 788–792 (1990). [CrossRef]

6. C. H. Kwak and S. J. Lee, “Approximate analytic solution of photochromic and photorefractive gratings in photorefractive materials,” Opt. Commun. **183**, 547–554 (2000). [CrossRef]

2. K. Sutter and P. Günter, “Photorefractive gratings in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane,” J. Opt. Soc. Am. B **7**, 2274–2278 (1990). [CrossRef]

6. C. H. Kwak and S. J. Lee, “Approximate analytic solution of photochromic and photorefractive gratings in photorefractive materials,” Opt. Commun. **183**, 547–554 (2000). [CrossRef]

*ϕ*against the intensity grating was just replaced by

_{g}*ϕ*+Ω

_{g}*t*, where Ω=

*K*is the frequency of the moving beam fringe pattern,

_{g}v*K*is the magnitude of the grating wave vector and

_{g}*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*by

_{g}*ϕ*+Ω

_{g}*t*can not be true and that the theory for the grating translation technique developed in [2

2. K. Sutter and P. Günter, “Photorefractive gratings in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane,” J. Opt. Soc. Am. B **7**, 2274–2278 (1990). [CrossRef]

6. C. H. Kwak and S. J. Lee, “Approximate analytic solution of photochromic and photorefractive gratings in photorefractive materials,” Opt. Commun. **183**, 547–554 (2000). [CrossRef]

7. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics22, 949–960 (1979), and *idem, ibid*22, 961–964 (1979). [CrossRef]

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

2. K. Sutter and P. Günter, “Photorefractive gratings in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane,” J. Opt. Soc. Am. B **7**, 2274–2278 (1990). [CrossRef]

8. Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi_{12}SiO_{20} crystals with moving grating: theory and experiments,” J. Appl. Phys. **58**, 45–57 (1985). [CrossRef]

*K*using a PZT transducer with a velocity

_{g}*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

*I*

_{0}=

*I*+

_{P}*I*is the total incident beam intensity,

_{S}*I*and

_{P}*I*are the pump and the signal beam intensities, respectively,

_{S}*I*

_{1}=

*mI*

_{0},

*K*is the grating wave vector, and

_{g}v, K_{g}*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

10. P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. **25**, 484–519 (1989). [CrossRef]

*n*is the background refractive index of the crystal,

_{b}*r*is the effective electro-optic coefficient depending on the incident beam polarization and crystal orientation, and |

_{eff}*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

**183**, 547–554 (2000). [CrossRef]

7. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics22, 949–960 (1979), and *idem, ibid*22, 961–964 (1979). [CrossRef]

8. Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi_{12}SiO_{20} crystals with moving grating: theory and experiments,” J. Appl. Phys. **58**, 45–57 (1985). [CrossRef]

10. P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. **25**, 484–519 (1989). [CrossRef]

14. J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi_{12}SiO_{20} crystals,” Opt. Commun. **38**, 249–254 (1981). [CrossRef]

*α*is the linear absorption coefficient,

*θ*is the half-angle between the two incident beams inside the crystal,

*I*=0) and

_{P}(z*I*=0) are the pump and signal beam intensities at entrance face of the crystal, respectively,

_{S}(z*β*=

*I*(0)/

_{P}*I*(0) is the incident beam intensity ratio, and

_{S}*z=L*is the medium thickness. Here, the time-dependent gain coefficient Γ(

*t*) is given by [7

7. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics22, 949–960 (1979), and *idem, ibid*22, 961–964 (1979). [CrossRef]

8. Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi_{12}SiO_{20} crystals with moving grating: theory and experiments,” J. Appl. Phys. **58**, 45–57 (1985). [CrossRef]

10. P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. **25**, 484–519 (1989). [CrossRef]

*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

*idem, ibid*22, 961–964 (1979). [CrossRef]

*idem, ibid*22, 961–964 (1979). [CrossRef]

*N*

^{+}

_{D}and

*N*are the ionized donor density and the donor density, respectively,

_{D}*N*is the acceptor density,

_{A}*N*is the electron density,

*J*is the current density,

*E*is the electric field,

*s*is the cross section of photoionization,

*γ*is the recombination constant,

_{R}*ε*

_{0}is the free space permittivity,

*ε*is the dielectric constant,

*µ*is the mobility,

*k*is the Boltzmann constant,

_{B}*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

*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

_{12}SiO_{20} crystals with moving grating: theory and experiments,” J. Appl. Phys. **58**, 45–57 (1985). [CrossRef]

11. C. H. Kwak, S. Y. Park, and E. H. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO_{3} crystal,” Opt. Commun. **115**, 315–322 (1995). [CrossRef]

*E*

_{1}(

*t*), we can get the differential equation for the space charge field of the moving grating of the form

*τ*=1/

*γ*is the photoelectron lifetime,

_{R}N_{A}*τ*=

_{d}*ε*

_{0}ε/eµN_{0}is the Maxwell relaxation time,

*E*is the diffusion field,

_{D}=k_{B}TK_{g}/e*E*is the drift field, and

_{M}=γ_{R}N_{A}/µK_{g}*E*=

_{q}*eN*is the limiting space charge field. In obtaining Eqs. (7), the average electron density

_{A}/εε_{0}K_{g}/e*N*

_{0}is assumed to be a constant with time and is given by

*N*

_{0}=

*sI*and the second-order time derivative term for

_{0}N_{D}/γ_{R}N_{A}*E*

_{1}(

*t*) is neglected by using the slowly varying amplitude approximation. Kwak

*et al*. [11

11. C. H. Kwak, S. Y. Park, and E. H. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO_{3} crystal,” Opt. Commun. **115**, 315–322 (1995). [CrossRef]

*∂E*. 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.

_{0}/∂t*i.e. ∂E*=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

_{0}/∂t*g*(Ω=0)≡

*g*

_{o}=1/

*τ*+

_{g}*jω*and

_{o}*g*(

*Ω*≠0)≡

*g*=

*g*-

_{o}*j*Ω, where the characteristic response time

*τ*and the characteristic frequency

_{g}*ω*of the photorefractive material are given by

_{o}*ω*=Im[

_{o}*g*] becomes zero when applied field is absent (i.e.

_{o}*E*=0).

_{o}### 3.1(i) First stage: Grating formations with no grating translation (i.e., Ω=0)

*E*

_{10}=

*h/g*

_{0}is the steady state value of the space charge field and is given by

*E*

_{10}| and the grating phase shift

*ϕ*with respect to the intensity grating are given by, respectively,

_{g}*τ*and a frequency

_{g}*ω*. The gain coefficient evolves no oscillatory behavior with time when an applied field is absent (i.e.

_{o}*ω*=0).

_{o}### 3.2 (ii) Second stage: Grating translations (i.e., Ω≠0) after first stage

*E*

_{1}(0)=

*mE*

_{10}at time

*t*=0 and is given by

*t*) is an additional phase shift induced by during a grating translation and

*g** is complex conjugation of

*g*=

*g*-

_{o}*j*Ω. 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

*b*=Ω

*τ*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

_{g}_{12}SiO_{20} crystals with moving grating: theory and experiments,” J. Appl. Phys. **58**, 45–57 (1985). [CrossRef]

*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

_{RES}=ω

_{0}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.

*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

*ϕ*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 by

_{g}*Y*(∞,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

_{12}SiO_{20} crystals with moving grating: theory and experiments,” J. Appl. Phys. **58**, 45–57 (1985). [CrossRef]

*r*→1), in which the parameters used in calculations are not optimized to have a maximum gain enhancement.

*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*=Ω

*τ*. The transient behaviors of

_{g}*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*=Ω

*τ*. It should be emphasized that the initial grating phase shift

_{g}*ϕ*relative to the intensity grating can not be replaced by

_{g}*ϕ*+Ω

_{g}*t*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}**7**, 2274–2278 (1990). [CrossRef]

*b*=Ω

*τ*≫1), retaining only the (Ω

_{g}*τ*)

_{g}^{2}terms in Eqs. (11) yields

*τ*, (i.e.,

_{g}*t*≪

*τ*), Eqs. (15) reduces to the usual

_{g}*undamped*expression of

*E*

_{1}(

*t*)=

*m*|

*E*

_{10}|exp[

*j*(

*ϕ*+Ω-

_{g}*ω*)

_{o}*t*)], so that the gain coefficient is Γ(

*t*)∝|

*E*

_{10}|sin[

*ϕ*+Ω-

_{g}*ω*)

_{o}*t*]. Note that in case of no external electric field, the characteristic frequency

*ω*=0, so the gain coefficient as well as the phase shift Ψ(

_{o}*t*)=Ω

*t*reduce to the result of the fast moving grating technique [2

**7**, 2274–2278 (1990). [CrossRef]

*b*=100 and within a very fast time

*t*≤0.2

*τ*, 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

_{g}*τ*, it is inevitable to gradual

_{g}*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

_{3}crystal (5×5.5×10 mm

^{3}, thickness

*L*=5

*mm*). 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}=200

*mW/cm*

^{2}and the beam diameter at the entrance face of the crystal was 2.5

*mm*. The incident half-angle between the two incident beams was

*θ*=10° in air (

_{air}*θ*≅4.1° in the crystal). The measured linear absorption coefficient was

*α*=0.46 cm

^{-1}.

*K*. The incident angles of two input beams with respect to the

_{g}*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*=0), the space charge field evolution is simply given by using Eq.(9a) as

_{o}*i.e*., Ω

_{OPT}=0 and Im[

*g*]=

_{0}*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.

*b*=Ω

*τ*. The experimental gain

_{g}*G(t*) is defined as [7

*idem, ibid*22, 961–964 (1979). [CrossRef]

_{12}SiO_{20} crystals with moving grating: theory and experiments,” J. Appl. Phys. **58**, 45–57 (1985). [CrossRef]

**25**, 484–519 (1989). [CrossRef]

*G(t)*is given by, using Eq. (4b)

*t*) is determined by the following formula with the experimental gain

*G(t)*

*πn*|

^{3}_{b}r_{eff}*E*

_{10}|/(

*λ*cos

*θ*) 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*=2.25 sec, the gain coefficient Γ(0)=2.57cm

_{g}^{-1}, the nonlinear refractive index change

*n*

_{1}(0)=2.1×10

^{-5}, and the initial photorefractive grating phase

*ϕ*=90°±10°. Note that the measured value of Γ(0)=2.57cm

_{g}^{-1}is relatively lower than Γ(0)=12.1 cm

^{-1}of previously published result [11

11. C. H. Kwak, S. Y. Park, and E. H. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO_{3} crystal,” Opt. Commun. **115**, 315–322 (1995). [CrossRef]

*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

_{12}SiO_{20} crystals with moving grating: theory and experiments,” J. Appl. Phys. **58**, 45–57 (1985). [CrossRef]

13. C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. **105**, 353–358 (1994). [CrossRef]

*m*. Refregier et

*al*. [8

_{12}SiO_{20} crystals with moving grating: theory and experiments,” J. Appl. Phys. **58**, 45–57 (1985). [CrossRef]

*et al*. [13

13. C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. **105**, 353–358 (1994). [CrossRef]

*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

_{12}SiO_{20} crystals with moving grating: theory and experiments,” J. Appl. Phys. **58**, 45–57 (1985). [CrossRef]

*f(m)*=

*m*/(1+

*cm*) [13

13. C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. **105**, 353–358 (1994). [CrossRef]

*a*and

*c*are fitting parameters depending on the experimental conditions. Note that for

*m*≪1,

*f(m)*→

*m*(i.e. linear modulation theory).

14. J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi_{12}SiO_{20} crystals,” Opt. Commun. **38**, 249–254 (1981). [CrossRef]

15. I. McMichael and P. Yeh, “Phase shift of photorefractive gratings and phase-conjugate waves,” Opt. Lett. **12**, 48–50 (1987). [CrossRef] [PubMed]

*ϕ*=90°, the gain coefficient becomes Γ(∞)=Γ(0)/(1+

_{g}*b*

^{2}) and reveals symmetric function of b, while for

*ϕ*≠90°, Γ(∞) is an asymmetric function. Figure 7 shows measured steady state normalized gain coefficients, Γ(∞)divided by Γ(0), against dimensionless moving frequency

_{g}*b*with theoretical curves for different grating phase shift

*ϕ*. From the best curve fittings with several grating phase shifts

_{g}*ϕ*, we conclude that the grating phase shift of our BaTiO

_{g}_{3}crystal is in-between

*ϕ*=90°±10°, implying a possibility to take into account the photovoltaic effect to the grating phase shift [15

_{g}15. I. McMichael and P. Yeh, “Phase shift of photorefractive gratings and phase-conjugate waves,” Opt. Lett. **12**, 48–50 (1987). [CrossRef] [PubMed]

## 5. Conclusion

*ϕ*to

_{g}*ϕ*+Ω

_{g}*t*can not be true and the theory for the grating translation technique developed in literatures [2

**7**, 2274–2278 (1990). [CrossRef]

**183**, 547–554 (2000). [CrossRef]

_{3}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

16. K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, C. H. Kwak, and J. E. Kim, “Effects of applied electric field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals,” Appl. Phys. Lett. **85**, 366–368 (2004). [CrossRef]

17. E. J. Kim, H. R. Yang, S. J. Lee, G. Y. Kim, and C. H. Kwak, “Orientational photorefractive holograms in porphyrin:Zn-doped nematic liquid crystals,” Opt. Express **16**, 17329–17341 (2008). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | M. Z. Zha, P. Amrhein, and P. Günter, “Measurement of phase shift of photorefractive gratings by a novel method,” IEEE Quantum Electron. |

2. | K. Sutter and P. Günter, “Photorefractive gratings in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane,” J. Opt. Soc. Am. B |

3. | R. S. Cudney, G. D. Bacher, R. M. Pierce, and J. Feinberg, “Measurement of the photorefractive phase shift,” Opt. Lett. |

4. | R. Hofmeister, A. Yariv, A. Kewitsch, and S. Yagi, “Simple methods of measuring the net photorefractive phase shift and coupling constant,” Opt. Lett. |

5. | D. G. Gray, M. G. Moharam, and T. M. Ayres, “Heterodyne technique for the direct measurement of the amplitude and phase of photorefractive space-charge field,” J. Opt. Soc. Am. B |

6. | C. H. Kwak and S. J. Lee, “Approximate analytic solution of photochromic and photorefractive gratings in photorefractive materials,” Opt. Commun. |

7. | N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics22, 949–960 (1979), and |

8. | Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi |

9. | S. I. Stepanov and M. P. Petrov, in |

10. | P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. |

11. | C. H. Kwak, S. Y. Park, and E. H. Lee, “Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO |

12. | G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. |

13. | C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, “An analytical solution for large modulation effects in photorefractive two-wave couplings,” Opt. Commun. |

14. | J. P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi |

15. | I. McMichael and P. Yeh, “Phase shift of photorefractive gratings and phase-conjugate waves,” Opt. Lett. |

16. | K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, C. H. Kwak, and J. E. Kim, “Effects of applied electric field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals,” Appl. Phys. Lett. |

17. | E. J. Kim, H. R. Yang, S. J. Lee, G. Y. Kim, and C. H. Kwak, “Orientational photorefractive holograms in porphyrin:Zn-doped nematic liquid crystals,” Opt. Express |

**OCIS Codes**

(190.4400) Nonlinear optics : Nonlinear optics, materials

(190.5330) Nonlinear optics : Photorefractive optics

(190.7070) Nonlinear optics : Two-wave mixing

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: August 1, 2008

Revised Manuscript: November 3, 2008

Manuscript Accepted: November 7, 2008

Published: November 12, 2008

**Citation**

Sangjo Lee, Hye Ri Yang, Eun Ju Kim, Yeung Lak Lee, and Chong Hoon Kwak, "Kinetics of two wave mixing gain for moving grating technique in photorefractive BaTiO_{3} crystal," Opt. Express **16**, 19615-19628 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-24-19615

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### References

- M. Z. Zha, P. Amrhein, and P. Günter, "Measurement of phase shift of photorefractive gratings by a novel method," IEEE Quantum Electron. 26, 788-792 (1990). [CrossRef]
- K. Sutter and P. Günter, "Photorefractive gratings in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane," J. Opt. Soc. Am. B 7, 2274-2278 (1990). [CrossRef]
- R. S. Cudney, G. D. Bacher, R. M. Pierce, and J. Feinberg, "Measurement of the photorefractive phase shift," Opt. Lett. 17, 67-69 (1992). [CrossRef] [PubMed]
- R. Hofmeister, A. Yariv, A. Kewitsch, and S. Yagi, "Simple methods of measuring the net photorefractive phase shift and coupling constant," Opt. Lett. 18, 488-490 (1993). [CrossRef]
- D. G. Gray, M. G. Moharam, and T. M. Ayres, "Heterodyne technique for the direct measurement of the amplitude and phase of photorefractive space-charge field," J. Opt. Soc. Am. B 11, 470-475 (1994). [CrossRef]
- C. H. Kwak and S. J. Lee, "Approximate analytic solution of photochromic and photorefractive gratings in photorefractive materials," Opt. Commun. 183, 547-554 (2000). [CrossRef]
- N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, "Holographic storage in electrooptic crystals. I. Steady state," Ferroelectrics 22, 949-960 (1979), and idem, ibid22, 961-964 (1979). [CrossRef]
- Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, "Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments," J. Appl. Phys. 58, 45-57 (1985). [CrossRef]
- S. I. Stepanov and M. P. Petrov, in Photorefractive materials and their applications I, P. Günter and J. P. Huignard, eds., (Springer, Berlin, 1988) Chap. 9.
- P. Yeh, "Two-wave mixing in nonlinear media," IEEE J. Quantum Electron. 25, 484-519 (1989). [CrossRef]
- C. H. Kwak, S. Y. Park, and E. H. Lee, "Intensity dependent two-wave mixing at large modulation depth in photorefractive BaTiO3 crystal," Opt. Commun. 115, 315-322 (1995). [CrossRef]
- G. C. Valley and M. B. Klein, "Optimal properties of photorefractive materials for optical data processing," Opt. Eng. 22, 704-711 (1983).
- C. H. Kwak, S. Y. Park, J. S. Jeong, H. H. Suh, and E. H. Lee, "An analytical solution for large modulation effects in photorefractive two-wave couplings," Opt. Commun. 105, 353-358 (1994). [CrossRef]
- J. P. Huignard and A. Marrakchi, "Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20 crystals," Opt. Commun. 38, 249-254 (1981). [CrossRef]
- I. McMichael and P. Yeh, "Phase shift of photorefractive gratings and phase-conjugate waves," Opt. Lett. 12, 48-50 (1987). [CrossRef] [PubMed]
- K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, C. H. Kwak, and J. E. Kim, "Effects of applied electric field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals," Appl. Phys. Lett. 85, 366-368 (2004). [CrossRef]
- E. J. Kim, H. R. Yang, S. J. Lee, G. Y. Kim, and C. H. Kwak, "Orientational photorefractive holograms in porphyrin:Zn-doped nematic liquid crystals," Opt. Express 16, 17329-17341 (2008). [CrossRef] [PubMed]

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