Orientational photorefractive holograms in porphyrin ... - OSA Publishing

Orientational photorefractive holograms in porphyrin:Zn-doped nematic liquid crystals Eun Ju Kim, Hye Ri Yang, Sang Jo Lee, Gun Yeup Kim and Chong Hoon Kwak* Department of Physics, Yeungnam University, Kyongsan 712-749, Korea * Corresponding author: [email protected]

Abstract: We investigated the diffraction properties of dynamic holograms recorded in porphyrin:Zn doped nematic liquid crystals (NLCs) under the influence of an applied dc electric field for various conditions of the grating period, the writing beam intensity and the applied electric field. We also derived an analytic expression for diffraction efficiency from NLCs material equations and torque balance equations and compared the experimental results with the theory, revealing excellent agreement. ©2008 Optical Society of America OCIS codes: (190.0190) Nonlinear optics; (160.3710) Liquid crystals; (190.7070) Two-wave mixing.

References and links 1. 2. 3. 4. 5.

6. 7.

8. 9. 10. 11. 12.

13. 14. 15. 16. 17.

P. Günter and J. P. Huignard, Photorefractive Materials and Their Applications (Springer, Berlin, 1989), Vols. I and II. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993). E. V. Rudenko and A. V. Sukhov, “Optically induced spatial charge separation in a nematic and the resultant orientational nonlinearity,” JEPT. 78, 875-882 (1994). I. C. Khoo, B. D. Guenther, M. V. Wood, P. Chen, and M.-Y. Shin, “Coherent beam amplification with a photorefractive liquid crystal,” Opt. Lett. 22, 1229-1231 (1997). L. Marrucci, D. Paparo, P. Maddalena, E. Massera, E. Prudnikova, and E. Santamato, “Role of guest-host intermolecular forces in photoinduced reorientation of liquid crystals,” J. Chem. Phys. 107, 9783-9793 (1997). H. Ono and N. Kawatsuki, “High-performance photorefractivity in high- and low-molar-mass liquid crystal mixtures,” J. Appl. Phys. 85, 2482-2487 (1999) K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, J. E. Kim, and C. H. Kwak, “Effects of applied field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals,” Appl. Phys. Lett. 85, 366-368 (2004). I. C. Khoo, “Holographic grating formation in dye and fullerene C60-doped nematic liquid crystal film,” Opt. Lett. 20, 2137-2139 (1995). I. C. Khoo, “Orientational photorefractive effects in nematic liquid crystal films,” IEEE J. Quantum. Electron. 32, 525-534 (1996). I. Janossy, A. D. Lloyd, and B. S. Wherrer, “Anomalous optical Freedericksz transition in an absorbing liquid crystal,” Mol. Cryst. Liq. Cryst. 179, 1-12 (1990). Y. -P. Huang, T. -Y. Tsai, W. Lee, W. -K. Chin, Y. -M. Chang, and H. –Y. Chen, “Photorefractive effect in nematic-clay nanocomposites,” Opt. Express 13, 2058-2063 (2005). M. Kaczmarek, M. -Y. Shin, R. S. Cudney, and I. C. Khoo, “Electrically tunable, optically induced dynamic and permanent gratings in dye-doped liquid crystals,” IEEE J. Quantum. Electron. 38, 451-457 (2002). H. Ono and N. Kawatsuki, “Orientational photorefractive effects observed in polymer-dispersed liquid crystals,” Opt. Lett. 22, 1144-1146 (1997). C. H. Kwak, J. Takacs, and L. Solymar, “Spatial subharmonic instabilities,” Opt. Commun. 96, 278-282 (1993). C. H. Kwak, M. Sharmonin, J. Takacs, and L. Solymar, “Spatial subharmonics in photorefractive Bi12SiO20 crystal with a square wave applied field,” Appl. Phys. Lett. 62, 328-330 (1993). I. C. Khoo and S. H. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, Singapore, 1993). S. -T. Wu and C. -S. Wu, “Experimental confirmation of the Osipov-Terentjev theory on the viscosity of nematic liquid crystals,” Phys. Rev. 42, 2219-2227 (1990).

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Received 1 Aug 2008; revised 12 Sep 2008; accepted 7 Oct 2008; published 14 Oct 2008

27 October 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 17329

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] ∂n ± 1 + γ R n + n − ± ∇ ⋅ J ± = αI e ∂t

,

(1a)

J ± = eμ ± n ± E ∓ k BTμ ± ∇n ± , ∇⋅E =

e

ε 0ε

(n

+

− n−

(1b)

),

(1c)

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 E1 . 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 I (r, t ) = I 0 (t )(1 + m cos q ⋅ r ) = I 0 (t ) +

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1 I 1 (t ) exp(iq ⋅ r ) + c.c. 2

(2)



where m = 2 I a I b (I a + I b ) is the modulation depth, I a and I b are the incident intensities of the writing beams, I1 (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.

We assume that the physical variables used in Eqs. (1) take the same periodic function with the intensity distribution I (r, t ) as: Y (r , t ) = Y0 (t )(1 + m cos q ⋅ r ) = Y0 (t ) +

1 Y1 (t ) exp(iq ⋅ r ) + c.c. , 2

(3)

where Yi (i = 0, 1) stands for the variables n ± , J ± and E , and Y1 (t ) = mY0 (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: ∂n0 + γ R n0 2 = αI 0 , ∂t J ±0 = eμ ± n0± E 0

(4a)

,

(4b)

n0 = n0 ≡ n0 ,

(4c)

+

−

and for the subscript 1:

(

)

(5a)

(

)

(5b)

∂n1+ + γ R n0 + iμ + q ⋅ E 0 + D + q ⋅ q n1+ + γ R n0 n1− + i μ + n0 q ⋅ E1 = αI 1 ∂t ∂n1− + γ R n0 − iμ − q ⋅ E 0 + D − q ⋅ q n1− + γ R n0 n1+ − i μ − n0 q ⋅ E1 = αI 1 ∂t E1 ⋅ q = −i

e

εε 0

Δn ,

(5c)

where n0 is the average (spatially uniform) value for the positive and negative charge carrier densities, Δn = n1+ − n1− and D ± = k B Tμ ± 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 #99704 - $15.00 USD

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large, Eq. (4a) with Eq. (4b) reduces to n0 = αI 0 γ R for the steady state. Eliminating n1+ (t ) and n1− (t ) from Eqs. (4) and (5), after some lengthy calculations, we obtain the following differential equation for the space charge field E1 . ⎡ d 2E

q⋅⎢

dt

⎢ ⎣

+ (a + b )

1

2

⎧⎪ emγ R n0 2 = q ⋅ ⎨i εε 0 ⎪⎩ ⎧

−q⋅⎨

)

(

)

(

k BT + ⎡ + − − ⎢ μ + μ E 0i + e μ − μ ⎣

⎡ 2 ⎢2n0 γ R ⎣

e

⎩ εε 0

(

⎤ dE1 + ab − c 2 E1 ⎥ dt ⎥ ⎦

(μ

+

)

+ μ − + q 2 n0

)q⎤⎥ ⎫⎪⎬⎪ ⎦⎭

(

)

⎡ en 2k B T + − ⎤ ⎫ dE1 ⎤ , μ μ ⎥ E1 ⎬ − q ⋅ ⎢ 0 μ + + μ − ⎥ εε e dt ⎦ ⎦ ⎭ ⎣ 0

(6)

where a = γ R n0 + iqz μ + E0 + q 2 D + , b = γ R n0 − iq z μ − E0 + q 2 D − , c = γ R n0 , q z = q sin β and β is the tilt angle. The applied dc field E 0 and q are related by

( )

q ⋅ E 0 = qqˆ || ⋅ [(E0 sin β )qˆ || + (E0 cos β )qˆ ⊥ ] = qE0 sin β

,

(7)

and the space charge field E1 has the same direction with q , where qˆ || and qˆ ⊥ 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 E1 . d 2 E1 dt 2

with

A=

B=

C=

+A

dE1 + BE1 = mC , dt

⎛ ⎜1 + τ d ⎜⎝

1

2

τ dτ

2τ d

⎛ ⎜1 + ⎜ ⎝

τ

+

(8)

E sin β ⎞⎟ , ED +i 0 ν ⎟ Eq Eq ⎠

E sin β ⎞⎟ , E 2 sin 2 β ED E ED2 + D + 0 + +i 0 ν ⎟ 2Eq EM 2Eq EM 2Eq EM 2 Eq ⎠

1 (iE Dν − E 0 sin β ) , τ dτ

where μ = μ + μ − (μ + + μ − ) , ν = (μ + − μ − ) (μ + + μ − ) = (D + − D − ) (D + + D − ) , τ = 1 (γ R n0 ) is the photoion lifetime, τ d = εε 0 [en0 (μ + + μ − )] is the Maxwell relaxation time, E D = k BTq e is the diffusion field, E M = γ R n0 (μ + + μ − ) (qμ + μ − ) = γ R n0 (qμ ) is the drift field, and E q = en0 qεε 0 is the saturating field. If we take the slowly-varying amplitude approximation of derivative, d 2 E1 dt 2 , can be neglected and then Eq. (8) becomes dE1 + gE1 = mh , dt

with

g=

h=

B 2 = A τ

⎛ ⎜1 + ⎜ ⎝

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in time, the second order

(9)

ED E E 2 sin 2 β E 2 E sin β ⎞⎟ ν + D + 0 + D +i 0 ⎟, EM 2Eq 2Eq EM Eq EM 2Eq ⎠ ⎛ ⎜1 + ⎜ ⎝

2τ d

τ

+

E sin β ED +i 0 Eq E qν

(iE Dν − E 0 sin β ) C 1 = A τ ⎛ 2τ E sin β ⎞⎟ E d ⎜1 + + D +i 0 ν ⎜ ⎟ Eq Eq τ ⎝ ⎠

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E1

⎞ ⎟ ⎟ ⎠

.



The transient solution for E1 (t ) =

E1

can then be written as

mh [1 − exp(− gt )] = E1 (∞ )[1 − exp(− gt )] . g

(10)

The steady state space-charge field E1 (∞) is given by E1 (∞ ) =

where

1 m [(E DνY − E0 X sin β ) + i(E DνX + E0Y sin β )] , 2 X 2 +Y2

(

)

(

)

(

X = 1 + E D E M + E D 2 Eq + E0 2 sin 2 β 2 Eq E M + E D 2 2 E q E M

(11)

) and Y = E0ν sin β (2Eq ) . Using Eq.

(11), we obtain the magnitude of the steady state space-charge field E1 (∞) and the phase shift φ between the space-charge field grating and the intensity grating as follows 1 2 2 m ⎛ E ν 2 + E 0 sin 2 β E1 (∞ ) = ⎜ D 2 ⎜⎝ X 2 +Y2

⎛

φ = tan −1 ⎜⎜

E DνX + E 0Y sin β − E0 X sin β

⎝ E DνY

⎞ ⎟ ⎟ ⎠

⎞2 ⎟ ⎟ ⎠

,

(12)

.

(13)

Λg=1.48μm

Λg=1.00μm

180

(a)

(b)

Im [E1(

1.5x10

E0

-2

1.0x10

-E0

135 φ (degree)

)] (V/μm)

-2

∞

Λg=0.50μm

|E1(∞ )|

-3

5.0x10

90 45

φ

0.0 -0.08

-0.04

0.00

0.04

Re [E1(∞ )] (V/μm)

0.08

0

-2

-1

0

1

2

E0 (V/μm)

Fig. 2. (a) Complex representation of the steady state space charge field E1(∞ ) as positively ( Eo ) and negatively ( − Eo ) increasing the applied dc field and (b) the phase shift variation φ against the applied dc field for various grating periods at I 0 = 220 mW/cm 2 .

The applied dc field E0 not only changes the magnitude of the space-charge field E1 (∞) , 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 E1 (∞) gradually increases to a maximum value and then rapidly diminishes, irrespective of the direction of the dc field E0 . Figure 2(b) represents the phase shift φ against E0 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 E0 = 0 in Eq. (13), the space charge gratings are spatially shifted by φ = 90o relative to the intensity gratings. As positively (negatively) increasing the applied dc field, however, the phase shift φ steeply approaches to 180o ( 0 o ) as shown in Fig. 2(b), and then the PR effect disappears. Similarly, Fig. 3(a) and 3(b) show the

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complex representation of space-charge field and the phase shift against E0 for various total input beam intensities, respectively. As the total beam intensity I 0 increases, the charge carrier densities n0 and E1 (∞) also increase and consequently PR effect is enhanced. In plotting Fig. 2 and Fig. 3, we have used the following parameters: Ed = 0.106 V/μ m , Ec = 1.44 V/μ m , Em = 0.270 V/μ m and E q = 0.295 V/μ m when Λ g = 1.48 μ m and I 0 = 250 mW/cm 2 . 2

2

I0 = 500 mW/cm

2

I0 = 100 mW/cm

I0 = 10 mW/cm

180 135

-2

2.0x10

E0

-E0

φ (degree)

)] (V/μm)

Im [E1(

∞

φ (degree)

(b)

(a)

|E1(∞ )|

-2

1.0x10

180

160 0

1

2

E0 (V/μm)

90 45

φ

0.0

-0.10

-0.05

0.00

0.05

0.10

0 -2

-1

Re [E1(∞ )] (V/μm)

0 E0 (V/μm)

1

2

Fig. 3. (a) Complex representation of the steady state space charge field E1(∞ ) as positively ( Eo ) and negatively ( − Eo ) increasing the applied dc field and (b) the phase shift variation φ against the applied dc field for various input intensities at Λ g = 1.48 μ m .

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