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A simple analytic solution for transient simultaneous four-wave mixing in photorefractive crystals
1 January 2002
Optics Communications 201 (2002) 191–196 www.elsevier.com/locate/optcom
A simple analytic solution for transient simultaneous four-wave mixing in photorefractive crystals F.A. Rustamov *, S.R. Muradov, V.K. Sharbatov Department of Physics, Semiconductor Physics Laboratory, Baku State University, Baku 370145, Azerbaijan Received 26 March 2001; received in revised form 23 July 2001; accepted 9 October 2001
Abstract Very simple solution of the coupled partial differential equations that leads to analytic formula for diffraction efficiency which describes the time dependence of four-wave mixing in photorefractive crystals is presented. The obtained results were used for analysis of simultaneous read–write dynamics in Bi12 SiO20 crystal. Ó 2002 Published by Elsevier Science B.V. PACS: 42.65.Hw Keywords: Holographic gratings; Four-wave mixing; Photorefractive crystals
The coupled partial differential equations which describe dynamics of two- and four-wave mixing in photorefractive crystals are not soluble in quadrature. A number of theoretical papers dealing with the dynamics of two-wave mixing is based either on numerical solution [1–4] or on power series expansion [5,6]. The dynamics of four-wave mixing using a power series expansion is also considered [7,8]. In this paper we propose a simple analytic method for solution of coupled partial differential equations describing the time dependence for simultaneous record–readout at four-wave mixing in transmission geometry in
*
Corresponding author. E-mail address: [email protected] (F.A. Rustamov).
photorefractive crystals. The analytic formula for diffraction efficiency is obtained. A typical scheme of experiments on four-wave mixture in transmission geometry is shown in Fig. 1. A holographic grating in photorefractive crystal is usually recorded by two linearly polarized beams g g (I1 and Iþ1 ), the wavelength (kg ) of which corresponds to intrinsic optical absorbtion region, but the readout is realized by the beam of weak inr r tensity (I1 and Iþ1 ) and the wavelength ðkr Þ far from the intrinsic optical absorption region. A two-level model of optical transitions taking into account the effect of readout beam on the dynamic holographic grating has been proposed [7–9]. According to this model, the two donor levels (with concentrations Nd1 and Nd2 ) are involved in optical transitions. The deeper Nd1 donor levels respond to the recording beams, and shallow Nd2 levels
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F.A. Rustamov et al. / Optics Communications 201 (2002) 191–196 g oCþ1 g ¼ idEþ2 C1 ; oz g oC1 g ¼ idEþ2 Cþ1 ; oz r oCþ1 r ¼ idEþ2 C1 ; oz r oC1 r ¼ idEþ2 Cþ1 ; oz oEþ2 Eþ2 g g r r þ ¼ aþ ½Cþ1 C1 þ SCþ1 C1 ; ot sþ
ð1Þ
where Fig. 1. Recording and readout geometry of the transmission g holographic grating: I 1 are the intensities of recording beams; r r is the intensity of difIþ1 is the intensity of readout beam; I1 fracted beam.
respond to the readout beam. Unlike the classical one-level model [1–6], this model allows one to take into account the effect of the reading beam on the holographic grating. The two incident recording beams (i.e., interference pattern) initiate the phototransitions of electrons from the Nd1 levels to the conduction band. Owing to diffusion and drift, these electrons move through the crystal and locate in dark sites of the interference pattern, recombining with free Nd1 and Nd2 donor levels. A periodic space charge (i.e., a periodic internal electric field) is formed and leads to a periodic change of refractive index through the photorefractive effect, that is to the formation of holographic grating. Switching on the readout beam satisfying the Bragg condition causes excitation of photoelectrons from the shallow Nd2 donor levels to the conduction band. Owing to diffusion and drift in the conduction band, electrons are redistributed again over the crystal and recombine with free donor levels. Thus initial distribution of electrons changes leading to a distortion of existing holographic grating. In the two-level model, the system of nonlinear coupled first-order partial equations describing recording and readout at four-wave mixing in photorefractive crystals has the following form [7–9]:
1 1 1 ¼ i s s0 sk E2 ET 1 þ Eq 1 þ EEMT þ Eq ETM ¼ 2 2 ET sM 1 þ EM þ EEM0 E E0 1 EMq Eq i 2 2 ; sM 1 þ EEMT þ EEM0 h i E2 E0 i ET 1 þ EEMT þ EM0 a ¼ a1 ia2 ¼ 2 2 ; g ET sM I0 1 þ EM þ EEM0 e0 re0 k 2 2pkb T ; ; d¼ ; ET ¼ eK 4peln0 2kz 2eKNa KNa cr Eq ¼ ; ; EM ¼ ee0 2pl g;r g;r I0g;r ¼ Iþ1 ð0Þ þ I1 ð0Þ; 0 at t < tr ; S¼ cr1 a2 Nd2 at t P t ;
sM ¼
cr2 a1 ðNd1 Na Þ
r
and tr is a switching moment of readout beam. g r Here C 1 and C 1 are the complex amplitudes of recording, readout and diffracted beams, respectively; Na , Nd1 and Nd2 are the concentrations of acceptor and donor levels, respectively; l is the mobility of the current carriers; T is the temperature; e is the electron charge; kb is the Boltzman constant; a1 , a2 and cr1 , cr2 are the coefficients of ionization and capture of the corresponding levels; e and e0 are the high-frequency and static dielectric constants; r is the linear electrooptical coefficient;
F.A. Rustamov et al. / Optics Communications 201 (2002) 191–196
ET is the ‘‘diffusion’’ field; EM is the ‘‘drift’’ field; Eq is the space charge field; E0 is the applied field; Eþ2 is the amplitude of electric field; K is the holographic grating period. To study polarization effects and selectivity properties of holographic gratings it is necessary to take into account anisotropy of crystals. But these problems are out of scope of this paper. We are interested only in dynamics of four-wave mixing, so application of scalar equations is reasonable. This system is not soluble in quadrature. Asg g suming that the interference term ðCþ1 C1 þ r r SCþ1 C1 Þ is more smooth function in respect to expðt=sÞ (s is a characteristic time of static state establishment) we can take this term in last equation as a constant. The criterion of interference term smoothness is fulfilled quite well for long times ðt P sÞ and, at starting stage, for low gain coefficient value cz < 1 and for small phase shift dAz < 1 [10]. As the values of c and A depend on the applied field E0 and the holographic grating period K through a2 a1 2 c ¼ 2djsþ j I0g ; s0 sk ð2Þ a2 2 g a1 A ¼ jsþ j I0 þ s0 sk then the conditions for the initial stage can be satisfied by the appropriate choice of E0 and K [6– 9]. Under these conditions, the last equation can easily be integrated, and taking into account the initial conditions Eþ2 ð0Þ ¼ 0 and
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we can rewrite the system of Eq. (1) as: oA1 þ iaþ sþ d½A1 A4 þ A2 A3 A4 oz oA4 þ iaþ sþ d½A1 A4 þ A2 A3 A1 oz oA3 þ iaþ sþ d½A1 A4 þ A2 A3 A2 oz oA2 þ iaþ sþ d½A1 A4 þ A2 A3 A3 oz Eþ2 ¼ aþ sþ ½A1 A4 þ A2 A3 :
¼ 0; ¼ 0; ¼ 0; ¼ 0;
The given system of the coupled first-order differential equations is similar to equations for the stationary state [9,11–13], and hence has a similar solution. Equations (5) have the following integrals A1 A2 A3 A4 ¼ c 1 ; A1 A3 þ A2 A4 ¼ c2 ; A1 A1 þ A4 A4 ¼ d1 ;
where d1 ¼ jU1 jI0g and d2 ¼ jU2 jI0r . The constants c1 , c2 , d1 , d2 have parametric dependence on time t and do not have dependence on space coordinate z. Substitution of the above integrals into the equation system (5) allows us to divide it into two independent systems oA1 ¼ l½A1 d1 A1 ðjU1 jI1 þ SjU2 jI3 Þ þ c2 A3 ; oz oA3 ¼ l A3 d2 A3 ðjU1 jI1 þ SjU2 jI3 Þ þ c2 A1 ; oz ð7Þ oA2
the following expression for the internal electric field amplitude is obtained t g g Eþ2 ¼ aþ sþ Cþ1 C1 1 exp sþ
t tr r r þ SCþ1 C1 1 exp : ð3Þ sþ Substituting U1 ¼ 1 expðt=sþ Þ; U2 ¼ 1 expððt tr Þ=sþ Þ; 1=2
1=2
ð6Þ
A2 A2 þ A3 A3 ¼ d2 ;
g g Eþ2 ¼ aþ sþ ½1 expðtr =sþ ÞCþ1 C1
g A1;4 ¼ U1 C 1 and A3;2 ¼ ðSU2 Þ
ð5Þ
r C 1 ;
ð4Þ
¼ l A2 d2 A2 ðjU1 jI4 þ SjU2 jI2 Þ þ c2 A4 ;
oz ð8Þ oA4 ¼ l A4 d1 A4 ðjU1 jI4 þ SjU2 jI2 Þ þ c2 A2 ; oz g and where the notations l ¼ idaþ sþ , I1;4 ¼ I 1 r I3;2 ¼ I 1 are introduced. Simple transformation bring the above two equation systems into the following form [11]: h i oA31 2 ¼ l c2 þ A31 ðd2 d1 Þ c2 ðA31 Þ ; oz ð9Þ h i oA24 2 ¼ l c2 þ A24 ðd2 d1 Þ c2 ðA24 Þ ; oz where A31 ¼ A3 =A1 and A24 ¼ A2 =A4 .
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where
Solving the last two equations, we have A31 ¼
L D1 eqz þ Lþ Deqz ; 2c2 ðD1 eqz þ Deqz Þ
ð10Þ
A24 ¼
L F 1 eqz þ Lþ F eqz ; 2c2 ðF 1 eqz þ F eqz Þ
ð11Þ
D ¼ d2 d1 ; 2 1=2
;
ð12Þ L ¼ D Q; lQ ; q¼ 2 where D and F are the integration constants determined from the boundary conditions. For further concretization of the problem, the boundary conditions should be taken into account: A2 ð0Þ ¼
A2 ð0Þ
I4 ð0Þ ¼ I40 ;
¼ 0;
I1 ð0Þ ¼ I10 ;
I3 ð0Þ ¼ I30 ;
f jU1 j ; ð1 þ mÞ 2
where the following notations are introduced Q ¼ ðD2 þ 4jc2 j Þ
g¼
I40 m¼ : I10
Then, from the integrals (6) and Eqs. (10) and (11) we have the following equation for the integration constants 1=2 1=2 L L 2d2 F ¼ ; D¼ : ð13Þ Lþ Lþ 2d2
R¼
qSjU2 j ; jU1 j 2
f ¼ ½ðR 1Þ ð1 þ mÞ þ 4Rð1 þ mÞ
1=2
:
The time dependencies of the intensities are included in f, g and R parameters through U1 and U2 . These analytic formulae at t ! 1 coincide with those of stationary case [11–13]. From Eq. (14) for the diffraction efficiency of the dynamic holographic grating taking into the account the reading beam effect, we have r I1 I2 ¼ I0r I2 þ I3 gcz h gcz i ¼ 4m exp cosh cosðgdAzÞ 2 2 .
g¼
f f½f þ Rð1 þ mÞ þ ð1 mÞ þ ½f Rð1 þ mÞ
ð1 mÞ expðgczÞg: The given expression at t ! 1 coincides with that of stationary case, and at I0r =I0g ! 0 with the expression for diffraction efficiency obtained by using one-level model of optical transitions [11]. The obtained expression for diffraction efficiency is applied to study the dynamic effects in Bi12 SiO20 crystal: r41 ¼ 3:4 1012 m=V, cr =l ¼
Combining Eqs. (6), (10), (11) and (13) after simple transformation, the following expressions are obtained for the intensities of the optical waves 2 I4 ¼ I40 ½ðR 1Þð1 þ mÞ f 2
þ ½ðR 1Þð1 þ mÞ þ f expðgczÞ gcz þ 4Rð1 þ mÞ exp 2 cosðgdAzÞÞ=f f½f þ Rð1 þ mÞ þ ð1 mÞ þ ½f Rð1 þ mÞ ð1 mÞ expðgczÞg; d1 I1 ¼ I4 ; jU1 j gcz h gcz i I2 ¼ 4qmd1 exp cosh cosðgdAzÞ 2 2 =f f½f þ Rð1 þ mÞ þ ð1 mÞ þ ½f Rð1 þ mÞ ð1 mÞ expðgczÞg; d2 I3 ¼ I2 ; SjU2 j ð14Þ
Fig. 2. Dependence of the diffraction efficiency on t at m ¼ 1, K ¼ 2 lm, E0 ¼ 0 and different values of q: (1) q ¼ 0; (2) q ¼ 10; (3) q ¼ 40; (4) q ¼ 100; (5) q ¼ 700.
F.A. Rustamov et al. / Optics Communications 201 (2002) 191–196
Fig. 3. Dependence of the diffraction efficiency on t at m ¼ 1, K ¼ 2 lm, E0 ¼ 0 and different values of tr : (1) tr ¼ 0 s; (2) tr ¼ 0:05 s; (3) tr ¼ 0:2 s; (4) tr ¼ 0:4 s; (5) tr ¼ 0:6 s.
5:33 1012 Vm, Na ¼ 1021 m3 , Nd1 ¼ 1025 m3 , Nd2 ¼ 1021 m3 , s ¼ 2 105 m2 =J, n0 ¼ 2:5, e ¼ 56, z ¼ 1 cm, I0g ¼ 100 W=m2 . As seen in Fig. 2,
195
increase of parameter q (i.e., increase of intensity of readout beam I0r at a fixed value of intensity of recording beam I0g ) leads to decrease of the stationary value of diffraction efficiency g. At a fixed value of q (Fig. 3) the final stationary value of g is independent of switching moment of readout beam and is determined only by the q value. As one would expect, in lack of applied field (E0 ¼ 0) the dynamics of read–write processes has a smooth character. In this case the diffusion mechanism is predominant and phase shift between the holographic grating and interference pattern remains constant, i.e., there is not movement of grating fringe, and the parameter change smoothly. The effect of readout beam is reduced only to a decrease of final steady-state values. With increasing E0 (Fig. 4) the drift mechanism becomes dominant and the grating fringe movement becomes relatively considerable. That leads to the time-dependent phase member gdAz increasing the transient energy transfer effects, just the mentioned effects
Fig. 4. The calculated time dependencies of diffraction efficiency at different grating periods and applied fields (m ¼ 1, q ¼ 10): (1) E0 ¼ 2 kV/cm; (2) E0 ¼ 4 kV/cm; (3) E0 ¼ 6 kV/cm; (4) E0 ¼ 8 kV/cm. (a) K ¼ 0:4 lm; (b) K ¼ 1:4 lm; (c) K ¼ 2 lm; (d) K ¼ 3 lm.
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are the oscillations on the read–write curves. As one can see these oscillations increase with increasing applied field and the grating period. It should especially be noted that the oscillations occur not only on the recording curves but also on readout curves. Such behavior of time dependencies of diffraction efficiency agrees well with experimental data [5,14]. Thus, the analytical expression obtained for diffraction efficiency describes satisfactorily all characteristic peculiarities in dynamics of recording–readout process indicating that the using of this approximation is acceptable.
References [1] L. Solymar, J.M. Heaton, Opt. Commun. 51 (1984) 76. [2] J.M. Heaton, L. Solymar, IEEE J. Quantum Electron. 24 (1988) 558.
[3] J. Goltz, T. Tschudi, Opt. Commun. 67 (1988) 414. [4] D.R. Erbschloe, T. Wilson, Opt. Commun. 72 (1989) 135. [5] N. Kukhtarev, V. Markov, S. Odulov, Opt. Commun. 23 (1977) 338. [6] F. Jariego, F. Aguilo-lopes, J. Mod. Opt. 36 (1989) 1321. [7] F.A. Rustamov, Opt. Quantum Electron. 25 (1993) 351. [8] F.A. Rustamov, Opt. Quantum Electron. 32 (2000) 1069. [9] F.A. Rustamov, Opt. Quantum Electron. 23 (1991) 613. [10] N. Kukhtarev, Pisma GTF 2 (1976) 1114. [11] F.A. Rustamov, F.N. Gadjiev, Opt. Quantum Electron. 24 (1992) 1165. [12] F.A. Rustamov, Opt. Quantum Electron. 27 (1995) 239. [13] F.A. Rustamov, E.A. Sadykhov, Opt. Quantum Electron. 27 (1995) 249. [14] M. Miteva, Opt. Commun. 48 (1983) 85.
Optics Communications 201 (2002) 191–196 www.elsevier.com/locate/optcom
A simple analytic solution for transient simultaneous four-wave mixing in photorefractive crystals F.A. Rustamov *, S.R. Muradov, V.K. Sharbatov Department of Physics, Semiconductor Physics Laboratory, Baku State University, Baku 370145, Azerbaijan Received 26 March 2001; received in revised form 23 July 2001; accepted 9 October 2001
Abstract Very simple solution of the coupled partial differential equations that leads to analytic formula for diffraction efficiency which describes the time dependence of four-wave mixing in photorefractive crystals is presented. The obtained results were used for analysis of simultaneous read–write dynamics in Bi12 SiO20 crystal. Ó 2002 Published by Elsevier Science B.V. PACS: 42.65.Hw Keywords: Holographic gratings; Four-wave mixing; Photorefractive crystals
The coupled partial differential equations which describe dynamics of two- and four-wave mixing in photorefractive crystals are not soluble in quadrature. A number of theoretical papers dealing with the dynamics of two-wave mixing is based either on numerical solution [1–4] or on power series expansion [5,6]. The dynamics of four-wave mixing using a power series expansion is also considered [7,8]. In this paper we propose a simple analytic method for solution of coupled partial differential equations describing the time dependence for simultaneous record–readout at four-wave mixing in transmission geometry in
*
Corresponding author. E-mail address: [email protected] (F.A. Rustamov).
photorefractive crystals. The analytic formula for diffraction efficiency is obtained. A typical scheme of experiments on four-wave mixture in transmission geometry is shown in Fig. 1. A holographic grating in photorefractive crystal is usually recorded by two linearly polarized beams g g (I1 and Iþ1 ), the wavelength (kg ) of which corresponds to intrinsic optical absorbtion region, but the readout is realized by the beam of weak inr r tensity (I1 and Iþ1 ) and the wavelength ðkr Þ far from the intrinsic optical absorption region. A two-level model of optical transitions taking into account the effect of readout beam on the dynamic holographic grating has been proposed [7–9]. According to this model, the two donor levels (with concentrations Nd1 and Nd2 ) are involved in optical transitions. The deeper Nd1 donor levels respond to the recording beams, and shallow Nd2 levels
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F.A. Rustamov et al. / Optics Communications 201 (2002) 191–196 g oCþ1 g ¼ idEþ2 C1 ; oz g oC1 g ¼ idEþ2 Cþ1 ; oz r oCþ1 r ¼ idEþ2 C1 ; oz r oC1 r ¼ idEþ2 Cþ1 ; oz oEþ2 Eþ2 g g r r þ ¼ aþ ½Cþ1 C1 þ SCþ1 C1 ; ot sþ
ð1Þ
where Fig. 1. Recording and readout geometry of the transmission g holographic grating: I 1 are the intensities of recording beams; r r is the intensity of difIþ1 is the intensity of readout beam; I1 fracted beam.
respond to the readout beam. Unlike the classical one-level model [1–6], this model allows one to take into account the effect of the reading beam on the holographic grating. The two incident recording beams (i.e., interference pattern) initiate the phototransitions of electrons from the Nd1 levels to the conduction band. Owing to diffusion and drift, these electrons move through the crystal and locate in dark sites of the interference pattern, recombining with free Nd1 and Nd2 donor levels. A periodic space charge (i.e., a periodic internal electric field) is formed and leads to a periodic change of refractive index through the photorefractive effect, that is to the formation of holographic grating. Switching on the readout beam satisfying the Bragg condition causes excitation of photoelectrons from the shallow Nd2 donor levels to the conduction band. Owing to diffusion and drift in the conduction band, electrons are redistributed again over the crystal and recombine with free donor levels. Thus initial distribution of electrons changes leading to a distortion of existing holographic grating. In the two-level model, the system of nonlinear coupled first-order partial equations describing recording and readout at four-wave mixing in photorefractive crystals has the following form [7–9]:
1 1 1 ¼ i s s0 sk E2 ET 1 þ Eq 1 þ EEMT þ Eq ETM ¼ 2 2 ET sM 1 þ EM þ EEM0 E E0 1 EMq Eq i 2 2 ; sM 1 þ EEMT þ EEM0 h i E2 E0 i ET 1 þ EEMT þ EM0 a ¼ a1 ia2 ¼ 2 2 ; g ET sM I0 1 þ EM þ EEM0 e0 re0 k 2 2pkb T ; ; d¼ ; ET ¼ eK 4peln0 2kz 2eKNa KNa cr Eq ¼ ; ; EM ¼ ee0 2pl g;r g;r I0g;r ¼ Iþ1 ð0Þ þ I1 ð0Þ; 0 at t < tr ; S¼ cr1 a2 Nd2 at t P t ;
sM ¼
cr2 a1 ðNd1 Na Þ
r
and tr is a switching moment of readout beam. g r Here C 1 and C 1 are the complex amplitudes of recording, readout and diffracted beams, respectively; Na , Nd1 and Nd2 are the concentrations of acceptor and donor levels, respectively; l is the mobility of the current carriers; T is the temperature; e is the electron charge; kb is the Boltzman constant; a1 , a2 and cr1 , cr2 are the coefficients of ionization and capture of the corresponding levels; e and e0 are the high-frequency and static dielectric constants; r is the linear electrooptical coefficient;
F.A. Rustamov et al. / Optics Communications 201 (2002) 191–196
ET is the ‘‘diffusion’’ field; EM is the ‘‘drift’’ field; Eq is the space charge field; E0 is the applied field; Eþ2 is the amplitude of electric field; K is the holographic grating period. To study polarization effects and selectivity properties of holographic gratings it is necessary to take into account anisotropy of crystals. But these problems are out of scope of this paper. We are interested only in dynamics of four-wave mixing, so application of scalar equations is reasonable. This system is not soluble in quadrature. Asg g suming that the interference term ðCþ1 C1 þ r r SCþ1 C1 Þ is more smooth function in respect to expðt=sÞ (s is a characteristic time of static state establishment) we can take this term in last equation as a constant. The criterion of interference term smoothness is fulfilled quite well for long times ðt P sÞ and, at starting stage, for low gain coefficient value cz < 1 and for small phase shift dAz < 1 [10]. As the values of c and A depend on the applied field E0 and the holographic grating period K through a2 a1 2 c ¼ 2djsþ j I0g ; s0 sk ð2Þ a2 2 g a1 A ¼ jsþ j I0 þ s0 sk then the conditions for the initial stage can be satisfied by the appropriate choice of E0 and K [6– 9]. Under these conditions, the last equation can easily be integrated, and taking into account the initial conditions Eþ2 ð0Þ ¼ 0 and
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we can rewrite the system of Eq. (1) as: oA1 þ iaþ sþ d½A1 A4 þ A2 A3 A4 oz oA4 þ iaþ sþ d½A1 A4 þ A2 A3 A1 oz oA3 þ iaþ sþ d½A1 A4 þ A2 A3 A2 oz oA2 þ iaþ sþ d½A1 A4 þ A2 A3 A3 oz Eþ2 ¼ aþ sþ ½A1 A4 þ A2 A3 :
¼ 0; ¼ 0; ¼ 0; ¼ 0;
The given system of the coupled first-order differential equations is similar to equations for the stationary state [9,11–13], and hence has a similar solution. Equations (5) have the following integrals A1 A2 A3 A4 ¼ c 1 ; A1 A3 þ A2 A4 ¼ c2 ; A1 A1 þ A4 A4 ¼ d1 ;
where d1 ¼ jU1 jI0g and d2 ¼ jU2 jI0r . The constants c1 , c2 , d1 , d2 have parametric dependence on time t and do not have dependence on space coordinate z. Substitution of the above integrals into the equation system (5) allows us to divide it into two independent systems oA1 ¼ l½A1 d1 A1 ðjU1 jI1 þ SjU2 jI3 Þ þ c2 A3 ; oz oA3 ¼ l A3 d2 A3 ðjU1 jI1 þ SjU2 jI3 Þ þ c2 A1 ; oz ð7Þ oA2
the following expression for the internal electric field amplitude is obtained t g g Eþ2 ¼ aþ sþ Cþ1 C1 1 exp sþ
t tr r r þ SCþ1 C1 1 exp : ð3Þ sþ Substituting U1 ¼ 1 expðt=sþ Þ; U2 ¼ 1 expððt tr Þ=sþ Þ; 1=2
1=2
ð6Þ
A2 A2 þ A3 A3 ¼ d2 ;
g g Eþ2 ¼ aþ sþ ½1 expðtr =sþ ÞCþ1 C1
g A1;4 ¼ U1 C 1 and A3;2 ¼ ðSU2 Þ
ð5Þ
r C 1 ;
ð4Þ
¼ l A2 d2 A2 ðjU1 jI4 þ SjU2 jI2 Þ þ c2 A4 ;
oz ð8Þ oA4 ¼ l A4 d1 A4 ðjU1 jI4 þ SjU2 jI2 Þ þ c2 A2 ; oz g and where the notations l ¼ idaþ sþ , I1;4 ¼ I 1 r I3;2 ¼ I 1 are introduced. Simple transformation bring the above two equation systems into the following form [11]: h i oA31 2 ¼ l c2 þ A31 ðd2 d1 Þ c2 ðA31 Þ ; oz ð9Þ h i oA24 2 ¼ l c2 þ A24 ðd2 d1 Þ c2 ðA24 Þ ; oz where A31 ¼ A3 =A1 and A24 ¼ A2 =A4 .
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where
Solving the last two equations, we have A31 ¼
L D1 eqz þ Lþ Deqz ; 2c2 ðD1 eqz þ Deqz Þ
ð10Þ
A24 ¼
L F 1 eqz þ Lþ F eqz ; 2c2 ðF 1 eqz þ F eqz Þ
ð11Þ
D ¼ d2 d1 ; 2 1=2
;
ð12Þ L ¼ D Q; lQ ; q¼ 2 where D and F are the integration constants determined from the boundary conditions. For further concretization of the problem, the boundary conditions should be taken into account: A2 ð0Þ ¼
A2 ð0Þ
I4 ð0Þ ¼ I40 ;
¼ 0;
I1 ð0Þ ¼ I10 ;
I3 ð0Þ ¼ I30 ;
f jU1 j ; ð1 þ mÞ 2
where the following notations are introduced Q ¼ ðD2 þ 4jc2 j Þ
g¼
I40 m¼ : I10
Then, from the integrals (6) and Eqs. (10) and (11) we have the following equation for the integration constants 1=2 1=2 L L 2d2 F ¼ ; D¼ : ð13Þ Lþ Lþ 2d2
R¼
qSjU2 j ; jU1 j 2
f ¼ ½ðR 1Þ ð1 þ mÞ þ 4Rð1 þ mÞ
1=2
:
The time dependencies of the intensities are included in f, g and R parameters through U1 and U2 . These analytic formulae at t ! 1 coincide with those of stationary case [11–13]. From Eq. (14) for the diffraction efficiency of the dynamic holographic grating taking into the account the reading beam effect, we have r I1 I2 ¼ I0r I2 þ I3 gcz h gcz i ¼ 4m exp cosh cosðgdAzÞ 2 2 .
g¼
f f½f þ Rð1 þ mÞ þ ð1 mÞ þ ½f Rð1 þ mÞ
ð1 mÞ expðgczÞg: The given expression at t ! 1 coincides with that of stationary case, and at I0r =I0g ! 0 with the expression for diffraction efficiency obtained by using one-level model of optical transitions [11]. The obtained expression for diffraction efficiency is applied to study the dynamic effects in Bi12 SiO20 crystal: r41 ¼ 3:4 1012 m=V, cr =l ¼
Combining Eqs. (6), (10), (11) and (13) after simple transformation, the following expressions are obtained for the intensities of the optical waves 2 I4 ¼ I40 ½ðR 1Þð1 þ mÞ f 2
þ ½ðR 1Þð1 þ mÞ þ f expðgczÞ gcz þ 4Rð1 þ mÞ exp 2 cosðgdAzÞÞ=f f½f þ Rð1 þ mÞ þ ð1 mÞ þ ½f Rð1 þ mÞ ð1 mÞ expðgczÞg; d1 I1 ¼ I4 ; jU1 j gcz h gcz i I2 ¼ 4qmd1 exp cosh cosðgdAzÞ 2 2 =f f½f þ Rð1 þ mÞ þ ð1 mÞ þ ½f Rð1 þ mÞ ð1 mÞ expðgczÞg; d2 I3 ¼ I2 ; SjU2 j ð14Þ
Fig. 2. Dependence of the diffraction efficiency on t at m ¼ 1, K ¼ 2 lm, E0 ¼ 0 and different values of q: (1) q ¼ 0; (2) q ¼ 10; (3) q ¼ 40; (4) q ¼ 100; (5) q ¼ 700.
F.A. Rustamov et al. / Optics Communications 201 (2002) 191–196
Fig. 3. Dependence of the diffraction efficiency on t at m ¼ 1, K ¼ 2 lm, E0 ¼ 0 and different values of tr : (1) tr ¼ 0 s; (2) tr ¼ 0:05 s; (3) tr ¼ 0:2 s; (4) tr ¼ 0:4 s; (5) tr ¼ 0:6 s.
5:33 1012 Vm, Na ¼ 1021 m3 , Nd1 ¼ 1025 m3 , Nd2 ¼ 1021 m3 , s ¼ 2 105 m2 =J, n0 ¼ 2:5, e ¼ 56, z ¼ 1 cm, I0g ¼ 100 W=m2 . As seen in Fig. 2,
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increase of parameter q (i.e., increase of intensity of readout beam I0r at a fixed value of intensity of recording beam I0g ) leads to decrease of the stationary value of diffraction efficiency g. At a fixed value of q (Fig. 3) the final stationary value of g is independent of switching moment of readout beam and is determined only by the q value. As one would expect, in lack of applied field (E0 ¼ 0) the dynamics of read–write processes has a smooth character. In this case the diffusion mechanism is predominant and phase shift between the holographic grating and interference pattern remains constant, i.e., there is not movement of grating fringe, and the parameter change smoothly. The effect of readout beam is reduced only to a decrease of final steady-state values. With increasing E0 (Fig. 4) the drift mechanism becomes dominant and the grating fringe movement becomes relatively considerable. That leads to the time-dependent phase member gdAz increasing the transient energy transfer effects, just the mentioned effects
Fig. 4. The calculated time dependencies of diffraction efficiency at different grating periods and applied fields (m ¼ 1, q ¼ 10): (1) E0 ¼ 2 kV/cm; (2) E0 ¼ 4 kV/cm; (3) E0 ¼ 6 kV/cm; (4) E0 ¼ 8 kV/cm. (a) K ¼ 0:4 lm; (b) K ¼ 1:4 lm; (c) K ¼ 2 lm; (d) K ¼ 3 lm.
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are the oscillations on the read–write curves. As one can see these oscillations increase with increasing applied field and the grating period. It should especially be noted that the oscillations occur not only on the recording curves but also on readout curves. Such behavior of time dependencies of diffraction efficiency agrees well with experimental data [5,14]. Thus, the analytical expression obtained for diffraction efficiency describes satisfactorily all characteristic peculiarities in dynamics of recording–readout process indicating that the using of this approximation is acceptable.
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