- Email: [email protected]
Effects of light modulation on grating phase shifts in photorefractive recording
15June
1997
OPT KS
COMMUNICATIONS ELSEVIER
Optics Communications
139 ( 1997) 8 I-84
Effects of light modulation on grating phase shifts in photorefractive recording J.G. Murillo a, L.F. Magafia a, M. Carrascosa b3*, F. AguWL6pez
b
” lash& de Fkica, Uthwsidad National de Mexico. Mexico, Mexico ’ Departumenfo de Fisica de Materiales. Unkvrsidad Autbnomu de Madrid, 28049 Madrid, Spain Received
12 November
1996; accepted
19 February
1997
Abstract Phase mismatches with regard to light of photorefractive gratings have been investigated for arbitrary modulation depths of the illuminating pattern. The time evolution of the phases during recording under an applied field as well as the steady state values have been calculated for various harmonics. A powerful numerical method has been used to solve the band transport model equations. A clear correlation between the oscillations of the phase shift and these for the grating amplitude have been found for low modulation depths m. They both are markedly damped for high m. The steady-state phase mismatch for the fundamental grating increases with m at variance with the linear prediction. This increase together with the superlinear behaviour for the amplitude gives rise to a marked enhancement of the photorefractive coupling coefficients with regard to the linear values. 1. Introduction The photorefractive effect [l] is a nonlinear mechanism that converts an inhomogeneous light intensity pattern (e.g. sinusoidal) into a correlated refractive index distribution. However, at variance with conventional nonlinearities the occurrence of a phase shift between light and index patterns is a peculiar feature of photorefraction. This phase shift is the origin of beam coupling and beam amplification effects with a high technological potential. The phase shift and consequently the beam coupling can be governed by imposing an adequate motion to the light fringes by either, using a piezomirror or applying electric fields. Moreover the amplitude of the index grating itself and also the erasure speed can be optimized by suitable modification of the phase shift of light fringes with regard to the index gratings [2,3]. In summary an adequate knowledge of the photorefractive phase shifts is very relevant to optimize either the recorded index grating or the beam coupling gain. Moreover, at a more fundamental level, the
* Corresponding
author. E-mail: [email protected].
0030-4018/97/$17.00 Copyright PII SOO30-4018(97)00114-4
evolution of the phase provides direct information on fringe motion and so on the microscopic details of the photorefractive kinetics. For low modulation depths m of the light profile I = I,,(1 + m cos Kn), the rate equations describing the photorefractive kinetics can be linearized and so, analytically solved. In other words, the time evolution of the amplitude and phase (with regard to light) of the recorded grating can be analytically predicted. For the steady state main results are that the amplitude is proportional to m and the phase mismatch is independent of m. However, since in many experiments one uses a high modulation (m - l), it is important to theoretically predict both the amplitude and phase behaviour for this case. A number of calculations for arbitrary m have been reported that mostly concern with the evolution of the grating amplitude [4-61. The information for the phases is more scarce, although some analytical and numerical results on the imaginary component E, sin 4 of the complex grating amplitude have been obtained when the light fringes are being moved by the use of a piezomirror [7]. For diffusion dominated transport the phase-mismatch is n/2, regardless of the modulation m and recording time. However when an electric field is applied the linear (low ml and
0 1997 Elsevier Science B.V. All rights reserved.
82 Table 1 Numerical
J.G. Murillo et al. /Optics Communications 139 (1997) 81-84
parameters
for the simulation
Average light intensity I,: 5 mW/cm’ Grating period A: 1 Frn Electrooptic coefficient r: 5.0X lo-” Dielectric constant E: 56 Refractive index n: 2.53 Acceptor density NA: 10” mm3 Donor density No: 10” m-’ Mobility CL:3 X 10m6 m*/Vs
m/V
nonlinear (high m> cases should lead to different phase behaviour. This paper presents some relevant results on the evolution of the grating phases for various harmonics during recording under an applied field at arbitrary modulation up to tn _ 1. These results are obtained solving the standard Kukhtarev equations using a numerical procedure based in a finite difference scheme. The details of the numerical method have been reported in a previous work [6]. The main differential features with regard to the linear results (m -=x 1) are remarked and discussed from the viewpoint of the microscopic grating dynamics. BSO has been used as an example of a non-photovoltaic material with the parameters commonly found in the literature (see for instance Refs. [5,6]) and summarized in Table 1.
2. Phase behaviour during recording 2.1. Fundamental
grating
Fig. la shows the time evolution of the fundamental grating amplitude E, for three modulation depths, m = 0.3, m = 0.6 and m = 0.9. The corresponding kinetics for the phase mismatch is plotted in Fig. lb. An external electric field E, = 5 kV/cm is applied to the crystal. Oscillations in the grating amplitude appearing for low m (and predicted by the analytical solution of the linearized equations 181) become smoother or even disappear for high m as observed by Soutar et al. in BSO 191. On the other hand, the phase-mismatch starts at a value different from zero and independent of m, which is correctly given by an analytical solution of the linearized material equations [lo] and reaches a steady-state value at the same time as the amplitude. By comparing Fig. la and 1b, a clear correlation between the amplitude and phase oscillations is observed. Minima in the phase oscillation are associated to maxima in the amplitude and vice versa. Moreover, an interesting outcome of these results is the occurrence of a damping in the amplitude and phase oscillations for the higher m values. In other words, fringe motion appears to be hindered by the nonlinear terms of the rate equations, so avoiding the fringe oscillations as well as the associated oscillations in the amplitude that appear for low m. This is
j&____:_
f
(
1
0
2 TIME
( set
3
)
Fig. 1. Temporal evolution of the amplitude (a) and phase mismatch (in radians) (b) of the fundamental grating amplitude corresponding to three modulation depths: (--_) m = 0.3; (- -) m = 0.6; (- - -) m = 0.9. A field of 5 kV/cm is applied. Average light intensity is 5 mW/cm’.
a very remarkable effect of strong modulation been sufficiently investigated.
that has not
2.2. Harmonic gratings When one is operating at high m values a substantial contribution of the harmonics sets in. The behaviour of the harmonics with m is essentially similar to that described above for the fundamental grating. For high m no oscillations appear in any of the monitored harmonics. The
0.6 fi 4
0_~
_----
0.5
2 -
1 /
i
.-
/
0.4 i
/
-_---------
O-OO*
2.0 TIME
( set
)
Fig. 2. Distance travelled by the fundamental (---I, second (--_) and third harmonic (---_) light fringes in units of the light grating period A as a function of time for m = 0.9 and applied field of 5 kV/cm.
J.G. Murillo
83
et al./ Optics Communicutions 139 (1997) 81-84
‘“1 (4
the fundamental and harmonic grating amplitudes on m. As for the phase, the amplitude values are markedly enhanced with regard to the linear prediction. This supralinear behaviour of the amplitudes for high m has been measured in BaTiO, [l l] and BSO [12] and it has been also previously inferred from numerical simulations [4,13].
3. Beam coupling coefficient
0: 0.0
0.2
0.4
MODULATION
0.0
0.2
0.4
MODULATION
0.6
0.8
RATIO
m
0.6
RATIO
0.8
1.0
1.0
m
The m-dependence of the amplitude and phase of the fundamental grating should have some relevant consequences on the wave coupling effects during recording. This influence appears in the coupling equations (see for instance Ref. [14]) through the coupling coefficient r sin 4 where r= l/m. c= 27rn3rE,/(Acos 01, E, being the amplitude of the fundamental grating, A the beam wavelength, 0 the incidence angle, r the effective electrooptic coefficient and n the refractive index. For high m both r and C$ depend on m according to the results obtained in the previous section. The coupling coefficient r sin 4 is plotted as a function of the light modulation m for no applied field (dashed curve) and E, = 0 (solid curve) in Fig. 4. In both cases one observes a marked enhancement of the gain coefficient for high m with regard to the linear case (m < 1). For m = 1, this enhancement factor is about 50% larger than in the diffusion case, when a 5 kV/cm is applied. In other words, the effects of high m on the gain coefficient are more relevant when recording is performed under an external field because of the variation of the phase-shift. In summary the coupling effects at high modulations under applied fields should be stronger than those predicted by the linear theory due to both the enhancement of the amplitude and the phase mismatch. It is to be noted that the enhancement of r with m under an applied field
Fig. 3. Steady state phase mismatches (in radians) (a) and amplitudes (b) of the fundamental ( n ), second ( +) and third ( A)
harmonic gratings for m = 0.9 and applied field of 5 kV/cm.
23 correlation between the temporal dynamics for the phase mismatch corresponding to the three harmonics is exemplified in Fig. 2 for rn = 0.9. The distances travelled by the fringes in units of the light grating period A are plotted as a function of time. The behaviour is similar for all gratings but the harmonic grating moves at higher speed for the higher harmonic orders. This effect contributes to the distortion of the overall fringe profile. Fig. 3a shows the stationary phase of the fundamental grating and of the two first harmonics as a function of m for an applied field of 5 kV. There is a marked increase of the phase-mismatch for m close to 1 at variance with the linear theory. To the best of our knowledge, no previous theoretical predictions on this behaviour have been reported. For comparison Fig. 3b shows the dependence of
,,-
r
,’
i
‘:
,a’
290
s
-+,5 e ._ iII, tot
, ..
,’
,I’
__I’ _/’
.
__--
--____-________.__.-------/ /= /
.
0,5 0,O
0,2
0,4
0.6
0,8
1 I,0
MODULATION RATIO, m Fig. 4. Value of the wave coupling parameter r sin do as a function of the light modulation m for no applied field (- - -) and E, = 5 kV/cm (--I.
84
J.G. Murillo et al. /Optics Communications I39 (1997) 81-84
has not been previously measured in fixed light fringes experiments. Anyhow these results are at variance with those found when the recording is optimized by the use of moving light fringes (i.e. with a piezomirror in one of the beams). In this case a reduction of r is observed [7]. This different behaviour with regard to the normal case, associated to the optimization effect of the piezomirror, should deserve more detailed attention.
Acknowledgements
This work has been carried out under European project No. CII-CT94-0039. Partial support from CICYT through grants TIC-950166 and MAT-951254-CE are also gratefuIly acknowledged.
4. Summary and conclusions The kinetics and steady state values of the photorefractive grating phase shifts have been investigated for arbitrary modulation depths m and under an applied field. Under these conditions (E, # 0 and m _ 1) phase mismatches are no more constant. Specifically, theoretical calculations carried out in this work for BSO show an enhancement of the steady state phase shift with regard to the linear case. This enhancement of the phase mismatch and consequently of the coupling efficiency should have an important bearing on two- and four-wave mixing experiments. Fortunately many amplification experiments are performed under m Z+ 1 and so measured gains are not affected by the nonlinear effects. Finally, one should remark another consequence of the strong modulation effects on phase mismatches. For low m the light and index fringes are bent but keep parallel due to the constancy of the phase-mismatch. On the other hand, for high m, this phase-mismatch depend on m (see Fig. 3) and consequently it varies with the propagation distance into the material. In other words, as a consequence of the m dependence of the coupling gain the parallelism between the light and index fringes is destroyed. This is expected to influence the width of the Bragg diffraction peak depending on the m value used in the experiment.
References
[I] Photorefractive Materials and their applications I, Eds. P. Ginter, J.P. Huignard (Springer, Berlin, 1989). [2] Maria Aguilar, E. Serrano, V. Lopez, M. Carrascosa and F. Agullb-L6pez, Opt. Mater. 4 (1995) 304, 461. [3] Maria Aguilar, E. Serrano, V. Lopez, M. Carrascosa, F. Agulh-Lopez, Optics Comm. 96 (1995) 116. [4] GA. Brost, Optics Comm. 96 (1993) 113. [5] E. Serrano, V. Lopez, M. Carrascosa, F. Agull&Mpez, J. Opt. Sot. Am. B 11 (1994) 670. [6] J.G. Murillo, L.F. Magaiia, M. Carrascosa, F. Agullo-Mpez, J. Appl. Phys. 78 (1995) 5686. [7] P. Refregier, L. Solymar, H. Rajbenbach, J.P. Huignard, J. Appl. Phys. 58 (1985) 45. [8] F. Jariego, F. Agullb-Mpez, Optics Comm. 76 (1990) 169. [9] C. Soutar, W.A. Gillespie, C.M. Cartwright, Optics Comm. 90 (1992) 329. [lo] M. Carrascosa, J.M. Cabrera, F. Agull&L6pez, Optics Comm. 69 (1988) 83. [I 11 Y.H. Lee, R.W. Hellwarth, J. Appl. Phys. 71 (1992) 916. [12] J.V. Alvarez-Bravo, M. Carrascosa, L. Arizmendi, Optics Comm. 103 (1993) 22. [13] E. Serrano, V. Lopez, M. Carrascosa, F. AgulM-Lopez, IEEE J. Quantum. Electron. 30 (1994) 875. [14] Photorefractive Materials and their applications, II, Eds. P. Ginter, J.P. Huignard (Springer, Berlin, 1989) Ch. 4.
1997
OPT KS
COMMUNICATIONS ELSEVIER
Optics Communications
139 ( 1997) 8 I-84
Effects of light modulation on grating phase shifts in photorefractive recording J.G. Murillo a, L.F. Magafia a, M. Carrascosa b3*, F. AguWL6pez
b
” lash& de Fkica, Uthwsidad National de Mexico. Mexico, Mexico ’ Departumenfo de Fisica de Materiales. Unkvrsidad Autbnomu de Madrid, 28049 Madrid, Spain Received
12 November
1996; accepted
19 February
1997
Abstract Phase mismatches with regard to light of photorefractive gratings have been investigated for arbitrary modulation depths of the illuminating pattern. The time evolution of the phases during recording under an applied field as well as the steady state values have been calculated for various harmonics. A powerful numerical method has been used to solve the band transport model equations. A clear correlation between the oscillations of the phase shift and these for the grating amplitude have been found for low modulation depths m. They both are markedly damped for high m. The steady-state phase mismatch for the fundamental grating increases with m at variance with the linear prediction. This increase together with the superlinear behaviour for the amplitude gives rise to a marked enhancement of the photorefractive coupling coefficients with regard to the linear values. 1. Introduction The photorefractive effect [l] is a nonlinear mechanism that converts an inhomogeneous light intensity pattern (e.g. sinusoidal) into a correlated refractive index distribution. However, at variance with conventional nonlinearities the occurrence of a phase shift between light and index patterns is a peculiar feature of photorefraction. This phase shift is the origin of beam coupling and beam amplification effects with a high technological potential. The phase shift and consequently the beam coupling can be governed by imposing an adequate motion to the light fringes by either, using a piezomirror or applying electric fields. Moreover the amplitude of the index grating itself and also the erasure speed can be optimized by suitable modification of the phase shift of light fringes with regard to the index gratings [2,3]. In summary an adequate knowledge of the photorefractive phase shifts is very relevant to optimize either the recorded index grating or the beam coupling gain. Moreover, at a more fundamental level, the
* Corresponding
author. E-mail: [email protected].
0030-4018/97/$17.00 Copyright PII SOO30-4018(97)00114-4
evolution of the phase provides direct information on fringe motion and so on the microscopic details of the photorefractive kinetics. For low modulation depths m of the light profile I = I,,(1 + m cos Kn), the rate equations describing the photorefractive kinetics can be linearized and so, analytically solved. In other words, the time evolution of the amplitude and phase (with regard to light) of the recorded grating can be analytically predicted. For the steady state main results are that the amplitude is proportional to m and the phase mismatch is independent of m. However, since in many experiments one uses a high modulation (m - l), it is important to theoretically predict both the amplitude and phase behaviour for this case. A number of calculations for arbitrary m have been reported that mostly concern with the evolution of the grating amplitude [4-61. The information for the phases is more scarce, although some analytical and numerical results on the imaginary component E, sin 4 of the complex grating amplitude have been obtained when the light fringes are being moved by the use of a piezomirror [7]. For diffusion dominated transport the phase-mismatch is n/2, regardless of the modulation m and recording time. However when an electric field is applied the linear (low ml and
0 1997 Elsevier Science B.V. All rights reserved.
82 Table 1 Numerical
J.G. Murillo et al. /Optics Communications 139 (1997) 81-84
parameters
for the simulation
Average light intensity I,: 5 mW/cm’ Grating period A: 1 Frn Electrooptic coefficient r: 5.0X lo-” Dielectric constant E: 56 Refractive index n: 2.53 Acceptor density NA: 10” mm3 Donor density No: 10” m-’ Mobility CL:3 X 10m6 m*/Vs
m/V
nonlinear (high m> cases should lead to different phase behaviour. This paper presents some relevant results on the evolution of the grating phases for various harmonics during recording under an applied field at arbitrary modulation up to tn _ 1. These results are obtained solving the standard Kukhtarev equations using a numerical procedure based in a finite difference scheme. The details of the numerical method have been reported in a previous work [6]. The main differential features with regard to the linear results (m -=x 1) are remarked and discussed from the viewpoint of the microscopic grating dynamics. BSO has been used as an example of a non-photovoltaic material with the parameters commonly found in the literature (see for instance Refs. [5,6]) and summarized in Table 1.
2. Phase behaviour during recording 2.1. Fundamental
grating
Fig. la shows the time evolution of the fundamental grating amplitude E, for three modulation depths, m = 0.3, m = 0.6 and m = 0.9. The corresponding kinetics for the phase mismatch is plotted in Fig. lb. An external electric field E, = 5 kV/cm is applied to the crystal. Oscillations in the grating amplitude appearing for low m (and predicted by the analytical solution of the linearized equations 181) become smoother or even disappear for high m as observed by Soutar et al. in BSO 191. On the other hand, the phase-mismatch starts at a value different from zero and independent of m, which is correctly given by an analytical solution of the linearized material equations [lo] and reaches a steady-state value at the same time as the amplitude. By comparing Fig. la and 1b, a clear correlation between the amplitude and phase oscillations is observed. Minima in the phase oscillation are associated to maxima in the amplitude and vice versa. Moreover, an interesting outcome of these results is the occurrence of a damping in the amplitude and phase oscillations for the higher m values. In other words, fringe motion appears to be hindered by the nonlinear terms of the rate equations, so avoiding the fringe oscillations as well as the associated oscillations in the amplitude that appear for low m. This is
j&____:_
f
(
1
0
2 TIME
( set
3
)
Fig. 1. Temporal evolution of the amplitude (a) and phase mismatch (in radians) (b) of the fundamental grating amplitude corresponding to three modulation depths: (--_) m = 0.3; (- -) m = 0.6; (- - -) m = 0.9. A field of 5 kV/cm is applied. Average light intensity is 5 mW/cm’.
a very remarkable effect of strong modulation been sufficiently investigated.
that has not
2.2. Harmonic gratings When one is operating at high m values a substantial contribution of the harmonics sets in. The behaviour of the harmonics with m is essentially similar to that described above for the fundamental grating. For high m no oscillations appear in any of the monitored harmonics. The
0.6 fi 4
0_~
_----
0.5
2 -
1 /
i
.-
/
0.4 i
/
-_---------
O-OO*
2.0 TIME
( set
)
Fig. 2. Distance travelled by the fundamental (---I, second (--_) and third harmonic (---_) light fringes in units of the light grating period A as a function of time for m = 0.9 and applied field of 5 kV/cm.
J.G. Murillo
83
et al./ Optics Communicutions 139 (1997) 81-84
‘“1 (4
the fundamental and harmonic grating amplitudes on m. As for the phase, the amplitude values are markedly enhanced with regard to the linear prediction. This supralinear behaviour of the amplitudes for high m has been measured in BaTiO, [l l] and BSO [12] and it has been also previously inferred from numerical simulations [4,13].
3. Beam coupling coefficient
0: 0.0
0.2
0.4
MODULATION
0.0
0.2
0.4
MODULATION
0.6
0.8
RATIO
m
0.6
RATIO
0.8
1.0
1.0
m
The m-dependence of the amplitude and phase of the fundamental grating should have some relevant consequences on the wave coupling effects during recording. This influence appears in the coupling equations (see for instance Ref. [14]) through the coupling coefficient r sin 4 where r= l/m. c= 27rn3rE,/(Acos 01, E, being the amplitude of the fundamental grating, A the beam wavelength, 0 the incidence angle, r the effective electrooptic coefficient and n the refractive index. For high m both r and C$ depend on m according to the results obtained in the previous section. The coupling coefficient r sin 4 is plotted as a function of the light modulation m for no applied field (dashed curve) and E, = 0 (solid curve) in Fig. 4. In both cases one observes a marked enhancement of the gain coefficient for high m with regard to the linear case (m < 1). For m = 1, this enhancement factor is about 50% larger than in the diffusion case, when a 5 kV/cm is applied. In other words, the effects of high m on the gain coefficient are more relevant when recording is performed under an external field because of the variation of the phase-shift. In summary the coupling effects at high modulations under applied fields should be stronger than those predicted by the linear theory due to both the enhancement of the amplitude and the phase mismatch. It is to be noted that the enhancement of r with m under an applied field
Fig. 3. Steady state phase mismatches (in radians) (a) and amplitudes (b) of the fundamental ( n ), second ( +) and third ( A)
harmonic gratings for m = 0.9 and applied field of 5 kV/cm.
23 correlation between the temporal dynamics for the phase mismatch corresponding to the three harmonics is exemplified in Fig. 2 for rn = 0.9. The distances travelled by the fringes in units of the light grating period A are plotted as a function of time. The behaviour is similar for all gratings but the harmonic grating moves at higher speed for the higher harmonic orders. This effect contributes to the distortion of the overall fringe profile. Fig. 3a shows the stationary phase of the fundamental grating and of the two first harmonics as a function of m for an applied field of 5 kV. There is a marked increase of the phase-mismatch for m close to 1 at variance with the linear theory. To the best of our knowledge, no previous theoretical predictions on this behaviour have been reported. For comparison Fig. 3b shows the dependence of
,,-
r
,’
i
‘:
,a’
290
s
-+,5 e ._ iII, tot
, ..
,’
,I’
__I’ _/’
.
__--
--____-________.__.-------/ /= /
.
0,5 0,O
0,2
0,4
0.6
0,8
1 I,0
MODULATION RATIO, m Fig. 4. Value of the wave coupling parameter r sin do as a function of the light modulation m for no applied field (- - -) and E, = 5 kV/cm (--I.
84
J.G. Murillo et al. /Optics Communications I39 (1997) 81-84
has not been previously measured in fixed light fringes experiments. Anyhow these results are at variance with those found when the recording is optimized by the use of moving light fringes (i.e. with a piezomirror in one of the beams). In this case a reduction of r is observed [7]. This different behaviour with regard to the normal case, associated to the optimization effect of the piezomirror, should deserve more detailed attention.
Acknowledgements
This work has been carried out under European project No. CII-CT94-0039. Partial support from CICYT through grants TIC-950166 and MAT-951254-CE are also gratefuIly acknowledged.
4. Summary and conclusions The kinetics and steady state values of the photorefractive grating phase shifts have been investigated for arbitrary modulation depths m and under an applied field. Under these conditions (E, # 0 and m _ 1) phase mismatches are no more constant. Specifically, theoretical calculations carried out in this work for BSO show an enhancement of the steady state phase shift with regard to the linear case. This enhancement of the phase mismatch and consequently of the coupling efficiency should have an important bearing on two- and four-wave mixing experiments. Fortunately many amplification experiments are performed under m Z+ 1 and so measured gains are not affected by the nonlinear effects. Finally, one should remark another consequence of the strong modulation effects on phase mismatches. For low m the light and index fringes are bent but keep parallel due to the constancy of the phase-mismatch. On the other hand, for high m, this phase-mismatch depend on m (see Fig. 3) and consequently it varies with the propagation distance into the material. In other words, as a consequence of the m dependence of the coupling gain the parallelism between the light and index fringes is destroyed. This is expected to influence the width of the Bragg diffraction peak depending on the m value used in the experiment.
References
[I] Photorefractive Materials and their applications I, Eds. P. Ginter, J.P. Huignard (Springer, Berlin, 1989). [2] Maria Aguilar, E. Serrano, V. Lopez, M. Carrascosa and F. Agullb-L6pez, Opt. Mater. 4 (1995) 304, 461. [3] Maria Aguilar, E. Serrano, V. Lopez, M. Carrascosa, F. Agulh-Lopez, Optics Comm. 96 (1995) 116. [4] GA. Brost, Optics Comm. 96 (1993) 113. [5] E. Serrano, V. Lopez, M. Carrascosa, F. Agull&Mpez, J. Opt. Sot. Am. B 11 (1994) 670. [6] J.G. Murillo, L.F. Magaiia, M. Carrascosa, F. Agullo-Mpez, J. Appl. Phys. 78 (1995) 5686. [7] P. Refregier, L. Solymar, H. Rajbenbach, J.P. Huignard, J. Appl. Phys. 58 (1985) 45. [8] F. Jariego, F. Agullb-Mpez, Optics Comm. 76 (1990) 169. [9] C. Soutar, W.A. Gillespie, C.M. Cartwright, Optics Comm. 90 (1992) 329. [lo] M. Carrascosa, J.M. Cabrera, F. Agull&L6pez, Optics Comm. 69 (1988) 83. [I 11 Y.H. Lee, R.W. Hellwarth, J. Appl. Phys. 71 (1992) 916. [12] J.V. Alvarez-Bravo, M. Carrascosa, L. Arizmendi, Optics Comm. 103 (1993) 22. [13] E. Serrano, V. Lopez, M. Carrascosa, F. AgulM-Lopez, IEEE J. Quantum. Electron. 30 (1994) 875. [14] Photorefractive Materials and their applications, II, Eds. P. Ginter, J.P. Huignard (Springer, Berlin, 1989) Ch. 4.