- Email: [email protected]
Dielectric relaxation time measurement in absorbing photorefractive materials
1 May 2000
Optics Communications 178 Ž2000. 251–255 www.elsevier.comrlocateroptcom
Dielectric relaxation time measurement in absorbing photorefractive materials Ivan de Oliveira, Jaime Frejlich ) ´ Laboratorio Instituto de Fısica, UniÕersidade Estadual de Campinas, Caixa Postal 6165, 13081-970 Campinas-SP, Brazil ´ de Optica, ´ Received 1 November 1999; received in revised form 14 February 2000; accepted 29 March 2000
Abstract We show that absorbing photorefractive materials characterized by a space-charge exponential relaxation time law exhibit an overall hologram optical erasure that is described by the so-called exponential integral function, as far as self-diffraction can be neglected. This fact is due to the bulk absorption producing an exponentially decreasing distribution of the erasure beam irradiance along the sample thickness that results in a correspondingly increasing dielectric relaxation time. The theoretical development in this paper is experimentally verified by the analysis of the holographic erasure in a nominally undoped Bi 12TiO 20 photorefractive crystal using the 514.5 nm laser wavelength where this material exhibits a relatively strong bulk absorption. Neglecting absorption in this experiment leads to a relaxation time that is about 4-fold larger than the actual value. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 42.65.Hw; 72.15.Lh; 42.40.Pa; 72.20.-i
1. Introduction The Maxwell Žor dielectric. relaxation time is inversely proportional to the conductivity and by this way it determines the movement of electric charges inside materials. The dynamics of holographic recording in photorefractive materials w1x that are electro-optic and photoconductive is also determined by electric charge movement so that it does also depend on the dielectric relaxation. In these materials charge carriers are excited by the action of light from photoactive centers into the conduction band where they move Žby diffusion andror by the action of an external field. until they are retrapped somewhere ) Corresponding author. Fax: q55-19-289-3137; e-mail: [email protected]
else. An equilibrium condition is reached after some time, so that the pattern of light projected onto the sample is mapped into a corresponding electric charge volume modulation. As a consequence, a spacecharge electric field modulation arises that induces an index-of-refraction modulation Žphase volume hologram. via electro-optic effect. Dielectric relaxation in photorefractive materials can be easily measured from the evolution of volume hologram buildup or erasure. From these data the photoconductivity and other parameters can be computed. This technique has already been w2–12x and is still largely employed for photorefractive materials research. It has already been shown w13–15x that in spite of the adequacy of an exponential law to describe the space-charge evolution at any position inside most photorefractive materials Žassuming one single active
0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 6 6 0 - X
I. de OliÕeira, J. Frejlichr Optics Communications 178 (2000) 251–255
252
center and no hole-electron competition., such a law is in general not adequate for the overall Žaccumulated effect along the sample thickness. hologram evolution as measured at the crystal output. This is due to the fact that the intensity of the light acting on the photoactive centers decreases along the sample thickness as a consequence of bulk absorption so that the response time does also vary. The presence of absorption should be taken into consideration in hologram erasure Žand recording. experiments because otherwise erroneous relaxation laws may be found out and non-realistic material parameters derived. Unfortunately many authors do not consider this effect and misleading conclusions usually follow. In this paper we show that for the case of hologram erasure experiments under illumination in conditions where self-diffraction effects can be neglected, the effect of bulk absorption can be quite accurately considered using a simple analytical formulation that strongly facilitates data analysis. As an application of this technique we measure the dielectric relaxation time in a Bi 12 TiO 20 photorefractive crystal exhibiting considerable bulk absorption and show that neglecting absorption leads to large errors.
2. Theory The decrease of the hologram diffraction efficiency h during the hologram erasure with an uniform illumination of irradiance I, in the absence of self-diffraction for h < 1 and for the simple model reported above, follows an exponential law w1,16–18x
h s h 0 ey2 t r t sc , 1 s
tsc
1 1 q K 2 l s2
t M 1 q K 2 L2D
L D s 'Dt , 1
tM
s
Ž 1.
Dsm
sd q sph e´ 0
,
,
Ž 2.
k BT q
sph s
,
Ž 3.
q mtFa I hn
,
Ž 4.
where t M is the Maxwell Žor dielectric. relaxation time, l s is the Debye screening length, e the dielectric constant, ´ 0 the electric permittivity of vacuum,
m and t the charge carrier mobility and lifetime respectively. The dark- and photo-conductivities are sd and sph respectively and K s 2prD with D being the grating spatial period. Because of the bulk absorption a the light intensity varies along the sample thickness z as I s I0 eya z so that tsc in Eq. Ž2. is not any more a constant but a function of z so that we can write
'h s
(h
0
d
d yt r t Ž z . sc
H0 e
dz,
Ž 5.
where d is the crystal thickness. If we assume that sd < sph the expression above becomes
'h s
(h
0
ad = Ei Ž ytrtsc Ž 0 . . y Ei Ž ytrtsc Ž 0 . eya d . ,
Ž 6. 1 s
tsc Ž 0 .
1 q K 2 l s2 q mtFa IR Ž 0 . 1 q K 2 L2D
IR Ž 0 . s IRo Ž 1 y R . cos u i ,
hne´ 0
,
Ž 7.
where Ei Ž x . is the so-called ‘exponential integral function’ w19x. It is easy to show that the Eq. Ž6. turns into a simple exponential function only for a d ™ 0. The incident pump beam of irradiance IRo is used to erase the hologram; tsc Ž0. and IR Ž0. are the values at the front plane Žz s 0. inside the sample. The terms Ž1 y R . and cos u i in Eq. Ž7. stand for the interface Fresnel reflection and the refraction effect respectively, where R is the intensity reflection coefficient and u i is the incident angle in the air. We are neglecting the back reflection at the output crystal plane because it represents a rather small effect Ž20% compared to a factor of about 10 for the light variation between the input and output crystal interfaces due to absorption. and would considerably complicate the calculations. Otherwise a non-analytically integrable function would result and the appealing simple formulation in Eq. Ž6. would be lost. Rough calculations had shown that neglecting back reflection in this experiment would lead to a 5 to 10% error in the relaxation time values that is not much compared to other experimental uncertainties involved here.
I. de OliÕeira, J. Frejlichr Optics Communications 178 (2000) 251–255
253
3. Experiment A standard two-wave mixing experiment Žusing a setup similar to that in Ref. w5x. was carried out using a 7.0 mm = 6.2 mm and 2.05 mm thick nominally undoped Bi 12 TiO 20 Žlabelled BTO-011. photorefractive crystal Žwith e s 47 w1x. grown by the Czochralski method w20x, with the 6.2 mm side along the holographic wavevector K, the 7.0 mm side along the w001x-axis and perpendicular to the incidence plane. A 514.5 nm wavelength line of an Arq laser was used and the corresponding absorption for this sample was a s 11.65 cmy1 w21x. The irradiance of one of the interfering beams Žpump IRo . is much larger than that of the other Žsignal ISo . one in order to produce a small pattern-of-fringes visibility to verify the so-called first spatial harmonic approximation w16x. The polarization of the input beams was chosen Žapproximately 458 q 128 from the vertical axis in the opposite sense of optical activity rotation. so as the diffracted pump and the transmitted signal beams behind the crystal are orthogonally polarized w22x in order to enable the separation of the weaker diffracted beam from the much stronger transmitted one, using a polarizer. Once the hologram is written and steady state is reached, the signal beam is switched off and the pump is used to erase the hologram and, at the same time, to measure the diffraction efficiency evolution. An independent experiment showed the overall response time of the detection system to be about 1 ms, probably due to the shutter. Because of the very low diffraction efficiency involved Žalways less than 0.1%. self-diffraction can be neglected and the pump beam can be used to erase without any sensible feedback w23x effect on the holographic dynamics. The evolution of the diffraction efficiency during erasure was thus measured for different intensities of the pump beam IRo , at a fixed hologram spatial frequency K, and the procedure repeated for 4 different values of K always keeping constant the pump-to-signal IRorISo ratio. Fig. 1 shows the result of one such experiment where the square root of the diffracted pump beam IRd A h is plotted as a function of time from the instant the recording is stopped and erasure starts. Because of the influence of environmental perturbations on the setup allied to the large variation of the relaxation time along the crystal thickness, it is in
(
'
( '
Fig. 1. Hologram erasure: Plotting of IRd A h Žarbitrary units. evolution Žthin curve connecting data points. using an erasure beam IRo f 254 Wrm2 and 514.5 nm wavelength, with K s10.07 mmy1 and IRo r ISo f 42. The exponential fit Ždashed curve. gives tsc s9.6=10y2 s whereas the fitting to Eq. Ž6. Žthick continuous curve. gives tsc Ž0. s 2.37=10y2 s. The non-zero asymptotic limit for the diffraction evolution seen in the figure is due to the scattering from defects in the crystal and from other sources in the setup that cannot be avoided and should be taken into account. The middle and bottom pictures show expanded views of the data and fitting curves.
general difficult to be sure that the hologram is uniformly recorded in the whole crystal thickness, as is assumed in our equations. In order to ensure the latter condition, the recording was always carried out using a light intensity high enough to produce small relaxation times to minimize environmental perturbations. Even in this case we selected only the experiments showing that a steady-state stable recording stage had been reached just before erasure started.
I. de OliÕeira, J. Frejlichr Optics Communications 178 (2000) 251–255
254
This sample did not exhibit sensible hole-electron competition as analysed following the techniques described elsewhere w24,25x. Our crystal showed a sensible light-induced absorption effect that saturates at an irradiance value much lower than that of the pump beam used to erase the holograms in our experiment. Therefore and whatever be the nature of the centers Ževentually shallow traps. responsible for this effect we may safely consider them to be saturated throughout the erasure process so that we may expect them not to interfere with this experiment. Shallow traps however may contribute to a non-zero limit of sph for I ™ 0 in Eq. Ž7.. Such a contribution however, if ever existing, was experimentally shown to be of the order of our experimental uncertainties Žlower than 10%. and is therefore not taken into consideration in the formulation of Eq. Ž7.. 4. Results The expanded views Žmiddle and bottom pictures. in Fig. 1 clearly show that Eq. Ž6. fits our data much better than the exponential function does. The latter leads to an overall time-constant tsc that is systematically about 4-fold larger than the tsc Ž0. computed at the crystal input plane Ž z s 0. from Eq. Ž6. in this experiment. This exponential time-constant does approximately describe the overall relaxation process from a phenomenological point of view as seen in Fig. 1, but is ill-defined in terms of the theoretical model and consequently is not adequate to compute material parameters and derive conclusions from the theory. From the experimental data fitting to Eq. Ž6. a value for tsc Ž0. is obtained that is processed in the usual way w5,26x: 1rtsc Ž0. is plotted vs IR Ž0. for four different spatial frequencies: K s 5.6 mmy1 , 8.36 mmy1 , 10.07 mmy1 and 11.3 mmy1 . These data are fit to Eq. Ž7. and the corresponding angular coeffiy1 cient represents the quantity tsc Ž 0 . IR Ž 0 . in Eq. Ž7.. The inverse of the latter parameter is plotted in Fig. 2 as a function of K 2 that is described by the equation
tsc Ž 0 . IR Ž 0 . s
1rL2D q K 2 e´ 0 hn k B T 1 q K 2 l s2
q2 F a
.
Ž 8.
Recent unpublished experiments carried out with this sample showed that l s s 0.05 " 0.005 mm instead of
Fig. 2. Plotting tsc Ž0. IR Ž0. vs K 2 for data computed from Eq. Ž6. Žcircles. and from an exponential law Žtriangles.. The dashed lines are the corresponding best fit to Eq. Ž8. with l s f 0.05 mm that give L D f 0.38 mm, F f 0.36 for the exponential integral and L D f 0.26 mm, F f 0.09 for the simple exponential.
the l s f 0.027 mm value previously published elsewhere w21x. Data fitting to Eq. Ž8. with this new value for l s leads to L D s 0.38 " 0.06 mm and F s 0.36 " 0.02 that are roughly within the expected range for these parameters as reported below. The uncertainty range reported for L D and F are estimated just from the uncertainty in l s without any other consideration. The corresponding photoconductivity coefficient is spho rIR Ž0. s Ž q mtFa .rŽ hn . f 10 = 10y1 0 mrŽWV .. Data derived using a simple exponential relaxation law are also plotted in Fig. 2a and show a much poorer fitting to Eq. Ž8.. This fit leads to quite different results Ž L D s 0.26 " 0.03 mm, spho rIRŽ0. f 1.1 = 10y1 0 mrŽWV . and F s 0.09 " 0.01 where the uncertainties are estimated as explained above. from those obtained from Eq. Ž6.. Note the unrealistically low value for F in disagreement with other data available for this sample as reported below. As a guideline for comparison we may refer to recently and yet unpublished results about holograms produced by running pattern of fringes under controlled conditions on the same sample at the same wavelength that gave F s 0.3 to 0.45, L D s 0.2 to 0.4 mm and l s s 0.045 to 0.055 mm. Already published data w27x for the same sample using the initial phase technique Žinsensitive to bulk absorption effects. gives L D s 0.14 mm, but for another Ž532 nm. wavelength. There are preliminary experimental results indicating that L D may actually vary with the wavelength so that the above results may not be surprising. The value of the spho rIR Ž0. parameter in
I. de OliÕeira, J. Frejlichr Optics Communications 178 (2000) 251–255
this paper is of the same order of magnitude than those already reported w28x for other sillenites like Bi 12 SiO 20 Ž7.3 = 10y1 0 mrŽWV .. and Bi 12 GeO 20 Ž5.7 = 10y 10 mrŽWV .., using non-holographic techniques.
w2x w3x w4x w5x
5. Conclusions w6x
We have shown that the overall hologram evolution in absorbing photorefractive materials that follow a space-charge exponential relaxation law is described by an exponential integral function, as far as self-diffraction can be neglected. The use of a simple exponential may, at first glance, reasonably describe the average hologram relaxation but it will certainly lead to erroneous values for the material parameters and to misleading conclusions about the nature of the process. We verify these conclusions for a Bi 12 TiO 20 crystal exhibiting strong absorption at the 514.5 nm wavelength. We also report some of the material parameters for this sample as measured in this experiment and show that the use of a simple exponential law actually leads to very different results. Acknowledgements This work was partially supported by Conselho Nacional de Desenvolvimento Cientıfico e ´ Tecnologico–CNPq, Fundac¸ao ´ ˜ de Amparo a` Pesquisa do Estado de Sao ˜ Paulo–FAPESP, Financiadora de Estudos e Projetos–FINEPrPADCT and Volkswagen-Stiftung ŽGermany.. We strongly acknowledge the B12 TiO 20 crystals produced at the Laboratorio ´ de Crescimento de CristaisrIFSC-USP that enabled this work. References w1x S. Stepanov, P. Petrov, in: P. Gunter, J.-P. Huignard ŽEds.., ¨ Photorefractive Materials and Their Applications I, Vol. 61
w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x
w20x
w21x w22x w23x w24x w25x w26x w27x w28x
255
of Topics in Applied Physics, Chap. 9, Springer, Berlin, 1988, pp. 263–289. J. Feinberg, D. Heiman, A.R.T. Hellwarth Jr., R.W. Hellwarth, J. Appl. Phys. 51 Ž1980. 1297. R.A. Mullen, R.W. Hellwarth, J. Appl. Phys. 58 Ž1985. 40. G. Pauliat, J.M. Cohen-Jonathan, M. Allain, J.C. Launay, G. Roosen, Opt. Commun. 59 Ž1986. 266. P.A.M. dos Santos, P.M. Garcia, J. Frejlich, J. Appl. Phys. 66 Ž1989. 247. F.P. Strohkendl, P. Tayebati, R.W. Hellwarth, J. Appl. Phys. 66 Ž1989. 6024. J. Li, X.H. Li, F.Y. Lu, H.F. Wang, S.Z. Yuan, Y. Zhu, X. Wu, J. Appl. Phys. 76 Ž1994. 7541. S.X. Dou, Y. Ding, H.J. Eichler, Y. Zhu, P.X. Ye, Opt. Commun. 131 Ž1996. 322. M. Kaczmarek, P. Hribek, R.W. Eason, J. Mod. Opt. 43 Ž1996. 1817. U. van Stevendaal, K. Buse, S. Kamper, H. Hesse, E. ¨ Kratzig, Appl. Phys. B 63 Ž1996. 315. ¨ F. Rickermann, S. Riehemann, K. Buse, D. Dirksen, G. von Bally, J. Opt. Soc. Am. B 1 Ž1996. 2299. J.Y. Chang, C.R. Chinjen, R.H. Tsou, C.Y. Huang, C.C. Sun, M.W. Chang, Opt. Commun. 138 Ž1997. 101. J. Baquedano, M. Carracosa, L. Arizmendi, J.M. Cabrera, J. Opt. Soc. Am. B 4 Ž1987. 309. E.S. Maniloff, K.M. Johnson, J. Appl. Phys. 73 Ž1993. 541. M. Jeganathan, M.C. Bashaw, L. Hesselink, J. Opt. Soc. Am. B 1 Ž1995. 1370. T.J. Hall, R. Jaura, L.M. Connors, P.D. Foote, Progress in Quantum Electronics 10 Ž1985. 77. K. Buse, Appl. Phys. B 64 Ž1997. 273. N.V. Kukhtarev, Sov. Tech. Phys. Lett. 2 Ž1976. 438. G.B. Arfken, H.J. Weber, Mathematical methods for physicists, 4th ed., Academic Press, San Diego, New York, London, Tokyo, Toronto, 1995. V.V. Prokofiev, J.F. Carvalho, J.P. Andreeta, N.J.H. Gallo, A.C. Hernandes, J. Frejlich, A.A. Freschi, P.M. Garcia, J. Maracaiba, A.A. Kamshilin, T. Jaaskelainen, Crystal Res. Tech. 30 Ž1995. 171. J. Frejlich, P.M. Garcia, K.H. Ringhofer, E. Shamonina, J. Opt. Soc. Am. B 1 Ž1997. 1741. A.A. Kamshilin, M.P. Petrov, Opt. Commun. 53 Ž1985. 23. M. Horowitz, D. Kligler, B. Fisher, J. Opt. Soc. B 8 Ž1991. 2204. J. Frejlich, J. Appl. Phys. 68 Ž1990. 3104. J. Frejlich, P.M. Garcia, Appl. Phys. A 55 Ž1992. 49. J. Feinberg, R.W. Hellwarth, Opt. Lett. 5 Ž1980. 519. A.A. Freschi, P.M. Garcia, J. Frejlich, Appl. Phys. Lett. 71 Ž1997. 2427. J.P. Huignard, F. Micheron, Appl. Phys. Lett. 29 Ž1976. 591.
Optics Communications 178 Ž2000. 251–255 www.elsevier.comrlocateroptcom
Dielectric relaxation time measurement in absorbing photorefractive materials Ivan de Oliveira, Jaime Frejlich ) ´ Laboratorio Instituto de Fısica, UniÕersidade Estadual de Campinas, Caixa Postal 6165, 13081-970 Campinas-SP, Brazil ´ de Optica, ´ Received 1 November 1999; received in revised form 14 February 2000; accepted 29 March 2000
Abstract We show that absorbing photorefractive materials characterized by a space-charge exponential relaxation time law exhibit an overall hologram optical erasure that is described by the so-called exponential integral function, as far as self-diffraction can be neglected. This fact is due to the bulk absorption producing an exponentially decreasing distribution of the erasure beam irradiance along the sample thickness that results in a correspondingly increasing dielectric relaxation time. The theoretical development in this paper is experimentally verified by the analysis of the holographic erasure in a nominally undoped Bi 12TiO 20 photorefractive crystal using the 514.5 nm laser wavelength where this material exhibits a relatively strong bulk absorption. Neglecting absorption in this experiment leads to a relaxation time that is about 4-fold larger than the actual value. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 42.65.Hw; 72.15.Lh; 42.40.Pa; 72.20.-i
1. Introduction The Maxwell Žor dielectric. relaxation time is inversely proportional to the conductivity and by this way it determines the movement of electric charges inside materials. The dynamics of holographic recording in photorefractive materials w1x that are electro-optic and photoconductive is also determined by electric charge movement so that it does also depend on the dielectric relaxation. In these materials charge carriers are excited by the action of light from photoactive centers into the conduction band where they move Žby diffusion andror by the action of an external field. until they are retrapped somewhere ) Corresponding author. Fax: q55-19-289-3137; e-mail: [email protected]
else. An equilibrium condition is reached after some time, so that the pattern of light projected onto the sample is mapped into a corresponding electric charge volume modulation. As a consequence, a spacecharge electric field modulation arises that induces an index-of-refraction modulation Žphase volume hologram. via electro-optic effect. Dielectric relaxation in photorefractive materials can be easily measured from the evolution of volume hologram buildup or erasure. From these data the photoconductivity and other parameters can be computed. This technique has already been w2–12x and is still largely employed for photorefractive materials research. It has already been shown w13–15x that in spite of the adequacy of an exponential law to describe the space-charge evolution at any position inside most photorefractive materials Žassuming one single active
0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 6 6 0 - X
I. de OliÕeira, J. Frejlichr Optics Communications 178 (2000) 251–255
252
center and no hole-electron competition., such a law is in general not adequate for the overall Žaccumulated effect along the sample thickness. hologram evolution as measured at the crystal output. This is due to the fact that the intensity of the light acting on the photoactive centers decreases along the sample thickness as a consequence of bulk absorption so that the response time does also vary. The presence of absorption should be taken into consideration in hologram erasure Žand recording. experiments because otherwise erroneous relaxation laws may be found out and non-realistic material parameters derived. Unfortunately many authors do not consider this effect and misleading conclusions usually follow. In this paper we show that for the case of hologram erasure experiments under illumination in conditions where self-diffraction effects can be neglected, the effect of bulk absorption can be quite accurately considered using a simple analytical formulation that strongly facilitates data analysis. As an application of this technique we measure the dielectric relaxation time in a Bi 12 TiO 20 photorefractive crystal exhibiting considerable bulk absorption and show that neglecting absorption leads to large errors.
2. Theory The decrease of the hologram diffraction efficiency h during the hologram erasure with an uniform illumination of irradiance I, in the absence of self-diffraction for h < 1 and for the simple model reported above, follows an exponential law w1,16–18x
h s h 0 ey2 t r t sc , 1 s
tsc
1 1 q K 2 l s2
t M 1 q K 2 L2D
L D s 'Dt , 1
tM
s
Ž 1.
Dsm
sd q sph e´ 0
,
,
Ž 2.
k BT q
sph s
,
Ž 3.
q mtFa I hn
,
Ž 4.
where t M is the Maxwell Žor dielectric. relaxation time, l s is the Debye screening length, e the dielectric constant, ´ 0 the electric permittivity of vacuum,
m and t the charge carrier mobility and lifetime respectively. The dark- and photo-conductivities are sd and sph respectively and K s 2prD with D being the grating spatial period. Because of the bulk absorption a the light intensity varies along the sample thickness z as I s I0 eya z so that tsc in Eq. Ž2. is not any more a constant but a function of z so that we can write
'h s
(h
0
d
d yt r t Ž z . sc
H0 e
dz,
Ž 5.
where d is the crystal thickness. If we assume that sd < sph the expression above becomes
'h s
(h
0
ad = Ei Ž ytrtsc Ž 0 . . y Ei Ž ytrtsc Ž 0 . eya d . ,
Ž 6. 1 s
tsc Ž 0 .
1 q K 2 l s2 q mtFa IR Ž 0 . 1 q K 2 L2D
IR Ž 0 . s IRo Ž 1 y R . cos u i ,
hne´ 0
,
Ž 7.
where Ei Ž x . is the so-called ‘exponential integral function’ w19x. It is easy to show that the Eq. Ž6. turns into a simple exponential function only for a d ™ 0. The incident pump beam of irradiance IRo is used to erase the hologram; tsc Ž0. and IR Ž0. are the values at the front plane Žz s 0. inside the sample. The terms Ž1 y R . and cos u i in Eq. Ž7. stand for the interface Fresnel reflection and the refraction effect respectively, where R is the intensity reflection coefficient and u i is the incident angle in the air. We are neglecting the back reflection at the output crystal plane because it represents a rather small effect Ž20% compared to a factor of about 10 for the light variation between the input and output crystal interfaces due to absorption. and would considerably complicate the calculations. Otherwise a non-analytically integrable function would result and the appealing simple formulation in Eq. Ž6. would be lost. Rough calculations had shown that neglecting back reflection in this experiment would lead to a 5 to 10% error in the relaxation time values that is not much compared to other experimental uncertainties involved here.
I. de OliÕeira, J. Frejlichr Optics Communications 178 (2000) 251–255
253
3. Experiment A standard two-wave mixing experiment Žusing a setup similar to that in Ref. w5x. was carried out using a 7.0 mm = 6.2 mm and 2.05 mm thick nominally undoped Bi 12 TiO 20 Žlabelled BTO-011. photorefractive crystal Žwith e s 47 w1x. grown by the Czochralski method w20x, with the 6.2 mm side along the holographic wavevector K, the 7.0 mm side along the w001x-axis and perpendicular to the incidence plane. A 514.5 nm wavelength line of an Arq laser was used and the corresponding absorption for this sample was a s 11.65 cmy1 w21x. The irradiance of one of the interfering beams Žpump IRo . is much larger than that of the other Žsignal ISo . one in order to produce a small pattern-of-fringes visibility to verify the so-called first spatial harmonic approximation w16x. The polarization of the input beams was chosen Žapproximately 458 q 128 from the vertical axis in the opposite sense of optical activity rotation. so as the diffracted pump and the transmitted signal beams behind the crystal are orthogonally polarized w22x in order to enable the separation of the weaker diffracted beam from the much stronger transmitted one, using a polarizer. Once the hologram is written and steady state is reached, the signal beam is switched off and the pump is used to erase the hologram and, at the same time, to measure the diffraction efficiency evolution. An independent experiment showed the overall response time of the detection system to be about 1 ms, probably due to the shutter. Because of the very low diffraction efficiency involved Žalways less than 0.1%. self-diffraction can be neglected and the pump beam can be used to erase without any sensible feedback w23x effect on the holographic dynamics. The evolution of the diffraction efficiency during erasure was thus measured for different intensities of the pump beam IRo , at a fixed hologram spatial frequency K, and the procedure repeated for 4 different values of K always keeping constant the pump-to-signal IRorISo ratio. Fig. 1 shows the result of one such experiment where the square root of the diffracted pump beam IRd A h is plotted as a function of time from the instant the recording is stopped and erasure starts. Because of the influence of environmental perturbations on the setup allied to the large variation of the relaxation time along the crystal thickness, it is in
(
'
( '
Fig. 1. Hologram erasure: Plotting of IRd A h Žarbitrary units. evolution Žthin curve connecting data points. using an erasure beam IRo f 254 Wrm2 and 514.5 nm wavelength, with K s10.07 mmy1 and IRo r ISo f 42. The exponential fit Ždashed curve. gives tsc s9.6=10y2 s whereas the fitting to Eq. Ž6. Žthick continuous curve. gives tsc Ž0. s 2.37=10y2 s. The non-zero asymptotic limit for the diffraction evolution seen in the figure is due to the scattering from defects in the crystal and from other sources in the setup that cannot be avoided and should be taken into account. The middle and bottom pictures show expanded views of the data and fitting curves.
general difficult to be sure that the hologram is uniformly recorded in the whole crystal thickness, as is assumed in our equations. In order to ensure the latter condition, the recording was always carried out using a light intensity high enough to produce small relaxation times to minimize environmental perturbations. Even in this case we selected only the experiments showing that a steady-state stable recording stage had been reached just before erasure started.
I. de OliÕeira, J. Frejlichr Optics Communications 178 (2000) 251–255
254
This sample did not exhibit sensible hole-electron competition as analysed following the techniques described elsewhere w24,25x. Our crystal showed a sensible light-induced absorption effect that saturates at an irradiance value much lower than that of the pump beam used to erase the holograms in our experiment. Therefore and whatever be the nature of the centers Ževentually shallow traps. responsible for this effect we may safely consider them to be saturated throughout the erasure process so that we may expect them not to interfere with this experiment. Shallow traps however may contribute to a non-zero limit of sph for I ™ 0 in Eq. Ž7.. Such a contribution however, if ever existing, was experimentally shown to be of the order of our experimental uncertainties Žlower than 10%. and is therefore not taken into consideration in the formulation of Eq. Ž7.. 4. Results The expanded views Žmiddle and bottom pictures. in Fig. 1 clearly show that Eq. Ž6. fits our data much better than the exponential function does. The latter leads to an overall time-constant tsc that is systematically about 4-fold larger than the tsc Ž0. computed at the crystal input plane Ž z s 0. from Eq. Ž6. in this experiment. This exponential time-constant does approximately describe the overall relaxation process from a phenomenological point of view as seen in Fig. 1, but is ill-defined in terms of the theoretical model and consequently is not adequate to compute material parameters and derive conclusions from the theory. From the experimental data fitting to Eq. Ž6. a value for tsc Ž0. is obtained that is processed in the usual way w5,26x: 1rtsc Ž0. is plotted vs IR Ž0. for four different spatial frequencies: K s 5.6 mmy1 , 8.36 mmy1 , 10.07 mmy1 and 11.3 mmy1 . These data are fit to Eq. Ž7. and the corresponding angular coeffiy1 cient represents the quantity tsc Ž 0 . IR Ž 0 . in Eq. Ž7.. The inverse of the latter parameter is plotted in Fig. 2 as a function of K 2 that is described by the equation
tsc Ž 0 . IR Ž 0 . s
1rL2D q K 2 e´ 0 hn k B T 1 q K 2 l s2
q2 F a
.
Ž 8.
Recent unpublished experiments carried out with this sample showed that l s s 0.05 " 0.005 mm instead of
Fig. 2. Plotting tsc Ž0. IR Ž0. vs K 2 for data computed from Eq. Ž6. Žcircles. and from an exponential law Žtriangles.. The dashed lines are the corresponding best fit to Eq. Ž8. with l s f 0.05 mm that give L D f 0.38 mm, F f 0.36 for the exponential integral and L D f 0.26 mm, F f 0.09 for the simple exponential.
the l s f 0.027 mm value previously published elsewhere w21x. Data fitting to Eq. Ž8. with this new value for l s leads to L D s 0.38 " 0.06 mm and F s 0.36 " 0.02 that are roughly within the expected range for these parameters as reported below. The uncertainty range reported for L D and F are estimated just from the uncertainty in l s without any other consideration. The corresponding photoconductivity coefficient is spho rIR Ž0. s Ž q mtFa .rŽ hn . f 10 = 10y1 0 mrŽWV .. Data derived using a simple exponential relaxation law are also plotted in Fig. 2a and show a much poorer fitting to Eq. Ž8.. This fit leads to quite different results Ž L D s 0.26 " 0.03 mm, spho rIRŽ0. f 1.1 = 10y1 0 mrŽWV . and F s 0.09 " 0.01 where the uncertainties are estimated as explained above. from those obtained from Eq. Ž6.. Note the unrealistically low value for F in disagreement with other data available for this sample as reported below. As a guideline for comparison we may refer to recently and yet unpublished results about holograms produced by running pattern of fringes under controlled conditions on the same sample at the same wavelength that gave F s 0.3 to 0.45, L D s 0.2 to 0.4 mm and l s s 0.045 to 0.055 mm. Already published data w27x for the same sample using the initial phase technique Žinsensitive to bulk absorption effects. gives L D s 0.14 mm, but for another Ž532 nm. wavelength. There are preliminary experimental results indicating that L D may actually vary with the wavelength so that the above results may not be surprising. The value of the spho rIR Ž0. parameter in
I. de OliÕeira, J. Frejlichr Optics Communications 178 (2000) 251–255
this paper is of the same order of magnitude than those already reported w28x for other sillenites like Bi 12 SiO 20 Ž7.3 = 10y1 0 mrŽWV .. and Bi 12 GeO 20 Ž5.7 = 10y 10 mrŽWV .., using non-holographic techniques.
w2x w3x w4x w5x
5. Conclusions w6x
We have shown that the overall hologram evolution in absorbing photorefractive materials that follow a space-charge exponential relaxation law is described by an exponential integral function, as far as self-diffraction can be neglected. The use of a simple exponential may, at first glance, reasonably describe the average hologram relaxation but it will certainly lead to erroneous values for the material parameters and to misleading conclusions about the nature of the process. We verify these conclusions for a Bi 12 TiO 20 crystal exhibiting strong absorption at the 514.5 nm wavelength. We also report some of the material parameters for this sample as measured in this experiment and show that the use of a simple exponential law actually leads to very different results. Acknowledgements This work was partially supported by Conselho Nacional de Desenvolvimento Cientıfico e ´ Tecnologico–CNPq, Fundac¸ao ´ ˜ de Amparo a` Pesquisa do Estado de Sao ˜ Paulo–FAPESP, Financiadora de Estudos e Projetos–FINEPrPADCT and Volkswagen-Stiftung ŽGermany.. We strongly acknowledge the B12 TiO 20 crystals produced at the Laboratorio ´ de Crescimento de CristaisrIFSC-USP that enabled this work. References w1x S. Stepanov, P. Petrov, in: P. Gunter, J.-P. Huignard ŽEds.., ¨ Photorefractive Materials and Their Applications I, Vol. 61
w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x
w20x
w21x w22x w23x w24x w25x w26x w27x w28x
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