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Selfenhancement in lithium niobate
Volume 72, number 3,4
SELFENHANCEMENT
OPTICS COMMUNICATIONS
IN LITHIUM
J. OTTEN, A. OZOLS, M. REINFELDE
15 July 1989
NIOBATE
and K.H. RINGHOFER
Fachbereich Physik, Universitiit, PF 4469, D-4500 Osnabtick, Fed. Rep. Germany
Received 28 March 1989
A simple linear theory for selfenhancement in transmission geometry is presented. The results are compared with measurements in lithium niobate. We obtain good overall agreement between theory and experiment.
1. Introduction
Selfenhancement is a dynamic effect in electrooptic crystals. It describes a possible increase of diffraction efficiency during the readout (under the Bragg angle) of a phase hologram by one of the recording beams. In transmission geometry the effect was first observed in 1972 by Staebler and Amodei [ 1 ] in lithium niobate. In this paper a linear theory is presented which describes the effect of selfenhancement in transmission geometry. The theory is compared with measurements of one of the authors [ 2,3 ] in lithium niobate. Additional measurements have been performed by us to allow a more detailed comparison with theory. Our measurements as well as the older ones have been done in the following way (fig. 1): a hologram is written by two interfering plane waves (signal- and reference beam) during a short time interval so that the diffraction efficiency always remains below 2%. The polarization used is parallel with polarization vectors es and eR and the intensities are Is and IR as shown in fig. 1. Then one of the writing beams is blocked and the effect of the other beam, which is at still the Bragg angle, is investigated. In many cases an increase of the diffraction efficiency results. For modelling this measurements we use a simple linear theory analogous to a theory proposed by Solymar and Heaton [4] which is expected to yield good results if the depletion of the reading beam can be neglected and if the angle between the writing beams is not too small so that higher Fourier orders
+ 2
Fig. 1. Geometry of the selfenhancement experiment. S refers to the signal beam and R to the reference beam, both with parallel polarization. The plane of incidence contains the crystal c-axis.
are of no importance. There is a history to this theory: to our knowledge, the method has first been proposed by Wang [ 51 in a different field.
2. Theory We separate the theoretical description into three parts. In the first part we depict the writing of the grating during a short time from t= 0 until t = t,, with two writing beams. Next we examine the situation at time t = to when one beam is blocked. The other beam is reading out the grating in the Bragg angle and we
0 030-4018/89/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
175
15 July 1989
OPTICS COMMUNICATIONS
Volume 72, number 3,4
can compute the initial diffraction efficiency q0 which is also given by our measurements. In the third part we investigate the time development of the process between to and the stationary state (when the hologram is erased). We are able to compute the increase of the diffraction efficiency (selfenhancement ) .
the Bragg angle. Reading out a volume grating has first been described by Kogelnik [ 81 using a coupled-wave formalism. Here we use a straightforward calculation for reading a volume grating at the Bragg angle [ 9 1, which, using eq. ( 3 ) may be written in the form
2. I. Writing during a short time with two beams
where ed is the amplitude of the diffracted beam, x is the crystal depth, ZG,is the vacuum wave number, and eR is the amplitude of the reading out reference beam. With the amplitude of the space charge field and with the abbreviation from eq. (2) Z= - f ikon 3r,ff Eefl we obtain
Recording a phase hologram in photorefractive crystals is described by the model of Kuktharev et al. [ 61. When modelling this part of the experiment, the feedback of the grating onto the distribution of light can be neglected. Only the equation for writing is to be taken into account. We write this equation in the form [7]
?,aE,/at+E,
= -Eefimw,
(1)
where the relaxation time during writing is r,, the amplitude of the modulated space charge field is El, the effective field is E,, and the modulation of the recording waves is m,. The relaxation time is defined by T, = tS’ec,/( ospZ) with the static dielectric constant es’, the vacuum permittivity eo, the specific photoconductivity oSp= a/Z and the intensity I. For our geometry the effective field given in ref. [ 71 reduces to Eeff= Eph’+iEdiff. The photovoltaic field and the diffusion field are defined by Ephv= cD~/osp with the donor density CDand the photovoltaic constant /I and by Ediff= k,TK/e with Boltzmann constant kg, the temperature T, the magnitude of the grating vector K and the unit charge e. For a short writing-time to << r, the solution of eq. ( 1) is linear in to, & (to) =&m
tolTw.
(2)
From E, we obtain the change of the refractive An, by the electrooptic effect, An,
(to)
=
-
In3re&(to)
,
-fibn3r,ffE,
eR,
(4)
&d/dx=hw(to/~w)eR. This equation beam
(5)
yields the amplitude
of the diffracted
(6)
cd(x) =Zxm,(tO/r,)eR,
so that the initial situation for the enhancement experiment is clearly defined. For a given diffraction efficiency o. after writing with two beams the writing-time to is obtained as to=&r,,JIZIDm,.,
,
(7)
with D the crystal thickness. With this equation and eq. (6) also the modulation ma(x) =ed(x)/eR with eR>> ed can be calculated as a function of C~Stal depth x, ma(x) =ZxJlrol
lrlDT
which we need for further developments section.
(8) in the next
2.3. Dynamic reading and writing with one beam
index
(3)
where n is the relevant refractive index and reffis the effective electrooptic coefficient. 2.2. Reading out the grating with one beam At time t = to. one of the writing beams is blocked so that the other beam is reading out the grating at 176
&d/ax=
Now we investigate the effect of the remaining recording beam. It reads out the grating already built up and together with the diffracted beam (which increases with crystal depth) it can enhance the existing grating. During this process there is no restriction on the diffraction efficiency. In fact, the maximum diffraction efficiency may become as high as 40 times the initial diffraction efficiency, so that the feedback of the grating onto the distribution of light has to be taken into account. This is done, as
Volume 72, number 3,4
OPTICS COMMUNICATIONS
has already been mentioned, by the theory of Solymar and Heaton [4]. To apply this theory we have to assume that the beam which has not been blocked, let us say the reference beam, is not depleted. Multiplying eq. (4) for reading by e; and eliminating the space charge field E, from the new equation with the eq. ( 1) for writing we obtain the following linear differential equation for the modulation m which depends on the crystal depth x and the time t:
Ta2m(x,t) + axat
amtx,t>=rm(x, ax
t) ,
with the initial conditions m (0, x) = m. (x) and m ( t, 0) =o.
The modulation m obtained is essentially an integral of the zero order Bessel function Jo over the crystal depth < m(t,x)=exp(-t/z) x
x s
am,(t)
at;
Jd2i,/m)
dt,
(10)
.
(11)
0
with Jo(2i,/m)=kEo
(trkF)k . .
In this equation we insert the initial modulation from eq. (8 ) and solve the integral
(12) With the modulation m(x, t)=ed(x, t)/eR (eRs%=ed) we finally can obtain the diffraction efficiency q(t)= led(D, t)/eR12 as a function of time t for a given crystal thickness D, q(t) =tlo exp(
-2tlr)
kzo [L$r, 2 . . a
(13)
If blocked and unblocked beam are interchanged, the following changes have to be performed K_, _K,
EPhv+Ephv*, E*ifi+ -Edif,
m+m*, (14)
15 July 1989
where K is the grating vector, Eph’=EphvK/K, and Ediff_-EdiffiYIK. 2.4. Parameters
and assumptions
Before we compare our theory with the experimental results we have to choose the measurements which conform to our assumptions. The assumptions are: 1. Our theory is restricted to transmission geometry. 2. The Bragg-angle must not be too small or too large. Otherwise one would be forced to take into account off-Bragg terms or birefringence. 3. The selfenhancement must not be too large, because the reading beam is assumed to be undepleted. All these restrictions are fulfilled by the measurement described by fig. 2(c) of ref. [ 21 which in fact combines two different measurements. Next we describe our choice of parameters. From the measurements of ref. [ 21 we get: initial diffraction efficiency qo= 0.47%, grating spacing n = 1.5 pm, wavelength &632.8 nm, crystal thickness D=3.5 mm and intensities Is = Z, = 0.1 W/cm2 of signal and reference beam. The concentration of iron ions is 0.03 wtl which corresponds to a particle density of cFe= 1.5x 1O25m-3. Our calculation needs some more parameters which have to be taken from the literature (see table 1). The photovoltaic constant /I has been recently measured by L. Holtmann (see the acknowledgement ) . Unfortunately there remain two parameters, the donor density CDand the specific photoconductivity rP, which are not exactly known. Only the ratio of the concentration of donors and acceptors is estimated to be 20-3OW. To obtain these parameters we tit the experiments with our theory to obtain the doTable 1 Material parameters 1.5x lo= Iron density c,. 2 x1O24 Donor density co Refractive index n 2.2 28.8x lo-‘* Electrooptic coefficient rev 30.0 Static dielectric constant e,, 4 x1O-34 Photovoltaic constant B Specific photoconductivity uap 3 x10-‘6
Im-‘I
121
[m--’
1101 [m/V]
[ 11)
[121 [m’/V]
[m/v*1 177
Volume 72, number 3,4
OPTICS COMMUNICATIONS
15 July 1989
nor density CD and the photoconductivity listed in the table 1. CD corresponds to a ratio of the concentration of donors and acceptors of 15Oh which is a little under our estimate. The value for the specific photoconductivity agrees well with ‘a measurement of R. Sommerfeld (see the acknowledgement). A crude estimate of the photoconductivity can also be obtained from fig. 2 (b) of ref. [ 21 which does not show selfenhancement so that the relaxation time can be read off directly from this figure.
3. Results In our measurements selfenhancement is described by r(t) =v( t)/qo, which is presented as a function of time in the diagrams. Theoretically, using eq. ( 13 ), we obtain
(15) At first we want to compare this result with some general statement of ref. [ 21: 1. The first statement is that a smaller initial diffraction efficiency leads to a larger maximum value of the selfenhancement. Our result is different from this statement and claims that the selfenhancement is independent of initial diffraction efficiency. 2. The second claim is that the position and the value of the maximum is changed when the intensity of the nonblocked beam is changed from the beginning. Our result is that for higher intensity the relaxation time is smaller and therefore the maximum appears at earlier times. However, there is no change in the value of the maximum. 3. The third statement is that selfenhancement can only be obtained up to a certain grating spacing of 38 urn. This statement cannot be checked by the present theory because the theory is not accurate enough to describe large grating spacings (see assumption 2 ) . Next, we try to reproduce the results of our measurements in detail: In fig. 2 both theory and measurements are shown. The full lines represent the calculations. The upper one corresponds to the situation when the signal beam is blocked. The corresponding experimental values are given by squares. The lower one corresponds to the situation when the reference 178
I
I
20.0
I
I
40.0 Time
60.0 t /
I
I
60.0
[min]
Fig. 2. Comparison between theory and measurements. Squares: the signal beam is blocked. Circles: the reference beam is blocked. Full lines: theory.
beam is blocked. The corresponding values are given by circles.
experimental
4. Summary and conclusions We have presented a linear theory to describe selfenhancement. The agreement with the experimental results can be called satisfying in view of the simplicity of the theory, although not all the details can be reproduced. Nevertheless, the overall trends are reproduced well by the theory, so that we can conclude that our physical picture about selfenhancement is correct. Also the material parameters which we had to obtain by lit agree well with the estimated values. It would be of interest to analyze also the other measurements of ref. [2] which could not be dealt with by our theory. Beam depletion, absorption, and higher Fourier orders would have to be included and transmission as well as reflection geometry would have to be analyzed.
Acknowledgement Two of the authors (J.O. and K.H.R.) wish to thank the Deutsche Forschungsgemeinschaft for
Volume 72, number 3,4
OPTICS COMMUNICATIONS
within the programme of the SFB 225 “Oxide Crystals for Electra- and Magnetooptical Applications”. Furthermore it is our pleasure to thank L. Holtmann for the measurement of the photovoltaic constant relevant for our problem and R. Sommerfeld for the measurement of the photoconductivity. support
References [ 1 ] D.L. Staebler and J.J. Amodei, Appl. Phys. Lett. 43 (1972) 1042. [2] K.K. Shvarts, A.O. 0~01s and M.J. Reinfelde, Ferroelectrics 63 (1985) 309.
15 July 1989
[ 31 K-K. Shvarts, A.O. Ozols, P. Augustov and M. Reinfelde, Ferroelectrics 75 (1987) 231. [4] L. Solymar and J.M. Heaton, Optics Comm. 5 1 (1984) 76. [ 5 ] Chen-Show Wang, Phys. Rev. 182 ( 1969) 482. [ 61 N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin and V.L. Vinetskii, Ferroelectrics 22 (1979) 949. [ 71 J. Marotz, K.H. Ringhofer, R.A. Rupp and S. Treichel, IEEE J. Quantum Electron. 22 (1986) 1379. [B] H. Kogelnik, The Bell System Technical Journal 48 (1969) 2409. [ 91 J. Marotz and K.H. Ringhofer, Optics Comm. 64 ( 1987) 245. [IO] D.F. Nelson and R.M. Mikulyak, J. Appl. Phys. 45 ( 1974) 3688. [ 111 I.P. Kaminow, E.H. Turner, R.L. Barns and J.L. Bet-stein, J. Appl. Phys. 5 1 ( 1980) 4379. [ 121 A. Savage, J. Appl. Phys. 37 (1966) 3071.
179
SELFENHANCEMENT
OPTICS COMMUNICATIONS
IN LITHIUM
J. OTTEN, A. OZOLS, M. REINFELDE
15 July 1989
NIOBATE
and K.H. RINGHOFER
Fachbereich Physik, Universitiit, PF 4469, D-4500 Osnabtick, Fed. Rep. Germany
Received 28 March 1989
A simple linear theory for selfenhancement in transmission geometry is presented. The results are compared with measurements in lithium niobate. We obtain good overall agreement between theory and experiment.
1. Introduction
Selfenhancement is a dynamic effect in electrooptic crystals. It describes a possible increase of diffraction efficiency during the readout (under the Bragg angle) of a phase hologram by one of the recording beams. In transmission geometry the effect was first observed in 1972 by Staebler and Amodei [ 1 ] in lithium niobate. In this paper a linear theory is presented which describes the effect of selfenhancement in transmission geometry. The theory is compared with measurements of one of the authors [ 2,3 ] in lithium niobate. Additional measurements have been performed by us to allow a more detailed comparison with theory. Our measurements as well as the older ones have been done in the following way (fig. 1): a hologram is written by two interfering plane waves (signal- and reference beam) during a short time interval so that the diffraction efficiency always remains below 2%. The polarization used is parallel with polarization vectors es and eR and the intensities are Is and IR as shown in fig. 1. Then one of the writing beams is blocked and the effect of the other beam, which is at still the Bragg angle, is investigated. In many cases an increase of the diffraction efficiency results. For modelling this measurements we use a simple linear theory analogous to a theory proposed by Solymar and Heaton [4] which is expected to yield good results if the depletion of the reading beam can be neglected and if the angle between the writing beams is not too small so that higher Fourier orders
+ 2
Fig. 1. Geometry of the selfenhancement experiment. S refers to the signal beam and R to the reference beam, both with parallel polarization. The plane of incidence contains the crystal c-axis.
are of no importance. There is a history to this theory: to our knowledge, the method has first been proposed by Wang [ 51 in a different field.
2. Theory We separate the theoretical description into three parts. In the first part we depict the writing of the grating during a short time from t= 0 until t = t,, with two writing beams. Next we examine the situation at time t = to when one beam is blocked. The other beam is reading out the grating in the Bragg angle and we
0 030-4018/89/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
175
15 July 1989
OPTICS COMMUNICATIONS
Volume 72, number 3,4
can compute the initial diffraction efficiency q0 which is also given by our measurements. In the third part we investigate the time development of the process between to and the stationary state (when the hologram is erased). We are able to compute the increase of the diffraction efficiency (selfenhancement ) .
the Bragg angle. Reading out a volume grating has first been described by Kogelnik [ 81 using a coupled-wave formalism. Here we use a straightforward calculation for reading a volume grating at the Bragg angle [ 9 1, which, using eq. ( 3 ) may be written in the form
2. I. Writing during a short time with two beams
where ed is the amplitude of the diffracted beam, x is the crystal depth, ZG,is the vacuum wave number, and eR is the amplitude of the reading out reference beam. With the amplitude of the space charge field and with the abbreviation from eq. (2) Z= - f ikon 3r,ff Eefl we obtain
Recording a phase hologram in photorefractive crystals is described by the model of Kuktharev et al. [ 61. When modelling this part of the experiment, the feedback of the grating onto the distribution of light can be neglected. Only the equation for writing is to be taken into account. We write this equation in the form [7]
?,aE,/at+E,
= -Eefimw,
(1)
where the relaxation time during writing is r,, the amplitude of the modulated space charge field is El, the effective field is E,, and the modulation of the recording waves is m,. The relaxation time is defined by T, = tS’ec,/( ospZ) with the static dielectric constant es’, the vacuum permittivity eo, the specific photoconductivity oSp= a/Z and the intensity I. For our geometry the effective field given in ref. [ 71 reduces to Eeff= Eph’+iEdiff. The photovoltaic field and the diffusion field are defined by Ephv= cD~/osp with the donor density CDand the photovoltaic constant /I and by Ediff= k,TK/e with Boltzmann constant kg, the temperature T, the magnitude of the grating vector K and the unit charge e. For a short writing-time to << r, the solution of eq. ( 1) is linear in to, & (to) =&m
tolTw.
(2)
From E, we obtain the change of the refractive An, by the electrooptic effect, An,
(to)
=
-
In3re&(to)
,
-fibn3r,ffE,
eR,
(4)
&d/dx=hw(to/~w)eR. This equation beam
(5)
yields the amplitude
of the diffracted
(6)
cd(x) =Zxm,(tO/r,)eR,
so that the initial situation for the enhancement experiment is clearly defined. For a given diffraction efficiency o. after writing with two beams the writing-time to is obtained as to=&r,,JIZIDm,.,
,
(7)
with D the crystal thickness. With this equation and eq. (6) also the modulation ma(x) =ed(x)/eR with eR>> ed can be calculated as a function of C~Stal depth x, ma(x) =ZxJlrol
lrlDT
which we need for further developments section.
(8) in the next
2.3. Dynamic reading and writing with one beam
index
(3)
where n is the relevant refractive index and reffis the effective electrooptic coefficient. 2.2. Reading out the grating with one beam At time t = to. one of the writing beams is blocked so that the other beam is reading out the grating at 176
&d/ax=
Now we investigate the effect of the remaining recording beam. It reads out the grating already built up and together with the diffracted beam (which increases with crystal depth) it can enhance the existing grating. During this process there is no restriction on the diffraction efficiency. In fact, the maximum diffraction efficiency may become as high as 40 times the initial diffraction efficiency, so that the feedback of the grating onto the distribution of light has to be taken into account. This is done, as
Volume 72, number 3,4
OPTICS COMMUNICATIONS
has already been mentioned, by the theory of Solymar and Heaton [4]. To apply this theory we have to assume that the beam which has not been blocked, let us say the reference beam, is not depleted. Multiplying eq. (4) for reading by e; and eliminating the space charge field E, from the new equation with the eq. ( 1) for writing we obtain the following linear differential equation for the modulation m which depends on the crystal depth x and the time t:
Ta2m(x,t) + axat
amtx,t>=rm(x, ax
t) ,
with the initial conditions m (0, x) = m. (x) and m ( t, 0) =o.
The modulation m obtained is essentially an integral of the zero order Bessel function Jo over the crystal depth < m(t,x)=exp(-t/z) x
x s
am,(t)
at;
Jd2i,/m)
dt,
(10)
.
(11)
0
with Jo(2i,/m)=kEo
(trkF)k . .
In this equation we insert the initial modulation from eq. (8 ) and solve the integral
(12) With the modulation m(x, t)=ed(x, t)/eR (eRs%=ed) we finally can obtain the diffraction efficiency q(t)= led(D, t)/eR12 as a function of time t for a given crystal thickness D, q(t) =tlo exp(
-2tlr)
kzo [L$r, 2 . . a
(13)
If blocked and unblocked beam are interchanged, the following changes have to be performed K_, _K,
EPhv+Ephv*, E*ifi+ -Edif,
m+m*, (14)
15 July 1989
where K is the grating vector, Eph’=EphvK/K, and Ediff_-EdiffiYIK. 2.4. Parameters
and assumptions
Before we compare our theory with the experimental results we have to choose the measurements which conform to our assumptions. The assumptions are: 1. Our theory is restricted to transmission geometry. 2. The Bragg-angle must not be too small or too large. Otherwise one would be forced to take into account off-Bragg terms or birefringence. 3. The selfenhancement must not be too large, because the reading beam is assumed to be undepleted. All these restrictions are fulfilled by the measurement described by fig. 2(c) of ref. [ 21 which in fact combines two different measurements. Next we describe our choice of parameters. From the measurements of ref. [ 21 we get: initial diffraction efficiency qo= 0.47%, grating spacing n = 1.5 pm, wavelength &632.8 nm, crystal thickness D=3.5 mm and intensities Is = Z, = 0.1 W/cm2 of signal and reference beam. The concentration of iron ions is 0.03 wtl which corresponds to a particle density of cFe= 1.5x 1O25m-3. Our calculation needs some more parameters which have to be taken from the literature (see table 1). The photovoltaic constant /I has been recently measured by L. Holtmann (see the acknowledgement ) . Unfortunately there remain two parameters, the donor density CDand the specific photoconductivity rP, which are not exactly known. Only the ratio of the concentration of donors and acceptors is estimated to be 20-3OW. To obtain these parameters we tit the experiments with our theory to obtain the doTable 1 Material parameters 1.5x lo= Iron density c,. 2 x1O24 Donor density co Refractive index n 2.2 28.8x lo-‘* Electrooptic coefficient rev 30.0 Static dielectric constant e,, 4 x1O-34 Photovoltaic constant B Specific photoconductivity uap 3 x10-‘6
Im-‘I
121
[m--’
1101 [m/V]
[ 11)
[121 [m’/V]
[m/v*1 177
Volume 72, number 3,4
OPTICS COMMUNICATIONS
15 July 1989
nor density CD and the photoconductivity listed in the table 1. CD corresponds to a ratio of the concentration of donors and acceptors of 15Oh which is a little under our estimate. The value for the specific photoconductivity agrees well with ‘a measurement of R. Sommerfeld (see the acknowledgement). A crude estimate of the photoconductivity can also be obtained from fig. 2 (b) of ref. [ 21 which does not show selfenhancement so that the relaxation time can be read off directly from this figure.
3. Results In our measurements selfenhancement is described by r(t) =v( t)/qo, which is presented as a function of time in the diagrams. Theoretically, using eq. ( 13 ), we obtain
(15) At first we want to compare this result with some general statement of ref. [ 21: 1. The first statement is that a smaller initial diffraction efficiency leads to a larger maximum value of the selfenhancement. Our result is different from this statement and claims that the selfenhancement is independent of initial diffraction efficiency. 2. The second claim is that the position and the value of the maximum is changed when the intensity of the nonblocked beam is changed from the beginning. Our result is that for higher intensity the relaxation time is smaller and therefore the maximum appears at earlier times. However, there is no change in the value of the maximum. 3. The third statement is that selfenhancement can only be obtained up to a certain grating spacing of 38 urn. This statement cannot be checked by the present theory because the theory is not accurate enough to describe large grating spacings (see assumption 2 ) . Next, we try to reproduce the results of our measurements in detail: In fig. 2 both theory and measurements are shown. The full lines represent the calculations. The upper one corresponds to the situation when the signal beam is blocked. The corresponding experimental values are given by squares. The lower one corresponds to the situation when the reference 178
I
I
20.0
I
I
40.0 Time
60.0 t /
I
I
60.0
[min]
Fig. 2. Comparison between theory and measurements. Squares: the signal beam is blocked. Circles: the reference beam is blocked. Full lines: theory.
beam is blocked. The corresponding values are given by circles.
experimental
4. Summary and conclusions We have presented a linear theory to describe selfenhancement. The agreement with the experimental results can be called satisfying in view of the simplicity of the theory, although not all the details can be reproduced. Nevertheless, the overall trends are reproduced well by the theory, so that we can conclude that our physical picture about selfenhancement is correct. Also the material parameters which we had to obtain by lit agree well with the estimated values. It would be of interest to analyze also the other measurements of ref. [2] which could not be dealt with by our theory. Beam depletion, absorption, and higher Fourier orders would have to be included and transmission as well as reflection geometry would have to be analyzed.
Acknowledgement Two of the authors (J.O. and K.H.R.) wish to thank the Deutsche Forschungsgemeinschaft for
Volume 72, number 3,4
OPTICS COMMUNICATIONS
within the programme of the SFB 225 “Oxide Crystals for Electra- and Magnetooptical Applications”. Furthermore it is our pleasure to thank L. Holtmann for the measurement of the photovoltaic constant relevant for our problem and R. Sommerfeld for the measurement of the photoconductivity. support
References [ 1 ] D.L. Staebler and J.J. Amodei, Appl. Phys. Lett. 43 (1972) 1042. [2] K.K. Shvarts, A.O. 0~01s and M.J. Reinfelde, Ferroelectrics 63 (1985) 309.
15 July 1989
[ 31 K-K. Shvarts, A.O. Ozols, P. Augustov and M. Reinfelde, Ferroelectrics 75 (1987) 231. [4] L. Solymar and J.M. Heaton, Optics Comm. 5 1 (1984) 76. [ 5 ] Chen-Show Wang, Phys. Rev. 182 ( 1969) 482. [ 61 N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin and V.L. Vinetskii, Ferroelectrics 22 (1979) 949. [ 71 J. Marotz, K.H. Ringhofer, R.A. Rupp and S. Treichel, IEEE J. Quantum Electron. 22 (1986) 1379. [B] H. Kogelnik, The Bell System Technical Journal 48 (1969) 2409. [ 91 J. Marotz and K.H. Ringhofer, Optics Comm. 64 ( 1987) 245. [IO] D.F. Nelson and R.M. Mikulyak, J. Appl. Phys. 45 ( 1974) 3688. [ 111 I.P. Kaminow, E.H. Turner, R.L. Barns and J.L. Bet-stein, J. Appl. Phys. 5 1 ( 1980) 4379. [ 121 A. Savage, J. Appl. Phys. 37 (1966) 3071.
179