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Recording of higher spatial harmonics of a photorefractive grating through the phase-locked detection mechanism
Optics Communications 93 ( 1992) 195-201 North-Holland
OPTICS COMMUNICATIONS
Recording of higher spatial harmonics of a photorefractive grating through the phase-locked detection mechanism A.V. Dooghin, P.N. Ilinykh, V.S. L i b e r m a n , O.P. Nestiorkin a n d B.Ya. Z e l ' d o v i c h Nonlinear Optics Laboratory, Technical University, 76 Lenin av., Chelyabinsk, 454080, Russian Federation
Received 14 April 1992
The peculiaritiesof higher harmonicsin the photorefractiveresponseare consideredtheoreticallyfor a grating recordedthrough the mechanism of phase-lockeddetection by an external ac field. The conditions for higher harmonic grating recording is found analytically. The dependenceof second spatial harmonics on spatial frequencyis measured in two different samplesof Bi~2TiO2o.
1. Introduction
The photorefractive holographic gratings recorded by a light interference pattern may be used for holographic information storage and for other applications [ l ]. The simples hologram may be recorded by a sinusoidal interference pattern with intensity l ( x , t) =Io{1 +½ [m exp(iqx) +c.c.]} performed by two plane waves. Here Io is the total intensity of those waves, q is the spatial frequency of fringes, m is the contrast of interference pattern. The redistribution of nonuniformly photoexcited carriers leads to the space-charge field formation. The phase hologram occurs as a result of the linear electrooptic effect. In this paper we investigate higher harmonic grating recording [2] with spatial frequency nq (n integer) when the fundamental harmonic is recorded through the mechanism of phase-locked detection of a running interference pattern by an external ac field [ 3 ]. The frequency, of the external ac field, t2 coincides with the detuning between the writing beams. For the beginning let us consider the grating formation qualitatively. If the crystal is illuminated by a static interference pattern, the fundamental grating of photoconductivity a~ ~ exp (iqx) + c.c. is excited there due to nonuniformity of the carrier photoexcitation. In a uniform externally applied dc field Eo that grating results in the spatially nonuniform photocurrentj~ ~ mEo exp (iqx) + c.c., which in turn leads to the space charge separation giving the periodical
field of the fundamental spatial frequency E~ ~ mEo exp(iqx) +c.c. In the absence of an external field the current j~ arises through the diffusion of nonuniformly excited carriers. The space-charge field of the fundamental harmonic with amplitude E~ ~ reED is formed in this case, where E D = q k T / l e l is the so called diffusion field, k T i s the temperature in energy units, e = - l el is the value of electron charge. The formed first harmonic field excites the photocurrent of the second harmonic spatial frequency .]'2 ~ trlEl exp(2iqx) +c.c. ~ m 2 E o e x p ( 2 i q x ) +c.c., which leads to second harmonic recording. Considering the higher harmonic grating formation in such a way one may see that their amplitudes decrease as m n for m < 1 and under small contrast m << 1 may be neglected. A time interval about the dielectric relaxation time TM= 4n~/tro is needed for the grating to be formed. Here ao is the average photoconductivity, is the dielectric permittivity of the crystal. If the exciting interference pattern I = Io{ 1 + ½[m exp(i(qx - t 2 t ) ) + c . c . ] } is formed by the beams with a frequency difference £2 which is much larger than the inverse build-up time ZM, then the fundamental space-charge field grating is not recorded effectively and hence the higher harmonics of the grating are also not recorded. Nevertheless the running interference pattern may record the static first harmonic grating if an external ac field of the frequency £2 is applied to the crystal
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[ 3]. In this case the running photoconductivity grating a , ~ m e x p [ i ( q x - [ 2 t ) ] +c.c. in the external ac field ~ Eo exp ( - i£2t) + c.c. gives the static spatially nonuniform current j~ ~ mEo exp(iqx) +c.c., which results in the accumulated space charge separation. The recorded space-charge field El ~ mEo exp(iqx) excites the running photocurrent grating of the second harmonic, j 2 ~ a l E l ~ m 2 E o e x p ( 2 i q x - i [ 2 t ) +c.c., which cannot write the grating due to its fast time oscillation/2>> zfi~. It is seen from this simple qualitative consideration that a phase-locked detection mechanism [3] records the grating without higher harmonics. Some conditions exist under which this statement is not valid. In fact the static interference pattern may record the grating through Stepanov's mechanism in an ac field with large frequency (I2>>rfi ~) if the electric drift length strongly exceeds the grating spacing [4,5 ]. We do not consider this mechanism in detail here. Phenomenologically that technique permits to record the static grating by moving the photocurrent grating. In the case of the original Stepanov's mechanism, that grating is formed by the static photoconductivity grating 0"1 m exp(iqx) + c.c. and ac external field ~Eo e x p ( - i g 2 t ) + c . c . ; j~ ~ mEo exp[i ( q x - [ 2 t ) ] + c.c. If a phase-locked detection mechanism is used for the fundamental harmonic grating formation, then the running higher harmonic photocurrent grating (see preceding section) may record their gratings through the Stepanov's mechanism in crystals with large electron drift length compared to fringes spacing. As was shown by Stepanov and Petrov [ 5 ] the grating recording in an ac field is more effective if a voltage of square-wave form is used. Below we analyze analytically the second spatial harmonic recording for both cases of sinusoidal and square-wave forms of an external field of frequency/2 if the crystal is illuminated by two plane waves with frequency difference/2. ~
2. Analytical theory The basic condition for our treatment are the material equations ON + / O t = S I ( N o - N + ) - ~ N ~ n ,
( 1)
On/Ot=SI(ND-N+)-yN~n-(I/e)divj,
(2)
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1 October 1992
and electrostatic equations div( e E ) = 4 n p ,
(3)
rotE=0.
(4)
Here ND (cm 3) is the total density of donors, N ~ (cm - 3) is the density of ionized donors (traps) and we assume that ND>>N~, n (cm 3) is the density of mobile electrons in the conductivity band, I (quants/cm2s) is the intensity of light, S (cm 2) is the cross section for the photoionization of donors, y is the recombination constant, e = - l el is the electron's charge, p a n d j are the charge density and density of the electric current, E is the static electric field responsible for the refractive index changes, and e is the static dielectric permittivity of the crystal. In the initial equilibrium state of the crystal N~ =NA, where N A (cm -3) is the density of the negatively charged acceptors. In the excited state N ~ (r, t) =Na + N ( r , t) ,
(5)
where N is the difference between actual density of the positively charged (ionized) donors and its equilibrium value NA. Then the charge density is p=eNa-eN
+ +en=e(n-N)
.
(6)
For simplicity we will neglect the diffusion and the photovoltaic effect so that j = elmE,
(7 )
where/1 > 0 is the electron mobility. Let us consider the crystal illuminated by the running interference pattern l(x,t)=Io+lo[mexp(iqx-ig2t)+c.c.]/2.
(8)
Suppose also that the external electric field Eo (t) is applied to the crystal. We will neglect the electron contribution to the charge density and therefore to the field E (n << N) which is correct for all realistic values of the light intensity o f a cw laser. We will neglect the saturation of the traps as well (N<
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settlement). Hence we can regard the electron concentration to be quasistatically driven b y / , N and E, i.e. we can neglect On/Ot in our equations. Being interested in the growth of the static electric field spatial grating in the crystal we can rewrite our equations ( 1 ) - (7) as follows: YNAn+/t ~X ( n E ) = S I ( N D - N 3 ) ,
OE / Ot= - 4 n e l m E ,
(9) (10)
where E=Ex, E= ¢xx. The time derivative of eq. (3) was used. We have used also all simplification mentioned above and it was taken into account that the problem had the 1D spatial nature. Considering the case of the small contrast of the interference pattern (m << 1 ) we will look for a solution expanded in the perturbation theory series
n(x, t) =no + ½[nl (t) exp (iqx) +c.c.] + ½[n2 (t) exp(2iqx) +c.c. ] + ....
( 11 )
E(x, t) = E o ( t ) + ½[E, (t) exp(iqx) +c.c. ] + ½[E2 (t) exp (2iqx) +c.c. ] + ....
If 2n/12<>r = (~NA) - 1 for not breaking our previous consider-
ation) i.e. Ul and F~ undergo several oscillations while the stationary value of Et is being settled, we can easily calculate that stationary value as El = ( E l ) / (
Ul ) •
(16)
Here ( F i ) and ( U~ ) are the time averaged values of F~ and U~, For the second spatial harmonic of the induced field E we have the equation of the same type
E2(t)+ U2(t) Ez(t) =F2(t) ,
(17)
with the coefficients (for t>> rM SO that the fundamental harmonic E I ( t ) is already settled to the stationary value)
U2(t)-
zM l + 2ixf( i2ot ) '
(18)
F2(t) = r f i ~[xE~/Eo +imE, exp( - it2t) ] × 2 1+i~f(~t)
(12)
Here Eo(t) is the external ac field applied to the crystal, no=SIoND/yNA is the number of electrons (per cm 3) created in the conduction band by the average intensity Io, n~ and El are of the order of m, n2 and Ez are of the order of m 2, etc. Substituting eqs. (11), (12) into eqs. (9), (10) and balancing the terms of the same order of magnitude we get
/~t (t) + U, (t) E, (t) =F, (t),
1 October 1992
(13)
where
_ noO + [l+ixf(t2t))][l+2ixf(t2t)]J"
The stationary value of Ez (t) can be readily obtained if we use the same averaging technique
E 2 = ( F 2 ) / ( U2) .
(20)
In fact we carried out the calculations for the two cases: sinusoidal and square-wave profile of Eo(t). In the first case
Eo(t) =Eo cos f2t,
Ul (t) -- 1 +ixf(I2t) '
(14)
(21)
we found
E~ = - ½ m E o [ 1 - ( l / x - x / l + x - 2 ) z ] F~ ( t ) = - m E o z ~ tf(s'-dt) e x p ( - i t 2 t ) l+ixf(t2t) '
(15)
(19)
,
(22)
with the asymptotical behavior
and we used the notations
E, = -½mEo,
x<< 1,
(23)
Eo( t ) = E o f ( ~2t),
El = - r n E o x -1,
x>> 1 .
(24)
x = L E q = ltqEo/yNA .
Here f is a periodical function of unit amplitude, LE=btEo/?NA is the drift length for the electron in the external field (the average distance covered by the electron up to the recombination), ZM= ~~4he#no is the Maxwell relaxation time.
We have got also the expression for Ez which is rather complicated so that we show only the asymptotical equation
E z = - ½ m 2 E o x,
x<
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and E2 =
-
½m2Eo/¢-2,
R'>>
1.
(26)
In the second case
Eo(t) =Eo sign(cos g2t),
(27)
we can show the exact expressions which are valid for arbitrary value of ~:
E~ = - ( 2 / n ) m E o ,
(28)
E2 = ( 4/n2)m2Eox .
(29)
It is obvious that we would expect a considerable contribution of higher harmonics to the photorefractive response only in the case of high contrast (m ~ 1 ). So that to estimate roughly the maximum values of the second harmonics in the E ( x ) grating we have extended our solutions ( 2 3 ) - ( 2 6 ) , (28), (29) up to m ~ 1. In figs. 1, 2 we show the relative contributions of the first IE~/Eol and the second IE2/go] harmonics to the photorefractive grating as a function of tC=LEqfor the case of contrast m = 1. We would like to emphasize that the higher harmonics of the photorefractive response for a grating recorded by the phase-locked detection mechanism (PLDM) have some special features which are dif-
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ferent from the ones of the other recording mechanisms. Here are these features for PLDM. For small x, i.e. for crystals with small drift length LE we get small values of the second spatial harmonics both for the sinusoidal and the square-wave profiles of external ac field Eo(t). In contrast to crystals with really large LE SO that x~> 1, the second harmonic (and therefore all higher harmonics) proved to be small for a sinusoidal external field and relatively large (of the order of the first one) for a square-wave profile of Eo(t). Thus the photorefractive grating is fairly sinusoidal in the case of a sinusoidal external field and quite unharmonical in the case of a square-wave field for the crystals with a large electron drift length.
3. Experiment The experimental setup for the investigation of the amplitude of the second spatial harmonic is presented in fig. 3. A HeNe laser of 1.5 mW power was used to record the phase grating in a photorefractive crystal Bi~2TiO2o(BTO). We have employed two samples of BTO for the experiments. The first sample had a high electron drift length in comparison with the period of the grating, and degenerate inter-
IGt
L 2
/
I
\
2 Fig. 1. Relative amplitudes of the first ( 1 ) and the second (2) spatial harmonics of the photorefractive grating for the sinusoidal external field Eo(t) =Eo cos 12t and for the 100% contrast of the interference pattern ( m = 1 ) as functions of the parameter X=ZEq.
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l October 1992
IE=,t
(2) /
Y
0
Z 2
ft)
~e
Fig. 2. Relative amplitudes of the first ( 1 ) and the second (2) spatial harmonics of the photorefractive grating for the square-wave external field Eo (t) = Eo sign (cos fit) and for the 100% contrast of the interference pattern (m = l ) as function of the parameter ~ = LEq.
do
i
i ~--~ 2
/ 9
Fig. 3. Experimental setup (see the text for details).
action in an ac external field (via Stepanov's mechanism [5] ) was possible. The second sample of BTO did not permit the realization of a degenerate interaction in an ac external field since the electron drift length was small. The beam of laser 1 was split into two beams. One of them was phase-modulated by piezomirror 6 with a saw-tooth voltage applied to it. The amplitude of the saw-tooth voltage was chosen to provide a 2~t phase shift. After that both beams were aligned to the crystal. The signal-to-pump ratio was equal to 1. The reason of such a choice is that the second spatial harmonic amplitude is proportional to the square of the interference pattern contrast m and has a maximum at m = 1. The polarizations of the interacting waves had an angle of 45 ° to the incidence plane in the middle of the crystal thickness (taking into account the optical activity p = 6 d e g / m m ) . The orientation of the crystal is shown in fig. 3. An external ac field, either of sinusoidal form or square-wave, was applied in the 110 direction. The applied field was phase-locked with the frequency shifting saw-tooth voltage (£2/27r= 50 Hz). The HeNe laser with I. 5 mW power was used for the reading of the grating of the second spatial har199
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Fig. 4. Scheme of adjustment of the reading beam to a Bragg angle for the second spatial harmonic grating.
monic. In order to align the reading beam on the crystal at Bragg's angle corresponding to the second spatial harmonic we have used the technique presented in fig. 4. An additional beam had a direction perpendicular to the crystal face and the angles with beams 1 and 2 were equal to 0/2. An auxiliary mirror was placed in front of the crystal face so that the beam 4 reflected from that mirror gave the direction
1 October 1992
for the Bragg angle for diffraction by the second spatial harmonic grating. Therefore the beam of laser 2 was to be aligned at the crystal parallel to this direction. The error in the Bragg angle was of the order of 02. Since 0< 0.05 in our experiment this technique was acceptable. We have investigated the diffraction efficiency of the second spatial harmonic grating and its dependence on the spatial frequency. The grating was recorded for a time 60-180 s. After that the writing beams from laser 1 were blocked and the reading beam from laser 2 was switched on. The intensity of the diffracted beam was registered by a photodetector and was measured just at the initial part of the erasure curve. The dependence of the diffraction efficiency of the second spatial harmonic grating r/2 on spatial frequency q of the fundamental grating is shown in fig. 5 for the case of a sinusoidal external field with the amplitude Eo= 11 kV/cm. The experimental results are in a reasonable agreement with the theory. The diffraction efficiency is small both for small and for large values of spatial frequency q. The maximum of the diffraction efficiency was equal to 0.07% at q~4X10
3 cm -1.
J 02
,i
0,1 +
. . . .
4"
.i J ~J I
,
2
L
3
~
5
"
iD'
~'SZ~
Fig. 5. Experimental values of the diffraction efficiency of the second spatial harmonic grating versus fundamental spatial frequency q. +: sinusoidal form of external field Eo= 11 kV/cm, ×: square-wave external field E o = 9 kV/cm.
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The same measurements were performed for a square-wave field applied to the crystal with amplitude E 0 = 9 kV/cm. The experimental dependence ~/2(q) for the second spatial harmonic grating is presented in fig. 5. The qualitative behaviour of this dependence is similar to the previous case, but the absolute value of the diffraction efficiency is greater by a factor of about 5: r/max~0.3%. In accord with the theoretical calculations, the diffraction efficiency decreases to zero at small values of spatial frequency q. Discrepancy between theory and experiment at large values of spatial frequency q may be explained by the fact that the theoretical expressions were derived in the nonsaturated trap approximation. The trap charge saturation, as we assume, explains the decrease of the diffraction efficiency at large values of spatial frequency q. The Raman-Nath diffraction was observed at small values of spatial frequency q. The wave arising as a result of this diffraction record secondary gratings with spatial frequency 2q with interacting primary beams supplied by laser 1. This process complicates the measurement of the diffraction efficiency of the second spatial harmonic grating. Similar investigations were performed with another sample of BTO. It had smaller drift length compared with the period of the fundamental grating. We had not observed diffraction of the reading beam for any of the investigated spatial frequencies. Therefore the recorded grating had only first spatial Fourier component. An explanation of this phenomenon is that the second spatial harmonic grating is recorded through Stepanov's mechanism [ 5 ], but in crystals which have a drift length smaller than the
1 October 1992
fringes spacing this mechanism is not effective.
4. Conclusion In summary, the theoretical analysis done in this paper and experiments performed show the following. In crystals with a small mobility-life-time product the phase-locked detection mechanism gives a photorefractive response without higher spatial harmonics under illumination by a sinusoidal interference pattern with large contrast. Exceeding of the electron drift length over the fundamental grating period A leads to the distortion of a sinusoidal form of the grating due to the recording of higher spatial harmonics. This distortion is more significant in the case of a square-wave form of the external ac field.
Acknowledgements The authors would like to thank N.D. Kundikova for valuable discussions.
References [ 1] P. Gunter, J.-P. Huignard, eds., Photorefractivematerialsand their applications,Vol. I, II (Springer, Berlin, 1988). [2 ] F. Vachssand L. Hesselink,J. Opt. Soc.Am. B 8 (1988) 1814. [3] P.N. Ilinykh, O.P. Nestiorkin and B.Ya. Zel'dovich, J. Opt. Soc. Am. B 8 (1991) 1042. [4] S.I.Stepanovand M.P. Petrov,OpticsComm. 53 ( 1985) 292. [5l S.I. Stepanovand M.P. Petrov, see ref. [ 1], Vol. I, chap. 9.
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Recording of higher spatial harmonics of a photorefractive grating through the phase-locked detection mechanism A.V. Dooghin, P.N. Ilinykh, V.S. L i b e r m a n , O.P. Nestiorkin a n d B.Ya. Z e l ' d o v i c h Nonlinear Optics Laboratory, Technical University, 76 Lenin av., Chelyabinsk, 454080, Russian Federation
Received 14 April 1992
The peculiaritiesof higher harmonicsin the photorefractiveresponseare consideredtheoreticallyfor a grating recordedthrough the mechanism of phase-lockeddetection by an external ac field. The conditions for higher harmonic grating recording is found analytically. The dependenceof second spatial harmonics on spatial frequencyis measured in two different samplesof Bi~2TiO2o.
1. Introduction
The photorefractive holographic gratings recorded by a light interference pattern may be used for holographic information storage and for other applications [ l ]. The simples hologram may be recorded by a sinusoidal interference pattern with intensity l ( x , t) =Io{1 +½ [m exp(iqx) +c.c.]} performed by two plane waves. Here Io is the total intensity of those waves, q is the spatial frequency of fringes, m is the contrast of interference pattern. The redistribution of nonuniformly photoexcited carriers leads to the space-charge field formation. The phase hologram occurs as a result of the linear electrooptic effect. In this paper we investigate higher harmonic grating recording [2] with spatial frequency nq (n integer) when the fundamental harmonic is recorded through the mechanism of phase-locked detection of a running interference pattern by an external ac field [ 3 ]. The frequency, of the external ac field, t2 coincides with the detuning between the writing beams. For the beginning let us consider the grating formation qualitatively. If the crystal is illuminated by a static interference pattern, the fundamental grating of photoconductivity a~ ~ exp (iqx) + c.c. is excited there due to nonuniformity of the carrier photoexcitation. In a uniform externally applied dc field Eo that grating results in the spatially nonuniform photocurrentj~ ~ mEo exp (iqx) + c.c., which in turn leads to the space charge separation giving the periodical
field of the fundamental spatial frequency E~ ~ mEo exp(iqx) +c.c. In the absence of an external field the current j~ arises through the diffusion of nonuniformly excited carriers. The space-charge field of the fundamental harmonic with amplitude E~ ~ reED is formed in this case, where E D = q k T / l e l is the so called diffusion field, k T i s the temperature in energy units, e = - l el is the value of electron charge. The formed first harmonic field excites the photocurrent of the second harmonic spatial frequency .]'2 ~ trlEl exp(2iqx) +c.c. ~ m 2 E o e x p ( 2 i q x ) +c.c., which leads to second harmonic recording. Considering the higher harmonic grating formation in such a way one may see that their amplitudes decrease as m n for m < 1 and under small contrast m << 1 may be neglected. A time interval about the dielectric relaxation time TM= 4n~/tro is needed for the grating to be formed. Here ao is the average photoconductivity, is the dielectric permittivity of the crystal. If the exciting interference pattern I = Io{ 1 + ½[m exp(i(qx - t 2 t ) ) + c . c . ] } is formed by the beams with a frequency difference £2 which is much larger than the inverse build-up time ZM, then the fundamental space-charge field grating is not recorded effectively and hence the higher harmonics of the grating are also not recorded. Nevertheless the running interference pattern may record the static first harmonic grating if an external ac field of the frequency £2 is applied to the crystal
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[ 3]. In this case the running photoconductivity grating a , ~ m e x p [ i ( q x - [ 2 t ) ] +c.c. in the external ac field ~ Eo exp ( - i£2t) + c.c. gives the static spatially nonuniform current j~ ~ mEo exp(iqx) +c.c., which results in the accumulated space charge separation. The recorded space-charge field El ~ mEo exp(iqx) excites the running photocurrent grating of the second harmonic, j 2 ~ a l E l ~ m 2 E o e x p ( 2 i q x - i [ 2 t ) +c.c., which cannot write the grating due to its fast time oscillation/2>> zfi~. It is seen from this simple qualitative consideration that a phase-locked detection mechanism [3] records the grating without higher harmonics. Some conditions exist under which this statement is not valid. In fact the static interference pattern may record the grating through Stepanov's mechanism in an ac field with large frequency (I2>>rfi ~) if the electric drift length strongly exceeds the grating spacing [4,5 ]. We do not consider this mechanism in detail here. Phenomenologically that technique permits to record the static grating by moving the photocurrent grating. In the case of the original Stepanov's mechanism, that grating is formed by the static photoconductivity grating 0"1 m exp(iqx) + c.c. and ac external field ~Eo e x p ( - i g 2 t ) + c . c . ; j~ ~ mEo exp[i ( q x - [ 2 t ) ] + c.c. If a phase-locked detection mechanism is used for the fundamental harmonic grating formation, then the running higher harmonic photocurrent grating (see preceding section) may record their gratings through the Stepanov's mechanism in crystals with large electron drift length compared to fringes spacing. As was shown by Stepanov and Petrov [ 5 ] the grating recording in an ac field is more effective if a voltage of square-wave form is used. Below we analyze analytically the second spatial harmonic recording for both cases of sinusoidal and square-wave forms of an external field of frequency/2 if the crystal is illuminated by two plane waves with frequency difference/2. ~
2. Analytical theory The basic condition for our treatment are the material equations ON + / O t = S I ( N o - N + ) - ~ N ~ n ,
( 1)
On/Ot=SI(ND-N+)-yN~n-(I/e)divj,
(2)
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1 October 1992
and electrostatic equations div( e E ) = 4 n p ,
(3)
rotE=0.
(4)
Here ND (cm 3) is the total density of donors, N ~ (cm - 3) is the density of ionized donors (traps) and we assume that ND>>N~, n (cm 3) is the density of mobile electrons in the conductivity band, I (quants/cm2s) is the intensity of light, S (cm 2) is the cross section for the photoionization of donors, y is the recombination constant, e = - l el is the electron's charge, p a n d j are the charge density and density of the electric current, E is the static electric field responsible for the refractive index changes, and e is the static dielectric permittivity of the crystal. In the initial equilibrium state of the crystal N~ =NA, where N A (cm -3) is the density of the negatively charged acceptors. In the excited state N ~ (r, t) =Na + N ( r , t) ,
(5)
where N is the difference between actual density of the positively charged (ionized) donors and its equilibrium value NA. Then the charge density is p=eNa-eN
+ +en=e(n-N)
.
(6)
For simplicity we will neglect the diffusion and the photovoltaic effect so that j = elmE,
(7 )
where/1 > 0 is the electron mobility. Let us consider the crystal illuminated by the running interference pattern l(x,t)=Io+lo[mexp(iqx-ig2t)+c.c.]/2.
(8)
Suppose also that the external electric field Eo (t) is applied to the crystal. We will neglect the electron contribution to the charge density and therefore to the field E (n << N) which is correct for all realistic values of the light intensity o f a cw laser. We will neglect the saturation of the traps as well (N<
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settlement). Hence we can regard the electron concentration to be quasistatically driven b y / , N and E, i.e. we can neglect On/Ot in our equations. Being interested in the growth of the static electric field spatial grating in the crystal we can rewrite our equations ( 1 ) - (7) as follows: YNAn+/t ~X ( n E ) = S I ( N D - N 3 ) ,
OE / Ot= - 4 n e l m E ,
(9) (10)
where E=Ex, E= ¢xx. The time derivative of eq. (3) was used. We have used also all simplification mentioned above and it was taken into account that the problem had the 1D spatial nature. Considering the case of the small contrast of the interference pattern (m << 1 ) we will look for a solution expanded in the perturbation theory series
n(x, t) =no + ½[nl (t) exp (iqx) +c.c.] + ½[n2 (t) exp(2iqx) +c.c. ] + ....
( 11 )
E(x, t) = E o ( t ) + ½[E, (t) exp(iqx) +c.c. ] + ½[E2 (t) exp (2iqx) +c.c. ] + ....
If 2n/12<
ation) i.e. Ul and F~ undergo several oscillations while the stationary value of Et is being settled, we can easily calculate that stationary value as El = ( E l ) / (
Ul ) •
(16)
Here ( F i ) and ( U~ ) are the time averaged values of F~ and U~, For the second spatial harmonic of the induced field E we have the equation of the same type
E2(t)+ U2(t) Ez(t) =F2(t) ,
(17)
with the coefficients (for t>> rM SO that the fundamental harmonic E I ( t ) is already settled to the stationary value)
U2(t)-
zM l + 2ixf( i2ot ) '
(18)
F2(t) = r f i ~[xE~/Eo +imE, exp( - it2t) ] × 2 1+i~f(~t)
(12)
Here Eo(t) is the external ac field applied to the crystal, no=SIoND/yNA is the number of electrons (per cm 3) created in the conduction band by the average intensity Io, n~ and El are of the order of m, n2 and Ez are of the order of m 2, etc. Substituting eqs. (11), (12) into eqs. (9), (10) and balancing the terms of the same order of magnitude we get
/~t (t) + U, (t) E, (t) =F, (t),
1 October 1992
(13)
where
_ noO + [l+ixf(t2t))][l+2ixf(t2t)]J"
The stationary value of Ez (t) can be readily obtained if we use the same averaging technique
E 2 = ( F 2 ) / ( U2) .
(20)
In fact we carried out the calculations for the two cases: sinusoidal and square-wave profile of Eo(t). In the first case
Eo(t) =Eo cos f2t,
Ul (t) -- 1 +ixf(I2t) '
(14)
(21)
we found
E~ = - ½ m E o [ 1 - ( l / x - x / l + x - 2 ) z ] F~ ( t ) = - m E o z ~ tf(s'-dt) e x p ( - i t 2 t ) l+ixf(t2t) '
(15)
(19)
,
(22)
with the asymptotical behavior
and we used the notations
E, = -½mEo,
x<< 1,
(23)
Eo( t ) = E o f ( ~2t),
El = - r n E o x -1,
x>> 1 .
(24)
x = L E q = ltqEo/yNA .
Here f is a periodical function of unit amplitude, LE=btEo/?NA is the drift length for the electron in the external field (the average distance covered by the electron up to the recombination), ZM= ~~4he#no is the Maxwell relaxation time.
We have got also the expression for Ez which is rather complicated so that we show only the asymptotical equation
E z = - ½ m 2 E o x,
x<
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and E2 =
-
½m2Eo/¢-2,
R'>>
1.
(26)
In the second case
Eo(t) =Eo sign(cos g2t),
(27)
we can show the exact expressions which are valid for arbitrary value of ~:
E~ = - ( 2 / n ) m E o ,
(28)
E2 = ( 4/n2)m2Eox .
(29)
It is obvious that we would expect a considerable contribution of higher harmonics to the photorefractive response only in the case of high contrast (m ~ 1 ). So that to estimate roughly the maximum values of the second harmonics in the E ( x ) grating we have extended our solutions ( 2 3 ) - ( 2 6 ) , (28), (29) up to m ~ 1. In figs. 1, 2 we show the relative contributions of the first IE~/Eol and the second IE2/go] harmonics to the photorefractive grating as a function of tC=LEqfor the case of contrast m = 1. We would like to emphasize that the higher harmonics of the photorefractive response for a grating recorded by the phase-locked detection mechanism (PLDM) have some special features which are dif-
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ferent from the ones of the other recording mechanisms. Here are these features for PLDM. For small x, i.e. for crystals with small drift length LE we get small values of the second spatial harmonics both for the sinusoidal and the square-wave profiles of external ac field Eo(t). In contrast to crystals with really large LE SO that x~> 1, the second harmonic (and therefore all higher harmonics) proved to be small for a sinusoidal external field and relatively large (of the order of the first one) for a square-wave profile of Eo(t). Thus the photorefractive grating is fairly sinusoidal in the case of a sinusoidal external field and quite unharmonical in the case of a square-wave field for the crystals with a large electron drift length.
3. Experiment The experimental setup for the investigation of the amplitude of the second spatial harmonic is presented in fig. 3. A HeNe laser of 1.5 mW power was used to record the phase grating in a photorefractive crystal Bi~2TiO2o(BTO). We have employed two samples of BTO for the experiments. The first sample had a high electron drift length in comparison with the period of the grating, and degenerate inter-
IGt
L 2
/
I
\
2 Fig. 1. Relative amplitudes of the first ( 1 ) and the second (2) spatial harmonics of the photorefractive grating for the sinusoidal external field Eo(t) =Eo cos 12t and for the 100% contrast of the interference pattern ( m = 1 ) as functions of the parameter X=ZEq.
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IE=,t
(2) /
Y
0
Z 2
ft)
~e
Fig. 2. Relative amplitudes of the first ( 1 ) and the second (2) spatial harmonics of the photorefractive grating for the square-wave external field Eo (t) = Eo sign (cos fit) and for the 100% contrast of the interference pattern (m = l ) as function of the parameter ~ = LEq.
do
i
i ~--~ 2
/ 9
Fig. 3. Experimental setup (see the text for details).
action in an ac external field (via Stepanov's mechanism [5] ) was possible. The second sample of BTO did not permit the realization of a degenerate interaction in an ac external field since the electron drift length was small. The beam of laser 1 was split into two beams. One of them was phase-modulated by piezomirror 6 with a saw-tooth voltage applied to it. The amplitude of the saw-tooth voltage was chosen to provide a 2~t phase shift. After that both beams were aligned to the crystal. The signal-to-pump ratio was equal to 1. The reason of such a choice is that the second spatial harmonic amplitude is proportional to the square of the interference pattern contrast m and has a maximum at m = 1. The polarizations of the interacting waves had an angle of 45 ° to the incidence plane in the middle of the crystal thickness (taking into account the optical activity p = 6 d e g / m m ) . The orientation of the crystal is shown in fig. 3. An external ac field, either of sinusoidal form or square-wave, was applied in the 110 direction. The applied field was phase-locked with the frequency shifting saw-tooth voltage (£2/27r= 50 Hz). The HeNe laser with I. 5 mW power was used for the reading of the grating of the second spatial har199
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Fig. 4. Scheme of adjustment of the reading beam to a Bragg angle for the second spatial harmonic grating.
monic. In order to align the reading beam on the crystal at Bragg's angle corresponding to the second spatial harmonic we have used the technique presented in fig. 4. An additional beam had a direction perpendicular to the crystal face and the angles with beams 1 and 2 were equal to 0/2. An auxiliary mirror was placed in front of the crystal face so that the beam 4 reflected from that mirror gave the direction
1 October 1992
for the Bragg angle for diffraction by the second spatial harmonic grating. Therefore the beam of laser 2 was to be aligned at the crystal parallel to this direction. The error in the Bragg angle was of the order of 02. Since 0< 0.05 in our experiment this technique was acceptable. We have investigated the diffraction efficiency of the second spatial harmonic grating and its dependence on the spatial frequency. The grating was recorded for a time 60-180 s. After that the writing beams from laser 1 were blocked and the reading beam from laser 2 was switched on. The intensity of the diffracted beam was registered by a photodetector and was measured just at the initial part of the erasure curve. The dependence of the diffraction efficiency of the second spatial harmonic grating r/2 on spatial frequency q of the fundamental grating is shown in fig. 5 for the case of a sinusoidal external field with the amplitude Eo= 11 kV/cm. The experimental results are in a reasonable agreement with the theory. The diffraction efficiency is small both for small and for large values of spatial frequency q. The maximum of the diffraction efficiency was equal to 0.07% at q~4X10
3 cm -1.
J 02
,i
0,1 +
. . . .
4"
.i J ~J I
,
2
L
3
~
5
"
iD'
~'SZ~
Fig. 5. Experimental values of the diffraction efficiency of the second spatial harmonic grating versus fundamental spatial frequency q. +: sinusoidal form of external field Eo= 11 kV/cm, ×: square-wave external field E o = 9 kV/cm.
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The same measurements were performed for a square-wave field applied to the crystal with amplitude E 0 = 9 kV/cm. The experimental dependence ~/2(q) for the second spatial harmonic grating is presented in fig. 5. The qualitative behaviour of this dependence is similar to the previous case, but the absolute value of the diffraction efficiency is greater by a factor of about 5: r/max~0.3%. In accord with the theoretical calculations, the diffraction efficiency decreases to zero at small values of spatial frequency q. Discrepancy between theory and experiment at large values of spatial frequency q may be explained by the fact that the theoretical expressions were derived in the nonsaturated trap approximation. The trap charge saturation, as we assume, explains the decrease of the diffraction efficiency at large values of spatial frequency q. The Raman-Nath diffraction was observed at small values of spatial frequency q. The wave arising as a result of this diffraction record secondary gratings with spatial frequency 2q with interacting primary beams supplied by laser 1. This process complicates the measurement of the diffraction efficiency of the second spatial harmonic grating. Similar investigations were performed with another sample of BTO. It had smaller drift length compared with the period of the fundamental grating. We had not observed diffraction of the reading beam for any of the investigated spatial frequencies. Therefore the recorded grating had only first spatial Fourier component. An explanation of this phenomenon is that the second spatial harmonic grating is recorded through Stepanov's mechanism [ 5 ], but in crystals which have a drift length smaller than the
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fringes spacing this mechanism is not effective.
4. Conclusion In summary, the theoretical analysis done in this paper and experiments performed show the following. In crystals with a small mobility-life-time product the phase-locked detection mechanism gives a photorefractive response without higher spatial harmonics under illumination by a sinusoidal interference pattern with large contrast. Exceeding of the electron drift length over the fundamental grating period A leads to the distortion of a sinusoidal form of the grating due to the recording of higher spatial harmonics. This distortion is more significant in the case of a square-wave form of the external ac field.
Acknowledgements The authors would like to thank N.D. Kundikova for valuable discussions.
References [ 1] P. Gunter, J.-P. Huignard, eds., Photorefractivematerialsand their applications,Vol. I, II (Springer, Berlin, 1988). [2 ] F. Vachssand L. Hesselink,J. Opt. Soc.Am. B 8 (1988) 1814. [3] P.N. Ilinykh, O.P. Nestiorkin and B.Ya. Zel'dovich, J. Opt. Soc. Am. B 8 (1991) 1042. [4] S.I.Stepanovand M.P. Petrov,OpticsComm. 53 ( 1985) 292. [5l S.I. Stepanovand M.P. Petrov, see ref. [ 1], Vol. I, chap. 9.
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