IBM Research Division, Almaden Research Center, San Jose, California 95120-6099. Received ... We have recently described a new class of photorefractive.
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et al. Moerner
1994
Orientationally enhanced photorefractive effect in polymers W. E. Moerner, S. M. Silence, F. Hache,* and G. C. Bjorklund IBM Research Division, Almaden Research Center, San Jose, California 95120-6099 Received May 17, 1993; revised manuscript
received August 9, 1993
We present experimental data that show that the greatly improved performance of a new class of photorefractive polymers [see Donckers et al., Opt. Lett. 18, 1044 (1993)] is too large to be explained by the simple electro-optic photorefractive effect alone. In these materials a photoconducting polymer host is doped with a small concentration of a sensitizer and a large concentration of a nonlinear optical chromophore that has orientational mobility at ambient temperatures. We present a theoretical model for a new orientational enhancement mechanism in which both the birefringence of the sample and the electro-optic coefficient are periodically modulated by the space-charge field itself. The predictions of this model for the size of the enhancement (which is greater than an order of magnitude in diffraction efficiency),the polarization anisotropy between p-polarized and s-polarized readout, and the presence of index modulation at twice the grating wave vector are in good agreement with the measured properties. This orientational enhancement mechanism should be important in any system in which the nonlinear optical chromophores have sufficient orientational mobility and dipole moment so as to be oriented by the space-charge field itself.
1.
INTRODUCTION
We have recently described a new class of photorefractive (PR) polymers,'` 3 based on doped poly(N-vinyl carbazole) (PVK), which exhibit greatly enhanced performance compared with the earlier reported PR polymers. 4 ' 1 For example, these new PR polymers show net internal two-beam coupling gain and diffraction efficiencies above 1% for samples of the order of only 100 um in thickness. This improvement in performance is due partly to the formation of larger space-charge fields and the higher nonlinearity that results from the application of larger poling electric fields. However, the performance of these new polymers is so improved that it cannot be explained by models based on the simple electro-optic PR effect that have worked so well for previously known PR polymers and crystals, even if it is assumed that the space-charge field actually approaches the externally applied field in value. We believe that a new orientational enhancement mechanism is responsible for the large PR effects observed in these materials, and our goal in this paper is to describe this new mechanism in detail. The enhancement relies on the ability of the nonlinear optical (NLO) chromophores to be aligned not only by the externally applied electric field but also in situ by the sinusoidally varying space-charge field itself during grating formation. The resulting periodic poling of the sample leads to a modulation of the birefringence of the material and to a modulation of the electro-optic response, the combination of which contributes favorably to the diffraction efficiency fields in the proper polarization. As is shown below, the nonlinear response of the material is quadratic in the total local electric field; hence the orientational enhancement effect may be regarded formally as a X(3) process. However, the orientational X(3) described here is different from the X(3) mechanisms previously 0740-3224/94/020320-11$06.00
reported for a variety of semiconductor and quantumwell systems 12 '4 based on Franz-Keldysh and band-edge effects and from the quadratic response in paraelectric crystals such as potassium tantalate niobate.15' 1 6 The orientational enhancement effect should be operative for any PR polymer in which the nonlinear chromophores have the ability to reorient appreciably in response to the local electric field; in contrast, it should not be important for permanently poled PR polymeric systems or for PR crystals, either inorganic or organic. In essentially all cases reported to date, the new materials are composed of the photoconducting PVK host matrix, a large concentration (=30 wt. %) of a relatively small NLO chromophore, and a small concentration of a sensitizing agent. 1 -3 For example, the first highefficiency example of this class' was composed of PVK, fluorinated diethylaminonitrostyrene, and trinitrofluorenone (PVK:FDEANST:TNF). This material may be taken as a model system for the purposes of this paper. Because of plasticization by the large concentration of the NLO chromophore and the resulting low glass-transition temperature of =40'C, application of an external bias electric field is required for maintenance of poling of the NLO chromophores at room temperature. The rest of this paper is organized as follows. In Section 2 we present evidence that an additional mechanism beyond the simple electro-optic PR effect is required. The new materials have such strong diffraction that the conventional PR model for the diffraction efficiency results in unphysically large estimates for the internal space-charge field. In Section 3 the theory of the diffraction efficiency and polarization anisotropy that is due to orientational enhancement is presented. In Section 4 the theory is compared with experimental measurements of the polarization anisotropy for the PVK:FDEANST:TNF model system. Gratings with modulation at twice the grating wave vector KG are ©1994 Optical Society of America
Moerneret al.
Vol. 11, No. 2/February
also expected and observed, and their dependence on the external field is explored. Two crucial parameters needed for comparison with experiment are the coefficients determining the birefringence and the x(2) induced by an electric field, and the technique that we have used to estimate these coefficients is described in Appendix A.
2. EVIDENCE FOR ENHANCEMENT MECHANISM In this section we show that use of the conventional electro-optic mechanism for the grating diffraction efficiency leads to unphysically large values for the spacecharge field in the material. As usual, two intersecting coherent light beams produce a sinusoidally varying space-charge field by a process of charge generation, transport, and trapping.' 7" 8 In this paper the details of the space-charge generation process are not treated; we simply assume that the result of irradiation by two coherent writing beams is a sinusoidal electric field given by E5c(r) = E5coKGexp(iKG r), which is directed along the grating wave vector with magnitude Esco. The effects of higher-order components of the space-charge field are neglected, as a more complicated analysis is required for description of the distortion that can occur at saturation when the writing beams have equal intensities.' As in earlier studies of photorefractive polymers, a tilted sample geometry (see Fig. 1) is used. The original rationalization for this choice was that it would promote charge motion and provide a projection of the spacecharge field along the poling, or 3 (=z), axis, which is normal to the plane of the sample. This is necessary, because in the C geometry of a poled polymer the electro-optic coefficients r 3 and r33 are nonzero, while the coefficient r3 l is zero. These statements are correct if the orientations of the chromophores are determined in advance solely by the external poling field along the 3 axis during the poling process. We now review the conventional electro-optic calculation of the diffraction efficiency expected from this unusual tilted grating geometry, assuming a permanently poled sample for which the electro-optic coefficients are fixed constants. For later reference we call this the simple electro-optic PR effect.20' 2' The sinusoidal spacecharge field leads to a susceptibility change by means of the linear electro-optic tensor in the usual fashion2 2 (ks units): AX = -e
(R
ESCKoG)
,
1994/J. Opt. Soc. Am. B
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a first-order grating. The diffraction efficiency resulting from a probing field with polarization unit vector e3 incident at the Bragg angle and diffracting into a scattered field with polarization unit vector 4 is given (for diffraction efficiencies much less than unity) by22
77=
H04*
AX * 3 )12 ,
* 3) (4*
(3)
where H = (rd)/[2nAo(cos 0 cos 02)"J'], with d the sample thickness, n the index of refraction, and Ao the vacuum wavelength; the dot product of the reading and the diffracted field polarization vectors is necessary to account correctly for the case of p-polarized as well as s-polarized diffraction.2 3 In this expression probing beam 3 has incidence angle 02 (counterpropagating to writing beam 2), the scattered beam has incidence angle 0 (counterpropagating to writing beam 1), and any static birefringence may be safely ignored. For the geometry of Fig. 1 we let G be the angle between the grating wave vector KG and the surface normal, which is taken to be the z axis. For s-polarized reading (and diffraction-no rotation of the plane of polarization occurs), 7
7s,simpleEO =
(Hn r13cos
GEsco)
2
(4)
For the p-polarized case, again no rotation of the polarization occurs, and the diffraction efficiency is 77p,simple EO
{Hn 4
cos(02
-
)Esco
X [r13 COS01 COS02 COS G
+ r13 sin(01 + 0 2 )Sin OG
+ r33 sin 0t sin
02 COS 00G}2-
(5)
For later reference, we remark that one test of the presence of the simple electro-optic mechanism is the polarization anisotropy, which is independent of the actual magnitude of ESCO and which is defined as the ratio
air
(1)
where R is the third-rank electro-optic tensor and e is the optical permittivity tensor. For C., symmetry, the electro-optic tensor in the contracted notation reduces to
= R R=
F0
0
r13
0
0
r13
0 0
0 r 13
r13
0
L0
0
r3
3
0
1 ,3'
(2)
0 O_
Under the assumption of a purely sinusoidal spacecharge field, the simple electro-optic effect leads to a modulation of the index of refraction at KG only, i.e.,
Fig. 1. Tilted grating geometry used for the PR measurements. The wave vector of the grating written by beams 1 and 2 (angles of incidence 01 and 02, defined inside the sample) is directed along KG, which forms an angle OG with the direction of the external bias field EB. PR gratings produced at 1KG are read out with beam 3 (diffracted into beam 4), and PR gratings produced at 2KG are read out with beam 3' (diffracted into beam 4').
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1994
, given by
(77P = CoO2(02 -01) )75simple EO
COS 01 COS02
+ sin(0, + 02 )tan OG
+(
)sin 0, sin 02]
r13
(6)
which was presented earlier 5 (with a complementary definition of the grating tilt angle 0s)In Section 4 it is shown from specific experimental measurements that the polarization anisotropy for the new orientational enhancement mechanism is quite different from that given by Eq. (6). Furthermore, higherorder modulations of the refractive index occur beyond the first-order grating even when the space-charge field is purely sinusoidal and has no higher-order distortion contributions. The primary evidence for the presence of a new enhancement mechanism is shown in Fig. 2. With a sample of PVK:FDEANST:TNF, steady-state diffraction efficiencies were measured for both s-polarized and p-polarized probing over a range of grating wave vectors KG. For variation of KG, the two writing beams were kept at a fixed external angle of 30°, and the sample was rotated to provide several different external angles of incidence and hence different values of the internal incidence angles 0 and 02 and the grating spacing. All the measurements were performed with a fixed external electric field of 40 V//gtm. Separately, at the same fixed field, the electro-optic coefficients were measured in a Mach-Zehnder interferometer (as described in Appendix A) to be n3r13 = -1.7 pm/V and n3 r3 3 = -5.0 pm/V. Then, from Eqs. (4) and (5), the resulting values for E,8C2 were computed. These values are plotted in Fig. 2 as filled circles (p-polarized case) and open circles (s-polarized case). The solid curve shows the square of the projection of the external poling field along the grating wave vector. In the standard Kukhtarev model of photorefractivity2 0' 2 the grating formation process at a high external electric field is dominated in steady state by a competition between the drift force from the external field and the restoring force produced by the space-charge field. In particular, the magnitude of the space-charge field is never larger than the external bias field. This should still be true on physical grounds, even if more-complex models of charge generation and transport more appropriate for organics are used.24 2 5 2 Thus the values of E8,C plotted in Fig. 2 are unphysically large by 1 order of magnitude or more, which cannot occur. This means that Eqs. (4) and (5) relating the spacecharge field and the diffraction efficiency are missing an important enhancement mechanism, which forms the subject of the rest of this paper.
3. THEORY OF THE ORIENTATIONAL ENHANCEMENT MECHANISM In this section we present a calculation of the diffraction efficiency for both p-polarized light and s-polarized light for the case in which both the space-charge field and the external electric field cause orientation of the nonlinear chromophores in the polymer. In this case the sinusoidal
space-charge grating produces a sinusoidal birefringence and a sinusoidal variation in the electro-optic coefficient, both of which contribute an orientational enhancement to the scattered optical field. The tilted sample geometry requires a somewhat complicated approach to the calculation of the orientational enhancement because the space-charge field E,8 is not parallel to the external bias field EB. The dc bias field has magnitude EB and is directed along the (laboratory) z axis as shown in Fig. 1, and without loss of generality the (laboratory) x axis is taken in the plane of incidence. The challenge in the analysis comes from the fact that the uniaxial symmetry axis produced by the poling varies sinusoidally in direction throughout the sample. So that this variation is correctly accounted for, the spatially varying susceptibility that is due to the orientational response of the sample is derived in the local frame, where the nonlinear chromophores see only the total field, which is a superposition of the internal space-charge field and the external dc bias field at the location of interest. It is clear that in this local frame the symmetry is rigorously uniaxial, and the local axes are the principal axes. The total field in the local frame is ET, which defines the local z axis and the local symmetry direction. This section is divided into five parts. In Subsection 3.A we discuss the calculation of the change in the firstorder susceptibility, or birefringence (in the molecular frame of reference), that is due to the orientation of molecules in response to the field. Since the resulting contribution to the PR signal is proportional to the square of the poling field, this birefringence effect will contribute to a modulation of the index of refraction at KG when the field that orients the molecules has both a dc component and a component at KG. In Subsection 3.B we discuss the calculation of the change in the second-order susceptibility that is due to the molecular orientation from the KG component of the field, again in the molecular frame of reference. Inclusion of both of these terms is necessary, as will be shown by direct measurement for the case of PVK:FDEANST:TNF (Section 4 and Appendix A). The total susceptibility change that is due to these two
In
5
"
KG (m)y 1 4.52 2.70 3.23 3.67 4.03 4.31 I I I I I I I
104
0
*
o
4.76
0
*
0
0
0 0
103
4.67 I
a
0
E 102 IcJ
w
10° 60 Fig. 2.
Eso
I
IIII
1o-,
2
65
determined
70
75 OG(degrees)
80
85
from qp (filled circles) and 77, (open
circles) for PVK:FDEANST:TNF with Eqs. (4) and (5) (nr 3 = -1.7 pm/V), shown as a function of G (bottom axis) and KG (top axis). The solid curve is the square of the projection of EB along the grating wave vector.
Moemneret
effects is calculated in Subsection 3.C, and the susciBptibility tensor is rotated back to the laboratory frar ne. Finally, the polarization anisotropies of the first-or ler and the second-order diffraction gratings are calculal ted in Subsections 3.D and 3.E, respectively. A. Change in First-Order Susceptibility (Birefringence) That Is Due to Orientation
The birefringence induced by orientation of nonlin Lar chromophores in an external field has been calculated byya number of authors, for example, Kuzyk et al.2 6 and Wt i.27 The result in leading order in the total poling field E,pole may be written AX(lBR)(_
as ; w)
( )-
()
2
/hEpole2)
CBREpole , =
A
( BR)(-CW; CO)= AXx BR)(-O; °) -
ABREpole
2
(7) (7)
A12iX(lBR)(_,r,,; A,)
= -(1/2)CBREpole 2 X
(8)
where AX(l1BR)is the change in first-order optical susc eptibility resulting from the poling-induced birefringence , i is the optical frequency, N is the density of chromophor-es, all and a are the (dressed) parallel and perpendicu nr optical polarizabilities of the (assumed linear) molectile, ,u is the (dressed) ground-state dipole moment assun led to be along the z axis of the
B. Change in Second-Order Susceptibility That Is Due to Orientation
An additional quadratic effect producing a susceptibility change is produced by the orientation of the (dressed) molecular hyperpolarizability, 6333, where we assume to good approximation that the NLO molecule has (positive) optical nonlinearity only along the molecular 3 axis. We treat this effect as a field-dependent second-order susceptibility x(, which has been presented by several authors.27 2 8 In fact, since three fields are involved in the nonlinear polarization, one power of the poling field, one external dc field, and one optical field, this is formally a X(3)effect, and a calculation similar to that of Kuzyk et al. 2 6 would be equivalent. The susceptibility components, again in the local frame, for a poling field along the local z axis are given by
o,0) = N/333I Epole \XZx~z-a ) 5kBT pole
AXz (-a'; xxz
AfZ,EO =
AnX,BR= -(1/4n)CBREpole2 = -(1/4n)CBRET2
(9)
this effect will produce
only first-order
diffraction from the space-charge field when a dc bias field is also present to produce a nonzero cross term. Implicitly assumed in Eqs. (7) and (8) is thermal equilibrium between the orienting force from the poling field and thermal excitations that tend to remove the poling. If thermal equilibrium is not strictly attained because of the inability of the NLO chromophores to rotate freely, then the actual birefringence should be lower than predicted by these microscopic expressions, but the sample still has C., symmetry with the uniaxial symmetry axis parallel to the poling field. Therefore the ratios
3 N15kBT a331-t Epole
, w,,2I j 0) 0
(12)
(1/2n)XET
(1/2n)CEoEpoleET 2
= (1/2n)CEoET ,
(10)
where to good approximation the unperturbed index n is used on the right-hand side of these expressions. Since we are concerned only with steady-state solutions, the poling field has now been identified as the total field ET in the local frame (the magnitude of the vector sum of the bias field EB and the space-charge field E8 c). The distinction between the poling field Epole and the total field ET is made for the sake of generality, since the total field may be time dependent and contain frequencies to which the chromophores cannot respond (see Appendix A). As expected, the birefringence is quadratic in the orienting
(11)
Here the usefulness of the local frame is clear: The offdiagonal terms vanish. The index change resulting from the total (dc) electric field ET is produced according to
given by An.Z,BR= (1/2)CBREpOle = (1/2n)CBRET,
323
of the susceptibility coefficients should still be given by ABR = -CBR/ 2 . For example, even though Kuzyk et al. calculated the effect of the competition between the orienting force of the external field and the restoring effect of a (previously poled) polymer matrix as expressed by the elastic modulus,2 6 the ratio of the two components of the susceptibility change is still - 1/2. When we compare calculations with specific measurements below, we assume this ratio to be correct and determine only one constant, CBR, by direct measurement.
molecule, kB is Bo]ltz-
mann's constant, and T is the absolute temperature. For concreteness in this paper we take arw - (') to be positive, as this is the most common case. Because the susceptibility and the permittivity tensors are diago nal in the local frame, the corresponding index changes are
field; therefore
1994/J. Opt. Soc. Am. B
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al.
Anx,EO= (1/2n)AX
(13)
ET = (1/2n)AEoEpoleET
= (1/6n)CEoET
= (/6n)CEOEpoleET
,
(14)
where, as expected, the constants AEO = CE013are in the usual relationship expected for linear electro-optic coefficients. Our essential assumption that the space-charge field itself orients the chromophores is implemented, as for the birefringence calculation, with the poling field Epoie taken to be equal to ET. [We do this for the same reason as in Eqs. (9) and (10); see the discussion following Eq. (10)]. These index changes may be equivalently regarded as changes in the first-order susceptibility, which now is quadratic in the total field. C.
Total Susceptibility Change
The total susceptibility change in the local frame is the sum of the contributions from birefringence and electrooptic effects:
A =
A 0 0 A
01 0
_O
C_-
0
ET2,
(15)
where the coefficients A and C are A = ABR + AEO and C = CBR + CEO- We define Opto be the spatially varying
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Moerneret al.
1994
angle between the local poling direction and the surface normal. With the scalar part of the space-charge field defined to be E6 c(r) = Escoexp(iKG r), the total internal electric field (in the lab frame) is then ET(r)
=
[E.c(r)sin
+ E.c(r)cos
G]AX+ [EB
tibility change,
AX1K =
2A cos OG
)sin
op = arctanF Ec(r)sin G 1(17) LE + E,~c(r)cosOG](7
77s,1K= [HEBECO(2A 7
r p,1K =
cos
U=
0
O
0
1
sin Op
0
-sin
Op
0
(23)
COSOG)]2
{HEBEsco COS(0 2 -
cos
P
01)
AXIab 2
Op 0
0
A
(C - A)cos Op sin Op
0
(C -
A)cos Op sin p 0
A sin 2 op
+
C cos2
p
X ET 2 .
(19)
Since p is a nonlinear function of Es5 (r), it is clear that the total susceptibility change tensor contains gratings at KG, 2KG, and higher orders. Here we concentrate only on the first-order (1K) and the secondorder (2K) gratings. We can simplify the expression for the susceptibility change by letting y(r) = E&:(r)/EB and expanding in powers of y. We keep terms to second order in y to generate the first-order and the secondorder gratings correctly (see below). With cos2 0p,. 2 2 2 1 - Y2 sin OG, sin 0p -y sin 2 OG, and cos p sin p y sin G - 2 sin OG cos G, the approximate expression for the susceptibility change in the lab frame becomes (without the subscript lab) A + (C - A)y2 sin2
AX =
OG
0 A
/2)sin
0
0 A)[y sin
G - (
-
A)sin(01 + 02 )sin
OG
+ 2C sin 01 sin 02 COS G]J2.
(24)
10
which corresponds to a rotation by Op about the y axis. The result of the matrix multiplication gives the susceptibility change in the laboratory frame:
op + C sin
+ (C
(18)
-
L(C-
(22)
X [2A cos Oi cos 02 COS G
-
r
*r).
G
The resulting diffraction efficiency can now be calculated with Eq. (3) for both s-polarized and p-polarized probing (and diffraction; as for the simple electro-optic case, no rotation of the plane of polarization occurs):
The susceptibility matrix can be transformed to the laboratory frame with the standard transformation AXIab = UTAXU, with the rotation matrix given by
A COS2
oG
0 2C cos
0
OG
x EBEs,co exp(iKG and the local tilt angle of the total poling field with respect to the laboratory z direction is
(C -A)sin
0
G
0
(C -.
(16)
0G]i,
[
2Ai -OS
2G]
(C - A)[y sin
A nonzero bias field EB is clearly required for generation of a nonzero diffraction efficiency. It is instructive at this point to consider the limit in which the birefringence modulation is ignored so that C = 3A; i.e., only the enhancement resulting from the modulated electrooptic coefficient is considered. The total field naturally includes the bias, or poling field, so that CEOEB can be identified with the corresponding electro-optic coefficient term, -n 4 r3 3 . Then the ratio of Eqs. (24) and (5) shows that a factor of 4 enhancement in the diffraction efficiency comes from the modulation of the electro-optic coefficient. Since this enhancement alone is not large enough to explain the measured results in Fig. 2, the birefringence term is also important and must be included. Conversely, if the hyperpolarizability were zero, so that the electro-optic effect were not present, a diffracted beam would still occur from the modulation of the birefringence. The size of the diffracted signal in this case depends on the actual value of the polarizability difference parallel and perpendicular to the molecular axis and the dipole moment [see Eq. (7)]. For more direct comparison with experiment, it is useful to calculate the polarization ratio, which should be independent of the actual size of the space-charge field and the bias field: OG - (y 2 /2)sin 20G]
0
ET2.
C - (C -A)y
2
sin 2
(20)
OG
The total field magnitude squared may be written as ET2 (r) =
E
2
+ 2EBEco
exp(iKG
r)cos OG-
=
(-)
exp(2iKG*r) + EB 2
7-
(21)
D. First-OrderGrating Polarization Anisotropy When we retain only terms varying as exp(iKG r), the first-order grating is given by diffraction from the suscep-
CoS2 (0 2 - 01) COS 01 COS02
1K
2 (- A
1 sin(01
+ A sin 0 sin 0 2 ]
+
2)tan
G
(25)
This expression is different from the polarization
Moemneret
Vol. 11, No. 2/February
al.
anisotropy expected for a simple electro-optic grating without the orientational enhancement [Eq. (6)] that would be encountered with a permanently poled sample. Two useful limits of Eq. (25) follow from ignoring bire-
fringence (C = 3A) and from ignoring the electro-optic effect (C = -2A). When birefringence is ignored, the polarization anisotropy for the geometry of Fig. 1 is 8.1, as was reported previously 5 (01 = 17.9°, 02 = 32.1°, and
OG= 65). Conversely, ignoring the electro-optic effect yields a polarization anisotropy of 3.7. In an actual material in which both effects are present, the fact that the two contributions add with particular signs that can cancel in the numerator or denominator of the ratio C/A makes the resulting anisotropy difficult to predict without of the coefficients
specific measurement
and CEO.
CBR
The resulting anisotropy can be much larger than either of the two limiting cases, as is shown below (Section 4)
for the PVK:FDEANST:TNF system. E.
Second-Order Grating Polarization anisotropy
Like that for the first-order grating, the modulation of the susceptibility at 2KG is given by AX2K
A + (C - A)sin2
OG 20
(C -A) (1/2)sin XE 5, 02
0
(C - A) (1/2) sin 20G
A
0 G
0
0
C)sin 2
C + (A-
OG
(26)
exp(i2KG * r).
For diffraction from this grating of twice the wave vector, the incidence angles 01 and 02 must be appropriately modified to satisfy the Bragg condition, as shown by the arrows labeled 3' and 4' in Fig. 1. With these new angles denoted 01' and 02', after simple algebraic steps, the diffraction efficiencies are given by ns,2K = (HESC0 2
2
77p,2K = cos (02
A) 2 ,
- 0){HE
2 8, 0 [A
cos 01' cos
+
=COS (02' 2K
(i
-
G. Specific values of C and A are needed for the
calculation, and we use the measured values of these parameters for the PVK:FDEANST:TNF material determined in Appendix A. Figure 3 shows the results for a range of grating tilt angles (or grating wave vectors) identical to those used in the measurements of Fig. 2. The open symbols show the values of 77, and the filled symbols the values of 77. The circles give the expected diffraction efficiencies for the simple electro-optic PR effect, and the squares the diffraction efficiencies including the orientational enhancement derived in Section 3. At essentially all angles the size of the enhancement is approximately a factor of 10 for 77, and more than a factor of -30 for 77p. Without specific knowledge of the size of EC, at each wave vector this calculation of the enhancement is in reasonable qualitative agreement with
G
(28)
103
orientatio aWy
10-4
simpleEO--
4.76
-enance-
-
10-5 ~
-0.
~
0
104 10-7
le4
I
I
75 OG (degrees)
80
I
I
I
60
65
70
R'. 85
Fig. 3. Illustration of the enhancement of the PR diffraction efficiency induced by dynamic orientation. The diffraction efficiencies -q, and 71ppredicted by the simple electro-optic theory (cir-
2
cles) from Eqs. (4) and (5) and the diffraction efficiencies a7 and
- 1)sin(01' + 02')sin2 6G
+ (-)sin 0' sin 02'].
4.67
10-2
2 0
01') COS01' COS02'
-
1 KG (m- ) 4.52 4.03 4.31
2.70 3.23 3.67
-)COS(01' + 02')sin OG
+ (-2) (
cos
1o-,
Note that these expressions are independent of EB. While the bias field is necessary during writing to overcome geminate recombination and form the spacecharge field, during reading the space-charge field itself can lead to second-order diffraction in the absence of an applied field. Finally, the polarization ratio is given by
77
There are three predictions that the theory makes for the PR response that differ significantly from what is expected from photorefractivity originating from the simple linear electro-optic effect alone. The first is the size of the effect: The diffraction efficiency when the orientational enhancement is operative is larger than that from the simple electro-optic mechanism. The second concerns the anisotropy of the diffraction efficiency of the grating when it is probed in a four-wave mixing experiment by a reading beam polarized parallel to the scattering plane (77,)and perpendicular to the scattering plane (71i). The third is the appearance of a grating at twice the grating wave vector KG even in the absence of higher-order distortions of the space-charge field. To illustrate the enhancement in diffraction efficiency, it is instructive to calculate the values of -q and Up for a hypothetical material in which the space-charge field EScohas reached the maximum allowed value, EB
02'
OG
+ (1/2) (C - A)sin(01 ' + 02 ')sin + C sin Oi' sin 021]}2.
(-)
COMPARISON WITH
PHOTOREFRACTIVE MEASUREMENTS
(27)
+ (C - A)cos(0 1 ' + 0 2 ')sin 2
2
4.
325
1994/J. Opt. Soc. Am. B
(29)
71p predicted by the orientational enhancement theory (squares) from Eqs. (23) and (24) are shown as a function of OG (bottom axis) and KG (top axis). The space-charge field E8 co is assumed = EB cos OG; A = 676 nm, to reach its maximum value Erco d = 125 ,.m, and the values of n 3 r33 , n 3 r13 = (1/3)n3 r33, C, and A are given in Appendix A.
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the enhancement required for removal of the contradiction seen from Fig. 2. We take this as evidence that the space-charge field in PVK:FDEANST:TNF is relatively large, i.e., EO has probably approached the maximum value that it can attain, which is given by the solid curve in Fig. 2. This is presumably because the trap parameters and transport in this material are superior to those for previous PR polymers.4'5 To test the predictions quantitatively for the polarization anisotropy, we used 125-/pzm-thicksamples of the model material PVK:FDEANST:TNF. The gratings were written with two continuous-wave s-polarized writing beams from a Kr+ laser overlapped in the sample with an external angle of 30°. The writing beam wavelengths were 676 nm, the beam diameters at the sample were 750 um, and the power in each writing beam was 6 mW (1 W/cm2). The grating was read out with a much weaker reading beam (150 ,/W, same wavelength and spot size as the writing beams) counterpropagating to one of the writing beams. The sample was tilted from the bisector of the writing beams as shown in Fig. 1. From the measured angles of the writing beams relative to the sample normal and the index of refraction of the material, the angle of the wave vector with respect to the surface normal G can be calculated. The polarization anisotropy is shown as a function of OG in Fig. 4. Both 7p and -q, have been corrected for power loss from surface reflections (which are different for each polarization) to give diffraction efficiencies defined inside the sample. The angle G is varied by rotation of the sample relative to the bisector of the writing beams. The range of G covered in Fig. 4 corresponds to a sample rotation of 40°. The grating wave vector KG varies with the angle of the sample as well, causing variations in the size of E,,cOwhich cancel when the polarization anisotropy ratio is computed at each angle. The polarization ratio for PVK:FDEANST:TNF is 25 ± 6 and is essentially independent of OG, as is shown in Fig. 4 (filled circles). The ratio is much larger than the ),/m7,found in other photorefractive polymers for which the NLO chromophore is covalently bonded to the polymer matrix. The ratios for bisphenol-A-diglycidylether 4-nitro1,2-phenylenediamine:diethylaminobenzaldehyde diphenylhydrazone
(Ref. 4) (bisA-NPDA:DEH;
Moerner et al.
1994
their ratio, C/A). From the values of CBR and CEO determined by the interferometric measurements in Appendix A [Eqs. (A7) and (A8)], the values of C and A for
PVK:FDEANST:TNF are, with the relations AEO= CEO/3 and ABR= -(1/2)CBR, C = CEO+ CBR = [(2.12 ± 0.06) + (4.11 ± 1.0)] x 10-'9 m 2 /V 2 , A = AEO + ABR = [(0.71 ± 0.02) + (-2.06 ± 0.5)] X 10-19 m 2 /V 2 . (31)
Therefore the ratio C/A = -4.6 ± 1.9. The theoretical value for 7/77s for this ratio is approximately 19 and is shown as a function of OG in Fig. 4. The theory agrees with the measured data reasonably well, with no adjustable parameters. Owing to the relatively large error in the determination of CBR(and ABR),the uncer-
tainty of C/A is quite large, -2.7 > C/A > -6.5. This gives a correspondingly large uncertainty in the polarization anisotropy ratio, 7 < 7p/s < 37. Another possible source of error in the comparison between theory and experiment is that the measurement of the C coefficients was done at 830 nm, whereas the diffraction efficiency measurements were performed at 676 nm. Even though the electro-optic coefficients should increase slightly at smaller wavelengths because of resonance enhancement, this would not affect the ratio C/A to a great extent unless the birefringence coefficients changed by a different amount than the electro-optic coefficients. As a final comment, we note that the observed polarization anisotropy is much larger than either of the limits in which one or the other of the electro-optic or the birefringence effect is neglected. It is the precise way in which the birefringence and electro-optic terms add to give the diffracted field that gives rise to the large value of the anisotropy. A third prediction of the theory is the presence of a grating with a wave vector of 2KG. However, this 50
6 + 2, open
40
triangle), a partially cross-linked nonlinear epoxy material with higher glass-transition temperature, and bisphenol-A-diglycidylether nitroaminotolan:diethylaminobenzaldehyde
diphenylhydrazone (Ref. 2) (bisANAT:DEH; 9.5 ± 1.5, open square), a permanently
poled linear epoxy polymer, are also shown in Fig. 4. For the simple electro-optic PR effect, the polarization anisotropy is given by Eq. (6). For r33 = 3r13, the predicted ratio is shown in Fig. 4 (dashed line). Equation (6) accurately predicts the observed anisotropies for bisA-NPDA:DEH and bisA-NAT:DEH, since the orientational enhancement should not be significant in systems with strongly hindered chromophore motion. When the orientational enhancement discussed in Section 3 is active, the predicted value for the polarization ratio 7p/X (for the 1KG grating) is given by Eq. (25). Apart from geometric factors, the only quantities that affect this ratio are the sums of the electro-optic and birefringence coefficients A and C (actually, only
(30)
30 0 *S
0.
*
I~~0
0~~~~~~~
20 10 01
60
I
65
I
70 OG (degrees)
I
75
80
Fig. 4. Ratio of the diffraction efficiency for p- and s-polarized probe beams shown as a function of G for PVK:FDEANST:TNF (filled circles), bisA-NAT:DEH (open square), and bisA-NPDA:DEH (open triangle). The dashed line is the predicted ratio from the simple electro-optic theory [Eq. (6), assuming that r33 = 3r13], and the solid line is the ratio predicted by the orientational enhancement theory for the 1KG grating [Eq. (25)] for the experimentally measured
value of C/A = -4.6.
Vol. 11, No. 2/February
Moerneret al.
1994/J. Opt. Soc. Am. B
327
orientation of the chromophores slowly disappears. By t = 40 s the 1KG signal has dropped by a factor of 103 from the peak value to the baseline noise. The diffraction efficiency for readout of a modulation of the index of refraction at 2KG (filled circles) also decays after the writing beams are blocked. In contrast to the 1KG signal, the 2KG signal drops by only a factor of 10 by t = 40 s. The theory of Section 3 predicts that ?72K
1 03
104
should be proportional 10 C00
0o
Ioo
1 06 0
10
20
30
40
50
Time (s)
Fig. 5. Diffraction efficiencyfor light phase matched for scattering from the 1KG component of the PR grating (open circles) and the 2KG component (closed circles) for PVK:FDEANST:TNF
at
EB= 40 V/Aim. The grating is written and the 1KG component probed at Ao = 676 nm and the 2KG component is probed at Ao= 782 nm. The writing beams are turned on at t = 0, and both the writing beams and the external field are turned off at t = 10 s. 71K decays to its pre-(t = 0) baseline after 30 s, while 772K remains well above baseline for much longer times.
grating must be distinguished from a grating at 2KG resulting from distortion of the space-charge field from a purely sinusoidal form; this distinction is not easily accomplished in general. Higher-order Fourier components of the space-charge field can contribute to gratings at 2KG, 3KG, etc. through the simple linear electro-optic effect19 and should grow in size with increasing writing time, especially for m = 1. Attempts to record at a much smaller modulation index m = 0.1 produced a greatly reduced signal-to-noise ratio in the measurements. Partially to discriminate against such gratings, we investigated the bias field dependence of the second-order diffracted signal. To probe the 2KG grating, we saw that the PR grating was written under the same conditions as those discussed above in measuring the polarization anisotropy. In the geometry in which the 2KG grating was probed, OG = 73.4°.As above, the 1KG grating
was probed with a
reading beam counterpropagating to one of the write beams at a wavelength of 676 nm. In addition, a second probe beam at A = 783 nm from a diode laser entered at a different angle chosen to Bragg diffract from the 2KG grating. In this way both the 1KG and the 2KG gratings were probed simultaneously with p-polarized light. The angles for the 2KG probing beam are shown in Fig. 1 as 3' and 4'. Using this arrangement, we also verified that both reading beams have no effect on the grating, by reading the 1KG(2KG) grating with the 2KG(1KG)reading beam both off and on. No difference in 71was detected. Data from this experiment are shown in Fig. 5 on a semilog scale to show the grating decay behavior more clearly. The writing beams were switched on at t = 0 and switched off at t = 10 s. The external bias field, turned on long before t = 0, was removed
at t = 10 s.
The
diffraction efficiency for 1KG readout (open circles) decays after writing because of the removal of EB [Eq. (24)]. In addition, the field itself decays because of dark conductivity and, with loss of the space-charge field, the modulated
to E&5
2
0
[Eq. (28)] and therefore
should be independent of the applied field if the spacecharge field itself does not decay. While the space-charge field does slowly decay in this material, it is clear from Fig. 5 that the 2KG signal lasts significantly longer than the 1KG signal. After a rapid initial decay, perhaps caused by the rapid decay of the 1KG grating distortion contribution to the 2KG signal, the decay of '72K has a smaller slope in the absence of the applied bias field. This is strong evidence that at least some portion of the 2KG signal is due to orientation of the chromophores in response to the space-charge field. Another distinguishing feature of the origin of the 2KG grating is the polarization anisotropy. A 2KG component that is due to distortion of the space-charge field would have the same polarization anisotropy as that of the 1KG grating, while one arising from orientational enhancement should have a different value predicted from Eq. (28) to be 13.5 for the geometry in use here. The measured value is 7.2, which is somewhat smaller than the predicted value but much smaller than the observed value for the 1KG grating. That is consistent with the origination of some fraction of the 2KG grating from space-charge field distortion as discussed above.
5.
CONCLUSION
This paper has presented experimental data that show that the greatly improved performance of a new class of PR polymers in which the NLO chromophores have orientational mobility is too large to be explained by the simple electro-optic PR effect alone. A model has been developed for an orientational enhancement mechanism in which both the birefringence of the sample and the electro-optic coefficient are periodically modulated by the space-charge field itself. The resulting modulation of the index of refraction of the sample gives rise to diffracted signals at the usual wave vector 1KG as well as to additional modulations at 2KG and at higher harmonics. The predictions of this model for the size of the enhancement (which is an order of magnitude in diffraction efficiency),the polarization anisotropy between p-polarized and s-polarized readout, and the presence of index modulation at twice the grating wave vector are in good agreement with the measured properties of the PVK:FDEANST:TNF material. This quadratic mechanism for grating formation is different from the previously reported quadratic effects operative for semiconductors and other inorganic materials, 1 2 '16 and it should be important in any system in which the NLO chromophores have sufficient orientational mobility and dipole moment to be oriented by the space-charge field itself. A similar analysis for the beam-coupling gain would predict that the direction of the gain will change when the sign of the applied bias field is changed, similar to the case
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1994
for quadratic inorganic PR materials, and this has been observed in the new PR polymers.' The modulation of poling produces a contribution to the diffraction efficiency even if the electro-optic effect is ignored and only the birefringence modulation is present. This implies that if the orientation were frozen in by cooling of the sample before the decay of the space-charge field, the eventual decay of the space-charge field would not cause the diffracted signal from the stored orientation to disappear completely. This phenomenon could permit nondestructive readout, where the long-term storage of information occurs by means of storage of a sinusoidal poling of the chromophores. It would also be useful to study the orientational enhancement effect in a PR polymer in which the traps have been altered to increase the lifetime of the space-charge field in the dark. It also would be useful to repeat these experiments with specifically synthesized NLO chromophores that exhibit high birefringence but negligible molecular hyperpolarizability and with NLO chromophores that exhibit negligible birefringence but high molecular hyperpolarizability. In both cases a large ground-state dipole moment is necessary to provide dynamic orientation by the spacecharge field. In this way both the electro-optic and the birefringent limits discussed in Subsection 3.D can be studied. Since the orientational enhancement effect depends on the ability of the NLO chromophores to orient dynamically during grating formation, the speed of the effect should be limited by the rotational mobility of the chromophores in the PR polymer.2 9 These molecular orientation times can range from picoseconds (observed in liquids3 0 ) to extremely long times of years or more observed in permanently poled polymers,3 depending on the viscosity of the material, the size of the chromophore,3 2 the presence of plasticizing agents, and other factors. The rotational orientation time should be varied over a large range in future materials to allow us to determine the actual limitations on speed of the orientational enhancement effect.
APPENDIX A: MEASUREMENT OF BIREFRINGENCE AN]) ELECTRO-OPTIC COEFFICIENTS To compare specific PR measurements quantitatively with the orientational enhancement theory, we must determine the magnitudes of the electro-optic response and birefringence induced by an electric field. The orientational mobility of the chromophores presents special problems, because both the birefringent and the electro-optical effects contribute to the observed signals. This appendix presents the details of the method that we used to separate the two effects and hence to measure the two coefficients CEO and CBR. The method determines the electric-field-induced change in the index of refraction with a conventional3 3 Mach-Zehnder interferometer operating at 830 nm (single-mode diode laser, 5 mW). The sample is mounted in one arm of the interferometer on a rotation stage, which varies the angle of incidence, and the incoming beam is polarized perpendicular to the axis of rotation of the sample (i.e., is s polarized). A Babinet-Soleil compensator is mounted in the second
Woerner et al.
arm of the interferometer to adjust the interferometer output signal. The sample is identical to that in which the PR properties were measured,' i.e., a 125-gm layer of PVK:FDEANST:TNF sandwiched between two glass plates coated with indium tin oxide. The voltage applied to the sample consisted of two components, an ac voltage, typically of 20 V (rms) with a frequency f ranging from 1 to 100 kHz, and a dc bias voltage of as high as 10 kV. The phase shift A induced by the ac field is measured with a lock-in amplifier, which detects a power modulation in the beam leaving the interferometer at the fundamental frequency f of the ac field (or harmonics of
f)
Aq = 2/2 Sac, rms
(Al)
Sfull, p p
where Sacrms is the modulation signal that is due to the ac field (at a given frequency) and Sfull,pp is the peakto-peak signal that is due to a full r phase shift of the beam in one arm of the interferometer. The change in the refractive index An is given by An(AO
Cos )A
(A2)
where A is the optical wavelength, 0 is the internal angle of incidence in the polymer measured from the sample normal, and d is the sample thickness. The sample thickness is assumed to be independent of the electric field. For this geometry the index change can be expressed in terms of the change of the principal indices as An = An. + (An - An,)sin2 0.
(A3)
In standard interferometric measurements of electrooptically active materials, 3 3 An is linearly proportional to both the electro-optic coefficient r (assumed to be independent of the applied field) and the applied electric field (neglecting quadratic and higher-order terms). Therefore detecting Sacrmslf gives a straightforward determination of r. In electro-optic crystals and permanently poled electro-optic polymers this is true for all frequencies of the applied field. In polymeric materials in which the chromophores can be partially oriented at room temperature2 ' 5 this is still true at sufficiently high frequencies. Orientation times in such materials, in which the chromophore is attached to the polymer backbone, are sufficiently large that frequencies of a few tens of kilohertz are generally sufficiently high to ensure that there will be no appreciable molecular orientation in response to the applied ac field. In PVK:FDEANST:TNF molecular orientation times are sufficiently small that molecular orientation by the ac field must be taken into account when the measurement is interpreted. As is discussed in Section 3, an electric field induces a change in the index of refraction through both the electro-optic and the birefringence effects. With both contributions
included [Eqs. (9), (10), (13), and (14)],
An is given by An= I
(1 + 2 sin20)
+ (3 sin
+ 01
2 23
EpoleET p 21
(A4)
Vol. 11, No. 2/February
Moerneret al.
where the total applied field ET = Edc + Eac sin(M) points in the 3 direction normal to the sample and fl = 2i7f is the ac modulation frequency. Unlike the steadystate PR measurements discussed in Section 4, these measurements are done at frequencies sufficiently high that the molecules can only partially respond to the ac component of the field. This effect is included by expression of the poling field as Epoie= Ed, + (f1)Eac sin(t). The frequency-dependent phenomenological function a(fQ) [O c (fQ)c 1] is inserted to quantify the degree to which the molecular orientation can follow the ac field. For dc, (il = 0) = 1, and the molecules can follow the ac field completely; at high frequency, S(fl= °°)= 0, and the molecules cannot follow the ac field at all. Substituting the expressions for ET and Epole into Eq. (A4) gives An
2 (1 + 2 sin 2 0) CEO 2n3
X [EdC+ 5(fl)Eac sin f1t](Edc + Eac sin ft) + (3sin 2 0 - 1) 2 [EdC + 8(fl)Eac sin 2
Ut]
(AS)
x [EdC+ a (f)Eac sin fit]I *
An important point to notice in Eqs. (A4) and (AS) is that the effect of the orientation is not the same for the birefringence and the electro-optic contributions; the birefringence term is proportional to Epole2, whereas the electro-optic term is proportional to ErEpole. For a(l) 0, Eq. (AS) reduces
to An proportional
CEO (1 +
3
(a)(a)*
cm2
+) CE
E
_\
1
C~~~~~~B
8) _ CBR (2])n 0
[ C
329
effect cannot explain the observed results. In particular, relaxation of the standard assumption that CEO = 3AEO (n3 r33 = 3n3 r13) would lead to an apparent nonzero value for CBR. If CBR is assumed to be zero and CEO is not assumed to be 3AEO,then one can carry through the analysis to determine the ratio CEO/AEO(or r33 /r13). The resulting ratio is -8, a value that is much larger than the expected value for linear chromophores2 8 and the value observed in other poled polymers.33 To remove the frequency dependence from the quantities plotted in Fig. 6(a) and to determine CEO and CBR, we estimated the frequency dependence of the molecular orientation separately by measuring the real part of the dielectric constant e'. This was done by measuring the capacitance of the sample on an LCR meter (HP 4284A). e' is shown in Fig. 6(b) on a log-log scale over the full range for which it could be measured and on a linear scale over the frequency range covered by the interferometric measurements [Fig. 6(b), inset]. Over the frequency range from 102 to 106 Hz, e' is described reasonably well as a power-law function of frequency (f)-a plus a constant, with a power-law exponent 0.17 ' a ' 0.29, with smaller values describing the full frequency range somewhat better and larger values describing the frequency range of the interferometric measurements (10-100 kHz) somewhat better. A power-law fit to e' shown in Fig. 6(b), with an exponent of a =
to an ac field-
independent coefficient times an ac field (plus a constant term), as discussed above. From Eqs. (A1)-(AS), CEO and CBR can be expressed in terms of the interferometric signals observed at the fundamental frequency f:
1994/J. Opt. Soc. Am. B
3
)
CB 2 (2 6)] sin 2 = 2nd
0
0
Ede
A cos 0 Sac,rms 1i 7TVac Sfullp-p
I I 60 40 Frequency (kHz)
I 80
100
8
(A6)
For PVK:FDEANST:TNF, Sacrms1f was measured as a function of 0 at room temperature with Edc = 16 V/pum. For frequencies above 10 kHz, the right-hand side of Eq. (A6) was observed to be a linear function of sin2 0,as the left-hand side requires. Below 10 kHz the data were complicated by other effects,2 6 such as piezoelectricity and electrode attraction, which should have a low-frequency resonance (below 10 kHz) for these 125-,um-thick samples. Therefore only data at higher frequencies were used. At each frequency, from the intercept and the slope of Eq. (A6) versus sin 2 0, the values of CEO(1 + 8) and CBR(28) were determined. The results are shown in Fig. 6(a) as a function of the frequency of the ac field. The measured values of CEO(1 + 8) and CBR(28) are frequency dependent, decreasing with increasing frequency. At any given frequency these coefficients were separately observed to be linear functions of the dc field up to 72 V/,um. It should be noted that an index-ofrefraction modulation that is due solely to the electro-optic
l 20
7 6 -w;
5
4
102
lo3
104 Frequency (Hz)
105
106
Fig. 6. Values of CEO(1 + ) (filled circles) and CBR(23)(open circles) determined by interferometric measurements shown as ac field. The solid a function of the frequency of the applied 23 curves are fits to the data with 6 = f-0. . (b) Value of E' for PVK:FDEANST:TNF determined by capacitance measurements shown on a log-log scale over the full measurement range and on a linear-linear scale over the frequency range of the interferometric measurements. The line is a fit to the data with a power-law function assumed for the frequency dependence with the exponent -0.23.
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J. Opt. Soc. Am. B/Vol. 11, No. 2/February
0.23, which is a good compromise.
1994
Moerner et al.
The phenomenological
function can therefore be approximated as 8 = (f)-023. Fits to CEo(1 + ) and CBR(23)using this behavior of 8 are shown in Fig. 6(a). These fits yield values of CEO= (2.12 ± 0.06) x
lo-1 9 m 2 /V 2 ,
CBR = (4.1 ± 1.0) x 10-19 m 2 /V 2 ,
(A7) (A8)
where the error ranges are estimates of the parameters for power-law fits to the data with the power-law exponents 0.20 and 0.25. The values of CEO and CBR may be improved somewhat with a more accurate choice of the fitting function for e'. For reference, in a field of Edc = 40 V/pum, the electro-optic coefficient is n3r33 = CEOEdc= -5.0 pm/V. (A9) n [The minus is required for consistency with Eq. (1).] The previously reported values of n3r,3 were in error because the interference from the modulation of the birefringence was not removed.12 An alternative method for removing the frequency dependence of the measured electro-optic and birefringence coefficients would be to measure the modulation in the refractive index at both f and 2f. It is apparent from Eq. (A4) that the quadratic effects discussed here give a modulation at twice the fundamental frequency of the ac field26 2f as well as f, and an expression analogous to Eq. (A6) can be derived for the interferometric signal Sacrms2 . Measuring both S,,rms'f and Sacrms2 f would permit a unique determination of CEO and CBRwith no assumptions regarding the frequency dependence of . Unfortunately, Scrms2f could not be measured in our samples at these high frequencies, primarily because of the relatively small ac fields that could be applied across our 125-gm-thick samples.
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ACKNOWLEDGMENTS We acknowledge partial support of this work by Advanced Research Projects Agency Defense Sciences Office contract DAAB07-91-C-K767 and thank D. M. Burland for helpful discussions. *Present address, Laboratoire d'Optique Quantique, Ecole Polytechnique, 91128 Palaiseau Cedex, France.
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