second-order susceptibilities of anisotropic liquid-crystal thin films

Second-Order Susceptibilities of Anisotropic Liquid- Crystal Thin Films

M. PINNOW, G. MAROWSKY, F. SIEVERDES Max-Planck-lnstitut fiir biophysikalische Chemie, Abt. Laserphysik, Am FaBberg, D-3400 Gbttingen, Germany

F. H. KREUZER, H. LEIGEBER, A. MILLER, and P. SPES Consortium fiir elektrochemische Industrie GmbH, Zentrale Forschungsgesellschaft der Wacker Chemie GmbH, ZielstattstraBe 20, D-80@ Miinchen 70, Germany

(Received: 15 October 1991; in final form: 11 December 1991)

Abstract. Using optical second-harmonic generation, we have measured the anisotropic orientation behavior of liquid crystal (LC) molecules in thin films. A high degree of in-plane anisotropy was obtained by depositing the LC films onto rubbed polyimide-covered substrates. The analysis with respect to the relevant second-order susceptibility tensor components is based upon the experimental results obtained in transmission geometry using different angles of incidence for the separation of isotropic and anisotropic contributions.

Key words. Second-harmonic generation, liquid crystals, orientation analysis.

1. Introduction Soon after the first demonstration of orientation analysis based upon the evaluation of nonlinear optical signals obtained by second-harmonic generation (SHG) in 1982 [l], the orientation behavior of liquid crystal (LC) thin films has gained increasing scientific interest. Both freely suspended films [2] as well as thin layers spread out on various interfaces [3] have been studied. These investigations re- vealed the mechanisms of bulk and surface LC alignment via interaction between substrate surface, surfactant, and organic thin film. Surfactants yielding homeo- tropic alignment, perpendicular to the surface, and substances yielding in-plane alignment have been identified. The perfect in-plane alignment of LC layers and other selected organic molecules, which exhibit a strong tendency for polar or- dering, is an excellent example of rotational anisotropy. In fact, organic monolayers usually show rotational isotropy around the normal to the surface [4], although small deviations from isotropy have been documented in the literature [5]. Aniso- tropic behavior is the rule for studies of crystalline surfaces of semiconductors or metals [6,7].

A quantitative characterization of the anisotropic coverage behavior requires a precise determination of the components of the relevant second-order susceptibility tensor. In this contribution we try to identify these components by polarization-dependent excitation and SH detection and by selected rotations of the thin film samples. Evaluation of the experimental SH data requires some functional modelling of the anisotropic sub-species, such as the assumption that the molecules

Molecular Engineering 1: 341-355, 1992. @ 1992 Kluwer Academic Publishers. Printed in the Netherlands.

342 M. PINNOW ET AL.

forming the anisotropic sub-species are stretched out flat in-plane as a result of the uniaxial rubbing procedure.

2. Theory

2.1. ISOTROPIC AND ANISOTROPIC TENSOR COMPONENTS: NONLINEAR PREFACTORS

We shall only consider monolayers and thin solid films, which may be described with respect to their second-order response in terms of isotropic tensor components xtl and anisotropic components xp2. The susceptibility tensor, responsible for SH emission, can be written in piezoelectric contraction in the following format for C, symmetry:

[

Xi2 xw XYY Xi% x$wx (a)

XXYX - - -

(0) XYXX

(a) XYYY

((1) Xrzz

(9 XYZY '

(a) XYXY (1) ---

X%X xG) ZYY xyiz (a) (a)

XZZY XZXZ -1 -- It should be noted that the number of isotropic components can be further

simplified by the condition ~$2~ = x$jy = xtiY = x$$. An average tilt angle 4i, describing isotropic orientation around the surface normal - the z-axis - of molecules of rod-like shape, may be defined by the relation [8]

& = arc tan(2x$~Jx~&). (2)

The total tensor according to (1) considers two special cases of C, symmetry: Components ~$2 underlined with solid lines, hence m I y, subsequently called ‘x- texture’; Components x$? underlined with wave lines, hence m I x, subsequently called ‘y-texture’.

As will be discussed in more detail in the experimental section, a rotation of the sample by (Y = 180” will lead to a sign reversal of the anisotropic components for both textures (cf. Fig. 2). Hence s-polarized signals from samples with y- texture will show SH signals proportional to ]x$iy ? (x$i + &,L + ,$‘$‘)]’ under such a rotation and for an oblique angle of incidence.

Prefactors Fijk = Fififk, comprising nonlinear (FJ and linear (fj,fk) Fresnel factors, need to be considered for different experimental conditions, such as transmission under various angles of incidence p (see Figure l), or total reflection geometry (see Figure 2). Normal incidence (/3 = 0”) and incidence under 45” from air onto a thin film covering a glass plate made from B-270 glass results in the following Fijk values for a fundamental wavelength h = 1064 nm [9]:

(34

SUSCEPTIBILITIES OF ANISOTROPIC LC FILMS 343

Transmission

R(p) m s

~~____---____----___-

I I

I I -_____----_-------_________

Fig. 1. Experimental setup for the transmission geometry. Polarization 0; exciting radiation is con- trolled by polarizer P1 and h/2 plates, polarization of SH response of sample (S) by P2. Inset: definition

of p- and s-polarization and angles a and p.

iljk xx YY zz YZ XZ YX x 0.184880 0.216331 0.270091 -0.241721 -0.223460 0.199989 y 0.199098 0.232968 0.290862 -0.260311 -0.240645 0.215368 (3b)

Z 0.225103 0.263397 0.328852 -0.294310 -0.272076 0.243498

Matrix (3a) shows that no contributions come from an isotropic coverage under normal incidence. Details of the calculation of the prefactors Fijk, including tables of the complex-valued set of prefactors for total reflection have been published elsewhere [lo]. We have discussed some details concerning the computation of these prefactors and application to isotropic adsorbates in this reference. Extension to anisotropic thin films is shown in this contribution for the first time.

2.2. CONTRACTED SH TENSOR NOTATION

All our susceptibility data will be derived from SH rotation patterns. These are obtained by p- or s-polarized SH detection and variation of the polarization of the fundamental by rotation of a A/2 plate by 360“. With respect to this experimental situation, it is convenient to introduce an appropriate system of coordinates, similar to the procedure of Ref. [ll]. According to Figure 3 the indices p and s denote field components of the fundamental parallel and normal to the plane of incidence and index k the coordinate in the propagation direction of E” = L%4 9 Es(41. TIJ e amplitudes E,(20) and E,(2w) of the SH fields can be derived from the following expression:


Total Reflection

h(P) g

Nd-Yag P w Laser

- cp

180° Prism Rotation

y-Texture L Y

--_ X

WPS)

Fig. 2. Total reflection geometry. Inset shows symmetry changes upon 180” prism rotation and examples of y- and x-texture.

(4)

Describing the angular position of the A/2-plate by the parameter cp as shown in Figures 1 and 2 and with p = 0” denoting the p-polarized excitation by a fundamental of amplitude Eo, the fundamental components E,(W) and E,(o) are given by the expressions:

EJw)= E,cos2qJ

E,(w) = E. sin 2~

For simplicity, the necessary projections of EJo) and E,(w) to yield {E,, Ey, E,} wiil usually be neglected. Subsequently, both coordinate systems will be used in parallel as far as identification of a particular tensor component in the Cartesian x-y-z system is concerned. According to Figure 3 the field components transform as follows:


Fig. 3. Cartesian x-y-z coordinate system, location of the thin film under consideration in the x-y plane, the definition of E,(2w) and &(2w) for reflection geometry.

E, = Ep(w) - cos /I

Ey = Ed4

E, = E,(w) . sin /3

(5b)

The set of coefficients a,,, . - - asps in the contracted tensor (4) represents the following sums of susceptibility components, together with the respective prefactors:

(6)

asps = 2F,(x%fX + x%fAf, For isotropic adsorbates, the coefficients aspp, asss, and a,,, vanish. With E,, - Ep(w) and E = E,(o) and the cpdependence according to Equation (5) we

zpect for the inte<sity Zp(2w) - /E,(~w)[~ with

IEp(2412 = lapppEp(42 + apJL(4212 (7)


LC-Siloxane

H3C\ ,CH3

H3C CH3

Fig. 4. Chemical structure of LC siloxane.

a four-fold rotation pattern with the actual shape depending on the uppplapss ratio. s-polarized SH detection in the presence of E,(W) and E,(o) results in

L&4 - lW41* = l~,sE,(4E&)l*, (8)

i.e. an eight-fold rotation pattern. In the latter case, addition of the anisotropic w susceptibility component xyXy, describing a pure x-texture, will change the ampli-

tude but not the periodicity. Other harmonics due to a,,,E,(o)* + asssEs(o)* will come into play upon consideration of these anisotropic contributions describing a pure y-texture.

3. Experimental Methods

The thin films were deposited onto a glass flat or onto the hypotenuse of a CaF2 prism. The substrates were covered with polyimide sublayers of about 100 nm thickness (Merck Liquicoat PI) which were structurized by rubbing to yield the required texture in the x- or y-direction. A liquid crystalline siloxane [12] of chemical structure as shown in Figure 4 was deposited onto the polyimide films from a lop3 molar solution in chloroform. As already mentioned, SH data were taken in transmission and total reflection geometry. The transmission configuration (Figure 1) facilitates the identification of the anisotropic tensor components since normal incidence delivers SH signals from these components only, and the isotropic components can be gradually introduced by rotating the sample around the y-axis by an angle j3, thus changing stepwise the angle of incidence of the fundamental onto the sample.

The total reflection geometry usually delivers SH signals which are larger by an order of magnitude in comparison with all other geometries and allows a convenient angular separation of fundamental and SH response of the adsorbate film due to the dispersion of the prism material (Figure 2). This configuration does not allow a continuous variation of the sample rotation angle (Y shown in Figure 1. The symmetry analysis is restricted to a prism rotation of 180”. Both geometries have in common excitation with 20 mJ pulses of 5 ns pulse duration form a Nd- YAG laser, rotation of the polarization of the fundamental (originally p-polarized) by a h/2 plate, p- of s-polarized detection with a monochromator photomultiplier- boxcar combination and signal processing by computer routines.


4. Experimental Results A complete experimental determination of the 12 unknown components of tensor (1) with 10 unknown anisotropic and 2 a priori unknown isotropic components is never unambiguous and by no means a trivial problem. Most authors limit the discussion to the determination of the respective tensor symmetry - and hence the number of independent components to be expected - rather than crucially iden- tifying the actual numerical values of the components [13]. We would like to identify both the tensor symmetry and, as far as possible, at least the degree of anisotropy of a given coverage distribution. To simplify the interpretation of the experimental results we shall drop the prefactors Fiik using the abbreviation Xe,ijk 3 Fijk * Xijk for both isotropic and aIliSotrOpic susceptibility components. In addition, we shall discuss directly the influence of anisotropies with respect to X$k rather than considering the manifold of parameters aPPP . - - aSPS as defined in (6). It should be pointed out, however, that the combinations of susceptibility components in these parameters are the quantities that can be derived directly from an SH experiment.

As a first example for the ‘deciphering’ of tensor (1) experimental data together with a theoretical simulation of the concomitant rotation patterns are presented for the analysis of an anisotropic film of x-texture prepared by simple wetting of the polyimide-coated total-reflecting prism with dye solution and subsequent evaporation of the solvent. The polyimide coating did not produce any background SH-signal. The left-hand side of Figure 5 shows two rotation patterns of the SH intensity Z,,,(2w) obtained by 180” rotation of the prism. The change in the se- quence of peaks is obvious and nicely documents the fact that SH data, unlike linear optical measurements, are direction-sensitive. A semiquantitative description of the respective SH amplitude E,,,(2w) obtained from an x-texture upon excitation with both fundamental fields E,(o) and E,(w) starts from Equation (4):

Es,,@4 = a,,E& + asp& + ass& (9)

With E,E, = E$2. sin 4~p and introducing an adjustable parameter c accounting for all possible anisotropies (pure x-texture and possible deviations towards y- texture) the s-polarized SH intensity Z,,,(20) - (Es(2~)12 can be expressed as

Z,,,(20) = 1x&,12 * Eo/4. (sin 49 * 2~)‘. (10)

The right-hand side of Figure 5 shows, for each prism orientation, three selected rotation patterns with 2c = 0.01,0.12, and 0.30. It is apparent from this simulation that the prism rotation experiment can be best described by the assumption that approximately 0.12/2 = 6% of the total second-order susceptibility is due to film anisotropy. Although this anisotropy is rather small, it strongly affects the pattern of Figure 5 which corresponds to

Z,,,(20) m (sin 4~)~ (11) in the absence of anisotropies. The modulations in the patterns of Figure 5 are entirely due to the product c . sin 4~, hence the nonlinear interaction between isotropy and anisotropy.

For the evaluation of all transmission experiments performed with rubbed,


2. 25

J...; . . . . :...: . . . :...; . . . . :...; . . :...; . . . :...; . . . . :...:....L:... -25~

2 ‘: 20

Ii

h ‘5 2

$ 10

*d 7”

F2 0 0

Rotation Angle (p

............... ... . ... .: ... . .... . ........ .: ..A.. . .: ... . ... .: ... . ... .: ... . ........ .: ... . ..~..r...:....:....~...l...:....:....r...~..~...~ ...

- 25 . . 4. . . .

. . . . .,., :...: . . . . . .,.,I . . . . :..,: ,... I...: ,.,, :...: .I.. ~..+....‘.... :... . . . . . . . . . . . . .

0 90 100 270 360 Rotation Angle Q Rotation Angle (p

Fig. 5. Experimental rotation patterns 1S,X(2w) obtained by 180” prism rotation together with simulation of three selected values of the parameter c of Equation (10).

polyimide-coated substrates carrying thin LC films prepared by the wetting technique, it was sufficient to assume the usual isotropic tensor components and one dominant anisotropic component: ,&“XX for x-texture, x$$,,~ for y-texture.

Allowing for small angular deviations A6 from perfect n- or y-texture results in the modified anisotropic tensor components X& with prefactors Fijk again in- cluded:

X-texture

Y-texture

(13)

SUSCEF’TIBILITIES OF ANISOTROPIC LC FILMS 349

Hence for both textures there exist only three non-zero tensor components which can be described in terms of the two independent parameters ,Y%; or J&L and A6. The situation is similar to the description of isotropic films, where five tensor components exist which may be described by xl’,‘, and the average tilt angle pi according to Equation (2). Consequently, one can derive a measure for the precision of both textures by the following relation

tan2 Aa = x~~yy~x&4?xx = x&Jx&~~~~ , (14)

indicating whether an alignment is sharply peaked or not. It has been tacitly assumed that the measurement of the angular spread results for identical textures in the same value for AI?. A different approach for testing details of the anisotropic orientational distribution has been considered in Ref. [14]. Using the expressions (12, 13) the anisotropic SH intensities are as follows for both textures:

X-texture

Y-texture

4,yGW = (2x%x . Ep . Ed2 (164

L,yW) = (x$;xx - E; + x$iyy - E:j2 WI

Insertion of E,(w) and E,(o) into Equation (15) and (16) and considering that xyLyy 4 x%ixx and x&4ixx -+ x$iyy will result in the following symmetries of the rotation patterns:

&fold for Z&20) and Z,,J2w),

4-fold for Z,,J2w) and Z,,(2w). (17)

Experimental data supporting these predictions for anisotropic films under normal incidence are presented in Figures 6-8. Figure 6 shows as a first example the SH intensities Z&20) and Z,,,(2w) for normal incidence and an angle of incidence p = 45” in comparison. Least-squares fits to the x-texture experimental data of Figure 6 for p = 0” are in good agreement wiht the predictions of expression (17). Deviation .from normal incidence requires consideration of isotropic components:

L(2w) = 4(x$;zy + xi’$xJ2E;E: (18)

The experimental data of Figure 6 show that Equation (18) describes the situation only approximately: for a sample orientation with x2;‘,, = - x&,, the (still prevailing) 8-fold symmetry of the rotation pattern must be modified by consideration of an additional constant term due to the neglected sum cx%x , , + ,Y$~& + J#,,,~E?, which accounts for a simultaneously present y-texture. Whereas the influence of isotropic components is less dramatic for x-textures as shown for Z,,,(2w) and Z,,,(20) in Figure 6, changes in the symmetry of the


h 0.16

2

i 0.12 0.06

2 T 0.04

3: 0.00 VI

Rotation Angle (p

Rotation Angle cp

Fig. 6. 1,,,(2w) and 1,,,(20) versus rotation angle cp of h/2 plate for normal and 45” incidence. Ratio of p- and s-polarized SH intensities, displayed in arbitrary units, can be derived from both parts of

this figure.

rotation patterns occur for y-textures (cf. Figures 7 and 8). Theoretical prediction (Figure 7a) and experimental results (Figure 7b) indicate for Z,,J2w) a gradual transition from an g-fold pattern for p = 0” to a 4-fold pattern for p = 45” ac- companied by an increase in intensity by one order of magnitude. All experimental curves of Figure 7b can be approximated by

As shown in Figure 7 p-polarized SH signals from y-textures change their symmetry from g-fold to 4-fold rotation patterns and are dominated for p > 10” by the isotropic contribution. The minute influence of anisotropic components on this signal is due to the small anisotropic contribution

(a) - (a) Xd,xyx - X&w *

r sin’ A19cos A6, hence modification of an eventually rather large

anisotropy xdq)yyy by the product sin2 A6 cos hi?, which may be rather small for nearly perfect in-plane alignment along the y-direction. Figure 8 represents angle- of-incidence studies of Z,,J2w) given by:

L,y(24 = <x&z . E; + x$;yy . E: + 2x?,~zyEpEJ2 (20)

With ~9’ .YYY = -x$,~~ for the sample orientation in the series of experiments displayed in Figure 8 the main anisotropy component interacts destructively with the incidence-dependent isotropic contribution. The rotation patterns Z,,,,(2w) cal- culated for p = O”, 5”, lo”, 15”, 30”, and 45” show continuously a 4-fold symmetry and a decrease in magnitude by approximately a factor of two upon an increase


- 3

0.24 c-4

- 0.20 a @=45’

- 0.16 h .di /3=15” 0.12

E 0.08 @=lO”

5 'i 0.04 /3=5”

x v] 0.00 0 90 l&l0 270 360

Rotation Angle p

b) T 0.24 1. .+ A. * r;r - 0.20

c1 2 0.18 h

.t: 0.12

i 2 o.oe

c 'i 0.04

X rn 0.00 _- 0

&atio?AngIeni 360

Fig. 7. Simulation according to Equation (16a) of SH intensity 1,,(20) versus rotation angle cp for five angles of incidence p (Figure 7a) together with experimental data points (Figure 7b).

of the angle of incidence p from 0” to 45”. A shift of the phase of the peaks of Z&2w) with increasing angle of incidence is also discernible in the simulation of Figure 8a. With ,&,, 4 x&y the term xy’

the dominant 4-fold pattern is entirely a result of ,yyy - E: = x(d) ,YyY . E’, . sin* 2~9. Figure 8a shows a simulation of Z,J2w)

according to Equation (20) under complete consideration of the angle of incidence- dependent prefactors Fjjk and field components. The experimental data are again in excellent agreement with the theoretical prediction according to Equation (20) in terms of shape, symmetry, and phase of the rotation patterns. An evaluation of all angle of incidence-dependent experimental data with inclusion of actual values of prefactors Fiik resulted in ,&i = 0.80x$,,. It is interesting to observe that upon consideration of the prefactors Fyy,, = 0.233 and Fyry = -0.260 of matrix (3b) for 45” incidence the dominant isorropic contribution (-2~~fi.~E,E,,) and the dominant anisotropic contribution (-xyYY y Y (‘) E E ) are nearly equal, but opposite in sign. Hence the amplitude due to anisotropic contributions exceeds the isotropic amplitude even for 45” incidence. A sample rotation by 180” with the concomitant sign reversal of the anisotropic components would increase the j3 = 45” SH intensity 1,,,,(20) by nearly one order of magnitude.

It is interesting to compare the angles & and A6, characterizing the angular distribution around the surface normal (& as defined by Equation (2)) and the degree of in-plane alignment (AIM as defined by Equation (14)). Evaluation of the


(4

6)

R x 0.02 2 d 0.01 ‘i 3: * 0.00

0 PI0 G&ion ‘I&e p

100

Fig. 8. Simulation according to Equation (16b) of SH intensity Zs,y(2w) versus rotation angle cp for five angles of incidence p (Figure Ba). Lower part (b) shows experimental data points for the selected

angles of incidence p = 0” and /3 = 45”.

p-polarized signals 1,,,(2w) and ZP,J2w) for 45” incidence and neglecting the influence of anisotropies results in I?~ = 42” 2 2”. Evaluating the respective SH intensities in terms of the susceptibility ratios ,&Jx~$,,~ and ,&,,Yl&~XX yields a value of 0.028 + 0.01 or AI? = 22” + 2”. As has been shown in some detail in Ref. [lo], the actual tilt angle 8ii,eff depends on the permittivity E, of the embedding medium of the chromophores responsible for the observed second-order nonlinearity. For the value of E,(O) and Q2w) as chosen in Ref. [lo] a typical tilt angle describing the isotropic distribution results in 8ii,eff = 21” ? 5”, hence a value close to 86, characterizing the in-plane angular distribution

5. Discussion and Conclusion Both, experimental data and theoretical description in the preceding section have shown that the tensor components governing SH emission from anisotropic thin LC films covering polyimide-coated substrates are now well understood. Within the framework of the various assumptions discussed earlier tensor (1) may be replaced for the present case by the following tensor:

SUSCEPTIBILITIES OF ANISOTROPIC LC FILMS

Fig. 9. Definition of angles $ and A9 for description of the thin film susceptibility ,&k.

[

xi% -(a) XXYY X% -(a)

a * XXXY

-(a) XYXX

-(a) XYYY a /ygy ’ f$$

-I_

(9 X ZXX (0 XZYY

(0 XZZZ 00.

353

(21)

It is apparent from tensor (21) that the assumption of pure in-plane alignment of part of the molecules eliminates all z-dependent anisotropic components and hence reduces the number of unknown components by four. Considering that a pure x-texture should be identical with a pure y-texture after 90” rotation of the sample the whole analytical problem reduces for these ‘pure’ cases to the determination of x’“’ or x’“’ xxx yyy together with A6 according to Equation (14) and x!~L)~ together with fii according to Equation (2). Furthermore for our example there exists the relation

Xfilm (2) = xt$d + x$l ) (22)

with no symmetry superposition as discussed in detail in Ref. [7], since the various assumptions and approximations leading to tensor (21) infer that the anisotropic ,$d components are only in-plane and the isotropic x$L components are by definition components surrounding the surface normal. This separation of both contributions is indicated in Figure 9. Following the results of Ref. [14], which showed that polyimide-coatings have an effective short-range orienting force on LC molecules, one may even subdivide the sum in Equation (22) into a spatially split second-order film susceptibility with x:$ governing the LC surface behavior and ~$1 the LC bulk behavior.


Yi- c?. 0.24

w. 0.20 2 3 0.10 ‘8 0.12 d a ‘; 0.02 7 0.04 u 0.00 C-r-x-l-I

0 270 260

Fig. 10. SH intensity Z,,J2w) versus rotation angle 6 of h/2 plate for unidirectional and bidirectional rubbing of the polyimide sublayer.

Based upon a comparison with the susceptibility of a hemicyanine dye [15] prepared with the Langmuir-Blodgett technique, showing under perfect polar alignment a x$i, value as large as 5 X 10 -13 esu for excitation with a fundamental of 1064 nm wavelength, we have derived an approximate value for

(0 (a) $= xzzz = xxxx XYYY w of 2 x lo-r4 esu for 1064 nm excitation and 1 x lo-l3 esu for excitation at resonance (800 nm). In any case, the component x’,i!, will govern the susceptibility behavior for the isotropic part and ,&.L (or x,$,, depending on the texture) the anisotropic part. Since the effective susceptibilities are xgiZZ = J& * COST 8i and x&4iXX = ~~22~~ * cos3 A6 and A6 < fii the influence of the anisotropic

tensor components on the susceptibility will be stronger due to their narrow in- plane distribution (cf. Figure 9) as compared to the three-dimensional distribution of the isotropic components.

As to the question, which texture is more efficient in terms of showing anisotropic behavior, it is obvious that y-textures upon consideration of relations (15, 16) and tensor (21) exhibit a larger influence of anisotropies. Figure 8 nicely shows that in particular the rotation pattern of the s-polarized SH-response is governed

@) by the component xrY,,. For both textures the influence of anisotropic components of p-polarized SH signals is less dramatic due to the presence of strong isotropic components having large prefactors Fijka

All results discussed so far were obtained with polyimide-coated substrates rubbed in one direction, e.g. the +x-direction. This sublayer texture determined the symmetry of the thin film, irrespective whether it was prepared by transfer of a LB monolayer onto the structurized substrate or by simple wetting from chloroform solution. Bidirectional rubbing, resulting in C,, symmetry [16], yielded SH signals which were one order of magnitude smaller (see Figure 10). A perfect in- plane alignment in both directions is centrosymmetric and yields no SH signal. The small signal of Figure 10 indicates ‘imperfect’ bidirectional in-plane orientation. The normal-incidence condition for this experimental result does not allow the consideration of isotropic contributions.

Finally, we would like to point out that the experimental procedure of changing the angle of incidence /3 from 0” to 45” seems to be a viable technique for unambiguously determining the anisotropic components. The shape and underly-


ing trend of the patterns in Figures 7-9 indicate that the assumptions which lead to the simplifications of the original tensor of C, symmetry are justified.

Acknowledgements

The experimental work has been supported by the Deutsche Forschungsgemein- schaft through the Leibniz Prize program. We thank J. Jethwa for a critical reading of the English version of this manuscript and E. Heinemann for expert technical assistance.

References

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10. F. Sieverdes, M. Pinnow, G. Marowsky, U. Felderhof, and A. Bratz: Nonlinear Optics (in press). 11. J.E. Sipe, V. Mizrahi, and G.I. Stegeman: Phys. Rev. B 35, 9091 (1987). 12. P. Spes, M. Hessling, and F.H. Kreuzer: European Patent EP-O-431-466-A2. 13. Y. Shimizu and M. Kotani: Opt. Comm. 74, 190 (1989). 14. W. Chen, M.B. Feller, and Y.R. Shen: Phys. Rev. Lett. 63, 2665 (1989). 15. G. Marowsky, L.F. Chi, D. Mobius, R. Steinhoff, Y.R. Shen, D. Dorsch, and B. Rieger: Chem.

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second-order susceptibilities of anisotropic liquid-crystal thin films

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