PHYSICAL REVIEW E SEPTEMBER 1998VOLUME 58, NUMBER 3

Optic models for short-pitch cholesteric and chiral smectic liquid crystals

Pascal Hubert,1 Pontus Ja¨gemalm,2 Claudio Oldano,1 and Mauro Rajteri3

1Dipartimento di Fisica del Politecnico di Torino and Istituto Nazionale di Fisica della Materia,Corso Duca degli Abruzzi 24, 10129 Torino, Italy

2Department of Microelectronics and Nanoscience, Chalmers University of Technology, S-41296 Go¨teborg, Sweden3Istituto Elettrotecnico Nazionale Galileo Ferraris, Settore Fotometria, Strada delle Cacce 91, 10135 Torino, Italy

~Received 31 July 1997; revised manuscript received 26 January 1998!

The optical properties of cholesteric and chiral smectic-C liquid crystals having a pitch shorter than the lightwavelength are studied, both theoretically and experimentally. A particular emphasis is placed on the opticalactivity. For smectics, the optical rotation is maximum for a tilt angle of 45° and for light propagatingorthogonally to the helix axis; for short-pitch cholesterics, the optical activity is, in any case, very small. Thelimits of validity of a recently proposed macroscopic model for such media are discussed, in the framework ofa more general discussion on optical models for gyrotropic media. It is shown that the macroscopic modelsgenerally work well for the bulk properties. However, for chiral smectics with the smectic planes parallel ornearly parallel to the boundary planes, no homogeneous model is able to account for the gyrotropic properties,independently of how small the pitch is and for any sample thickness. Our experimental data are in agreementwith these theoretical findings.@S1063-651X~98!05509-3#

PACS number~s!: 61.30.2v, 78.20.Ek, 78.20.Bh

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I. INTRODUCTION

In most liquid crystal~LC! compounds the molecules astrongly elongated, and some periodic LC phases are kn

where the average directionn ~nematic director! of the longmolecular axes rotates uniformly along a given directiodescribing a helix. If the helix pitch is smaller than the ligwavelength, the periodic medium can be convenientreated in optics as macroscopically homogeneous, as ufor crystals. Recently, a macroscopic optical model for shpitch chiral smectic-C liquid crystals (Sc* ) has been suggested@1#, which also is valid for other LC phases having tsameC2 element of symmetry (SI* , SF* , andSK* ), and in-cludes as a particular case the cholesteric (N* ) phase. Themacroscopic effective medium is uniaxial, with the optic aalong the helix axis of the periodic structure, and is opticaactive.

Here, a critical discussion of the model is given. It will bshown that with minor changes concerning its constitutequations, the model can be successfully applied to the aally available short-pitchN* andSc* compounds, except fosome very particular optical geometries. This fact could itsmotivate our investigations, owing to the increasing interof such compounds in applications@2,3#. However, our mainmotivation goes beyond the optics of LCs and refers to baoptics. In fact, the search of a macroscopic model for ocally active crystals is still an open problem, since this proerty is hardly incorporated in the framework of homogneous models, in particular for crystals with a long ranorder. Here we recall some of the difficulties found in amacroscopic description of optical activity.

~1! A first difficulty comes from the fact that optical activity is only displayed by dissymmetric~chiral! objects. Themodels used to represent optical activity are therefore racomplicated and require heavy numerical computations.different molecular models suggested different constitut

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equations, relating the vectorsE, D, B, andH, and definingthe macroscopic properties of the medium. A surprisinlarge number of constitutive equations are indeed foundthe literature and actually used. This fact is perhaps the msource of confusion for nonspecialists~and for specialiststoo!. For a review, see Refs.@4–6#.

~2! A more substantial difficulty comes from the fact thoptical activity is a nonlocal property, depending expliciton the molecular sizea. In particular, the circular birefrin-gence of isotropic media~which is proportional to the opticarotation along a path of one wavelength! scales asa/l, asfirst recognized by Boltzmann@7# on the basis of a moleculamodel. The optical rotation per unit length, i.e., the rotatopower, scales therefore asa/l2. The higher order powers othe Boltzmann ratioa/l are generally negligible, sincea!l. For anisotropic media the nonlocality of the dielectrproperties gives rise to spatial dispersion, namely, to thependence of the gyrotropic parameters on both the modand the direction of the light wave vector. The usual frquency dispersion, which is neglected here for the sakesimplicity, gives an additionall dependence to the simpla/l2 law for the rotatory power.

For gyrotropic crystals we have a further characterislength, namely, the spatial periodicityp, and hence a contri-bution scaling asp/l can appear. For the actually availabchiral smectics, this contribution is dominant, sincep@a,and the higher order powers of the Boltzmann ratiop/l playa non-negligible role. For the helical-shaped LC phasesoptical model is universally used where the molecular strture is neglected and the optical properties are definedmeans of a local dielectric tensor«(r ). Since we consideronly short-pitch compounds, this model plays in our theothe role of amesoscopic model.

~3! For periodic media the main difficulty comes from thfact that the eigenwaves within the medium are Bloch wavwhose Fourier components represent plane waves ha

3264 © 1998 The American Physical Society

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wave vectorsk1q, where q is a vector of the reciprocalattice. A macroscopic model is obtained by considering othe zeroth-order Fourier component and neglecting thefects of the higher order ones. Unfortunately, the amplituof the neglected components is not negligible for actual crtals @8,9#. Despite this fact, no experimental evidence opossible role of the neglected Fourier components has bfound to date. To explain this, we recall that the higher orFourier components give rise to a fine structure of the etromagnetic field within the crystal cells, which often hnegligible influence on optical experiments, where onlyaverage~macroscopic! field is of interest.

The above quoted problems are well known and hbeen discussed for a long time in the literature~see, for in-stance, Refs.@8,9#!. We approach here the same problemsthe basis of the fact that a mesoscopic and a macroscmodel for the same medium are now available, both defiby fully analytic and very simple constitutive equations. Afar as we know, nothing similar can be found in the literture, at least for gyrotropic media, and all the previous dcussions concerning the limits of validity of macroscopmodels are based on qualitative or semiquantitative aments. Here we give, instead, a fully quantitative analyOur attention will be concentrated on the possible failurethe macroscopic homogeneous model, considering as exacthe mesoscopicone ~periodic!. As already stated, in somparticular geometries strong discrepancies are indeed foIn all these cases our experiments confirm the full validitythe mesoscopic model.

II. THE MODELS

A. Mesoscopic model

In the mesoscopic model, the chiral smectic is assumebe a continuous periodic medium that is locally uniaxinonmagnetic, and nongyrotropic. The optic axis is parallethe nematic directorn, which is uniformly rotating alongz insuch a way that its components are given by

nx5sin a cos~qz1w0!,

ny5sin a sin~qz1w0!, ~1!

nz5cosa,

wherea is the tilt angle,q52p/p, and the phase constanw0 defines the direction ofn at the planez50, which in ourexperiments is considered as the first boundary ofsample. In the two limiting casesa50 anda5p/2 we ob-tain a nongyrotropic homogeneous medium and a cholesliquid crystal, respectively.

The optical properties of the mesoscopic model are fudefined by the real dielectric tensor

«5«o1«an~z!n~z!, ~2!

where «a is the local dielectric anisotropy, defined as tdifference«e2«o between the principal values of«. Thecomponents of the tensor«(z) are periodic functions ofz,with period equal to the helix pitch.

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The magnetic polarizability of actually available compounds is negligibly small, whereas their local biaxiality agyrotropy are in general non-negligible, having the sameder of magnitude as for usual crystals. However, their inflence on the optical properties of the smectic samples conered here is very small. The intrinsic optical activity~comingfrom the chirality of the individual molecules! gives non-negligible effects only in the limit where the pitch becomcomparable to the atomic size. Even in this limit, it is inteesting to find out and discuss the optical activity comifrom the helical arrangement of the molecules.

B. Landau-type constitutive equationsfor the macroscopic model

A macroscopic model for short-pitch chiral smectics hbeen derived in Ref.@1# in the framework of the theory ospatial dispersion, by making use of the Landau-Lifshitz fmalism @10#. For harmonic plane waves the constitutivequations are written as

D5 ««0E, ~3a!

B5m0H, ~3b!

where the vectorsE, D, B, andH are the effective values othe fields. In a perturbation expansion where only lineterms of the parameters«a

2 and p/l are considered, the effective permittivity tensor is given by

«5S «o 0 0

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«e5«o1«a cos2a, «o5«o~11«e / «e!/2, ~5!

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andm5k/k0 , wherek is the wave vector of the considereplane wave within the sample,k052p/l, and l is thevacuum wavelength. The fact that« explicitly depends onkmeans that the medium displays spatial dispersion.

The effective homogeneous medium defined by Eqs.~4!–~6! is uniaxial and its optic axis is coincident with the helaxis. The real part of the permittivity tensor« displays thetypical structure of uniaxial media with only two scalar prameters«e and «o . A similar behavior is in general exhibited by the imaginary part of uniaxial gyrotropic media~andmore precisely by the crystal classesCn , Dn , with n.2),with two parametersgi andg' related to the components othe vectork parallel and orthogonal to the optic axis, respetively. The absence ofmz5kz /k0 in Eq. ~4! means that, forwhat concerns the considered homogeneous model, theallel componentgi is equal to zero. The optical activity ithus zero for light propagating along the helix axis areaches its maximum value for light propagatingorthogo-nally to the helix axis.

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3266 PRE 58HUBERT, JAGEMALM, OLDANO, AND RAJTERI

The gyrotropy parameterg' is maximum fora545°, andzero fora50° anda590°. Within the approximation of themodel, the optical activity of cholesteric LCs is therefozero, since their tilt angle is exactly 90°.

C. Post-type constitutive equations

The constitutive equations~3! are indeed very simple, butheir simplicity has rather heavy counterparts. In particuwithin gyrotropic media a new term must be added toE3H to obtain a Poynting vectorS having the meaning opower flux density@11,12#. The continuity of the normacomponent of this vector at the sample boundaries impliediscontinuity of the tangential components of the field vetors E or H. New boundary-type equations are thereforequired to fully define the optical properties of the sample

A discontinuity ofB @and, from Eq.~3b!, also ofH] im-plies the presence of a surface density currentj s. The pres-ence of such a current, related to a bulk magnetizationM , isstrongly suggested by the fact that within any dielectric hethe polarization current gives rise to a magnetic dipole mment. The simplicity of the Landau formalism is due to tfact that the magnetization currentjM5¹3M is incorpo-rated into the main currentj P5]P/]t. In the bulk, this isalways allowed, since Maxwell’s equations remain uchanged under the transformation]P/]t5]P/]t1¹3M , ifwe accordingly define new vectorsH andD. Obviously, thistransformation gives a different set of constitutive equatioOne of the difficulties quoted in the Introduction is relatedthis fact, namely, to the difficulty in separating the effects]P/]t and¹3M .

A possible way to escape the above difficulty is to impothe continuity of the Poynting vector at the boundary surfaIt is, however, preferable to make use of a different setconstitutive equations. In the following we make use of tequations derived by Post@13# under the requirement of relativistic covariance of all the equations used to describeelectromagnetic field, without using any microscopic moels. For nonmagnetic lossless materials the Post equacan be written as

D5««0E1x«0cB, ~7a!

H52x†«0cE1m021B. ~7b!

In the bulk, the equation systems~3! and ~7! becomeequivalent by assuming

«5 «1R2, ~8a!

x5 iR, ~8b!

where

R5S 0 0 0

0 0 0

0 0 g'D . ~9!

For nonchiral mediax50 and «5«. The two sets ofequations then become identical, and in the absence

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losses, the permittivity« is real and symmetric. The opticaactivity of gyrotropic lossless media is therefore relatedthe pseudotensorx and to the imaginary part of the tensor«,which in this case is complex and Hermitian. The bulk costitutive Eq. ~7!, in contrast to Eqs.~3!, are self-consistenwith the usual assumptions of continuity for the componeof the vectorsB, E, D, andH ~tangential continuity forE andH and normal continuity forB andD!.

III. VALIDITY LIMITS OF THE MACROSCOPIC MODEL:BULK EFFECTS

To test the validity of the macroscopic model, we haconsidered the transmittance and the reflectance of a LCconfined between two parallel planes, and compared theues given by the macroscopic and mesoscopic models,particular attention to the polarization properties relatedthe optical activity. For the mesoscopic model, we have csidered samples with the helix axis orthogonal to the bouary planes~homeotropic geometry! and samples with the helix axis parallel or obliquely oriented with respect to thboundary planes~phase grating geometry!. In the first case,the periodic structure can be considered as an anisotrstratified medium and the computations are performedmaking use of the basic Berreman formalism@14#. In theother cases, an extended Berreman formalism, developefind out the efficiencies of anisotropic gratings@15,16#, hasbeen used. Forp.l the structure indeed acts as a phagrating, giving rise to diffracted beams. By decreasingpitch, the diffraction angle of the higher order diffractewaves increases, and when the angle reaches 90° twaves become evanescent. For small enoughp values onlythe zeroth-order transmitted and reflected beams survHowever, the presence of evanescent waves cannot beglected in computations.

The most important discrepancies between the mecopic and macroscopic models are related to boundaryeffects and will be considered in Sec. IV. In this sectionconsider some bulk effects, related to the fact that our mroscopic model neglects higher order terms inp/l and«a .

We recall that gyrotropic crystals display pure optical rtation only for light propagating along the optic axis. In another direction the medium transforms linear into elliptic plarization, but the presence of the gyrotropy has only smeffects on most optical properties.

Let us first consider the case of a light beam parallel tooptic axis. The macroscopic model does not give opticaltation, sincegi50. This property is quite unexpected, sinfor p.l, cholesterics and chiral smectic LCs display a hupseudorotatory power for light propagating along to the heaxis, according to de Vries’ equations@17#. A residual circu-lar birefringence for light propagating along the helix axsurvives forp,l, but it decreases quickly by decreasing tratio p/l. More precisely, the rotatory power along the hedecreases two orders of magnitude if the internal wavelenchanges fromp ~namely at the Bragg reflection band! to 2p.For higherl values, it decreases asl24, and becomes negligible to any practical purpose forl.5p. In the limit p!l, the quantitygi is in any case not exactly zero, buscales as (p/l)3.

To test the effects ofg' , we must consider light beam

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PRE 58 3267OPTIC MODELS FOR SHORT-PITCH CHOLESTERIC . . .

orthogonal or obliquely oriented with respect to the opaxis. These cases require a more complex analysis. Wecomputed the ellipticity and the directions of the ellipse axof the transmitted and reflected beams for linear input poization. Only for thin enough samples the polarization of ttransmitted beam is nearly linear and the rotation of the mjor axis is simply related tog'. It is therefore convenient toconsider rather thin samples, a fact that, however, doesallow for a good separation of the bulk properties from tboundary effects.

The bulk properties are better evidenced by making usthe phase grating geometry. Therefore, we have considthis geometry to compare the optical rotations given bymacroscopic and mesoscopic models. The computathave been performed for different values of the optic paraeters (p/l, «a , tilt angle, input polarization, and incidencangle!. The results can be summarized as follows.

For the bulk properties, the macroscopic model wowell for any realistic value of«a , despite the fact that onlyterms up to«a

2 have been considered. For what concernspitch, the range of validity of the macroscopic model in tgrating mode geometry is unexpectedly large, going upthe p value corresponding to the first Bragg peak, as shoin Fig. 1. It is obvious that in the Bragg regime, where tBloch waves of the periodic medium represent standwaves, the macroscopic~homogeneous! models lose anymeaning. Near the Bragg bands the Boltzmann scaling lano longer valid and the role of the higher order terms inpower series expansion of the function«(p/l) becomes im-portant~similar behavior is found in crystals displaying ci

FIG. 1. Rotation angles of the major axis of the output ellippolarization for normally incident light linearly polarized along~a!and orthogonally~b! to the helix axis, and corresponding ratiobetween the intensity along the minor and the major axes ofellipse @~c! and ~d!#, for the helical periodic medium~solid lines!and for its macroscopic model~dotted lines!. The sample is in thegrating mode geometry~helix axis parallel to the boundary plane!with tilt angle a545° and thicknessd52l. Here, and in all thesubsequent figures, the principal refractive indices arene51.66 andno51.5.

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cular dichroism around an absorption line!. At lower p val-ues, our computations suggest that the first neglected terthe expression ofg' is the one scaling as (p/l)3, whichgives again negligible effects forp,l/5.

IV. VALIDITY LIMITS OF THE MACROSCOPIC MODEL:BOUNDARY EFFECTS

In the following we will consider two types of boundareffects, having different origins.

~a! Matter is intrinsically discontinuous and a sample hnever a defined ‘‘boundary surface.’’ This is a trivial obsevation, but must not be neglected at least for what concernonlocal property as optical activity. In our macroscopic dscription it is convenient to define a thin boundary ‘‘transtion layer’’ where the optical properties are changing gradally.

~b! The macroscopic models for periodic media neglthe higher order Fourier components of the Bloch wavwhose effects are in principle very long ranged and cannoaccounted for by simply considering a thin transition layerthe boundaries~moreover, it is not self-evident that a description in terms of bulk properties and boundary conditiois allowed for crystals!.

In the framework of our theory a quantitative accountboth these effects can be done. The computations haveperformed for different values of the parameters of intereincident angle, sample thickness, pitch, and wavelength.have found that effects of type~b! are indeed present if theboundary layers are orthogonal or nearly orthogonal tohelix axis, i.e., in the homeotropic or nearly homeotropgeometry.

A. Phase grating geometry and transition layer

The presence of boundary transition layers is well edenced by Fig. 2~a!, which refers to a sample with the heliaxis parallel to the boundary planes. The homogenemodel gives an optical rotation that increases linearly wthe sample thickness. The mesoscopic model, instead, gpractically no rotation if the sample thickness is small wrespect to the helix pitch. To account for this effect in tframework of a macroscopic description, we must assuthat g' is gradually changing from zero~at the sampleboundary! to the bulk value. Nothing similar has been foun

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FIG. 2. Rotation angles~a! and axes ratios~b! for the sameoptical geometry of Fig. 1, as a function of the normalized samthickness, for the periodic medium~solid line! and its homogeneousmodel ~dotted line!, for a530°, p5l/3, and TM input polariza-tion.

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3268 PRE 58HUBERT, JAGEMALM, OLDANO, AND RAJTERI

for the parameters«e and «o . For thin samples, large discrepancies between the mesoscopic and the macroscmodels are only found for the optical properties related toparameterg' , as, for instance, the output ellipticity for inpuTM polarization@Fig. 2~b!#.

This fact has interesting consequences for what concthe thin film interference patterns. We expect that the prence of the transition layers at the sample boundarieslower the amplitudes of the interference fringes. In fact,the phase grating mode geometry the interference pattcomputed on the basis of the exact mesoscopic modelgenerally attenuated and disappear completely in somecal conditions@18#. We have not been able to experimentatest these effects because the surface interactions caushelix to unwind in thin cells and because our samples arany case not perfectly monodomain.

B. Homeotropic geometry and breaking downof homogeneous model

The phases of the Fourier components of the Bloch waare at any point related to the phasew5w01qz of the peri-odic helical structure of the medium at the same point. Tphase constantw0 plays therefore the role of a control parameter for these phases. In the homeotropic geometryphasew is constant at each boundary plane and a changthe phasew0 has the same effect as a rotation of the whsample with respect to the incidence plane of the light.therefore expect a dependence of the optical properties oactual periodic medium onw0 , whereas the properties of thhomogeneous model are independent of the parameterw0 ,which does not appear in its constitutive equations~3!–~9!.

The dependence of the rotation angle onw0 given by themesoscopic model is shown in Fig. 3 for an incidence anof 40°. This dependence is very strong for any incidenangle. Surprisingly,it does not decrease by increasing ththickness of the sample, as shown in Fig. 4. This dependenmust therefore be considered as a true breaking down ohomogeneous model, not as a simple boundary effect. It

FIG. 3. Rotation angle in the homeotropic geometry as a fution of the normalized sample thickness and of the phase conw0 ~which defines the direction of the local optic axis at the inpplane!, for p5l/3, a530°, and incidence angle of 40°.

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‘‘boundary’’ only in the sense thatw0 defines the direction ofthe local optic axis at the boundary planez50 ~and at thesecond boundary plane if the sample contains an intenumber of pitches!. By decreasing the ratiop/l we obtaincurves practically identical to the ones plotted in Figs. 3 a4, but with smaller rotation angles. The optical rotation aits derivative with respect tow0 scale asp/l, according tothe Boltzmann law.

Discrepancies between the mesoscopic and macroscmodels are also found in the reflectance curves from a seinfinite sample. For a TM~or TE! polarized incident beamthe homogeneous model gives a TM~or TE! polarized re-flected beam since the macroscopic optic axis lies inincidence plane. The mesoscopic model, instead, givegeneral an elliptic polarization, since the local optic axnear the boundary planes do not lie in the incidence plaThe dependence of the reflectance onw0 appears here aobvious if we consider that a change ofw0 is equivalent to arotation of the local optic axes at the boundary planes. Inestingly, aw0 change has strong effects on the optical proerties related to optical activity, whereas it has generasmall effects for the other optical properties. In fact, the tointensity of the reflected beam is practically independentw0 , as shown by Fig. 5~a!, where the curves computed witdifferent values ofw0 are practically coincident. On the contrary, the polarization properties depend strongly onw0 , asshown by the curves plotted in Figs. 5~b! and 5~c!, whichagain scale asp/l. Figure 5~d! shows a neww0-dependentboundary effect, consisting of a small dephasing betwreflected and incident beams~we recall that the dephasing fohomogeneous lossless media is equal to zero or to 180°

The failure of the homogeneous model can also be unstood on the basis of the following argument: considesample in the phase grating oblique geometry. The graconstant~i.e., the spatial period of the grating! is equal top/sind, whered is the angle between the helix axis and tboundary normal. For small enoughd values, the gratinggives a set of diffracted beams~which do not satisfy theBragg law since the grating works in the Raman-Nathgime!. Obviously, no homogeneous model can be usedapproximate a grating.

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PRE 58 3269OPTIC MODELS FOR SHORT-PITCH CHOLESTERIC . . .

A failure of the homogeneous model is expected for aperiodic structure as, for instance, for three-dimensiocrystals. The fact that no homogeneous model is able to faccount for the optical properties of crystals is in a seobvious. Here, we focus the conditions in which the homgeneous models fail.

It is, however, to be observed that the failure of the mroscopic model involves rather small effects and is partilarly evident only in the cases where both optical activ~i.e., a nonlocal property! and the higher order Fourier components play an important role. Further, we expect thataveraging of the reflectance and transmittance overw0 ,which is equivalent to average over the phase constantthe neglected Fourier components, could restore the valiof the homogeneous model. Theaveraged curvesare in factrather well fitted by our homogeneous model, as shownFigs. 4 and 6~indeed, in Ref.@1# the homogeneous modehas been defined by averaging overw0 the transfer matrix ofplanar samples, i.e., the matrix relating the field vectorsEand H at the sample boundaries!. This fact is of the mostimportance from the experimental point of view. Mansources of averaging are always present in actual expments~as, for instance, nonperfect flatness of glasses, inmogeneous anchoring of the smectic structure, and mgenerally nonperfect periodicity due to various defec!,which are expected to become more efficient by decreathe pitch of the helical structure or the spatial period of otperiodic media. So, we draw the conclusion that homoneous models for crystals could work well in actual expements, and even in optical geometries where they are cceptually wrong. The main aim of our experiments is to tthis important point.

Finally, we notice that the boundary effect related to tpresence of a transition layer does not appear in the hom

FIG. 5. Polarization parameters~intensity, phase, rotation of thmain ellipse axis, and axes ratio! of the beam reflected by a seminfinite sample with the helix axis orthogonal to the boundaplanes, as a function of the incident angle for input TM polarizatand for different values ofw0: ~a! 30°, ~b! 90°, ~c! 150°, ~d! 210°,~e! 270°, and~f! 330° (p5l/4, a545°, andd5l).

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tropic geometry~Fig. 6!. This can have important implications for nonperfect crystals if the long range helical ordelost ~owing to some source of randomness!, with differentcorrelation lengths in different directions. By comparinFigs. 2~a! and 6, we expect that the optical activity is pratically unchanged if the order is lost along the helix diretion, whereas it decreases if the order is lost in the orthogodirection.

V. EXPERIMENTS

The experiments were performed with the ZKS-25compound. This compound is in the chiral smectic-C phaseat room temperature, with a pitch of about 0.17mm, tiltangle a526°, and effective optical anisotropyDn50.16.The cells were prepared by enclosing the liquid crystaltween two identically treated glass plates, which were serated with either evaporated SiO films or Mylar spaceSamples of different thickness were prepared (d54, 2, and1.6 mm) in both the homeotropic and the phase gratingometries, i.e., with the helix axis perpendicular or parallelthe boundary planes, respectively. The homeotropic orietion was obtained by depositing a lecithin layer on the glplates. To obtain the phase grating orientation, a unidirtional mechanical shear of the cell substrates was appliepolarization microscope was used to control the qualitythe samples and the direction of the helix axis. For themeotropic geometry, the helix axis is orthogonal to tboundaries only where the sample appears as uniforblack between crossed polarizers. A more complex analis required to find out the direction of the helix axis, i.e., tmacroscopic optic axis, in the phase grating geometry.have first looked for the direction of the input polarizatiowhere the transmittance of the sample between crossedlarizers reaches a minimum. The parallelism between thetic axis and the boundary planes was then ensured bysidering the symmetry of the transmittance curves undesign change of the incidence angle. The measurementsperformed with a 10 mW He-Ne laser (l50.6328 mm),which was linearly polarized with a prism polarizer. Th

FIG. 6. Rotation angles as a function of the normalized samthickness averaged overw0 ~full line! for the same optical geometrand the same parameters as in Fig. 3, and the corresponding agiven by the homogeneous model~dotted line!.

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3270 PRE 58HUBERT, JAGEMALM, OLDANO, AND RAJTERI

outcoming light from the sample was analyzed with a GlaThomson polarizer, which could be rotated by a step mowith a resolution of 0.05°. The light was detected withsilicon photodiode whose output signal was sent to a variagain amplifier. The position of minimum and maximum itensities, obtained by rotating the analyzer, correspond todirections of the minor and major axis of the output polization ellipse, respectively, and the intensity ratio givessquare of the axes ratio. The angle between the main axthe ellipse and the polarization direction of the incident ligis considered as the rotation angle.

Curves giving the square of the axes ratio~ellipticity! andthe rotation angle as a function of the incident angle andinput polarization direction were obtained and comparedthe theoretical curves given by the mesoscopic model~ex-act!. A best fit procedure of these measurements is not ontest of the validity of the model, but also a way to obtainthe optical parameters of the liquid crystal cell~thickness,pitch, tilt angle, birefringence, and phase constantw0). Infact, our computations reveal that the output ellipticity shoa strong dependence on all these parameters. An evidenthis property is shown in Fig. 7, which is a three-dimensioplot of the ellipticity as a function of the phase constantw0and of the normalized thicknessd/p for light at normal in-cidence on a perfect homeotropic sample. In this caschange inw0 can be obtained by rotating the input polariztion or the sample around its normal, equivalently. Howevfurther experiments and an improvement of the preparaprocedure of the samples are required for a precise detenation of the material parameters. Our homeotropic samare instead good enough to evidence the important efdiscussed in Sec. IV B, i.e., the fact that a simple rotationthe sample around its macroscopic optic axis drasticchanges the transmittance. This is shown in Fig. 8, whrefers to a sample with nominal thicknessd54 mm. Theexperimental points obtained by a sample rotation~circles!are rather well fitted by the theoretical curves computedthe basis of the mesoscopic model, by assuming a perfe

FIG. 7. 3D representation of the ellipticity~squared ratio of theellipse axes! as a function of the phase constantw0 and of thenormalized thickness. The parameters are the same as in Fig.

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monodomain sample withd54.14 mm and nominal valuesof the other parameters~solid lines!. The estimated experimental errors~mainly related to the sample defects! are ofthe same order of magnitude as the differences betweenoretical and experimental curves. We have therefore atsame time proved the rather good quality of the sampreparation procedure~remember that any source of defecalways gives some type of averaging overw0), the validityof the mesoscopic model and the failure of the macroscomodels.

VI. CONCLUSIONS

A great emphasis has been given in the literature tooptical properties of cholesteric and chiral smectic liqucrystals for light propagating along the helix axis, since thoproperties are at the same time very interesting and canexpressed by simple analytic expressions@17#. The opticalproperties of the now available short-pitch cholesteric achiral smectics are even simpler for any direction of the ligbeam, since the periodic helical structure can be consideas a homogeneous uniaxial medium~here referred to as themacroscopic model!, whose optical properties can be dscribed by the same constitutive equations valid for uscrystals@1#. Two different and equivalent sets of constitutivequations have been considered, making use of the Lanand Post formalism. In the first set, the bulk propertiesdescribed by a complex tensor« and the tangential compo

nents ofH are discontinuous. So we have preferred to mause of the Post equations, where the bulk properties arescribed by two real tensors« andx, and the boundary conditions are expressed by the usual requirement of continfor the tangential components of the vectorsE andH and forthe normal components of the vectorsB andD. In both sets

. FIG. 8. Polarization parameters~ellipticity and rotation angle! ofthe transmitted beam as a function of the orientationw0 of a ho-meotropic sample, for incident TM polarization: experimental d~circles! and theoretical fit~solid lines! for nominal values of thematerial parameters and with a best fit thickness of 4.14mm.

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only three scalar parameters appear, which are simply relto the parameters defining the periodic structure~referred toas the mesoscopic model, which in our theory plays theof the microscopic models for usual crystals!. In general, thesymmetry class requires four parameters, which are thedielectric constants«e5 «' and «o5 « i appearing in«, andthe two corresponding gyrotropic parametersgi andg' ap-pearing inx. In our case,gi is equal to zero. This means ththe well known pseudorotatory power of cholesteric and cral smectic-C liquid crystals for light propagating along thhelix axis goes to zero in the limit of short pitches. Toptical rotation is only given byg' , and is therefore maxi-mum for light propagating orthogonally to the helix axiThe dependence ofg' on the pitchp and on the tilt angleais such that the rotation rate scales asp/l2 and is propor-tional to sin2(2a). The optical activity is thus practically absent in short-pitch cholesterics, wherea590°.

The central point of our research is the discussion oflimits of validity of macroscopic models, by consideringexact the mesoscopic one. The main interest of the reseis connected to the difficulty in inserting optical activityany macroscopic description of matter. In fact the optiactivity is one of the optical effects~the most important one!arising from the molecular structure of matter, and is thefore hardly accommodated in the framework of a ‘‘cotinuum’’ theory, which neglects the molecular structure. Whave shown that this difficulty is something more thanhypothetical paradox and focused on the point wheremacroscopic description can fail. We recall here the mimportant points.

The macroscopic model has been derived by considethe limit p!l, but for the nongyrotropic properties it workwell in any optical geometry up top'l/5. Instead, it failsseriously for the gyrotropic properties of chiral smecticsthe homeotropic geometry,for any value of the pitch and foany thickness of the sample. According to any homogeneoumodel, a rotation of the sample around the macroscopic oaxis ~which is orthogonal to the boundary planes! has noeffect on the transmittance and reflectance. On the contra rotation of the actual periodic sample changes the amtudes of the transmitted and reflected beams and couldchange the sign of the optical rotation~this important facthas been proved both theoretically and experimentally!. Thismeans that no homogeneous model is able to describeoptical activity of the periodic medium. It is already know@19# that different macroscopic models for gyrotropic medcan give different reflection coefficients, a fact that reflethe difficulty of finding out the correct boundary conditionfor such models. According to our computations, the discrancies among the different models are generally smalcompared to the discrepancies between any one of the mels and the actual periodic medium.

Interestingly, by averaging the amplitudes of the transmted and reflected waves given by the mesoscopic modelthe rotation anglew0 , we obtain practically the same ampltudes as those given by the macroscopic model. The fai

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of the macroscopic model is due to the fact that it neglethe higher order Fourier components, since the rotation shthe phases of all these components. An averaging overrotation angle is therefore equivalent to an averaging othe phases of the neglected components, a procedurerestores the validity of the macroscopic model. It is impotant to observe that other sources of phase averagingalways present in actual~nonperfect! crystals, which againcould restore the validity of the macroscopic model. Onbasis of the above discussion we can presume that the mroscopic models universally used in optics for usual cryscould similarly fail if the crystal boundaries are coincidewith the lattice planes and the surfaces are perfectly flat wout too many defects.

We conclude with some further comments. A few yeaafter the discovery of optical activity, Fresnel was ableexplain the effect by simply assuming circular birefringenci.e., different refractive indices for left and right circular polarizations. By analogy with the helical morphology of cicularly polarized waves, he suggested a helixlike structfor optically active media@20#. The advent of x-ray diffrac-tion techniques offered the possibility to experimentally tFresnel’s hypothesis for usual crystals. A different waytest the same hypothesis is given by cholesteric liquid crtals, whose morphology is exactly the same as the onecircularly polarized light. As well known, a cholestersample is indeed able to rotate the polarization plane of ligbut this effect is not the same as for usual optically actmedia@21,22#. Further, the rotation becomes practically zeif we extrapolate the cholesteric pitch to molecular dimesions. Only in the last years it has been shown that the hesmecticlike phases display true rotatory power. Fresnel’spothesis is now confirmed, but certainly not in the senseexpected, since the helical structure gives optical rotationlight propagating orthogonally to the helix. Once again, tstudy of liquid crystals greatly helps our understandingbasic physics.

We have considered chiral smectic-C liquid crystals,which represent the first example of media whose optrotation is described by simple analytic expressions. Wepect that similar expressions will be found for other medand in particular~1! for smecticlike structures, similar to thcholestericlike structures, recently studied by Lakhtakia aco-workers@23,24#; ~2! for helixlike structures appearing innematics undergoing a Meyer@25# or Pikin @26# type Freed-ericksz phase transition, where the direction of the helix aand the grating wave vector are orthogonal~whereas inSc*they are parallel!; ~3! for the recently discovered twist graiboundary~TGB! phases@27,28#, which display discrete rotation of the optic axis along a given direction;~4! for two- andthree-dimensional helixlike structures, as, for instance,ones appearing in the Blue@29# and in the TGBC* @27#phases. The existence of a second helical ordering in thtwo last structures gives rise therefore to an intermedcomplexity between the one-dimensional and the thrdimensional periodicity, allowing for a further step towarthe optics of usual crystals.

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3272 PRE 58HUBERT, JAGEMALM, OLDANO, AND RAJTERI

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