New nonlinear optical processes in liquid crystals

15 June 2000

Optics Communications 180 Ž2000. 349–359 www.elsevier.comrlocateroptcom

New nonlinear optical processes in liquid crystals S.K. Srivatsa 1, G.S. Ranganath) Raman Research Institute, Bangalore 560 080, India Received 22 February 2000; accepted 11 April 2000

Abstract We have worked out some consequences of the optical nonlinearities due to laser induced changes in the order parameter of a liquid crystal. The change in the order parameter can be affected by laser induced suppression of the director fluctuations in liquid crystals and or changes in the tilt angle of smectic liquid crystals. Both the processes lead to well known nonlinear optical effects like self-focusing, self-divergence, self-phase modulation, wave mixing, and so on. In addition, we predict some new phenomena like self-iridescence and new types of optical spatial solitons. In the case of chiral liquid crystals in the short wavelength limit the laser beam induces a change in the twist and at the long wavelength edge of the Bragg band it leads to temporal oscillations in the twist and the transmitted intensity. In smectic liquid crystals interesting periodic structures in a standing laser wave are to be expected. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 61.30.Gd; 61.30.Jf; 42.70.Df Keywords: Liquid crystals; Director fluctuations; Director reorientation; Tilt angle; Optical solitons

1. Introduction Nonlinear optics of liquid crystals has attracted a great deal of attention recently. The director reorientation results in a large optical nonlinearity w1–3x. It appears to have not been realised by workers in the field that there can be additional nonlinear processes in a liquid crystal. It is well established that an external static magnetic or electric field can suppress director fluctuations in nematic liquid crystals w4x. It is natural to expect the same thing to happen even in a suitably polarised laser field. We point out here )

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that laser induced suppression of the director fluctuations can lead to considerable optical nonlinearities in liquid crystals. This nonlinearity is entirely independent of that due to director reorientation and it affects the order parameter. Its magnitude is not so large as the one due to director reorientation but is still much larger than the classical optical Kerr nonlinearity found in crystals and liquids. Further, on its own it not only results in most of the familiar nonlinear effects but also leads to some new effects. To highlight this process we consider situations wherein the usual director reorientation process is strictly absent. It is well known that a majority of liquid crystals are transparent and they absorb only in the deep ˚ .. Hence, the absorption is ultraviolet Ž l - 2500 A

0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 7 0 6 - 9

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S.K. SriÕatsa, G.S. Ranganathr Optics Communications 180 (2000) 349–359

negligible in the visible part of the spectrum. Therefore we do not consider the laser absorption and its associated thermal effects in this study. It may be mentioned here that in the study of optical Kerr effect in the isotropic phase of the liquid crystals the laser absorption is again not considered. A partially polarised light beam propagating in a nematic with its more intense component parallel to the director also suppresses the director fluctuations. We obtain in this case a self-focusing of this component and a self-divergence of the incoherent orthogonal component. In the presence of a standing wave we get Bragg reflection or iridescence due to the laser induced periodic structure. We have also worked out optical spatial solitons permitted by this new nonlinearity. We find that only a bright soliton solution is permitted, with an entirely different structure. In fact this is in reality a solitary wave and not a soliton. In chiral liquid crystals in the short wavelength limit, a permitted eigen mode for propagation along the helix axis has its electric vector parallel to the local director. Hence, locally we again get suppression of the director fluctuations. This in turn leads to an increase in the pitch. The pitch increases as the square root of the intensity. Understandably, in this geometry a standing wave leads to a modulation of the twist. When the wavelength of the laser is at the long wavelength edge of the Bragg band, the same process comes into play since the net field is along the local director though of diminishing strength as we go down the twist axis. This results in temporal oscillations in the pitch and the transmitted intensity. The effect of laser suppression of the director fluctuations on light scattering from nematics and cholesterics has also been considered. In smectic liquid crystals there can be an entirely different nonlinear process. The laser beam can change the molecular tilt with respect to the layer normal. The refractive index change due to this effect grows linearly with intensity. This process will be very dominant near a smectic A to chiral or achiral smectic C phase transition. In these liquid crystals this process will have to be considered along with the process of laser suppression of in-plane director fluctuations. This new process may enhance or bring down the effects due to the latter. Further, on its own it can lead to some new effects. For example, in a standing wave with polarisation paral-

lel to the layers the laser induced periodic structures can consist of alternating smectic A and chiral or achiral smectic C blocks. On the other hand, if the polarisation were to be parallel to the layer normal in chiral systems we obtain a two dimensional periodic structure. In chiral smectics near such a transition, for propagation along the twist axis, the pitch variation with laser intensity need not be monotonic w5x. Finally in smectic C different types of optical solitons are allowed since both the nonlinear processes can operate.

2. Laser induced suppression of director fluctuations At a finite temperature, in a liquid crystal, there are always thermal fluctuations in the preferred direction of alignment of molecules also called the director. These thermal fluctuations can be reduced by lowering the temperature but this may often lead to even a change of phase in the liquid crystal. We can consider other means of suppressing the director fluctuations. It was pointed out long ago by de Gennes w4x that the director fluctuations in a nematic liquid crystal can be suppressed by a static magnetic field applied along the director. Expectedly, this enhances the dielectric anisotropy, the increase being proportional to the strength of the applied field. One of the consequences of the reduction in director fluctuation is that the order parameter of the system increases. Due to an increase in order parameter we find a decrease in light scattering from nematics. This prediction was experimentally verified later by Malraison et al. w6x. A similar process also operates in cholesterics. In this case the enhancement in dielectric anisotropy implies an increase in the twist elastic constant. Hence, the cholesteric pitch increases. This was experimentally established by Belyaev et al. w7x. These authors observed a red shift in the Bragg band of a cholesteric, with negative dielectric anisotropy, by the application of a static electric field along the twist axis. Naturally, suppression of director fluctuations can be expected even in the electric field of an intense laser beam. We first address ourselves to this effect in the non-absorbing nematic and cholesteric liquid crystals.

S.K. SriÕatsa, G.S. Ranganathr Optics Communications 180 (2000) 349–359

We consider an aligned nematic sample with the undisturbed director parallel to the z-axis i.e., n 0 s Ž0, 0, 1.. Following de Gennes w4x it is easy to show that the director fluctuations, of a wavevector q, in a linearly polarised laser beam with its electric vector parallel to n 0 is given by: ² n2x Ž q . : s ² n2y Ž q . : s

k BT 2

K Ž q q jy2 .

.

Ž 1.

In real space this leads to: ² n 2x : s ² n 2y : s

k BT

1

2p 2 K

l

p y 2j

,

² n 2z : s 1 y 2² n2x : .

Ž 2.

(

Here j s 8p KcrIea , l is of molecular dimensions, c is the velocity of light, K is the curvature elastic constant, I is the intensity of a laser beam, eaŽs e I ye H . is the optical dielectric anisotropy, T is the absolute temperature, and k B is the Boltzmann’s constant. From equation Ž2. we can easily calculate the modified optical dielectric constant parallel to the director and it is given by: ² e I : s e I0 y

2 K BT

p Kl

q

ea3r2 k B T 2'16p 3 cK 3

'I ,

Ž 3.

where e I0 is the optical dielectric constant along the director in the absence of fluctuations. Thus in the presence of a laser beam the correction term is dependent on the square root of the intensity. Further, though the third term linearly increases with T, yet at any temperature the second term is greater than the third term. Hence, an increase of temperature always reduces this dielectric constant. These and similar considerations apply to smectic liquid crystals as well. Smectic A liquid crystals have molecules packed in layers with their preferred direction of alignment along the layer normal. In this system only splay fluctuations in the director are suppressed. In smectic C liquid crystals, the molecules are again packed in layers but the director is at an angle to the layer normal. Here the in-plane fluctuation of the c-director- which is the projection of the director on to the layers, are suppressed by the laser field. Now we consider the same effect in a cholesteric liquid crystal which can be looked upon as a nematic

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with a uniform director twist along a direction perpendicular to the nematic director. The various regions in the wavelength regime for light propagating parallel to the twist axis are: Ži.: l < D m P Ž D m s local birefringence, P s Pitch of the structure. In this limit the eigen states have their electric vectors either parallel or perpendicular to the local director eÕerywhere in the medium. This limit is also known as the Mauguin limit or the adiabatic limit w4,8x. Žii.: D m P - l - m P Ž m s average refractive index. In this limit the eigen states are right and left circular polarised waves propagating with different velocities. Here the system exhibits a very large optical rotation. This limit is known in the standard literature as the de Vries limit w4,8x. Žiii.: m P y Ž D m Pr2 . - l - m P q Ž D m Pr2 . In this case the wavelength is in the Bragg band. Eigen states are again right and left circularly polarised waves. The circularly polarised wave that has the same sense as the helix suffers total reflection. The other circularly polarised wave goes through the medium w4,8x. Živ.: l 4 m P Here also the right and left circularly polarised waves are the eigen states. Again the system exhibits optical rotation but of opposite sign and is very small compared to that found under Žii. w4,8x. In this letter we have considered the first case i.e., the pitch of the cholesteric satisfies the limit Ži.. Also we consider the eigen state for which the electric vector is everywhere parallel to the local director. This can be easily realised experimentally. In other words the electric vector follows the director twist inherent in the cholesteric. It may be mentioned in passing that it is in this limit that the twisted nematic displays work. In cholesteric liquid crystals there are two modes of fluctuations viz., Ži. the in-plane fluctuations also called the twist mode, and Žii. the out-of-plane fluctuations also called the umbrella mode. The amplitude of the fluctuations of a wavevector q in the first mode is given by: ² d nf2 Ž q . : s

k BT 2

K Ž q q jy2 .

,

Ž 4.

S.K. SriÕatsa, G.S. Ranganathr Optics Communications 180 (2000) 349–359

352

while for the second mode it is given by: ² d n 2z Ž q . : s

k BT 2

K Ž q q q02 q jy2 .

.

Ž 5.

Here d nf and d n z are respectively the amplitudes of director fluctuation in the two modes and q0 is the wavevector of the helix. We note that the twist mode is similar to a mode in a nematic but in the umbrella mode the fluctuations are sensitive to the inherent pitch. In case of smectic C liquid crystals only the suppression of in-plane fluctuations in the c-director are relevant. This is given by equation Ž1. or Ž4.. It is important to estimate the magnitude of this effect before we embark upon its implications. From the statistical theory of the nematic state we know that the change D S in the order parameter, D ea in the optical dielectric anisotropy and D K in the elastic constants are related to the relative change in the order parameterŽ S .. In fact, D SrS s D earea s D Kr2 K. Assuming, K s 10y1 2 N, T s 300 K, e I s 2.89, e H s 2.25 and the laser intensity of I s 10 5 kWrm2 Ž10 kWrcm2 ., we find that the relative change in ea or K is of the order of 10y4 . The correction to optical dielectric constant thus is quite high. Only an applied magnetic field as high as 10 5 G can effect the same amount of change in ea or K. Further, at these intensities the optical nonlinearity is greater by three orders of magnitude compared to the normal Kerr effect in isotropic phase of liquid crystals w9x. It is thus meaningful to work out the nonlinear optical effects due to this process. 2.1. Nematic liquid crystals 2.1.1. Self-focusing and self-diÕergence We first consider effects of laser suppression of the director fluctuations in a nematic. Let the sample be illuminated by a parallel beam of linearly polarised light with electric field parallel to the nematic director. Further, the beam has, say, an intensity profile with a central peak. As a consequence of suppression of the director fluctuations, the refractive index along the director increases in proportion to the local electric field strength. Thus the medium acquires a greater refractive index at the center of the beam than at its periphery. Hence, the incident plane wavefront gets distorted to a concave shape so that

the beam is self-focused on propagation through the medium. It is easy to show that we get self-focusing in many other geometries and even when ea is negative. We get a very interesting result when a partially polarised light beam is incident on the medium. It is well known that a partially polarised light can be decomposed into two completely polarised orthogonal but incoherent beams. Let the more intense component have its electric vector parallel to the director. The weak incoherent orthogonal component is ineffective in suppressing the director fluctuations. Then for a parallel beam with a central peak intensity profile the refractive index for the intense component is again a profile with a central peak. Hence, this component is self-focused on propagation through the medium. On the other hand, the refractive index profile for the orthogonal incoherent electric field will have a central dip. Therefore, this lateral component exhibits self-divergence as it propagates through the medium.

2.1.2. Self-iridescence We now work out the effects of the boundary in the case of a finite sample. We assume the laser beam to propagate normal to the boundary with its electric vector parallel to the director. The rear boundary reflects part of the incident light. The reflected light interferes with the forward propagating light. This sets up a standing wave inside the medium. The intensity of the standing wave at the antinodes is four times the incident intensity of the reflected component and zero at the nodes. As the intensity of incident light increases the intensities at the antinodes also increase. When the intensity is sufficiently high the director fluctuations are considerably suppressed at the antinodes. Thus we get a periodic change in the refractive index due to a periodic variation in the intensity of the standing wave. Interestingly, the induced periodicity also satisfies the Bragg condition for reflection of the incident wave. This process further increases the reflected component. When the intensity of the laser beam is increased further the suppression of fluctuation becomes all the more effective and the Bragg reflection from the induced periodic structure increases leading to an almost complete reflection of the incident laser beam, if the sample size is compa-

S.K. SriÕatsa, G.S. Ranganathr Optics Communications 180 (2000) 349–359

rable to the penetration depth of the Bragg reflected wave. This phenomenon can be termed as selfiridescence. It must be stated here that the standing wave induced periodicity through Kerr nonlinearity has been studied theoretically in the isotropic phase of cholesterics near the isotropic-cholesteric transition by Ye et al. w10x. Though they propose a helical structure induced by counter propagating circular polarised light beams, yet they have not realised the importance of this inevitable self-iridescence discussed here. In fact this process will completely wipe out the effect predicted by these authors. 2.1.3. Solitary waÕes In a nonlinear medium, in the slowly varying envelope approximation the Maxwell’s wave equation reduces to a nonlinear Schrodinger equation ¨ ŽNLS. w11,12x. This equation allows for optical spatial soliton solutions. The soliton results from the fact that the self-focusing effect is exactly balanced by the imminent diffraction that the beam undergoes due to self-focusing. At low intensities diffraction dominates while at high intensity self-focusing predominates. Hence at a particular intensity the two opposing effects annul each other. This leads to self-trapping of the laser beam with an unchanging profile. The one-dimensional nonlinear Schrodinger ¨ equation is given by: 2i k 0 n 0

E E Ž X ,Z . EZ

q

E 2 E Ž X ,Z . EX2

q 2 k 02 n 0 n nl Ž I . E Ž X ,Z . s 0 ,

d 2c Ž X . dX2

q 2 k 02 n 0 n nl Ž I . c Ž X . s 0 .

In case of the usual Kerr nonlinearity the function n nl Ž I . takes a simple form n nlŽ I . s n 2 I, where n 2 is a constant and the permitted soliton state is:

cŽ X.s

2a

1

b

cosh Ž 'a X .

ž /

,

Ž 8.

where a s 2 n 0 k 0 h and b s 2 n 0 k 02 n 2 . Here h is a real positive parameter and 1r h is directly related to the soliton width. In practise the width of the soliton depends on the full solution of the differential equation subject to the boundary condition that at Z s 0, the solution matches the input pulse profile. This parameter also determines the amplitude and the period of the soliton along the propagation direction. The nonlinear process of suppression of the director fluctuations gives for the nonlinear function n nl Ž I .:

'

(

n nl Ž I . s k B T

ea3 2p 3 K 3

'I

and the corresponding soliton solution is given by

cŽ X.s

3A

ž / 2B

1 cosh

2

Ž 'A Xr2.

,

Ž 9.

w here A s 2 n 0 k 0 h and B s 2 k B Tn 0 k 02 ea3r2p 3 K 3 . This solution has to be compared with Ž8.. The above nonlinearity is of non-Kerr type and such nonlinearities are not completely integrable w12x, and thus equation Ž9. describes only a solitary wave. We have assumed here a local response of the medium to the laser field i.e., changes in the order parameter at different points are uncorrelated.

( Ž 6.

where E is the envelope of the electric field, Z is the distance along the direction of propagation, X is the transverse coordinate, n 0 is the linear refractive index and k 0 is wavevector of the laser beam. The local nonlinearity is introduced by the function n nl Ž I .. Assuming a solution of the form E Ž X,Z . s c Ž X . expŽih Z ., where c Ž X . is a function of X only, the equation becomes y2 k 0 n 0 h c Ž X . q

353

Ž 7.

2.1.4. Light scattering Finally, we consider the effect of laser induced suppression of the director fluctuations on light scattering. The scattering of light due to the director fluctuations is a well studied subject w4x. In the presence of the electric field of the laser beam whose polarisation is parallel to the director and selecting

S.K. SriÕatsa, G.S. Ranganathr Optics Communications 180 (2000) 349–359

354

an outgoing field orthogonal to initial polarisation, the scattering cross-section can be shown to be:

sŽ q . s V

ea2p 2

ž / l

4

k BT 2

Kq q ea Ir16p c

Ž 10 .

with positive ea . It is obvious from this equation that the scattering cross-section is reduced on increase of the laser intensity and this in turn should lead to an increase in transmitted intensity for polarisation parallel to the director. 2.2. Cholesteric liquid crystals Now we discuss the nonlinear optical effects to be expected in cholesteric liquid crystals. Let a linearly polarised light in the Mauguin limit be incident on the structure along the helix axis as shown in Fig. 1a. The electric field is assumed to be parallel to the director and hence the director fluctuations are suppressed globally. This increases the order parameter, as said earlier, everywhere which in turn increases the twist elastic constant. This results in a linear dependence of the pitch on the square root of intensity with a slope of 4 P0 k B T ear Ž 8p cK . rŽ S0 p K ., where P0 is the uniform pitch and S0 is the zero field order parameter. A typical calculation is shown in Fig. 1b. As a result of this the azimuth of the electric field vector of the emergent beam will be different from that when the laser is absent or the intensity is very low. As the laser intensity is increased, say to I s 10 5 kWrm2 Ž10 kWrcm2 . the azimuth changes by 18 to 28. This change is easy to detect by optical techniques. It is not difficult to see that we can get in this limit all the effects like self-focusing, self-divergence, self-iridescence, and solitary waves predicted in the case of nematic liquid crystals. The only difference is that as we go along the twist axis the local electric vector twists with the director. Further, in the standing wave geometry, the electric field suppresses the director fluctuations more at the antinodes. As the twist changes locally the pitch of the cholesteric also increases in these regions. Thus we find a modulation of the twist in the presence of standing wave. This modulation is nonuniform and is on a scale of the wavelength of light. If the wavelength of incident light is comparable to the optical period l0 Žs m P . of the cholesteric

(

Fig. 1. Ža. Linearly polarised laser beam propagating along the twist axis, with the wavelength l < P0 m. The electric vector is everywhere parallel to the local director. Žb. The relative change in pitch of the cholesteric helix as a function of the square root of laser intensity. The parameters used here are T s 300 K, e I s 2.89, e H s 2.25, k 0 s 4p P10 6 my1 , D m s 0.64, K s10y1 2 N, P0 s 20 mm.

then we get Bragg reflection at normal incidence for a circularly polarised wave of the same sense as the helix. This reflection takes place in a band of width of PD m centered at l0 . Inside the Bragg band we get a linearly polarised standing wave. At the long wavelength edge of the Bragg band, the standing wave will have its electric vector parallel to the director while at the short wavelength edge it is perpendicular to the director w15x. In both the cases this electric field decays over a finite length called the penetration depth. We consider here the long wavelength edge only. Due to local but non-uniform suppression of the director fluctuations there is a non-uniform change in pitch. This leads to a phase mismatch between the waves reflected from different regions of the cholesteric. Hence, the strength at the antinodes of the standing wave reduces with a conse-

S.K. SriÕatsa, G.S. Ranganathr Optics Communications 180 (2000) 349–359

quent increase in the transmitted intensity. When the field inside the medium is thus reduced the structure relaxes towards the original uniform structure resulting again in an enhancement of the Bragg reflected wave. This process repeats indefinitely leading to temporal oscillations in the twist of the structure and the transmitted intensity. When a laser beam of wavelength l 4 m P is propagating perpendicular to twist axis with its electric vector parallel to the twist axis, we get a different result. Here we assume the boundary to anchor the twist axis. In-plane director fluctuations are unaffected and the out-of-plane director fluctuations are enhanced. Hence, the elastic constants decrease leading to a decrease in the pitch of the structure. However, in chiral smectic C there is no such effect, since only in-plane c-director fluctuations, which are relevant to this system, are unaffected by the laser beam. 3. Laser induced change in tilt order parameter Now we consider the smectic C liquid crystals. We assume the laser beam to be propagating perpendicular to the layers with its electric vector along the c-director. Then the laser field not only suppresses the in-plane fluctuations of the c-director but it also affects the tilt angle u , of the molecules relative to the layer normal. The free-energy density is given by: FAC s a u 2 y a XX Iu 2 q bu 4 q higher order terms q coupling terms .

Ž 11 .

Here a s a 0 ŽT y TAC ., and b Ž) 0. are phenomenological constants in the free-energy density, a XX s ear16p c, and ea is the dielectric anisotropy. Thus the laser intensity affects the order parameter u . It may be pointed out here that in smectic C liquid crystals, when the electric field is at an angle to the c-director we get c-director reorientation. Ong and others have worked out the nonlinear optics due to this process w16x. But they have ignored the change in u which is important near a transition point and must be considered along with the c-director reorientation. The tilt angle u may increase or decrease depending upon the sign of ea and the polarisation of the laser beam. For example, for propagation

355

perpendicular to layers with positive ea and the electric field parallel to c-director, u increases while for propagation along the c-director with electric field perpendicular to the layers, u decreases. Hence, we expect corresponding changes in the elastic and the optic dielectric constants of the medium. The relative changes in the elastic constant, and the optical dielectric constant is related to the relative change in the tilt angle. In fact, as D KrK s D ere s 2D uru s ea Ir8p cu 2b . The change D u is very large near the transition point at which u is small. In the absence of a laser field at a temperature T ) TAC , u s 0 and the medium is in smectic A phase. At this same temperature let a linearly polarised laser beam propagate along the layer normal. It is easy to show that at a certain threshold intensity given by Ith s 16p c a 0 ŽT y TAC .rea , a transition from smectic A to smectic C is induced leading to a non-zero value of u . For a s 0.1, b s 0.2, Ith is of the order of 5 = 10 3 kWrm2 Ž0.5 kWrcm2 .. As the intensity is increased and at twice the threshold intensity the induced tilt u is as much as 308! If the medium is made of chiral molecules, at any non-zero u we get a chiral smectic C. In some chiral smectics the tilt angle and the pitch are decoupled while in others they are coupled. Hence a given u corresponds to a unique pitch. Now the evolution of the structure depends on the pitch of the helix. We note that we can get all the effects already referred to in nematic and cholesterics, but here they will be due to suppression of c-director fluctuations. We point out now the existence of some new nonlinear optical effects due to the process of tilt change in a laser field. 3.1. Chiral smectic C In a chiral smectic C with the tilt angle and the pitch decoupled we find, in the Mauguin limit, that an increase in the laser intensity leads to an increase in the tilt angle which in turn increases the twist elastic constant. Therefore, the pitch increases. In this sense this process is no different from the effect due to suppression of the director fluctuations. The order of magnitudes are given by similar expressions as in the case of director fluctuation suppression. For I s 10 4 kWrm2 Ž1 kWrcm2 . the relative change in the elastic constant is of the order of 0.6 which is quite high compared to 10y4 in cholesterics. The

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S.K. SriÕatsa, G.S. Ranganathr Optics Communications 180 (2000) 349–359

change in the refractive index is of the order of 10y3 . However, in some chiral smectic C’s as the transition to smectic A is approached the pitch increases to start with, reaches a maximum and then sharply falls to zero at the transition point w5x. As is to be expected the tilt angle monotonically decreases to zero as the transition point is approached. If we assume that the tilt angle and the pitch are coupled in this fashion then we obtain an interesting behaviour in the Mauguin limit. Increase of laser intensity increases the tilt angle. This may either increase or decrease the pitch depending upon the inherent tilt angle and pitch. 3.2. New periodic structures We take up next a smectic A Ži.e., at T ) TC A . with light propagating perpendicular to the layers. Then in the standing wave geometry, the electric field is periodic inside the medium. Therefore we expect a periodic variation in the director tilt. It has already been said that the director is tilted only in the regions of field strength above a threshold. Thus we get smectic A and smectic C type blocks to alternate along the direction of propagation. This can happen even when the incident intensity is only one-fourth of the threshold intensity. Just above this intensity, the thickness of the smectic A blocks will be much more than that of smectic C. With increasing intensity the smectic C block thickness increases and at very high intensities it becomes much more than that of smectic A blocks. A possible evolution of the structure is depicted schematically in Fig. 2. Very much the same can be expected in smectic A which

Fig. 2. Ža. Smectic A liquid crystal in the standing wave geometry. The laser beam propagates parallel to the layer normal and the electric vector is parallel to the layers. Žb. When the intensity of the laser beam is below the threshold intensity for the smectic A to achiral or chiral smectic transition we get only smectic A. Žc. Smectic A and smectic C blocks when the intensity of the laser beam is just above the threshold for smectic A to smectic C transition. The smectic A block is much thicker than the smectic C block. The tilt in the smectic C block is non-uniform. Žd. Smectic A and smectic C blocks when the intensity of the laser beam is very high. The smectic C block is now thicker than the the smectic A block. The tilt is again non-uniform in smectic C block. Insets show the smectic layers with the molecular tilt in A and C regions.

S.K. SriÕatsa, G.S. Ranganathr Optics Communications 180 (2000) 349–359

has a low temperature chiral smectic C phase. If the pitch is very large and so coupled to tilt angle that it varies inversely with the tilt angle then above a threshold intensity we will be in the Mauguin limit. Thus we obtain thick smectic A blocks alternating with thin chiral smectic C blocks. This structure is rather reminiscent of the twist grain boundary phases. With increase of intensity thickness of the chiral smectic C blocks increases. On the other hand, let us consider a chiral smectic C near the chiral smectic C–smectic A transition. The incident light whose wavelength is very large compared to the pitch is propagating parallel to the layers with polarisation parallel to the twist axis. In this situation the incident polarisation is an eigen mode. Then in the standing wave geometry we obtain periodic variation of the tilt angle, along the layers leading to a novel two dimensional periodic structure. This periodic structure consists of alternating smectic A and chiral smectic C blocks in a direction perpendicular to the inherent twist axis.

357

w13,14x. In liquid crystals, the nonlinear coefficient being million times large compared to the usual Kerr nonlinearity the soliton formation length is very small. In fact, it can be of the order of few mm. We consider three geometries as depicted in Fig. 3 which

3.3. Effect of third harmonic generation In smectic liquid crystals the symmetry allows the generation of a third harmonic polarisation. The electric vector of third harmonic can be parallel to that of the fundamental. This happens if the laser beam is propagating along the layer normal and the electric vector is parallel to the c-director. Invariably, there will not be a perfect phase matching between the fundamental and the harmonic due to optical dispersion. If the smectic is near a smectic A to chiral or achiral smectic C transition then a sufficiently intense electric field induces a tilt as discussed earlier. The tilt will be periodic and the period is given by prŽ k 3 v y k v ., where the k v and k 3 v are the wavevectors for the fundamental and the third harmonic respectively. 3.4. Solitary waÕes We now consider solitary waves in smectic liquid crystals. Here both the process of suppression of the c-director fluctuations and the tilt angle can exist in some geometries and in others only the second process exists. Usually the soliton formation length is about a few kilometers length in the nonlinear media

Fig. 3. Ža. Geometry showing smectic A liquid crystal with the laser propagating parallel to the layer normal and the electric vector parallel to the layers. The molecular tilt is induced only beyond a particular intensity leading to a threshold type nonlinearity. Žb. Geometry showing smectic C liquid crystal with the laser propagating parallel to the layers and the electric vector perpendicular to the layers. The molecular tilt becomes zero beyond a particular intensity leading to a saturable type nonlinearity. Žc. Geometry showing smectic C liquid crystal with the laser propagating parallel to the layer normal and the electric vector parallel to the layers along the c-director. The molecular tilt increases with increasing intensity and also the in-plane c-director fluctuations are suppressed.

S.K. SriÕatsa, G.S. Ranganathr Optics Communications 180 (2000) 349–359

358

yield three different types of nonlinearities leading to three different types of soliton solutions. For a selfconsistent solution of the Maxwell’s equation, the tilt angle should be obtained by minimising the Landau free-energy density given by Ž11.. In all these cases the soliton solutions are due to non-Kerr type nonlinearity and thus are only solitary waves. But if we make the parameter a in the free-energy density expression Ž11. zero by going to T s TAC Žwe can effectively get the same result by an applied static magnetic field., then the nonlinear function n nl Ž I . is the same as Kerr type nonlinearity which possesses a true soliton solution. Ži.: In smectic A, near the smectic A to smectic C transition only the second nonlinear process is relevant. As the tilt angle is induced only beyond a particular intensity, this is of the threshold type nonlinearity w12x. The nonlinear function n nl Ž I . is given by:

n nl

°0 , ~ e e I Ž I. s ¢2 n b ž 8p c y a / , a 2 e

I ) Ith

The soliton solution with this nonlinearity is given by:

cŽ X.s

(

(

2 A2

1

B2

cosh A 2 X

1

B1

cosh A1 X

ž(

,

(

n nl Ž I . s k B T

q

where A 1 s 2 k 0 n 0 h q b and B1 s k 02 n 0 ea2r8p cn2e b . Note that the width of the soliton is a function of both the parameter h and the material parameters unlike the case discussed earlier in Section 2. Žii.: In smectic C liquid crystals as shown in Fig. 3b again only the second nonlinear process is relevant. We observe that as the intensity is increased the tilt reduces and beyond the intensity, Ith s 8p c < a
2 n20 b

,

Ž 14 .

/

'Irc q n 0 y n Ž u 0 .

2p 3 K 3

2 n 2e b

q

ea2 n 0 16p cn2e b

I.

Ž 15 .

3 A 3rB3

cŽ X.s

)ž

9 A23 C3 2 B32

/

,

ž(

q 1 cosh A 3 X

/ Ž 16 .

ae a r2 n 2e

ea n e < a <

ea3

ea n 0 < a <

Ž 12 .

/

ž(

where A 2 s 2 k 0 n 0 h y w n e y nŽ u 0 . y e a n e < a
1q

2 A1

n nl Ž I . s n e y n Ž u 0 . y

cŽ X.s

Here the first term is due to the process of suppression of c-director fluctuations and the others are contribution from the second process. The soliton solution is like that worked out in Section 2. It is given by:

I - Ith

a

where nŽ u 0 . is the refractive index without the laser field. The soliton solution is given by:

q

ea2 n e 16p cn20 b

where A 3 s 2 k 0 n 0 h q 2 k 02 n 0 w n 0 y n Ž u 0 . q ea n 0 < a
(

h G y 29 a 3 y b 3 ,

Ž 17 .

where a 3 s Ž2 ea n e k B T . 2brp 2 n20 K 3 and b 3 s k 0 w n 0 y nŽ u 0 . q ea n 0 < a
4. Conclusions I,

Ž 13 .

We point out that two new nonlinear optical processes can exist in transparent liquid crystals.

S.K. SriÕatsa, G.S. Ranganathr Optics Communications 180 (2000) 349–359

These processes lead to changes in the order parameter of these systems. The nonlinearity in one case is due to suppression of the director fluctuations by a laser beam resulting in the change of refractive indices as seen by the light itself. The change in the refractive index for an intensity of 10 5 kWrm2 Ž10 kWrcm2 . could be as high as 10y4 . We have worked out a few consequences of this nonlinearity. We find both self-focusing and self-divergence in the case of a partially polarised light beam. Self-induced Bragg reflections or self-iridescence is to be expected in a standing wave. The optical spatial bright soliton solution due to this new nonlinearity has a structure very different from that due to the classical Kerr effect. We have shown that the scattering cross-section is reduced due to suppression of the director fluctuations. In chiral liquid crystals with wavelength of light very small compared to pitch and for the propagation along the twist axis the pitch of the structure increases as the square root of laser intensity and leads to a non-uniform modulation of the twist in a standing wave. At the long wavelength edge of the Bragg band we predict temporal structural oscillations. In smectic liquid crystals we have an additional nonlinear optical process due to laser induced change in the molecular tilt. The threshold intensity required to induce the smectic A to smectic C transition can be about 5 = 10 3 kWrm2 Ž0.5 kWrcm2 ., which is quite low. The change in the refractive index is of the order of 10y3 for an intensity of about 10 4 kWrm2 Ž1 kWrcm2 .. Thus this process results in a giant optical nonlinearity. In a standing wave, this process leads to new periodic structures. Some of the periodic structures are reminiscent of the twist grain boundary phases. Smectic liquid crystals exhibits a rich class of optical solitons

359

due to the presence of both the nonlinearities. At the smectic A to smectic C transition point or in a suitably applied magnetic field of appropriate strength a true soliton is permitted while in other geometries we get only solitary wave solutions.

Acknowledgements Our thanks are due to K.A. Suresh for helpful comments.

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