Light-induced modulated structures, intrinsic optical multistability and instabilities for the competitive wave interactions in liquid crystals

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Light-induced modulated structures, intrinsic optical multistability and instabilities for the competitive wave

interactions in liquid crystals

S.M. Arakelian, Yu. S. Chilingarian, R.B. Alaverdian, G.L. Grigorian, A.S.

Karaian, S. Ts. Nersissian, V.E. Drnoian

To cite this version:

S.M. Arakelian, Yu. S. Chilingarian, R.B. Alaverdian, G.L. Grigorian, A.S. Karaian, et al..

Light-induced modulated structures, intrinsic optical multistability and instabilities for the com- petitive wave interactions in liquid crystals. Journal de Physique, 1989, 50 (12), pp.1393-1415.

�10.1051/jphys:0198900500120139300�. �jpa-00211004�

1393

Light-induced ^modulated structures, ^intrinsic optical multistability ^and instabilities for the competitive ^wave

interactions in liquid crystals

S. M. Arakelian (1), ^{Yu. S.} Chilingarian (1), ^{R. B.} Alaverdian (2), ^{G. L.} Grigorian (1),

A. S. Karaian (1), ^{S. Ts.} Nersissian (1) ^{and V. E. Drnoian} (1)

(1) Department ^of Physics, Yerevan State University, Yerevan, Armenia, 375049, ^U.S.S.R.

(2) Nagomiy Kharabakh, Armenia, 375000, ^U.S.S.R.

(Reçu le 15 avril 1988, révisé le 7 décembre 1988, accepté ^{le 14} février 1989)

Résumé. 2014 On ^{étudie à la fois} théoriquement ^et expérimentalement la multistabilité et les instabilités temporelles de processus ondulatoires ^non linéaires dans les conditions de transfert

d’énergie ^{et de} compétition ^{entre les} différentes composantes ^de polarisation, pour ^un milieu

anisotrope inhomogène présentant ^une non-linéarité à seuil. Dans de telles interactions, ^{il se} produit ^un couplage rétroactif lorsque ^le rayonnement ^{laser induit un réseau} d’indice de réfraction dans le milieu, ^à cause de la réponse ^non locale du milieu élastique anisotrope ^au champ ^extérieur.

Abstract.

The multistability ^and temporal instabilities of nonlinear wave process with ^an energy transfer and competition between different light polarization components ^{have been} studied both theoretically ^and experimentally ^{for a} nonhomogeneous anisotropic medium with threshold nonlinearity. ^A feedback for such wave interactions, when the laser radiation induces the bulk-gratings of refractive index, arises because of the non-local response of the anisotropic

elastic medium on the external field.

J. Phys. ^{France 50} (1989) ^{1393-1415 15 JUIN} 1989,

Classification

Physics ^Abstracts

42.65P - 61.30 - 64.70M

1. Introduction.

The non-steady ^and stochastic wave processes and instabilities which arise due ^to propagation

of light ^{in a} highly nonlinear medium are a subject of intense study ^{for the} present ^time.

Although ^the problem ^of instabilities of nonlinear wave interactions in condensed matter has been discussed repeatedly ⁱⁿ early ^nonlinear optics, such studies were based on the principle

of a weak local nonlinear response of medium. In fact, ^the strong nonlinear interactions in these cases arise because of the accumulation of coherent processes along the thickness of a

nonlinear medium.

At present ^the nonlinear materials with very high nonlinearity have been already

discovered. Liquid crystals (LC) ^{are one of the} examples of such medium. Because of the high anisotropy ^{as well as the} collective behavior of molecules under any extemal field LC ^are

unique nonhomogeneous anisotropic objects ^{for nonlinear} optics ^{and their use leads to} qualitatively ^new phenomena [1]. ^{Let us enumerate a} few of them.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500120139300

First, the intrinsic optical bistability (multistability) ^{in wave} phenomena without any external feedback. The multivalued regimes ^occur because of the propagating light ^waves

induce the dynamic gratings of refractive index inside the highly nonlinear medium. These effects are manifested in two aspects : (i) the threshold high nonlinearity of LC leads to real laser-induced structural phase transitions without any temperature variation of the sub- stance [2], ^and (ii) the variations of the Bragg ^resonance conditions due to laser intensity ^lead

to self-action effects of the light ^{in a} spatial periodic nonlinear medium [3]. ^{The first case}

concerns nematic LC (NLC), the second one cholesteric LC (CLC). Both of these cases are a

realization in nonlinear optics of distributed feedback systems [8]. For NLC different spatial

modulated structures arise in the medium [4]. In CLC the spatial ^{scale of} periodicity ^is changed. This intrinsic optical multistability ^{is due to the} physical properties ^{of the} developed

nonlinear phenomena. ^A feedback arises even when ^no back reflected wave exists from these structures, and it is determined by ^a nonlocal nonlinear response of the medium ^to the laser field because of the elastic forces [5]. ^The experimental study ^of these effects has been carried out in [6, 7].

Second, ^the temporal instabilities and stochastic processes for the light-induced ^reorien-

tation in LC (at ^CW pump radiation) [5]. ^These dynamic self-diffraction effects due to the

anisotropy of the medium because of two waves of different polarizations, travelling through

the medium, ^create laser-induced gratings along the thickness (d) ^{of the} sample (z-direction).

An energy interchange (over time) ^{occurs between two} polarization components (1) ^{in this}

laser-distorted nonhomogeneous anisotropic medium with spatial modulation of the optical

axis n (z ), which is determined by ^a non-adiabatic deformation [4]. Experiments ^{have been}

carried out in [7, 9-11].

Third, the nonlinear phenomena ^{on a surface as well as on a} boundary between LC and other media (metals, semiconductors, dielectrics). ^From the power optics point ^{of view} (i.e.

when the variations of the surface properties ^of the medium occur) ^an optical bistability ^also

takes place ^{for these cases} [12]. Experimental demonstrations for LC have been presented ⁱⁿ [13].

And finally, fourth, the fluctuations and ^a nonlinear light scattering ^at phase transitions

(both temperature ^and light-induced). Experiments ^have been carried out e.g. in [14] ^and [15]

for these two cases accordingly. ^{On the one} hand, namely the fluctuations in non-linear

dynamic system determine the development of the processes in it, ⁱⁿ particular the transition to the chaotic regime [16]. On the other hand, the fluctuations can lead to noise-induced

phase transitions [24].

Note that the correct description of the above mentioned effects in LC strictly speaking

have to be based on the conception ^{of a} non-equilibrium ^statistic thermodynamics for open systems [17] ^{but this} approach ^{is still not} developed ^at present [18].

The study of all these (and other) processes in LC (in particular ^{in an} experiment) ^{is very} important ^{from the} physical point of view for understanding ^{of the} phenomena in nonlinear

dynamic systems ^of quite ^common type ^{with a} threshold behavior. Some of them we shall discuss in this paper.

The organization ^{of the material in the} presented paper is the following :

the main part ^{consists of two} divisions. First one (Sect. 2) ^concerns steady-state phenomena

due to laser-induced bulk gratings ^{in a NLC, in} particular, ^{to the} arising of non-adiabatic

(over ^the space) structures. A new effect of enhancement but not suppression ^{of the} light-

induced reorientation in two crossing polarization components is examined in this section both experimentally ^and theoretically. Temporal instabilities under condition of competition

(1) Description of these processes with analogy ^{of two} coupled (orthogonal) oscillators is useful.

1395

of two orthogonal components ^{of the} light polarization ^{in the nonlinear medium are discussed}

in the second division (Sect. 3). ^New experimental ^{results are} described. A complete theory

of transient processes occurring ^{in a field of} circularly polarized light ^{in NLC is} presented ^for

the first time. The role of fluctuations when the reorientation effects occur in LC are finally

discussed in conclusion.

2. Laser-induced bulk gratings ^{in NLC.} Adiabatic and non-adiabatic structures. Steady-state description.

The laser-field action on LC leads to a spatial instability of the initial equilibrium ^{state of the}

medium (homogeneous ^{over the} sample ^thickness d), ^and reorientation effects, ^i.e. light-

induced structural phase-transitions, ^{occur. The} parameter changing under laser radiation is the angle of reorientation w, which shows the deviation of the orientation of the local optical

axis (director no) from the initial (unperturbed) ^direction (no//z ) ⁱⁿ accordance with the

configuration ^{of the} light-wave field in the LC.

On the contrary, the distorted structure, produced ^{in the LC} by ^{a laser} radiation, ^{has an} opposite ^{effect on the} propagating light ^wave.

The most interesting ^{of these} self-action effects for the optical range ^are the schemes of

dynamic self-diffraction of two or more waves when the light ^{field induces the} gratings ^{of the}

refractive index np ^on which the incident waves are diffracted [1]. These processes correspond

to four-wave interactions in usual terms of non-linear optics.

The standard schemes of self-diffraction when two or more waves intersect (under ^{a small} angle 0) ^{are a} typical ^{case for an} isotropic nonlinear medium (the ^{different waves have the}

same polarizations inside of the medium). ^The period ^{of the recorded} grating ^{is :}

11., À /nr 0, where À is the wave length.

In the LC, ^as in any anisotropic ^{medium, a two-wave} propagation ^{condition can also occur} (with ^a single input ^wave E), ^{because two waves of} ordinary (A) ^and extraordinary (B) polarizations with refractive indexes nro and iZre, respectively, exist inside of the medium. As

long ^{as two} orthogonal components ^of polarization ^{of the} light ^field propagate along ^the

thickness of the medium (z-axis), ^{we have a} continuous variation of the phase retardation between them. Thus, ^a non-homogeneous (along z) polarization ^{of the} transmitted wave occurs. This means that the light ^{field acts on the NLC} molecules with various forces along ^the

thickness. Therefore, rotational moments affecting the molecules are not equal. ^{This leads} (with ^a sufficient intensity) ^{to the} non-homogeneous reorientation of the molecules inside the medium and so a bulk grating, which is determined by ^the polarizations of the transmitted waves, is induced in NLC.

Thus, ^{the wave} propagation in NLC is accompanied by ^a writing ^of refractive-index

gratings, which results in self-modulation of the transmitted light ^waves. Essentially, ^{this case}

may also be ascribed to the scheme of dynamic self-diffraction. Here As À/(nre - nro ). (A

more precise expression ^is given ^{below - see formula} (10)). ^{In an} experiment ^with LC, nr . 0 nre - nro usually, therefore, this grating ^{has a} much smaller period than in the scheme with intersecting ^{waves of identical} polarizations.

We confine ourselves here to the comment that such induced gratings ^{are not} phase gratings only. ^An expression describing ^the relationship between the amplitudes ^{of waves with} orthogonal polarizations ^not only ^{includes a term} corresponding ^{to the nonlinear} phase retardation 0 NL but also a term including ^the changes ^{in the} amplitude parameters. ^A characteristic spatial scale for this energy-interchange in the NLC is determined by ^{the value}

1/vA/, where Ai ^{is the} intensity ^{of each} component ^{of the} light-field (i = 1, 2 ),

v

qz -1 Ea/ 16 ^{7T’K, qz is the} z-component ^{of the} wave-vectors difference q

ke - ko ^between

to the director) ^{and K are the} optical anisotropy ^{and elastic} parameter ^{of the} NLC, consequently [4]. ^For CLC, ^the periodic energy redistribution between the waves leads to the so-called pendulum beatings (over ^the space) [1].

We begin by considering ^the simplest ^case where the two waves have orthogonal polarizations ^and compete in the nonlinear medium, ^{but that occurs in the same} plane ^where

the reorientation takes place.

2.1 ENHANCEMENT AND SUPPRESSION OF REORIENTATION EFFECTS IN LC UNDER TWO WAVES WITH CROSSING LINEAR POLARIZATIONS.

In this paragraph ^{we shall} discuss the detailed picture ^{of a new} effect of ^a dramatic increasing, ^{but not} suppressing, ^{of the}

reorientation in NLC under a condition of two waves at oblique ^incidence (but symmetrically

with respect ^to the initial orientation of the director) ^with different linear polarizations ^{in the} plane of incidence.

Experiment.

^An experimental setup ^{is shown in} the insertion of figure ^{1. The radiation}

of a CW YAG : Nd3+-laser (a

^1.06 u) ^{was used as two} strong pump beams (the intensities

Il ^and 12) ^{which were focused in NLC-cell} (5CB) of thickness d

125 w (the ^initial

orientation of the director was homeotropic - no//z). The ratio of the intensities

12/ Il ^is equal ^to 1.5 and 2.7. As a weak probe beam, ^we used the He-Ne laser radiation

(je

^0.633 p,) ^{for the} measurements of the pump field-induced nonlinear phase retardation OE ^NL (a ring-pattern ^{for the} passing light).

In our experiment ^{one beam was blocked} (e.g. 12) ^{from the} beginning ^and only ^the steady-

state reorientation in one beam (Il) ^was observed. Then the second beam (12) ^{was also}

turned on (indicated by ^{arrows in} Figs. la, b) and the reorientation in two beams was

measured. A steady-state ^{as well as a} transient characteristics of the 0 ^NL are shown in figure ¹

for different conditions. A very important ^{result is that in contrast to the} geometric factor, ^a

new effect of a dramatic increase (but ^not suppressing) ^{of the 0 NL due to} the molecules reorientation has been observed when the second field E2 ^{was switched on} (the reorientation due to the first field El ^has already been saturated in the steady-state

Fig. la). This effect possesses ^a threshold on the sum intensity Il ⁺ 12.

Theory and calculations.

Basic equations.

^The geometric optics approximation has been used for the theory ^of

nonlinear interaction of the laser radiation with a nonhomogeneous anisotropic medium. The standard procedure ^{for a} NLC is described in [4, 27]. The initial equation ^is

Fig. 1. Nonlinear phase retardation cp NL ^vs. time t (for ^a probe beam) ^at Il

²⁸⁰ W/cm2, ^@ I2

⁷⁰⁰ W/cm2 (a) ^and Il

¹⁵⁷ W/cm2, 12

⁴⁴⁰ W/cm2 (b) ^{as well as vs.} intensity (Il ⁺ 12 ) of incident

light ^for steady ^state (c). ^The points ^an experimental, the line is theoretical (the calculations have been done for ol ^{« 1} ( cp NL /2 ^{’TT «} 36 ). ^The experimental setup is shown in the insertion. Values of parameters : (i) ^one beam switched on a) A ^{= - 0.01, B}

^{0.03, C} = - 0.029, ^D = - 0.035 ; b)

A

0.0075, B

0.017, C

0.05, ^D

0.02 ; (ii) ^two beams switched on (indicated by ^the arrows) a) A

0.05, ^B

0.046, C

0.08, ^D

0.05 ; b) A

^{0.02, B}

^{0.006, C}

^0.01,

D

0.008 ; (iii) ^for c), ^{two cases are shown :} 12/I1 ^{= 1.5} (1) ^and 12/I1

^{2.7 (2).} The theoretical curve

corresponds ^to (2)-case only. ^-

INSTABILITIES FOR THE COMPETITIVE WAVE INTERACTIONS 1397

where _Ea

_EII - £1 ; K33, ^x

(xl l - K33)/ K33 ^{are the elastic constants of} NLC ; y is ^the viscosity ; ^{c is the} light velocity ; ^{the constant values} (2) Il,, I2 z ^{are the} z-components ^{of the} Poynting ^vector for each of the _{waves ;}

ao is the angle of incidence (in air) (3). ^The boundary conditions are rigid : w(z

0) = w (z

d )

^{0. The} approximation ^{of the} light-induced adiabatic deformation (see ^next item)

for each of the waves in NLC are used for this experimental geometry :

~

03C8 = 03A3 ^{C p sin (} 03C0zl/d). ^{As usual we take into account the first term} only (i

1 ) ^{because of}

i = 1

the energetic advantage

The approximate expression ^{with an} accuracy - 03C83 may be obtained for gl, 2 (/1) ^from equation (1) in the form ( 03A8 1) :

Qualitative analysis. ^{- A} few of the qualitative ^{results can} be obtained by ^an analysis ^of equation (3). ^In fact, e.g. if 12 z

Il,, ^{then the} terms - 03C8 and Ip 3are ^left only ⁱⁿ equation (3).

That means that equation (1) ^{describes the} ordinary ^{case of the threshold} light-induced

reorientation in NLC at the normal incidence of one linearly polarized ^{wave with an} intensity Iz = (I1z + 12z)(2 - h) hl/2.

The threshold intensity ^is determined by ^the expression

Thus, ^the reorientation occurs under condition 2 - h > 0 which is always satisfied. It is

important ^to emphasize that for this case the two fields act both in tendency ^{to increase gl}

When Il z = I z 1 ^{the most} interesting ^{effect is} determined by ^{the first term in} equation (3)

with the difference (12 z - Il,) (zero ^{order on} Ji). For the weak light intensities this term plays

the principal ^{role and two fields work} against ^each other, ^{i.e. the} suppression of reorientation

occurs in contrast to the case of one field action. For the high intensities (large ^{value of} w)

both fields work in the same way and the reorientation is enhanced (the ^{linear term on w in} Eq. (3) is taken into account). ^This enhancement of ip ^takes place ^if

(2) ^We neglected ^an interference between the ^waves in the medium because of the averaging along

the thickness of the sample.

(3) ^For simplicity ^{we assume that « o is the same for two beams} (see Fig. 1).

1399

For a further increase of the intensity, the influence of the next terms in equation (3) (i.e.

,..., .p2 and .p3) is essential and the mutual suppression ^or enhancement of the reorientation in two fields occurs accordingly (4).

Thus, ^{we have obtained a} qualitative explanation ^{for the} experimental ^{data in} figure ^1.

From a microscopic point ^of view, ^the possibility of mutual enhancement of tkin ^two crossing linearly polarized ^waves is determined by ^{the fact} that, ^{for an} anisotropic system ^{with the} external E field-induced polarization, the essential parameter ^{is the} projection ^{of E on the}

direction of the maximal polarizability (on ^n, e.g.) only. ^{This means} that in the case of two

orthogonal fields, ^{the effective} field, ^{which acts on the} anisotropic ^{center, is directed not} along ^{these fields but} along ^{the n} direction, ^{i.e. the centre senses these two} crossing ^{fields as} parallel ^fields (cf. ^the magnetic systems [23]). ^{For the sum of the} propagating ^{waves it is}

necessary ^to take into account the phase ^{terms also} (- e’k") ^{in contrast to} the static fields. This results in the short-scale (with respect ^to the thickness d) spatial interferometric terms, ^but these terms disappear ^{because of the} spatial averaging. ^{That is} why the reorientation effects

are determined only by ^the intensity ^sum (Ei + E 2 2) ^{for this case.}

General description.

^{Let us} carry ^out the obvious analysis ^{of the} possible ^{states of the}

examined system ^from quite general conceptions.

Equation (1), taking ^{into account} equation (2), ^{can be rewritten} in the form

where the coefficients of the series in terms of t/1m powers ^are equal ^to

and the parameters

Let the initial value t/1m, ^{in =} t/1in (5) correspond ^to the maximal angle of n-reorientation in the centre of thé sample when the field El ^is only ^{switched on ; at t}

0 the second field

E2 ^{is switched on and both two fields act on NLC.}

By changing ^X

t/1m - t/1in equation (5) ^{can be} given ^{as an} expression

where F2(Pin) is rewritten as

and FI is determined by ^the right hand side of equation (5).

(4) ^More opportunities arise when two beams have different angles of incidence.

(5) ^It is determined by the fact which field (El ^or E2) ^{is switched on first.}

If X « 1 the first term in equation (7) ^is neglected ^{and the} integration ^of equation (7)

results in the equation :

One can see that when F2( .pin):> 0 ^then X = .pm - «/Jin is increased vs. t (i. e. 03C8m ^is increased also), ^{and when} F2(.03C8in) : 0 ^then .pm is decreased. Which regime ^{is realized} concretely depends ^on the values of the coefficients at the .pm power.

The first case is when A 0. The dependence F2 (.pin) is ^{shown in} figure ^{2 for two cases :}

one real root 03C81 (a ) and three real roots .pl, 2, 3 (b) (6). ^The stability analysis ^{of these solutions,}

which correspond ^{to the} steady-state solutions of equation (5), ^{can be} easily ^done.

In factor. increases because of F2( .pin) > 0 (see Eq. (8)), ^{and then at} .pin ^« 4, .pm increases and approaches 03C8 1. Under the condition 03C8in > 03C8 1 ^{we have} F2( .pin) ^{0 and} 03C8m decreases down to the value .03C81 (.03C8m -+ .03C81). ^{Thus, this case is} very simple ; it doesn’t

depend ^on the initial value .pin and the director n always ^turns (over time) ^to the direction which is determined by 1.

A more interesting ^case corresponds ^to figure 2b. Three real solutions .pl, 2, 3 do exist if the

sum intensity Il ⁺ I2 is above the threshold value (see Eq. (4)). ^The wi ^and 03C83 ^solutions correspond ^{to two stable states, but} .p2 ^{- to an unstable} state ; ^if Iii. > 03C82 ^the angle .03C8m -+.03C83 (by increasing), ^{but if} 4ri. ^« .p2, ^then 03C8m -03C81 1 (by decreasing). ^{Thus, the}

.

Fig. ^2.

Dependence ^of F2( .pin) ^{for one} (.pl): (a)

^{A «:c 0 and} (c) ^A > 0 ; and for three

(.p 1,2,3): (b) ^{- A 0} steady-state solutions of equation (5) (see text).

(6) ^{For two real roots we have the same case as} figure ^2a does ; .p2

⁰ corresponds ^to the unstable

state.

1401

reorientation angle 03C8m can ^{increase or} decrease but this fact does not depend ^{on the relative}

values of Il ^and 12.

Note that the transition between stable states of the system ^can define the optical bistability, ^{so a} hysteresis loop arises for this transition (cf. [2]).

We shall discuss very briefly the second case, when A > 0. When there is one solution

(03C81), ^it corresponds ^{to an unstable state}

figure 2c ; when there are three solutions

(03C8l, 2,3) ^{we have one stable} ( 03C82 ) and two unstable (03C81, 3) states - cf. figure 2a. The first case

is of ^a special interest, because the absence of the stable steady-state (7) ^{means that the} temporal instabilities, ^{i.e. the} oscillations, ^and probably ^a stochasticity ^can arise in the system

(s).

Thus, the obtained experimental results described above can be explained by ^the given analysis.

Estimations.

Using ^the following values of the material parameters of NLC 5CB

K33 = ^{4.4 x} 10-’ dyn, ^{e --} 3.02, £.1 2.31 [1] ^{and if d} = 125 U ^{we have from} equation (4) ^the

estimation : (1 1 z + 1 2 z )th = 320 W Icm 2. ^The values |03C8|> ^0.12 correspond ^to enhancement of reorientation in the condition of the crossing ^waves.

In our experiment ^{the case A 0 was realized} (see ^the caption ^to Fig. 1). ^A geometry ^with

one beam switched _on, corresponds ^{to one} (03C81/(1) ⁱⁿ Fig. ^{la and} tpl(l), ⁱⁿ Fig. 1b), ^{but with two}

beams - to three 03C81(2), 3 (Fig. ^la - high fields) ^{or to one} 03C81, (2 )’ (Fig. ^{1b - weak} fields), ^real

roots of equation (5). ^The steady-state is determined by the stable solutions tPl,3. ^{A numerical}

calculation gives ^{the values}

The theoretical curves are shown also in figure ^{1. One can see a} good agreement ^between theory ^and experiment for |03C8l ^{1, when the} theory ^{does work.}

Finally ^note, that the threshold behavior is sharper for the close values I1, 2 (Fig. lc) ⁱⁿ

agreement ^{with the} analysis which has been done above.

2.2 SPATIAL INSTABILITIES AND INTRINSIC OPTICAL MULTISTABILITY IN AN ANISOTROPIC MEDIUM WITH A THRESHOLD NONLINEARITY. In this item we shall discuss the case of nonlinear interaction of two waves with orthogonal polarizations ^{in an} anisotropic ^medium

when the reorientation is described by ^two reorientation angles ⁱⁿ orthogonal planes.

Let us consider more specifically ^a homeotropically oriented NLC cell (this ^{case is the most}

convenient for an experiment) of thickness d confined between the planes ^z

^{0 and}

z = d of a Cartesian laboratory system (xyz ) - figure ^{3a. The NLC are assumed to be a} nonconducting ^and nondissipative medium ; ^the dependence ^{on one} coordinate (z ) ^{is taken}

into account (the optical ^{field E} (oblique incidence) is unbounded and the distribution of the orientation of the director over the transverse (cross) ^section (x, y ) ^is homogeneous).

Because the two orthogonal components ^{of the} light polarization (A ^and B) ^{act in the} medium, ^the orientation of the director n(r, t) ^{is described} by ^two angles ^cp and

( Q , 03C8 « 1 ) ^{in two} orthogonal planes. We select them in the following way (Fig. 3a) :

n

(sin If, cos Ji ^sin Q, cos w ^cos Q ).

(7 ) To which the system ^{must reach} finally.

(8) It may be shown in the theory ^{that even for this} geometry ^{a director can come out from the} El E2-plane ^{and so two} reorientation angles ⁱⁿ orthogonal planes ^arise (cf. ^Sect. 3). ^But special

measurements showed that this effect doesn’t occur in our experiment.

Fig. ^3.

^Cartesian laboratory systems of the coordinates and experimental geometries ^{for the} light-

induced reorientation of NLC at oblique ^incidence (a) ^{and at} normal incidence of circularly polarized light (b).

This means the angle f/1 determines the deviations of n from the yz-plane, ^the angle ^(P

determines the azimuthal rotation of n in the yz-plane.

After the standard procedure of variation of the free energy density ^{of a} NLC in the presence of ^an optical ^{field E} (the variations over two variables, f/1 ^and cp, have to be done in this case) ^{we can determine all the} optical characteristics of a light ^beam emerging ^{from NLC}

in the geometrical optics approximation ^{for a} given rigid boundary condition 03C8 (z

0) = 03C8(z = d) = ~ (z = 0 ) = ~ (z = d) = 0[4].

A different configuration ^of light-induced gratings in NLC arises as a function of the orientation of the applied ^{field E} concerning the NLC-cell. We shall consider two cases :

orientational effects with and without threshold.

Thresholdless reorientation. - A relatively simple ^case corresponds ^{to the} thresholdless

~

reorientation of NLC (E, no # 90°, ^two components ^{of E :} Ell (xz-plane ) ^and E1 (yz-plane )

exist inside the medium - Fig. 3a). ^The steady-state solution may easily ^{be found} [4] :

where K33 is the elastic constant of NLC, Cij ^{are found from the boundary} conditions, ^{A and}

B are the ordinary (o) ^and extraordinary (e) polarization components ^{of the} light ^field 6, respectively, in the local coordinate system associated with the director rotation (in ^this

coordinate system ^the light propagates ^{in an uniaxial} homogeneous anisotropic medium),

e

Pe exp (ikc r) ⁺ Po exp (i ko r),

1403

the angles coe., and f3 0, e ^define the orientation of wavevectors in a local coordinate system

(which ^are determined by the refraction laws, e.g. by ^the continuity ^{of the wavevector} tangential components ^{at the} boundary : kx

kox = kex , ky

koy

key) , the connection between incident (E) ^{and 8} fields is defined by ^the operator R, ^{E -} R&, which describes the orientation of the local system coordinate with respect ^{to the} laboratory ^one. Because of the

rigid boundary conditions for NLC-molecules E

e on the boundaries, ^{that means the} input- output parameters ^{of the} light, which induces the grating ^{in the} medium, may be obtained without using ^the R-operator.

For simplicity, ^the parameters A, B, _ae,o ⁱⁿ equations (9) ^{are assumed to be constants} (first approximation - 03C8 Q, when the geometric phase retardation of the propagating ^{waves is}

taken into account only ^{but not the} direct energy transformation between the amplitudes ^of components ^{with o- and} e-polarizations). Without loss in

generality ^{we also assume that the wave vector k} of the incident light lies in the xz-plane (k = (kx’ 0, kz)).

As it is obvious from equations (9), ^{two types of the} light-induced structures arise. The first

ones (~ z2) ^{are the adiabatic} (smooth) distortions over the sample ^thickness (A ~ d). ^The

second ones (~ e"q’z) ^are the non-adiabatic (sharp) distortions (A «: d) (9). ^{For the} geometry under study, the adiabatic gratings only arise in the vertical plane (xz ) ^and only non-adiabatic

gratings arise in the horizontal plane (yz ). The reorientation occurs ( Q, 03C8 = 0 ) for any values of A, B # ⁰ (see Eqs. (9)). ^{An obvious} physical meaning ^{of this} property is that the forces

moments (M) are nonzero from the very beginning :

The steady-state reorientation finally ^occurs when the rotational moments of the light ^field

which act on the director are balanced by the elastic moments arising in NLC because of the

rigid boundary.

The non-adiabatic gratings ^{are a} specific optical effect ; they don’t reduce to the spatial

distorted structures of higher ^order (cf. ^formula (2)), ^{which can} also arise in NLC but usually

for ^a large value of the field. These high ^{order terms are} included in the first equation ^{of the} system (9) ; ⁱⁿ fact, ^the development ^{as a} series in sin 7 fl , _d ^{where f} = 0, 1, 2 ^..., gives ^the high modes. But they ^are still adiabatic by ^our classification, ^because they ^{occur when even} only ^{one wave} (e-wave) propagates inside the distorted NLC (in ^a static field (e.g. magnetic)

these deformations do also exist). ^In contrast, the non-adiabatic deformations certainly

demand two waves (o- ^and e-polarizations) inside the medium and consequently ^new phenomena ^{occur. In} particular, temporal oscillations exist for the light-induced reorientation at CW incident radiation. We shall analyze ^this problem in detail in the ^next section. But even

for steady-state, ^{new effects occur} when the non-adiabatic gratings ^{arise. We} discuss them for the case of reorientation with threshold.

Reorientation with threshold. - Let us consider a geometry ^{when E 1} no for an oblique

incidence of the laser radiation (Fig. ^{3a, E -} E1 ). ^{Because of the} competition ^{of two} forces,

(9) ^{In both cases} A > À (the geometric optics approximation).

namely ^the light field and the elastic force, ^a reorientation with threshold occurs in this case

(the free energy density of NLC under external field has ^a potential ^barrier separating ^two

states of the system 03C8, Q

⁰ and Q, ç = 0).

The solution of the threshold reorientation problem ^can be obtained by separation ^of

variables in the second ( ~ t/J2, Q2, ^cf. Eqs. (9)) approximation (the procedure ^{is described in} [4]). The formation of a grating within the medium, ^induced by ^the light-field, ^{is described} by

the following ^formula (we display ^the expression ^{for the} angle ^Q (z, t ) only ^{as a} function of the

spatial (z ) ^and temporal (t ) variables) :

where

Abound ^{is the} boundary magnitude of the field amplitude A, y is the viscosity, Co is ^the

constant, the eigenvalues v ^are determined by ^the boundary ^condition cp (z

d )

^0.

It should be noted that in the considered approximation (cf. Eqs. (9)) ^the grating period ⁱⁿ equation (10) ^is governed by ^the parameter X ^{which itself} depends ^on the incident intensity

but not only ^on qz. This is ^a very important point ^{because the} dependence X on A determines the feedback in the system (1°), which leads to an intrinsic optical multistability (see below).

Stable (over time) reorientation occurs only for v > 0, ^so that the threshold intensity I tn ^is found from the condition v - 0. The expression determining I th ^can be written in the

following ^form (for simplicity, ^at the interface inside the NLC layer)

where

and _Xlh is the minimal positive solution of the transcendental equation

When kx --> ⁰ (normal incidence, ^«

0) ^we have qz ---> 0, y --> 7T Id, ^and equation (11) ^is

transformed to the well-known expression ^{for the threshold} intensity Ith ⁼ (c/8 7r).

ef2lEl- ( «

0)|2 th for the orientation with ^an adiabatic deformation [1]. ^In

this case ( « = 0 ) the value of Ith ^{is a} minimum for the different angles of incidence. For the

typical ^NLC I th (a

0) - ^{100 W/cm 2} (when d - ²⁵⁰ )JL), ^{so this} magnitude ^is ordinary ^{for the}

CW lasers.

(lo) ^{Like the} dependence ^{of an} oscillation eigen-frequency ^{on the} amplitude ^{for an anharmonic}

oscillator (non-isochronism property) [16].

(11) ^Because of f03C8, Q; ^{« 1} kx ~ 1 ko, e (- sin ao,e + 03C8 ^cos « o, e ), 13 0, e ===: - q;. ^Note, the nonlinear

phase retardation for the passing ^{waves in the} given approximation may nevertheless be sufficiently

large ^here (up ^{to 20} 7r).

1405

The dependence Ilh ^{vs. a as well as the} picture ^{of the} arising gratings ^{are shown in} figure ^4.

Two results follow from these figures : (i) Ith depends ^{on a} (Ith ^{increases vs. « for our} case)

and (ii) Ith changes discontinuously (by jumps) ^{at certain a}

a m.

The physics ^{of these} phenomena ^is quite obvious. When the value of ^{an effective} anisotropy

qZ ( - ^a grating period) ^varies (because of the incident angle a) e.g. increases, this leads to a

reduction in the deformation period A - 1 /q, (see Fig. 4a), ^and consequently, ^the ith ^increases - owing ^to the elastic properties ^{of NLC} being ^{easier to} induce the smooth distortions rather than the sharp ^ones. On the other hand, the number of periods ^{of a}

distorted modulated structure, ^{which can} be fitted within a thickness d subject ^to rigidly specified conditions on the boundary, discretely changes ^{into a}

« m (Fig. 4b) ^{- for this}

case an integrated energy for the formation of the distortions throughout the thickness of a

cell is important. ^{When a « 1 the} discontinuity ^{in the} Ith ^{occurs at the} angles satisfying ^the equation

where m is the number of the discontinuity, « m + > « m . In the given approximation ^{m is not}

too large, and therefore the right-hand ^{side of} equation (13) is less than unity ^{for a} typical

NLC. For example, ^{for MBBA} (El/2 1.6, Ea = 0.8 when k - 0.6 _iL, d

100 U ) ai

5.7°, a2 = 7. 5 °, a3 , 7.9° and so on.

Fig. ^4. - (a) ^{The structure of the} light-induced distortions in NLC (the ^threshold reorientation, oblique

incidence of _o-wave, MBBA, d

100 u) ^{at different a: 1}

^a

0, 03C8 - [sin (7rz/d)] ; ²

a

a = 5.7° (first jump), .p ’" [sin (3 irz/2 d ) ⁺ z/d ] ; ³

^«

^{a2 - 7.5°} (second jump), Q - [sin (5 ^-rzl2 d ) ⁺ z/d ] ; 03C8 is the reorientation angle. ^{The curves on} the left and right ^{hand sides}

determine the modulated structures on upper and lower branches ^at the Ith-jumps, consequently.

(b) Multistability ^of I th ( a ) . ^The dashed vertical line shows the different ^states of the system ^{for a fixed}

value of ^« when I varies (the ^static magnetic ^{field H//E have to be} applied ^{to NLC in a real}

experiment [4].

The Ith jumps ^are given by

As it is evident from equation (13), the increase in /th(a

a m ) ^is sufficiently large.

We shall point ^{out the} possibility ^of strongly reducing Ilh ^{this can be done} using ^a quasi-

static magnetic ^{field H//E} [4]. ^{An effective} reduction in Ith obtained in this case makes it

possible ^to observe the jumps-effect ^{even in a weak CW laser} field, ⁱⁿ particular ^{when the} I th ~ I th ( a = 0 ) is fixed but the H-magnitude ^varies (the ^{H-fields ten times} larger ^{than the}

threshold value may quite easily ^be obtained).

This _means, that the different states through ^{which the} system passes when a varies ^can be

really achieved in an experiment. The successive increase and decrease of the operating parameter (a) ^{lead to a} specific ^intrinsic multistability (the multiple branches in Fig. 4b) (12).

In fact, ^{when a}

a o is fixed (dashed ^{line in} figure 4b) ^the system ^can be in different states for

a given ^I (or H) ^{above the} I th ( « o ) (or Hth ( «o )) ^on the upper branch (see Fig. 4b). Because the

angular distances between two successive jumps ^{as well as the} magnitudes differences of

I th (or Hth) ^{on the} neighbouring branches decrease when m increases, the random fluctuations

can even throw-over the system ^over the different states. What branch precisely is followed

finally ^{when a} is decreased does depend ^{on random} factors. This characterizes the specific

stochastic behavior in the system.

3. Transient _processes for nonlinear wave interactions in a non-homogeneous anisotropic

médium. Oscillations over time.

3.1 EXPERIMENTS FOR DIFFERENT GEOMETRIES. QUALITATIVE DESCRIPTION. - Temporal

instabilities occur in any experiment ^{with the LC when two waves of} orthogonal polarizations

propagate through the nonlinear medium and as a result a reorientation arises in two

orthogonal planes [5]. ^{In fact, the} different schemes of the dynamic self-diffraction of the light

in an anisotropic medium have been realized in experiments ^{for the} present ^{time and} temporal pulsations ^have been obtained (for ^CW input) both in the NLC (two ^linear orthogonal polarizations ^{of the} light propagate through ^the medium) and in the CLC (two

circular polarizations ^{in the} medium) [7, 9-11, 27].

For the first _case, the oscillations (of ^the ring-pattern arising ^{for the} passing ^{laser beam} [1])

which are due to oscillations of the reorientation angle, ^{have been observed in the} following experiments : (a) ^the non-adiabatic deformations excitation in a threshold reorientation of NLC for an oblique ^incidence - figure ^3a [9-11] ; (b) reorientation in a hybrid aligned ^NLC [10] ; (c) ^two counterpropagating ^{coherent waves} with different linear polarizations ^{in NLC}

at normal incidence [10] ; (d) ^an elliptically (circularly) polarized light ^at normal incidence

[10, 11, 27]. Except these, intensity oscillations, others due to polarization ^{rotation have been}

observed in [11] ^{for a} passing light.

In the CLC the oscillations of the intensity ^{for a} passing ^{radiation were obtained} (e) ^{for the}

self-action in the Bragg reflection condition of light [7] ^and (f) ^{for an} oblique ^incidence (o- ^or e-wave) ^{in a} homeotropically (no//z ^{on the} boundaries) aligned sample [10].

The most obvious results are obtained by ^{us in the} (b), (c), (f) geometries (are ^{shown in} Figs. 5, 6, 7). ^{The common behavior of these} systems ^{reduces to the} following : ^{there is a}

smooth transition from the steady-state reorientation via the damped oscillations to the oscillations which are unlimited over time, ^{when a} governing parameter ^varies.

(12) Interesting, that the numbers of the jumps up and down ^{are not} equivalent.

1407

Fig. ^5. Temporal oscillations of 03A6 NL at oblique ^incidence (CW input, Ar+-laser, ^À

0.515 U) ^{of e-}

wave E (1

² kW/cm 2 ) for different angles (a) ; homeotropic aligned ^CLC (mixture ^{of NLC 5CB and}

chiral addition (0.017 % by weight)), d

⁵⁰ u. The director n goes ^out of the Ek-plane ^{due to the} reorientation, because of the chiral addition.

Qualitative explanations ^{of the} temporal instabilities in the LC can be given, ^{based on an}

energy transfer and ^{on a} competition process between the ^waves of different polarizations passing through the nonlinear medium [5,19]. ^This approach is very close ^to the regenerative pulsations discussed e.g. in [21].

In fact, e.g. in the (a)-case, Ith does exist for an o-wave only, ^{but and e-wave} arises due to the reorientation, ^{and an} energy transfer (13) ^{to the e-wave from the} o-wave can decrease the

intensity ^{of o-wave below the threshold} magnitude. ^{Then no} reorientation occurs in the LC and the molecules retum to the initial nonperturbated ^{state. In this case the e-wave} disappears, ^the intensity ^{of the o-wave is} again ^{above the} Ith magnitude ^{and so, all the} cycle repeats, i. e. oscillations take place (14). ^This explanation ^{does work near} the reorientation threshold only ^{and is in} agreement ^{with the} experimental situation, where the oscillations arise in the light field with an intensity exactly ^near Ith.

Note that the solution (10) shows itself a formal possibility of oscillations : it is easy ^to show that v may be complex ^{and therefore the} temporal pulsations ^occur. (A necessary

approximation ^{for accurate} calculations is to take account of the quantities - Q3).

(13) Because of the transient processes ^as well as of the induced shifted gratings ^{due to the} dependence ^{of the} grating period ^{vs. I} (see Eq. (10)).

(14) ^The dependence Ith (a ) ⁱⁿ figure ^{4b leads to a similar case. In fact, because of the} reorientation

the parameters ao.e, 13o.e vary in the way discussed above.

Fig. ^6.

Oscillations of cp ^NL for a hybrid aligned ^{NLC MBBA at different} magnitudes ^{of the} light- intensity ^I (CW Ar+-laser, A

^0.515 f.L), a

90°. The director n is going ^{out of the} yz-plane ^{due to the} reorientation, E//x, d

^{100 it. When} I > 5.3 kW/cm2, ^unstable regimes ^occur.

In the (b)-, (c)-, (e)- ^and (f)-experiments the second component ^of polarization arises inside of the medium also as a result of the laser radiation action on the LC and so the oscillations

can be explained by ^the same reasons as in the (a)-case. ^The period of the oscillations for all these cases is determined by ^the efficiency of the energy transfer between the different components ^{of the} polarizations.

An important point ^to mention is the development of the oscillations in an opposite phase

for two waves in the (c)-geometry ; ^this strongly supports ^the conjecture ^{about the} important

rôle of the energy interchange between the amplitude ^{of the waves} (for ^the travelling, ^{but not} counterpropagating ^{waves, the} phase-retardation ^effects play ^a principal role, because of the interference of the waves, but not of the direct energy interchange).

The nonlinear phase-retardation effects, ^which give the variation of the polarization ^{of the}

wave passing through the nonlinear anisotropic ^{medium, can also lead to} oscillations. In fact,

a linear (on ^the input plane) polarization ^{of the} light ( 03BE

Ey’lEx’ 2 2 ^{= 0 ) becomes elliptical}

1409

Fig. 7. - Oscillations of 03A6 1NL2 in the field of two counterpropagating ^waves with linear polarizations El ^and E2 (CW Ar+-laser, ^À

0.515 u) for different angles ^{a =} Oz, ^{El 1} xz-plane. The director n

goes ^out of the yz-plane ^{due to the} reorientation, ^NLC 5CB, d

²⁰⁰ u, Il

12

²³⁰ W/cm2. The oscillations for two waves (El-solid ^{curve and} E2-dashed curve) ^have opposite phases (indicated by ^the

dashed lines for a

90°).

( 03BE= 0 ) inside the medium when 1 > I th (because ^{of the} reorientation). According ^{to the} experiment [10] ((d)-geometry) ^{the value} Ith ( 03BE ) ^increases (15) ^{and so} when from the

beginning ^the intensity Io > 1 th (g

o ) ^{is fixed,} after reorientation this magnitude 10 may be less than Ith(g). ^{Thus, we have} again ^{a similar case as the} (a)-geometry ^does.

Another type of instabilities (see [22]) which determine the stochastic behavior of a

dynamic system ^{can also occur} in the LC. The important point ^here is that the characteristic time of feedback ( T f), ^{which has to be more than a} relaxation time [22], is determined by ^a spatial scale of the induced grating. This time is quite long (10- 5 ^s up ^to 1 s and even more) [1]. ^{That means that this} type of instabilities in the LC can arise in the all-optical experiments (but ^not only ^{in a} hybrid (opto-electronic) schemes) ^{in contrast to the} ordinary ^resonator

schemes where _{T f} is the round trip time, ^which usually is very small (Tf ^{10- 8} s ).

An exact analysis ^{of these} problems ^meets serious difficulties even for ^a numerical calculation. But for a normal incidence of circularly polarized light ((d)-geometry) ^the

(15 ) ^{This means} that the effective anisotropy of the medium varies because of the reorientation.

analysis ^can be carried ^out analytically. We shall discuss it in the next item (the preliminary

results have been presented ⁱⁿ [19]).

3.2 TEMPORAL INSTABILITIES INDUCED BY A CIRCULARLY POLARIZED LIGHT IN A THRESHOLD REORIENTATION OF NLC. THEORY. - For this case the reorientation of n is characterized by ^two angles 03C8, ^{and cp in the} orthogonal planes. ^We select them in the following

way (Fig. 3b) : ⁿ

(sin Q ^{cos cp ,} sin Ji ^sin cp , cos 03C8 ). ^{This means that the} angle Q determines

as above the deviations of ⁿ from the initial homogeneous molecular orientation (i.e.

nollz) ^{and describes the} oscillations of the ring pattern. ^The angle ^cp determines the azimuthal rotation of n in the xy-plane ^{1 z} (the oscillations of polarizations). Physically, ^{this last case in} principle ^is determined by ^a well known effect, i.e. the rotation momentum arising ^{for the}

molecules which are irradiated by ^an elliptically polarized light (Sadowskii effect), ^{and as a} result, ^the precession ^occurs [23].

Since the input circularly polarized light ^becomes elliptical inside the medium (because ^of

the reorientation) ^the angular ^momentum transfers ; this leads to an energy transfer ^{from the} light ^to the medium and the rotation of the polarization ellipse ^occurs (the ^rotation frequency

is 03A9) [11]. ^{This means that the two} circularly polarized components ^{of the} elliptical polarization have different frequences (w ^and w’) ^{and a} stimulated light scattering process takes place [20]. This four-wave mixing approach (2 il = w’ - w ) is alternative to another one, when the temporal instabilities are explained by ^a phase modulation of the passing

waves. We develop this last approach ^below.

Our study explains all the results of the experiment [11], ^{i.e. the} temporal pulsations ^both

of the light polarization ^{and of an} arising ring pattern ^{as well as} the exhibition of a hysteresis.

Earlier the rotation of the light polarization ^{has been} only interpreted [11].

The procedure ^{of our} calculations is based also on the approach describes in [4, 27] (see

Sect. 2.2).

Basic equations.

Omitting the details we write the basic equations, ^{which are} reduced, firstly, ^{to the} transport equations (the coupling ^{of the} orthogonal components ^{of the} light ^field

inside of NLC) ^and, secondly, ^{to the} equations of motion for a director under the light-field.

The first ones yield in the form

where

Al

A, ^and Ag ^is the eikonal difference for the waves with orthogonal polarizations :

The second ones reduce to a form (the bulk non-adiabatic gratings)