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Front propagation transition induced by

diffraction in a liquid crystal light valve

ALEJANDRO J. ÁLVAREZ-SOCORRO,1CA MILA CAS TILLO-PINTO,1

MARCEL G. CL ERC,1GRE GORIO GON ZÁLEZ -CORTES,1AND MARI O

WILSON2,*

1Departamento de Física and Millennium Institute for Research in Optics, FCFM, Universidad de Chile,

Casilla 487–3, Santiago, Chile

2CONACYT–CICESE, Carretera Ensenada-Tijuana 3918, Zona Playitas, CP 22860, Ensenada, México

*mwilson@cicese.mx

Abstract:

Driven optical systems can exhibit coexistence of equilibrium states. Traveling waves

or fronts between diﬀerent states present complex spatiotemporal dynamics. We investigate the

mechanisms that govern the front spread. Based on a liquid crystal light valve experiment with

optical feedback, we show that the front propagation does not pursue a minimization of free

energy. Depending on the free propagation length in the optical feedback loop, the front speed

exhibits a supercritical transition. Theoretically, from ﬁrst principles, we use a model that takes

it into account, characterizing the speed transition from a plateau to a growing regime. The

theoretical and experimental results show quite fair agreement.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Front propagation occurs in a wide range of physical contexts such as optics, liquid crystals,

granular matter, combustion, population dynamics, chemical reactions, industrial deposition

processes, among others [1]. Since the seminal works of Fisher [2] and Kolmogorov, Petrovsky,

and Piskunov [3] in genetics and population dynamics, respectively, on traveling fronts (called

FKPP fronts) there has been an increasing interest in the study of this phenomenon. The FKPP

front solutions are peculiar of connecting a stable state with an unstable one. The propagation

speed of these fronts depends on the initial conditions. When the disturbance of the unstable state

is bounded, the fronts always propagate with a minimal speed [1]. In liquid crystals, these fronts

have been subject of intense research [4

–

12], since they play a fundamental role in understanding

and applicating average molecular reorientations through light.

Theoretically, the interface dynamics is well understood for variational systems, i.e., systems

whose dynamics is described in terms of the minimization of a physical quantity (free energy,

entropy, and so forth). In contrast, nonvariational systems do not pursue a minimization of a free

energy. Indeed, front propagation into an unstable state does not follow a minimization principle

and its dynamics is less explored. However, front propagation between two stable states in

nonvariational systems has been analyzed by Álvarez-Socorro et al [11]. The non-variationality

is a generic characteristic of nonequilibrium systems [13,14].

This work aims to investigate the eﬀect of diﬀraction in the front propagation between two

domains of average molecular orientations in a liquid crystal light valve (LCLV) with optical

feedback. The diﬀraction produced by the free propagation length

L

governs the nonvariational

eﬀects. Based on a LCLV subjected to optical feedback experiment, front propagation into an

unstable state is studied. Depending on diﬀraction, front speed exhibits a supercritical transition.

Theoretically, using the nonlinear elasticity and optics theory, a model that accounts for this

transition is inferred. The front speed exhibits a transition from a plateau to a growing regime.

The theoretical and experimental results show a quite fair agreement.

Vol. 27, No. 9 | 29 Apr 2019 | OPTICS EXPRESS 12391

#360003

https://doi.org/10.1364/OE.27.012401

© 2019

Received 12 Feb 2019; revised 14 Mar 2019; accepted 14 Mar 2019; published 17 Apr 2019

t= 15.4 s

t= 28.8 s

1 mm

PC L < 0

V0

L > 0

He-Ne

Laser

CCD

SLM PBS

f

BS

M

LCLV

f

f

f

FB

t= 21.8 s

(a) (b)

x

y

z

Fig. 1. Liquid crystal light valve with optical feedback. (a) Schematic representation of the

experimental setup. The LCLV is composed of a nematic liquid crystal ﬁlm sandwiched

in between a glass and a photoconductive plate-over with a dielectric mirror. The light is

injected through a He-Ne laser beam, fstands for lenses with a focal length of 25 cm, PBS

represents a polarizer beam splitter, BS a beam splitter, and SLM is a spatial light modulator

controlled by a computer (PC). The feedback loop is closed by an optical ﬁber bundle (FB).

The free propagation length is denoted by

L

and the image in the LCLV is captured through

a CCD camera. (b) Temporal snapshots sequence of the front propagation showed in the

LCLV taken at

L=

0mm,

ν=

1KHz, and

V0=

2

.

62

Vr ms

. Dark and light area account

for diﬀerent average molecular orientations, respectively. The dashed rectangles mark the

illuminated region.

2. Experimental setup

A simple optical system that presents multistability and nonvariational dynamics is the LCLV

with optical feedback [5

–

11]. Figure 1 schematically shows the used experimental setup. This

setup consists in a liquid crystal cell with a photo-sensitive wall inserted in an optical feedback

loop closed by an optical ﬁber bundle (FB). This experimental array has been designed in

order to have coexisting diﬀraction and polarization interferences. The LCLV structure is

composed by a nematic liquid crystal ﬁlm between a glass with transparent electrodes (ITO)

and a photoconductive plate with a deposited dielectric mirror. The liquid crystal ﬁlm under

consideration is a nematic LC-654, produced by NIOPIK, planar aligned, with thickness

d=

15

µm

. It is a mixture of cyanobiphenyls, with a positive dielectric anisotropy. The optical

free propagation length

L

drives the nonvariational eﬀects. ITO electrodes are used to apply

an external voltage

V0

across the nematic layer. The photoconductor resistance is inversely

proportional to applying illumination [6].

Light suﬀers a phase shift while crossing the LCLV depending on the nematic director state

(i.e., the average liquid crystal molecular orientation), which, in its time, modulates the eﬀective

local voltage applied to the nematic sample. Once a critical voltage is passed, the director tends

to orient along the direction of the applied electric ﬁeld, this reorientation changes local and

dynamically, following the spatial distribution of light present in the photoconductor. Another

eﬀect of this molecular change, due to the liquid crystal birefringent nature, is an induced eﬀective

refractive index change. Thus, the LCLV can be seen as an active Kerr medium, causing a phase

variation

φ=βcos2θ=

2

kd∆ncos2θ

in the reﬂected beam proportional to the incoming beam

intensity

Iw

on the photoconductive side,

θ

stands for the longitudinal average of the molecular

reorientation [6] and

k=

2

π/λ

represents the wavenumber. An expended He-Ne laser beam,

λ=

633 nm and power

Iin =

6

.

5mW/cm

2

, linearly polarized along the vertical

y

axis is used as

light source to illuminate the LCLV. Quasi-one-dimensional conditions are reached thanks to a

computer controlled spatial light modulator (SLM) placed in the input beam. All experiments

were conducted at a working temperature of 28

◦

C. The voltage

V0

and free propagation length

L

are the control parameters.

Vol. 27, No. 9 | 29 Apr 2019 | OPTICS EXPRESS 12392

(a)

0

1

2.2 2.35

rms]

2.55 2.7

V0[V

Iw[a.u.]

0.5 mm

L [mm]

18

22

26

Speed [px/s]

Exp

(b)

-4 -2 0 2 4

VFT

Fig. 2. Experimental characterization of the bifurcation diagram and the front propagation

transition. (a) Bifurcation diagram observed in the LCLV with optical feedback constructed

at

L=

0mm. The points account for the intensity of the reﬂected light by the LCLV as a

function of the applied voltage

V0

. The system exhibits three regions, two monostable and

one bistable between the planar and reoriented state.

VFT

accounts for the critical value

of the reorientation instability, the Fréedericksz transition. The insets stand for respective

snapshots obtained in the indicated voltages. (b) Front speed as a function of free propagation

length

L

at

V0=

2

.

62

Vr ms

. The points account for the front speed measured in pixels

per second. The dashed line is the union between consecutive experimental points. The

continuous curve stands for the trend line of the experimental points.

2.1. Experimental characterization of front propagation into an unstable state

Thanks to the use of the spatial light modulator, a bidimensional channel is illuminated on the

liquid crystal light valve of dimensions 6 mm long by 0.9 mm wide [cf. Fig. 1(b)]. By changing

the voltage

V0

applied to the liquid crystal ﬁlm and monitoring the evolution of light intensity

that goes through the LCLV employing a CCD camera, we characterize the bifurcation diagram

of the director reorientation transition. Figure 2(a) shows the bifurcation diagram obtained. For

small voltage

V0<VFT

, when the molecules are not reoriented, a little light is transmitted in the

optical feedback, which corresponds to the channel being dark [see inset in Fig. 2(a)]. The critical

voltage from which the molecules begin to reorient is designated by

VFT

. On the contrary, when

the director is reoriented, the transmitted light increases and then the channel turns light gray.

Note that the transition of average molecular reorientation of the LCLV with optical feedback is

of the ﬁrst order type [4,8]. Indeed, the transition exhibits an abrupt color change. Besides, when

the voltage is varied, a hysteresis loop is observed between the average molecular conﬁgurations.

The hysteresis region is between the two monostable regions.

To study the front propagation into an unstable state, we follow the strategy: initially applied

voltage is small (

V0VFT

). Hence, the initial conﬁguration is planar, and it is stable.

Subsequently, the applied voltage is increased above a critical value of reorientation bifurcation

(

V0=

2

.

62

Vrms >VFT

). Then the planar state becomes unstable, and the reoriented alignment is

stable. The reoriented state (light color) starts to invade the planar alignment from the edges or

imperfections of the channel. Figure 1(b) shows a sequence of snapshots of the observed front

propagation. From the recording of the front propagation, its speed is determined. Subsequently,

by changing the position of the optical ﬁbers bundle, we can change the value of the free

propagation length

L

, which is the distance where light diﬀraction occurs in our experimental

setup. Figure 2(b) shows the front speed as a function of the free propagation length

L

at ﬁxed

applied voltage

V0

. Unexpectedly, we infer that for small and negative

L

, the front speed is

modiﬁed slightly, but for

L

positive this speed increases and is signiﬁcantly modiﬁed. Therefore,

we experimentally observe that the front speed exhibits a transition between a plateau and a

growing regime. It is worthy to note that

L

does not change the relative stability between the

director conﬁgurations, but rather changes the coupling between the molecular arrangements.

Vol. 27, No. 9 | 29 Apr 2019 | OPTICS EXPRESS 12393

The origin of the front propagation transition will be elucidated in the next section.

3. Theoretical model of the LCLV with optical feedback

Based on the elastic theory, dielectric eﬀects, and optical feedback, close to the Fréedericksz

transition

VFT

, the average molecular reorientation is given by the dimensionless model [4,8, 11]

∂tu=µu+βu2+γu3−u5+∂xx u+bu∂x x u+c(∂xu)2,(1)

where

x

and

t

, respectively, account for the spatial transverse coordinate and time. The order

parameter

u(x,t)

is the amplitude of the critical mode of the average molecular reorientation.

µ

is the bifurcation parameter,

µ

1, that accounts for the competition between the electric

and elastic force, which is proportional to (V0−VFT )/VF T .βis a phenomenological parameter

that accounts for the pretilt induced by the anchoring in the walls of the liquid crystal layer. The

cubic and quintic terms stand for the competition between elastic and electrical forces induced

by optical feedback [8]. The diﬀusion term

∂xx u

describes the transverse elastic coupling. The

coeﬃcients

b

and

c

account, respectively, for the diﬀusion and the nonlinear advection. These

two terms are proportional to the free propagation length

L

and have the same sign. Indeed,

when

L=

0,

b=c=

0. Higher-order terms in Eq. (1) are ruled out by the scaling analysis, since

u∼µ1/4

,

γ∼µ1/2

,

β∼µ3/4

,

∂x∼µ1/2

, and

b∼c∼

0. The previous model (1) satisﬁes an

equation that is governed by the minimization of free energy F[u, ∂xu]at L=0, that is,

∂tu=−∂F

∂u,(2)

where

F=∫dx[−µu2/

2

−βu3/

3

−u4/

4

+u6/

6

+(∂xu)2/

2

]

. However, the diﬀraction eﬀect

generates that the diﬀusion and the nonlinear advection allow the emergence of permanent

dynamics, such as spatiotemporal chaos [15] or oscillatory behaviors [7]. This type of behaviors

is incompatible with a dynamic governed by a principle of minimization. The methodology of

how to derive the parameters

{µ, β, b,c}

and the relation with the physical parameters are given

by Clerc et al [8,11].

The term proportional to

β

breaks the reﬂection symmetry of the amplitude

u

. This eﬀect

always renders the reorientation transition into a discontinuous instability with a small hysteresis.

Note that positive and negative equilibria exist for

β, µ >

0. Besides, the negative values of the

amplitude

u(x,t)

has no physical sense. Figure 3 shows the bifurcation diagram of model Eq. (1).

This model is characterized by exhibiting a ﬁrst-order bifurcation when

µ=

0. Then the system

presents a hysteresis region between two monostable regions. Note that this bifurcation diagram

is qualitatively similar to that observed experimentally [cf. Fig. 2(a)].

An ideal region to study fronts into an unstable state is

µ >

0. In this region of parameter

space, there are fronts between the planar unstable

up

and stable reoriented state

u+

. Figure 4

shows the front propagation for

µ >

0. In order to study the eﬀects of nonvariational terms, we

consider a front solution initially with

b=c=

0and at a given time (

t=

20) we activate the

nonvariational eﬀects (

b=

0and

c=−

30). In the variational regime, the minimal front speed

vmin

is determined by the linear terms, marginal criterion [1], which has the explicit expression

vmin =

2

√µ

. Indeed, if the values of the nonlinear parameters are changed, the front speed does

not change. From Fig. 4, we infer that when the nonvariational terms are included the proﬁle

of the front is modiﬁed. The front solution exhibited a readjust of the spatial proﬁle where the

front suﬀers a back propagation, and after this readjust, the front solution acquires a form with

which it spreads with the marginal speed. Figure 4 depicts the front proﬁles without and with the

inﬂuence of non-variational terms. Unexpectedly, although the front proﬁle is markedly modiﬁed,

the front speed remains constant.

To analyze how the front speed is modiﬁed as a function of the non-variational terms, we

have numerically measured the front speed as a function of

b=c=L

. This is consistent with

Vol. 27, No. 9 | 29 Apr 2019 | OPTICS EXPRESS 12394

the functional dependence of the parameters as a function of the free propagation length

L

.

Figure 3(b) summarizes the front speed as a function of the free propagation length

L

. We

observe that for free negative propagation lengths the front speed is constant and as it increases

the numerical precision it tends to the front speed predicted by the marginal criterion [see

Fig. 3(b)]. It is well-known that numerical discretization eﬀects modify this speed [12]. For

free positive propagation lengths, we observed that the speed of the front grows linearly with

L

.

Hence, we observe that the front speed presents a transition between a plateau and a growing

regime, which is consistent with experimental observations [cf. Figs. 2(b) and 3(b)]. Indeed,

the system exhibits a transition between fronts where its speed is determined by the marginal

criterion (pulled front [1]) to fronts where the nonlinear terms determine the speed, nonlinear

criterion (pushed front [1]). A pulled-pushed transition of fronts, with a speed transition diagram

similar to that shown in Fig. 3(b), is well-known in a cubic reaction-diﬀusion model when the

nonlinear terms are modiﬁed [16]. To ﬁgure out how the front speed is modiﬁed by the presence

of the nonvariational terms a perturbative analysis can be performed.

4. Analytical and numerical analysis of the front speed

Due to the transition between pulled-pushed fronts occurs at free propagation length

L=

0, we

can consider the nonlinear diﬀusion and advection terms as perturbatives (

b,c

1). Let us

consider

uf(x−v0t)

as the front solution for the unperturbed problem of Eq. (1) with

b=c=

0,

where

v0>

2

√µ

is the front speed. To calculate the front speed for the perturbed problem, we

consider the following ansatz,

u(x,t)=ufz≡x−v0t−Û

P(t)+w(x−v0t−p(t)),(3)

where

z

is the coordinate in co-moving system,

Û

P

and

w

account for the correction of the front

speed and the proﬁle function, respectively. Moreover,

Û

P

and

w

are of order of

b∼c∼

, where

1is a small control parameter. Introducing the ansatz (3) in Eq. (1) and leaving only the

-0.4 0.0 0.4 0.8

u0

0.0

0.6

1.2

(a)

=

0.1 (b)

-0.5 0.0 0.5 1.0 1.5

2.00

2.20

2.40 Numerical

Marginal Criteria

Linear Fit

L

Front speed

-1.0

u+

u-

up

Fig. 3. Characterization of bifurcation diagram and front speed of model Eq. (1). (a)

Bifurcation diagram of Eq. (1). Equilibrium amplitude

uo

as a function of the parameter

µ

for ﬁxed

β

. The continuous and dashed curves account for stable and unstable equilibrium,

respectively. These curves were obtained by solving the algebraic equation 0

=µu0+

βu2

0+u3

0−u5

0

;

up

,

u−

, and

up

account for the upper, middle, and lower equilibrium branch,

respectively. The system exhibits three regions, two monostable and one bistable. (b) Front

speed as a function of free propagation length

L

. The continuous curve shows the front

speed of model Eq. (1) obtained numerically with

µ=

1

.

0,

β=

0

.

1, and

b=c=L

. The

dashed horizontal curve accounts for the minimal front speed using the marginal criterion

vmin =2√µ.

Vol. 27, No. 9 | 29 Apr 2019 | OPTICS EXPRESS 12395

070

0

10

20

30

40

0.0 1.2

t=10

t=35

Space

Time

u(x,t)

0.0

1.2

070

0.0

1.2

(b)

(c)

(a)

u(x,t) u(x,t)

x

Fig. 4. Front propagation into an unstable state of model Eq. (1). (a) Spatiotemporal

evolution of amplitud of critical model

u(x,t)

of model Eq. (1) by

µ=

1

.

0,

β=

0

.

1, and

b=

0. Temporal evolution of the front propagation after (

c=

0,

t<

20) and before (

c=−

30,

t>

20) consider the nonvariational advection term. The arrows show the direction of front

propagation in the respective periods. Front proﬁles at

t=

35 (b) and

t=

10 (c), respectively.

terms up to order, after straightforward calculations, we get the linear equation

Lw=−Û

p(t)∂zuf−buf∂zz uf−c(∂zuf)2,(4)

where the linear operator has the form

L ≡ [µ+

2

βuf+

3

u2

f−

5

u4

f+v∂z+∂zz +b(uf∂zz +

∂zz uf)+

2

c∂zuf∂z]

. To solve this linear equation, we use the Fredholm alternative or solvability

condition [17] and obtain

Û

p(t)=vnv ≡ −bhφ|uf∂zz ufi

hφ|∂zufi−chφ|(∂zuf)2i

hφ|∂zufi,(5)

where the symbol

hf|gi ≡ ∫∞

−∞ f(z)g(z)dz

and the function

φ(z)

belong to the kernel of the

adjoint operator of

L

, which is independent of diﬀraction eﬀect. The

φ

function is only accessible

numerically. As a matter of fact, the correction of the front speed of nonvariational origin is

proportional to the free propagation length

L

. When

L>

0(L<0), the previous integrals are

negative (positive) then

vnv

is positive (negative). Hence the front speed has two contributions,

one of variational origin given by the linear criterion and another nonlinear one given by the

nonvariational eﬀects, i.e.,

v=v0+vnv .(6)

Therefore, from this perturbative analysis, it is expected that the front speed increases or decreases

with the free propagation length

L

. However, numerically only for positive diﬀraction, the

front speed increases linearly with the free propagation length [cf. Fig. 3(b)]. Figure 5 shows

how the speed and the proﬁle of the front are modiﬁed when the free propagation length

L

is

changed. Therefore, for

L>

0, the system exhibits an excellent qualitative agreement with Eq.

(5). Despite the above calculation, for

L<

0, the speed of the front remains at the minimum

speed in contradiction with Eq. (6). This behavior can be understood in the following way,

the front modiﬁes its asymptotic proﬁle (cf. Figs. 4 and 5), which increases its propagation

speed given by the linear criterion [18], so that it cancels the speed decrement induced by the

nonvariational eﬀects. Then, the previous perturbative analysis cannot be valid because the base

Vol. 27, No. 9 | 29 Apr 2019 | OPTICS EXPRESS 12396

Fig. 5. Spatiotemporal propagation of front solution into an unstable state for diﬀerent free

propagation lengths. Top panels account for front propagation in the experiment by

L=−

0

.

4

cm (a),

L=

0

.

0cm (b), and

L=

0

.

4cm (c), respectively. Bottom panels stand for the front

propagation of model Eq. (1) by

µ=

1

.

0,

β=

0

.

1, and free propagation length

L=−

1

.

0

(d),

L=

0(e), and

L=

4

.

0(f). The insets account for the front proﬁle at a given instant

experimentally and numerically, respectively.

solution is modiﬁed and it is not a small correction. This mechanism explains the origin of the

pull-pushed transition of fronts, when the disturbance tries to decrease the fronts speed, it adapts

their shape to maintain the minimum speed. Also, when the disturbance increases the speed

of propagation, the system responds by increasing the speed. Therefore, the system exhibits

a pull-pushed transition of fronts at zero free propagation length in the optical feedback loop,

L=

0, when the disturbance begins to increase the minimum speed. Spatiotemporal diagrams

are an adequate tool to characterize this type of transitions. When the parameters are modiﬁed,

if the speed remains unchanged, pulled fronts will not produce any noticeable change in the

spatiotemporal diagram, however when fronts are pushed, the spatiotemporal diagram presents a

change in the front position slope (cf. Fig. 5).

5. Conclusions and remarks

Based on a liquid crystal light valve experiment with optical feedback, we have investigated a

mechanism of speed control in interfaces connecting a stable with an unstable state based on

varying the free propagation length, showing a supercritical transition from a plateau speed to an

increasing speed regime, pulled-pushed front transition. Experimental and theoretically, we have

characterized this supercritical transition, which result shows quite fair agreement.

The possibility of having diﬀerent molecular domains with varying eﬀective refractive indices,

being able to manipulate the speed between these domains, allows the opportunity of having

switches between electronic and optical elements. The presented results open the possibility of

Vol. 27, No. 9 | 29 Apr 2019 | OPTICS EXPRESS 12397

novel photonic devices in such direction.

Funding

Comisión Nacional de Investigación Cientíﬁca y Tecnológica (2015-21151618, 2017-21171672);

Fondo Nacional de Desarrollo Cientíﬁco y Tecnológico (1180903).

Acknowledgments

A. J. Álvarez-Socorro, C. Castillo, M. G. Clerc and G. González-Cortés gratefully acknowledge

the ﬁnancial support from the Millennium Institute for Research in Optics (MIRO).

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Vol. 27, No. 9 | 29 Apr 2019 | OPTICS EXPRESS 12398