Noise and transverse flow effects on spatio- temporal instabilities in a liquid crystal optical system

Noise and transverse flow effects on spatio- temporal instabilities in a liquid crystal optical

system

Gonzague Agez

Directors:

Pierre Glorieux Eric Louvergneaux

Christophe Szwaj

Spontaneous pattern formation

Homogeneous system Spatial organization

(ordered or disordered, dynamical or stationary)

Examples of natural patterns

clouds zebra skin

gator cuirass

spiral galaxy sand ripples leopard skin

Outline

pattern formation mechanism

I. Pattern formation in the liquid crystal device

II. The noise effects below and near the threshold

III. The effects of a transverse flow

model

The noise in the system

noisy precursors

phase localization at the onset of the 1D instability

Speckle analysis to determine dynamical constants of the system

Introduction to the convective and absolute instabilities

Experimental evidence of noise sustained structures in optics

The properties of 1D and 2D pattern in presence of a transverse flow Noise sustained superlattice and quasicrystals

IV. Conclusion

The set up : 2D configuration

Laser

Mirror Kerr Medium

n=f(I)

x y

Input beam y

x

Camera

x y

Output beam Hexagonal lattice

Laser

Mirror Kerr Medium

n=f(I)

x y

Input beam y

x

Camera

x y

Output beam Spot line

The set up : 1D configuration

The mechanism of selective amplification: the Talbot effect

-A.M -P.M +A.M +PM

+P.M

z

 Homogeneous

front plane

z=0 z=l_T/4 z=2l_T/4 z=3l_T/4 z=l_T

Auto-reproduction Talbot length

l_T=2²/λ₀

The distance of feedback sets the periodicity

Mirror

Spatial selective amplification

Kerr medium P.M :Phase modulation

A.M :Amplitude modulation

The model

Kerr medium n=f(I)

Laser

Mirror (R)

F

B

 



 



 ^  ⁰ ₀

2

0 e F

e R B

shift phase

Ln i

n diffractio

k i d

 





L d

2 2 2

2

n F B

t n n

l

  



 







with ^lD

0

Medium properties:

: diffusion length : relaxation time : Kerr coefficient Device properties:

d : feedback distance L : sample width

k : wave number of light R : reflectivity of the mirror

W.J.Firth, J. Mod. Opt. 37, 151 (1990)

the model

d<0 changes the sign of the

Kerr nonlinearity

4f Image of the mirror

Linear stability analysis

the model

Solution for the refractive index ^{n(r,t)  exp}





 

Marginal curve of stabilty

I₀

I₀

k

k

I

I

 0



diffusion n /

diffractio

2 

 k l

 d

Positive nonlinearity

 0



Positive nonlinearity

 

2

sin 2

1

k R

I_th k





 

 





 2 with k

Theoretical bifurcation diagram Theoretical bifurcation diagram

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

0.9 0.95 1 1.05 1.1 1.15 1.2

Modulation amplitude U

I/Ith

Homogeneous

the model

H

H^- S R

+

-

Positive hexagons

Rolls

Square

Negative hexagons

D’Alessandro et al., Phys. Rev. A, 46(1). (1992)

The liquid crystal layer

2 2 2

2 ( , ) ( , ) ( ) ( , )

) ,

( n r t F r B r t

t t r t n

r n

l_D   



 



 _ 

nˆ

Additive Gaussian white noise

) , ( r t







Fluctuation of the director axis around his mean value nˆ

Homeotropic nematic cell

= Kerr medium Stochastic term

the model

Thermal fluctuations

II.

The noise effects below

and near the threshold

with noise without noise

Above threshold (µ=1.05µ_c) Below threshold (µ=0.95µ_c)

Near field intensity (at the ouput of the LC)

Far field intensity (optical FT)

Numerical simulations Experiment

80 W/cm² 85 W/cm²

0 0.5 1 1.5 -0.5

-1 -1.5

k (µm^-1)

1

-1 0

x/w

0.5 0.5 -1 0.5 0 0.5 1

x/w

0 0.5 1 1.5 -1 -0.5

-1.5

k (µm^-1) k_x

x

x x

y y

k_x k_x

k_y k_y

Noise effects on pattern formation

Noise needs to be taken into account for achieving a realistic description

Analytical expression for the noisy precursors

Linearized expression of the evolution equation of the index fluctuations Δn in presence of noise:

 





) ,

( _{2 2 2}

t r t

r n t l

t r n

d   

        











Experimentally observable quantity: the far field intensity I_FF(k,t)

 





) ,

(k t RI₀ C ^{2 2} n k t n k t k²

I_FF      ^   









      

n~ (k,t) n~(k,t t) n~ (k,t) n~(k,t t)dt

Only the auto-correlation function of Δn can be analytically written :

Analytical expression for the time-averaged far field intensity: I_FF (k,t)

Analytical results

0) t

for ) (

sin(

1

1

4 ^{2 2}  



 

k

k  





  2RI₀ :pump parameter

Properties of the precursors

  ^_^









 

 ^{2 2} ₂ ² ₂

sin 1

cos 2 1

) 0 2 (

) ,

( k k

C k t

k I_FF





 



 



),(tkI FF(u.a.)

) (l_d^1 k

ty) nonlineari (negative

15

 

Analytical results

Marginal

stability curves

k_c:: critical wave number

),(tkI FF(u.a.)

ty) nonlineari (positive

15

 

) (l_d^1 k Analytical results

Marginal

stability curves

k_c:: critical wave number

Properties of the precursors

  ^_^









 

 ^{2 2} ₂ ² ₂

sin 1

cos 2 1

) 0 2 (

) ,

( k k

C k t

k I_FF





 



 



),(tkI FF(u.a.)

15

 

) (l_d^1 k

Analytical results Properties of the precursors

The noisy precursors anticipate the wave number that appear at threshold

  ^_^









 

 ^{2 2} ₂ ² ₂

sin 1

cos 2 1

) 0 2 (

) ,

( k k

C k t

k I_FF





 



 



Experimental results

analytical expression

time-averaged optical FT intensity (u.a.)

+k_c -k_c

5 .

14

 

60 80 100 120 140 160 180 200

55 65 75 85

I₀ (W/cm²)

Evolution of the fondamental Fourier

component

  ²¹

) ,

( _c  _c  ^

FF k t I

Evolution of the time- average experimental optical

FT intensity with input intensity

Experiments

2D configuration: concentrical rings six spots

The crossing of the threshold

Needs a criterion to localize the threshold in the 1D configuration Below threshold Above threshold

1D configuration: no qualitative difference

Intensity profile

k_c k_c

Intensity profile

Experiments

Spatial phase localization

t

x

Temporal evolution of a 1D pattern

below above

Time average

Crossing the threshold = phase localization Experiments

instantaneous Spatial phase

t

x

) (t₁

 ) (t₂



t1

t2

Spatial phase dispersion

)

(t

The crossing of the threshold

threshold without noise

0 20 40 60 100

95 130 164 199 0,5 0,7 0,9 1,1

0 0,1 0,2 0,3 0,4 0,5 80

Indicator for the level of noise

08 .

0

Standard deviation of the spatial phase (degrees) 

Input intensity (W/cm²) Input intensity (u.a.)

Experiment Numerical simulation

The localization of the spatial phase can be used to determine a threshold in presence of noise

Experiments

Inflexion point

Speckle analysis to determine dynamical constants

Application

 



t t r

n ( , ) ^l

^

^{n ( r , t )  f}  ^{n ( r , t ); } 

 ^l



relaxation time diffusion length

Quantitative comparisons between

theory and experiments Experimental measurement of et

 l

Problem :

No direct measurement of the intrinsic relaxation time

(only response time measurements)

New method

Standard diffusive equation:

 

 ^{1 sin 2}  ^{n(r,t) (r,t)}

f    _   

In our case:

Speckle analysis to determine dynamical constants

x y

k_x

k_y

Near field Far field

Analytical expression:

in

E LC

Eout

nˆ

),(tkIFF(u.a)

k (µm^-1)

Ω (s^-1)

2 2 ),(ikE(u.a)

) 1

) ( ,

( _{2 2}⁰ ₂

2

k l t I

k I

d FF L

 





Time-averaged far field intensity

Analytical expression:

 

 ^{2 2}^{2 2 2}

2 2 2

1 ) sin ,

( 









 



i d

i p

i l k

k k E

Square modulus of the double Fourier transform of the output intensity :

s 18 . 0 27

.

2 

 

m l_d,x  9.95 0.31

Diffusion length:

Relaxation time:

experiment fit

experiment fit

Application

III.

Effects of a transverse flow

(non local interaction)

The system with nonlocal interaction

Time t

Liquid crystal

Laser

mirror Liquid

crystal

Laser

mirror

Theory

Transverse flow

See Ramazza et al.

Vorontsov et al.

Ackemann et al.

The system with nonlocal interaction

Time t

Liquid crystal

Laser

mirror Liquid

crystal

Laser

mirror

Theory

Time t

Absolute and convective regimes

Liquid crystal

Laser

mirror

With transverse flow

Convective instability

Absolute instability

(the pattern grows fighting the drift upstream) (the pattern grows but is

advected away by the drift)

Competition between spatial amplification and drift

Analysis of temporal evolution of an initial local perturbation

Theory

Transverse coordinate x Transverse coordinate x

Time t

The impulse response of the system

time

Convective instability

Absolute instability

Local perturbation Local perturbation

x t

x t

x t

x t

x t

0 0 0 0 0

(x/t)

C

L

Evolution R

of the wave packet

Convective threshold

(x/t)

(x/t)

(x/t)R

(x/t)R

(x/t)L

Absolute threshold

x x

Only one critical mode k_c with zero growth rate

A mode k_a with zero group velocity and zero growth rate

0 ) (k_c 

 ⁽ka^{)  0}

Theory

Dispersion relation : (k) with =^r+iⁱ and k=k^r+ikⁱ Spatial mode

x t

x t

x t

x t

x t

0 0 0 0 0

(x/t)

C

L

Evolution R

of the wave packet

Convective threshold

(x/t)

(x/t)

(x/t)R

(x/t)R

(x/t)L

Absolute threshold



x/t



x/t



x/t



x/t



x/t

(x/t)L (x/t)R (x/t)R

(x/t)R

(x/t)L

C

0 A

growth rate λ

(x/t)

C

Determination of convective and absolute thresholds

^{kx k t} ^{k tx k t i} ^{k x k t} ^{k t i} ^{k x k t}

i ^{r r r r}

i i

e e

e e

e ^{( ) ( )} . ^{  ( ) ( )} . ^{  ( )}

 



 



  

 ^

c 

k ^{Spatial amplification:k}_a ^{ k}_a^{r ik}_a^{i }

Theory

Conditions of threshold

Convective Absolute

Unstable wavenumber

Wavenumber defined by

Threshold defined by

k c r

k i

t x k

k c c



 









 





 0; 0;  0





 







a

a r k

r k

r i

k k

0 )

( 

ⁱ k_c ⁱ(k_a)  0

c 

k k_a 

Theory

The dispersion relation Ω(k)

Liquid crystal

Laser

mirror

) , , ( 1

) , , (

1 ^{( , , ) 2}

2

2 n x h y t RI e ² e x y t

l_d  t ^{i i n x y t}   



 



 









 



 



 



 _ ^{ }





i

k) 1 sin( ).

(    ²    ²



n  variation of the refractive index

l_d  diffusion length

 ^{decay time}

n  diffusion length

R  mirror reflectivity

 

x,y,t  gaussian white noise

  noise amplitude

I=|F|^{2 with} F=F₀ e^-(x/w)2  gaussian pumping field

= d/k₀ ^with k₀  laser wave number

h Theory

Evolution equation of the refractive index Δn in presence of noise and with a tilted mirror:

The 1D configuration

Experimental evidence of convective structures

2 .

 4



convective region

Analytical prediction for the liquid crystal device with tilted mirror

Absolute threshold

Convective threshold

Transverse coordinate x

Time

Transverse coordinate x

No pattern Absolute pattern

Without noise

Transverse coordinate x Transverse coordinate x

With noise

Absolute pattern Noise sustained

pattern

Local perturbation → finding a new criterion

noise sustained structures

Time

Experimental evidence of noise sustained structures

(W/cm) I0 ²

(arb. units)

F0 2

1.22

1.02 1.12 100 110 120 130

C

A A

H C

u f H

Numerical simulations Experiments

H: homogeneous state C: convective regime

(noise sustained structures) A: absolute regime

uf d_f

Edge detection

Time

x

Experiments

Conditions of threshold

Analytical results

0 ,

15 



 h





+ - + - + - +

) (l^1 k

Convective threshold



Marginal stability curves µ_c=f(k)

  _feedback distance h : nonlocality

Conditions of threshold

Analytical results

Large convective region

0 ,

15 



 h





) (l^1 k

Convective threshold

 ^Absolute _threshold

] 1 [

c ] 2 [

c

] 4 [

c

] 1 [

c ] 2 [

c ] 4 [

c

Properties of 1D noise sustained patterns

] [p

c

Convective threshold of the 5 first tongues

(p=1 to 5)

] [p

kc

Critical wavenumber of the

5 first tongues (p=1 to 5)

tongue n°:

1 2

3 4

5

Analytical results

10



Experimental data

15

 

Theory h

0 3 6 9 12

Properties of 1D noise sustained patterns

group velocity V_g(k_c) phase velocity

V_φ(k_c)

] [p

c

Convective thresholds of the 5

first tongues (p=1 to 5)

] [p

kc

Critical

wavenumbers of the 5 first tongues

(p=1 to 5)

tongue n°:

1 2

3 4

5

Analytical results

Stationary noise sustained patterns for local minimum in the threshold curve

Experimental stationary noise sustained pattern

Experiments

Generator of stationary patterns with discrete wavelengths ajustable with the drift strength (h)

The 2D configuration

The different types of 2D convective structures

Experiments

 0

cy

k

Convective conditions with a non locality along the x-direction

The 1D type:

The 2D type:

h k_{c n} n

x

 

) (

Vertical rolls

(as in the 1D case)

Horizontal rolls n = 0

Rectangular lattice n > 0

Near field

Far field

Ramazza et al, Phys. Rev. A 54(4), 3472(1996)

2D type convective thresholds

Analytical results

h 1

1.2 1.4 1.6 1.8 2

0 2 4 6 8 10 12 14

p=1 p=2 p=3

p=1 p=2 p=3 n=4

n=3 n=2

n=1 n=0

E D

B’ C’

1.09 1.28 1.46 1.65 1.83 2.02

p=1 p=2 p=3

A B C





A B,B’

E

C,C’

D

n=0n=1n=2

p: tongue index h k_{c n} n

x

 

)

n: from (

1D type convective thresholds

Analytical results

Convective threshold



Convective threshold for the vertical rolls

Convective threshold of 1D type patterns (vertical rolls) Convective threshold of 2D type patterns

(horizontal rolls and rectangular lattices)

Experimental stationary noise sustained pattern

Near field Far field Near field Far field Near field Far field

Horizontal rolls

Vertical rolls Rectangular lattice

experiments

numerical simulations

Dynamical properties of the 2D structures

Convective and absolute regime

Drifting or stationary Vertical rolls

Purely convective structures

(no absolute threshold)

Stationary at the convective threshold (null phase velocity) Horizontal rolls and

rectangular lattice

+ +

Noise sustained superlattice and quasicrystals

Patterns composed of at least 2 different wavelengths

(i.e. composed of 2 previous modes- vertical, horizontal rolls and rectangular lattices)

k_x k_y

Resonance condition :

0

3 1



 



ki

k1

k2 k³

x/w y

0

-1 -0.5 0.5 1

Near field

Experimental superlattice

Experiments

0 0.5

-0.5

t(s)

x/w

0 200 0

-0.1 0.1 0.2

-0.2

k_x (µm^-1)

k_y

stationarity Far

field

k_y

k_x krect

krect k^rollsH

5 17





 h



Noise sustained superlattice

Numerical simulations

With noise Without noise

µ=1.05 ^µ=1.05

Far field

Near field

No structures at long time 5

17





 h



Noise sustained quasicrystal

Analytical results

Numerical simulations

Near field

Pattern

composition Far field filter

Far field

Examples of superlattices

Numerical simulation

0 1

-1

x/w

0 1

-1

-2 2

n=1 n=2

h=12 =-17

n=1

n=2 -1 0 1

x/w

0 1

-1

-2 2

h=21 =-17

0 1

-1

x/w

0 1

-1

-2 2

n=1

h=21.8 =+17

n=0 n=1

B

D

n=1 0

0 1

-1

x/w

0 1

-1

-2 2

h=5.7 =-17 k (en l_d^-1)

k (en l_d^-1) k (en l_d^-1)

k (en l_d^-1)



Conclusion

I. Noisy precursors

II. Transverse flow effects

Evidence of noise effect below the threshold

Complete analytical characterization and very good matching with experiments Spatial phase localization during the onset of the 1D pattern

Speckle analysis to determine experimentally dynamical constants of the system

Determination of convective and absolute threshold

Experimental evidence of noise sustained structures in optics

Dynamical study of the 1D patterns (stationary, drifting, different wavenumber)

Three different families of 2D pattern (horizontal and vertical rolls, rectangular lattice) No absolute threshold for horizontal rolls and rectangular lattice

Resonance condition to build noise sustained superlattice and quasicrystals

Perspectives

Noise sustained pattern properties away from threshold

Pattern nonlinear interaction (between 2 structures either convective or absolute) Experimental evidence of quasicrystals

Thank you for your attention ! And others effects… one funny example

The 2D patterns precursors

Experimental results

Vertical rolls Horizontal rolls Rectangular lattices

Experiments

Analytics