Grid States and Nonlinear Selection in Parametrically Excited Surface Waves

Tamir Epstein and Jay Fineberg

The Racah Institute of PhysicsThe Hebrew University of Jerusalem

+ = ?

Uns

tabl

e m

ode

Uns

tabl

e m

ode

NL

Stable mode

Grid State – Spatial mode locking new family of Superlattice structures

Talk outline:

• Motivation – why study parametrically excited waves (especially with a lot of excitation frequencies)

• Linear stability and the 2 frequency phase diagram

• What exactly is a Grid State?

• Control of the system using a 3rd frequency

• Grid states in nonlinear optics…

Motivation: What structures are selected when a few distinctly different nonlinear modes

are allowed to interact?

Selection criteria / interaction mechanisms?

Routes to complexity in both space and time

Well -studied nonlinear systems:

Patterns: Single excited spatial mode + secondary instabilities

e.g. R-B, Couette-Taylor, Faraday Instability...

Chaos: Order - Disorder in the temporal domain

Turbulence: A large number of concurrently excited modes

Superlattice states Localized States

H. Arbell and J.F. PRL. 85, 756-60 (2000)

H. Arbell and J.F.PRL 84, 654 (2000) )

H. Arbell and J.F.

PRL 81, 4384 (1998)

Propagating dissipative solitons – single frequency driving with high dissipation

O. Lioubashevski, H. Arbell, and J. F., PRL 76, 3959 (1996 ).

O. Lioubashevski and J. F., Phys. Rev. E63, 5302 (2001).

Experimental System

Gradient lit cylinder

Shaker PC

Temp.

control

CC

D

12 lamps

Imaging system

1.2 meter

Reflecting surface wave

Gradient lit cylinder

CCD

Single Frequency Forcing: a() = a·cos(t)

The Faraday Instability:

a<ac

h

Squares

½

↔ k

a>ac

Why is the fluid response subharmonic (/2) ?

The most effective forcing occurs when the amplitude is maximal

=> geff oscillates at twice the response frequency

geffgeff

geff

d2dt2 = geff(t)

(Mathieu Eqn.)

= [g + (a.sin(t)]+N.L

geff = g + a sin( t)

g

Two-frequencies Forcing:

a(meven, modd) = aevencos(mevent) + aoddcos(moddt)

Two frequency forcing: Controlled mixing of two different modes

meven↔ keven

modd ↔ kodd

meven modd

W. S. Edwards and S. Fauve, Phys. Rev. E47, 788 (1993)

H. W. Muller, Phys. Rev. Lett. 71, 3287 (1993).

H. Arbell and J. F., Phys. Rev. Lett. 81, 4384 (1998).

A. Kudrolli, B. Peir and J. P. Gollub, Physica 123D, 99 (1998).

Superlattice

keven

kodd

kdamped

Spatially• Each tongue describes a well-defined wave number.

Temporally• Each tongue is described by an infinite series of harmonics having a

well-defined parity when t → t + 2/:

Linear Stability Analysis

Single Frequency Forcing: a () = a·cos(t)

k (cm -1)

a c (g)

~½~ ~1½

~2~2½

~3

~4

~3½ -odd ↔ (p +½)

-even ↔ p·

K. Kumar and L. S. Tuckerman , JFM 279, 49 (1994).

Linear Stability Analysis

Two-frequency Forcing: a(meven, modd) = aecos(mevent)+aocos(moddt)

~2½~2

~½~

~1½

koddkeven

4 0 – 5 0 driving

As for single frequency forcing, each tongue has: • A well-defined critical wave number • A well-defined temporal parity (odd – (p+½) , even –p)

An important difference:• new stable tongues corresponding to multiples of /2

Fluid response:

modd = (p +½)

A -A ↔ A3 dominant

2

Besson, Edwards, Tuckerman Phys. Rev E54, 507 (1996)

t t + 2

2

t t + 2

meven = p ·

A A ↔ A2 dominant

0 1 2 30

1

2

3

4

a odd

)g(

aeven )g(

Squares

Squares

Flat

2MS

STC

Hexagons

Linear Stability Analysis

Two-frequencies Forcing: a(meven, modd) = aecos(mevent)+aocos(moddt)

Strategy: Work in the even parity dominated regime to isolate effects due to quadratic interactions

4 /5 driving

Are hexagonal states the only possible states when quadratic interactions are possible?

M. Silber, M. R. E. Proctor, Phys. Rev. Lett. 81, 2450 (1998)

Grid States:• Two co-rotated sets of critical wavevectors, |Ki| = kc

• There exist basis vectors, such that:

121 2 1 0n n n

iK

2gridk2n1

gridk1n

n1: n2 = 3:2

2gridk

1gridk

2221

21

2221

21 22

(cos)nnnn

nnnn

2121

21 nnnnkk cgrid

iK

kgrid= kc / 19kgrid= kc / 7kgrid= kc / 13kgrid= kc / 31kgrid= kc / 37kgrid= kc / 43

1 26, 5

42

n n

kd=2kgrid

1 27, 4

9

n n

kd=kgrid

1 27, 6

22

n n

kd=2kgrid

1 24, 3

28

n n

kd = 2kgrid

1 25, 3

13

n n

kd= kgrid

1 23, 2

22

n n

kd=kgrid

Examples of Grid States

3:2 Grid States in parametrically forced surface waves

kc

kgrid= kc/7

A. Kudrolli, B. Pier and J P. Gollub, Physica D123, 99(1998)

H. Arbell and J. Fineberg, Phys. Rev. Lett. 84, 654 (2000)

3/2 driving ratio 7/6 driving ratio

Experimentally observed states

Questions:

1. Where do kgrid or kd come from?

Are these wavenumbers physically significant?2. Is the correct description of the 3:2 grid state fortuitous – or is this description more general ?3. Can we control the selection of a given state?

Conjectures:• kd may be related to one of the linearly stable tongues

• Perturbing the system with a third frequency, 3, may selectively stabilize a desired nonlinear coupling – choosing a desired kd via the dispersion relation (k)• The phase of the driving frequencies is important:

symmetry generalized phase

C. M. Topaz and M. Silber, Physica D172, 1 (2002).

M. Silber, C. M. Topaz and A. C. Skeldon, Physica D143, 205 (2000).J. Porter, C. M. Topaz and M. Silber, PRL 93, 034502 (2004); C. M. Topaz, J. Porter and M. Silber, Phys. Rev. E 70, 066206 (2004).

“Unperturbed” phase diagram for 7/6 forcing =14Hz

h = 0.3cm.

= 18 cSt..

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

a odd

(g)

aeven (g)

Flat

Square

Hexagons

3:2 grid state

5:3 grid state

Square2MS

STC

3:2 Grid State

5:3 Grid State

Statekd (theory)

kd (expt)

(theory)

(expt)

3:21/7 kc~ 0.38kc

0.38 ± 0.02 kc

21.822

5:31/19 kc~ 0.23kc

0.23 ± 0.01 kc

13.213

At least one additional type of grid state exists!

Can these states be controlled by means of a 3rd frequency?

2 3 4 5

5

6

7a od

d(g

)

aeven

(g)

F l a t

Sq u are

Hex ago n

3 :2

5 :3

2M

S

ST

C

Square

Unperturbed phase diagram

3

Test case: 7/6/2 forcing

=14Hz

h = 0.3cm.

= 18 cSt.

Both the a3 and the phase have aprofound effect on pattern selection!

a = aecos(mevent +1)+aocos(moddt +2) + a3cos(m3t +3)

3

7/6/2 forcing

F l a t

H e x a g o n s

2 3 4 55

6

7

a odd(g

)a

even(g)

3 : 2 G r i d S t a t e s

5 : 3 G r i d s t a t e s

F l a t

H e x a g o n s

2 3 4 55

6

7

a odd

(g)

aeven

(g)

The choice of phase entirely changes the phase diagram but… ONLY where quadratic interactions dominate!

Generalized phase, Cro

ss-C

oupl

ing

Coe

ffic

ient

Cro

ss-C

oupl

ing

Coe

ffic

ient

Coupling angle,

Theoretical predictions for 6/7/2 forcing

J. Porter, C. M. Topaz and M. Silber, PRL 93, 034502 (2004); C. M. Topaz, J. Porter and M. Silber, Phys. Rev. E 70, 066206 (2004).

3:2 Grid State• max = 90 (3:2 Grid State)• No 5:3 grid state predicted

= even – odd – 3

Is the predicted generalized phase, , relevant?

0 60 120 180 240 300 360

-0.2

0.0

0.2

0.4

0.6

Ck(2

2O)

(degrees)0 60 120 180 240 300 360

-0.2

0.0

0.2

0.4

0.6

Ck(1

3O)

(degrees)

To quantify the degree of symmetry of a given k Angular Correlation Function, Ck()

k

Hexagon State

0 30 60 90 120 150 180-1.0

-0.5

0.0

0.5

1.0

Ck (

)

kodd

2-ModeSuperlattice

state

keven

0 30 60 90 120 150 180-1.0

-0.5

0.0

0.5

1.0

Ck (

)

2- -

-

k k k k

k

k k

f f f fC

f f

3:2 Grid State (6/7/2 driving) 5:3 Grid State (6/7/2 driving)

Predicted values of work well for both types of Grid States!(the analysis actually uses only temporal symmetries…)

Do other grid states exist?

28°

Upon increase in dissipation (h=3mm h=2mm)3:2 Grid states are replaced by 4:3 grid states

Predicted: kd = 2kc/13 ~ 0.56 kc; = 27.8

Observed: kd = 0.55 ± 0.02 kc; = 28

k5:3 k3:2 k4:3 kc

Cri

tica

l acc

eler

atio

nh = 0.3 mmh = 0.29 mmh = 0.28 mmh = 0.27 mmh = 0.25 mmh = 0.21 mmh = 0.26 mmh = 0.24 mmh = 0.23 mmh = 0.22 mm

Selection of grid states

Even parity tongues Odd parity tongues

• 3:2 and 4:3 grid states are selected when the corresponding kd approaches the minimum of an even parity tongue• The kd of 5:3 grid states does not correspond to any tongue

h = 0.20 mm

How general are grid states?

We might expect to see them when a nonlinear system has: • The possibility of exciting many different modes (i.e. spatially extended system)

• The dominant (allowed) non-linear interactions are quadratic

E. Pampaloni, S. Residori, S. Soria, and F. T. Arecchi, Phys. Rev. Lett. 78, 1042 (1997).

Patterns in Nonlinear Optics with feedback

• Quadratic nonlinearity• Many discrete (transverse) nonlinear modes that can interact

Laser

Kerr Medium (LCLV)

Feedback + /3 twist

Detector

2D Pattern in the transverse direction

Laser

=28kc

kd

kd /kc = 0.54 ~ 2/13 4:3 Grid State!

kd

kd /kc = 0.25 ~ 1/19 5:3 Grid State!

Grid States in Optical Systems

E. Pampaloni, S. Residori, S. Soria, and F. T. Arecchi,Phys. Rev. Lett. 78, 1042 (1997).

kc

~13

Real Space Fourier Space

Summary

• Quadratic (three-wave) nonlinear interactions + nearby stable modes Grid States ↔ sublattice spanned by linearly stable wavevectors.

Spatial mode locking is preferred by the system

• All of the possible grid states for 6/7 driving can be excited

• Selection of different grid states by the nearest nearly stable modes with exceptions…

• 3rd frequency selection of nonlinear wave interactions. Phase of the perturbation a significant influence on pattern selection.

• Universality: Grid states in nonlinear optics.

The disorder is due to competition between degenerate modes that possess different temporal and spatial symmetries

Squares

2-Mode

SuperlatticeH

exagons

Spatio-T

emporal

Chaos

SSSFlat

1.6 1.8 2.0 2.2 2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

a odd(g

)

aeven (g)

Properties:•Bifurcates directly from the featureless state •No well-defined symmetry in space or time•Both keven and kodd present

Spatio-Temporal Chaos

Fluctuations of Ck()•Decrease of Ck-odd(90°) ↔ Increase of Ck-even(60°)

and vice versa

Transition from ordered state to the Spatio-Temporal Chaos State:

0 30 60 90 120 150 180-1.0

-0.5

0.0

0.5

1.0

Ck (

)

0 30 60 90 120 150 180-1.0

-0.5

0.0

0.5

1.0

Ck (

)

Ck-even(60°)Ck-odd(90°)

•Increase as we reach the STC state

0 1000500-0.5

1

0.5

0Ck()

Control of Spatio-Temporal Chaos

Use of symmetries (temporal parity) of the systems for both control of this type of Spatio-Temporal Chaos and selection of states

Open-loop method:Break the degeneracy by a 3rd controlling frequency – :

a(neven, nodd) = aevencos(nevent)+aoddcos(noddt)+Acos(t)

– even/odd multiple of

The principle:The parity of the perturbation frequency – will induce the selection of the same-parity state.

T. Epstein and J. F., PRL 92, 244502 (2004)

0.00 0.05 0.10 0.15 0.200.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

<C

k-od

d (

90O)>

aa-c

•Locking occurs at small a amplitudes (a< 0.5 acrit.).•The Chaotic state rapidly becomes “controlled”

Locked 2MS state

a-lock

Controlling with odd 3rd frequency of =0

Rapid SwitchingSwitching the parity of the control frequency will trigger a rapid transition between states of different temporal symmetries

This switching is achieved within time in order of a single period.

Odd → Even

(= 0 → = 2 0)

Even → Odd

(= 2 0 → = 0)

Conclusions:

• Characterization of Spatio-Temporal Chaos.

Competition between degenerate modes that possess different temporal and spatial symmetries

• Open loop control of STC.

Small periodic perturbations having well-defined temporal symmetries

A general method.relevant for any parametrically forced system:

e.g. nonlinear optics, forced reaction diffusion