sodium vapor in a single- mirror feedback scheme: a paradigm of self-organizing systems in optics w....

Sodium vapor in a single-mirror feedback scheme: a paradigm of self-organizing

systems in optics

W. LangeInstitut fuer Angewandte Physik

Univ. of Muenster (Germany)w.lange@uni-muenster.de

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“Single-mirror“ system: basic setup

laser beam

nonlinear medium mirror

Firth 1990,d’Alessandro Firth

1991,1992

•spatial coupling via diffraction and reflection•nonlinearity and spatial coupling spatially separated

)()(2 200 rEkrE

zik

)(rE

2)(E

Talbot effect

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Choice of nonlinear medium

Theory: Kerr medium n = n0 + n2I

Experiment:

liquid crystals

Liquid Crystal Light Valves (LCLV)

Photorefractive crystals

alkali vapors, esp. Na

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Coupling between photon spin and atomic spin: production of “orientation” w in atomic ground state (Zeeman pumping)

Nonlinearity in Na vapor: spin-1/2 model

mj =-1/2 mj =+1/2

1

PN2

1

Nonlinear (complex) susceptibility:

(1 – w(E))(1 + w(E))

No Zeeman pumping in linearly polarized light – but polarization instabilityPolarization very critical – add polarizing element in feedback loop

Orientation very sensitive to magnetic field – introduce longitudinal and transverse components

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Self-induced patterns

Stripes (“rolls”) Squares

Hexagons (pos. and neg.)

Transitions between pos. andneg. hexagons via rolls and squares

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Quasipatterns

I

FT

8-fold 12-fold

Aumann et al., Phys. Rev. E66, 046220 (2002)

R. Herrero et al., PRL 82, 4657 (1999)

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Superstructures

hexagonal subgrid

square subgrid

Two slightly different wave numbers involvedE. Große Westhoff et al., Phys. Rev. E67, 025203 (2003)

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Self-induced patterns

• Observed phenomena reproduced in simulations semiquantitatively

• Linear stability analysis available

• Weakly nonlinear analysis in most cases

• Gaussian beam reduces “aspect ratio”, but usually has little influence on patterns

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Polarization instability

(perfect) pitchfork bifurcation

very low threshold

angle between input polarization and main axis of /8-plate

two equivalentstates

obs.

polarizer

medium mirror

analyzer

plate

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Rotated polarizer (

30` 5o

perturbed pitchfork bifurcation

Increased threshold of bistability“Negative branch” preferred

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The complementary case (

“Positive branch“ preferred

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Polarization fronts

In switch-on experiments spontaneous formation of polarization fronts

Analyzer adjustedto suppress input beam

Analyzer adjusted for minimum intensity in region with (a) negativeor (b) positive rotation

Dark line indicatesIsing front

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Circular domains

System is locally brought to

complementary state by “address

beam” of suitable polarization, i. e.

domains are ignited.

Evolution after switching off the

address beam?

In “holding beam” system sits on“disadvantaged branch”

pump rate of“holding beam”

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Front dynamics

• straight fronts are stable• circular domains contract:

“curvature driven contraction” (not in 1D)

Case of equivalent states

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Domain contraction

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Fronts between nonequivalent states

The ‘preferred‘ state expands:

“pressure driven expansion”

nonvanishing

Simulation

i()

(also determined experimentally)

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Evolution of circular domains (simul.)

=-5°=-0°

=5°

=9°

=10°

Expansion and contraction can balanceBut: Equilibrium is not stable

Stabilization of a domain requires additional mechanism

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Circular domains: switching experiment

• “domain” can be switched on and off by an addressing beam• direction of switching determined by the polarization of addressing beam• bistable behavior

intensity ofaddressing beam

time

polarization of addressing beam

In detection: projection on linear pol. state such that holding beam is suppressed

stable stationary “domain”

“domain”extinguished

“domain” ignited

Transverse (feedback) soliton

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Repetition of the experiment

second soliton observed much easier!)

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(Unexpected?) result: family of solitons

backgroundsuppressedwith LP

family of solitons*)

“higher order solitons”“excited states of soliton”

S1 S2 S3 S4

Note: Observed quantity (intensity) is not the state variable!

*) Many predictions for 1D-systems

M. Pesch et al., PRL 95, 143906 (2005)

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Spatially resolved Stokes parameters

Rotation represents orientation! (for low absorption)

M. Pesch, PhD thesis, Muenster 2006 (unpublished)

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Positive Solitons

“target state”“initial (background) state”

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Negative Solitons

“initial state” “target state”

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Comparison with simulations

numerical simulations for Gaussian beam

experiment

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Comparison: medium power – high power

Soliton “sits” on modulated background – homogeneous background not required

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Dynamics of domain wall

low power high power

M. Pesch et al., PRL 99, 153902 (2007)

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Shape of initial domain

Strong diffraction patterns for high power!

Solitons occur when pronounced diffraction patterns are present: self-interaction of circular front by diffraction prevents contraction

Fronts interact with intensity and phase gradients

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Bifurcation diagram

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The mechanism

Curvature-driven contraction+ (pressure- driven expansion)+ diffraction= transverse soliton

Enhancement of diffraction by modulation insta-bility or its precursors required

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High power behavior

pa

tte

rn f

orm

ati

on

Zero crossing of c?

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Labyrinths

• “Negative contraction”• Distances determined by Talbot effect• Limitations by Gaussian beam

J. Schüttler, PhD thesis, Muenster 2007 (unpublished),J. Schüttler et al. (submitted)

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Target patterns and spirals

Occurs in oblique magnetic field, but only in phase gradient produced by self-induced lens (Gaussian beam)

Spirals = azimuthally disturbed target patterns

(observed by sampling method)

F. Huneus et al., Phys. Rev. E 73, 016215 (2006)

F. Huneus, PhD thesis, Muenster 2006 (unpublished)

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Coexistence between spirals and solitons

Solitons do not need a stationary background

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Simulation

E. Schöbel, diploma thesis, Münster 2006 (unpublished)

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Conclusions

• System displays vast variety of phenomena• (Relatively) simple (microscopic) model• Simulations agree with (nearly) all observations

semiquantitatively• Some analysis, but more in-depth theoretical

work welcome• Small aspect ratio• New phenomena due to phase and intensity

gradients in Gaussian beam; beam divergence and convergence need attention

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The team and its supporters

• Thorsten Ackemann(- 2005; now: Strathclyde Univ.)

• Andreas Aumann(-1999; now: consultant)

• Edgar Große Westhoff(-2001; now: product manager)

• Florian Huneus(-2006; now: optical engineering)

• Matthias Pesch(-2006; now: optical engineering)

• Burkhard Schäpers(-2001; now: banking, risk analysis)

• Jens Schüttler(-2007; now: optical engineering)

• Several diploma students

Support by Deutsche Forschungsgemeinschaft

Guests:

• Ramon Herrero (Barcelona) • Yurij Logvin (Minsk)• Igor Babushkin (Minsk/Berlin)

Cooperations:

• Damian Gomila (Palma)• Willie Firth (Glasgow)• Gian Luca Oppo (Glasgow)

Stimulus by Pierre Coullet

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Plane wave simulations of w (large int.)

Hex.up

Hex.down

S 1

S 2

S 3

S 4

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Three-dimensional plot (low input power)

Direct comparison with experiment not possible!

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Contraction of domains

Parameter: Input power

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New type of soliton

exp.

sim.

unobserved

New family

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Time-dependence of domains

i() c()

c(P)

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Time-dependence of domains

i() c()

c(P)

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Time-dependence of domains

i() c()

c(P)

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Contracting domains (simulation)

patt

ern

form

atio

n

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SuS2,1+SiS AS2,1+SiS Experiment

unstable stable

Phase selection on the square grid

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The great mystery

• Angle in the compressed grid: 41.9o (exp.)• Wave vectors have equal length for 41.4o

• Occurs far above threshold• Requires slightly divergent laser beam (phase gradient)

General problem: structures in nonplanar situations

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PP

c

qc q

PP

q q

eine Wellenzahl Wellenzahlband

ij AA2

ikj AAA *ikj AAA *

Origin: phase sensitive cubic coupling

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Patterns on polarized branches

Intensity ofFourier mode

Input power Waveplate rotation

Patterns + Patterns -

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Variable: rotation of waveplate

Bistable behavior

threshold rotation of polarizationPositivebranch

Negativebranch

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Experimental access to Fourier space

ff f f d

“Fourier filter”

mirror

Fourier space

real space

image of nonlinearmedium

nonlinear medium

“far field”or

“near field”

Camera

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Marginal stability curve

linear stability analysis experiment

M. Pesch et al., Phys. Rev. E68, 016209 (2003).