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)[email protected]
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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).