Spontaneous Patterns in Nonlinear Optics

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Spontaneous Patterns in Nonlinear Optics
Scottish Universities Physics Alliance
Netwon Institute, Cambridge August 2005 William
J Firth Department of Physics, University of
Strathclyde, Glasgow, Scotland willie_at_phys.strath.
ac.uk
Acknowledgements Thorsten Ackemann, Gonzague Agez
many colleagues/collaborators Funding FP6
FunFACS 2005-08 Leverhulme Trust 2005-08
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Spontaneous Patterns in Nonlinear Optics
Abstract
Spontaneous patterns in optics usually involve
diffraction, rather than diffusion, as the
primary spatial coupling mechanism. The simplest
and most successful system involves a nonlinear
medium with a single feedback mirror. The basic
theory and experimental status of that system
will be reviewed, along with discussion of other
systems such as semiconductor micro-resonators,
and the closely related topic of dissipative
solitons in such systems.
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Traditional Nonlinear Optics
nonlinearity very fast, but very weak
Modulational Instability in 2nd Harmonic
Generation
Type II phase matching for SHG in KTP fundamental
only input input beam ellipticity of 111 input
peak intensity of 57 GW/cm2 Fuerst et. al., Phys.
Rev. Lett, 78, 2760 (1997)
Input Output

• need to accumulate or concentrate nonlinearity
• e.g. use material excitation
• then material determines bandwidth
• and the light has to be essentially
  monochromatic
• leading to envelope patterns (and solitons)

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Inertial NLSE

• Envelope E of a quasi-monochromatic optical
  field,
• coupled to a material excitation N(r) evolves
  like
• N is a refractive index perturbation. Suppose it
  diffuses and relaxes, and is driven by E2
  (optical intensity)
• In steady state, NL Schrödinger type equation
• Strength of nonlinearity scaled by ??(good rule
  of thumb)
• Spatial and temporal bandwidth scaled by 1/lD,
  1/?.

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Scaling Confirmation Nematic Liquid
CrystalLille Group
Using noise speckle pattern for the measurements
of director reorientational relaxation time and
diffusion length of aligned liquid crystals, G.
Agez, P. Glorieux, C. Szwaj, and E. Louvergneaux,
Opt. Comm. 245, 243 (2005)
Other materials and response times
photorefractive ms Na vapour µs semiconductor
ns glass fs.
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Kerr-like Nonlinearity of Nematic Liquid Crystal
Lille Group
Refractive index change of 1 (large!) at
intensity levels eight orders of magnitude lower
than in SHG modulational instability
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Spontaneous Pattern Formation
Needs NONLINEARITY and SPATIAL COUPLING
In NL Optics coupling usually diffractive.
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Mechanism of instability
homo- geneous phase and amplitude
d
length scale L ( d l ) 0.5
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Patterns in Feedback Mirror System

• Instability lobes at Talbot intervals
• Diffusion raises high-K threshold
• Interleaved lobes for Ngt0 and Nlt0.

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Liquid Crystal Patterns Lille
Quasi-pattern due to effect of higher lobe.
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Liquid Crystal Patterns Lille
Tilting mirror Hexagons give way to drifting
rolls, then to static rolls via squares, then
diamonds.
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Self-organization phenomena in nonlinear optical
systems High-order spatial solitons and
dynamical phenomena(ENOC, Aug 8-12 2005)
T. Ackemann, M. Pesch, F. Huneus, J. Schurek, E.
Schöbel, W. Lange
Institut für Angewandte PhysikWestfälische
Wilhelms-Universität Münster
Department of PhysicsUniversity of Strathclyde
Glasgow, Scotland, UK
Email thorsten.ackemann_at_strath.ac.uk
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Feedback mirror patterns in Na vapour

• medium sodium vapor in nitrogen buffer
  gas
• pumping in vicinity of D1-line
• nonlinearity optical pumping between Zeeman
  sub-levels

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Theoretical model

• modeled as homogeneously broadened J1/2 -gt J1/2
  transition
• optical pumping by circularly polarized light
• optical properties (absorption coefficient and
  index of refraction) dependent on z-component of
  magnetization

precession
collisions
pumping
saturation
thermal diffusion
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Length scales

• scaling of length like square-root of
  cell-to-mirror distance expected for
  single-mirror scheme
• size of solitons related to pitch of hexagons
  ? indicates relationship between solitons and
  modulational instability

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Targets and spirals
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Multistability

• switch-on experiments power is switched from
  zero to a value beyond threshold and a snapshot
  is taken (200 cycles)
• dynamical targets and spirals with opposite
  chiralities and different numbers of arms are
  observed for one set of parameters
• most frequent number of arms is obtained from
  histogram

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Na vapor feedback scheme polarization-sensitive.
T. Ackemann et al, Münster
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Soliton Clusters in Na Vapour Feedback Mirror
System Schäpers et al PRL 85 748 (2000)

• Circular polarisation holding beam
• Spontaneous over a small range
• Clusters show preferred distances

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Experimental confirmation that CS exist as
stable/unstable pairs (LCLV feedback system)
Unstable branch identified with marginal
switch-pulse
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Propagating Dissipative Solitons Ultanir et al,
PRL 90 253903-1 (2003)
Peak field of solitons versus gain in alternate
gain/loss waveguide (inset). Current assumes 300
µm width contact patterns on a 1 cm long device.

• Images from output facet when the measured input
  is 160 mW and 16.5 µm FWHM.
• (b) Numerical simulation of the output profile

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Spontaneous Pattern Formation
Needs NONLINEARITY and SPATIAL COUPLING
In NL Optics coupling usually diffractive.
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Optical Cavity Basics
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Experimental Cavity Patterns
VCSEA, external injection, two different
wavelengths (Nice)
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Patterns in a Saturable Absorber Cavity
Using exact numerical techniques, we have
traced existence and stability of stripes as a
function of wavevector and driving.
Eckhaus unstable. White region stable.
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Fourier Control of Optical Patterns

• Natural patterns are imperfect
• May also have wrong symmetry
• Both problems fixed by Fourier feedback control

• Negative feedback of unwanted Fourier components
  (mask)
• Stabilizes existing but unstable states by
  "subtle persuasion"

Martin et al PRL (1997), Harkness et al PRA 58
2577(1998)
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Fourier Control of Optical Patterns
Numerics
Experiment (LCLV)
"Optical turbulence" stabilized to any of three
unstable steady patterns
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Cavity Solitons
Seems possible to create and control regular
optical patterns. For image and informatic
applications of patterns, it should be possible
to selectively create or remove any single
element of the pattern. Requires that a single
isolated spot be stable. Such a structure now
called a CAVITY SOLITON.
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Practical Definition of a Cavity Soliton A cavity
soliton

• is exponentially self-localized transverse to
  its propagation direction
• can be present or absent under the same
  conditions - sub-critical
• has freedom of movement in the localization
  dimension(s)
• IS BOUNDARY-LOCALIZED IN PROPAGATION DIRECTION
• has losses, needs driving, hence has fixed
  amplitude (is an attractor)

VCSR device for cavity solitons in
semiconductors. PIANOS 1998-2002 FunFACS
2005-08.
Experiment INLN (Nice)
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Cavity Soliton Pixel Arrays
Stable square cluster of cavity solitons which
remains stable with several solitons missing
pixel function.
John McSloy, private commun.
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Cavity Solitons linked to Patterns
Coullet et al (PRL 84, 3069 2000) argued that
n-peak cavity solitons generically appear and
disappear in sequence in the neighbourhood of the
locking range within which a roll pattern and a
homogeneous state can stably co-exist.
We have verified this in general terms (in both
1D and 2D), but find much more complexity than
Coullet et al imply.
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Bifurcation Structure of Kerr Cavity Solitons
D. Gomila, W.J. Firth, and A.J. Scroggie

• We have tested Coullets theory for the
  bifurcation structure
• of Kerr cavity solitons.
• This theory seems to properly describe the
  bifurcation structure, but is incomplete
• We find a much higher level of complexity than
  predicted
• Additional homoclinic and heteroclinic
  intersections between the manifolds of fixed
  points and periodic orbits should be considered
• As a consequence new types of localized states
  are found
• Existence of arbitrary soliton-bit sequences
  not proven.

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Applications of Cavity Solitons?

• binary soliton-1, no-soliton -0 logic
• but not viable vs Intel
• transverse mobility may be the key
• e.g. optical buffer memory for serial-parallel.

normal beam also moves, but diffracts away.
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Pinning of Cavity Solitons
Experiment (left) and simulation (right) of
solitons and patterns in a VCSEL amplifier agree
provided there is a cavity thickness gradient
and thickness fluctuations. Latter needed to stop
the solitons drifting on gradient.
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A cavity soliton is self-localized transverse to
its propagation direction, but not self-located

• What determines its location?
• boundary/background effects then at best a
  dressed CS
• control beam informatics, tweezers
• other CS interactions and dynamics
• local imperfections (as in experiments) CS
  microscope?

• CS may move (due to any of above)
• parameter gradients couple to, and excite,
  translational mode
• velocity proportional to gradient force
• no force, no motion CS normally has no inertia

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Coherent/Incoherent Switching and Driving
In Nice VCSEL experiment (left), CS were created
and destroyed with a coherent address pulse,
resp. in and out of phase with Ein. In other
systems switching (both on and off) has been
achieved with incoherent pulses.

• CS are usually composite light/excitation
  structures
• can create/destroy CS through the excitation
  component
• why not DRIVE CS through the excitation?
• such a drive incoherent
• e.g. current drive - Cavity Soliton Laser
• basis of new FunFACS EU project 2005-08

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www.funfacs.org

• main FunFACS aims relate to cavity solitons in
  semiconductor laser systems
• related to pattern formation in these systems
• links to other work by Thorsten Ackemann (while
  at Münster)

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Tilted Waves

• if resonator is too long for emission in
  gain maximum, L gt m l/2 ? tilted wave favored,
  since projection of tilted waves fits into
  resonator, effective wavelength leffgtl

Change in temperature shifts gain curve and
resonance ?detuning
Emission wavelength lower than longitudinal
resonance, off-axis emission
in VCSELs temperature controls detuning
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Length scales
experiment
theory
w/o dispersion
with dispersion
scaling exponent 0.5
scaling exponent 0.49

• confirmation of predicted scaling behavior
• good qualitative agreement of length scales

(cold cavity theory propagation through spacer
layer and Bragg reflectors)
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Patterns and tilted waves
Coordinate space (near-field)

• Infinite laser traveling wave, homogeneous
  emission
• Laser with boundaries reflection creates
  standing wave, line pattern
• Four wave vectors stripe-like, wavy pattern
• ... and more complex cases possible!

Fourier space (far-field)
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Spatial structures
I 12 mA
I 17 mA
I 20 mA
I 15 mA
I 18 mA
I 23 mA
T -10.3C
T 18.3C
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Other aspects Quantum billiard

• For low temperatures patterns with a very high
  wave number, well defined wave vectors
• Pattern bears resemblance to trajectory of a
  quantum particle in a 2d potential well

270-280K
260-270K
240-260K
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G. Robb (Strathclyde) co-workers
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Spontaneous Patterns in Nonlinear Optics
CONCLUSIONS

• Spatial patterns and cavity solitons can be
  found in many nonlinear optical media
• Potential CS applications as pixel arrays, but
  more likely using their transverse mobility
• Micro/nano structured media, and time domain,
  are interesting future directions

Some useful references and material from this
talk on www.funfacs.org