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Spring School on Solitons in Optical
Cavities Cargèse, May 8-13, 2006
Introduction to Cavity Solitons and Experiments
in Semiconductor Microcavities
Luigi A. Lugiato Dipartimento di Fisica e
Matematica, Università dellInsubria, Como (Italy)

• Collaborators
• F. Prati, G. Tissoni, L. Columbo (Como)
• M. Brambilla, T. Maggipinto, I.M. Perrini (Bari)
• X. Hachair, F. Pedaci, E. Caboche, S. Barland,
  M. Giudici, J.R. Tredicce, INLN (Nice)
• R. Jaeger (Ulm)
• R. Kheradmand (Tabriz)
• M. Bache (Lingby)
• I Protsenko (Moscow)

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Program
- Science behind Cavity Solitons Pattern
Formation (Maestoso)
- Cavity Solitons and their properties (Andante
con moto)

• Experiments on Cavity Solitons in VCSELs
  (Allegro)

Future the Cavity Soliton Laser (Allegro
vivace)
- My lecture will be continued by that of
Willie Firth

• The lectures of Paul Mandel and Pierre Coullet
  will elaborate
• the basics and the connections with the general
  field of
• nonlinear dynamical systems

- The other lectures will develop several closely
related topics
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Optical Pattern Formation
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A recent review LL, Brambilla, Gatti, Optical
Pattern Formation in Advances in Atomic,
molecular and optical physics, Vol. 40, p 229,
Academic Press, 1999
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Nonlinear Optical Patterns 1
? The mechanism for spontaneous optical pattern
formation from a homogeneous state is a
modulational instability, exactly as e.g. in
hydrodynamics, nonlinear chemical reactions
etc ? Modulational instability a random initial
spatial modulation, on top of a homogeneous
background, grows and gives rise to the formation
of a pattern ? In optical systems the
modulational instability is produced by the
combination of nonlinearity and diffraction.
In the paraxial approximation diffraction is
described by the transverse Laplacian
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Nonlinear Optical Patterns 2
? Optical patterns may arise ? in
propagation ? in systems with
feedback, as e.g. optical
resonators or single feedback
mirrors ? Optical patterns arise for many
kinds of nonlinearities (?(2), ?(3),
semiconductors, photorefractives..) ? There
are stationary patterns and time-dependent
patterns of all kinds
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Optical Pattern Formation
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MEAN FIELD MODELS
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MEAN FIELD MODELS as simple as pattern
formation models in nonlinear chemical reactions,
hydrodynamics, etc. The ideal configuration
for mean field models (mean field limit, plane
mirrors) has been met in broad area VCSELs
(Vertical Cavity Surface Emitting Lasers).
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• Crossing the Kerr slice, the radiation undergoes
  phase modulation.
• In the propagation from the slice to the mirror
  and back, phase modulation
• is converted into an amplitude modulation
• Beautiful separation between the effect of the
  nonlinearity and that of
• diffraction, only one forward-backward
  propagation ? Simplicity
• - Strong impact on experiments

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Encoding a binary number in a 2D pattern??
Problem different peaks of the pattern are
strongly correlated
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The solution to this problem lies in the concept
of Localised Structure

• The concept of Localised Structure is general in
  the field of pattern formation
• it has been described in Ginzburg-Landau models
  (Fauve Thual 1988)
• and Swift-Hohenberg models (Glebsky Lerman
  1995),
• it has been observed in fluids (Gashkov et al.,
  1994), nonlinear chemical
• reactions (Dewel et al., 1995), in vibrated
  granular layers (Tsimring
• Aranson 1997 Swinney et al, Science)

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Solution Localised Structures
Spatial structures concentrated in a relatively
small region of an extended system, created by
stable fronts connecting two spatial structures
coexisting in the system
Theory 1D P. Coullet, C. Riera, and C. Tresser,
Phys. Rev. Lett. 84, 3069-3072 (2000)
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Solution Localised Structures
Spatial structures concentrated in a relatively
small region of an extended system, created by
stable fronts connecting two spatial structures
coexisting in the system
Theory 1D P. Coullet, C. Riera, and C. Tresser,
Phys. Rev. Lett. 84, 3069-3072 (2000)
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Localised Structures
Tlidi, Mandel, Lefever
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• - Localised structure a piece of a pattern
• The scenario of localised structures corresponds
  to a pattern
• broken in pieces
• E.g. a Cavity Soliton corresponds to a single
  peak of a hexagonal pattern
• (Firth, Scroggie PRL 76, 1623 (1996))

• WARNING there is a smooth continuous transition
  from a pattern
• (in the rigid sense of complete pattern or
  nothing at all) to a scenario
• of independent localised structures (see e.g.
  Firths lecture)

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Program
- Science behind Cavity Solitons Pattern
Formation (Maestoso)
- Cavity Solitons and their properties (Andante
con moto)

• Experiments on Cavity Solitons in VCSELs
  (Allegro)

Future the Cavity Soliton Laser (Allegro
vivace)
- My lecture will be continued by that of
Willie Firth

• The lectures of Paul Mandel and Pierre Coullet
  will elaborate
• the basics and the connections with the general
  field of
• nonlinear dynamical systems

- The other lectures will develop several closely
related topics
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CAVITY SOLITONS
Holding beam
Output field
Nonlinear medium ?nl
Intensity profile
The cavity soliton persists after the passage of
the pulse. Each cavity soliton can be erased by
re-injecting the writing pulse.
Intensity
x
y

• Cavity solitons are independent of one another
  (provided they are not too
• close to one another) and of the boundary.
• - Cavity solitons can be switched on and off
  independently of one another.
• - What is the connection with standard solitons?

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Solitons in propagation problems
Solitons are localized waves that propagate (in
nonlinear media) without change of form
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Cavity Solitons are dissipative !
E.g. they arise in the LL model, which is
equivalent to a dissipative NLSE
dissipation
diffraction
Dissipative solitons are rigid, in the sense
that, once the values of the parameters have been
fixed, they have fixed characteristics (height,
radius, etc)
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Typical scenario spatial patterns and Cavity
Solitons
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On/off switching of Cavity Solitons

• Coherent switching the switch-on is obtained by
  injecting a writing beam
• in phase with the holding beam the switch-off
  by injecting a writing beam
• in opposition of phase with respect to the
  writing beam

• Incoherent switching the switch-on and the
  switch-off are obtained
• independently of the phase of the holding beam.
• E.g. in semiconductors, the injection of an
  address beam with a frequency
• strongly different from that of the holding
  beam has the effect
• of creating carriers, and this can write and
  erase CSs.
• (See Kuszelewiczs lecture)

The incoherent switching is more convenient,
because it does not require control of the phase
of the writing beam
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Motion of Cavity Solitons

• KEY PROPERTY Cavity Solitons move in presence of
  external gradients, e.g.
• Phase Gradient in the holding beam,
• Intensity gradient in the holding beam,
• temperature gradient in the sample,
• In the case of 1) and 2) usually the motion is
  counter-gradient, e.g. in the case
• of a modulated phase profile in the holding beam,
  each cavity soliton tends to
• move to the nearest local maximum of the phase

A complete description of CS motion, interaction,
clustering etc. will be given in Firths lecture.
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• Review articles on Cavity Solitons
• L.A.L., IEEE J. Quant. Electron. 39, 193 (2003).
• W.J. Firth and Th. Ackemann, in Dissipative
  solitons, Springer Verlag
• (2005), p. 55-101.

• Experiments on Cavity Solitons
• in macroscopic cavities containing e.g. liquid
  crystals,
• photorefractives, saturable absorbers
• - in single feedback mirror configuration (Lange
  et al.)
• - in semiconductors

• The semiconductor case is most interesting
  because of
• miniaturization of the device
• fast response of the system

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Program
- Science behind Cavity Solitons Pattern
Formation (Maestoso)
- Cavity Solitons and their properties (Andante
con moto)

• Experiments on Cavity Solitons in VCSELs
  (Allegro)

Future the Cavity Soliton Laser (Allegro
vivace)
- My lecture will be continued by that of
Willie Firth

• The lectures of Paul Mandel and Pierre Coullet
  will elaborate
• the basics and the connections with the general
  field of
• nonlinear dynamical systems

- The other lectures will develop several closely
related topics
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The experiment at INLN (Nice) and its
theoretical interpretation was published in
Nature 419, 699 (2002)
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Experimental Set-up S. Barland, M. Giudici and J.
Tredicce, Institut Non-lineaire de Nice (INLN)
L L
aom
Holding beam
aom
M
M
Tunable Laser
Writing beam
BS
BS
C
L L
CCD
VCSEL
BS
BS
C
Detector linear array
BS beam splitter, C collimator, L lens, aom
acousto-optic modulator
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The VCSEL Th. Knoedl, M. Miller and R. Jaeger,
University of Ulm
p-contact
Bottom Emitter (150?m)
Bragg reflector
Active layer (MQW)
Bragg reflector
GaAs Substrate
n-contact
Features 1) Current crowding at borders (not
critical for CS) 2) Cavity resonance detuning
?(x,y) 3) Cavity resonance roughness (layer
jumps) See R.Kuszelewicz et al. "Optical
self-organisation in bulk and MQW GaAlAs
Microresonators", Phys.Rev.Lett. 84, 6006 (2000)
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Experimental results
Interaction disappears on the right side of the
device due to cavity resonance gradient (400
GHz/150 ?m, imposed by construction)
In the homogeneous region formation of a single
spot of about 10 ?m diameter
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Experimental demonstration of independent writing
and erasing of 2 Cavity Solitons in VCSELS below
threshold, obtained at INLN Nice
S. Barland et al, Nature 419, 699 (2002)
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The Model M. Brambilla, L. A. L., F. Prati, L.
Spinelli, and W. J. Firth, Phys. Rev. Lett. 79,
2042 (1997). L.Spinelli, G.Tissoni, M. Brambilla,
F. Prati and L. A. L., Phys.Rev.A 58 , 2542 (1998)

• E normalized S.V.E. of the intracavity field
• EI normalized S.V.E. of the input field
• N carrier density scaled to transp. value
• cavity detuning parameter
• ? linewidth enhancement factor
• 2C bistability parameter

Where
?(x,y) (?C - ?0) / ? ??(x,y)
Broad Gaussian (twice the VCSEL)
Choice of a simple model it describes the basic
physics and more refined models showed no
qualitatively different behaviours.
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Theoretical interpretation
The vertical line corresponds to the MI boundary
CS form close to the MI boundary, on the red side
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Pinning by inhomogeneities
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7 Solitons a more recent achievement
X. Hachair, et al., Phys. Rev. A 69, 043817
(2004).
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CS can also appear spontaneously ...........
Numerics
Experiment
In this animation we reduce the injection level
of the holding beam starting from values where
patterns are stable and ending to homogeneous
solutions which is the only stable solution for
low holding beam levels. During this excursion we
cross the region where CSs exist. It is
interesting to see how pattern evolves into CS
decreasing the parameters. Qualitatively this
animation confirms the interpretation of CS as
elements or remains of bifurcating patterns.
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VCSEL above threshold
Depending on current injection level two
different scenarios are possible (Hachair et al.
IEEE Journ. Sel. Topics Quant. Electron., in
press)
5 above threshold
20 above threshold
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Despite the background oscillations, it is
perfectly possible to create and erase solitons
by means of the usual techniques of WB injection
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Program
- Science behind Cavity Solitons Pattern
Formation (Maestoso)
- Cavity Solitons and their properties (Andante
con moto)

• Experiments on Cavity Solitons in VCSELs
  (Allegro)

Future the Cavity Soliton Laser (Allegro
vivace)
- My lecture will be continued by that of
Willie Firth

• The lectures of Paul Mandel and Pierre Coullet
  will elaborate
• the basics and the connections with the general
  field of
• nonlinear dynamical systems

- The other lectures will develop several closely
related topics
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Cavity Soliton Laser

• A cavity soliton laser is a laser which may
  support cavity solitons (CS)
• even without a holding beam simpler and more
  compact device!

• A cavity soliton emits a set of narrow be18ams
  (CSs), the number and
• position of which can be controlled

CS are embedded in a dark background maximum
visibility.
- In a cavity soliton laser the on/off switching
must be incoherent
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The realization of Cavity Soliton Lasers is the
main goal of the FET Open project FunFACS.
LPN Marcoussis INLN Nice INFM Como, Bari USTRAT
Glasgow ULM Photonics LAAS Toulouse
- CW Cavity Soliton Laser - Pulsed Cavity Soliton
Laser (Cavity Light Bullets)

• Approaches
• Laser with saturable absorber
• Laser with external cavity or external grating

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Conclusion
Cavity Solitons are interesting !
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Control of two independent spots
Spots can be interpreted as CS