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Spring School on Solitons in Optical Cavities Cargèse, May 8-13, 2006PowerPoint Presentation

Spring School on Solitons in Optical Cavities Cargèse, May 8-13, 2006

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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 dell’Insubria, 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)

- 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

- Optical pattern formation: old history
- J. V. Moloney
- A huge and relevant Russian literature
- (A.F. Sukhov, N.N. Rosanov, I. Rabinovich, S.A. Akhmanov,
- M.A. Vorontsov etc.)
- In particular, N.N. Rosanov introduced and studied “Diffractive Autosolitons”,
- precursors of Cavity Solitons

A recent review:

LL, Brambilla, Gatti, Optical Pattern Formation

in Advances in Atomic, molecular and optical physics, Vol. 40,

p 229, Academic Press, 1999

The mechanism for spontaneous optical pattern formation from a homogeneous

state is amodulational 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:

Optical patterns may arise

in propagation

in systems with feedback, as e.g.

optical resonatorsor 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

Nonlinear Medium

Nonlinear Medium

c

c

nl

nl

Input

Cavity

Output

)

)

Pattern

(

Plane Wave

(

Hexagons

Honeycomb

Rolls

Optical Pattern Formation

Mean field limit thin sample, high cavity finesse

The purely dispersive case (L.L., Lefever PRL 58, 2209 (1987))

cavity damping rate (inverse of lifetime of photons in the cavity)

input field of frequency 0

normalized slowly varying envelope of the electric field

cubic, purely dispersive, Kerr nonlinearity

diffraction parameter

cavity detuning parameter , c = longitudinal cavity frequency nearest to 0

MEAN FIELD MODELS

The purely absorptive case (LL, Oldano PRA 37, 96 (1988) ;

Firth, Scroggie PRL 76, 1623 (1996))

saturable absorption, C = bistability parameter

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).

Kerr slice with feedback mirror(Firth, J.Mod.Opt.37, 151 ( 1990))

B | F

F

thin Kerr slice

Plane Mirror

- 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

1

0

0

0

1

1

1

0

Encoding a binary number in a 2D pattern??

Problem: different peaks of the pattern are strongly correlated

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)

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)

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)

Tlidi, Mandel, Lefever

- - 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. Firth’s lecture)

- 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

pulses

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?

Temporal Solitons: no dispersion broadening

“Temporal” NLSE:

z

propagation

dispersion

Spatial Solitons: no diffraction broadening

x

1D

“Spatial” NLSE:

diffraction

z

2D

y

Solitons in propagation problems

Solitons are localized waves that propagate(in nonlinear media)without change of form

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)

Roll pattern

Honeycomb pattern

Typical scenario: spatial patterns and Cavity Solitons

~2ns

CS on

CS on

CS off

CS off

On/off switching of Cavity Solitons

- Coherent switching: the switch-on is obtained by injecting a writing beam
- in phasewith the holding beam; the switch-offby 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 Kuszelewicz’s lecture)

The incoherent switching is more convenient, because it does not require

control of the phase of the writing beam

realisation of reconfigurable

soliton matrices, serial/parallel

converters, etc

Phase profile

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 Firth’s lecture.

- 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

- 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

and its theoretical interpretation

was published in

Nature 419, 699

(2002)

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

L L

BS

C

VCSEL

CCD

C

BS

BS

Detector linear array

BS: beam splitter, C: collimator, L: lens, aom: acousto-optic modulator

E In

E R

The VCSEL

Th. Knoedl, M. Miller and R. Jaeger, University of Ulm

p-contact

Bottom Emitter (150m)

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)

no injection (FRL)

Below threshold,

injected field

x

x

Intensity (a.u.)

Intensity (a.u.)

Frequency (GHz)

Frequency (GHz)

x (m)

x (m)

Observationof different

structures (symmetry and

spatial wavelength)

in different spatial regions

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

Experimental demonstration of independent writing and erasing of 2 Cavity Solitons in VCSELS below threshold, obtained at INLN Nice

S. Barland et al, Nature419, 699 (2002)

The Model erasing of 2 Cavity Solitons in VCSELS below threshold,

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.

x ( erasing of 2 Cavity Solitons in VCSELS below threshold, m)

0 37.5 75 112.5 150

Patterns (rolls, filaments)

Cavity Solitons

-2.25 -2.00 -1.75 -1.50 -1.25

Theoretical interpretation

The vertical line corresponds to the MI boundary

CS form close to the MI boundary, on the red side

Experiment erasing of 2 Cavity Solitons in VCSELS below threshold,

Numerics

(x,y)

Broad beam only

Add local perturbation

Cavity Solitons

appear close to the MI boundary,

Final Position is imposed by roughness

of the cavity resonance frequency

Broad beam only

Pinning by inhomogeneities

7 erasing of 2 Cavity Solitons in VCSELS below threshold, Solitons: a more recent achievement

X. Hachair, et al., Phys. Rev. A 69, 043817 (2004).

CS can also appear spontaneously ........... erasing of 2 Cavity Solitons in VCSELS below threshold,

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”.

VCSEL above threshold erasing of 2 Cavity Solitons in VCSELS below 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

Despite the background oscillations, it is perfectly possible to create and erase solitons by means of the usual techniques of WB injection

Program possible to create and erase solitons by means of the usual techniques of WB injection

- 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

CSL possible to create and erase solitons by means of the usual techniques of WB injection

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

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

50 the W writing beam

(WB) in b,d. WB-phase

changed by in h,k

All the circled states

coexist when only the broad

beam is present

Control of two independent spots

Spots can be

interpreted

as CS

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