Spring School on Solitons in Optical Cavities Cargèse, May 8-13, 2006 Introduction to Cavity...

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

Introduction to Cavity Solitons and Experiments Introduction to Cavity Solitons and Experiments in Semiconductor Microcavitiesin Semiconductor Microcavities

Luigi A. LugiatoLuigi A. LugiatoDipartimento di Fisica e Matematica, Università dell’Insubria, Como (Italy)Dipartimento di Fisica e Matematica, Università dell’Insubria, Como (Italy)

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

ProgramProgram

- Science behind Cavity Solitons: Pattern Formation (Maestoso)- Science behind Cavity Solitons: Pattern Formation (Maestoso)

- Cavity Solitons and their properties (Andante con moto)- Cavity Solitons and their properties (Andante con moto)

- Experiments on Cavity Solitons in VCSELs (Allegro)Experiments on Cavity Solitons in VCSELs (Allegro)

Future: the Cavity Soliton Laser (Allegro vivace)Future: the Cavity Soliton Laser (Allegro vivace)

- My lecture will be “continued” by that of Willie Firth- My lecture will be “continued” by that of Willie Firth

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

- The other lectures will develop several closely related topics- The other lectures will develop several closely related topics

Optical Pattern FormationOptical Pattern Formation

x

y

z

Optical pattern formation: old historyOptical pattern formation: old history

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

A recent review:LL, Brambilla, Gatti, Optical Pattern Formationin Advances in Atomic, molecular and optical physics, Vol. 40, p 229, Academic Press, 1999

The mechanism for spontaneous optical pattern formation from a The mechanism for spontaneous optical pattern formation from a homogeneoushomogeneous state is astate is a modulational instability, exactly as , exactly as e.g. e.g. in hydrodynamics, in hydrodynamics, nonlinear chemical reactions etcnonlinear chemical reactions etc Modulational instability: a random initial spatial modulation, on top of: a random initial spatial modulation, on top of a homogeneous background, grows and gives rise to the formation of a homogeneous background, grows and gives rise to the formation of aa patternpattern

In optical systems the modulational instability is produced by the In optical systems the modulational instability is produced by the combination of nonlinearity and diffraction.combination of nonlinearity and diffraction. In the paraxial approximation diffraction is described by the In the paraxial approximation diffraction is described by the

transverse Laplacian:transverse Laplacian:2

2

2

22

yx

0 23

Nonlinear Optical Patterns Nonlinear Optical Patterns 11

Nonlinear Optical Patterns Nonlinear Optical Patterns 22 Optical patterns may ariseOptical patterns may arise

in propagationin propagation

in systems with feedback, as in systems with feedback, as e.g. e.g.

optical resonatorsoptical resonators or single feedback mirrorsor single feedback mirrors

Optical patterns arise for many kinds of nonlinearities (Optical patterns arise for many kinds of nonlinearities ((2)(2), , (3)(3), semiconductors,, semiconductors, photorefractives..)photorefractives..)

There are stationary patterns and time-dependent patterns of all kindsThere are stationary patterns and time-dependent patterns of all kinds

InputInput

Nonlinear Medium

nl

CavityCavity OutputOutput(Plane Wave) (Pattern)

Nonlinear Medium

nl

Nonlinear media in cavitiesNonlinear media in cavities

HexagonsHexagons HoneycombHoneycomb RollsRolls

Optical Pattern FormationOptical Pattern Formation

MEAN FIELD MODELSMEAN FIELD MODELS

EaiEEiEiEEt

EI

22

normalized slowly varying envelope of the electric fieldnormalized slowly varying envelope of the electric field

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

input field of frequency input field of frequency 00

cavity detuning parameter , cavity detuning parameter , cc = longitudinal cavity frequency nearest to = longitudinal cavity frequency nearest to 00 0

c

cubic, purely dispersive, Kerr nonlinearitycubic, purely dispersive, Kerr nonlinearity

diffraction parameter

Mean field limit Mean field limit thin sample, high cavity finesse thin sample, high cavity finesse

The purely dispersive case The purely dispersive case (L.L., Lefever PRL (L.L., Lefever PRL 5858, 2209 (1987)), 2209 (1987))

The purely absorptive case The purely absorptive case (LL, Oldano PRA (LL, Oldano PRA 3737, 96 (1988) ;, 96 (1988) ; Firth, Scroggie PRL Firth, Scroggie PRL 7676, 1623 (1996)), 1623 (1996))

EaiEE

CEiEE

t

EI

22

1

2

saturable absorption, C = bistability parametersaturable absorption, C = bistability parameter

MEAN FIELD MODELS as “simple” as pattern formation models in nonlinearMEAN FIELD MODELS as “simple” as pattern formation models in nonlinearchemical reactions, hydrodynamics, chemical reactions, hydrodynamics, etc.etc.

The “ideal” configuration for mean field models (mean field limit, plane mirrors)The “ideal” configuration for mean field models (mean field limit, plane mirrors)has been met in broad area VCSELs (Vertical Cavity Surface Emitting Lasers).has been met in broad area VCSELs (Vertical Cavity Surface Emitting Lasers).

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

FF

thin Kerr slicethin Kerr slice

B | FB | F

Plane MirrorPlane Mirror

- Crossing the Kerr slice, the radiation undergoes phase modulation.Crossing the Kerr slice, the radiation undergoes phase modulation.- In the propagation from the slice to the mirror and back, phase modulationIn the propagation from the slice to the mirror and back, phase modulation is converted into an amplitude modulationis converted into an amplitude modulation- Beautiful separation between the effect of the nonlinearity and that of Beautiful separation between the effect of the nonlinearity and that of diffraction, only one forward-backward propagation diffraction, only one forward-backward propagation Simplicity Simplicity- Strong impact on experiments - Strong impact on experiments

1 1

1 1

1

0

00

0

Encoding a binary number in a 2D pattern??Encoding a binary number in a 2D pattern??

Problem: different peaks of the pattern are strongly correlatedProblem: different peaks of the pattern are strongly correlated

The solution to this problem lies in the concept of The solution to this problem lies in the concept of Localised StructureLocalised Structure

The concept of Localised Structure is general in the field of pattern formation:The concept of Localised Structure is general in the field of pattern formation:

- it has been described in Ginzburg-Landau models (Fauve Thual 1988)it has been described in Ginzburg-Landau models (Fauve Thual 1988) and Swift-Hohenberg models (Glebsky Lerman 1995),and Swift-Hohenberg models (Glebsky Lerman 1995),

- it has been observed in fluids (Gashkov it has been observed in fluids (Gashkov et alet al., 1994), nonlinear chemical ., 1994), nonlinear chemical reactions (Dewel reactions (Dewel et alet al., 1995), in vibrated granular layers (Tsimring., 1995), in vibrated granular layers (Tsimring Aranson 1997; Swinney et al, Science) Aranson 1997; Swinney et al, Science)

1D case1D case

Spatial structures concentrated in a relatively small regionSpatial structures concentrated in a relatively small regionof an extended system, created by stable fronts connecting of an extended system, created by stable fronts connecting

two spatial structures coexisting in the systemtwo spatial structures coexisting in the system

Solution: Localised StructuresSolution: Localised Structures

Theory 1D: P. Coullet, C. Riera, and C. Tresser, Phys. Rev. Lett. Theory 1D: P. Coullet, C. Riera, and C. Tresser, Phys. Rev. Lett. 8484, , 3069-3072 (2000)3069-3072 (2000)

Localised StructuresLocalised Structures Tlidi, Mandel, LefeverTlidi, Mandel, Lefever

- Localised structure = a piece of a pattern - Localised structure = a piece of a pattern

- The scenario of localised structures corresponds to a pattern The scenario of localised structures corresponds to a pattern “ “broken in pieces”broken in pieces” E.g.E.g. a Cavity Soliton corresponds to a single peak of a hexagonal pattern a Cavity Soliton corresponds to a single peak of a hexagonal pattern

((Firth, Scroggie PRL Firth, Scroggie PRL 7676, 1623 (1996)), 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 scenarioof independent localised structures (see e.g. Firth’s lecture)

ProgramProgram








Inte

nsit

y

x y

CAVITY SOLITONSCAVITY SOLITONS

The cavity soliton persists after the passage of the pulse.The cavity soliton persists after the passage of the pulse.Each cavity soliton can be erased by re-injecting the Each cavity soliton can be erased by re-injecting the writing pulse. writing pulse.

Intensity profile

Nonlinear medium nl

Holding beamHolding beam Output Output fieldfield

WritingWritingpulsespulses

- Cavity solitons are independent of one another (provided they are not too Cavity solitons are independent of one another (provided they are not too close to one another) and of the boundary.close to one another) and of the boundary.

- Cavity solitons can be switched on and off independently of one another.- Cavity solitons can be switched on and off independently of one another.

- What is the connection with standard solitons? - What is the connection with standard solitons?

Solitons in propagation problemsSolitons in propagation problems

Temporal Solitons: no dispersion broadeningTemporal Solitons: no dispersion broadening

z

““Temporal” NLSE:Temporal” NLSE: 02

22

t

uuu

z

ui

dispersiondispersionpropagationpropagation

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

Spatial Solitons: no diffraction broadeningSpatial Solitons: no diffraction broadening

““Spatial” NLSE:Spatial” NLSE:

02

22

x

uuu

z

ui 1D

02

2

2

22

y

u

x

uuu

z

ui 2D

x

y

z diffraction

Cavity Solitons are dissipative !Cavity Solitons are dissipative !

E.g.E.g. they arise in the LL model, which is equivalent to a “dissipative NLSE” they arise in the LL model, which is equivalent to a “dissipative NLSE”

022

injuiuuiuut

ui

diffractiondiffractiondissipationdissipation

Dissipative solitons are “rigid”, in the sense that, once the valuesDissipative solitons are “rigid”, in the sense that, once the valuesof the parameters have been fixed, they have fixed characteristics of the parameters have been fixed, they have fixed characteristics (height, radius, etc)(height, radius, etc)

Typical scenario: spatial patterns and Cavity Typical scenario: spatial patterns and Cavity SolitonsSolitons

0,0 0,5 1,0 1,5 2,00,0

0,5

1,0

1,5

2,0

2,5

Stable hom. branch Unstable hom. branch

| E |

| EI |

Honeycomb patternHoneycomb patternRoll patternRoll patternCavity SolitonsCavity Solitons

On/off switching of Cavity SolitonsOn/off switching of Cavity Solitons

- Coherent switchingCoherent switching: the switch-on is obtained by injecting a writing beam : the switch-on is obtained by injecting a writing beam in phasein phase with the holding beam; the switch-offwith the holding beam; the switch-off by injecting a writing beamby injecting a writing beam in opposition of phasein opposition of phase with respect to the writing beam with respect to the writing beam

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

(See Kuszelewicz’s lecture)(See Kuszelewicz’s lecture)

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

~2nsCS onCS on

CS offCS off CS offCS off

~5 nsCS onCS on

Motion of Cavity SolitonsMotion of Cavity Solitons

KEY PROPERTYKEY PROPERTY: Cavity Solitons move in presence of external gradients, : Cavity Solitons move in presence of external gradients, e.g.e.g.

1)1) Phase Gradient in the holding beam,Phase Gradient in the holding beam,2)2) Intensity gradient in the holding beam,Intensity gradient in the holding beam,3)3) temperature gradient in the sample,temperature gradient in the sample,

In the case of 1) and 2) usually the motion is counter-gradient, In the case of 1) and 2) usually the motion is counter-gradient, e.g.e.g. in the case in the caseof a modulated phase profile in the holding beam, each cavity soliton tends toof a modulated phase profile in the holding beam, each cavity soliton tends tomove to the nearest local maximum of the phasemove to the nearest local maximum of the phase

A complete description of CS motion, interaction, clustering A complete description of CS motion, interaction, clustering etc. etc. will be givenwill be givenin Firth’s lecture.in Firth’s lecture.

Phase profile

Possible applications:Possible applications:realisation of reconfigurablerealisation of reconfigurablesoliton matrices, serial/parallelsoliton matrices, serial/parallelconverters, etcconverters, etc

ExperimentsExperiments on Cavity Solitons 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

Review articlesReview articles on Cavity Solitons 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.

ProgramProgram








Nature 419, 699 (2002)

The experiment at INLN (Nice) The experiment at INLN (Nice) and its theoretical interpretationand its theoretical interpretation

was published inwas published in

Tunable Laser

CCD

Holding beamHolding beam

Writing beamWriting beam

Detector linear arrayDetector linear array

VCSEL

BS

BS BS

BS

aom

aom

C

L L

L L

C

M M

Experimental Set-upExperimental Set-upS. Barland, M. Giudici and J. Tredicce, Institut Non-lineaire de Nice (INLN)S. Barland, M. Giudici and J. Tredicce, Institut Non-lineaire de Nice (INLN)

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

Active layer (MQW)

E R

Bottom Emitter (150Bottom Emitter (150m)m)

FeaturesFeatures1) Current crowding at borders (not critical for CS)1) Current crowding at borders (not critical for CS)2) Cavity resonance detuning 2) Cavity resonance detuning (x,y)(x,y)3) Cavity resonance roughness (layer jumps) 3) Cavity resonance roughness (layer jumps) See R.Kuszelewicz et al. See R.Kuszelewicz et al. "Optical self-organisation in bulk and MQW GaAlAs Microresonators", Phys.Rev.Lett. "Optical self-organisation in bulk and MQW GaAlAs Microresonators", Phys.Rev.Lett. 8484, , 6006 (2000)6006 (2000)

n-contact

The VCSELThe VCSELTh. Knoedl, M. Miller and R. Jaeger, University of UlmTh. Knoedl, M. Miller and R. Jaeger, University of Ulm

Bragg reflector

Bragg reflector

GaAs Substrate

E In

p-contact

Experimental resultsExperimental results

In the homogeneous region: In the homogeneous region: formation of a single spot of about formation of a single spot of about

10 10 m diameterm diameter

Observation of differentstructures (symmetry and spatial wavelength) in different spatial regions

Interaction disappears on the right side Interaction disappears on the right side of the device due to cavity resonance of the device due to cavity resonance gradient (400 GHz/150 gradient (400 GHz/150 m, imposed m, imposed by construction)by construction)

Intensity (a.u.)

x (m)

Fre

quen

cy (

GH

z)

x

Above threshold,Above threshold,no injection (FRL)no injection (FRL)

Intensity (a.u.)

x (m)

Fre

quen

cy (

GH

z)x

Below threshold,Below threshold,injected fieldinjected field

Experimental demonstration of independent writing and erasing Experimental demonstration of independent writing and erasing of 2 Cavity Solitons in VCSELS below threshold, of 2 Cavity Solitons in VCSELS below threshold,

obtained at INLN Niceobtained at INLN Nice

S. Barland et al, S. Barland et al, NatureNature 419419, 699 (2002), 699 (2002)

E = E = normalized S.V.E. of the intracavity fieldnormalized S.V.E. of the intracavity fieldEEII = = normalized S.V.E. of the input fieldnormalized S.V.E. of the input field

NN = carrier density scaled to transp. value = carrier density scaled to transp. value = cavity detuning parameter= cavity detuning parameter = linewidth enhancement factor= linewidth enhancement factor2C 2C = bistability parameter= bistability parameter

1 NiN

,2),(),(1 2EaiENCiyxEEyxit

EI

NdyxIENNt

N 22),(Im

WhereWhere

Choice of a simple model: it describes the basic physics and more refined models Choice of a simple model: it describes the basic physics and more refined models showed no qualitatively different behaviours. showed no qualitatively different behaviours.

(x,y) = (C - 0) / + (x,y)

),( yxEI Broad Gaussian (twice the VCSEL)Broad Gaussian (twice the VCSEL)

The ModelThe ModelM. Brambilla, L. A. L., F. Prati, L. Spinelli, and W. J. Firth, Phys. Rev. Lett. M. Brambilla, L. A. L., F. Prati, L. Spinelli, and W. J. Firth, Phys. Rev. Lett. 7979, 2042 (1997)., 2042 (1997).

L.Spinelli, G.Tissoni, M. Brambilla, F. Prati and L. A. L., L.Spinelli, G.Tissoni, M. Brambilla, F. Prati and L. A. L., Phys.Rev.Phys.Rev.A A 5858 , 2542 (1998) , 2542 (1998)

Theoretical interpretationTheoretical interpretation

-2.25 -2.00 -1.75 -1.50 -1.250

1

2

3112.537.5

x (m)150750

|ES|

x (m)0 37.5 75 112.5 150

-2.25 -2.00 -1.75 -1.50 -1.25

Patterns (rolls, filaments)Patterns (rolls, filaments)

Cavity SolitonsCavity Solitons

The vertical line corresponds to the MI boundaryThe vertical line corresponds to the MI boundary

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

Pinning by inhomogeneitiesPinning by inhomogeneities

Broad beam onlyBroad beam only

ExperimentExperiment

Add local perturbationAdd local perturbation

Broad beam onlyBroad beam only

Cavity Solitonsappear close to the MI boundary, Final Position is imposed by roughnessof the cavity resonance frequency

NumericsNumerics

(x,y)

77 Solitons: a more recent achievementSolitons: a more recent achievement

X. Hachair, et al., Phys. Rev. A X. Hachair, et al., Phys. Rev. A 6969, 043817 (2004)., 043817 (2004).

CS can also appear spontaneously ...........CS can also appear spontaneously ...........

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”.

ExperimentExperiment NumericsNumerics

Depending on current injection level two different scenarios are possible(Hachair et al. IEEE Journ. Sel. Topics Quant. Electron., in press)

5% above threshold

0,0 0,2 0,4 0,6 0,8 1,0 1,20,0

0,4

0,8

1,2

1,6

Turing unstable

Hopf unstable

CS

= - 2, = 3, J = 1.01, d = 0.052, / p = 0.01, ||/p = 0.0001

EI

|ES|

20% above threshold

0,0 0,5 1,0 1,5 2,00,0

0,5

1,0

1,5

2,0

2,5 = - 2, = 3, J = 1.2, d = 0.052, / p = 0.01, ||/p = 0.0001

Hopf unstable

Turing unstable

CS on unstable background

EI

|ES|

VCSEL above thresholdVCSEL above threshold

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

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Cavity Soliton LaserCavity Soliton Laser

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

CS are embedded CS are embedded in a dark background: in a dark background: maximum visibilitymaximum visibility..

- In a cavity soliton laser the on/off switching must be - In a cavity soliton laser the on/off switching must be incoherentincoherent

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

CSLCSL

The realization of Cavity Soliton Lasers is the main goal of the The realization of Cavity Soliton Lasers is the main goal of the FET Open project FunFACS.FET Open project FunFACS.

- CW Cavity Soliton Laser- CW Cavity Soliton Laser- Pulsed Cavity Soliton Laser (Cavity Light Bullets)- Pulsed Cavity Soliton Laser (Cavity Light Bullets)

Approaches:Approaches:- Laser with saturable absorberLaser with saturable absorber- Laser with external cavity or external gratingLaser with external cavity or external grating

LPN MarcoussisLPN Marcoussis INLN Nice INLN Nice INFM Como, BariINFM Como, BariUSTRAT GlasgowUSTRAT GlasgowULM Photonics ULM Photonics LAAS ToulouseLAAS Toulouse

ConclusionConclusion

Cavity Solitons are Cavity Solitons are interesting !interesting !

Control of two independent spotsControl of two independent spots

Spots can beSpots can be interpreted interpreted

as CSas CS

50 50 W writing beamW writing beam(WB) in b,d. WB-phase (WB) in b,d. WB-phase changed by changed by in h,k in h,k

All the circled statesAll the circled statescoexist when only the broad coexist when only the broad

beam is presentbeam is present

Spring School on Solitons in Optical Cavities Cargèse, May 8-13, 2006 Introduction to Cavity...

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