• General contextGeneral context Physics and nonlinear dynamics of semiconductor lasers

Introduction2

• GoalGoalTo understand and identify the physical

mechanisms governing the optical instabilities

• MethodologyMethodology Physical models with adequate level of description

Electromagnetic problem

Semiconductor response

Motivation3

Evolution of compound-cavity modes

FeedbackMutual coupling

• Longitudinal Structures

• Vertical Structures

Light polarization Transverse modes

Free-running

EEL

VCSEL

~1 m

Activelayer

–

Part I: Compound-cavity edge-emitting semiconductor lasers+

+ Part II: Polarization and transverse mode dynamics in vertical-cavity surface-emitting lasers

Contents

+ Perspectives

Part I: Compound-cavity edge-emitting semiconductor lasers+

–

Part I: Compound-cavity edge-emitting semiconductor lasers–

+ Part II: Polarization and transverse mode dynamics in vertical-cavity surface-emitting lasers

+ Perspectives

+

+

Semiconductor lasers with optical feedback

Bidirectionally coupled semiconductor lasers

+ Semiconductor lasers with optical feedback

Contents

• Low frequency fluctuationsweak to moderate feedback, and injection current close-to-threshold

Low Frequency Fluctuations6 Semiconductor lasers with optical feedback

D. Lenstra et al., IEEE J. Quantum Electron. 21, 674 (1985)

C. H. Henry et al., IEEE J. Quantum Electron. 22, 294 (1986)

J. Mørk et al., IEEE J. Quantum Electron. 24, 123 (1986)

J. Sacher et al., Phys. Rev. Lett. 63, 2224 (1989)

T. Sano, Phys. Rev. A 50, 2719 (1994)

M. Giudici et al., Phys. Rev. E 55, 6414 (1997)

T. Heil et al, Phys. Rev. A 58, 2672 (1998)

G. van Tartwijk and G. Agrawal, Prog. Quantum Electron. 22, 43 (1998)

• Power dropouts (slow dynamics)Tn-1 Tn Tn+1 ···

0 200 400 600 800 1000

10080604020

0

Time [ns]

Inte

nsity

[a

rb. u

nits

]

T. Heil et al, PRA 58, R2674 (1998)

Distributed Feedback Lasers (DFB)7 Semiconductor lasers with optical feedback

• Contribution: Statistical characterization of the time T

between consecutive power dropouts

Comparison between experiments and simulations

Experiments DFB lasers Strong side-mode suppression

Modeling Lang-Kobayashi model Single longitudinal mode approximation

T. Heil, et al. Opt. Lett.18, 1275 (1999)

solitaryfeedback

Lang-Kobayashi Model (LK)8 Semiconductor lasers with optical feedback

Weak feedback conditions

Monochromatic solutions: External-cavity modes G.H.M van Tartwijk et al., IEEE JSTQE 1, 446 (1995)

.||1

)()(

,|)(|)()(

)()()1(21)(

2

2

EsNNg

tG

tEtGNeI

dttdN

tEtGidt

tdE

t

e

Extensive numerical simulation of the LK model Long time intervals (~ms) ~ 106 external roundtrips

~ 104 power dropouts

),(0 tEe if

SVA electric field:

Carriers:

Gain: R. Lang and K. Kobayashi, IEEE JQE 16, 347 (1980)

Results: Probability Density Functions9 Semiconductor lasers with optical feedback

• Transitions among regimes– Stable operation

– LFFs

– CC

• Control parameter

Injection current I/Ith

Experiment

LK model

I=0.98 Ith

=2.3 ns, R=5.4%, R16

T. Heil, et al. Opt. Lett.18, 1275 (1999)

Results: Probability Density Functions10 Semiconductor lasers with optical feedback

Experiment Numerics

I=0.98 Ith

I=1.04 Ith

I=1.08 Ith

=2.3 ns, R=5.4%, R16

• Distribution of power dropouts

– Dead time: refractory time– One side exponential decay

• Control parameter

Injection current I/Ith

• Transitions among regimes– Stable operation

– LFFs

– CC

• Transition from Stable LFF regimeT scales with the injection current

Results: Scaling Laws11 Semiconductor lasers with optical feedback

• Power dropouts ~ Intermittent process

12 LFFII

Normalization LFF onset

Power law

=2.3 ns, R=5.4%, R16

J. Mulet et al., Phys. Rev. E 59, 5400 (1999)T. Heil et al., Opt. Lett. 18, 1275 (1999)

2~T –1.0

+ Bidirectionally coupled semiconductor lasers

• Natural generalization of the feedback systemPassive mirror Active semiconductor sectionNonlinear feedback effect

Motivation13 Bidirectionally coupled semiconductor lasers

–L– l – l lz

I1

r

0 L+l

I2

E2

r’ rr’E

1

Synchronization of distant oscillators

Modeling: Electromagnetic problemTasks J. Mulet et al., PRA 65, 063815 (2002)

T. Heil et al., PRL 86, 795 (2001)J. Mulet et al., Proc. SPIE 4283, 293 (2001)

Generalize unidirectional or lateral coupling

• Phenomenological model weak coupling, no detuning

Dynamical Properties14 Bidirectionally coupled semiconductor lasers

.||1

)()(

,|)(|)()(

)()()1(21)(

22,1

2,12,1

22,12,12,1

2,12,1

2,12,12,1

EsNNg

tG

tEtGNe

Idt

tdN

tEtGidt

tdE

t

e

),(1,2 ci

c tEe

c

• Experiments Twin Fabry-Perot lasers

Mutual injectionwith delay

• Monochromatic solutions compound-cavity modesSymmetric: In-phase, anti-phase locking

J. Mulet et al., PRA 65, 063815 (2002)A. Hohl et al., PRL 78, 4745 (1997)

Results: Synchronization Scenario15 Bidirectionally coupled semiconductor lasers

=0 and Ilong coupling times: c ~ 4 ns

Symmetric conditions

1. Onset of coupling-induced instabilities Irregular pulsations with small correlation

2. Transition to correlated dynamics

• Twofold threshold behavior upon coupling increases

)( )(

)( )()(

22

21

21

tPtP

ttPtPtS

Normalized cross-correlation1 2

J. Mulet et al., Proc. SPIE 4283, 293 (2001)

• thsol Correlated power dropouts with a time shift

Results: Dynamics in regime 216 Bidirectionally coupled semiconductor lasers

T. Heil et al. PRL 86, 795 (2001)

Experiment Numerics

Inte

nsity

Inte

nsity

400 450 500 550 600 400 450 500 550 600Time / ns Time / ns

LASER 1

LASER 2

c c

• Synchronized subnanosecond pulsations with a time shift thsol

Inte

nsity

Inte

nsity

Time / ns

Experiment Numerics

0 2 4 6 8 10 0 2 4 6 8 10 Time / ns

c

• Isochronal state + small perturbation Achronal state

Results: Achronal Synchronization17 Bidirectionally coupled semiconductor lasers

Intensity

• Within phase locking regime although do not occur dynamically

ttct

ttct

Phase

Deterministicsimulation

18 Conclusion to Part I

• Power law <T>~(I/ILFF-1)–1 associated with the transition from stable operation to LFFs. Deterministic mechanisms

• Phase-locked compound-cavity modes of two mutually coupled semiconductor lasers

• Twofold threshold behavior: i) coupling-induced instabilities ii) transition to synchronization

• Achronal synchronization persists in symmetrically coupled lasers

• Feedback-induced instabilities appear in singlemode lasers

–

Part I: Compound-cavity edge-emitting semiconductor lasers

Part II: Polarization and transverse mode dynamics in vertical-cavity surface-emitting lasers

+ Perspectives

+

+

+

Part II: Polarization and transverse mode dynamics in vertical-cavity surface-emitting lasers

Contents

+

–

Part I: Compound-cavity edge-emitting semiconductor lasers

Part II: Polarization and transverse mode dynamics in vertical-cavity surface-emitting lasers

Perspectives

–

+

+

Polarization resolved intensity noise in VCSELs

Spatiotemporal optical model for VCSELs

+

Polarization resolved intensity noise in VCSELs

+

Contents

21 Polarization resolved intensity noise in VCSELs What does Determine the Light Polarization State?

x y

z

Oxide layerActive region

Top contact

EyEx

Fundamental mode

Bottom contact

Linear effect

Cavity anisotropies p, a

Preferential directions x (HF), y (LF) Passive material

Two different contributions

Active material (QWs)

Light – matter

Nonlinear effect

No preferential direction imposed by the geometry

M. San Miguel, In semiconductor quantum optoelectronics, 339 (1999)

22 Polarization resolved intensity noise in VCSELs Spin Dynamics and Light Polarization State

Population inversion per spin channel: N Ne – Nh

e e

j

E–E+

+1/2 –1/2

Jz=+3/2 Jz= –3/2

Ne+

Nh+

Electrons CB

Holes HHB

Ne–

Nh–

Four-level system:magnetic sublevels

Spin-flip reverseelectron’s spin

)(tFN

)(tF

noise

20 ||)( )()(

ENNNNNdt

dNeje

spontaneousrecombination rate injection rate spin-flip rate

M. San Miguel, Q. Feng, J.V. Moloney, PRA 54, 1728 (1995)Spin-Flip ModelSpin-Flip Model

EiENNidt

dEpa )( ]1)[1( 0

stimulated recombination

Nonthermal polarization switching and optical bistability– J. Martín-Regalado et al., APL 70, 3550 (1997) – M. B. Willemsen, et al. PRL 82, 4815 (1999)

Nonlinear anisotropies in the spectra of the polarization components– M.P. van Exter, et al. PRL 80, 4875 (1998)

Anticorrelated polarization fluctuations– E. Goodbar et al., APL 67, 3697 (1995) – C. Degen et al., Electron Lett. 34, 1585 (1998)

VCSELs in magnetic fields (Larmor oscillations)– S. Hallstein et al. PRB 56, R7076 (1997) – A. Gahl et al. IEEE JQE 35, 342 (1999)

23 Polarization resolved intensity noise in VCSELs Spin Dynamics and Light Polarization State

)(tFN

)(tF

noise

20 ||)( )()(

ENNNNNdt

dNeje

spontaneousrecombination rate injection rate spin-flip rate

M. San Miguel, Q. Feng, J.V. Moloney, PRA 54, 1728 (1995)Spin-Flip ModelSpin-Flip Model

EiENNidt

dEpa )( ]1)[1( 0

stimulated recombination

24 Polarization resolved intensity noise in VCSELs Anticorrelated Polarization Fluctuations

Effective birefringence

1||2

thspCOs I

I

ROs

ROs

)()(2)()()(

)(

BA

BABAAB SS

SSSC

J. Mulet et al., PRA 64, 023817 (2001)

Spin-flip rate

Birefringence

j

E–E+

=3, p=1 ns–1,s=100 ns–1, I/Ith=1.04

+ Spatiotemporal optical model for VCSELs

• Spatiotemporal model Large signal dynamics Mechanisms that affect the selection of

Transverse modes and Polarization modesTransverse modes and Polarization modes

Transverse Effects in VCSELs26 Spatiotemporal optical model for VCSELs

• Motivation • Joint interplay of transverse and polarizationtransverse and polarization instabilitiesinstabilities

C. Degen et al. J. Opt. B 2, 517 (2000)T. Ackemann et al, J. Opt. B 2, 406 (2000)H. Li et al., Chaos 4, 1619 (1994)

0º 90º

current

• Polarization in the fundamental transverse mode Spin-flip model M. San Miguel et al, PRA 54, 1728 (1995)

Dressed spin-flip model S. Balle et al, Opt. Lett. 24, 1121 (1999)

Spatiotemporal Optical Model27 Spatiotemporal optical model for VCSELs

);( )(

);(2

ˆ);(

trDAAi

trPaiAiAtrA

sppa

t

L

• Transverse dependence of SVA electric fields

cavity losses QW Material Polarization

linear anisotropies spontaneous emission

2yx iAA

A

J. Mulet and S. Balle. IEEE J. Quantum Electron. 38, 291 (2002)

• Material polarization

);( ,,);( trADDAAitrP t

Instantaneous frequency

• Passive waveguiding

thermal lensing

Arnn

cnncA e

ge

);(22

ˆ2

22 L

diffraction

g

ggtl

rrrrrrn

rn

0

)/(1)(

2

Material Model28 Spatiotemporal optical model for VCSELs

Normalized frequency:

Detuning:

/3/1DDu

t S. Balle. Phys. Rev. A 57, 1304 (1998)

J. Mulet and S. Balle. IEEE J. Quantum Electron. 38, 291 (2002)

• Optical susceptibility to circular light

iub

iuDD

iuD

DD 1ln1ln2

1ln),,( 0

APAPi

aD

DDBDADrCttrD jt

**2

2

2

)(2

)();(

D

carrier diffusion stimulated recombination(Spatial Hole Burning)

current profile spin flip for e-spontaneousrecombination

tNND /

• Carrier dynamics (Spin-Flip)

/ , )(exp)( 26crrnC

Results: Transverse Mode Selection at Threshold29 Spatiotemporal optical model for VCSELs

• Analytical theory: Stability analysis “off” solution

• Relevant factors when ( I Ith )- Material gain: Detuning- Modal gain : Confinement thermal lensing & current profile- However SHB neglected

J. Mulet and S. Balle. IEEE JQE 38, 291 (2002)

• Structures

Parameters: c=15 m, g=18 m

ntl=5·10–3 ntl=5·10–4

ntl=10–3

ntl=10–2

30 Spatiotemporal optical model for VCSELs

Parameters: c=15 m, g=18 m, =0.25

Numerical simulations

LP12

sin - cos

LP10

Results: Transverse Mode Selection at Threshold

Subnanosecond Electrical Excitation31 Spatiotemporal optical model for VCSELs

Excitation current pulses Experimental findings

O. Buccafusca, et al., IEEE JQE 35, 608 (1999) M. Giudici, et al., Opt. Comm. 158, 313 (1998) O. Buccafusca, et al., APL 68, 590 (1996)

Delayed onset of higher order modes

8290

8288

8286

82840 400 800 1200 1600

time (ps)

wav

elen

gth

()

on

1ns

thb

curr

ent on= 1 th 9 th

b= 0.85 th

1s 1nstime

Subnanosecond Electrical Excitation32 Spatiotemporal optical model for VCSELs

Evolution of the near fields (Both LP)Results: Bottom-Emitteron = 4 th

12 12 mm 22 22 mm0º 90º0º 90º

Excitation current pulses Experimental findings

O. Buccafusca, et al., IEEE JQE 35, 608 (1999) M. Giudici, et al., Opt. Comm. 158, 313 (1998) O. Buccafusca, et al., APL 68, 590 (1996)

Delayed onset of higher order modes

8290

8288

8286

82840 400 800 1200 1600

time (ps)

wav

elen

gth

()

on

1ns

thb

curr

ent on= 1 th 9 th

b= 0.85 th

1s 1nstime

J. Mulet et al., Proc SPIE 4283, 293 (2001)

Turn-on Delay33 Spatiotemporal optical model for VCSELs

TTT - Fundamental mode

c=12 m c=22 m

O. Buccafusca et al., IEEE JQE 35, 608 (1999)

400

300

200

100

0

0 2 4 6 8 10Ip/Ith

Turn

-on

dela

y (p

s)

c=22 m

J. Mulet et al., Proc. SPIE 4283, 139 (2001)

• Physical mechanisms defining the onset

Spatial hole burning

Blue-shift gain peak (band filling)

Ng

progressive enhance of the gain

of higher-order modes

D

x

y

t

Turn-on Delay versus Thermal Lensing34 Spatiotemporal optical model for VCSELs

Single mode operation: i) Moderate thermal lensing ii) Detuning at the blue side of the gain peak

Turn-on delay when thermal lensing (TL)

StrongTL

WeakTL

cm =4th, p=30 ns– 1, a=0.5 ns–1, J=25 ns–1,

Near Fields

ntl=10–2

ntl=5·10–4

Time [ns]

0º

90º

0º

90º

Optical spectra

Carrier-Induced Index of Refraction35 Spatiotemporal optical model for VCSELs

Gain-guided VCSELsGain-guided VCSELs passive guiding = thermal lensing

• single mode favored by weak thermal lensing

• passive guiding carrier-induced refractive index

• Dynamical modes spatiotemporal model

Thermal lensingThermal lensing?

Spatiotemporal model

Modal expansion

n=10–2 Discb= th, on= 4th,=1.0

SpatiotemporalModal expansion

Results Intense thermal lensing

36 Optical modal expansion Large-signal Current Modulation I

Large-signal modulation

A. Valle et al, JOSAB 12, 1741 (1995)Secondary Pulsations

hole filling

th

th

curr

ent

time

turn-off transients

Small devices (6 m single mode)

Good agreement

J. Mulet and S. Balle. PRA 66, 053802 (2002)

n=3·10–3 Discb= th, on= 4th,=1.0

SpatiotemporalModal expansion

Results weak thermal lensing

37 Optical modal expansion Large-signal Current Modulation II

Large-signal modulation

A. Valle et al, JOSAB 12, 1741 (1995)Secondary Pulsations

hole filling

th

th

curr

ent

time

turn-off transients

Small devices (6 m single mode)

Worse agreement

J. Mulet and S. Balle. PRA 66, 053802 (2002)

Origin of the discrepancies between the models?38 Optical modal expansion

• Optical profiles from the spatiotemporal model during turn-on

(weak TL, n=9·10–4)

intensity

turn-on

• Optical Susceptibility Evolution

Profile shrinkage

Carrier antiguiding (Extra waveguide!)

Spatial hole burning

Initial

Final

J. Mulet and S. Balle. PRA 66, 053802 (2002)

39 Conclusions to Part II

• Selection of transverse modes Close-to-threshold: Onset in a higher-order mode in top-emitters material gain & optical confinement

Large-signal excitation Well defined onset of transverse modes Secondary pulsations

spatial-hole burningcarrier diffusion

band filling

• Relevance of spin determining light polarization Anticorrelated polarization fluctuations Selection of polarization modes

• Optical modal expansion Strong TL: Validity of the modal expansion ntl3·10–3

Weak TL : Distortion of the optical profiles

Spatial redistribution of carrier-induced refractive index

+ Perspectives

–

Part I: Compound-cavity edge-emitting semiconductor lasers+

+ Part II: Polarization and transverse mode dynamics in vertical-cavity surface-emitting lasers

Contents

+ Perspectives

41 Perspectives

• Novel applications of semiconductor lasersNovel applications of semiconductor lasers Encoded communication systems using chaotic carriers

• Nonlinear Optical Feedback • CSK – On-off Phase Shift Keying C. Mirasso et al, IEEE PTL 14, 456 (2002) – T. Heil et al, IEEE JQE 38, 1162 (2002)

• VCSEL with Saturable Absorber – Vectorial Chaos A. Scirè et al, Opt. Lett. 27, 391 (2002)

• Polarization Encoding• Device designDevice design

Spatiotemporal model for VCSELs - Range of single mode operation - Realistic large-signal modulation conditions

• Self-consistent solutions • VCSEL arrays • VCSEL with optical injection / feedback • Mode locking in VCSELs

Extension

+ Acknowledgments

• Technical University of Darmstadt (Germany)

T. Heil and I. Fischer

• Institut Mediterrani d’Estudis Avançats S. Balle and A. Scirè

• Instituto de Física de Cantabria

A. Valle and L. Pesquera

• Universidad de la República Uruguay C. Masoller

C. Mirasso and M. San Miguel

• Vrije University of Brussels J. Danckaert