• General contextGeneral context Physics and nonlinear dynamics of semiconductor lasers
Introduction22
• GoalGoal
To understand and identify the physical mechanisms governing the optical instabilities
• MethodologyMethodology
Physical models with adequate level of description
Electromagnetic problem
Semiconductor response
Motivation33
Evolution of compound-cavity modes
Feedback
Mutual 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 Fluctuations
66 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
100
80
60
40
20
0
Time [ns]
Int
ensi
ty
[arb
. un
its]
T. Heil et al, PRA 58, R2674 (1998)
Distributed Feedback Lasers (DFB)
77 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 approximationT. Heil, et al. Opt. Lett.18, 1275 (1999)
solitaryfeedback
Lang-Kobayashi Model (LK)
88 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(2
1)(
2
2
Es
NNgtG
tEtGNe
I
dt
tdN
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 Functions
99 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 Functions
1010 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 Laws
1111 Semiconductor lasers with optical feedback
• Power dropouts ~ Intermittent process
12 LFFI
I
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
Motivation
1313 Bidirectionally coupled semiconductor lasers
–L– l – l l
z
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 Properties
1414 Bidirectionally coupled semiconductor lasers
.||1
)()(
,|)(|)()(
)()()1(2
1)(
22,1
2,12,1
22,12,12,1
2,12,1
2,12,12,1
Es
NNgtG
tEtGNe
I
dt
tdN
tEtGidt
tdE
t
e
),(1,2 ci
c tEe
c
• Experiments Twin Fabry-Perot lasers
Mutual injectionwith delay
• Monochromatic solutions compound-cavity modes
Symmetric: In-phase, anti-phase locking
J. Mulet et al., PRA 65, 063815 (2002)A. Hohl et al., PRL 78, 4745 (1997)
Results: Synchronization Scenario
1515 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 2
1616 Bidirectionally coupled semiconductor lasers
T. Heil et al. PRL 86, 795 (2001)
Experiment Numerics
Inte
nsi
ty
Inte
nsi
ty
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
nsi
ty
Inte
nsi
ty
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 Synchronization
1717 Bidirectionally coupled semiconductor lasers
Intensity
• Within phase locking regime although do not occur dynamically
ttct
ttct
Phase
Deterministicsimulation
1818 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
2121 Polarization resolved intensity noise in VCSELs
What does Determine the Light Polarization State?
x y
z
Oxide layer
Active 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)
2222 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 reverse
electron’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)
2323 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
2424 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 VCSELs
2626 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)
0º90º
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 Model
2727 Spatiotemporal optical model for VCSELs
);( )(
);(2
ˆ);(
trDAAi
trPa
iAiAtrA
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
);( ,,);( trADDA
AitrP t
Instantaneous frequency
• Passive waveguiding
thermal lensing
Arnn
cnn
cA e
ge
);(22
ˆ2
22
L
diffraction
g
ggtl
rr
rrrrnrn
0
)/(1)(
2
Material Model
2828 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
iu
b
iu
DD
iu
DDD 1ln1ln
21ln),,( 0
APAPi
aD
DDBDADrCt
trD 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 Threshold
2929 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
3030 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 Excitation
3131 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
8284
0 400 800 1200 1600time (ps)
wa
vele
ng
th ()
on
1ns
thb
curr
en
t
on= 1 th 9 th
b= 0.85 th
1s 1nstime
Subnanosecond Electrical Excitation
3232 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
8284
0 400 800 1200 1600time (ps)
wa
vele
ng
th ()
on
1ns
thb
curr
en
t
on= 1 th 9 th
b= 0.85 th
1s 1nstime
J. Mulet et al., Proc SPIE 4283, 293 (2001)
Turn-on Delay
3333 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
Tu
rn-o
n d
ela
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 Lensing
3434 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 Refraction
3535 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
Spatiotemporal
Modal expansion
Results Intense thermal lensing
3636 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
en
t
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
Spatiotemporal
Modal expansion
Results weak thermal lensing
3737 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
en
t
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?
3838 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)
3939 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
4141 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