Multistability and chaotic transients in semiconductor lasers with delayed feedback
J. Zamora-Munt, C. Masoller, J. García-OjalvoDepartament de Fisica i Enginyeria Nuclear,
Universitat Politecnica de Catalunya,
Terrassa, Barcelona, Spain
4th Rio de la Plata Workshop on Laser Dynamics and Nonlinear PhotonicsDecember 8 – 11, 2009, Piriapolis, Uruguay
mailto:[email protected]
Outline of the talk
Motivation: overview of feedback-induced chaotic dynamics in semiconductor lasers
Theoretical framework: Lang-Kobayashi model
Multistability of coexistent attractors
LFF transients
Summary and conclusions
Outline of the talk
Motivation: overview of feedback-induced chaotic dynamics in semiconductor lasers
Theoretical framework: Lang-Kobayashi model
Multistability of coexistent attractors
LFF transients
Summary and conclusions
Optical feedback induced dynamics in semiconductor lasers
Laser diodes are widely used in optical fiber communication systems
Also in: laser printers, scanners, CDs, DVDs, sensors, etc.
Optical feedback (due to external reflections): is unavoidable in many applications.
Controlled optical feedback can be used to optimize the laser characteristics (lower the threshold, reduce the linewidth and the intensity noise) but spurious optical reflections can induce a variety of instabilities and a chaotic output.
5
Feedback-induced instability: Low Frequency Fluctuations (LFFs)
Photo-detector + oscilloscope signal
Streak camera signal
I. Fischer et al., PRL 76, 220 (1996)G. Vaschenko et. Al, PRL 81, 5536 (1998)Torcini et al, PRA 74, 063801 (2006)
The laser intensity displays dropouts that
are the envelope of fast pulses
LFFs occur when the laser is biased close to threshold
Two questions
Are the LFF dropouts deterministically or stochastically generated?
Is the recovery a deterministic or a stochastic process?
LFFs, coherence collapse & coexistence of LFFs and stable emission
Heil et al, PRA 58, R2672 (1998)
Injection current
Coherence collapse
(fully developed
chaos)
Heil et al, PRA 60, 634 (1999)
Outline of the talk
Overview of feedback-induced dynamics in semiconductor lasers
Theoretical framework: Lang-Kobayashi model
Multi-stability of coexistent attractors
LFF transients
Summary and conclusions
)()(1)||,(1 02 tDetEEENGikdt
dE ifd
Carrier injection
Spontaneous recombination
Lang and Kobayashi, IEEE JQE 16, 347 (1980)
22 |||| EEN,GNJdt
dNN
Optical feedback
Stimulated recombination
Spontaneous emission noise
Time-delayed rate equationsMain approximations: 1) single mode emission (at frequency 0)
2) single reflection in the external cavity
The Lang-Kobayashi Model
Solitary laser
2
2
||1)||,(
E
NENG
Outline of the talk
Overview of feedback-induced dynamics in semiconductor lasers
Theoretical framework: Lang-Kobayashi model
Multi-stability of coexistent attractors
LFF transients
Summary and conclusions
ti
sseEtE
)( 0)(
sNtN )(
fb=2
fb=5
Feedback-induced fixed points:(external cavity modes, ECMs)
Multi-stability of steady state solutions
Stable ECMs
Solitary laser mode
unstable ECMs(antimode)
When the laser operates above threshold and with moderate feedback: coexistence of attractors
C. Masoller PRA (1994); Masoller and Abraham, PRA (1999)
=10 nsJ/Jth =2
Feedback strength, fb
Chaotic attractors develop from the stable ECMs:
N (t)- (t- ) (rad)
s-
0(r
ad)
E
E
time
With stronger feedback: attractor merging
C. Masoller PRA (1994); Masoller and Abraham, PRA (1999)
|E|2
Time (ns)
Deterministic (not noise-induced) attractor switching
The dimension of the chaotic global attractor increases with the delay and the feedback strength
C. Masoller Chaos (1997)
Dynamics when the laser operates close to threshold
T. Sano, PRA 50, 2719 (1994);Van Tarwijk et al, IEEE JSTQE 1, 466 (1995)
LFFs dropouts:
|E|2
Time
Stable ECM
Solitary laser stateJ/Jth 1
LFFs also in model for two-mode lasers (e.g., VCSELs)
C. Masoller and N. B. Abraham, PRA 1999
Outline of the talk
Overview of feedback-induced dynamics in semiconductor lasers
Theoretical framework: Lang-Kobayashi model
Multi-stability of coexistent attractors
LFF transients
Summary and conclusions
Influence of phase-amplitude coupling (alpha-factor)
“Stationary” LFF dynamics: Transient LFF dynamics:
TLFF > 10 to 100 ms TLFF
Transient LFF dynamics
Stochastic initial condition: the stable state of the solitary laser
Noisy initial condition
Parameters and definition of the LFF “lifetime”
Solitary laser stable
solution
Moving window ~1.8 s
When the noise strength is not too large, the duration of the transient is not affected by noise.
Influence of spontaneous emission noise
Black: J=1.02, k=30Red: J=0.98, k=30Blue: J=1.02, k=15
Zamora Munt et al, PRA submitted (2009)
Two maxima Exponential tail as expected in
chaotic transients Noise independent
Statistics of transient times
Probability distribution
of TLFF
Red: D=0Black: D=10-4
With large enough noise the system can escape the
stable ECM (D=10-2 ns-1)
Interplay of stronger noise and gain saturation
=0
=0.05Nonlinear gain: increases the probability of noise-induced
escapes
Zamora Munt et al, PRA submitted (2009)
In good agreement with observations of coexistence LFF – stable emission
Hohl and Gavrielides, PRL 82, 1148 (1999)
Injection current
factor
Gain saturation
Influence of laser parameters
Torcini et al, PRA 74, 063801 (2006)
Zamora Munt et al, PRA submitted (2009)
Feedback phase
Time delay
Influence of optical feedback parameters
Feedback strength
Outline of the talk
Overview of feedback-induced dynamics in semiconductor lasers
Theoretical framework: Lang-Kobayashi model
Multi-stability and coexistence of attractors
LFF transients
Summary and conclusions
Summary and conclusions
We analyzed the statistical properties of transient LFFs for experimentally realistic parameters.
Spontaneous emission noise does not affect transient time statistics (but with large enough noise: noise-induced escapes).
Nonlinear gain increases the transient time and the probability of noise-induced escapes.
Feedback parameters can also tune the transient time.
Our results suggest that the interplay of noise and gain nonlinearities induce the power drop-outs, and then the laser recovers in a deterministic fashion.
Future work: influence of multimode emission.
Acknowledgments: work supported in part by AFOSR grant FA9550-07-1-0238
Thank you for your attention!
