Lecture 2
Synchronization of chaotic oscillators:
Focus on laser diodes with timedelayed feedback
D. RONTANI* and D. S. CITRIN
citrin@gatech.edu
School of Electrical and Computer Engineering
Georgia Institute of Technology
Atlanta, Georgia 30332-0250
and
Unité Mixte Internationale 2958 Georgia Tech-CNRS
Georgia Tech Lorraine
Metz Technopôle, 2 rue Marconi
57070 Metz, France
*Now at Department of Physics, Duke University, Durham, North Carolina
Outline
•
Review
•
Chaos in Time-Delay Systems
•
Introduction to Synchronization
•
Chaos Synchronization
•
Optical Chaos Cryptography
•
Conclusion
Visualizing Chaos: Strange Attractors
INTRODUCTION TO CHAOS THEORY
▶ Representations of chaotic states
▶
▶
The evolution of the state variable can be represented as 1D time
series
Evolution of the state variable can be represented simultaneously in a nD
phase space.
▶
When the system is chaotic, the trajectory is called a ‘‘strange attractor
▶ Lorenz Attractor (3D nonlinear system)
Lorenz’s Model
▶
▶
Fractal trajectory confined in phase space with a chaotic attractor
Unpredictable time series confined in the phase
Digression: Key Ingredients for Chaos
INTRODUCTION TO CHAOS THEORY
▶ Ingredients
Nonlinearity
▶ Dimension (lower bound)
Poincaré-Bendixon
▶
Theorem
y
y
Given a differential equation dx/dt = F(x) in the plane
(2D). Assume x(t) is a solution curve which stays in a
bounded region. Then either x(t) asymptotically
converges to an equilibrium point where F(x) = 0,
or it converges to a single periodic cycle.
What if the assumptions are not
▶
satisfied?
Consider a system time-continuous,
x
x
, and be sure to have the
system’s state dimension >2 (or a number of degree of freedom >2) and
trajectories are bounded. Adjust the system’s parameters (upcoming slides)
and the result follows for large t.
▶ Some words on maps (discrete-time systems)
Maps
are not subject to the same rules. For instance, a
simple scalar nonlinear map can exhibit chaos.
S. Strogatz, “Nonlinear Dynamics and Chaos with application to physics, biology, chemistry and engineering’’, Perseus
Visualizing Chaos: Lyapunov Exponents
INTRODUCTION TO CHAOS THEORY
▶ Lyapunov exponent (LE)
▶
▶
▶
▶
Basic idea: to measure the average rate of divergence for neighboring
trajectories on the attractor in phase space.
A small sphere centered on the attractor. With time, the sphere
becomes an ellipsoid. The principal axes are in the direction of
contraction and expansion.
Lyapunov exponents (LE): average rate of these
contractions/expansions
For chaos (SIC), one LE (hyperellipsoid) must be positive.
trajectory in phase-space
Mathematical
formulation
See next slide
and :
http://en.wikipedia.org/wiki/Lyaponov_expon
ent
deformation of the ith principal axis
S. Strogatz, Nonlinear ‘‘Dynamics and Chaos with application to physics, biology, chemistry and engineering,’’ Perseus
Lasers: A Dynamical Point of View
APPLICATION TO OPTICAL SYSTEMS
▶ Maxwell-Bloch equations
▶
Coupled nonlinear PDEs for the slowly-varying envelope of the electric
field E, the polarization (coherence between upper and lower state) P,
and the population difference (inversion) W=Nupper-Nlower between the
upper and lower state.
Tph = cavity-photon lifetime
T1 = upper-state lifetime
T2 = dephasing time
c = in-vaccuo speed of
light
= drive frequency
k = freespace
= transition frequency
propagation
= propagation
constant
constant
W0= inversion at
= dipole moment
equilibrium
Lasers: A Dynamical Point of View
APPLICATION TO OPTICAL SYSTEMS
▶ Lorenz-Haken equations
▶
Simplification of Maxwell-Bloch equations (PDE becomes ODE)-integrate out spatial (z) dependence:
with
H. Haken, Phys Lett A 53, 77–78 (1975)
▶
Laser equations are identical to those of Lorenz:
,
,
, and
Lasers: A Dynamical Point of View
APPLICATION TO OPTICAL SYSTEMS
▶ Arecchi’s classification of lasers
3 Classes (A, B, or C) depending on the values of 3
characteristic times:
▶ Class C Laser (only intrinsically chaotic lasers):
Ne-Xe, infrared He-Ne)
▶
▶
(ruby, Nd, CO2, edge-emitting single-
Class A Laser:
(visible He-Ne, Ar, Kr, dye lasers,
quantum cascade lasers)
In Class B and A lasers, the short-timescale quantities can be
integrated out, effectively reducing the dimensionality of the system:
Class C - 3D
Class B - 2D
Class A - 1D
▶
▶
Class B Laser:
mode laser diodes)
(NH3,
Chaos in Semiconductor Lasers
APPLICATION TO OPTICAL SYSTEMS
▶ Semiconductor laser diodes: class-B lasers
▶
Adapted From M.
Sciamanna
Rate equations to describe the laser--polarization (coherence) P has been
eliminated
▶ One equation for the field amplitude (E) coupled to one equation for the carrier
inversion (N). One equation for the field phase which is independent!
with
= linewidth enhancement factor (gives coupling between amplitude and
phase
of E--feature for semiconductor lasers)
G = G(N(t)) = gain coefficient roughly proportional to N(t)
= carrier recombination rate (other than stimulated emission)
= cavity-photon lifetime
J = injection current
Outline
•
Review
•
Chaos in Time-Delay Systems
•
Introduction to Synchronization
•
Chaos Synchronization
•
Optical Chaos Cryptography
•
Conclusion
Chaos in Semiconductor Lasers
APPLICATION TO OPTICAL SYSTEMS
▶ How can we add dimensions (degrees of freedom)?
Time-delayed feedback
The number of dimensions is equal to the number of initial
conditions needed to specify the subsequent dynamics t > 0.
For an ordinary particle in 3D, the number of dimensions is 6.
For a time-delay t system, the subsequent dynamics t > 0
require a knowledge of x(t) and v(t) for –t < t < 0.
Infinite number of values infinite dimensional.
Chaos in Semiconductor Lasers
APPLICATION TO OPTICAL SYSTEMS
▶ Configurations exploiting internal nonlinearities
▶
Optoelectronic
feedback
S. Tang and J.-M. Liu, IEEE J. Quantum Electron. 37, 329336 (2001)
▶
optical feedback (external cavity
laser)
R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347355 (1980)
▶ Configurations exploiting external nonlinearities
▶
Optoelectronic feedback
J.-P. Goedgebuer et al., IEEE J. Quantum
Electron. 38, 1178-1183 (2002)
▶
Erbium-doped fiber ring laser
(EDFRL)
G.D. VanWiggeren and R. Roy, Phys. Rev.
Lett. 81, 3547-3550 (1998)
DEFINITION OF A TIME-DELAY SYSTEM
INTRODUCTION
▶ Mathematical definition
▶
▶
Delay-differential equation (DDE)
Delays can be constant, state-dependent, or distributed according to a
memory kernel, i.e.,
is replaced by
DEFINITION OF A TIME-DELAY SYSTEM
INTRODUCTION
▶ Main properties
▶
Infinite-dimensional dynamical systems: specification of a function over
one finite delay interval as the initial condition--different from typical ODEs
▶
Multistability at large delays: different initial conditions leads to different
attractors
▶
Finite (fractal) dimension of the strange attractor in chaotic regimes
▶
▶
Extremely high dimensions
In some cases, the dimension is proportional to the time delay
J. Foss, A. Longtin, B. Mensour and J. Milton, Phys. Rev. Lett. 76, 708 (1996)
V. Kolmanovskii and A. Myshkis, Mathematics and its applications 85 , (Kluwer Acadernic Publishers
Dordrecht, 1992)
TYPICAL EXAMPLE OF TIME-DELAY
SYSTEMS
INTRODUCTION
▶ Mackey-Glass systems (not laser diode)
▶
mathematical definition
▶
describes the production of blood cells
M.C. Mackey and L. Glass, Science 197, 287
(1977).
TYPICAL EXAMPLE OF TIME-DELAY
SYSTEMS
INTRODUCTION
▶ Ikeda systems
▶
mathematical definition
▶
describes the behavior of ring
lasers
K. Ikeda and K. Matsumoto, Physica D 29, 223 (1987
TYPICAL EXAMPLE OF TIME-DELAY
SYSTEMS
INTRODUCTION
▶ Lang-Kobayashi systems
▶
▶
mathematical definition
G is proportional to N
describes the behavior of laser diodes with external
cavity
R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347
LASER DIODES WITH TIME-DELAY
SYSTEMS
EXAMPLES
WAVELENGTH CHAOS GENERATOR
COMMUNICATION WITH TIME-DELAYED OPTOELECTRONIC SYSTEMS
▶ Theory and experimental setup
LD: DBR laser
diode
DL: Delay line
RF: RF low-pass
filter
PD: Photodiode
OI: Optical
isolator
BP:
Birefringent
Courtesy of University of Franche Compté, FEMTO
plate
PC: Polarization
controller
J.-P. Goedgebuer, L. Larger, H. Porte, Phys. Rev. Lett. 80, 2249 (19
▶ Mathematical model
▶
Scalar delay differential equation (x represents wavelength):
▶ Principle
System with wavelength modulation of DBR laser diode. Nonlinearity due
to birefringent crystal in external loop. 1/T ~ cutoff of low-pass filter.
INTENSITY CHAOS GENERATOR
COMMUNICATION WITH TIME-DELAYED OPTOELECTRONIC SYSTEMS
▶ Theory and experimental setup
LD: CW laser diode
DL: Optical delay
line
RF: RF band-pass
filter
PD: Photodiode
MZ1:
Mach- coupler
OC: Optical
Zehnder
RF:
RF band-pass
interferometer
filter
Courtesy of University of Franche Compté, FEMTO
▶ Mathematical model
▶
J.-P. Goedgebuer, P. Levy, L. Larger, C. Chang, W.T. Rhodes,
IEEE J. Quantum Electron. 38, 1178 (2002)
Delay integro-differential equation:
▶ Principle
MZ in feedback loop chaotically modulates intensity of a CW laser diode.
Nonlinearity due to the MZ--it is external to the laser. 1/T ~ upper cutoff of pass
band.
PHASE CHAOS GENERATOR (PCG)
COMMUNICATION WITH TIME-DELAYED OPTOELECTRONIC SYSTEMS
▶ Theoretical setup
LD: CW laser diode
DL: Optical delay
line
RF: RF band-pass
filter
PD: Photodiode
PC: Polorarization
PM:
Phase
controller
VA:
Variable
modulator
attenuator
R. Lavrov, M. Peil, M. Jacquot, L. Larger, V. Udaltsov, and J. Dudley, Phys. Rev. E
80, 026207 (2009)
▶ Mathematical model
▶
Delay integro-differential equation:
▶ Principle
PM in feedback loop chaotically modulates phase of CW laser diode.
Nonlinearity due to interferometer. Again, nonlinearity external to laser.
EXTERNAL-CAVITY LASER DIODES
COMMUNICATION WITH TIME-DELAYED OPTOELECTRONIC SYSTEMS
▶ Theory and experimental setup
EEL LD: Edge emitting
laser diode
Mf : Mirror
VAm: Variable
attenuator
CS : Current source
Courtesy of UMI 2958 Georgia Tech - CNRS
▶ Mathematical model
▶
Vectorial DDE:
▶
Two time scales: relaxation oscillation period
Three operational parameters: pumping current
and external-cavity roundtrip time
.
▶
and time delay
, feedback strength
Outline
•
Review
•
Chaos in Time-Delay Systems
•
Introduction to Synchronization
•
Chaos Synchronization
•
Optical Chaos Cryptography
•
Conclusion
A BRIEF HISTORY OF SYNCHRONIZATION
INTRODUCTION TO SYNCHRONIZATION
C. Huygens reported the first observation of
1665 synchronization (mutual synchronization) of
two pendulum clocks. He wrote on the
‘‘sympathy of two clocks.’’ Importance of
weak coupling.
Lord Rayleigh on identical pipes to sound at unison and the
1870 effect of quenching (oscillation damping in interacting
systems).
1945 - E.V. Appleton and B. van der Pol on the synchronization of
triode generators using weak synchronization signals
A. Pikovsky et al., ‘‘Synchronization an universal concept in nonlinear sciences,’’ Cambridge University Press (2
SYNCHRONIZATION EXPERIMENT @ HOME
INTRODUCTION TO SYNCHRONIZATION
Finally add two metronomes and set them with approximately
identical frequencies and with different initial conditions
Put a rule or thin plate of wood on the top
Use two empty beer cans (empty works better and is more fun)
DEFINITIONS OF SYNCHRONIZATION
INTRODUCTION TO SYNCHRONIZATION
▶ Fundamental understanding and key concepts
▶
Synchronization comes from the greek words syn (with) and chronos
(time): occuring at the same time
▶
Synchronization refers to an adjustment of rhythms of oscillators due
to weak interactions
Oscillator (self-sustained): active system with internal source of
▶
energy. Mathematically described by an autonomous system (ODE,
map).
▶ Rhythms: frequency or period of oscillations
▶
Coupling: interaction or transmission of information between system:
unidirectional (forcing) or bidirectional (mutual interaction).
One
oscillator
single
Two
oscillators
interaction
spring
solid bar
Coupling has to be weak
in
MECHANISMS OF SYNCHRONIZATION
INTRODUCTION TO SYNCHRONIZATION
▶ Synchronization of periodic oscillators by external forcing
▶
▶
When forced, the oscillator’s internal frequency is shifted.
Existence of a frequency-locking region that becomes larger as coupling is
increased.
Arnold Tongue
frequency locking region
▶
The explanation of such behavior originates in the phase dynamics of the
driven oscillator (beyond the scope of this introduction)
▶ Synchronization of mutually coupled periodic oscillators
1
▶
▶
2
Each oscillator tries to drive the
frequency of the other.
1
2
The two oscillator end up oscillating at
an identical frequency but different
1
2
from their natural ones. (CoupledOscillator 1 Oscillator 2
mode
theory)et al., ‘‘Synchronization an universal concept in nonlinear sciences’’, Cambridge University Press (2
A. Pikovsky
SYNCHRONIZATION IN NATURE
INTRODUCTION TO SYNCHRONIZATION
▶ Example: (Phase) Synchronization of fireflies
TYPES OF SYNCHRONIZATION
INTRODUCTION TO SYNCHRONIZATION
▶
Complete synchronization (CS)
▶
▶
Previous example: phase synchronization (amplitude
unaffected)
Existence of a type of synchronization for both amplitude and phase,
and more generally for all state variables xi of a dynamical system.
▶ Complete Synchronization (CS)
then
asymptotically
K1 and K2, the mathematical descriptions of coupling 1/2 and 2/1
▶
Generalized synchronization (GS)
▶
▶
Existence of functional relationship between state variables of systems 1
and 2
depending on the smoothness of we distinguish
▶
weak or strong GS.
Lag synchronization
▶
Synchronization of two systems at different
times
TYPES OF SYNCHRONIZATION
INTRODUCTION TO SYNCHRONIZATION
The foregoing ideas are well known for
periodic oscillators.
What about chaotic oscillators?
Outline
•
Review
•
Chaos in Time-Delay Systems
•
Introduction to Synchronization
•
Chaos Synchronization
•
Optical Chaos Cryptography
•
Conclusion
SYNCHRONIZATION OF CHAOS
INTRODUCTION TO SYNCHRONIZATION
▶
Complete synchronization (CS) of chaotic systems
▶
Involving two identical chaotic oscillators (physical twins)
▶
Long thought it was not possible that chaotic systems could
synchronize because of SIC
Pecora and Carroll, proved that it was possible under particular
coupling conditions using Lorenz-like systems. They proved it
theoretically, numerically, and experimentally.
▶
L.M. Pecora and T. Carroll., Phys. Rev. Lett. 64, 821-824 (1990)
emitter/master
L.M. Pecora and T. Carroll., Phys. Rev. A, 44, 2374-2383 (1991)
L.M. Pecora and T. Carroll., IEEE Trans. Circ. Syst. 38, 453-456 (1991)
receiver/slave
SYNCHRONIZATION OF CHAOS in LASERS
INTRODUCTION TO SYNCHRONIZATION
▶ Observations in a gas laser
▶ Observations in a semiconductor laser
OPEN-LOOP CONFIGURATION
SYNCHRONIZATION OF EXTERNAL CAVITY SEMICONDUCTOR LASERS
▶ Open-loop configuration for unidirectional synchronization
Master
Slave
▶ Model
EEL LD: Edge emitting laser
diode
Mf : Mirror
VAm: Variable
attenuator
CS : Current source
OI : Optical Isolator
delayed feedback
delayed injected field
▶ Index m and s for master and slave and with
CLOSED-LOOP CONFIGURATION
SYNCHRONIZATION OF EXTERNAL CAVITY SEMICONDUCTOR LASERS
▶ Closed-loop configuration for unidirectional synchronization
Master
Slave
EEL LD: Edge emitting laser
diode
Mf : Mirror
VA : Variable attenuator
CS : Current source
OI : Optical Isolator
▶ Model
master delayed feedback
slave delayed feedback
▶ Index m and s for master and slave and with
delayed injected field
Outline
•
Review
•
Chaos in Time-Delay Systems
•
Introduction to Synchronization
•
Chaos Synchronization
•
Optical Chaos Cryptography
•
Conclusion
PHYSICAL LAYER SECURITY & CHAOS
OPTICAL CHAOS CRYPTOGRAPHY
▶ Layer structure of a communication network (optical)
Alice
Bob
Application
Application
Transport
Transport
Network
Network
Data Link
Eve
▶
▶
Data Link
(eavesdropper)
▶
Physical
Physical
Different method to secure each high
layer of the protocol
Recent interest in additional security at
the physical layer: chaos cryptography
or QKD
Special interest in optoelectronic
devices because of their large
bandwidth and speed
▶ Generic principles of optical chaos cryptography
▶
Alice
Bob
Physical
Physical
▶
Alice injects her message in the
dynamics of a chaotic laser.
Bob has an identical laser that
synchronizes chaotically with
Alice’s
laser.
Using
“substraction,”
he
recovers
Alice’s message.
ENCRYPTION & DECRYPTION
OPTICAL CHAOS CRYPTOGRAPHY
▶ Chaos masking (CMa)
▶
▶
Encryption: the message is added at the output of the chaotic system.
Decryption: the message is an additional pertubation. The receiver will
detect it through a loss of synchronization
▶ CMa encryption/decryption using lasers
original message
encrypted message
decrypted message
After A. Sanches-Dıaz, C.R. Mirasso, P. Colet, P. Garćıa-Fernandez, IEEE J Quantum Electron. 35, 292–296 (19
ENCRYPTION & DECRYPTION
OPTICAL CHAOS CRYPTOGRAPHY
▶ Chaos Shift Keying (CSK)
▶
Encryption: The message m controls a switch. Depending on the bit (”0” or
”1”), Each emitter feed alternately the communication channel.
▶
Decryption: performed by monitoring synchronization errors: eE1/R1 = 0
(eE2/R2 = 0) which corresponds to m = 0 (m = 1).
▶ CSK encryption/decryption using lasers
▶
Original square message and error of
synchronization at the output of one of the
receiver eE1/R1.
V. Annovazzi-Lodi, S. Donati, A. Scire, IEEE J Quantum Electron.
33,1449–1454 (1997)
ENCRYPTION & DECRYPTION
OPTICAL CHAOS CRYPTOGRAPHY
▶ Chaos Modulation (CMo)
▶
▶
Encryption: Similar to the CMa technique except that the message m
also participates in the system dynamics.
Decryption: Similar to the CMa technique, except that the message m
does not disturb the synchronization.
▶ CMo encryption/decryption using lasers
encrypted message
receiver’ output
decrypted
message
original message
▶ Encoding
▶
at 2.5 Gb/s
Decryption with an additional lowpass filtering effect
After J.-M. Liu, H.F. Chen, S. Tang, IEEE Trans Circuits Syst I 48,
REAL FIELD EXPERIMENT
OPTICAL CHAOS CRYPTOGRAPHY
▶
Recently tested on real fiber-optic network in Athens (2005)
▶
Actual Gb/s encryption/decryption using a chaos masking
(CMa)
A. Argyris et al., Nature 438, 343-346, (2005)
Conclusion
▶ On synchronization
▶
▶
Synchronization is a universal concept in nonlinear sciences. It
describes the behavior of oscillators interacting with each other.
Synchronization was known for a long time for periodic oscillators, but
was demonstrate in chaotic systems only recently.
▶ Optical chaos-based physical-layer security
▶
▶
▶
▶
Chaos is used to encrypt the data--chaos synchronization to
decrypt it.
Different methods exist to mix the message: CMa, CSK or CMo are the
most popular.
Optical systems are used because of their large bandwidth and speed.
Real-field experiments proved potential for practical optical
telecommunication.