Transcript [Seminar]"Synchronization of chaotic oscillators: Focus on
Lecture 2
Synchronization of chaotic oscillators: Focus on laser diodes with time delayed feedback
D. RONTANI
*
and D. S. CITRIN
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 Unpredictable
in
phase space
with a chaotic attractor time series confined in the phase space
Digression: Key Ingredients for Chaos
INTRODUCTION TO CHAOS THEORY ▶
Ingredients
▶ ▶ ▶ Nonlinearity Dimension (lower bound)
Poincaré-Bendixon Theorem
y Given a differential equation (2D). Assume x(t) dx/dt = F(x) in the plane is a solution curve which stays in a bounded region. Then either converges to an x(t)
equilibrium point
asymptotically where F(x) = 0 , or it converges to a single
periodic cycle
.
y x x What if the assumptions are not satisfied?
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 Book (1994)
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):
contractions/expansions average rate of these For chaos (SIC), one LE (hyperellipsoid) must be positive. trajectory in phase-space http://en.wikipedia.org/wiki/Lyaponov_expon ent deformation of the
i
th principal axis
S. Strogatz, Nonlinear ‘‘Dynamics and Chaos with application to physics, biology, chemistry and engineering,’’ Perseus Book, (1994)
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
=
N
upper -
N
lower between the upper and lower state.
T ph
= cavity-photon lifetime
T 1
= upper-state lifetime
T 2
= dephasing time
c
= in-vaccuo speed of light
=
drive frequency
=
transition frequency = propagation constant
k =
freespace propagation constant
W 0
= 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): ( NH 3 , Ne-Xe, infrared He-Ne ) ▶ Class B Laser: ( ruby, Nd, CO 2 , edge-emitting single mode laser diodes ) ▶ ▶ 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
Chaos in Semiconductor Lasers
APPLICATION TO OPTICAL SYSTEMS ▶
Adapted From M. Sciamanna
Semiconductor laser diodes: class-B lasers
Rate equations to describe the laser--polarization (coherence)
P
▶ eliminated has been ▶
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 J
=
G
(
N
(
t
)) = gain coefficient roughly proportional to = cavity-photon lifetime = injection current
N
(
t
) = carrier recombination rate (other than stimulated emission)
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
require a knowledge of
x
(
t
) and
v
(
t
) for – t <
t
< 0. Infinite number of values infinite dimensional.
> 0
Chaos in Semiconductor Lasers
APPLICATION TO OPTICAL SYSTEMS ▶ ▶
Configurations exploiting internal nonlinearities
Optoelectronic optical feedback (external cavity ▶ feedback laser)
S. Tang and J.-M. Liu, IEEE J. Quantum Electron. 37, 329 336 (2001) R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 355 (1980)
▶ ▶
Configurations exploiting external nonlinearities
Optoelectronic feedback ▶ Erbium-doped fiber ring laser (EDFRL)
J.-P. Goedgebuer et al., IEEE J. Quantum Electron. 38, 1178-1183 (2002) 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 describes the behavior of laser diodes with external
N
cavity
R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980).
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
Courtesy of University of Franche Compté, FEMTO
▶ controller
J.-P. Goedgebuer, L. Larger, H. Porte, Phys. Rev. Lett. 80, 2249 (1998)
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 band-pass filter
PD
: Photodiode
OC
: Optical coupler Zehnder 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 band-pass filter
PD PC
: Photodiode : Polorarization 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
M f VA m
: Mirror attenuator
CS
: Current source ▶
Courtesy of UMI 2958 Georgia Tech - CNRS
Mathematical model
▶ Vectorial DDE: ▶ ▶ Two time scales: relaxation oscillation period and time delay Three operational parameters: pumping current , feedback strength and external-cavity roundtrip time .
Outline
• Review • Chaos in Time-Delay Systems • Introduction to Synchronization • Chaos Synchronization • Optical Chaos Cryptography • Conclusion
A BRIEF HISTORY OF SYNCHRONIZATION
INTRODUCTION TO SYNCHRONIZATION
1665
-
1870
-
C. Huygens
reported the first observation of synchronization (mutual synchronization) of two pendulum clocks. He wrote on the ‘‘sympathy of two clocks.’’ Importance of
weak coupling
.
Lord Rayleigh
effect of on identical pipes to sound at unison and the
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 (2001
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
(time):
occuring at the same time
(with) and
chronos
▶ Synchronization refers to an
adjustment of rhythms
▶ map).
Rhythms
: frequency or period of oscillations of
oscillators
due to
weak interactions Oscillator
(self-sustained): active system with internal source of ▶ energy. Mathematically described by an autonomous system (ODE, ▶
Coupling
: interaction or transmission of information between system:
unidirectional
(forcing) or
bidirectional
(mutual interaction).
One oscillator single Two oscillators interaction in spring solid bar Coupling has to be
weak
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 Each oscillator tries to drive the 2 frequency of the other.
The two oscillator end up oscillating at 1 2 an identical frequency but different 1 2 from their natural ones. (Coupled Oscillator 1 Oscillator 2 mode theory)
A. Pikovsky et al., ‘‘Synchronization an universal concept in nonlinear sciences’’, Cambridge University Press (2001
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 ▶ and more generally for all state variables
x i
▶
Complete Synchronization (CS)
of a dynamical system.
then asymptotically
K 1
and K 2, 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) L.M. Pecora and T. Carroll., Phys. Rev. A, 44, 2374-2383 (1991)
emitter/master
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
EEL LD
: Edge emitting laser diode : Mirror
VAm
: Variable
OI
: Optical Isolator ▶
Model
delayed feedback ▶ Index
m
and
s
for master and slave and with delayed injected field
CLOSED-LOOP CONFIGURATION
SYNCHRONIZATION OF EXTERNAL CAVITY SEMICONDUCTOR LASERS ▶
Closed-loop configuration for unidirectional synchronization
Master Slave
EEL LD
: Edge emitting laser diode : Mirror
VA
: Variable attenuator
CS
: Current source
OI
: Optical Isolator ▶
Model
master delayed feedback slave delayed feedback delayed injected field ▶ Index
m
and
s
for master and slave and with
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 Application Transport Network Data Link Physical Eve (eavesdropper) Bob Application Transport Network Data Link 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 Physical Bob Physical ▶ Alice injects her message in the dynamics of a chaotic laser.
▶ Bob has an identical laser that synchronizes Alice’s chaotically laser.
“substraction,” Alice’s message.
he with Using recovers
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 (1999
▶
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: e E1/R1 = 0 (e E2/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 e E1/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 low pass filtering effect
After J.-M. Liu, H.F. Chen, S. Tang, IEEE Trans Circuits Syst I 48, 1475 –1483 (2001)
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.