Synchronization of chaotic oscillators: Focus on laser diodes with time- delayed feedback Lecture 2 D. RONTANI * and D. S. CITRIN [email protected] School

Synchronization of chaotic oscillators: Focus on laser diodes

with time-delayed feedback

Lecture 2

D. RONTANI* and D. S. [email protected]

School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, Georgia 30332-0250

and Unité Mixte Internationale 2958 Georgia Tech-

CNRSGeorgia 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 AttractorsINTRODUCTION TO CHAOS THEORY

▶

▶

The evolution of the state variable can be represented as 1D time series

Representations of chaotic states ▶

▶ Evolution of the state variable can be represented simultaneously in a nD phase space.

Lorenz Attractor (3D nonlinear system)

When the system is chaotic, the trajectory is called a ‘‘strange attractor’’▶

Lorenz’s Model

Fractal trajectory confined in phase space with a chaotic attractor

▶Unpredictable time series confined in the phase space

▶

Digression: Key Ingredients for ChaosINTRODUCTION TO CHAOS THEORY

▶ Ingredients

Poincaré-Bendixon TheoremGiven 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. x x

y y

▶What if the assumptions are not satisfied?

▶Some words on maps (discrete-time systems)

Dimension (lower bound)▶

Nonlinearity▶

S. Strogatz, “Nonlinear Dynamics and Chaos with application to physics, biology, chemistry and engineering’’, Perseus Book (1994)

Maps are not subject to the same rules. For instance, a simple scalar nonlinear map can exhibit chaos.

Consider a system time-continuous, , 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.

Visualizing Chaos: Lyapunov ExponentsINTRODUCTION TO CHAOS THEORY

▶Lyapunov exponent (LE)

S. Strogatz, Nonlinear ‘‘Dynamics and Chaos with application to physics, biology, chemistry and engineering,’’ Perseus Book, (1994)

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

▶

deformation of the ith principal axis

trajectory in phase-space

Mathematical formulation :

For chaos (SIC), one LE (hyperellipsoid) must be positive. ▶

See next slide and http://en.wikipedia.org/wiki/Lyaponov_exponent

http://en.wikipedia.org/wiki/Lyaponov_exponent

http://en.wikipedia.org/wiki/Lyaponov_exponent

▶ Maxwell-Bloch equationsCoupled 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.

▶

Lasers: A Dynamical Point of ViewAPPLICATION TO OPTICAL SYSTEMS

Tph = cavity-photon lifetimeT1 = upper-state lifetimeT2 = dephasing timec = in-vaccuo speed of light

= drive frequency = transition frequency = propagation constant = dipole moment

k = freespace propagation constantW0= inversion at equilibrium


▶ 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, , ,


▶Arecchi’s classification of lasers3 Classes (A, B, or C) depending on the values of 3

characteristic times:▶

▶ Class C Laser (only intrinsically chaotic lasers): (NH3, Ne-Xe, infrared He-Ne)

▶ Class B Laser: (ruby, Nd, CO2, 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 LasersAPPLICATION TO OPTICAL SYSTEMS

▶Semiconductor laser diodes: class-B lasersRate 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

Adapted From M. Sciamanna

= 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





• Conclusion


▶ How can we add dimensions (degrees of freedom)?Time-delayed feedbackThe 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 system, the subsequent dynamics t > 0 require a knowledge of x(t) and v(t) for – < t < 0. Infinite number of values infinite dimensional.


▶Configurations exploiting internal nonlinearities

▶Configurations exploiting external nonlinearitiesOptoelectronic

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)

R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347-355 (1980)

S. Tang and J.-M. Liu, IEEE J. Quantum Electron. 37, 329-336 (2001)

Optoelectronic feedback

▶optical feedback (external cavity laser)

▶

DEFINITION OF A TIME-DELAY SYSTEMINTRODUCTION

▶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 SYSTEMINTRODUCTION

▶ Main properties

Infinite-dimensional dynamical systems: specification of a function over one finite delay interval as the initial condition--different from typical ODEs

▶

▶

Finite (fractal) dimension of the strange attractor in chaotic regimes

▶

V. Kolmanovskii and A. Myshkis, Mathematics and its applications 85 , (Kluwer Acadernic Publishers Dordrecht, 1992)

Multistability at large delays: different initial conditions leads to different attractors

J. Foss, A. Longtin, B. Mensour and J. Milton, Phys. Rev. Lett. 76, 708 (1996)

▶In some cases, the dimension is proportional to the time delay

▶Extremely high dimensions

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).


INTRODUCTION

▶ Ikeda systems

mathematical definition ▶

describes the behavior of ring lasers▶

K. Ikeda and K. Matsumoto, Physica D 29, 223 (1987).


INTRODUCTION

▶

▶ Lang-Kobayashi systems

mathematical definition

describes the behavior of laser diodes with external cavity

▶

R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980).

G is proportional to N

LASER DIODES WITH TIME-DELAY SYSTEMS

EXAMPLES

WAVELENGTH CHAOS GENERATORCOMMUNICATION WITH TIME-DELAYED OPTOELECTRONIC SYSTEMS

▶

▶▶

Theory and experimental setup

Mathematical model

LD: DBR laser diodeDL: Delay lineRF: RF low-pass filterPD: Photodiode

Courtesy of University of Franche Compté, FEMTO

Scalar delay differential equation (x represents wavelength):

▶ PrincipleSystem with wavelength modulation of DBR laser diode. Nonlinearity due to birefringent crystal in external loop. 1/T ~ cutoff of low-pass filter.

OI: Optical isolatorBP: Birefringent plate

J.-P. Goedgebuer, L. Larger, H. Porte, Phys. Rev. Lett. 80, 2249 (1998)

PC: Polarization controller

INTENSITY CHAOS GENERATORCOMMUNICATION WITH TIME-DELAYED OPTOELECTRONIC SYSTEMS

▶

▶ Mathematical model ▶


Courtesy of University of Franche Compté, FEMTO

LD: CW laser diodeDL: Optical delay lineRF: RF band-pass filterPD: PhotodiodeOC: Optical coupler

Delay integro-differential equation:

▶ PrincipleMZ 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.

MZ1: Mach-Zehnder interferometer

J.-P. Goedgebuer, P. Levy, L. Larger, C. Chang, W.T. Rhodes, IEEE J. Quantum Electron. 38, 1178 (2002)

RF: RF band-pass filter

PHASE CHAOS GENERATOR (PCG)COMMUNICATION WITH TIME-DELAYED OPTOELECTRONIC SYSTEMS

▶


Theoretical setup

LD: CW laser diodeDL: Optical delay lineRF: RF band-pass filter

PD: PhotodiodePC: Polorarization controller

Delay integro-differential equation:

▶ PrinciplePM in feedback loop chaotically modulates phase of CW laser diode. Nonlinearity due to interferometer. Again, nonlinearity external to laser.

PM: Phase modulator

R. Lavrov, M. Peil, M. Jacquot, L. Larger, V. Udaltsov, and J. Dudley, Phys. Rev. E 80, 026207 (2009)

VA: Variable attenuator

EXTERNAL-CAVITY LASER DIODESCOMMUNICATION WITH TIME-DELAYED OPTOELECTRONIC SYSTEMS

▶


▶


Courtesy of UMI 2958 Georgia Tech - CNRS

Vectorial DDE:

Two time scales: relaxation oscillation period and time delay▶ Three operational parameters: pumping current , feedback

strength and external-cavity roundtrip time .

EEL LD: Edge emitting laser diodeMf : MirrorVAm: Variable attenuatorCS : Current source

Outline

• Review





• Conclusion

A BRIEF HISTORY OF SYNCHRONIZATIONINTRODUCTION TO SYNCHRONIZATION

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.

1665 -

1945 -E.V. Appleton and B. van der Pol on the synchronization of triode generators using weak synchronization signals

Lord Rayleigh on identical pipes to sound at unison and the effect of quenching (oscillation damping in interacting systems).

1870 -

A. Pikovsky et al., ‘‘Synchronization an universal concept in nonlinear sciences,’’ Cambridge University Press (2001)

SYNCHRONIZATION EXPERIMENT @ HOMEINTRODUCTION TO SYNCHRONIZATION

Finally add two metronomes and set them with approximately identical frequencies and with

different initial conditions

Use two empty beer cans (empty works better and is more fun)

Put a rule or thin plate of wood on the top

DEFINITIONS OF SYNCHRONIZATIONINTRODUCTION 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).

▶

Coupling has to be weak

One single oscillator

Two oscillators in interaction

solid bar spring

MECHANISMS OF SYNCHRONIZATION INTRODUCTION TO SYNCHRONIZATION

▶Synchronization of periodic oscillators by external forcingWhen forced, the oscillator’s internal frequency is

shifted. ▶

frequency locking region

Existence of a frequency-locking region that becomes larger as coupling is increased.

▶Arnold Tongue

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

21

21

21

Oscillator 1 Oscillator 2

Each oscillator tries to drive the frequency of the other.

▶

The two oscillator end up oscillating at an identical frequency but different from their natural ones. (Coupled-mode theory)

▶

A. Pikovsky et al., ‘‘Synchronization an universal concept in nonlinear sciences’’, Cambridge University Press (2001)

SYNCHRONIZATION IN NATUREINTRODUCTION TO SYNCHRONIZATION

▶Example: (Phase) Synchronization of fireflies

TYPES OF SYNCHRONIZATIONINTRODUCTION TO SYNCHRONIZATION

▶ Complete synchronization (CS)

▶Generalized synchronization (GS)

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)

▶Existence of functional relationship between state variables of systems 1 and 2

then asymptotically

K1 and K2, the mathematical descriptions of coupling 1/2 and 2/1

▶ depending on the smoothness of we distinguish weak or strong GS.

▶ Lag synchronization

▶Synchronization of two systems at different times

TYPES OF SYNCHRONIZATIONINTRODUCTION TO SYNCHRONIZATION

The foregoing ideas are well known for periodic oscillators.

What about chaotic oscillators?

Outline

• Review





• Conclusion

SYNCHRONIZATION OF CHAOSINTRODUCTION TO SYNCHRONIZATION

▶ Complete synchronization (CS) of chaotic systems

Long thought it was not possible that chaotic systems could synchronize because of SIC

▶

Involving two identical chaotic oscillators (physical twins)▶

▶ Pecora and Carroll, proved that it was possible under particular coupling conditions using Lorenz-like systems. They proved it theoretically, numerically, and experimentally.

emitter/master receiver/slaveL.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)

L.M. Pecora and T. Carroll., IEEE Trans. Circ. Syst. 38, 453-456 (1991)

SYNCHRONIZATION OF CHAOS in LASERSINTRODUCTION TO SYNCHRONIZATION

▶ Observations in a gas laser

▶Observations in a semiconductor laser

OPEN-LOOP CONFIGURATIONSYNCHRONIZATION OF EXTERNAL CAVITY SEMICONDUCTOR LASERS

▶Open-loop configuration for unidirectional synchronization

▶Model

EEL LD: Edge emitting laser diodeMf : MirrorVAm: Variable attenuatorCS : Current sourceOI : Optical Isolator

▶Index m and s for master and slave and with

delayed feedback

delayed injected field

Master Slave

CLOSED-LOOP CONFIGURATIONSYNCHRONIZATION OF EXTERNAL CAVITY SEMICONDUCTOR LASERS

▶Closed-loop configuration for unidirectional synchronization

▶Model

EEL LD: Edge emitting laser diodeMf : MirrorVA : Variable attenuatorCS : Current sourceOI : Optical Isolator

▶Index m and s for master and slave and with

delayed injected fieldslave delayed feedback

master delayed feedback

Master Slave

Outline

• Review





• Conclusion

PHYSICAL LAYER SECURITY & CHAOSOPTICAL CHAOS CRYPTOGRAPHY

▶Layer structure of a communication network (optical)

Network

Transport

Application

Alice Bob

Application

Transport

Network

Data Link (eavesdropper)

Physical

Data Link

Physical

Eve

Physical Physical

Alice Bob

▶Generic principles of optical chaos cryptography

Different method to secure each high layer of the protocol

▶

▶Recent interest in additional security at the physical layer: chaos cryptography or QKD

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.

▶

▶Special interest in optoelectronic devices because of their large bandwidth and speed

ENCRYPTION & DECRYPTIONOPTICAL CHAOS CRYPTOGRAPHY

▶ Chaos masking (CMa)Encryption: the message is added at the output of the chaotic system.

▶

▶CMa encryption/decryption using lasers

original message encrypted message decrypted message

Decryption: the message is an additional pertubation. The receiver will detect it through a loss of synchronization

After A. Sanches-Dıaz, C.R. Mirasso, P. Colet, P. Garcıa-Fernandez, IEEE J Quantum Electron. 35, 292–296 (1999)

▶


▶ 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.

▶

▶ CSK encryption/decryption using lasers

Decryption: performed by monitoring synchronization errors: eE1/R1 = 0 (eE2/R2 = 0) which corresponds to m = 0 (m = 1).

▶

V. Annovazzi-Lodi, S. Donati, A. Scire, IEEE J Quantum Electron. 33,1449–1454 (1997)

Original square message and error of synchronization at the output of one of the receiver eE1/R1.

▶


▶ Chaos Modulation (CMo)Encryption: Similar to the CMa technique except that the message m also participates in the system dynamics.

▶

▶ CMo encryption/decryption using lasers

Decryption: Similar to the CMa technique, except that the message m does not disturb the synchronization.

▶

original message

encrypted message

decrypted message

receiver’ output

After J.-M. Liu, H.F. Chen, S. Tang, IEEE Trans Circuits Syst I 48, 1475–1483 (2001)

▶Encoding at 2.5 Gb/s▶Decryption with an additional low-pass filtering effect

REAL FIELD EXPERIMENTOPTICAL 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

▶ Optical chaos-based physical-layer securityChaos 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.

▶

▶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.

Documents

Synchronization of chaotic oscillators: Focus on laser diodes with time- delayed feedback Lecture 2 D. RONTANI * and D. S. CITRIN [email protected] School