Control and Synchronization of Chaotic Dynamics in Laser Systems

E. AllariaSincrotrone Trieste

Relatore:Prof. F.T. ArecchiDipartimento di Fisica Universita’ di FirenzeeIstituto Nazionale di Ottica Applicata

Correlatori:Dr. R. MeucciIsitituto Nazionale di Ottica Applicata

Dr. G. De NinnoSincrotrone Trieste

•Nonlinear dynamics and chaos in laser systems

•The CO2 laser with feedback

•Synchronization and noise effects

•Networks properties of chaotic systems

•The CO2 laser with modulated losses

•Synchronization of two coupled nonautonomous systems

•The Elettra Storage Ring Free Electron Laser

•FEL stabilization through a delayed feedback

•Conclusions

Outline

•The basic model for the dynamics is given by the three coupled equations for

laser field (E), polarization (P) and population inversion () of the laser medium

Nonlinear dynamics and chaos in lasers

gPE

EPP

gPkEE

40||

k ||

k ||

k ||

Class A laser

Class B laser

Class C laser

In class B lasers different setups may lead to chaotic dynamics:•Longitudinal multi mode emission•Spatial multi mode emission,•Adding a third variable to the system by means of

Feedback External forcing

k,, are decay rates and g a coupling constant.

1- Laser mirror2- CO2 laser tube3- Brewster window4- Electro-optic modulator

5- Power meter6- Detector7- Beam Splitter8- Amplifier9- Power supply

CO2 laser with feedback A CO2 laser has been developed at INOA for studies of nonlinear dynamics

and chaos

Control parameters: R and B0 gain and bias on the feedback loop

Zero level

Chaos in the CO2 laser with feedback

With the chosen parameter the laser intensity shows large peaks occurring erratically in time.

Saddle focus

Noise induced synchronization - Setup

The possibility of a common noise source to induce a synchronized regime between two uncoupled chaotic system has been investigated.

Instead of using two systems driven by a common noise source we apply twice the same noise signal to one chaotic laser with different initial conditions

C.S. Zhou, E. Allaria, F.T. Arecchi, S. Boccaletti, R. Meucci and J. Kurths “Constructive effects of noise in homoclinic chaotic systems” Phys. Rev. E 67, 66220 (2003).

High Voltage

Start of the common noise signal

Noise induced synchronization – Experimental results

Experiments show that for a suitable noise strength two uncoupled chaotic lasers can reach a common behavior if driven by the same noise signal

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21

II

IIE

Noise induced synchronization – Numerical results

Numerical results Experimental results

Largest Lyapunov Exponent (1) and Synchronization Error (E) for a systems without (-) and with () intrinsic noise

Experimental results are confirmed by numerical simulations if the effect of the intrinsic noise signal is taken into accountI1,2 : laser intensities

Noise Enhance SynchronizationThe effect of noise on the single chaotic laser is investigated by looking at the synchronization properties of the system

Depending on the noise value the synchronization region can be enlargedC.S. Zhou, J. Kurths, E. Allaria, S. Boccaletti, R. Meucci and F.T. Arecchi, “Noise enhanced

synchronization of homoclinic chaos in a CO2 laser ” Phys. Rev. E 67, 015205(R) (2003).

0.14%0.3%1.0%

A

A: Amplitude of the external periodic modulation

: Detuning of the external frequency with respect to the natural frequency of the chaotic laser

: Frequency mismatch between the external signal and frequency of the modulated laser

Positive feedback Negative feedback

Synchronization in phase Synchronization in antiphase

Unidirectional coupled network of chaotic oscillators

Using the model of the laser with feedback we investigate the synchronization properties of networks of chaotic elements

Depending on the sign of the coupling between elements two possible regimes of synchronization are possible in phase and out of phase.

I. Leyva, E. Allaria, F.T. Arecchi and S. Boccaletti, “In-phase and antiphase synchronization of coupled homoclinic chaotic oscillators” Chaos 14, 118 (2004).

time (a.u)

Lase

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a.u

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ve

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ste

rsl

ave

Lase

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a.u

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time (a.u)

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i

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Delayed Self Synchronization

Unidirectionally coupled array

i

Tim

e (a

.u)

Tim

e (a

.u)

i

Delayed Self Synchronization and unidirectional coupled array

i

i

The space-time representation of the dynamics of a closed chain of unidirectional coupled systems shows results similar to the ones obtained with the delayed self synchronization on the laser with feedback.

The equivalent of the delay in delayed self synchronization for the closed chain is the number of elements.

timespace-time representation

Synchronization patterns in arrays of homoclinic chaotic systems

= 0.0; 0.05; 0.1; 0.12; 0.25

ForcingForcing

Using the model of the laser with feedback we look at the synchronization properties of a network of bidirectional coupled chaotic elements.

Increasing the coupling strength, clusters of phase synchronized elements are first shown; the dimension of cluster increases up to a complete synchronized network

For larger values of the coupling strength the repetition rate of spikes is decreased

I. Leyva, E. Allaria, F.T. Arecchi and S. Boccaletti, “Competition of synchronization patterns in arrays of homoclinic chaotic systems” Phys. Rev. E 68, 066209 (2003).

Site index i i i i i

= 0.13 and 0()=0.02. Forcing:

=0.015; =0.042.

Response of the network to an external periodic forcing

Information penetration depth vs. for different coupling strengths

= 0.12, 0.15, 0.2, 0.25

In the case of an external modulation applied to one side of the network the information relative to the frequency of the external signal can be propagated through the network depending on the coupling strength between elements and on the signal frequency

Site index i i

Laser with modulated losses A different setup able to produce chaotic dynamics in CO2 laser is the one with an external modulation of the cavity losses

The setup has been implemented in order to be able to study the synchronization between two lasers in a master-slave configuration.

The master is realized by recording a time series of the unperturbed laser; the laser becomes the slave when the recorded signal is used for controlling the amplitude modulation of the external modulation

Transition to chaos

Due to the presence of the external modulation the system shows periodic oscillations.

Depending on the strength of the modulation the amplitude of those oscillations can reach a chaotic behavior.

The regime we are considering is characterized by large chaotic pulses occurring almost periodically in time.

kT

Tk

kdtx

yx

Tkerror

1

2

1

11 ,...2,1,1

Coupling between two chaotic lasersuncoupled coupled

If the phase of the external modulation of master and slave lasers is the same, the occurrence of pulses in both system is synchronized also without the coupling

When applying the coupling also the amplitude of pulses of both systems becomes synchronized

I.P. Marino, E. Allaria, M.A.F. Sanjuan, R. Meucci, F.T. Arecchi, “Coupling scheme for complete synchronization of periodically forced chaotic CO2 lasers” Phys. Rev. E 70, 036208 (2004).

1) Relativistic electron beam

2) UndulatorUndulator

3) Electromagnetic field co-propagating with the electron beam and getting amplifiedgetting amplified to the detriment of electrons’ kinetic energy

A Free-Electron Laser (FEL) is a light source exploiting the spontaneous and/or induced emission of a relativistic electron beam “guided” by the periodic and static magnetic field generated by an undulator

Free Electron Laser and Storage Ring FEL

Free Electron Laser and Storage Ring FEL

In a SRFEL electrons bunches are circulating in the storage ring and photons are oscillating in the optical cavity

A crucial parameter is the timing between electrons and photons that should isochronous pass into the ondulator

Derivative feedback

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Las

er in

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arb.

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0.200.150.100.050.00Time (s)

2.2

2.0

1.8

1.6

1.4

1.2

no

rma

lize

d s

td

300025002000150010005000derivative gain (a.u)

Without feedback With feedback

Tim

e se

ries

Pow

er s

pect

ra

The use of a derivative feedback on the SRFEL can partially remove the oscillation due to the residual detuning on the system

Depending on the gain of the feedback loop it is possible to reduce but not completely eliminate the oscillation

C. Bruni et.al. “Stabilization of the Pulsed Regimes on Storage Ring Free Electron Laser: The Cases of Super-ACO and Elettra” 5-9 July 2004 European Particle Accelerator Conference, Lucerne (CH)

2delay feedback

1 2

- -

+

A feedback based on 2 delay lines can be used to stabilize the unstable fixed point of the system

The method shows good numerical results and could be experimentally implemented by means of a FPGA

E. Allaria et al.  “Stabilization of the Elettra storage-ring free-electron laser through a delayed feedback control method”, 27th International Free Electron Laser Conference, Stanford, California.

Comparison between derivative and 2delay feedbackBifurcation diagram for the FEL maxima as a function of the detuning

Free running

Derivative feedback

2Delay feedback

A comparison between the two method shows the advantage of using the 2delay method in the region of interest for the Elettra SRFEL

The thesis work concentrated on nonlinear dynamics studies carried out on laser systems.

Gas lasers, solid state lasers and FEL have been studied

In particular, we have addressed control and synchronization of chaos, noise-induced effects and properties of networks of chaotic elements

Further research•Optimization studies for the FERMI project

Stabilization of fluctuations in a single pass FELOptimizing the FEL schemes

•Experimental activities on the storage ring FEL Realization of new feedback methodsSeeded FEL on a storage ring

•Numerical studies on complex networksProperties of chaotic elements useful for the synchronization

Conclusions

Effects of the EOM on the laser dynamics

CO2 laser with feedback – numerical results

TT

Rvar

Evidence of stochastic resonance

Numerical results Experimental results

For a fixed modulation frequency and amplitude the laser show the stochastic resonance similar to excitable systems.Numerical results are confirmed by similar experimental results

Delayed self synchronizationThe use of a very long delay feedback can simulate the coupling between two chaotic lasers

Data are analyzed in the spatiotemporal representation: intensity is mapped by using a grayscale, time between the delay time is plotted in Y while the number of delays in X

The activation of the long delay feedback can stabilize periodic patterns of spikes sequences

Competition between spatial synchronization regimes induced by two external forcing

(a) 1 = 0.020; 2 = 0.0210; = 0.13 (b) 1 = 0.038; 1 = 0.0420; = 0.12 (c) 1 = 0.040; 1 = 0.0405; = 0.11

In the case of two external modulations to the end of the network different synchronization pattern are produced depending on the relation between the used frequencies

Coupling between two chaotic lasers (2/2)

Master-slave correlation (exp) Lyapunov exponents (num)

The synchronization is confirmed by the numerical simulations that show a transition from positive to negative of one of the Lyapunov exponents