Control and Synchronization of Chaotic Dynamics in Laser Systems E. Allaria Sincrotrone Trieste...
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Transcript of Control and Synchronization of Chaotic Dynamics in Laser Systems E. Allaria Sincrotrone Trieste...
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
r in
tens
ity (
a.u
)m
ast
er
sla
ve
ma
ste
rsl
ave
Lase
r in
tens
ity (
a.u
)
time (a.u)
21..
i
N
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
140
120
100
80
60
40
20
0
Las
er in
tens
ity (
arb.
uni
ts)
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