Chaotic Behavior Mechanisms in the Bloch-Maxwell Laser

Chaotic Behavior Mechanisms in the
Bloch-Maxwell Laser Model
Avadis Hacinliyan ∗ Orhan Ozgur Aybar ∗∗
Ilknur Kusbeyzi ∗∗∗ E. Eray Akkaya ∗∗∗∗
∗
Department of Information Systems and Technologies,Yeditepe
University, Kayisdagi,Istanbul,Turkey (e-mail:
[email protected]).
∗∗
Department of Information Systems and Technologies,Yeditepe
University, Kayisdagi,Istanbul,Turkey (e-mail: [email protected])
Departments of Mathematics, Gebze Institute of Technology, Turkey
∗∗∗
Department of Information Systems and Technologies,Yeditepe
University, Kayisdagi,Istanbul,Turkey (e-mail:
[email protected])
Departments of Mathematics, Gebze Institute of Technology, Turkey
∗∗∗∗
Department of Physics,Yeditepe University,
Kayisdagi,Istanbul,Turkey (e-mail: [email protected])
Abstract: Regions of the full parameter space for which chaotic behavior in laser models based
on the Maxwell Bloch equation occurs is studied. The range in the parameter space where
a maximal positive Lyapunov exponent occurs corresponds to that for Type B lasers. Hopf
bifurcation is confirmed for the He-Ne case.
Keywords:
Maxwell-Bloch equations, Lyapunov stability, Hopf Bifurcation, Chaotic behaviour, Optical
Nonlinearity, Magnetic Resonance
The pure Bloch model is known to be a linear model, so
that chaotic behavior is not ordinarily expected. However,
for strong coupling of the AC component, trajectories
that are dense in the phase space can result in seemingly
chaotic like regions in the trajectory plot, as reported by
(6). Chaotic behavior as evidenced by positive Lyapunov
exponents requires the addition of nonlinear terms. One
way to add this nonlinearity is to couple the Bloch system
ro the Maxwell equations in the presence of material
media. Such systems serve as laser models including the
He-Neon laser. Transition to chaos is based on considering
the resonance between the laser cavity frequency and
atomic cavity TEM modes. The Maxwell-Bloch equations
as given by (1) and (7) involve the coupling of the
fundamental cavity mode, E with the collective variables
P and ∆, that represent the atomic polarization and the
population inversion and are represented by the following
equations.
Ė = −kE + gP
Ṗ = −γ⊥ P + gE∆
˙ = −γk (∆ − ∆o ) − 4gP E
∆
For the parameter values k = σ, γ⊥ = g 2 /k = 1,
g 2 ∆o /k = r, γk = b, the system can be transformed into
the Lorenz system about the equilibrium point ∆ = ∆o by
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The basic operating point given by the MINUIT Analysis showing “Throw
and Catch” chaotic behaviour similiar to LORENZ System.
P
7.5
5
2.5
E
-2
-1
1
2
3
-2.5
-5
-7.5
k=11.75, g=6.06, gper=2.66, gpar=2.75, dl0=28
7
Fig. 1. The Strange Attractor for basic operating point
with given parameters
setting x = E, y = gP/k,z = ∆o − ∆. The meaning of the
parameters in the original equations are given by Arrechi,
while σ, r, b are the Lorenz parameters.
The Maxwell-Bloch equations have more parameters than
the Lorenz system; this justifies a more detailed parameter
study. Chaotic behavior has been experimentally observed
in laser systems (1; 7) and studying the range of parame-
The prohibitively long computation time for a full parameter space study led us to locate the above mentioned
basic operating point by minimizing the negative of the
maximal Lyapunov exponent using the CERN MINUIT
package. The dominant chaotic behavior exhibited by the
system is the so-called Throw and Catch mechanism, in
which two linearized equilibrium points with a pair of
complex conjugate eigenvalues with slightly positive real
parts surround an unstable third equilibrium point with
eigenvalues (+ − −). The chaotic behaviour is predictably
similiar to the LORENZ System in Figure 1(8; 13).
P
P
10
7.5
7.5
5
5
2.5
2.5
E
-3
-2
-1
1
2
E
3
-3
-2
-1
1
2
3
-2.5
-2.5
-5
-5
-7.5
-7.5
k=5.4
P
k=7.0
P
7.5
7.5
E
-3
-2
5
5
2.5
2.5
-1
1
E
2
-2
-1
1
2
-2.5
-2.5
-5
-5
-7.5
-7.5
-10
k=24.5
k=10.0
9
Fig. 3. The Strange Attractor for varied parameter k
Liapunov exponent
ters for which chaos can occur is important for controlling
chaos in possible applications. A parameter study that
would reveal the range of parameters for which chaotic
behavior characterized by the well known invariant, a positive maximal Liapunov exponent is of interest and such a
study using the Wolf algorithm.(11) will be reported. At
the very early stages the system has a spectrum consisting
of negative or zero Liapunov Exponents for values of the
parameter k less than 6. Whenever k exceeds 6.75, chaotic
behavior, as evidenced by the presence of a positive Liapunov exponent occurs for a range of parameter values,
between k = 6.75 and k = 11.50, γ⊥ = 2.66, γk = 2.75,
∆o = 28, g = 6.06. These values are close to the values
suggested from a correspondance with the Lorenz model
and the closest laser system that they characterize would
be a Type B laser.
Liapunov exponent
Variation of each parameter showing regions where chaotic behaviour
continues as identified positive Lyapunov Exponents.
g
k=11.75, g=varied, gper=2.66, gpar=2.75, dlo=28
10
Fig. 4. Variation of parameter g vs Liapunov Exponents
Regimes
k=varied, g=6.06, gper=2.66, gpar=2.75, dlo=28
10
0.06
k
10
0.05
5
8
0.04
8
6
0.03
-2
-1
1
2
4
0.02
-5
2
0.01
Fig. 2. Variation of parameter k vs Liapunov Exponents
Regimes
The variation of the parameter k shows the behaviour of
the different regimes of the Liapunov Exponents both positive and negative values in Figure 2. For different values
of parameter k, the shape of the attractor are plotted in
Figure 3. For values of the parameter k that are greater
than 23, the system reverts to a regime with three negative
Liaupunov Exponents (3; 9).
0.5
0.01
0.02
0.03
1
1.5
2
-10
0.04
g=2.86
g=1.01
g=3.85
7.5
6
5
4
5
2
2.5
-3
-2
-1
1
2
-3
-2
-5
-1
1
2
-2
3
1
-2.5
-2
-5
-4
-10
2
3
-6
-7.5
g=4.5
6
-1
g=10.34
g=11.3
3
4
2
2
The variation of parameter g for the values less than 3,
k = 11.75, γ⊥ = 2.66, γk = 2.75, ∆o = 28 does not give
rise to positive Liapunov exponents. For the values greater
than 3, the strange attractor is plotted for different values
of g until the sink reverts to an attractor with unique
chaotic properties(8).In Figure 4 and 5 the limit cycle is
-2
-1
1
1
2
-2
-1
1
2
-1
-2
-2
-4
g=12.4
-3
11
g=15.7
Fig. 5. The Strange Attractor for varied parameter g
observed as orbitally stable with the negative Liapunov
In Hopf bifurcation, a small amplitude limit cycle bifurcates from the fixed point. The first Liapunov coefficient is
important for Hopf bifurcation since it determines whether
the bifurcation supercritical or subcritical. The validity
of the Hopf bifurcation approximation is investigated numerically by comparing the bifurcation diagrams of the
original laser equations and the generated data in Figure
6,7 and 8. Determination of the analytical expressions for
all branches of periodic solutions from Hopf bifurcation
points are studied for numerically different cases of different regimes. In these figures, The attractor is changing
from a limit cycle to a saddle point between these points
where Hopf and Branch points are observed for e, d and p
values(2; 4; 5).
coefficient, and therefore the bifurcation is supercritical.
40
35
30
BP
25
d
20
15
10
5
H
0
-5
-10
-20
-10
0
10
20
30
40
12
50
Fig. 6. Bifurcation Plot of variation of parameter g vs d
40
Liapunov exponent
g
35
30
gper
25
k=11.75, g=6.06, gper=varied, gpar=2.75, dlo=28
20
p
15
15
10
5
Fig. 9. Variation of parameter gper vs Liapunov Exponents
Regimes
H
BP
0
H
-5
7.5
-10
-20
-10
0
10
20
30
40
4
5
1350
2
2.5
g
-2
-2
-1
1
Fig. 7. Bifurcation Plot of variation of parameter g vs p
2
-1
1
-2
3
-2.5
-4
-5
-6
-7.5
gper=1.5
15
gper=5.18
2.36
6
2.34
4
2.32
10
2
-2
5
-1
1
2
1.16
1.17
1.18
1.19
-2
2.28
-4
2.26
-6
2.24
e
2.22
H
gper=6.86
BP
0
gper=12.8
16
H
Fig. 10. The Strange Attractor for varied parameter gper
-5
-10
-10
-5
0
5
g
10
15
2014
Fig. 8. Bifurcation Plot of variation of parameter g vs e
The variation of the parameter gper behaves like the parameter gpar. At the initial stages, both of the paramters
have a maximal positive Liapunov Exponents then after
a value, that is nearly the same for both parameters,
negative Liapunov Exponents are obtained in Figure 9
and 11. The plots of the attractors are consistent with the
Liapunov exponent
Liapunov exponent
'0
gpar
k=11.75, g=6.06, gper=2.66, gpar=2.75, dlo=varied
k=11.75, g=6.06, gper=2.66, gpar=varied, dlo=28
20
17
Fig. 11. Variation of parameter gpar vs Liapunov Exponents Regimes
4
Fig. 13. Variation of parameter dl0 vs Liapunov Exponents
Regimes
10
3
4
3
2
5
2
2
1
1
-3
-1
-0.5
0.5
1
-2
-1
1
2
3
-1.5
1.5
-1
-1
-0.5
0.5
1
-1.5
-1
-0.5
0.5
1
1.5
-1
-5
-2
-2
-2
-10
-3
-3
gpar=4.5
gpar=0.5
10
10
5
5
-4
dl0=11.75
dl0=10.2
20
10
10
5
-3
-2
-1
1
2
3
-3
-2
-1
1
-5
-5
-10
-10
2
3
-3
-2
-1
1
2
3
4
-4
-2
2
4
-5
-10
-10
gpar=6.18
gpar=6.24
dl0=68
30
dl0=116
5.939
20
5.9388
10
5.9386
dl0=165
5.9384
-4
-2
2
4
6
5.9382
3.06273 3.06275 3.06278
3.06283 3.06285 3.06288
gpar=16.34
-10
18
21
-20
Fig. 12. The Strange Attractor for varied parameter gpar
Fig. 14. The Strange Attractor for varied parameter dl0
positive and negative Liapunov Exponents. In Figure 10
and 12, a family of limit cycles are observed in the negative
Liapunov Exponent regime where the Hopf point occurs .
In conclusion, the Maxwell-Bloch system that is usually
used as a model for the Helium-Neon Laser has been studied. This system has a rich structure as a function of its parameters. We have shown that this system exhibits several
types of instability. The Liapunov spectrum is compatible
with the eigenvalues of the linearized system, which consist
of a negative value plus a pair of complex conjugate eigenvalues with slightly negative real part. These parameters
correspond to far infrared lasers where Lorenz like chaos
10
5
e
The parameter dl0 does have the same scale with respect
to the other parameters. The initial movement is starting
with positive Liapunov Exponents then it goes into sink
part after 120. In Figures 13 and 14, it can be easily observed that behaviour of the strange attractor is changing
into the saddle point where the sink appears. In Figures
15 and 16 the branch and Hopf points around dl0=15 can
be observed.
15
H
BP
0
H
-5
-10
0
5
10
15
20
25
22
30
d0
Fig. 15. Bifurcation Plot of variation of parameter dl0 vs
e
[10] A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and B. Sautois,. MATCONT and
CL MATCONT: Continuation toolboxes in matlab.
Ghent and Utrecht Universities Preprint, 2006.
[11] A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano.
Routes to chaotic emission in a cw He- Ne Laser.
Physics Letters A, Vol. 28,2, 1983.
[12] A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano.
Determining Lyapunov exponents from a time series.
Physica D, pages 285–317, 1985.
[13] A. Umur. Local Nonperturbative Methods for Algebraic Calculation of Liapunov Exponents. M. S.
Thesis, Department of Physics, Boğaziçi University,
1997.
30
25
20
15
p
10
5
H
BP
0
H
-5
-10
-15
-10
0
10
20
d0
30
40
50 23
Fig. 16. Bifurcation Plot of variation of parameter dl0 vs
p
has been observed(10; 12). The bifurcation mechanism
that characterizes the possible transition to chaos from
this operating point is studied by the MATCONT(6; 10)
package and Hopf bifurcation is identified in several instances. Emphasis has been placed on one operating point
where chaotic nature as evidenced by a positive maximal Liapunov exponent is shown while the change of the
varying parameters are plotted for each item. However,
this behavior is sensitive to variations in the parameters
and rapidly changes to steady state. As each parameter
is varied, transition to/from chaotic behavior can be seen.
We have identified several instances of Hopf bifurcation at
this operating point, the operating point is within limits of
the parameter range for which chaotic behaviour has been
studied in the He-Ne laser(2; 3; 4; 5).
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