REGULAR AND CHAOTIC BEHAVIOR IN THE VECTOR COMPLEX GINZBURG-LANDAU EQUATION

Antoni Amengual, Emilio Hernández-García,
Raúl Montagne, Maxi San Miguel, and Daniel Walgraef

Departament de Física, Universitat de les Illes Balears
E-07071 Palma de Mallorca, Spain
E.H-G e-mail address: dfsehg4@ps.uib.es

On leave from Université Libre de Bruxelles.

ABSTRACT

The complex Ginzburg-Landau equation describes generically the amplitude of the unstable modes near a homogeneous Hopf bifurcation in spatially extended systems. It has been recently shown that a vector generalization of this equation is adequate for describing the dynamics of light polarization in some nonlinear optical systems [1]. An exploratory analysis of the behavior of this vectorial complex Ginzburg-Landau equation in the one-dimensional case will be presented here. Travelling and standing wave patterns, localized pulses, regimes of spatio-temporal intermitency and the consequences of a breaking of rotational symmetry will be described.

References

• [1] San Miguel M., Phys. Rev. Lett 75, 425 (1995); Gil L., Phys. Rev. Lett, 70, 162-165, (1993).

The VCGL Equation and its Physical Origin

• On symmetry grounds, the VCGL Equation is the generic equation for the complex envelope of self-oscillations of a vector field with internal rotational symmetry near a Hopf bifurcation at zero wavenumber.

   

  (1)

• In particular, it can be deduced from a model for a wide-aperture single-longitudinal-mode ring laser with transverse flat mirrors in wich the laser transition occurs between two upper degenerate levels of angular momentum J=1 and a lower level of J=0. Polarization of light requires a vector description of the electric field:

   

  (2)

  
  
  

  

  

  For negative detuning, the equations for the complex envelopes of the two unstable oscillating vector modes at the lasing bifurcation of the above equations read: , Which, after using (2), is of the form (1) with . and are the same as for the scalar two-level model, and . turns-out to be positive (Benjamin-Feir stable regime).

Simple Solutions (FIGURE 1)



Dynamics of Polarization-Phase Instabilities

• The phase of the TW solutions is linearly unstable for .
• The phases and of SW solutions are simultaneously linearly unstable for .
• The depolarized solutions have also a region of the -plane centered in the origin beyond which they are phase-unstable. Our numerical analysis shows that TW and SW unstable solutions evolves to other solutions of the same class and smaller wavenumber through transient depolarized states (FIGURE 2a) . Unstable depolarized solutions evolve in time to other states, which generically are also depolarized, through a mechanism in which the circularly polarized component of larger wavenumber reduces its wavenumber while the other component remains stable (FIGURE 2b) .

Spatio-Temporal Intermittency Regimes

• As for the scalar equation, there are parameter regimes in which the solutions of the VCGL consist of regions close to the "simple" solutions above, separated by turbulent regions in which localized objects can be identified (FIGURE 3).
• For close to 1, the main localized objects are circulary polarized spots in a linearly polarized background.
• Increasing the coupling the chaotic dynamics of the two components becomes strongly correlated (syncronized) (FIGURE 4) .

Breaking of the internal rotational symmetry

• The introduction of terms breaking the -phase symmetry (birrefringence, imperfections,...) leads to new phenomenology.

   

  (3)

• The anisotropy fixes the direction of polarization of the TW, and modifies the SW:

  

• For ( if ) these solutions destabilize, giving rise to TW with oscillating propagation direction (FIGURE 5) .
• Spatio-temporal intermittency regimes are also present in (3) (FIGURE 6) .




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