1. Phase synchronization and polarization ordering
IMEDEA
Palma de Mallorca, Spain
Phase synchronization and polarization ordering
in VCSELs arrays
Alessandro Scirè, Pere Colet and Maxi San Miguel

2. Motivation.I
Synchronization, phase transitions and cooperative effects
IMEDEA

3. Motivation.II IMEDEA K Kuramoto model:
analogy between synchronization and phase transition
IMEDEA
Order parameter is a measure of the degree of synchronization
Critical behavior
2nd order phase
transition K

4. Motivation.III IMEDEA Phase transitions in optics 1. Laser oscillation
“A complete analogy of the laser light distribution function to that of the Ginzburg-Landau theory of superconductivity is found mathematically which allows us to interpret the laser threshold as a quasi-second-order phase transition”.
[Cooperative phenomena in systems far from thermal equilibrium and in non-physical systems
H. Haken Rev. Mod. Phys –121 (1975)]
2. Many-modes laser dynamics
Introduction of a theory for the ordering of many interacting modes in lasers is presented. By exactly solving a Fokker-Planck equation for the distribution of waveforms in the laser in steady state, equivalence of the system to a canonical ensemble is established, where the role of temperature is taken by amplifier noise.
Passive mode locking is obtained as a phase transition of the first kind.
[Phase Transition Theory of Many-Mode Ordering and Pulse Formation in Lasers
Ariel Gordon and Baruch Fischer, Physical Review Letters (2002)]

5. Lasers with polarization degree of freedom
IMEDEA
Amplitude equation for lasers with polarization degree of freedom
m net gain
w detuning
b nonlinear frequency shift.
g cross saturation term. g<1 => Linearly Polarized solutions are stable.
ga dichroism gp birefringence d preferential polarization direction
M. San Miguel, Phys. Rev. Lett. 75, 425 (1995).
A. Amengual, D. Walgraef, M. San Miguel & E. Hernández-García, PRL, 76, 1956 (1996).
Array of lasers with global coupling:

6. Phase model.I IMEDEA Disregarding polarization (2yj(t)=dj=d0, gp=0):
Kuramoto model for limit cycle oscillators.
Y. Kuramoto, Chemical Oscillations, Waves & Turbulence, Springer (1984).
G. Kozyreff, A.G. Vladimirov & P. Mandel, PRL 85, (2000),
PRE 64, (2001), Europhys. Lett. (2003)

7. Phase model.II
Mean field vs local coupling
IMEDEA
Kuramoto model in a square lattice: with local coupling.
The synchronization process is shown to be the same,
but for a rescaling of the coupling K
[arXiv:Cond-mat ] MF
3rd
2nd
1st

8. Order parameters.I IMEDEA Pw distribution of frequencies
definitions
IMEDEA
Pw distribution of frequencies
Pd distribution of polariz. angles
Two order parameters to characterize the degree of collective synchronization:
phase synchronization
polarization ordering

9. Uncoupled case. IMEDEA C=0 Stationary solutions: Then
Continuous limit:
P(d): distribution of natural polarization angles

10. Global phase synchronization transition
IMEDEA
sw<< sd
Take yj=dj/2 then:
Small disorder: define an effective coupling as:
Finally:
Effective Kuramoto model => Apply standard techniques.
Pw distribution of natural frequencies

11. Polarization ordering transition
IMEDEA
sw<< sd
Assume phase synchronization
Not a Kuramoto-like model! However, we can apply similar techniques.
Using order parameter definition:
Implicit eq. for stationary values
Introducing stationary values in order parameter definition provides a self-consistent eq.
P(d): distribution of natural polarization angles

12. Order parameters.II IMEDEA sw=0.01 sd=0.91 numerical results
Coupling term in polarization eq.

13. Synchronization transitions
Coherence lowering due to polarization ordering
IMEDEA
sw=0.01
sd=0.91

14. Synchronization transitions
IMEDEA
sw~ sd
sw=0.12
sd=0.91

15. Phase synchronization and polarization ordering in VCSELs arrays
IMEDEA
Palma de Mallorca, Spain
Phase synchronization and polarization ordering in VCSELs arrays
Conclusions
We introduced a prototype model to study phase synchronization and polarization ordering in oscillators with polarization degree of freedom, with a possible application to VCSELs arrays.
We derived a self-consistent theory to provide the order parameters describing the transitions to phase synchrony and polarization ordering.