1. Geometric Frustration in Large Arrays of Coupled Lasers Near Field Far Field Micha Nixon Eitan Ronen, Moti Fridman, Amit Godel, Asher Friesem and Nir Davidson Weizmann Institute of Science

2. Introduction What is phase locking ? Coupling

4. How to phase lock lasers ? (Diffraction coupling.) Output coupler Mirror GAIN

5. Degenerate cavity Mirror GAIN Mask f1f1 f2f2 Lens (f 2 ) Lens (f 1 ) Output coupler f2f2 f1f1 300µm

6. Near field Far field Degenerate cavity square array Near field

7. Degenerate cavity Mirror GAIN MaskLens Output coupler Far Field No coupling Far Field With coupling

8. Negative coupling 0 π π 0 0 0 0 π π

9. 0π π0 XY model of anti ferromagnetic interactions

10. Negative coupling 0π π0

11. Triangular array

13. Near Field Many longitudinal modes. Far Field

14. Kagome array

15. Near Field Far Field Moessner R and Chalker J T “Low-temperature properties of classical geometrically frustrated antiferromagnets”, Phys. Rev. B 58 12049 (1998)

16. Honey-comb (Grafin) array Near Field Far Field

17. No φ phase ordering !3φ phase ordering

18. Is this a 3φ “condensate” ? 3φ condensate simulations: Ground states using a Monte Carlo simulation that minimizes the spins energy. XY model

19. Next Nearest Neighbor Coupling 0 -+ -+

20. Kagome array with next nearest neighbor coupling z Mirror GAIN Mask f1f1 f2f2 f2 f2 f1 f1 Output coupler f2f2 f1f1

21. Kagome array with next nearest neighbor coupling

23. External “field” Degenerate cavity External laser 0 π 0 0 0 0 ? Gain

24. External Field for 1D Near field spiral Far field

25. External disordered field Degenerate cavity 0 π 0 ? ? ? ? External degenerate laser cavity

26. Effects of finite “temperature” in square array

27. Summary Frustration and other ground states of coupled systems (XY model) can be demonstrated experimentally with large arrays of coupled lasers. Coupling strength, range and sign can be easily controlled. Study effects of external fields, noise and quenched disorder.

29. Fourier coupling Mirror GAIN Mask f1f1 f2f2 Lens (f 2 ) Pinhole Lens (f 1 ) Output coupler f2f2 f1f1

30. “Phase transitions” Near field square lattice No coupling Positive coupling Negative coupling

31. Sharp phase transitions

32. Direct measurement of phase decoherence 50% fringe visibility 100% fringe visibility

33. Distance 123 4 56 Fringe visibility 0.5 0.6 0.7 0.8 0.9 1 0.4 0.3 0.2 0.1 Short range phase ordering

34. Numerical model Laser rate equations Cavity transfer matrix.

35. Phase Locking Fourier plain f1f1 f1f1 f2f2 f2f2 (f 2 ) (f 1 )

36. Many Longitudinal modes A single lasers spectral lines are separated by