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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

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Introduction What is phase locking ? Coupling

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

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

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Near field Far field Degenerate cavity square array Near field

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

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Negative coupling 0 π π 0 0 0 0 π π

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0π π0 XY model of anti ferromagnetic interactions

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Negative coupling 0π π0

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Triangular array

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Near Field Many longitudinal modes. Far Field

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Kagome array

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

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Honey-comb (Grafin) array Near Field Far Field

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No φ phase ordering !3φ phase ordering

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

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Next Nearest Neighbor Coupling 0 -+ -+

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

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Kagome array with next nearest neighbor coupling

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

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External Field for 1D Near field spiral Far field

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

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Effects of finite “temperature” in square array

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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.

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

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

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Sharp phase transitions

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

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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

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Numerical model Laser rate equations Cavity transfer matrix.

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

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Many Longitudinal modes A single lasers spectral lines are separated by

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