geometric frustration in large arrays of coupled lasers near field far field micha nixon eitan...
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Geometric Frustration in Large Arrays of
Coupled LasersNear FieldFar Field
Micha NixonEitan Ronen, Moti Fridman, Amit Godel,
Asher Friesem and Nir DavidsonWeizmann Institute of Science
IntroductionWhat is phase locking ?
)()()( ttt 12 const
Coupling
How to phase lock lasers ? (Diffraction coupling.)
Output coupler Mirror
GAIN
Degenerate cavity
Mirror
GAIN
Mask
f1f2
Lens (f2)
Lens (f1)
Output coupler
f2 f1
),( yxE
),( yxE
300µm
),( yxE
Near field
Far field Far field
Degenerate cavity square array
Near field
Degenerate cavity
Mirror
GAIN
Mask Lens Lens Output coupler
Far Field
No coupling
Far Field
With coupling
Negative coupling
21 EE
E
0 ππ
0
00
0 π
π
Negative coupling
0 π
π 0
XY model of anti ferromagnetic interactions
n
ji
jiJ Η
Negative coupling
0 π
π 0
Triangular array
32
32
0
32 3
2
32
32
32
32
Triangular array
0
32
32
32
323
2
32
Triangular array
Near FieldMany longitudinal modes.
Far Field
Kagome array
32
32
0
32
0 32
0
Kagome array
Near Field
Far Field
Moessner R and Chalker J T “Low-temperature properties of
classical geometrically frustrated antiferromagnets”, Phys. Rev. B 58 12049 (1998)
Honey-comb (Grafin) array
Near Field
Far Field
No φ phase ordering !
32
32
0
?2 ? ?
?
??
?
??
?
??
?
??
?
??
?
??
?
??
?
??
?
??
?
??
2
22
2
22
2
22
2
22
2
22
2
22
2
22
2
22
2
22
2
22
2
2
2
22
2
22
?
? ?
3φ phase ordering
Is this a 3φ “condensate” ?
3φ condensate simulations: Ground states using a Monte Carlo simulation that minimizes the spins energy.
XY model
Next Nearest Neighbor Coupling
0
-+
-+
Kagome array with next nearest neighbor coupling
z
Mirror
GAIN
Mask
f1f2
f2
f1
Output coupler
f2 f1
T
nm
zzif
x
nm e4
22
2
NN
NN
nmnm
df
ddx
0
Kagome array with next nearest neighbor coupling
2
2
),(
),(
yxI
yxIIPR
Kagome array with next nearest neighbor coupling
External “field ”
Degenerate cavity
External laser
0 π 0 0 0 0?
Gain
External Field for 1D
Near field spiral Far field
External disordered field
Degenerate cavity
0 π 0 ? ? ??
External degenerate laser cavity
Effects of finite “temperature” in square array
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.
Fourier coupling
Mirror
GAIN
Mask
f1f2
Lens (f2) Pinhole
Lens (f1)
Output coupler
f2 f1
E }{EF
}{EF
U
U}{UFE
“Phase transitions”
Near field square lattice
No coupling
Positive coupling
Negative coupling
Sharp phase transitions
Direct measurement of phase decoherence
0 0 0 0,, 0,,
50% fringe visibility 100% fringe visibility
0
Distance1 2 3 4 5 6
Frin
ge v
isib
ility
0.5
0.6
0.7
0.8
0.9
1
0.4
0.3
0.2
0.1
Short range phase ordering
Numerical model
• Laser rate equations
• Cavity transfer matrix.
}{
)sin(innj
iji
dt
d
ni
i
e
e
1
M
ni
i
e
e
1
Phase Locking
Fourier plain
f1f1f2
f2
(f2)
(f1)
Many Longitudinal modesA single lasers spectral lines are separated by
LK
1