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
Introduction
What is phase locking ?
Coupling
 (t )   2 (t )  1 (t )  const
How to phase lock lasers ?
Output
coupler
(Diffraction coupling.)
Mirror
GAIN
Degenerate cavity
Output Mask
coupler
Lens
(f2)
Lens
(f1)
Mirror
GAIN
f2
f2
f1
f1
E ( x, y )
E (  x , y )
E ( x, y )
300µm
Degenerate cavity square array
Near field
Far field
Near field
Far field
Degenerate cavity
Output Mask
coupler
Lens
Lens
Mirror
GAIN
Far Field
No coupling
Far Field
With coupling
Negative coupling
E1   E2    
0
π
0
π
0
π
0
π
0
E

Negative coupling
XY model of anti
ferromagnetic
interactions
n
Η  J  i 
i j
0
π
π
0
j
Negative coupling
0
π
π
0
Triangular array
 2
2
3
0
3

 2
2
3
3
 2
 2
3
3
Triangular array
2
 2
3
3
3
2
0
3
2
 2
 2
3
3
Triangular array
Near Field
Many longitudinal modes.
Far Field
Kagome array
0
 2
3
2
3
0
 2
 20
3
3
Kagome array
Near Field
Moessner R and Chalker J T
“Low-temperature properties of
classical geometrically
frustrated antiferromagnets”,
Phys. Rev. B 58 12049 (1998)
Far Field
Honey-comb (Grafin) array
Near Field
Far Field
No3φ
φ phase ordering !
2?
2
?
2?
2
?
2?
2?
2?
2?
2?
2?
2?
 22
3
2?
 22
3
2?
2?
2?
2?
2?
2?
2?
2?
2
?
2?
2?
20
2?
2?
2?
2?
2?
2?
2
?
2?
2?
2?
2?
2?
2?
Is this a 3φ “condensate” ?
XY model
3φ condensate simulations: Ground states using a Monte Carlo
simulation that minimizes the spins energy.
Next Nearest Neighbor Coupling
+
-
0
+
-
Kagome array with next nearest
neighbor coupling
Output Mask
coupler
f2
Mirror
f1
GAIN
z
f2
 nm  e
f2
2
2 x nm
 f 2 4 i z
f1
f1
xnm 
zT
f 
d nm
0
d NN
d NN
Kagome array with next nearest
neighbor coupling
IPR 
I ( x, y)
I ( x, y)
2
2
Kagome array with next nearest
neighbor coupling
External “field”
External laser
Gain
Degenerate cavity
0
π
0
?
0
0
0
External Field for 1D
Near field spiral
Far field
External disordered field
External degenerate laser cavity
Degenerate cavity
0
π
0
?
?
?
?
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
Output Mask
coupler
Lens
(f2)
Lens
Pinhole (f1)
Mirror
GAIN
f2
f2
E
F{E}  U
E F{U }
F{E}  U
f1
f1
“Phase transitions”
Near field
square lattice
Negative coupling
No coupling
Positive coupling
Sharp phase transitions
Direct measurement of phase
decoherence

0

0

50% fringe visibility

0
100% fringe visibility
0

, ,0 , ,0
Short range phase ordering
1
0.9
Fringe visibility
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1
2
3
Distance
4
5
6
Numerical model
• Laser rate equations
• Cavity transfer matrix.
 e i 1 
 e i 1 
d i    
  j
    sin(
dt e i n j nn{ie} i n 
M

i )
Phase Locking
Fourier
plain (f )
1
(f2)
f2
f2
f1
f1
Many Longitudinal modes
A single lasers spectral lines are separated by
K 
1
L