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Department of Physics of Complex SystemsDepartment of Physics of Complex Systems The Weizmann Institute of ScienceThe Weizmann Institute of Science
RehovotRehovot, Israel, Israel
The Ultrafast Optics GroupThe Ultrafast Optics Group
YES.YES.OUI.OUI...כןכן
HanburyHanbury BrownBrown--TwissTwiss
InterferometryInterferometry with interacting photonswith interacting photons
Department of Physics of Complex SystemsDepartment of Physics of Complex Systems The Weizmann Institute of ScienceThe Weizmann Institute of Science
RehovotRehovot, Israel, Israel
Yaron Bromberg, Yoav
Lahini, Eran
Small and Yaron
Silberberg
The Ultrafast Optics GroupThe Ultrafast Optics Group
19561956HB&T invent intensity correlations interferometry
to measure angular size of distant stars
THG images of biological specimenTHG images of biological specimenHanburyHanbury
BrownBrown--TwissTwiss
InterferometryInterferometry
•
Source size determines the speckle size
A source of incoherent emitters generates a (time varying) speckle field
•
Spatial intensity fluctuations carry information on the source
2
Δ
ΔxSLW // λδθ ≈=
S
HBT and QOHBT and QO
HBT was a key point in the development of quantum optics.
It is often discussed
today in terms of particle interference.
HBT reflects quantum statistics -
bunching
(bosons) or anti-bunching (fermions).
1999
The HBT analysis is valid only for non-interacting particles
How interactions affect HBT Correlations?
HanburyHanbury BrownBrown--TwissTwiss
InterferometryInterferometry with interactionswith interactions
THG images of biological specimenTHG images of biological specimenPhoton interactions via material nonlinearityPhoton interactions via material nonlinearity
Intensity dependent index of refraction induces photon interactions
n2
> 0 : Focusing nonlinearity -
attractive interactions
n2
< 0 : Defocusing nonlinearity -
repulsive interactions
•
HBT correlations in 2d (1+1) •
Attractive
•
Repulsive•
HBT correlations in 3d (2+1)
•
Experiments•
Intensity Histograms
HanburyHanbury BrownBrown--TwissTwiss
InterferometryInterferometry with interacting photonswith interacting photons
The propagation of a speckle field in a nonlinear medium
1+1 speckle evolution with attractive 1+1 speckle evolution with attractive interactionsinteractions
n2
=0n2
=0.01n2
=0.02n2
=0.04
SolitonsSolitons
in 1+1 Focusing Mediumin 1+1 Focusing Medium
Solitons
will be formed once the speckle size spreads enough. The correlation function width then reflects the average soliton
width.
There is no minimal power for soliton
formationHigh power
beams generate narrow solitons
Weaker beams generate broader solitons
In 1+1, there will be always a distance at whichsolitons are formed.
Beyond this distance, the correlation function nolonger carries information on the source
Nonlinear Schrödinger equation:
111 ~)/(cosh~)( −−− aPaxaxE
THG images of biological specimenTHG images of biological specimennumerical simulations numerical simulations ––
2D (1+1)2D (1+1)
linear
n2 =0
focusing
n2 >0
x
z
Nonlinear Schrödinger equation:
The speckle size is determined by the nonlinearity.
“solitons”
0 0.5 1 1.5 2 2.5 3x 10-3
1
2
3
4
5
n2
wid
th-1
-200 -100 0 100 2000
2
4
6
8
Δ
g(2) (Δ
)
n2=0n2=0.0006n2=0.0013n2=0.0019n2=0.0025
Width of correlation function
1+1 Defocusing Medium
n2
=0n2
= -0.01n2
= -0.02n2
= -0.04
1+1 Defocusing Medium
Speckles become flatter, forming dark solitons between them.
As the background spreads, the solitons broaden with distance, as z1/2
The correlation width is related to the soliton
width
)/tanh(~)( 1 axaxE −
HBT with defocusing nonlinearity HBT with defocusing nonlinearity
n2 <0
defocusingx
z
“dark solitons”
-100 -50 0 50 1000
0.5
1
1.5
2
Δ
g(2) (Δ
)
n2=0
n2=-0.0006
n2=-0.0013
n2=-0.0019
n2=-0.0025
The correlation function width is related to the dark solitons. Its width grows as z 1/2, and its peak is reduced below the 2:1 ratio of thermal light.
Nonlinear HBT correlations in 1+1
Linear propagation –Ideal 2:1 corr. Peak.Width is linear with distance
Attractive interactions: Correlations attain a fixedwidth, no longer relate tothe source!
Repulsive interactions:width increases as z1/2,peak decays
2+1 speckles 2+1 speckles ––Linear spaceLinear space
2+1 Defocusing Space
Dark solitonsdominates the structure also in 2+1 geometry.
However, since the background spreads as z-1, the soliton
width
grow linearly with z
2+1 Focusing Space
When the power in a single speckle surpasses the critical power for self focusing, correlation peaks increase dramatically
Experiments in GaAs
waveguides
AlGaAs
slab waveguide, Kerr nonlinearity
Experiments with focusing nonlinearity (1+1)Experiments with focusing nonlinearity (1+1)
CCDdiffuser
x
x
2
-200 -100 0 100 2000
0.5
1
1.5
2
2.5
Δ (CCD pixel)
g(2) (Δ
)
Linear
-200 -100 0 100 2000
0.5
1
1.5
2
2.5
Δ (CCD pixel)
g(2) (Δ
)
LinearP=35mW
-200 -100 0 100 2000
0.5
1
1.5
2
2.5
Δ (CCD pixel)
g(2) (Δ
)
LinearP=35mWP=60mWP=90mW
-200 -100 0 100 2000
0.5
1
1.5
2
2.5
Δ (CCD pixel)
g(2) (Δ
)
LinearP=35mWP=60mWP=90mWP=120mW
Observed:Narrowing
and growing
of correlations function
“1D”
diffuser
Experiments with thermal nonlinearity (n2
<0)
Increasing Intensity…
propagation through liquid + absorbing dye
P=10mWP=300mWP=600mWP=1000mW
diffuser
Experiments with thermal (defocusing) nonlinearityExperiments with thermal (defocusing) nonlinearity
-60 -30 0 30 600
0.5
1
1.5
2
Δ
g(2) (r)
P=20mWP=250mWP=500mWP=1200mW
P=800mWP=1500mWP=2000mWP=100mW
Speckle StatisticsSpeckle Statistics
1
10
100
1000
0 200 400 600
W
δx
g(2)
Correlations Intensity Distribution
0.0001
0.001
0.01
0.1
1
0 1 2 3 4 5 60.0001
0.001
0.01
0.1
1
0 1 2 3 4 5 60.0001
0.001
0.01
0.1
1
0 1 2 3 4 5 610-5
0.0001
0.001
0.01
0.1
1
0.1 1 10 100
Intensity, I/<I>
Log
(Pro
babi
lity)
Intensity Histograms (1+1)Intensity Histograms (1+1)
IIe /−Linear,
Attractive2/3−I
Intensity histogramsIntensity histogramsThe intensity histograms strongly depend on the nonlinearity
numerics, n2
=0
0 2 4 610
-2
10-1
100
I/<I>
P(I)
n2=-0.002
numerics, n2
<0
0 5 10 15
10-4
10-2
100
I/<I>
P(I)
LinearP=120mW
experiment, n2
>0 experiment, n2
<0
0 5 10 15 20
10-2
100
I/<I>
P(I)
n2=0.002
100 10210-4
10-2
100
numerics, n2
>0
0 5 10 15
10-4
10-2
100
I/<I>
P(I)
LinearP=1.5W
0 2 4 610-3
10-2
10-1
100
I/<I>
P(I)
n2=0
ConclusionsConclusions
• Interactions strongly affects HBT interferometry
• HBT measurements can probe the nature of interactions between particles
Atom-matter waves Electron interferometry Heavy ion collisions
• A “soliton picture” is fruitful for analyzing the effects of interactions. Statistical optics Nonlinear optics
HBT with defocusing nonlinearity HBT with defocusing nonlinearity
n2 <0
defocusingx
z
• In defocusing nonlinearity, dark solitons are formed
“dark solitons”
-100 -50 0 50 1000
0.5
1
1.5
2
Δ
g(2) (Δ
)
n2=0
n2=-0.0006
n2=-0.0013
n2=-0.0019
n2=-0.0025