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