A.A. Chabanov, Abe Pena (UT-San Antonio)

Jing Wang, A.Z. Genack (Queens College of CUNY)

Speckle Fluctuations and Correlation

Speckles

Wave propagation in disordered media

mean free path

wavelength

Field

Intensity

Average intensity:

Gaussian statistics: only the pairs of identical paths have the same phase and thus give a contribution to the average intensity

Wave diffusion in a disordered medium

wavelength

mean free path

Diffusion equation for the average intensity:

Wave diffusion in a disordered medium

(This equation would yield the Ohm’s law for a disordered conductor)

Wave interference

A

A* 22*222* 2,4 AAAAAA Probability of return:wave particle

• transport reduction

• nonlocal correlation

• weak localization

• non-Gaussian statistics

Transmission coefficients

a′ b

:abt

a ababa

abba

abab

TTT

TT

tT2

Transmitted intensity = speckle intensity

Total transmission = brightness

Transmittance = conductance

a b′

Transmission coefficients

,

0

...

01

VUt

N

N

tt

0

...

01†

n n

nn ana

nm mbnbmnamanab

nbnn anab

ttTrT

uT

vvuuT

vut

)( †

2

**

i.e., Beenakker, RMP (1997)

Statistics of tab and Tab

ban bn nannbnanab vTvuvut 2

0

22

/exp

1)(

/),(

NT

irTP

NT

dTirP

aa

a

a

22

baabab vTtT

0 /exp)(

/)(

NT

TTP

NT

dTTP

a

aba

a

aab

Kogan & Kaveh, PRB (1995)

AAC & Genack, PRA (2005)

Alumina sample

d=0.9 cm

n=3.14

f=0.068

alumina sphere:

copper tube: D=7.3 cm

L=60 cm, 10,000 sample configurations

A: ν=14.7-15.7 GHz, var(sab)=1.18, diffusive wave

B: ν =9.95-10.15 GHz, var(sab)=6.18, localized wave

C: t=740 ns, var[sab(t)]=20.1, strongly localized wave

Transmission in alumina samples

a

aa

ab

abab T

Ts

I

Is var21var

7 10 13 16 19-40

-30

-20

-10

0

<I a

b>

(dB

)

Frequency (GHz) -500 0 500 1000 150010-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

time (ns)

<I ab

(t)>

AB C

σ = 5 MHz

Transmitted field distribution

-10 -8 -6 -4 -2 0 2 4 6 8 10

P

10-5

10-4

10-3

10-2

10-1

100

101

AB C

abab T

i

T

r

2exp1

)(

P

Gaussian statistics:

Characteristic and distribution functions of total transmission

sa0 1 2 3 4 5

P(s

a )

0.0

0.5

1.0

1.5

A

B

C

A

B

C

z0 2 4 6 8 10

F(z

)

10 -2

10 -1

100

)()exp()2cos( zFzszasa

)var(3/2)],//1(lnexp[)( 2as sggzgzgzF

a

Nieuwenhuizen & vanRossen (1995)

Stoytchev & Genack (1999)

Factorizing of statistics of the field and intensity

nan

na

n

bn

ab

kakkn

an

bn

ab

TN

nTvT

kn

knTN

kTvt

!

120

22

!)!12()()(

2

2/

Ea

bbabaab FN

TvvTtt

**

2

2

2222 1 E

a

bbabaab FN

TvvTTT

Fluctuations:

Correlations:

Correlation with polarization

0 15 30 45 60 75 90

C

0.0

0.5

1.0

1.5

0 15 30 45 60 75 90

Re

FE

0.0

0.2

0.4

0.6

0.8

1.0

=0.29

=0.24

=0.32

a

a

T

Tvar

cos*

*

bbab

baab

E vvNT

ttF 22

2 11 EE

ab

baab FFT

TTC

AAC, Hu & Genack (2004)

Statistics of total transmission

n nana uT 2

TT

T

T

TTuvvTT

TuuT

a

a

ababbaab

aaa

222

2

1

2

1 In localized regime (only one open channel):

Statistics of transmission quantities in localized regime

0 /exp)(

/)(

NT

TTP

NT

dTTP a

a

)//(2)(/

2/

exp)(/

)( 20

02

0

NTTKTPNT

dT

NT

TTP

NT

dTTP ab

a

aba

a

aab

22

2

2222

'

22 11 outE

inEbbaabaab FF

N

TvvuuTTT

)1()(1 32'' outinoutinoutinoutinbaab FFFFAFFAFFss Pnini (2001)

)/var()1)(1)(1(,1 33''2 TTAFFAssA outinbaab

Correlation with wave polarization

Ei(S )E yi

E xi

S Dsample z

E x

E y E(D )

Intensity correlation of localized waves

D

0 15 30 45 60 75 90

C

0

2

4

6

8

10

S = 0o

(a)

(b)

S = 90o

S D

S D

S D

0 15 30 45 60 75 90

C

0

5

10

15

20

S D

4L

Intensity correlation of localized waves

S = 0o

S = 90o

S D

S D

D

0 15 30 45 60 75 90

C /

(1+

cos2

D)

0

2

4

6

8

10

12

S DS D

]1[2,1 3AFin

31,0 AFin

)1)(1()1( 3

''in

out

baab FAF

ss

04.038.4)var(3 sA

• In a given random configuration, the statistics of transmitted field is Gaussian for both diffusive and localized waves; non-Gaussian mesoscopic field statistics arise in ensemble of configurations due to mesoscopic fluctuations of transmission

• In localized regime, the transmitted intensity can be written as a product of three statistically independent variables; two of them have Rayleigh distribution

• Future work:

Conclusions

In diffusive regime (many channels):

TT

TTTP a

aa , ?