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 , ?
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