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Statistics of microwave radiation in the approach to localization

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STATISTICS OF MICROWAVE RADIATION
IN THE APPRO ACH TO LOCALIZATION
by
MARIN STOYTCHEV
A dissertation subm itted to the G raduate Faculty in Physics in p a rtia l fu lfillm e n t
o f the requirem ents for the degree o f D octor o f Philosophy, The C ity U niversity o f
New York.
1998
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UMI Number: 9830766
Copyright 1998 by
S toytchev, Marin
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M A R IN S. S T O Y T C H E V
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iii
This manuscript has been read and accepted for the Graduate Faculty in Physics in
satisfaction of the dissertation requirement for the degree of Doctor of Philosophy.
*//W
ML
Date
•hair o f Examining Committee
s'
Date
Executive Officer
Joel I. Gersten, City College
Alexander Lisyansky, Queens College
Harold Metcalf, SUNY Stony Brook.
Steven A. Schwarz, Queens College
Supervisory Committee
THE CITY UNIVERSITY OF NEW YORK
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To Narciso Garcia,
a teacher and a friend whom I d e a rly miss.
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Acknowledgments
I wish to th a n k m y supervisor D r. A z rie l Genack for the generous help and his
guidance th ro u gh o u t th is w ork. His expertise has been cru cia l fo r the successful
com pletion o f th is research. I am g ra te fu l to D r. Narciso G arcia fo r being always
there for me when I needed su p po rt and encouragement the m ost.
to thank D r.
1 w ould like
A lexander Lisyansky fo r the valuable help a t some stages o f this
w ork. 1 w ould lik e to th a n k D r. M ichael Kem pe, D r. D m itry L ivd a n , M r. A ndrey
Chabanov, and A ir. V ic to r P odolsky fo r the fru itfu l discussions and frie n d ly advice.
1 w ould like to th a n k M r. W a lte r P olkosnik and M r. D m itry Zaslavsky fo r helping
me learn enough about com puters and com puting so th a t I managed to accom plish
th is p a rt o f the w ork on acceptable level. 1 w ould like to thank M r. G rig o ry German.
M r. Ed Kuhner, and M r. Z iggy O zim kovski fo r th e ir expert technical assistance.
I w ould like to th a n k D r. Joel Gersten, D r. H arold M etcalf. D r. Steven Schwartz,
and Dr.
A lexander Lisyansky fo r being k in d to serve in m y thesis Supervisory
C om m ittee.
1 th a n k m y w ife N ora and m y son Stoytcho fo r bearing w ith me d u rin g the many
years o f study.
It is m y pleasure to exte n d m y g ra titu d e to a ll members o f the F aculty and S taff
at the D epartm ent o f Physics a t Queens College.
F inally, I w ould like to acknowledge the co n tribu tio n , in the form o f discussions
or com m unications o f results and ideas, o f D r. M a rk van Rossum, D r. P ie t Brouwer,
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D r. Eugene Kogan, D r. B oris Shapiro, D r. Reuven P n ini, D r. P atrick Sebbah. D r.
B a rt van Tiggelen, D r. Costas Soukoulis, and D r. M ichalis Sigalas.
T h is w ork is p a rt o f the ongoing research on wave propagation in random m edia
w hich is supported b y the N ational Science F oundation and by PSC-CUNY.
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Contents
A cknow ledgm ents
L ist o f Figures
v
ix
1 In trod u ction
1
2 W ave T ransport in M esoscop ic S ystem s
9
2.1 Key transm ission quantities - th e ir averages, fluctuations and correlations 9
2.2 Theoretical approaches
15
2.3 iVlicrowave experim ents - po ssib ilitie s and problem s
23
3 T otal T ransm ission D istrib u tio n in A bsorbing
R andom W aveguides
26
3.1 Background
26
3.2 Samples and measurements
28
3.3 Results and discussions
30
3.4 Conclusions
33
4 In ten sity D istrib u tion in th e A p p roach to L ocalization
35
4.1 Background
35
4.2 Samples and measurements
37
4.3 Results and discussions
38
4.4 Conclusions
45
5 F ield and In ten sity C orrelations in R andom W avegu id es 46
5.1 F ie ld -fie ld correlation fu n ctio n w ith frequency sh ift
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46
V lll
5.2 In te n s ity -in te n s ity sp a tia l correlation
fu n c tio n
47
5.3 T o ta l transm ission measurements o f long- a nd infin ite -ra ng e in te n s ity corre­
la tio n
50
6 M icrow ave T ransm ission T hrough a P erio d ic 3D
M etal W ire N etw o rk C ontaining R a n d o m S catterers
53
6.1 Background
53
6.2 Samples and measurements
55
6.3 Results and discussions
56
6.4 Conclusions
61
Sum m ary
63
Figures
65
R eferences
98
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List of Figures
F ig u re 1. R epresentation o f inte nsity, to ta l transm ission, and conductance in term s
o f transm ission coefficients for incom ing and o u tgoing modes.
...65
F ig u re 2. T y p ic a l spectra o f the in te n s ity and to ta l transm ission norm alized to
th e ir ensemble average values, sab and sa, respectively for a polystyrene sam ple w ith
L / £ ~ 0 . 1.
...66
F ig u re 3. T o ta l transm ission spectra fo r tw o d iffe re nt sample co n figu ra tio ns. The
b o tto m graph in the figure represents the re la tive difference between tw o spectra
taken fo r a single sam ple configuration.
...67
F ig u re 4. D iagram m atic representation of: (a) the fu ll G reen’s fu n c tio n in term s
o f single scattering events; (b) the t-m a trix o f an in d iv id u a l scatterer: (c) the fu ll
G reen’s fu n ctio n using t-m atrices. The th ic k solid lin e denotes the fu ll G reen’s func­
tio n , the th in n e r so lid fine denotes the bare G reen’s function, the circles represent
single scattering process, and the d o tte d curves ind ica te (repeating) s c a tte rin g from
the same p a rticle .
...68
F ig u re 5. D iagram m atic representation of: (a) th e in te n sity propagator; (b) ’’ la d ­
der” diagram s; (c) ’’ m ost crossed” diagram s.
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...69
F ig u re 6. Schem atic representation o f a disordered conductor as a single sca tte rin g
center.
...70
F ig u re 7. P ro b a b ility d is trib u tio n o f th e eigenvalues o f the transm ission m a trix
p{r).
...71
F ig u re 8. T o ta l transm ission measurements - experim ental set up.
...72
F ig u re 9. P ro b a b ility d is trib u tio n o f the norm alized transm ission, P ( s a), fo r three
samples w ith dim ensions (a) d - 7.5 cm , L = 66.7 cm, (b) d
- 5.0 cm , L = 50.0
cm, and (c) d = 5.0 cm, L = 200 cm.
...73
F ig u re 10. S em i-logarithm ic p lo t o f th e transm ission d is trib u tio n s fo r the same
samples as in Fig. 9.
...74
F ig u re 11. C om parison o f the calculated (circles) and measured (squares) moments
o f the transm ission d is trib u tio n fo r samples w ith (a) g' = 1 0 .2 (L /£ ~ 0.1) and (b)
g' = 3 .0 6 ( L /£ ~ 0.4).
...75
F ig u re 12. Dependence o f var(sa) upon sam ple dimensions.
...76
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F ig u re 13. Measurements o f the tra n s m itte d field, consequently its in te n s ity - ex­
perim ental set up.
...77
F ig u re 14. P ro b a b ility d is trib u tio n o f in te n s ity for samples w ith L / £ ~ 0.1 and
0.4; the samples dimensions are (a) d = 7.5 cm, L = 100 cm, and (b) d = 5.0
cm, L = 200 cm, respectively. The so lid lines represent d is trib u tio n s obtained from
measured transm ission d istrib u tio n s [43] using Eq. (21).
...78
F ig u re 15. F it o f a negative stretched exponential o f power 1/2 to the ta il o f the
inte nsity d is trib u tio n (T /£ ~ 0.4).
...79
F ig u re 16. C om parison between the m oments o f in te n sity and to ta l transm ission
( £ / f ~ 0.4): • moments obtained fro m the measurements, o mom ents calculated
from the extended d istrib u tio n s.
...80
F ig u re 17. R elationship between var(sab) and var(sa) - com parison between the­
ory (Eq. (22)) and experim ent.
...81
F ig u re 18. In te n sity d is trib u tio n fo r the sample w ith L = 520 cm and d = 5.0 cm
(L /£ ~ 1.0). The curves represent the d is trib u tio n obtained from a transform of
the to ta l transm ission d is trib u tio n fo r th is sample calculated using the expressions
w ith absorption (Ref. [79])(solid line) and the expressions w ith o u t a b sorp tio n (Ref.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[45]) (dashed lin e ). The insert shows the calculated to ta l transm ission d is trib u tio n s
w hich are transform ed to give the corresponding in te n s ity d is trib u tio n s .
...82
F ig u re 19. P ro b a b ility d istrib u tio n s P(ln(sab) fo r samples w ith : (a) L/E, ~ 1.0 and
( b ) L / e ~ 0 .1 .
F ig u re 20.
...83
In te n s ity spectra in frequency and tim e dom ain as measured (th ic k
line) and a fte r correction for absorption (th in lin e ).
...84
F ig u re 21. In te n s ity d istrib u tio n s fo r L /L a ~ 5 obtained from the m easured spec­
tra ( • ) and from the spectra corrected fo r absorption (o ). The in se rt shows the
d is trib u tio n w ith o u t absorption which is com pared to a transform o f the to ta l trans­
m ission d is trib u tio n fo r this sample calculated using the expressions fro m Ref. [45]
(solid lin e ).
...85
F ig u re 22.
Dependence o f var(s a) upon L. The different sym bols represent: •
results o b ta in e d fro m the measurements, o results from the d a ta corrected fo r ab­
sorption, □ values calculated using the th e o re tica l expressions in Ref.
solid line represents the ra tio £var(sa) / L ca lculated in Ref.
[79]. The
[79] in the diffusive
lim it.
...86
F ig u re 23. Dependence o f C(L) = < s ^ > c / < s 2> 2 upon L. The d iffe re n t sym bols
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represent: • results obtained fro m the measurements, □ values ca lcu la te d using the
th e o retica l expressions in Ref. [79] w hich include loca lizatio n corrections. T he solid
line gives the behavior o f C(L) in the diffusive lim it as calculated in Ref. [79]. ...87
F ig u re 24. F ie ld a u to co rre la tio n fu n c tio n w ith frequency s h ift for the sam ple w ith
L/£ ~ 0.1. The so lid line represents a f it o f the theory to the data.
...88
F ig u re 25. In te n s ity -in te n s ity co rre la tio n fu n ctio n w ith sp a tia l displacem ent for
the same sam ple as in Fig. ‘21. The so lid lin e represents a f it o f the th e o ry to the
data.
...89
F ig u re 26. T o ta l transm ission a u to co rre la tio n fu n ctio n w ith frequency s h ift fo r the
same sam ple as in Fig. 21. T he d iffe re n t curves represent ca lculations using the
theoretical expression from Ref. [85] in w hich the sample param eters are used: solid
line - b o th absorption and in te rn a l reflection are included, dashed lin e - w ith o u t
absorption and in te rn a l re fle ctio n , d o tte d lin e - o n ly absorption inclu d ed .
...90
F ig u re 27. T o ta l transm ission cross-correlation fu n ctio n w ith frequency s h ift:
(a) obtained from m easurements fo r the same sample as in Fig. 21; (b ) th e o retica l
results from Ref. [87] - from upper to low er curves a = 0, 1 /L , 2 / L.
...91
F ig u re 28. Transm ission o f m icrowave ra d ia tio n as a fu n ctio n o f frequency through
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XIV
the m etal netw ork: (a) em pty, (b ) fille d w ith Teflon spheres.
...92
F ig u re 29. Average o f the real p a rt o f the field tra n sm itte d th ro u g h samples of
Teflon spheres contained in a copper tube, the fie ld a m p litu d e is norm alized to
u n ity.
...93
F ig u re 30. Phase ve lo city in samples o f Teflon spheres contained in a copper tube
w ith fillin g fraction / = 0.60.
...94
F ig u re 31. Transm ission spectrum fo r a single sample co n fig u ra tio n fo r m ixtures o f
Teflon-alum inum spheres at a fillin g fra ctio n o f alu m in um spheres f ai = 0.05. The
ve rtica l dashed line indicates th e band-gap edge for the system fille d w ith Teflon
spheres.
...95
F ig u re 32. Average transm ission spectra obtained from 200 sam ple configurations
o f Teflon-alum inum m ixtures a t fillin g fractions o f alum inum spheres o f 0.05 and
0.10.
...96
F ig u re 33. Frequency dependence o f the second moment o f the n orm alized inten­
sity, <s^b> for the stru ctu re fille d w ith m ixtures o f T e flo n -a lu m in u m spheres. The
ve rtica l dashed line indicates th e band-gap edge for the system fille d w ith Teflon
spheres.
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...97
1
1 Introduction
The propagation o f classical wave th ro u g h random media affects many aspects
o f everyday life . Inform ation ca rrie d b y sound, lig h t, radio and microwave radia­
tio n is tra n sm itte d through m edia w hich, in general, vary ra n d o m ly in tim e and
space so th a t the am plitude and the phase o f the waves flu ctu a te random ly in tim e
and space, [ l j E xtra ctin g in fo rm a tio n fro m th is ’’ random ” signal and investigating
the o rig in o f these fluctuations has been o f inte rest through the ages. [*2,3] In this
thesis we stu d y steady state tra n sp o rt in m u ltip ly scattering m edia consisting o f
random ly positioned scatterers. Interference between scattered waves leads to large
flu ctu a tio n s in key transm ission q u an titie s - inte nsity, to ta l transm ission, and to ta l
transm ittance, w hich corresponds to the dim ensionless conductance g. Because o f
the pervasiveness o f fluctuations, a fu ll d e scrip tio n o f tra n sp ort m ust provide not
o n ly the averages b u t the variances and th e fu ll d istrib u tio n s o f these transm ission
quantities. [4|
Interest in the problem has in te nsifie d w ith the recognition o f the analogy be­
tween classical and quantum waves in disordered systems. In the m id-eighties flu c­
tu a tio n s o f the electronic conductance o f order o f u n ity were observed in disordered
conductors cooled below 1 K . [5,6] These flu ctu a tio n s appeared to be independent o f
sample param eters such as the conductance its e lf, the size o f the sam ple or the trans­
p o rt mean free p a th t o f the electrons in th e sam ple and were therefore named u n i­
versal conductance fluctuations (U C F ). Such flu ctu a tio n s were explained by consid­
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ering the Interference processes between electron waves. [7,8j The discovery o f UCF
and the predictions and th e observations o f coherent backscattering [9,101: short[11-14] and long-range [15—25] in te n s ity correlation, photon lo c a liz a tio n [26-32], and
the photonic band gap [33] have been o f p a rtic u la r im portance fo r the developm ent
o f a new fie ld now know n as Mesoscopic Physics. It deals w ith tra n s p o rt in random
systems in w hich the waves are te m p o ra lly coherent th ro u g h o u t the sample.
In 1958, Anderson proposed th a t electrons could become localized by a random ly
varying p o te n tia l. [34] In m u ltip le -sca tte rin g media, the interference processes may
become so stro n g th a t n o rm a l diffu sion vanishes w ith the dim ensionless conduc­
tance g becom ing sm aller th a n u n ity . Follow ing the analogy between electrons and
classical waves in mesoscopic systems, the p o s sib ility o f lo ca lizin g classical waves in
stro n g ly sca tte rin g m edia was suggested. [26-30] In a p ro o f o f p rin c ip le experim ent
o f microwave transm ission in a three dim ensional (3D) p e rio d ic m etal w ire network,
i t was shown th a t classical wave lo ca liza tio n due to strong sca tte rin g is possible [35].
Recently, lo ca liza tio n o f in fra re d lig h t in stro n g ly sca tte rin g samples o f pure GaAs
particles was observed by W iersm a et. a ll. [36] Localization due to stro n g scattering
was achieved also in m icrowave experim ents in q u a si-ID samples o f alu m in a spheres
by Chabanov and Genack [37].
Strong in trin s ic sca tte rin g can be evaluated in term s o f the p ro d u ct o f the wave
num ber k and the tra n s p o rt mean three path I o f the wave in the system . The
con ditio n fo r lo ca liza tio n due to strong scattering is given b y the loffe-R iegel c ri­
te rio n kl < 1 [38], w hich im plies th a t the wave scatters in a distance o f order or
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3
sm aller th a n the w avelength. T h is strong condition m ust be m et in the case o f a
unbounded 3D system, b u t is n o t necessary fo r samples on w hich s p a tia l boundaries
are im posed.
It was firs t suggested by Thouless th a t electrons w ill alw ays be localized in suf­
fic ie n tly long wires a t low tem peratures. [39] This approach to lo c a liz a tio n can be
viewed as a p a rtic u la r re a liz a tio n o f the loca lizatio n co n d itio n g < 1. In connection
w ith th is, we note th a t the dim ensionless conductance, w hich is a central param eter
in the description o f the s ta tis tic s o f mesoscopic tra n sp o rt [40], can be used in a
diffe re nt context in the d e scrip tio n o f wave tra n sp o rt. Here, we consider three ways
in w hich the dimensionless conductance is a central param eter in the description o f
the s ta tis tic a l properties o f transm ission quantities. The d e fin itio n o f the conduc­
tance o f a p a rtic u la r system comes from the scattering th e o ry approach pioneered
by Landauer [41]. A ccording to th is approach, the conductance equals the to ta l
transm ission when a ll in p u t modes are present. In th is case, the average value o f
the conductance g is given as g = N i j L . where N is the to ta l n u m b e r o f in p u t
modes, L is the sample length, and i is the tra n sp ort mean free p a th . The average
value o f to ta l transm ission fo r one in p u t mode is i/ L . The lo c a liz a tio n threshold is
reached a t g = 1 w hich determ ines the localization length o f the sam ple £ = NL
For L >
the transm ission th ro u g h the sample decreases e xp o n e n tia lly w ith in ­
creasing L. [40] A n o th e r way to describe the tra n sitio n to lo c a liz a tio n is provided
by considering the conductance o f the system as the ra tio between the level w id th
SE o f energy states (modes) and the level spacing A E , g = 8 = 8 E / A E (8 is known
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4
as the Thouless num ber) [42J. W hen th is ra tio is much greater th a n u n ity, different
modes (states o f the system ) overlap creating a conductance band and the system
is in m e ta llic regime. If, however, the Thouless num ber is sm aller th a n one, the
modes are localized and the sam ple is an insulator. The lo c a liz a tio n c o n d itio n is
again given by g = 1. F inally, in th e p ro b a b ilistic w orld, the re cip ro ca l value o f the
conductance l /g gives the p ro b a b ility fo r a wave to re tu rn to a coherent volume.
The firs t two aspects are associated w ith fin ite systems and re fle ct boundary condi­
tions and indicate th a t lo c a liz a tio n can be achieved when g ~ 1 even when k£ > I.
The crite rio n k l < 1 is analogous to the condition for lo c a liz a tio n associated w ith
the p ro b a b ilistic m eaning o f g in th e case o f a 3D system. It is im p o rta n t, however,
n ot o n ly to achieve classical wave lo ca liza tio n , but also to give a fu ll sta tis tic a l
description o f wave tra n sp o rt in th is process.
In th is thesis, we investigate the changing character o f the s ta tis tic s o f wave
tra n sp o rt in the approach to the lo c a liz a tio n and provide a q u a n tita tiv e description
o f the d istrib u tio n s o f in te n s ity and to ta l transm ission o f m icrowave ra d ia tio n near
the loca lizatio n threshold.
The thesis is organized as follow s. In c h a p te r 2, we give an overview o f wave
tra n sp o rt in mesoscopic systems.
The key transm ission q u a n titie s are presented
using the scattering theory approach o f Landauer. [411 The d iffe re n t term s con­
trib u tin g to the in te n sity co rre la tio n fu n ctio n , which determ ine th e m agnitude o f
flu ctu a tio n s in transm ission q u a n titie s are discussed. In th is chapter, we describe
b rie fly the two theoretical approaches - diagram m atic fie ld th e o ry and random ma­
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tr ix th e o ry (R M T ) - the results o f w hich are heavily used in the analysis o f the
measured d istrib u tio n s o f in te n s ity and to ta l transm ission. The diagram m atic ap­
proach uses an expansion in the sm all param eter 1jg and is expected to provide an
adequate description o f wave tra n s p o rt o n ly in the d iffusive regim e o f propagation
(g
1). It appears, however, th a t th is approach, when m odified appropriately, can
describe fluctuations in samples w ith sm all values o f g as w ell. [43] R M T does not
require the condition g
1 and is p a rtic u la rly useful fo r stu d yin g the tra n s itio n
to lo ca liza tio n . This approach, however, uses the iso tro p ic approxim ation which
assumes perfect m ixing o f modes. [44| It can be used in describing wave tra n sp ort in
q u a si-ID samples in w hich the sam ple length is much greater th a n its cross-sectional
diam eter d. A t the end o f chapter 2 we discuss the p o ssib ilitie s and the d ifficu ltie s
w hich e xist in microwave experim ents.
In c h a p te r 3. we study the s ta tistics o f to ta l transm ission in quasi-ID random
samples w ith sm all values o f the dim ensionless conductance. Recently, an expression
for the p ro b a b ility d is trib u tio n o f to ta l transm ission in term s o f g fo r non-absorbing
samples was obtained by Nieuwenhuizen and van Rossum using diagram m atic tech­
niques com bined w ith R M T [45] and subsequently by Kogan and Kaveh using the
R M T approach [46]. The d is trib u tio n was proposed to be Gaussian w ith a nonGaussian ta il. In o p tical experim ents w ith samples w ith g ~ 103, de Boer e t. al
observed sm all but clear deviations from Gaussian d is trib u tio n . [47| In microwave
measurements in samples w ith sm all values o f g (g ^ 10), we observe d is tin c tiv e ly
non-G aussian d istrib u tio n s. [43] S urprisingly, we fin d th a t the measured transm is­
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6
sion d istrib u tio n s are adequately described by the th e o ry [45.46] when the measured
variance o f the d is trib u tio n , w hich equals the degree o f nonlocal in te n s ity correlation,
is used as a ch aracteristic param eter in the theoretical expression. The variance o f
the to ta l transm ission d is trib u tio n norm alized to its ensemble average value, how­
ever, is found to increase su b lin e a rly w ith sample length. T h is is in contrast to the
superlinear increase expected from the th e o ry in the absence o f absorption. This
result is associated, therefore, w ith the affect of absorption on the s ta tistics o f wave
tra n sp o rt and raises the question o f w hether loca lizatio n can be achieved in strongly
absorbing samples.
In c h a p te r 4, we present measurements o f in te n s ity d is trib u tio n s fo r the same
samples as those used in th e measurements o f to ta l transm ission. Here, however,
measurements were made fo r sample lengths as large as the lo c a liz a tio n length £
(m ore than tw ice as large as the m axim um length used in the measurements o f to ta l
transm ission). [48] These measurements allows us to inve stig a te fundam ental issues
o f the sta tistics o f mesoscopic tra n sp o rt. Recently, Kogan and Kaveh proposed a
relationship between the m om ents o f in te n s ity and to ta l transm ission. Using this
re la tio n , the in te n sity d is trib u tio n can be obtained as a tra n sfo rm o f the to ta l trans­
m ission d is trib u tio n . [46] We confirm experim entally the re la tio n sh ip between the
moments o f these transm ission qu an titie s and the re la tio n sh ip between the corre­
sponding d is trib u tio n s. T h is allow s us to investigate the scaling o f the variance o f
to ta l transm ission using measurements o f the in te n s ity d is trib u tio n s for lengths up
to the loca lizatio n length. W e fin d th a t the variance o f the n o rm alized tr a n sm is sion
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increases sup erline a rly w ith len g th near the lo ca liza tio n le n g th even in the pres­
ence o f stro n g absorption. T h is re su lt indicates th a t a b so rp tio n does not destroy
lo ca liza tio n . We also dem onstrate th a t fie ld measurements p rovide means fo r sta­
tis tic a lly e lim in a tin g the influence o f absorption on wave tra n s p o rt. T h is fa cilita tes
the d e te rm in a tio n o f £ and allow s us to cla rify the in te rp la y between absorption and
lo ca liza tio n . O u r results, show th a t the variances o f in te n s ity and to ta l transm ission
are reliable measures o f the closeness to the lo ca liza tio n threshold.
In c h a p te r 5, we o b ta in the absorption length and the d iffu sio n constant o f the
samples discussed in chapters 3 and 4 from the fie ld a u to c o rre la tio n function w ith
frequency s h ift.
We show also th a t the variance o f the to ta l transm ission equals
the degree o f nonlocal in te n s ity correlation across the o u tp u t face o f the sample.
F in a lly, we present measurements o f the long- (C 2) and in fin ite -ra n g e (C 3) terms o f
the in te n s ity co rrela tio n fu n c tio n w ith frequency s h ift.
In c h a p te r 6, an a lte rn a tive approach to lo ca liza tio n - p h o to n ic band gap (P B G )
m aterials - is discussed. In th is approach, lo ca liza tio n can be achieved by t u n in g
the frequency o f the ra d ia tio n th ro u gh the edge o f the p h o to n ic band gap, in con­
tra s t to the approach em ployed in chapters 3 and 4 in w hich the sam ple dimensions
were changed in order to achieve localization. The p o s s ib ility o f creating periodic
d ie le ctric structures w hich possess a photonic band gap (P B G ) was firs t discussed
b y E. Y ablonovitch [33] and S. John [49] in 1987. Since then, the band structure
fo r various 3D die le ctric [50-53] and m etallic [54-56] p e rio d ic stru ctu re s possessing
band gaps have been calculated and dielectric [50,57-59] and m e ta llic [60] systems
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8
possessing band gaps have been constructed. I f disorder is intro du ce d in to a PBG
m aterial it is possible to create a localized state w ith in the band gap w hich has
been dem onstrated in m any w orks. [50] The sta tistics o f lo ca liza tio n have not been
explored, however, fo r states associated w ith defects. In a p ro o f o f p rin cip a l exper­
im ent, we use a 3D cubic m etal stru ctu re possessing p h oto nic band gap and show
th a t the sta tistics o f the lo c a liza tio n tra n sitio n can be stu d ied for an ensemble o f
strong scatterers random ly po sitio n e d inside the photonic crysta l. [35]
We conclude the thesis w ith a s u m m a ry o f the m ain results o f th is work.
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9
2 Wave Transport in Mesoscopic System s
2.1 K ey tra n sm issio n quantities - th eir averages, fluctuations
and correlations
The term mesoscopic systems has its o rig in from the subm icron size o f the disor­
dered conductors in w hich below
1
EC EJCF have been observed [5,6J. The subm icron
dimensions o f the samples (in the case o f electronic tra n s p o rt) are in the interm e­
diate regime between th e atom ic (m icroscopic) scale and the everyday macroscopic
scale, hence the name ’’ mesoscopic” . Mesoscopic systems are su fficie n tly sm all th a t
electrons (o r waves) m a in ta in th e ir phase coherence, so th a t the classical description
o f transport is inadequate. On the other hand, th e y are s u ffic ie n tly large th a t a sta­
tis tic a l description is m eaningful. In order to m a in tain phase coherence, the sample
m ust be sm aller than th e characteristic phase coherence len g th Lv determ ined by
the strength o f ine la stic scattering.
The sta tis tic a l behavior o f electron transport encountered in mesoscopic samples
is o n ly quantum m echanical in th a t it derives from the w avelike behavior o f electrons
and analogous s ta tis tic a l behavior is present in any system in w hich waves propa­
gate by m u ltip le coherent scattering. The analogy between quantum and classical
wave tra n sp ort emerges clea rly from the scattering th e o ry approach pioneered by
Landauer [41]. In th is approach, the electronic conductance can be expressed solely
in term s o f the transm ission coefficients o f the sample considered as a single, com­
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10
plex scattering center. Here we present b rie fly the argum ents leading to the sim plest
expression o f th is typ e as th e y appeared in the w ork o f Stone e t. al on random ma­
tr ix theory and m axim um e n tro py models in disordered conductors [61). A n ideal
tw o-probe measurement is considered in which the sam ple is attached between two
perfect reservoirs w ith electrochem ical potentials Hi and H2 = H l + eV , respectively,
where V is the applied voltage. These reservoirs serve b o th as current source and
sink and as voltage te rm in a ls. The to ta l current w hich flow s through the sample
can be obtained from a ’’ co u n tin g argum ent” . In the energy in te rv a l eV between H2
and Hi electrons are in je c te d in to righ t-g o ing states em erging fro m reservoir
1.
but
none are injected in to le ft-g o in g states emerging from reservoir 2. Thus there is a
net right-going cu rren t p ro p o rtio n a l to the num ber o f states in the interval Hi ~~1^2 given by
( 1)
where lV is the num ber o f pro pa g a ting channels in the sam ple. Vi is the lo n g itu d in a l
ve lo city for the ith m om entum channel at the Ferm i surface. Tjj- is the transm ission
p ro b a b ility from i to j, and g is the dimensionless conductance.
For a quasi-ID
system the density o f states is driijde = l/ZiU j. *
*In the description o f classical wave transport, the letters a and b are used as subindexes
to indicate different modes instead o f the letters i and j as given in Eq. (1). The ’’classical
wave” notation w ill be used throughout the rest of this thesis.
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11
T h is approach can be d ire c tly applied to classical wave transport in random
media.
The key transm ission quantities in order o f increasing spatial averaging
are the inte nsity, Tab, w hich is the transm ission coefficient for incom ing m ode a
in to mode
6,
the to ta l transm ission for inco m in g mode a, Ta =
to ta l tra n sm itta n ce T = J2ab^ab-
Tab) and the
The to ta l tra n sm itta n ce is equivalent to the
dimensionless conductance in electronic systems, T = g. Schematic presentation o f
these q u an titie s is g ive n in Fig.
1.
In describing wave tra n sp o rt in random media, there are im po rta n t ch a racteristic
lengths w hich define th e regim e o f propagation. A mean free path is a ch a racteristic
length scale w hich is generally introduced to describe the scattering process.
In
m u ltip le sca tte rin g processes, the system is characterized by the transport mean
free p a th £, w hich is th e average distance in w hich the d irection o f propagation o f
the wave is random ized. I f the sample le n g th L is much larger than the tra n s p o rt
mean free path , L
£, and away from the lo ca liza tio n threshold, g = £ / L
I.
wave tra n sp o rt is described by diffusion theory. The diffusive regime is then given
by f C L <
W hen L becomes com parable to £, loca lizatio n corrections m ust be
taken in to account.
A no th e r length scale o f p a rtic u la r im portance is the absorption length La w hich
is defined as the tra ve le d len g th over w hich the in te n s ity is reduced by a fa cto r o f 1 /e
due to loses in the m edium . In the diffusive regim e o f propagation La = (Dra)1/2.
where
1jr a
is the a b so rp tio n rate and D is the d iffu sio n coefficient. A b so rp tio n is
often neglected in th e o re tica l models. In re a lity, however, absorption is generally o f
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12
im portance to an extent determ ined by the ra tio L f La. Systems then can be consid­
ered as w eakly absorbing (L < La) or stro n g ly absorbing (L
L). In the la te r case
the influence o f absorption upon the behavior o f ce rta in tra n sp o rt parameters could
be significant. T h is is fo r exam ple the case o f the average values o f the transm ission
quantities w hich fa ll e xp o n e n tia lly w ith sample le n g th due to the presence o f ab­
sorption. Because lo ca liza tio n also leads to an exponential decay o f transm ission it
can be d iffic u lt to disentangle the influence o f ab sorp tio n and localization on tra n s­
mission. In th is w ork, we fin d , however, th a t a b so rp tio n affects o n ly m arginally the
sta tistics o f in te n sity and to ta l transm ission and th e variances o f these quantities
serve as reliable measures to the closeness to the lo ca liza tio n threshold.
In the diffusive regim e the sta tistica l averages o f inte nsity, to ta l transm ission,
and transm ittance are given by <Tab>— Z/NL, < T a >= Z/L. and < T > = g =
iVZ/L. Because o f wave interference these transm ission quantities e xh ib it significant
fluctuations from th e ir ensemble average values. T y p ic a l spectra o f inte nsity and
to ta l transm ission norm alized by th e ir ensemble averages, sab = Tab/ <Tab > and
sa = Ta/ <Ta>, measured in samples w ith g « 10 are shown in Fig. 2.
The in te n s ity is the least sp a tia lly averaged q u a n tity and e xh ib its the largest
fluctuations. Even fo r g
1, its variance equals one, whereas the variance o f to ta l
transm ission is o f order o f l/g . We note also the difference between the w id th o f
the peaks observed in the in te n s ity spectrum and th a t in the case o f to ta l transm is­
sion. This illu stra te s the w ell known fact th a t the w ith o f the in te n s ity correlation
function w ith frequency s h ift is determ ined by the short-range correlation (C \ term )
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13
and is much sm aller th a n the correlation frequency in to ta l transm ission w hich is
determ ined by long-range correlation ( C2 te rm ). In Fig. 3, we show to ta l trans­
m ission spectra fo r tw o different sample configurations w hich illu s tra te the large
sample to sample flu c tu a tio n s o f the tra n sm itta n ce quantities. On th e contrary, the
spectrum fo r a given sam ple configuration is h ig h ly reproducible. T h is can be seen
from the difference between the two spectra taken fo r the same sam ple configuration
w hich is shown in th e b o tto m part o f the figure. S im ila r flu ctu a tio n s are observed
when one moves th e detector across the o u tp u t face o f the sample w hile keeping
the frequency o f the ra d ia tio n fixed. A typ ica l exam ple o f spatial flu ctu a tio n s is the
spectrum observed w hen the laser beam is tra n s m itte d through a slab containing
random scatterers.
The m agnitude o f th e spectral and spatial flu ctu a tio n s in the transm ission quan­
titie s is determ ined b y the spectral and sp a tial co rrela tio n o f in te n s ity inside the
sample. In a waveguide geom etry in which modes are p e rfe ctly m ixed, the ensem­
ble average o f Tab. < T ab>, does not depend on a o r b. The co rre la tio n m a trix.
Cabaibi =<STab8Ta,bi > / < Tab > ( 5Tab = Tab— < T ab > ), is then the sum o f three
d is tin c t term s, [17,19,62]
C = C\ + C2 + Cz = Ai5aai8bb’ -j- A 2(5aai + 6bbi) -I- A 3 ,
(2)
where, to second o rd e r in the sm all param eter l/g , the coefficients A j, A 2, and A 3
are given by A L =
7 (1
+ ^ - ) , A2 =
^3
=
7
^ a w ith
7
=
. [63] The firs t te rm describes shot-range co rre la tio n and dom inates flu ctu a tio n s o f
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14
Tab■ The second term, describes long- range correlation and determ ines flu ctu a tio n s
o f Ta. The last term represents in fin ite -ra n g e correlation and dom inates the flu ctu a ­
tions in T. [ t is the degree o f nonlocal co rre la tio n (C2 + C3) w hich is the source o f the
deviations o f in te n sity from negative exponential statistics and o f to ta l transm ission
from a norm al d is trib u tio n w hich have been predicted [45,46,64] and e xperim entally
observed [22,43,47,65] in studies o f th e s ta tis tic o f wave tra n s p o rt in random me­
dia. As the extent o f sp a tia l averaging increases, the variances o f th e norm alized
transm ission quantities , sab, sa and s (= T / < T > ), are reduced. However, these
q u an titie s do n o t self average, as th e y w ould i f spatial co rrela tio n were absent. To
leading order in l/g, the variances o f sa&, sa, and s arising from nonlocal correlation
for d iffu sin g waves are 1, '2/3g, and 2 /1 5 g2 [17,19] corresponding to enhancem ent o f
I, L / i [16], and ( L f i ) 2 [7,8], respectively.
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15
2.2 T h eo retica l ap p roach es
rn th is section, we b rie fly introduce two theoretical approaches, the results o f
which are used in the analysis o f the measured in te n s ity an d transm ission d is trib u ­
tions presented here, nam ely the diagram m atic approach an d the R M T approach. *
In considering the d ia g ra m m a tic Green’s fun ctio n approach we w ill tre a t the
sta tio n a ry fie ld E (r). T h e e le ctric fie ld satisfies the tim e -in d e pe n d e nt Helm holtz
equation
V 2 £ ( r ) + ^ - < r ) £ ( r) = 0,
(3)
where E ( r) denotes one o f the fie ld ’s components, e (r) is th e (s p a tia lly random )
dielectric constant o f the system , u is the frequency, and c is the speed o f lig h t in
vacuum. The wave equation can be re w ritte n as
S7 2E( r ) + ^ E ( r ) = V ( r ) E ( r ) ,
(4)
where V(v) is the sca tte rin g p o te n tia l defined as F"(r) = —{ui/ c)2[e(r) — 1 ], For a
iThe reader should not expect to see here development o f these approaches in order to
solve particular problems and the solutions of such. Great deal o f work has been done
on wave propagation in random media (mesoscopic systems) using both approaches. The
authorof this thesis is most
W. vanRossum
fa m ilia r w ith the work of Th. M. Nieuwenhuizen and M. C.
on Green’s functions diagrammatic approach (see Ph. D. thesis of M.
C. W. van Rossum [6 6 ] and references therein) and the work o f C. W. J. Beenakker and
co-authors on R M T approach (fo r an updated review on R M T see Ref. [67]).
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16
collection, o f p o in t-like scatterers w ith p o la riz a b ility
£*0
in a surrounding m edium
w ith die le ctric constant o f u n ity, the scattering p o te n tia l is given by
V(r) = - ^ - Y ^ S ( r - r t ),
(5)
w ith Tf denoting the p o sition o f th e scatterers.
In tro d u cin g the Green’s fu n c tio n G!o (r 1 , r 2) as the so lu tio n o f the equation
2
V ^ G 'o (ri,r2) -I- — G 'o (ri,r2) = —6(r1 — r 2),
(6 )
one can w rite the solution for the fie ld in Eq. (4) fo rm a lly as
£ ( r i) = Ein(r x) - J d v 2GQ{vx, r 2 ) K ( r 2 ) £ ( r 2),
(7)
where Ein (r i) represents the incom ing coherent wave. G'0 ( r 1. r 2) is also referred
to as a bare Green’s fu n ctio n and. describes propagation o f the field, in a m edium
w ith o u t scatterers. It is given by
r (r _ n _ e x p ( - i k | r x — r 2 |)
C o (r i' r 3 ) -
4?r | r i — r2 [
'
(8)
where k — u /c is the wave num ber.
To describe the propagation o f the fie ld in the m edium independent o f Ein. we
use the to ta l Green’s function G (r l5 r 2) w hich is defined as a so lu tio n of
V 2 G (r1; r 2) -b ^ e ( r ) G ( r i, r 2) = - S f a - r 2),
(9)
The G reen’s function G (rL, r 2) describes the fie ld at any p o in t r x in the medium,
due to a source a t r 2. It can be presented as p e rtu rb a tio n series as
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17
^ ( r i j r2) —
+
J
— Jdr^G o ivi, r a)Vr( r a)Gr0 ( r a, r 2)
y r f r arfrbGo(r1, r a) K ( r a)Go(ra, r b) K ( r b)6 ro (rb, r 2)
(10)
.
The firs t te rm o f Eq. (10) describes propagation w ith o u t sca tte rin g, the second term
equals the stun o f a ll single sca tte rin g contributions, th e th ird te rm - the sum o f a ll
double sca tte rin g co n trib u tio n s, etc. To s im p lify the n o ta tio n , Feynman diagrams
are used. The diagram m atic representation o f the above series is shown in Fig. 4a.
The lines represent the bare G reen’s fu n ctio n Gfo (r 1 , r 2) and th e circles represent
the sca tte rin g p o te n tia l o f an in d iv id u a l scatterer, —a 0(uj/c)26(rx — r 2).
lines connect id e n tica l scatterers.
Dashed
The above series can be fu rth e r sim plified by
intro du cin g the single p a rtic le t-m a trix £(r1; r 2, o>) (d ia g ra m m a tic representation:
x ) defined as the sum o f a ll repeated scattering events fro m one scatterer (Fig. 4b).
The re su ltin g diagram m atic presentation o f the to ta l G reen’s fu n ctio n using the
t-m a trix o f a single scatterer is shown in Fig. 4c. The p e rtu rb a tio n series fo r the tm a trix is called the B om series. The physical in te rp re ta tio n o f th is series is th a t the
incom ing fie ld induces an e le ctric p o la riza tio n (firs t te rm ). T h is p o la riza tio n changes
the fie ld around the scatterer, and th is change in tu rn affects the p o la riza tio n again
(second te rm ), etc. The t-m a trix o f a p o in t scatterer lo ca te d a t r t is to firs t order
t(r-L, r 2 ,w ) = a0(w/c)2S(v2 - r 1)<5(r1 - r t).
If one considers the fie ld and takes average over the p o sitio n s o f the scatterers to
obtain s ta tis tic a l param eters, a ll interference processes are lo st. In order to account
for wave interference the in te n s ity o f the fie ld m ust be considered. In term s o f the
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18
to ta l Green’s fu n ctio n , the in te n s ity is given by
f(r) = E(r)E*(r) =
J
y ’dr1dr2G (r,ri)C T(r,r2)E in(r1)E ^ (r 2 ).
(11)
The product GG* defines th e in te n s ity propagator R{ri,ro',r3,r 4) = <S'(r: , r 2) x
G{r3,r±) w hich describes th e in te n s ity a t any p o in t in the system due to the p ro d u ct
o f the incom ing waves Ein E'n . The diagram m atic expansion o f R is presented in
Fig. 5a. The upper lin e corresponds to C ? (ri,r 2 ) and the low er fine to the com plex
conjugate G * (ri, r 2 ). Dashed fines again connect id e n tica l scatterers. The s ta tis tic a l
properties o f tra n s p o rt th e n are given by the ensemble average < /? (rl ; r 2; r 3, r 4)> .
where angular brackets denote averaging over disorder. T h e goal o f th is approach
is to obtain < /? (rL, r 2; r 3, r 4) > . To lowest order in scatterers density, the in te n s ity
propagator can be a p pro xim ate d as b u ilt up o f the so called ’’ ladder” diagram s (F ig .
5b). Higher order corrections are included when various term s o f the ’’ most crossed”
diagram s are taken in to account some o f which are shown in Fig. 5c.
W hile the d ia g ra m m atic approach attem pts to b u ild th e sca tte rin g m a trix o f
the system as a whole fro m th e scattering m a trix o f an in d iv id u a l scatterer, R M T
considers the whole system as a single scatterer. In our presentation we fo llo w a
recent w ork by Beenakker [67]. A lth o ug h , th is w ork deals sp e cifica lly w ith quantum
tra n sp o rt in mesoscopic conductors it provides an updated overview o f mesoscopic
wave tra n sp ort in general.
A mesoscopic conductor is m odeled by an e la stica lly sc a tte rin g disordered region
connected by ideal leads (w ith o u t disorder) to tw o electron reservoirs (see Fig.
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6 ).
19
A ll in e la stic scattering is assumed to take place in the reservoirs, w hich are in equi­
lib riu m a t zero tem perature. The ideal leads are ” electron waveguides” . in tro d u ce d
to define a basis for the sca tte rin g m a trix o f the disordered region.
The wave fu n ctio n 'k ( r ) o f an electron in a lead at energy E f separates in to
a lo n g itu d in a l and a transverse p a rt, ^ “ (r ) = <$n(y, z)exp(±zknx ) . The integer
n — 1 ,2 ,
N labels the propagating modes, also referred to as sca tte rin g chan­
nels. M ode n has a real wave num ber kn > 0 and transverse wave fu n c tio n (&R.
(For s im p lic ity o f n o ta tio n , the two leads are assumed to be ide n tica l.) T he n o r­
m a liza tio n o f the wave fu n ctio n vfr is chosen such th a t it carries u n it cu rre n t.
A
wave incident on the disordered region is thus described by a vector o f coefficients
cm = ( a f , a<T,..., a j-, b^
,b% ,
bjf) describing th e am plitudes o f the incid e nt modes.
The firs t set o f N coefficients refers to the le ft lead and the second set o f coefficients
to the rig h t lead as shown in Fig.
6.
S im ila rly, the reflected and tra n s m itte d wave is
a vector w ith coefficients c°“ £ = (a f, a-J,..., a_y, b{ , bo , ..., bj/-). The sca tte rin g m a trix
S is a 2lV x 'IN m a trix w hich relates these tw o vectors,
( 12)
It has the block stru ctu re
S=
(13)
\
t F
/
w ith N x iV reflection m atrices r and F (re fle ctio n from le ft to le ft and fro m rig h t to
rig h t) and transm ission m atrices t and F (transm ission from le ft to rig h t and from
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20
rig h t to le ft).
C urrent conservation requires th a t S is a u n ita ry m a trix: S "
1
consequence o f u n ita rity th a t the four H e rm itia n m atrices t t *
1
= Sh
1
[t is a
— n ri. and
—r 'r d have the same set o f eigenvalues 7 \, T 2 , .... TV - Each o f these N transm ission
eigenvalues is a real num ber between 0 and 1. The sca tte rin g m a trix can be w ritte n
in term s o f T n:s by mean o f the polar decom position [17,68|
s=
(U
0
\ ( -V T = T
Vt
\
(
U'
0
\
(14)
0 V
Vt
y/l= T
/
0 V'
/
Here U, V. U'. V are fo u r N x N u n ita ry m atrices and T is a N x N diagonal m a trix
w ith the transm ission eigenvalues on the diagonal. *
The scattering matrix relates incom ing to o u tg o in g states. The transfer matrix
relates states in the le ft lead to states in the rig h t lead. A wave in the le ft lead can
be represented by a vector cle^1 = ( a f , a ^ ,..., a%, a f ,
, —, a^) w ith the firs t set
o f N coefficients re fe rrin g to incom ing waves and the second set o f coefficients to
outgoing waves. S im ila rly, a wave in the rig h t lead can be represented by a vector
cm = (b f , b f , ..., 6 J-, bf, b f , ..., bjf). The transfer m a trix M is a 2N x 2N m a trix th a t
relates these two vectors,
(15)
^For the constraints imposed on the scattering m a trix depending on the sym m etry present
see the work by Beenakker [67]
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21
The sca tte rin g a nd tra n sfe r m atrices are equivalent descriptions o f sca tte rin g from
the disordered region. A convenient property o f the tra n sfe r m a trix is the multi­
plicative co m p ositio n ru le - the transfer m a trix o f a num ber o f disordered regions in
series (separated b y ideal leads) is the product o f the in d iv id u a l transfer m atrices.
Thus, the tra n sfe r m a trix can be re a dily used to investigate the scaling o f mesoscopic
transport. The sca tte rin g m a trix , in contrast, has a m ore com plicated com position
rule (in vo lvin g m a trix inversion) and cannot be d ire c tly used in a scaling approach.
By expressing th e elem ents o f M in terms o f the elements o f S one obtains the p o la r
decom position o f th e tra n sfe r m a trix [17],
/
V
0
\
/
vr -1- 1
M =
0
V 't
/
V
VT~l - 1
VT^
/
U'
0
\
(16)
0
IP
in term s o f the same N x N m atrices used in Eq. (14).
The transm ission eigenvalues determ ine a va rie ty o f tra n s p o rt properties. F irs t
o f a ll is the conductance G = lim ^ _ o 7/F7 defined as th e ra tio o f the tim e-averaged
electrical current I th ro u g h the conductor and the voltage difference V between the
two electron reservoirs in the lim it o f vanishingly sm all voltage. A t zero tem perature,
the conductance is given by
N
G = GQJ 2 T n ,O 0 = rf!
h
(17)
The fa cto r o f tw o in G q accounts fo r the tw ofold spin degeneracy.
As an illu s tra tio n o f the description o f wave tra n s p o rt in term s o f transm ission
eigenvalues we give the e xp la n a tion o f UCF made by [m ry [69]. Im ry :s argum ent con-
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22
trasts ’’ closed” and ’’ open” scattering channels. M ost transm ission eigenvalues in a
disordered conductor are e xp onentially sm all. These are the closed channels. A frac­
tio n £ /L o f the to ta l num ber N o f transm ission eigenvalues is o f order o f u n ity . These
are the open channels. O n ly the open channels co n trib u te effectively to the conduc­
tance g ivin g a value o f th e dimensionless conductance g — G /G q = N eff « N£/L.
F luctuations in the conductance can be in te rp re te d as flu ctu a tio ns in the num ber
N eff o f open channels. T he picture o f ’’ closed” and ’’ open” channels o f propaga­
tio n is described m a th e m a tica lly by the bim o d a l (D orokhov [70|) d is trib u tio n o f the
transm ission eigenvalues p{r) which is shown in Fig. 7 in the case o f nonabsorbing
systems. The d is trib u tio n p(r) has been used to calculate the p ro b a b ility d is trib u ­
tio n o f to ta l transm ission [45,46,73,79|. Consequently, the relationship between the
sta tistics o f in te n s ity and to ta l transm ission is used in order to o b ta in the p ro b a b ility
d is trib u tio n o f in te n s ity . [461
To sum m arize, R M T o f mesoscopic tra n s p o rt addresses the fo llo w ing tw o ques­
tions. W h a t is the ensem ble o f scattering m atrices fo r a p a rtic u la r system? How
does one o b ta in the s ta tis tic s o f transport p ro pe rtie s from this ensemble?
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•23
2.3 M icrow ave exp erim en ts - p ossib ilities an d d ifficu lties
In classical wave experim ents it is possible in prin cip le to measure a ll three trans­
m ission q u an titie s. In o ptical experim ents one measures re a d ily in te n s ity in the far
fie ld and to ta l transm ission. In m icrowave experim ents one u su a lly measures the
near fie ld . Measurements o f conductance, however, in classical wave experim ents
have n o t been reported yet. In contrast, one d ire c tly measures the conductance in
studies o f electronic tra n sp ort. In studies o f the statistics o f wave tra n sp o rt, the
a b ility to create an ensemble o f s ta tis tic a lly equivalent sample configurations is o f
p a rtic u la r im portance. An ensemble o f s ta tis tic a lly equivalent sam ple configurations
can be easily realized in the case o f classical waves by physically changing the posi­
tio n o f the scatterers inside the sample. In the case o f electrons one does not have
an ensemble o f different sample configurations. Instead, a va rying m agnetic field
is used to sim ulate different sample configurations. In this case, however, the tim e
reversal sym m etry o f the system is broken. T h is can lead to some differences in the
s ta tis tic s o f tra n sp o rt as compared to systems in w hich tim e reversal sym m etry is
preserved.
M icrow ave experim ents are p ra c tic a lly w ell suited for stu d ying fundam ental is­
sues o f the sta tistics o f wave tra n sp o rt in random media among experim ents in
w hich classical waves are used (lig h t in p a rtic u la r). Because o f the re la tiv e ly large
w avelength o f the radiation, A ~ 1 cm, appro p ria te samples can be easily realized.
Here, one can re a d ily achieve different regimes o f propagation e ith e r by changing
the properties o f the in d iv id u a l scatterers or b y varying the m acroscopic dimensions
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24
o f the system - its length a n d /o r cross-sectional area for example. The large size
o f scatterers makes it possible to produce a s ta tis tic a l ensemble o f random sample
configurations. U sing microwave ra d ia tio n one can measure b o th the a m p litu d e and
the phase o f the fie ld tra n sm itte d th ro u g h the sample. Thus, such measurements
provide the com plete info rm a tio n necessary to describe wave tra n sp o rt. T h is makes
it possible to investigate different aspects o f the sta tistics o f tra n sp o rt in mesoscopic
systems in d iffe re n t regimes o f p ropagation in clu d in g b a llistic, diffusive, c ritic a l,
and localized regimes. Also, very accurate measurements can be carried o u t so th a t
adequate com parison w ith theoretical p re d ic tio n can be made. In th is thesis, we re­
p o rt studies o f the statistics o f wave tra n s p o rt in the approach to lo ca liza tio n using
microwave ra d ia tio n . Sm all values o f the dimensionless conductance are achieved
by increasing the length o f samples w ith q u a si-ID geom etry and by tu n in g the fre­
quency o f ra d ia tio n through the edge o f the band gap o f a '’ photonic c ry s ta l” in
which localized states are created by in tro d u c in g disorder.
Am ong the realities th a t microwaves experim ents encounters the presence o f ab­
sorption is p ro b a b ly the most sign ifica nt. A triv ia l problem arising from the presence
o f absorption is the attenuation o f the tra n s m itte d signal, thus m aking d iffic u lt mea­
surements w ith low level signals. A n o th e r problem w hich concerns wave lo ca liza tio n
is the fa ct th a t b o th absorption and lo ca liza tio n give rise to exponential decay o f
transm ission w hich makes d iffic u lt to d istin g u ish localization from ab sorp tio n af­
fects on transm ission in stro n g ly absorbing samples. It is therefore im p o rta n t th a t
the influence o f absorption upon s ta tis tic a l properties o f wave tra n s p o rt be c la ri­
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25
fied. In o u r w ork we tackle th is question by stu d ying th e scaling o f inte nsity and
transm issioa d is trib u tio n s in stro n g ly absorbing samples. We fin d th a t even strong
absorption does not affect su b sta n tia lly the p ro b a b ility d is trib u tio n s o f in te n sity and
to ta l transm ission. Furtherm ore, we dem onstrate th a t i t is possible to s ta tis tic a lly
e lim ina te th e influence o f absorption upon the sta tistics o f in te n s ity w hich allows us
c la rify the in te rp la y between absorption and lo ca liza tio n .
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26
3 Total Transmission Distribution in Absorbing
Random Waveguides
F lu ctu a tion s in tra n sm itta n ce quantities increase d ra m a tic a lly as the ensemble
average o f the dimensionless conductance, g, approaches u n ity . Low values o f g can
be achieved in quasi-one dim ensional samples such as c o n d u ctin g wires o r m u ltimode waveguides w ith lengths m uch greater than the transverse dim ensions. In this
chapter, we report m easurem ents o f to ta l transm ission o f m icrowave ra d ia tio n in
long waveguides fille d w ith ra n d o m ly positioned scatterers in w hich values o f the
dim ensionless conductance as low as g ss
3
are achieved. T h e d is trib u tio n s are ob­
ta in ed accurately fo r values o f sa sig n ifica n tly larger th a n u n ity w hich allows us to
adequately compare th e o ry and experim ent and investigate the num ber o f indepen­
dent param eters needed to ca p tu re the character o f the d is trib u tio n as the sample
moves tow ard the lo ca liza tio n tra n s itio n .
3.1 B ackground
N onlocal correlation in th e flu x tra n sm itte d through mesoscopic samples leads
to enhanced fluctuations o f in te n s ity and sp a tia lly averaged transm ission for both
classical and quantum waves. [71,72] In order to examine th e scaling and the uni­
ve rsa lity o f transport, therefore, i t is im p o rta n t to measure the fu ll d is trib u tio n o f
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27
key transm ission q u a n titie s as the sample size, and hence g, changes. [4] In previous
studies, nonlocal co rre la tio n has been shown to lead to higher p ro b a b ilitie s a t large
values o f the intensity, leading to a deviation from negative e xponential sta tistics for
polarized microwave ra d ia tio n in samples w ith g ~ 10 [22,65|, as w ell as to discem able deviations from a Gaussian d istrib u tio n and enhanced variance fo r the to ta l
o ptical transm ission when g > 103 [47].
Recently, an expression fo r P (sa) in terms o f g for nonabsorbing samples was
obtained by Nieuwenhuizen and van Rossum using diagram m atic techniques com­
bined w ith random m a trix th e o ry [45] and subsequently by Kogan and Kaveh w ith in
the fram ework o f random m a trix theory. [46] The diagram m atic calculations neglect
some term s o f order hig h e r th a n l/g, whereas com putations based on random ma­
tr ix theory neglect sam ple-to-sam ple fluctuations in the p ro b a b ility d is trib u tio n s o f
eigenvalues o f the transm ission m a trix and are expected to be accurate o n ly to order
I Jg. More recently, van Langen, Brouwer and Beenakker ca rrie d o u t a nonperturbative calculation o f the to ta l transm ission d is trib u tio n in the absence o f absorption.
[73| A n a n a lytic so lu tio n is obtained for the case in w hich tim e reversal invariance
is broken (/3 = 2 ) b u t n o t fo r the case o f tim e reversal sym m etry (/? =
1)
considered
here. However, good agreem ent is found between the ,/?-independent result fo r P(sa)
obtained in Refs. [45,46] and the result for (3 = 2 in Ref. [73] fo r g > 10.
The d is trib u tio n o f to ta l transm ission was firs t measured by de Boer et al. in
o p tical measurements in slabs o f tita n ia particles. [47] Samples w ith g > 103 were
studied and the d is trib u tio n was found to be Gaussian to w ith in
1 %.
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A measure o f
*28
the deviations o f the d is trib u tio n from a Gaussian is th e value o f the th ird cum ulant
< s ^ > c which gives the skewness o f the d is trib u tio n and vanishes for a Gaussian
d is trib u tio n . For the samples studied, < s „ > c was o f o rder o f 10-6 . In th is w ork,
we present measurements o f the to ta l transm ission d is trib u tio n in samples in w hich
values o f g as sm all as g « 3 are achieved. The d is trib u tio n s obtained from these
measurements are m arkedly non-Gaussian reaching a value o f the th ird cum ulant o f
approxim ately
0 .1
fo r the sample w ith the m inim um value o f g.
3.2 Sam ples and m easurem ents
In the present w ork, low values o f g are achieved by placing the sample in a
c y lin d ric a l copper tube in order to re strict transverse d iffu sio n and thus the num ber
o f modes N. The samples consist o f polystyrene spheres w ith diam eters o f 1.27 cm
random ly positioned inside a cylin d rica l copper tube so th a t transverse diffusion is
re stricte d leading to a lim ite d num ber o f modes iV. Sam ple tubes w ith diam eters
d = 5.0 and 7.5 cm and various lengths up to 520 cm are used. Because o f the
difference in the tube diam eters, the samples have s lig h tly d iffe re nt fillin g fractions
o f 0.52 and 0.55 fo r d = 5.0 and 7.5 cm, respectively. M easurem ents are made in the
frequency range 16.8 - 17.8 GHz using frequency steps o f 4 M H z. The sample tube is
ro ta te d between successive measurements to produce new scatterer configurations.
In order to e lim inate the in stru m e n ta l response, the spectra are norm alized to th e ir
ensemble average to give sa from which the d is trib u tio n P (sa) is obtained. The
microwave ra d ia tio n is coupled to the sample by a 0.4 cm w ire antenna placed 0.5
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cm from the fro n t surface o f the sample and the to ta l transm ission is measured by
use o f a S ch o ttky diode detector positioned inside an in te g ra tin g sphere ro ta tin g
about the sample axis a t 2 Hz. A schematic diagram o f the experim ental set up is
shown in Fig.
8.
The in te g ra tin g sphere has a diam eter o f 40 cm and is com prised
o f tw o concentric p la s tic spherical shells separated by
them fille d w ith m ovable scatterers.
2
cm w ith the space between
The outer shell is covered w ith alum inum
fo il to form an irre g u la r refle cting surface. The region between the shells is fille d
w ith alum inum cylinders w ith diam eters o f 0.75 cm and ty p ic a l lengths o f
1
cm. The
cylinders tum ble as the in te g ra tin g sphere rotates re s u ltin g in a flu c tu a tin g in te n sity
at the detector w ith a co rre la tio n tim e of ~
2
ms fo r th e sam ple w ith a length o f
100 cm. The signal is averaged fo r 1 s at each frequency g iv in g an u n certain ty o f
\
2.5 % in the measurem ent o f transm ission. A t the m axim um length o f 200 cm a t
w hich to ta l transm ission measurements were made, a tra v e lin g wave tube a m p lifie r
(T W T A ) w ith an o u tp u t power o f 40 W is used. The transm ission d is trib u tio n s are
obtained by using the data from a t least
1000
sam ple configurations.
In the frequency range o f the measurements, i ~ 5 cm [74]. A f it o f measure­
ments o f the fie ld a u to co rre la tio n function w ith frequency s h ift to th e o ry [63] gives
La = 3 4 ± 2 cm and D = (3.03 ± 0 .2 1 ) x 10LO cm 2 /s . [75] T h e lo c a liza tio n length for
the samples w ith d = 5 cm is found to be £ = 551 ± 18 cm from fie ld measurements
[48] which w ill be discussed in the ne xt chapter.
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30
3.3 R esu lts an d d iscu ssion s
From the m easurem ents o f to ta l transm ission made, we o b ta in the p ro b a b ility
d is trib u tio n P (sa). T h e d is trib u tio n s for three samples w ith dim ensions (a) d = 7.5
cm, L = 66.7 cm, (b) d — 5.0 cm, L — 50.0 cm, and (c) d - 5.0 cm, L = 200 cm are
shown in Fig. 9. [n th e absence o f absorption, the dim ensionless conductance for
these samples w ith o u t lo c a liz a tio n corrections, g = N £ /L , w ou ld be approxim ately
15.0, 9.0, and 2.25 fo r samples a, b, and c, respectively. The d is trib u tio n broadens
and the deviation fro m a Gaussian becomes more pronounced as e ith e r the sample
length increases o r the cross-sectional area decreases. A value o f < s ^ > c as large as
0.112 ± 0.035 is observed fo r sample c. D eviations fro m a Gaussian d is trib u tio n in
the ta il o f the d is trib u tio n fo r th is sample can be seen in the sem ilog p lo t o f P (sa)
in Fig. 10. For values o f sa > 2 the d is trib u tio n is e sse ntia lly exponential.
We compare the th e o re tica l results from Refs. [45,46| to the measured trans­
m ission d is trib u tio n s . T he fu ll d istrib u tio n s are given by the theory as functions
o f g fo r nonabsorbing sam ples. In the present case o f stro n g absorption, the pho­
to n num ber is not conserved and g cannot be defined in term s o f the steady state
transm ission, w hile se rving as a useful measure o f th e p ro x im ity to the localization
tra n sitio n . T h is can be seen b y no ting th a t the re d u ctio n o f the average transm is­
sion due to the presence o f absorption is associated w ith a decrease rather than an
increase o f the degree o f co rre la tio n in the sample and to push the system farther
from the loca lizatio n th re sh old . A param eter w hich characterizes the transm ission
d is trib u tio n as w ell as th e closeness to the loca lizatio n th re sh old is the degree o f cor-
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3
1
relation o f in te n sity in d iffe re n t coherence areas o f the tra n s m itte d speckle p a tte rn .
<8sab8sabi > . Were this co rre la tio n to vanish, flu ctu a tio n s in different coherence
areas w ould be independent and the transm ission d is trib u tio n would be Gaussian
as required by the central lim it theorem , w ith v a r (s a) = <s2>c= l/N . As a result
o f nonlocal correlation, however, the variance o f the transm ission is enhanced. It is
given by v a r(s a) = ( v a r (s ab) — 1)/2 = <6sab6.sab'> ■ [17,19,46,62] The last e q u a lity
is consistent w ith the results o f Ref. [62] when the cum ulant in te n s ity co rrela tio n
function is properly norm alized, to the renorm alized average transm ission. [63] In
th a t case, the crossing param eter found by Shnerb and Kaveh [64] which determ ines
the inte nsity d is trib u tio n is found experim entally to be equal to <5sab8sab>>. [62.65]
In independent measurements w ith the sample w ith L — 100 cm and d = 7.5 cm. we
fin d th a t the in te n sity co rre la tio n fu n ctio n w ith space s h ift A x has a constant value
o f 0.0646 ± 0.0012 for A x > 3 w hich equals w ith in experim ental error the value o f
v a r(s a)
= 0.0656 ±0.0020 fo r th is sample. [75] The connection between <8sab8sab'>
and the fu ll transm ission d is trib u tio n can be seen by considering the expression o f
Refs. [45,46] for P (s a) in th e absence o f absorption in the lim it g
/
-H o o
dx
— exp[xsa - 4>(x)],
• to o
1,
(18)
21TZ
where
<E>(x) = gln2(\Jl + x /g + yjx/g)
is the generating function. From Eq. (18) one obtains the expression fo r var(sa) in
term s o f g,
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32
v a r (s a)
2
= —.
39
(19)
From these expressions, a general relation for P ( s a) in term s o f v a r ( s a)- o r equiv­
a le n tly < 5 s ab6sab> > , can be found by using Eq. (19) to define a new param eter
g ' = 2 /3 v a r(s a) w hich is su b stitu te d fo r g in to Eq. (18). P lots o f P ( s a) obtained by
follow ing th is procedure w ith g ' determ ined from the measured values o f v a r { s a),
are shown as the so lid lines in Figs. 9 and 10. We fin d th a t P (s a ) is accurately
given even for the low est value o f g ' o f 3.06 (sample c). T he d is trib u tio n o f Eq. (18)
w ith g' su b stitu te d fo r g gives the exponential ta il, P (s a) ~ e x p (—g 'sa) in the lim it
sa »
1. For sa > 2.0, th e linear f it to the log a rith m o f the measured transm ission
d is trib u tio n fo r sample c gives a slope of —(2.71 =h 0.06) in accord w ith the fit o f
an exponential fu n ctio n to the theoretical curve gives a slope o f - 2.70 in this range
and is close to its p re d icte d asym ptotic value o f 3.06 for sa
1.
The extent o f the agreem ent o f Eq. (18) when g ' is s u b s titu te d fo r g can also
be gauged from the com parison between the calculated (circles) and the measured
(squares) moments o f th e transm ission d is trib u tio n shown in Fig. 11 for samples
w ith g ' = 10.2 ± 0.1 and g ' = 3.06 ± 0.04. The moments calculated from the theory
are close to those o b ta in e d from the measured d is trib u tio n s . A t n = 10, these d iffe r
by approxim ately 10 % w h ich is w ith in the experim ental e rro r. Thus i t appears th a t
P (s a) can be expressed as a fu n ctio n o f the param eter v a r ( s a).
The agreement between theory and experim ent indicates th a t the ra tio o f mo­
ments is accurately reflected in Eq. (18).
The dependence o f the variance its e lf
R
e
p
ro
d
u
c
e
dw
ithp
e
rm
is
s
io
no
fth
ec
o
p
y
rig
h
to
w
n
e
r. F
u
rth
e
rre
p
ro
d
u
c
tio
np
ro
h
ib
ite
dw
ith
o
u
tp
e
rm
is
s
io
n
.
33
upon sample dimensions is shown in Fig. 12. In the lim it o f g
o f absorption and in te rn a l re fle ctio n var(sa) = 2 L /3 N I.
1 in the absence
T h e s tra ig h t line in the
figure is drawn through the firs t data p o in t and represents var(sa) ~
As
<7 —*■ 1, the scaling theory o f lo ca liza tio n [40] suggests th a t g fa lls e xp on e n tia lly and
hence var(sa) should increase superlinearly w ith sam ple le n g th . Instead, we find,
th a t var(s a) depends su b lin e a rly upon L. T h is is presum ably due to the presence o f
absorption which dim inishes the degree o f nonlocal c o rre la tio n . T h is raises the ques­
tio n o f w hether the transm ission d is trib u tio n continues to broaden as L increases or
instead it reaches a lim itin g d is trib u tio n for p a rtic u la r sam ple param eters.
3.4 C onclusions
In conclusion, we have m easured the to ta l transm ission d is trib u tio n o f microwave
ra d ia tio n in quasi-one dim ensional absorbing samples w ith sm a ll values o f g. We
fin d th a t the d is trib u tio n can be described using an expression o rig in a lly derived for
nonabsorbing samples in the lim it g »
1 when th is expression is reform ulated as
a fu n ctio n o f the single param eter g' = 2 /3 var(sa) d e te rm in e d fro m the measure­
ments. The v a lid ity o f the expression for values o f g' as s m a ll as 3, w ell beyond the
lim its assumed in the calculations, may w ell be associated w ith th e id e n tifica tio n
o f var(s a) w ith <Ssab5sab>>, th e degree o f spatial c o rre la tio n in th e sample, w hich
is the key m icroscopic param eter in mesoscopic physics. T h e su b lin e a r increase o f
var(sa) w ith L in stro n g ly absorbing samples, however, raises th e question as to
R
e
p
ro
d
u
c
e
dw
ithp
e
rm
is
s
io
no
fth
ec
o
p
y
rig
h
to
w
n
e
r. F
u
rth
e
rre
p
ro
d
u
c
tio
np
ro
h
ib
ite
dw
ith
o
u
tp
e
rm
is
s
io
n
.
whether ra d ia tio n can be loca lized in the presence o f a b so rp tio n .
R
e
p
ro
d
u
c
e
dw
ithp
e
rm
is
s
io
no
fth
ec
o
p
y
rig
h
to
w
n
e
r. F
u
rth
e
rre
p
ro
d
u
c
tio
np
ro
h
ib
ite
dw
ith
o
u
tp
e
rm
is
s
io
n
.
35
4 Intensity D istribution in the Approach to
Localization
In. th is chapter, we focus on the in te n s ity d is trib u tio n , w hich is the key d is tri­
b u tio n in sta tistica l optics. [3| We dem onstrate its re la tio n sh ip to the d is trib u tio n
o f to ta l transm ission, fin d the scaling o f the variance o f the in te n s ity and to ta l
transm ission up to L = <f, and determ ine the extent to w hich absorption influences
loca lizatio n .
4.1 B ackground
In the diffusive lim it, the degree o f long-range in te n s ity co rre la tio n is sm all and
the in te n s ity d is trib u tio n is w ell approxim ated by the R ayleigh d is trib u tio n . [3,22,76|
For polarized detection, this corresponds to negative exponential statistics. P(-sab) —
exp(—sab). In previous w ork, deviations from negative e xponential behavior have
been observed and ascribed to long-range inte nsity co rre la tio n [22,64,65]. In these
studies [22,65], fluctuations as large as sab ~ 10 ~ g were observed. In the present
w ork, fluctuations as large as fifty tim es g are observed in samples w ith lengths
The in te n sity d is trib u tio n is studied in a quasi-ID geom etry, w hich is equivalent
to the electronic case o f a th in w ire.
Thouless argued th a t, the level w id th 5u
R
e
p
ro
d
u
c
e
dw
ithp
e
rm
is
s
io
no
fth
ec
o
p
y
rig
h
to
w
n
e
r. F
u
rth
e
rre
p
ro
d
u
c
tio
np
ro
h
ib
ite
dw
ith
o
u
tp
e
rm
is
s
io
n
.
36
in. a wire a t T = 0 should become sm aller than the level spacing A u since Su is
pro po rtio n al to the inverse o f the travel tim e and so fa lls as
1/ L 2
in the diffusive
lim it, whereas A v is th e inverse o f the density o f states and so falls as
1 jL .
As a
result, the modes in adjacent sections o f the w ire should n o t overlap, and electrons
should become localized. [39]
The question arises as to w hether ra d ia tio n can be localized in the presence
o f absorption. In this case, the level w id th falls as l / L , ju s t as the level spacing
does. In the previous chapter, we showed th a t the variance o f the norm alized to ta l
transm ission, var(s a), scales sublinearly w ith L in the presence o f absorption. If the
attenuation len g th due to absorption, La. serves as a c u to ff len g th fo r localization
[26,32] then var(sa) w ould approach an asym ptotic lim it as L increases. On the other
hand, if loca lizatio n can be achieved in absorbing samples, then var(sa). w hich is
essentially the degree o f co rre la tio n in the in te n s ity o f d iffe re n t outgoing modes,
should increase su p e rlin e a rly as L approaches £. T h is m ig h t occur, since the wave
remains te m p o rally coherent in the presence o f absorption. [77,78] Weaver has shown
in a 2-D sim ula tio n th a t the in tro d u c tio n o f absorption does not d is ru p t the spatial
localization o f acoustic waves in closed systems, though th e ove rall energy decreases
exponentially w ith tim e . [77] In recent calculations, B rouw er found th a t for diffusive
waves the prefactor m u ltip ly in g L/f; in the expression fo r var(sa) drops from | to
^ as the ra tio L I L a increases. [79] The behavior o f th is q u a n tity , however, was o n ly
considered fo r lengths considerably less than the lo c a liz a tio n length.
Here we re p o rt measurements o f in te n sity tra n s m itte d th ro u g h random waveg­
R
e
p
ro
d
u
c
e
dw
ithp
e
rm
is
s
io
no
fth
ec
o
p
y
rig
h
to
w
n
e
r. F
u
rth
e
rre
p
ro
d
u
c
tio
np
ro
h
ib
ite
dw
ith
o
u
tp
e
rm
is
s
io
n
.
37
uides w ith L < £, b u t > L a. We expect th a t modes in th is sam ple are com pletely
m ixed and the degree o f in te n s ity correlation between d iffe re n t modes is constant.
Wave propagation in th is sam ple should therefore be described by random m a trix
th e o ry (R M T ). [61] R ecently, Kogan and Kaveh used R M T to o b ta in a relationship
between the moments o f in te n s ity and to ta l transm ission in nonabsorbing quasi-ID
samples. [46] They fin d ,
(20)
<Sab>= n\ < s > ,
T h is leads to a re la tio n sh ip between the d is trib u tio n s o f in te n s ity and to ta l trans­
mission, [46,80]
(21)
Since the d is trib u tio n o f to ta l transm ission can be ca lcu la te d fro m the d is trib u tio n
o f the eigenvalues o f th e transm ission m a trix, [45,46,73] these relations provide a
basis for calculating th e in te n s ity moments and d is trib u tio n s fro m R M T.
4.2 Sam ples and m easu rem en ts
The samples used in th e measurements o f the in te n s ity d is trib u tio n s are the
same as those described in the previous chapter. Here, however, measurements o f
in te n sity were made fo r sam ple lengths as large as 520 cm ( ~ £ ). D ifferent sam­
ple configurations are created as in the case o f to ta l tra n sm issio n measurements by
b rie fly ro ta tin g the sam ple tu b e between successive m easurem ents.
A t least two
thousand sample co n figu ra tio ns were used fo r each d is trib u tio n . F ie ld spectra were
R
e
p
ro
d
u
c
e
dw
ithp
e
rm
is
s
io
no
fth
ec
o
p
y
rig
h
to
w
n
e
r. F
u
rth
e
rre
p
ro
d
u
c
tio
np
ro
h
ib
ite
dw
ith
o
u
tp
e
rm
is
s
io
n
.
38
taken from. 16.8 to 17.8 GHz in steps o f
1
M Hz using a H ew lett-P ackard 8722C
netw ork analyzer. T he ra d ia tio n is coupled in to and o u t o f the sample by 0.4 cm
wire antennas placed 0.5 cm from the ends o f the sam ple. Schem atic diagram o f the
experim ental set up is shown in Fig. 13. In order to ensure th a t the d is trib u tio n s
were not d isto rte d by noise, it was necessary to use an a m p lifie r w ith an o u tp u t
power o f 40 W fo r samples w ith lengths greater th a n 200 cm so th a t the average
inte nsity was a t least three hundred times the noise.
4-3 R esu lts an d d iscu ssion s
In Fig. 14, we present the in te n s ity d is trib u tio n s fo r tw o samples w ith L/t f ~ 0.1
and 0.4. C alculations fo r diffu sive waves in the absence o f absorption have pre­
dicted th a t fo r sab ^
9
, the in te n s ity d is trib u tio n fa lls as exp(—2^/gsab). [451 For
the samples measured, we fin d P (sab) ~ exp(—2^/'ysab) in the ta il o f the in te n s ity
d is trib u tio n . The values o f 7 obtained from a fit to th e ta il o f the measured d is trib u ­
tions is w ith in
20%
o f th e param eter g1 = ^Var(~ ) >
w hich equals g in the absence
o f absorption . The f it o f a negative stretched exponential to the d is trib u tio n fo r
£■/£ = 0.4 in the range o f sab fro m 10 to 18 is shown in Fig. 15 and gives
7
= 2.9,
w hich is close to value o f g' o f 3.06 for this sample.
The measured in te n s ity d is trib u tio n s are com pared to the transform o f the mea­
sured transm ission d is trib u tio n s [43] for the corresponding samples using Eq. (21).
The transform s shown as so lid lines in Fig.
14, are in good agreement w ith the
R
e
p
ro
d
u
c
e
dw
ithp
e
rm
is
s
io
no
fth
ec
o
p
y
rig
h
to
w
n
e
r. F
u
rth
e
rre
p
ro
d
u
c
tio
np
ro
h
ib
ite
dw
ith
o
u
tp
e
rm
is
s
io
n
.
39
measured in te n s ity d is trib u tio n s . A com parison o f th e moments o f in te n s ity and
transm ission is shown in F ig.
16. We fin d an increasing deviation o f the ra tio s
< sab> / n - < sa > fro m the value o f u n ity expected from Eq. (20) as n increases.
We fin d , however, th a t agreement w ith Eq. (20) is d ra m a tic a lly im proved w hen the
moments are calculated using the asym ptotic expressions in the diffusive lim it for
the in te n sity and transm ission d istrib u tio n s beginning from the p o in t a t w hich the
measured d is trib u tio n have th e ir firs t zero. T he a sym p to tic expressions fo r th e in ­
te n sity d is trib u tio n exp(—2 ^/g'sab) is su b stitu te d fo r the measured d is trib u tio n for
values o f sab between20 and 150, whereas the a sym p to tic exponential expression
exp(—g'sa) [43,45,46] is su b stitu te d for the measured transm ission d is trib u tio n for
values o f sa between 5 and 25. T he im proved agreem ent indicates th a t the extent to
w hich the measured ra tio o f moments is in accord w ith Eq. (20) is largely lim ite d
by the range o f in te n s ity and transm ission values measured, w hich depends on the
num ber o f configurations on w hich measurements were made.
A p p lyin g Eq. (20) to the second moments gives
var(s ab) = 2 var(sa) + 1 .
(2 2 )
The same expression can be obtained from a p e rtu rb a tio n calculation up to order
l/g [17,46]. O ur measurements confirm the p re d ic tio n o f R M T th a t Eq. (22) is
independent o f the value o f g and is correct up to o rd e r
1 /N
. In Fig. 17 we p lo t the
ra tio var(sab)f[ 2 var(sa) + 1 ] obtained from the m easurements o f to ta l tr a n s m is s io n
and intensity. For a ll samples fo r w hich th is com parison is possible agreem ent to
R
e
p
ro
d
u
c
e
dw
ithp
e
rm
is
s
io
no
fth
ec
o
p
y
rig
h
to
w
n
e
r. F
u
rth
e
rre
p
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d
u
c
tio
np
ro
h
ib
ite
dw
ith
o
u
tp
e
rm
is
s
io
n
.
40
w ith in 3% is fo u n d between experim ent and th e o ry (E q. (22)).
In te n sity s ta tis tic s a t th e localization th re sh old are studied in measurements a t
L = 520 cm and d. = 5 cm ( L /£ ~ 1.0) and shown in Fig. 18. Values o f sab as
large as 50 are observed. The d is trib u tio n fo r th is sam ple is seen in Fig.
19a to
be nearly log-norm al, in agreement w ith p re d ictio n s fo r localized ra d ia tio n [73,79|.
A com parison to th e d is trib u tio n obtained fo r a sam ple w ith L /£ ~ 0.1 (F ig. 19b)
indicates the tra n s itio n to a log-norm al d is trib u tio n as L /£ —* 1 Using Eq. (21), we compare these measurements o f the in te n sity d is trib u tio n
to random m a trix calculations o f the transm ission d is trib u tio n in the presence o f
absorption [79]. Tn o rd er to compare the in te n s ity d is trib u tio n to theory, however.
£ must be determ ined. We recall, th a t fa r from the lo ca liza tio n threshold, in the
absence o f a b so rp tio n and in te rn a l reflection, v ar(s a) and
are related by Eq. (19)
[17.45.46,79].
To fin d £ in sam ples in w hich corrections due to absorption, loca lizatio n and
internal reflection cannot be ignored, we firs t o b ta in the in te n s ity d is trib u tio n fo r the
equivalent samples w ith o u t absorption using the m easured spectra in our absorbing
samples.
We firs t o b ta in the tim e response E ( t ) to a narrow Gaussian pulse in
tim e by Fourier tra n s fo rm in g the product o f the measured spectrum and a broad
Gaussian in the frequency dom ain. The tim e dependent fie ld is then m u ltip lie d by
exp(t/'2ra), where ra = La2/ D = (3 .8 l± 0 .7 0 ) x 10- 8 s is the exponential a tte n u a tio n
tim e due to a b sorp tio n . The m odified tim e spectra are then transform ed back in to
the frequency dom ain. T he spectra in frequency and in tim e dom ain as obtained
R
e
p
ro
d
u
c
e
dw
ithp
e
rm
is
s
io
no
fth
ec
o
p
y
rig
h
to
w
n
e
r. F
u
rth
e
rre
p
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d
u
c
tio
np
ro
h
ib
ite
dw
ith
o
u
tp
e
rm
is
s
io
n
.
41
from the m easurem ents (th ic k line) and as corrected fo r absorption (th in lin e ) are
shown in Fig. 20. T h e peaks in the frequency spectrum obtained a fte r co rre ctin g
fo r absorption are c le a rly narrow er as com pared to those observed in the measured
spectrum . T h is is consistent w ith the affect o f absorption which reduces th e average
dw ell tim e o f photons and thus broadens th e level w id th o f modes. From th e fie ld
spectra corrected fo r absorption, we now o b ta in in te n s ity d is trib u tio n s . In F ig. 21.
we compare these w ith the measured d is trib u tio n s . A t high in te n s ity values, the
measured d is trib u tio n appears lower than th e one obtained from the d a ta corrected
for absorption. T h is is consistent w ith the influence o f the presence o f a b sorp tio n
upon in te n s ity s ta tis tic s .
The in te n sity d is trib u tio n s obtained in th is way are in
good agreement w ith transform s o f the d iffu sive re su lt fo r the d is trib u tio n s o f to ta l
transm ission ca lcu la te d in Ref. [45,46] (see th e in se rt in Fig. 21) and give values
for the param eter g' [43] equal to the value o f g. as expected in the absence o f
absorption. We also fin d th a t the average transm ission obtained from th e spectra
corrected fo r a b so rp tio n is consistent w ith th e expected scaling as (L -r 2z6)~ l . .
where Zf, is the d iffu s io n e xtra p o la tio n length due to in te rn a l reflection [81]. These
results co n firm the a b ility o f th is approach to s ta tis tic a lly elim inate a b sorp tio n in the
diffusive lim it. We do n o t expect the procedure to be effective fo r localized waves.
In th is case, the presence o f gain is p re dicte d to lead to an overall suppression
o f transm ission, [82,83] whereas an extension o f the present approach w ou ld lead
to enhanced transm ission. T h is indicates the im portance o f dispersion as w ell as
a m p lifica tio n o r loss in th is case. The influence o f in te rn a l reflection upon var(sa)
R
e
p
ro
d
u
c
e
dw
ithp
e
rm
is
s
io
no
fth
ec
o
p
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rig
h
to
w
n
e
r. F
u
rth
e
rre
p
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u
c
tio
np
ro
h
ib
ite
dw
ith
o
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tp
e
rm
is
s
io
n
.
42
in the absence o f absorption can be accounted fo r by su b s titu tin g L = L + 2zb for
L in Eq. (19). We next account fo r th e leading order correction to var(sa) due to
nonlocal co rre la tio n . In the absence o f absorption, the variance is increased by an
a d ditio n al fa cto r o f (1 ± |£ ) [63,84] to yie ld ,
f
'2L
AL2
varM = ^ +
(23)
A fit o f Eq. (23) to the data corrected fo r absorption using f and zb as fittin g
parameters gives £ = 551 ± 1 8 cm and zb = 5.25 ± 0 .3 1 cm. The value o f zb obtained
is consistent w ith the values o f th is param eter fo r the same samples in the frequency
range between 18 and 19 GHz. [74]
The dependence o f var(s a) upon L w ith and w ith o u t absorption is shown in
Fig. 22. The so lid curve represents th e re su lt o f the calculations in the diffusive
regime (L /£ <C 1) by Brouwer [79] w hich account for absorption and th e dashed
curve shows the f it o f Eq. (23) to the d a ta corrected for absorption. The values
o f var(sa) are calculated from var(sab) using Eq. (22). For lengths up to 200 cm,
the result from the measurements, show sublinear behavior which is consistent w ith
the results from to ta l transm ission measurements [43] and the ca lcu la tion s in Ref.
[79]. The d e via tio n from the so lid fine increases fo r larger lengths and m ay reflect
localization corrections th a t were n o t in clu d e d in the theory. For s tro n g ly absorbing
samples (L
La), Brouwer finds a log -n o rm al d is trib u tio n for the to ta l transm ission
w ith <LnTa>= —L / L a — 3 fr/4 £ — In N and var{lnT a) = L / 2£. [79] T h is allow s us
to calculate the values o f var(sa) fo r th e tw o samples w ith L > 10Ta fo r w hich
R
e
p
ro
d
u
c
e
dw
ithp
e
rm
is
s
io
no
fth
ec
o
p
y
rig
h
to
w
n
e
r. F
u
rth
e
rre
p
ro
d
u
c
tio
np
ro
h
ib
ite
dw
ith
o
u
tp
e
rm
is
s
io
n
.
43
measurements were made. These are presented as the squares in the figure and
indicate sig n ifica n t corrections due to lo ca liza tio n in q u a lita tive agreem ent w ith the
results from th e measurements. Thus, we can associate the increase o f £var(sa)/ L
for these samples w ith the tra n sitio n to lo ca liza tio n .
We now com pute P (sab) for the sample w ith L = 520 cm using the transm ission
d is trib u tio n calculated in Ref. [791 and the values o f £ and La found here.
The
calculated in te n s ity d is trib u tio n is presented as the solid lin e in F ig. 18 and is in
good agreem ent w ith the measurements.
We also te st the extent to which the th e o retica l calculations in the diffusive
lim it and in the absence o f absorption [45,46] agree w ith the measurements for this
sample for w hich L /£ ~ 1 and L /L a
15.
Using Eq. (18), we calculate P (sa)
w ith g' o b ta in e d from the measured variance o f intensity. The calculated transm is­
sion d is trib u tio n is then transform ed to give the corresponding in te n s ity d is trib u tio n
which is shown by the dashed line in Fig. 18. S urprisingly, even in the presence
o f strong ab sorp tio n and a t the loca lizatio n len g th the diffusion th e o ry in the ab­
sence o f a b sorp tio n when properly m odified (th e d is trib u tio n obtained from theory
is calculated so th a t it has the rig h t variance) s till provides good agreement w ith the
experim ent. A com parison w ith the calculations in which absorption and localiza­
tio n corrections are included (the solid line in the figure) shows d e via tio n between
these two results o n ly fo r large values o f in te n sity. In contrast, the transm ission
d istrib u tio n s fro m w hich the inte nsity d is trib u tio n s are calculated can be v is ib ly set
apart (see the in s e rt).
In order to describe q u a n tita tiv e ly the difference between
R
e
p
ro
d
u
c
e
dw
ithp
e
rm
is
s
io
no
fth
ec
o
p
y
rig
h
to
w
n
e
r. F
u
rth
e
rre
p
ro
d
u
c
tio
np
ro
h
ib
ite
dw
ith
o
u
tp
e
rm
is
s
io
n
.
44
these tw o results, one has to consider hig h e r mom ents in to ta l transm ission (o r in ­
te n s ity ). The diffusion theory in the absence o f absorption predicts th a t the ra tio
C (L ) = < s „ > c fvar(sa)2, where < s^> c is the th ird cum ulant o f the d is trib u tio n ,
has a constant value o f 12/5 in the case o f an in cid e n t plane wave. [451 In R M T
calculations in the diffusive lim it in w hich absorption is included, B rouw er found
th a t the value o f th is param eter changes from 12/5 in the absence o f absorption to
3 in s tro n g ly absorbing samples. [79] In F ig . 23, we p lo t the values o f C (L ) obtained
from the in te n s ity measurements via Eq. 20 w hich show a q u a lita tiv e agreement
w ith the th e o re tica l result o f Brouw er (so lid curve). Because o f the sm all values o f
<s2>c fo r samples w ith sm all L (large values o f g), the error bar in the results from
the measurements is very large and a q u a n tita tiv e com parison w ith th e o ry is not
possible. For L ~
however, the value o f the th ird cum ulant increases and the
accuracy o f the experim ental results is im proved significantly. For these samples,
we fin d values o f C (L ) greater th a n th e th e o re tic a lly predicted value o f 3. T his
can be associated w ith the approach to lo ca liza tio n . We calculate C(L) using the
expressions fo r the to ta l transm ission from Ref. [79] in which lo ca liza tio n corrections
are included. The results are shown as squares and, as in the case o f var(s a), show
the same tre n d w hich is found in the experim ent.
R
e
p
ro
d
u
c
e
dw
ithp
e
rm
is
s
io
no
fth
ec
o
p
y
rig
h
to
w
n
e
r. F
u
rth
e
rre
p
ro
d
u
c
tio
np
ro
h
ib
ite
dw
ith
o
u
tp
e
rm
is
s
io
n
.
45
4.4 C on clu sion s
In conclusion, we fin d th a t the in te n s ity d is trib u tio n fo r sab ^
a negative stretched exponential to pow er
is close to log -n o rm al.
1 /2
g is described by
and th a t the d is trib u tio n fo r L ~ £
We confirm e xp e rim e n ta lly the re la tio n sh ip s obtained by
Kogan and K aveh between the m om ents and fu ll d istrib u tio n s o f in te n s ity and to ta l
transm ission. These relations u n ify th e s ta tis tic a l description o f lo ca l and sp a tially
averaged tra n sm itta n ce quantities. O u r measurements dem onstrate th a t the statis­
tics o f wave tra n s p o rt is o n ly m a rg in a lly affected by absorption a nd th a t absorption
does not s u b s ta n tia lly in h ib it lo c a liz a tio n .
The a b ility to reach th e loca lizatio n
threshold using a quasi-one-dim ensional sam ple is an extension to classical waves of
the suggestion b y Thouless th a t electrons w ill always be localized in s u ffic ie n tly long
wires a t low tem peratures. These re su lts show th a t the variances o f the in te n s ity or
transm ission are re liable measures o f th e im p a ct o f lo ca liza tio n u p o n tra n s p o rt in
random media.
R
e
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u
c
e
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ithp
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is
s
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no
fth
ec
o
p
y
rig
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to
w
n
e
r. F
u
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e
rre
p
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u
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tio
np
ro
h
ib
ite
dw
ith
o
u
tp
e
rm
is
s
io
n
.
46
5 Field and Intensity Correlations in Random
Waveguides
In th is ch a pte r we o b ta in the absorption length and the d iffu sio n constant o f the
samples discussed in chapters 3 and 4 from the fie ld auto co rrela tio n function w ith
frequency s h ift.
We show also th a t the variance o f the to ta l transm ission equals
the degree o f nonlocal in te n sity co rre la tio n across the o u tp u t face o f the sample.
Finally, we present measurements o f the long- (C 2) and in fin ite -ra n g e (C 3 ) terms o f
the in te n s ity co rrela tio n fu n ctio n w ith frequency s h ift. To the best o f o u r knowledge,
measurements o f the frequency dependence o f the C 3 co rrela tio n te rm are reported
for the firs t tim e in this work.
5.1 F ield -field correlation fu n ction w ith freq uency shift
The fie ld
autocorrelation fu n c tio n w ith frequency s h ift
6
u is defined as
Alj)>=<E(uj)E*(uj - f Acj)>. It reflects the short-range in te n s ity correlation
which is s im p ly Ci(u. Auj) =\<I(u), A c j ) >
|2
Inside the sam ple
< /( r ,c j, A o j)> = Sinh[ir<* ~ iri- )(L ~ z )l
1 , 1
where
7^
=
;
sin h[('Y +-i'y-)L]
(24)
’
\ f a A +(3A ± a 2, a = L~ l is the absorption coefficient, and
(A u / D ) 1/2. [63] For samples for w hich a£, ^
<
1
K
8
}
=
and Zf, w I the fie ld auto corre­
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u
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rig
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to
w
n
e
r. F
u
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np
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ib
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ith
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n
.
47
la tio n fu n c tio n a t a p o in t at the o u tp u t face o f the sample is given b y
< I( u A u )> =
1
K
— i’7 _ )L [
( 7+
s in /i[(7 i — Z7_)£]
,25,
W hen the fu n c tio n is norm alized to the outgoing average in te n s ity
—
sinh(aL)
<1 (w, Au; ) > = -A-------- '---— ■
a
s z n h [(7 + — i 7 _ ) L |
,, .
26
We o b ta in <I{u>, A u)> from the measurements from w hich th e in te n s ity d is tri­
butions (discussed in chapter 4) were obtained. In Fig. ‘24 we present its real part
for the sam ple w ith L /£ w O .l. From a f it o f Eq. (26) to the data , using a and D as
fittin g param eters, we obtain La — 34 ± ‘2 cm and D = (3.03 ± 0.‘2 l) x 101Q cm’2 /s.
The result fro m the f it is shown on the p lo t as a solid line.
5.2 In ten sity -in ten sity sp a tia l correlation fu n ctio n
In th is section we dem onstrate experim entally th a t the variance o f the to ta l
transm ission d is trib u tio n equals the degree o f nonlocal co rre la tio n across the output
face o f the sam ple. In microwave measurements o f the in te n s ity d is trib u tio n and the
in te n s ity -in te n s ity correlation fu n ctio n w ith frequency sh ift. Genack and G arcia have
shown th a t th e variance o f the d is trib u tio n is d ire c tly related to the C x(= C2 + C 3)
term o f the c o rre la tio n function a t zero frequency s h ift w hich represents the degree
o f nonlocal in te n s ity correlations in the system. [65,62] Here we co n firm th is relation
in a more d ire c t way by m easuring the in te n sity-in te n sity co rre la tio n fu n ctio n w ith
displacem ent.
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u
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ithp
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rm
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s
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ec
o
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rig
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to
w
n
e
r. F
u
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e
rre
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u
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np
ro
h
ib
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dw
ith
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n
.
48
T h is is done by m easuring the tra n s m itte d fie ld as the d e te ctin g antenna is
moved to different positions along th e diam eter o f the sample o u tp u t. The in te n sityin te n s ity correla tio n fu n ctio n , C w ( r , r ') = < 6 I{J(r) 6 I<
Jj(r,)>, consists o f three con­
trib u tio n s C = Ci + C2 + C 3 . For tw o observation points a t the o u tp u t plane w hich
are separated by a distance A R. the short-range part C\ is given b y [63|
sin(k 0 AR)
k0A R
exp(—A R /£ ),
(27)
where ko is the wave num ber in a ir and A R is the distance between the two points
o f detection. N orm alizing to the outg o in g average inte nsity and ta k in g u = u>' leads
to
Ci{ R ,R ) =
sin(k 0 A R )
k$AR
exp(—A R /i).
(28)
The long-range term , C 2 , in the in te n s ity -in te n s ity correlation fu n ctio n also depends
on the displacem ent. In the q u a s i-ID geom etry, however, th is te rm is dom inated
by the zero-mode c o n trib u tio n w hich is s p a tia lly independent and is equal to the
relative flu ctu a tio ns in to ta l transm ission ( var(s a)). [63| The C 3 term is o f order
1 /<7 2(<C 1 )
and is considered to be constant.
The in te n sity co rrela tio n fu n c tio n w ith
polystyrene sample w ith L /£ ~ 0.1.
displacement is obtained fo r the
The tra n sm itte d fie ld is measured as the
p o sitio n o f the detecting antenna is changed in steps o f
1
mm to cover a distance
o f 5 cm across the o u tp u t face o f th e sam ple. Field spectra between 16.8 and 17.8
GHz are taken for each sample co n fig u ra tio n and for each p o s itio n o f the detecting
antenna. Measurements are made fo r 600 d iffe re nt sample configurations. From the
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ithp
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is
s
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rig
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to
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n
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tio
np
ro
h
ib
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dw
ith
o
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is
s
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n
.
49
data we o b ta in the in te n s ity -in te n s ity c o rre la tio n fu n ctio n w ith displacement (F ig .
25) We f it the expression fo r C (A i? ) = C\ + C2 w ith C\ given by Eq. (28) and
Co = constant to the experim ent. The te rm C<i represents the s p a tia lly independent
p a rt o f C<i and includes the in fin ite -ra n g c c o rre la tio n (C 3 ) term as w ell. We use ko,
I, and C 2 as fittin g param eters. The re su lt o f the f it is presented on the p lo t by
a so lid fine. We note th a t the value o f ko = 4.22 obtained from the fit gives an
index o f re fra ctio n n = 1.16 w hich is between the value o f n in a ir and the effective
index o f re fra ctio n in the sam ple o f n
1.35.
T h is can be associated w ith the
fact th a t close to the o u tp u t face o f the sam ple, the speckle p a tte rn has already
begun to ” broaden1’ as com pared to the p ic tu re inside and its characteristic len g th
is d iffe re nt from the w avelength in the sam ple. The value o f the constant term ,
however, is not affected in any way. We o b ta in C 2 = (6.46 ± 0.12) x 10-2 w hich,
w ith in experim ental error, equals the value o f the variance o f the to ta l transm ission
for th is sample, var(s a) = (6.56 ± 0 .2 0 ) x 10- 2 . The excellent agreement between
the results o f these independent measurements (we recall th a t var(sa) is obtained
from the measurements o f to ta l transm ission d is trib u tio n s ) confirm s th a t var(sa) is
given by the degree o f nonlocal co rrela tio n. In fig h t o f the results presented in the
previous chapter, th is indicates th a t the degree o f nonlocal in te n sity correlation is
a key param eter w hich determ ines the s ta tis tic s o f mesoscopic tra n sp ort.
R
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p
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d
u
c
e
dw
ithp
e
rm
is
s
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fth
ec
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rig
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to
w
n
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r. F
u
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rre
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np
ro
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ib
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dw
ith
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is
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n
.
50
5.3 T otal transm ission m easurem ents o f long- and in fin iterange in ten sity correlation
Here we present measurements o f the C<i and C 3 term s o f the in te n sity correla­
tio n fu n ctio n w ith frequency s h ift. A lthough the analysis o f the results from these
measurements are s till in progress, we present o u r results, because o f the sign ificant
interest tow ard these quantities. C 3 is o f p a rtic u la r in te re st, since it is the analog
o f U C F in the case o f classical waves. The frequency dependence o f the long- (C->)
and infinite-range (C 3 ) co rrela tio n terms has been considered in various theoretical
works (fo r a comprehensive review on the subject we re fe r to the w ork o f van Rossum
and Nieuwenhuizen [8 6 ] and the references therein) and the frequency dependence
o f C2 has been observed experim entally [62,65]. In these experim ents, however, the
co n trib u tio n o f C3 in the in te n s ity correlation fu n c tio n was found to be sm all as
com pared to the other term s and in the analysis o f the d ata was considered to be
frequency independent. In measurements o f to ta l transm ission we obtain the crossco rre la tio n function between tw o in p u t modes w ith frequency sh ift in w hich th e C 3
term is dom inant. [75]
T o ta l transm ission measurements w ith a polystyrene sample w ith L/E, ~ 0.1 are
made as described in chapter 3. In order to observe the infinite-range correlation,
however, two identical w ire antennas are used as independent sources o f e xcita tio n .
These were placed 0.5 cm in fro n t o f the in p u t face o f th e sam ple and were separated
by a distance o f 3.0 cm w hich is approxim ately tw ice th e wavelength. Transm ission
spectra for a single sample configuration are taken separately for each antenna by
R
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rig
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to
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n
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r. F
u
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np
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ib
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ith
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n
.
51
using a microwave sw itch w hich couples ra d ia tio n to one o f the antennas o n ly w hile
isolating the other.
M easurem ents for four thousand sample configurations were
made. From these m easurem ents, we obtain the a u to co rre la tio n fu n c tio n in to ta l
transm ission w ith frequency s h ift using spectra taken w ith the same antenna and
the cross-correlation fu n c tio n between spectra taken w ith different antennas.
We firs t tu rn our a tte n tio n to the to ta l transm ission au to co rre la tion fu n ctio n
which is shown in F ig. ‘26. Tt is much broader as com pared to the in te n s ity auto
correlation fu n ctio n (w hich is dom inated by the short-range correlation te rm ) which
explains the difference betw een the average peak w id th in in te n s ity and to ta l trans­
mission as discussed in th e beginning o f chapter ‘2 .
To order 1 /iV , the to ta l transm ission autocorrelation fu n ctio n equals the G> term
of the in te n s ity auto c o rre la tio n function. In the presence o f absorption th is term is
given by [85]
Q;2 _ 0(7^
3 tT
Co(i1) =
A cos (2b L) — cosh(2aL))~l x [—— --------— sinh(2aL )
2 a rtA
' a
a {a * — o r )
b(a2 + b 2)
~sin(2bL ) +
Q
o ^ , 2, sinh{2aL)\,
(a2 — a 2)(a 2 + b2)
(29)
where Q — A u /D , a — (a 4 + Q 2 ) 4 cos(9? /2 ), b = (aA+ Q.2)*sin((p/2), and tamp = Or.
As before, the role o f in te rn a l reflection is accounted fo r by s u b stitu tin g L = L + 2zb
for L in th is expression.
We calculate C2 (Q) from Eq.
(29) using the values o f
La and D determ ined fro m the fie ld autocorrelation fu n ctio n . The result from the
calculations is presented b y the solid fine in Fig. 26 and shows good agreement
w ith experim ent. O n ly fo r large frequency sh ift s lig h t deviations between theory
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u
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rig
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to
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np
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dw
ith
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n
.
52
and experim ent are observed. We p lo t also the th e o re tica l curves in the absence
o f absorption and in te rn a l refle ction (dashed line) and when absorption is inclu d ed
o n ly (d o tte d lin e ). The com parison shows th a t a b so rp tio n affects C2 in tw o ways:
firs t, it decreases the m agnitude o f the degree o f lon g range co rrela tio n (Co(0 )) w hich
is consistent w ith the results fo r var(s a) in chapters 3 and 4
second, the presence
o f absorption broadens the frequency w id th o f th e co rre la tio n w hich is associated
w ith the reduction o f th e average dwell tim e o f photons in the sample.
Finally, in Fig. 27a, we p lo t the cross-correlation fu n ctio n w ith frequency s h ift
obtained from these m easurements. We fin d th a t its value a t zero frequency s h ift o f
is comparable to the value o f ‘2 /log 2 as predicted b y th e o ry [62,87|. The behavior
o f the to ta l transm ission cross-correlation fu n ctio n measured is in q u a lita tive agree­
m ent w ith the th e o re tica l calculations by van Rossum et al. [87] in the presence o f
absorption (F ig. 27b). A d e tailed q u a n tita tive analysis considering the c o n trib u tio n
o f long-range co rre la tio n w hich are expected to be com parable to the C 3 c o n trib u tio n
in the measured cross-correlation function are in order.
sWhen we used Eq. (29) to calculate C3(0) in the presence o f strong absorption, we
found the same reduction from 2/3 to 1/2 as predicted by Brouwer [79 [.
R
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.
53
6 Microwave Transmission Through a Periodic 3D
M etal W ire Network Containing Random Scatterers
6.1 B ackground
Band gaps exist in the electrom agnetic spectrum in a va rie ty o f p e rio d ic stru c­
tures in analogy w ith the electronic band gaps in crystals [33,49,50]. In the p h oto nic
band gap (P B G ), electrom agnetic waves are evanescent. W hen disorder is in tro ­
duced in a PBG stru ctu re , modes can be created in to the gap b u t th e y are localized
and again propagation is in h ib ite d . Such localized states could be associated w ith a
single defect in the photonic crysta l w hich are analogous to localized e lectron states
on isolated im p u ritie s in sem iconductors [8 8 ] o r to localized v ib ra tio n a l modes asso­
ciated w ith defects in crystals [89,90]. T h e y m ay also be associated w ith sca tte rin g
from a s ta tis tic a lly homogeneous random d is trib u tio n o f scatterers. John p re dicte d
the existence o f a m o b ility edge separating localized from propagating electrom ag­
netic waves w ith in the pseudogap fo r n e a rly p e rio d ic systems [491. He proposed th a t
intro d u cin g disorder w ould lead to stro n g Anderson localization fo r photons [34]. In
the present study, we measure transm ission th ro u gh a nearly periodic copper w ire
netw ork w ith a stru ctu re close to sim ple cubic. A PBG is found in th is stru c tu re
both when it is em pty and when it is fille d w ith Teflon spheres producing a Teflona ir m edium . B y s u b s titu tin g a lum inum spheres fo r some o f the Teflon spheres, we
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rig
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to
w
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r. F
u
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.
54
create a succession o f random, scatterer configurations inside the stru ctu re .
The
transm ission is measured in an ensemble o f random scatterer configurations fo r var­
ious degrees o f disorder introduced w ith in the periodic m etal m a trix . The band
structure fo r various 3D dielectric [50-53] and m e ta llic [54-56] p e rio d ic structures
possessing band gaps have been calculated and die le ctric [50,57-59] and m e ta llic [60]
systems possessing band gaps have been constructed. Localized states have been
created by adding to, rem oving or displacing p a rt o f the stru ctu re in d ie le ctric PBG
m aterials and by c u ttin g selected wires in m e ta l systems [58,60]. These defects are
introduced in a co n tro lle d manner so th a t i t is possible to associate specific modes
w ith p a rtic u la r defects. O ur interest here is to create an ensemble o f equivalent ran­
dom configurations o f scatterers w ith in the P BG structure. We introduce disorder
in a nearly p e rio d ic m etal w ire netw ork by fillin g i t w ith m ixtures o f Teflon and
alum inum spheres. The Teflon and alum inum spheres have diam eters o f 0.47 cm.
This is approxim ately h a lf the la ttic e constant o f the w ire netw ork o f a =
1
cm and
the spheres re a d ily penetrate the in te rio r o f th e w ire structure. The Teflon sphere
medium has a sca tte rin g length w hich is m uch greater than the dimensions o f the
m etal structure. Consequently, the Teflon spheres serve as an ideal support m a trix
in to which scatterers can be substituted.
Sigalas e t al.
calculated the band stru ctu re s fo r 3D periodic m etal systems
consisting o f isolated m etal scatterers and o f m etal wares w hich form a continuous
netw ork [56]. P eriodic systems composed o f isola te d m etal scatterers show behavior
sim ila r to th a t o f d ie le ctric PBG m aterials and do not e x h ib it a c u to ff frequency
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to
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r. F
u
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tio
np
ro
h
ib
ite
dw
ith
o
u
tp
e
rm
is
s
io
n
.
below w hich transm ission is sharply reduced. C alculations fo r m etal netw orks w ith
sim ple cubic and diam ond type structures p re d ic t a PBG in b o th systems. The
cubic stru ctu re considered is in fin ite along the x and y directions and has a th ic k ­
ness along the r-a x is o f L — 4a w ith a la ttic e constant a o f 1.27 [i m . The volum e
fraction occupied by the m etal wires is 0.03. A band gap is found below a c u to ff
frequency uc = "fc/a. where c is the speed o f lig h t in vacuum and
7
= 0.4445. A t
uc/2, the a tte n u a tio n is approxim ately 15 dB p e r u n it cell.
6.2 Sam p les and m easurem ents
The m etal s tru ctu re used in our experim ent is a netw ork o f 1 mm diam eter
copper wires. The system has a geom etry close to sim ple cubic stru ctu re w ith a
la ttice constant o f
1
cm. The sample has a le n g th o f
8
cm on each side. Square
wire meshes are created by overlaying one set o f p a ra lle l wires onto another set o f
parallel wires o rie n te d in the perpendicular d ire ctio n . S tra ig h t wires are extended
through p a ra lle l 2D w ire meshes in the lo n g itu d in a l d ire ctio n . W ires oriented in the
three d ire ctio n are connected by a lig h t solder applied a t the vertices o f the u n it
cells. The la ttic e is not perfectly periodic because o f slig h t bending o f the wires and
because o f variations in the solder jo in ts . The volum e o f the copper wires is
6
% of
the to ta l volum e o f the system. The fro n t and back o f the stru ctu re are open and
copper plates are placed on the sides.
The transm ission is measured in the frequency range between 4.0 and 10.4 GHz
using a H ew lett-P ackard 8722C netw ork analyzer. Antennas, consisting o f
R
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rig
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to
w
n
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r. F
u
rth
e
rre
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np
ro
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ib
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dw
ith
o
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rm
is
s
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n
.
1 .5
cm
56
pins, coupled th e ra d ia tio n to th e m e ta l netw ork and detected th e tra n s m itte d ra­
d ia tio n . The in p u t antenna and th e fro n t surface o f the n e tw ork are enclosed in
an absorbing cham ber and the d e te cto r and the o u tp u t surface o f th e sample are
w ith in another cham ber w ith absorbing w alls. The antennas were placed 0.5 cm
from the in p u t and o u tp u t faces o f th e sam ple. They are placed close to the sample
to enhance the signal so th a t the m easurem ents can be perform ed w ith an adequate
dynam ic range. As a result, however, the incid e nt and detected waves cannot be
approxim ated b y plane waves.
6.3 R esu lts an d discussions
The results o f the measurements in an em pty m etal netw ork are shown in Fig.
28a. There is a sharp reduction in th e transm ission coefficient a t a c u to ff frequency
o f 9.33 GHz. T h is gives
uc =
7 c/a.
7
= 0.31 fo r th e constant o f p ro p o rtio n a lity in the relation
T here is a difference betw een the value o f
7
observed and the value
calculated th e o re tic a lly [56] w hich m aybe due to the difference in the fillin g fractions
of m etal wires in the sample used in th e experim ent and in the th e o re tic a l model.
For frequencies below uc, the transm ission is sig n ifica n tly suppressed. The average
value o f the a tte n u a tio n in the gap fro m
6
GHz to uc is a p p ro xim a te ly 30 dJ3. A t
frequencies aro un d 5 GHz, however, a transm ission peak occurs in w h ich a ttenuation
is reduced to a p pro xim ate ly 10 dB . T h is peak in transm ission m ay arise because o f
deviations o f the stru ctu re from sim ple cubic sym m etry o r it m ay be associated
w ith surface sates [91]. The sharp peaks in the spectrum observed above the cu to ff
R
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dw
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rm
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io
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rig
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to
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r. F
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rth
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ith
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rm
is
s
io
n
.
frequency are a result o f the interference between p a rtia l waves p ro p a g a tin g through
the system and reaching the d e te ctin g antenna. When the antennas are fa r from
the sample only modes w ith sm all transverse components o f the wave vector are
detected and the spectrum appears re la tiv e ly fla t above the gap.
W hen the netw ork is fille d w ith T e flo n spheres at a volum e fillin g fra c tio n o f 0.57
w hich is measured as a fra ctio n o f th e to ta l volum e free o f m etal, the c u to ff frequency
sh ifts to 7.58 GHz (F ig. 28b). T he frequency o f the transm ission peak in the gap
undergoes the same fra ctio n a l s h ift. T h is confirm s th a t the peak is in trin s ic to the
m etal structure. Since vc ~ vphja, where vPh is the phase v e lo city in the medium
. we expect th a t the fra ctio n a l s h ift o f the c u to ff frequency is p ro p o rtio n a l to the
ra tio o f the phase velocities in a ir and in the m edium o f random T eflo n spheres and
a ir.
In order to determ ine the phase ve lo city in the medium o f T e flo n spheres and
a ir. we have measured the fie ld tra n s m itte d through a sample o f Teflon spheres
contained in a 7.5 cm diam eter copper tube w ith a length L = 10 cm.
In these
measurements, the fillin g fra ctio n o f the Teflon spheres is 0.60 w hich is s lig h tly
higher than it is in the m etal n e tw ork. We firs t determ ine the in d e x o f refraction
n fo r th is sample at a frequency uq = 6.45 GHz to be n = 1.26, b y m easuring the
frequency s h ift o f a resonant c a v ity mode in a m edium o f Teflon spheres relative
to th a t in air. The absolute phase o f th e fie ld tra n sm itte d th ro u g h a sam ple w ith
len g th L is
= kL = 27rvL /vph. Here vph is the phase ve lo city in a copper tube
co n ta in in g the Teflon spheres and is g ive n by v'ph = vpk[l — (u^ 9 / u ) 2 ] ~ l / 2 [92|, where
R
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p
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d
u
c
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dw
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rm
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ec
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p
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rig
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to
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e
r. F
u
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np
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ib
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is
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.
58
vph = c/ n is th e phase velocity in an unbounded m edium o f T eflon spheres and
is the c u to ff frequency o f the cylin d rica l waveguide. T h e value o f
— ‘2.06 GHz
in th is case was determ ined experim entally from measurements o f the transm ission
through the copper tube fille d w ith Teflon spheres fro m 1 to 4 GHz. Using these
relations and the ind e x o f refraction measured a t uq we determ ine the absolute phase
ifo = 'IttvqL / v'ph. Then, sta rtin g from th is frequency, we measure the frequency
dependence o f the fie ld tra n sm itte d through the sam ple o f T eflon spheres contained
in a copper tube.
Transm ission spectra were obtained fo r 1000 different sample
configurations. The sm all variation between configurations o f less than
1%
o f the
field a m p litu d e indicates th a t the scattering due to the T eflon spheres is extrem ely
weak. T h is explains the persistence o f the band gap when the netw ork is fille d w ith
Teflon spheres. T he real part o f the ensemble average com plex fie ld <Ecosip> is
shown in F ig. 29. T he well defined sinusoidal frequency dependence o f <Ecosip>
allows us to fo llo w the phase ro ll-u p w ith frequency. B y fo llo w in g the increase in the
phase o f the tra n s m itte d field when the frequency is changed in sm all increments
of 2.5 M H z, we are able to o btain the phase A (p accum ulated between 6.45 and
10.0 GHz.
B y adding the accum ulated phase A cp to the absolute phase
we
determ ine th e absolute phase in th is frequency range. From th is phase, we calculate
the phase ve lo c ity vPh in a m edium o f Teflon spheres and a ir contained in a copper
tube. T h e frequency dependence o f vPh is shown in Fig. 30. U sing M axw ell-G arnet
ap proxim ation [93], sm all corrections to i iph are made to determ ine the phase velocity
in the T e flo n -a ir m edium w ith in the m etal netw ork. A t a frequency equal to the
R
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dw
ithp
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rig
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to
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r. F
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.
59
c u to ff frequency for the m etal s tru c tu re fille d w ith Teflon spheres, th e phase ve lo city
is 2.39 x 1010 cm /s. T h is gives vvh/ c = 0.80 w hich is close to the ra tio o f the c u to ff
frequencies o f the netw ork in Teflon and in a ir o f 0.81.
W hen some o f the Teflon spheres are replaced by alum inum spheres, transm ission
peaks appear below the band edge. Measurements have been made fo r tw o fillin g
fractions o f alum inum spheres f aL = 0.05 and f ai = 0.10. A ty p ic a l transm ission
spectrum a t a fillin g fra ctio n o f the alum inum spheres o f 0.05 is show n in Fig. 31.
The peaks below the c u to ff frequency are rem iniscent o f modes associated w ith
donor” defects which have been observed in 3D dielectric PBG m aterials [58|.
However, the radius o f the sca tte rin g spheres in the present case is a t least an order
o f m agnitude sm aller than the w avelength. Thus these transm ission peaks are more
lik e ly associated w ith random configurations o f m any m etal spheres ra th e r th a n w ith
in d iv id u a l scatterers.
We characterize the strength o f the disorder o f the sca tte rin g m edium a t an
alum inum fillin g fra ctio n o f 0.10 by m easuring the scattering length £s o f the Teflon alum inum m ixtu re apart from the m e ta l la ttic e . The scattering len g th is determ ined
by m easuring the in te n s ity associated w ith the average fie ld [cofl =\< E >\2 as a
fu n ctio n o f the sample len g th L. In a sample tube w ith a diam eter w hich is larger
th a n the wavelength in the m edium , the coherent in te n sity fa lls e xp o n e n tia lly w ith
an a tte n u a tio n length w hich is close to the scattering length, once the sam ple is long
enough th a t a single waveguide m ode predom inates o r has a large enough diam eter
so th a t the incident wave can couple effectively to a large num ber o f modes w ith
R
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dw
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to
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r. F
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.
60
sm all com ponents o f transverse m om entum .
In this case, 7co/t ~
[0exp(—L/£s).
The average fie ld fo r ‘2000 sam ple configurations in the frequency range from 4 to
10 G H z was o b ta in e d fo r various thicknesses. B y averaging over an ensemble of
samples, the d iffu sive com ponent o f the fie ld tends towards its average value o f zero.
From measurements w ith sam ple le n g th from 7.5 to '25 cm we fin d th a t l 3 > ‘20cm
a t 4 GHz and th a t i t decreases to l s = 4.5 ± 0.5 cm a t 10 G H z. Since i s > L
where i is the tra n s p o rt mean free p a th , the measurements show th a t k i is greater
th a n 10 over th e e n tire frequency range. Since localization in sam ples w ith o u t longrange order is expected when ki. ~ 1 th is indicates th a t m icrowave ra d ia tio n in the
m ixtures o f T eflon and a lu m in um spheres ap art from the m etal n e tw o rk is fa r from
the lo ca liza tio n threshold. The values o f k used in the estim ate are calculated using
the phase ve lo city in a random m edium o f Teflon spheres.
fn Fig. 32, we present th e average transm ission spectra fo r *200 sam ple config­
u ra tion s for the tw o concentrations o f a lu m in um spheres. In these measurements,
d iffe re n t sample co n figu ra tio ns were created shaking the sam ple by hand. In the
average spectrum the band edge is broadened relative to th a t o f th e m etal netw ork
in a ir and w ith Teflon spheres inside. The band gap becomes less pronounced as the
density o f m etal scatterers increases. The frequency o f the peak in the transm ission
a t low frequencies, however, is n o t affected. T his indicates th a t the phase velocity
is n o t changed s u b s ta n tia lly by the in se rtio n o f m etal spheres in to th e netw ork. The
flu ctu a tio n s in th e phase ve lo c ity due to the presence o f m etal scatterers does not.
therefore, appear to cause a sm earing o f the band edge.
R
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p
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dw
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.
61
We recall th a t var(s ab) was fo u n d to be a reliable measure o f the approach to
lo ca liza tio n . We investigate the behavior o f the second moment. <sab> = var(sab) +
1, o f the in te n sity tra n sm itte d th ro u g h the structure fille d w ith m ixtu re s o f Teflona lum inum spheres as the frequency is tim e d through the band gap edge. A sharp
increase is observed w ith a m axim um value o f approxim ately 4.5 w hich indicates
lo ca liza tio n tra n sitio n .
6 .4 C onclusions
In conclusion, we have observed a band o f attenuated m icrowave transm ission
th ro u g h a 3D periodic m etal netw ork.
W e have shown th a t fillin g the m etallic
n etw ork w ith dielectric scatterers. w hich are sm all compared to the electrom agnetic
w avelength, shifts the band edge to a value p ro po rtio n al to the phase ve lo city in
the m edium . T his dem onstrates th a t a w ell defined band gap can be produced in a
s tru c tu re which deviates from perfect p e rio d ic ity and suggests th a t useful photonic
devices can be produced using stru ctu re s w hich are not p e rfe c tly periodic.
The
extent o f disorder th a t can be in tro d u ce d before the band gap is s ig n ific a n tly altered
can be studied by in tro d u cin g a va rie ty o f defects in to the p e rio d ic la ttic e .
In
th is study, we have measured transm ission through a periodic m e ta l w ire stru ctu re
w ith various concentrations o f ra n d o m ly positioned m etallic spheres. We observe a
broadening o f the band edge w hich is associated w ith the presence o f localized states
in the gap. Because the systems stu d ie d are sm all, the transm ission peaks observed
m ay n o t be associated w ith sh a rp ly defined localized states w hich can be seen in
larg e r structures. Studies in larg e r systems w ould make it possible to investigate the
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.
62
lo ca liza tio n tra n s itio n in this system . For such samples th e a b ility to observe sharp
transm ission peaks associated w ith in d iv id u a l localized modes w ould be lim ite d by
the absorption o f the sample.
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.
63
Summary
In a series o f independent m icrowave experim ents, we s tu d y the s ta tis tic s o f in ­
te n sity and to ta l transm ission in mesoscopic systems in the approach to localization.
In measurements using lon g waveguides w ith volum e disorder, we were able to
reach the loca lizatio n threshold. T h is is the realization using classical waves o f the
Thouless idea o f lo ca liza tio n o f electrons in long wires. We fin d th a t, near the lo­
ca lization length, the p ro b a b ility d is trib u tio n s o f in te n s ity a n d to ta l transm ission
deviate sig n ifica n tly from negative exponential statistics and fro m a n o rm al d istrib u ­
tio n , respectively. For large values o f these quantities, the corresponding d is trib u tio n
decay instead as a negative stretched exponential to the pow er 1 /2 and as a simple
negative exponential.
We confirm e xp e rim e n ta lly the relationships between the m om ents o f inte nsity
and to ta l transm ission and th e ir fu ll d istrib u tio n s derived fro m R M T calculations
for samples w ith q u a si-ID geom etry. These relations u n ify the s ta tis tic a l description
o f local and s p a tia lly averaged tra n sm itta n ce quantities.
The results o f these measurements show th a t the d is trib u tio n s o f in te n s ity and
to ta l transm ission are not s ig n ific a n tly affected even by the presence o f strong ab­
sorption. In contrast, the ensemble average value o f the transm ission falls exponen­
tia lly as a result o f absorption. The presence o f absorption o n ly postpones somewhat
the approach to lo ca liza tio n , b u t does n o t destroy lo ca liza tio n . We fin d th a t the
R
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p
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d
u
c
e
dw
ithp
e
rm
is
s
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no
fth
ec
o
p
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rig
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to
w
n
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r. F
u
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rre
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.
64
variances o f in te n s ity and to ta l transm ission are d ire c tly re la ted to the degree o f
nonlocal in te n s ity co rre la tio n and serve as a reliable measure o f the approach to
lo ca liza tio n .
We have obtained the a u to - and cross-correlation fu n c tio n w ith frequency s h ift
o f th e to ta l transm ission. These functions are dom inated b y the long- and in fin ite range co rre la tio n term s o f the in te n s ity correlation fu n c tio n and allow us to experi­
m e n ta lly determ ine the frequency dependence o f the in fin ite -ra n g e correlation term
fo r the firs t tim e.
In a p ro o f o f p rin cip le experim ent o f microwave transm ission in a 3D periodic
m etal w ire netw ork possessing a p h o to n ic band gap. we dem onstrate th a t, by creat­
in g an ensemble o f random sca tte re r configurations w ith in the ” p hotonic c rysta l",
i t is possible to investigate the s ta tis tic s o f the lo ca liza tio n tra n s itio n .
In conclusion, in th is thesis we have established the n a tu re o f in te n s ity and to ta l
transm ission sta tistics and show how these can be used to s tu d y the localization
tra n s itio n .
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.
65
Fig. 1
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.
66
10
8
6
4
2
0
2.0
1.5
a
oo
l. o
0.5
0.0
16.8
17.0
17.2
F re q u e n c y
17.4
17.6
(G H z )
Fig. 2
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.
17.8
67
VO
1 -0
16.8
17.0
17.2
F re q u e n c y
17.4
17.6
(G H z )
Fig. 3
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17.8
68
(a)
(b)
—
O
+
O—“ O
-+- O— O -“O
-+-
-f-
(c)
4-
X "
"t-
X
X
-I-
X
Fig. 4
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.
X
4-
69
(a)
-Xill"""
-X — # .
“X_ - X
- X-
).<
V
X
■X- - X
-X
-X —x — X -
-X — H r
-x -x- -x
(C )
Xt
T*
+
X - -X
x - -X-
*
- V - - V
Fig. 5
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.
-v
70
-►b+
a+
ss B S 8 S ^ ^ ra a g
a- «-
Fig. 6
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.
71
20.0
Distribution of eigenvalues
of transmission matrix
0.0
0.0
0.2
0.5
0.8
Fig. 7
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.
1.0
72
PC
Microwave
Generator
(Lock-in |
• TWTA. •
Fig. 8
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.
(a)
o experiment
theory
(b)
co
1.0
Fig. 9
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74
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98
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