# Enhancement of Localization in One-Dimensional Random Potentials with Long-Range Correlations Page: 2

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PRL 100, 126402 (2008)

PHYSICAL REVIEW LETTERS

week ending

28 MARCH 2008The strength U, of a single scatterer is associated with

its length embedded inside the waveguide, and the length is

varied by micrometer screws. When U, = const, the wave-

guide is fully transparent in the Bloch band. For the white-

noise potential 100 numbers e, = U, - (U,) were drawn

as uncorrelated random numbers with variance 02. The

corresponding localization length was much larger than the

length L of the waveguide; thus, the eigenstates are delo-

calized within the waveguide. However, if the white-noise

potential E, is replaced by a correlated sequence, the

localization length inside a prescribed window of energy

is strongly reduced and the effects of localization in the

transmission and reflection become observable.

The experiment can be well described by a one-

dimensional equation of Kronig-Penney type [13],i/ "(x) + k2 i/(x)

n= -o

Here U, is the amplitude of the nth delta scatterer located

at x = xn = nd. Therefore Eq. (1), which assumes an

electron with parabolic dispersion, is replaced for an elec-

tromagnetic waveguide by [13]1 '-1(E)

02k2 sin2(kd)

8 W(2sin).

8 sin2/'Here -2 = (U2) _(Un)2. The phase ji is given by the

Kronig-Penney dispersion relation,(U sin(kd),

2 cosyi= 2 cos(kd) + sin(kd),

k0 < < .

The function W(2 p) is the Fourier transform of the binary

correlator )(s) (en+,s ,,)/2,

W(2u) = 1 + 2 t(s) cos(2s/). (5)

s=l

It follows from Eq. (5) that W(2 ,) is symmetric with

respect to the band center = 7r/2.

For any white-noise potential the correlator c(s) 0,

apart from (0)= 1, leading to W(2 ),) 1. To observe

the effect of enhancement of localization it is necessary to

have the function W(2) = W > 1 within some interval

A = 2 - l 1. Because of the normalization condition

=()-V W(2/)d = 1, (6)

the width A0 and the enhancement factor W are related,

2Wo = rA0, providing that W(2 p) vanishes outside the

interval [ tl, f 2]. Thus, the correlation-induced enhance-

ment of localization within the interval A is accompanied

by full transparency of the waveguide for all other frequen-

cies [18]. The Fourier coefficients of the step function, that

is W(2 )= Wo within the interval A and W(2 )= 0

otherwise, is given by1 sin(2sg2) - sin(2sg1)

S s) = 2s2 -- 1(7)

The inverse-power-law decay of i(s) is a signature of the

long-range correlations. A correlated sequence was gener-

ated using the algorithm proposed in Ref. [13]. The two

random sets used in the experiments are shown in Fig. 2,

including their correlations.

For both sets the elements S12 = S21 and S22 of the

scattering matrix were measured as functions of antenna

position. Here S12 (S22) is the transmission (reflection)

amplitude of the scattering process when the fixed antenna

is in front of the first scatterer (n = 1) and the moving

antenna is located between the nth and (n + 1)st scatterer.

The moving antenna emits and receives the signal while

measuring S22.

The single-mode transmission patterns I S121 are shown

in Fig. 3 for the purely random (upper) and correlated

(lower) sequence (see also Fig. 2) as a function of antenna

position (vertical axis) and wave number k (horizontal

axis). In addition, on top of each figure, we present the

dependence of transmission value IS12 through the whole

waveguide. As one can see, for the uncorrelated disorder

there is a gap close to the edge of the Brillouin zone

kd/7r = 1. It originates from the periodic spacing between

the scatterers. For small wave numbers the transmission is

small because of the weak antenna coupling to the wave-

guide, whereas for high wave numbers it is small due to

large absorption. In case of the correlated disorder there are

additional gaps located at kd/r - 0.25, 0.65, 1.25, and

1.65. These gaps originate from enhancement of localiza-

tion due to long-range correlations with lt1 = 0.2 and

t2 = 0.3 [see Eq. (7)]. It is important that inside these

gaps the transmission is practically zero, since the local-

ization length is reduced by a factor of Wo - 15.7 and it

becomes much less than the length of the waveguide.

Outside the gaps the transmission is 5%-10%. This is a

bit less than within a band for a periodic arrangement. At

random sequence 0

0 10 20 30 40 50

correlated sequence o

-IZ

0 10 20 30 40 50

FIG. 2 (color online). Profile of intrusion of all 100 microme-

ter screws into the waveguide for uncorrelated and correlated

random sequence Un. The insets show the corresponding corre-

lation function calculated from the micrometer screw depths

[Eq. (7)].126402-2

I I z

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Kuhl, Ulrich; Izrailev, Felix M. & Krokhin, Arkadii A. Enhancement of Localization in One-Dimensional Random Potentials with Long-Range Correlations, article, March 28, 2008; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc103275/m1/2/: accessed April 25, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.