Photon localization in disorder-induced periodic multilayers

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Photon localization in disorder-induced periodic multilayers

B. van Tiggelen, Adriaan Tip

To cite this version:

B. van Tiggelen, Adriaan Tip. Photon localization in disorder-induced periodic multilayers. Journal

de Physique I, EDP Sciences, 1991, 1 (8), pp.1145-1154. �10.1051/jp1:1991196�. �jpa-00246400�

J. Phys. I France 1 (1991) l145-l154 AOOT1991, PAGE l145

Classification

Physics

Abstracts

05.60 63.20P

Photon localization in disorder-induced periodic multilayers

Bart A. _van

Tiggelen

and

Addaan Tip

FOM-Institute for Atomic and Molecular

Physics,

Kruislaan4O7, 1098SJ Amsterdam, The Netherlands

(Received

28

February

1991,

accepted

26

April 1991)

Rksumk. Nous _avons examink l'influence du ddsordre _{Sur un}

Systdme

Stratifid et

diklectrique,

I

une dimension. Le

Systdme

est

d'origine p6riodique.

Des simulations

numkriques

ont ktk ef§ectudes _et devraient dtre confrontdes I des

prkdictions thkoriques. L'exposant

de

Lyapunov

est non

analytique

aux bords du spectre. ^Nous avons constatk _que les anomalies

qui pourraient

_se

trouver au centre des bandes sont

beaucoup plus

nettes que celles dans le moddle d'Anderson.

Allstract. We studied the influence of randomness _{on a}

periodic

dielectric lD Stratified system.

Numerical simulations _are

performed,

and _are

compared

to theoretical

predictions. Non-analytic

behaviour is found at the band

edges.

The anomalies that _can arise in the middle of _a band _tum out to be far _more

pronounced

than the _ones in the Anderson-model.

1. Inwoducfion.

Localization is the

phenomenon

that _{a wave} becomes

spatially

localized in the presence of disorder. Since its introduction

by

Anderson

[Ii

in 1958 there has been much _progress in its

understanding.

Criteria

[2J

^and

scaling

theories [3J were

developed, predicting

the _occurrence of localization in _one and two dimensions

regardless

of the

degree

of

disorder,

and the

presence of _a

mobility edge

in three dimensions

[4J. Later,

Vollhardt and W61fe

[5J,

presented

_a

microscopic

localization

theory,

where the constructive interference of time- reversed _waves, known _{as «} weak localization _{», was} taken to be

responsible

for localization.

In favour of their _{treatment was} its

agreement

^with

scaling theory,

and the fact that the existence of weak localization _was

reported

in

light

reflection

experiments [6J.

On the other

hand, techniques

first

developed by Furstenberg [7j

have been used to treat localization in

one dimension in _a mathematical

rigorous

_way

[8, 9J,

without any reference to

perturbation theory.

This

approach

does not work in three

dimensions, although

_some results have been

obtained for this _case

[10J.

The exact conditions for the _occurrence of localization in _one dimension _are

subtle, but, loosely speaking,

^the statement is that states _are localized for

« almost _{any energy} and _« almost _any realization of the

system, provided

the

probability

measure,

determining

nature and

strength

of the

disorder,

is

ergodic.

This makes lD

localization _very suitable for numerical

simulation,

for the

ergodicity

demand is

justified

in

most _cases

[I lJ.

l146 JOURNAL DE PHYSIQUE I M 8

The

physical quantity

that determines the

exponential decay

of the _wave

function~

is the

(upper) Lyapunov

exponent. Given _{a set} of random matrices

(M,(w j)

it is defined _as

y w lim

log

ⁱ

Mj (w M~(w My

(w ⁱ

,

(1)

~

- w

N

and

is,

for almost any energy

[12],

a

positive quantity

almost

surely independent

of _{w, w}

indicating

_a realization of the system. A very useful result is the Thouless formula [13]~

relating

the

Lyapunov

exponent to the

density

of states

(DOS) by

_means of _a

dispersion

relation. It finds its

origin

in the fact that both

quantities

_are real and

imaginary

_part of _one

analytic

function. This fundamental relation may be taken _to be

responsible

^{for anomalous}

behaviour of the

Lyapunov

_{exponent on} band

edges

of the parent system

[14].

In _more

dimensions,

_a

simple

Thouless formula _no

longer holds~

but, nevertheless~ the DOS may be

an

appealing pathway

to find criteria for localization

[3, 15~16].

To describe the _one dimensional localization of

electrons,

most workers _treat the Anderson Model with

diagonal disorder,

but the concept

applies

_{to any}

problem involving

random transfer

matrices,

and thus also for the transmission of

light

^{in random}

layers. Consequently~

a

sufficiently large

stack of dielectric

layers,

with random thickness

and/or

index of refraction will act _{as a}

perfect

mirror _ii almost

surely

_», that is

always except

^for a countable number of

energies

and _a set of realizations with

probability

zero, such _as

periodic

stacks.

Recently,

the

study

of

quasi-periodic stacks,

within the _context ob both electron

[17,

18] ^and

light

transmission

[19,

20~

21],

^has become _very

popular.

It has been realized that these

deterministic structures _can also

give

rise _to localized states, and _as

such,

form _an

important regime

between order and disorder.

This _paper deals with the influence of

periodicity

of the parent system on the

Lyapunov

exponent. The band _structure of the parent system ^{will have} a drastic influence _on this

quantity.

The system we describe is of the form

ABj AB~AB~..

where the indices of refraction of the

B,'s

will be

given

random distributions of different kind. The

non-analytic

behaviour at different band

edges

is

investigated,

and

large dips

_are discovered _near

special energies,

for which the

wavelength

matches with both A and

(B,).

In

principle~

this

phenomenon

_can be used to construct

sharp

filters. In both _cases, the numerical simulations show universal behaviour of the

Lyapunov

_exponent. Excellent agreement ^{is found with the}

theory developed by Kappus

^and

Wegner

[22]~ ^Lambert

[23]

and Derrida and Gardner

[J4].

2. The model.

We consider the

propagation

of

electromagnetic

_waves in _a stack of dielectric

layers,

and confine ourselves to the _case of normal incidence of

linearly polarized light.

To this end _we define the _{state vector} Cm

(E

_'~~E~,

B~),

^with E= ^and B~ ^the components ^{of the electric and}

magnetic

field

along

the z-axis and

y-axis.

The x-axis is located

along

the

slab,

in the direction of

propagation;

_F is the dielectric constant. Maxwell's

equations imply

the

following

dynamics:

dc

_i~~~ .j

^o

F~pj

^{~ ~}

pi-I

o

Here is _pm I

d/dx,

F

m

F"~

and _our units _are such that the

speed

^of

light

ⁱⁿ _vacuum, co = I. If _we insert modes

exp(- iEt)

the solution of

equation (2)

in the n-th

layer

_is

C~(x)

_{= a~}

e~'~~~"~

^~~ ₊

_tin e'~~~~~~~

bt 8 PHOTON LOCALIZATION IN MULTILAYERS l147

with

k~

_m

Er~.

The

continuity

of

Ez

^and

_B~

on the interface of _two

layers

fixes the transfer matrix:

~~~~ I= M(n). (~~

,

n+i _n

M(n)~~

=

M(~)i

=

~~

_~

~~

+ i

~~ikn+

I(Xn+I ~Xn)

2

F~

^'

~

~~ )

~ ~~

)j ^~~

^{+ i}

^l~n

^{~- lkn+I(Xn+1 Xn)}

~~ 2

r~

The

layers

with _{even n are} taken identical and

nixed,

and for

simplicity

referred _{to as vacuum.}

From _{now on} the index _n counts the _vacuum

layers only.

The _wave function at site _{» n}

determines the

electromagnetic

field in the n-th _vacuum

layer

_;

r~

^{is the}

reciproke speed

of

light

inbetween the n-th and

(n +1)-th

_vacuum

layer.

The transfer matrix

mapping

_one

vacuum

layer

onto the next _one is

given by

M(n

_{- n +} I

)

=

f~ _~~/

,

(3a)

lq~ fl~

where a~ and q~ are

given by:

rj

₊ I

a~ = e~ ^Y cos

yr~

I sin

yr~

_;

2

r~

i

r]

q~ = e ^iY

~ sin

yr~ (3b)

~

n

All

layers

_are assumed to have _common width

d;

_ym Ed is _a dimensionless _energy parameter. ^The exponent ⁱⁿ q~ can be eliminated

by

_a

(n-independent) unitary

transform- ation. The matrix has determinant I. If all the _non-vacuum

layers

have the _same value for ra

superlattice

structure

[24J

^is present, ^{and the} «

photon

» band structure is determined

by

the

criterium that both

eigenvalues

have

unity

_norm,

which,

in turn,

implies iReai

_w I.

Although

in the

perfect periodic

_case it is

possible

to write down _{an «}

Anderson-type

_» of

equation

lint

_~

_iii-i1

_{~ ~~}

~

Iii1

~

~~~

this formula _no

longer

holds if the Bloch

symmetry

is

broken,

which is true if disorder is

introduced,

^and

r~

m

ro

+ 8

~.

We do not introduce randomnes in the width of the

layers _[25J.

The

8~'s

are assumed to be

mutually independent, identically

distributed stochastic variables with _{zero mean,} and deviation _«. This guarantees ^{the stochastic} process ^to be

ergodic [I Ii.

Consequently,

the

i~yapunov

_{exponent can} be calculated

numerically

_on the computer

by generating only

_one realization

(r~),

and then evaluate the

product

in

equation (I). Many

authors made _use of the

stationarity

of the

underlying probability

_measure, to find the behaviour of both the

Lyapunov

exponent and the

DOS,

often in the limit of low disorder.

We

investigated

the _range of

validity

of these

predictions,

and find

good agreement. Special

interest is devoted to

energies corresponding

to the band

edges

of the

parent system

and

special

_«

Fabry-Perot

_»

energies.

J148 JOURNAL DE PHYSIQUE I M 8

2.I THE BAND EDGES.

Many

workers have

predicted

the anomalous behaviour of the

Lyapunov

exponent on a band

edge [26-28].

A _very

elegant

renormalization treatment was

given by

Bouchaud and Daoud

[?9], resulting

in _{y «} ^~ _at the band

edges.

A

straightforward application

of their method _{to our} random matrix

yields,

in the limit

w, ~

- 0

yjA,«) =w~~F~( j.

₍₅₎

w'~

Here Am _{_v} y~d~~. ^{On the band side of the}

edge

_we expect ^the anomalous behaviour to

disappear

and to go over into _y w~. Hence

F~(7~ j l/7~.

On the gap side the

Lyapunov

exponent

approaches

the

logarithm

of the

largest eigenvalue

of the transfer

matrix,

which

implies F~(7~) ~,,~.

Ho~&>ever, ^this does _not fix

F~(y/ ) completely. Using

the

degenerate perturbation theory

of

Kappus

and

Wegner [22],

Derrida and Gardner

[14]

calculated the

function

F~(y/

for the Anderson model:

~

dt t~

~e~~~~~~~' Fb(7~

~

~~

~~~

(~

^{~~ ~- l,2 ~- t~6 ~ 2 ~i}

0

If _we compare our model with the Anderson _{model, with energy E and}

diagonal

disorder

V,,, identifying

_as in

equation (4)

E ₊

V~

₌ ^{2 Re} a (_v,

F~)

₌ 2 Re

al

_y,

To

j + 2 ~~~ _~

(_i,

To) 3~ (7)

then _{we can} show

(see

the

Appendix) that,

within the

scaling assumption

of Derrida and

Gardner, equation (6)

is valid for _our model _as well. This is not trivial, for _we have

already

indicated that _a

rigorous mapping

onto the Anderson model breaks down in the presence of

randomness. The

figures

_compare the

scaling prediction

to numerical simulations.

performed

on a VAX 3100 station. The distribution of the 3's _was taken _to be

Gaussian,

and every data

point

was

generated independently

from all the others. The agreement between the

scaling

result and the simulations is very

good,

_over three decades in

disorder,

and _we

conclude that

scaling theory provides

_an accurate

description

of the

Lyapunov

_{exponent near}

the

edges

of _a band.

2.2 FABRY-PEROT RESONANCES. In this section _we let the average index of refraction

To

of the random

layers

be rational. In that _case the band _structure of the parent system itself becomes

periodic

in energy.

Special energies

E

exist,

to which _we shall refer _as

Fabry-Perot energies,

for which _a

multiple

of the

wavelength

^fits

exactly

^{in both} the _vacuum

layer

and the average of the random

layer.

Such _{an energy} is

guaranteed

to be located in the band of the

parent

system ^{since then Re} a = ± I.

Using again

the

procedure

of Bouchaud and Daoud

[29]

it is

easily

established that the

scaling

limit of small _w and A takes the form:

y

(A,

_«

=

w~F~

_~~

(8)

w-

Now,

A

m _v nw. We have taken

To

=

m/n.

The function

F~(y/

_can be calculated

using

the

degenerate perturbation theory developed by Kappus

and

Wegner [22].

Because the transfer matrix has determinant I _we have _a conservation law _a~ ^~

li~

^~ ₌ constant. For _{a semi-} infinite system it is

easily

seen that this _constant is _{zero, so} that _{we can} take

a,~lli~

=

I

e'~~

bt 8 PHOTON LOCALIZATION IN MULTILAYERS l149

al o 0.00006

~A,a)ll~

^{° °'°°°°~}

° o.oooi

x o.ooi

DwWa& Gadner

r ₌ 02

edge: 5~457339x

60 40 -20 0

~ 20 40 60

- A/~a

a)

ioo

2~

~

~~

~ ~

l~fi,O)fO _edge: 5.l6ll103163x

~go.e~.6+'fi~°o

~d6 ^uuauaa

0 ^° D

/~

/

Dwda Gwdn«

o/

^~ ~ ~ ~~~

D o.oi

~~ _~

o o.ooi

o ( ^#

x o.oool

+ o.ooos o.oi

60 40 20 0

~~ 20 40 60

- &Ka

b)

Fig.

I. Numerical simulation of the

Lyapunov

_{exponent near a} band

edge, using

5 _x 10~

layers.

A Gaussian distribution with deviation _{« was} used. The

scaling

behaviour of equation

(5)

is confirmed.

The dotted line is the

theory

of reference

[14].

Equation (3)

results in

e'~~~

=

'lln~l~li~li

>

~9)

where _we defined S~ m

q~/

_a~ and

e'°"

w

a~/

_a~ With

p~

m

(e'J' )

and a~t w

(S~ ^e'~° )

it follows

[23]

that

~

~j Wtj pj

= art,

j =1

°iJ=3fJ- I

(-iY~~~~~i_)(ii~jfi~l'i)ia2k~J-f,f~J ^(i°)

~~i-j

1150 JOURNAL DE

PHYSIQUE

I K 8

(the

_paper of Lambert

[23]

^contains an erroneous factor I-I instead off in the nominator of

Eq. (10)).

We search for _a

scaling

solution of the form of

equation (8).

To this end _we write

_i' = nw +

~s

_{and r}

=

naj'n ₊ At, with A small. The variables _a and _q in equation

(3j

_can be

expanded

_as follows

a =

(- 1)"'~

^"

[l

iAA iBA ^~ CA _~]

, q =

(- 1)~ [DA

+ EA _~]

with A

= w

(na

^~ _{+ n~)}

t/2

ni, B

=

(na

^~ ₊ 3 _n ~) s/2

n~

₊

w (na^~ n~)

t~

_n/?

na~. C

= n

~

w

t~/?.

D

= w

(na

^~ _{n ~)}

t/2

na and E

=

(na

^~ _n~)

s/2

^na~ _{+ w} ^na~ _{+ fi}~)

t~

_n'2

na ~. We find _to order

~

ao~ =

(-1)~~~+"~ ^l k~(A~)

₊ ik

(B) ~j

2

aj _~ = (-

l)~+~~~+~~ (E)

^ik

(AD) ~,

_a~j

=

(-1/~~+~~ (D~)

A ^~ (ll)

All other at~ are

higher

order _m and _are

ignored.

The matrix elements Wt~,

again

to order A~, can now be obtained from

These elements _are seen to be

proportional

to

A~

so that _we write Wt~ m

Qij ^~;

^{all other}

elements _{are zero to} order A_~. Furthermore _aj

w vi ^A~, a22 W vi ^A~, and r, =

0 for

i m 3. The moments

(p

j, ~1~, w p are seen to follow from the

equation: Q

_lC

= v Since

m p~ ~ 0 _as

j

~ cc it _can be solved

accurately, choosing

_a

sufficiently large

dimensionah- ty ^{M for the matrix}

Q.

We find, with

7~ =

si't~

F~(7~

)

= ~

(log (a+ qe~'"(~)

=

~,+

Re

~~lj ~~/1~lj

~~~lzj

^{+ dlA ).}

^(l~j

2«

~-

1- t-

~-

The

figures

2

display

the result of

simulations,

for different

disorders,

_as well _{as a} calculation based _on

equations (12)

^and

(13).

The remarkable

dip

in the

Lyapunov

_{exponent is} known _as

an

anomaly,

and _turns out to be

slightly

redshifted with respect to the

Fabry-Perot

_energy

nw. From the simulations _we infer that the

dip disappears

for

large

disorders and the

scaling

result becomes inaccurate. Because the

theory

still involves the

diagonalization

of _{a matrix} of infinite

dimensionality,

it is _not clear _{to us} whether the

scaling

^result

exactly

^vanishes in the center of the

anomaly. Clearly,

_{we expect a very}

large stack, generated according

to a similar

procedure,

to act as a narrow

bandpass

filter. This

anomaly

thus may have _an

important application [30].

The

theory

demonstrates the irrelevance of the _nature of the

probability

distribution.

provided

the

scaling hypothesis

is satisfied, that

is,

the deviation

« must exist and be

sufficiently

small. This is also true for _a distribution of the kind r

=

na/n _{± «,} both with

probability 1/2.

Numerical results for _a

stack, generated according

to such _a distribution, _are

presented

ⁱⁿ

figure

3.

They

_are found _to coincide

nicely

with the

scaling

solution.

3.

Binary

distribution.

The _one dimensional Anderson model with

diagonal binary

disorder _was studied

extensi;ely by

Nieuwenhuizen

[31]

and others. A

singular power-law

behaviour of the

density

of _states

was

predicted,

_as well _as the existence of

extremely

narrow and

pronounced dips

in the

Lyapunov

exponent. ^These

dips

were referred to as islands _» in order _to

distinguish

them

bt 8 PHOTON LOCALIZATION IN MULTILAYERS llsl

1o

y(A,a)11

al ° o.oo8 o-I

° o.oos

r~=^{o.5 ° o-o'}

0.0< ~~~~6 ^{x 0.03}

theory (M~<0) energy: 2K

o.oo<

60 40 20 0 ₂ 40 60 80 <00

- A/Ka

a)

ioo

y~A,a)la~ ^~

l,

al ° o.oos

° o.oo8

° o.ol

0.01 ~~°'~

x 0.03

~"~~~

theory (M~20) SK

io'~

~~ ~~ ~~ ~ A/xa~ ^{~~ ~~ ~~ ~~~}

b)

Fig. 2. Numerical simulation of the

Lyapunov

exponent near 2 special « Fabry-Perot

energies

_»

using

a Gaussian distribution for the index of refraction. The line is the

application

of

equations (10)-(13).

The dimension of the matrix Q is indicated

by

M, the number of

layers by

N.

Every data-point

_{represents an}

independent

simulation.

from anomalies which _are not

specific

for

binary

distributions.

Very recently,

Crisanti

[32]

investigated

_a

binary optical multilayer

and found _zero

Lyapunov

_{exponents on}

special Fabry-

Perot

energies,

for _any disorder. We demonstrate the

applicability

of the

theory

of section 2.2 for any dielectric

binary multilayer.

We consider

again

the sequence All

i

AB~...

and _assume that

B;

= A

(r

= I

)

with

probability

I p and

B;

=

Bo(r

=

r(B))

with

probability

p. Furthermore _a and q are the

parameters

^{defined in}

equation (3) corresponding

to

r(B).

With the notation of section 2.2 it is

easily

shown that

~kf "

30k(1~P)

_~ ^~~'~ _+P

~

)~ ~ ^~ _(l~)

152 JOURNAL DE PHYSIQUE I tS 8

lo

~A,a)la~

0

al ° 0 005

Q 0 008

0 01

energy ^2x o 001

r(Bh0 5

x 0 03

0 00) N=10~

theory (M=10)

4

~° ~° ~° ~° '°

- AJna~ ^{~° ~°}

Fig. ³ As in

figure

2, but now the F,, are distributed according to1) 5 _{± «} both with probabilii, 2

(~;%, ;~j

p=09

.'

»

~~~

P"° ~

j _~

,

_~

~ , j~ ,'

10

p-o

I

_,, ^{~ i ~ ~ ~, ~}

~ fl

~~

_

~ -2

~/

0

~

~

~

'

1~~3

'

10 -~

_~

, lo

~ ~ ~ ~

ENERGY Y

(~)

~ 5 5

Fig. ^4. The Lyapunov exponent for the

optical multilayer

with binary distribution. The lines represent e~aluation of equations (lsj and (10). The matrix W has been _given dimension At 2. the

data-points

_are simulations _on 5 _x 10~ layers The dashed line _is the exponent for the period>c ^~tacL

ABC ABn (p ₌ ).

bt 8 PHOTON LOCALIZATION IN MULTILAYERS l153

The various _{p, can} be found

using equation (10).

The

Lyapunov

_exponent is then calculated

as follows

y =

log _I

= Re

(log (a

₊

qe~~'))

= p

log (a( jj ~~_~~

Re

[qj pj] (15)

"n

j i J

Figure

4 shows _a

comparison

of _a

simple

version of the

theory (M

=

2)

with _a numerical simulation _on 5 _x 10~

layers.

On the vertical axis _we

plotted

_y

~p )/p(

I _p

), knowing

that the denominator is

proportional

to the variance of the

binary

distribution. Perfect

agreement

with

theory

was established for M

=

10,

but the outcome for M

=

2 is still

surprisingly satisfactory.

In addition _we

plotted

the

Lyapunov exponent

^{for the}

periodic system BOABOABO..., corresponding

to p = I. For

energies

in the spectrum ^of this system, we infer _y

«~

_and

anomalous behaviour _seems absent. We have not found evidence for the presence of

islands,

but this _may well be caused

by

insufficient _energy resolution. Because the

binary

distribution is attractive from _an

experimental point

of

view,

random

multilayers, generated accordingly,

may find _an

important application

in

X-ray optics [33].

Acknowledgment.

We

acknowledge

Ad

Lagendijk,

Theo

Nieuwenhuizen,

J. Luck and Jan Verhoeven for useful discussions. This work is part of the research _program of the

Stichting

_voor Fundamenteel

Onderzoek der Materie

(FOM)

and _was made

possible by

financial

support

^{from the}

Nederlandse

Organisatie

_voor

Wetenschappelijk

Onderzoek

(NWO).

Appendix.

We

give

an outline of the

proof

that the

analytical scaling

result for the

Lyapunov exponent, equation (6), applied

to the _one dimensional Anderson

model,

also holds _near the band

edges

of the dielectric

multilayer.

To this end _we show that the random matrices

Mi

=

i~ ~~

and

M~

=

(~ ~

^~~

^~

lqn ~n ,

have the _same

Lyapunov _exponent

_near the band

edge,

in the

scaling

limit of Derrida and Gardner

[14]. M2

is in «Anderson _» form if the identification

equation (7)

^is made. The

change

of base

U~, mapping Mi

onto

M~

U~.M~.Uj~=Mi

is

given by

~~

l

~s~

z~

)/~~

where

s~ =

~/2 _iq~/(a~

_{z~ +} I

)

, z~ = I Im a~ + I

(Im a~)~.

If _we denote the _wave function

corresponding

to the transfer matrix

M~ by

the vector

(&~, $~)

and

R~

_w

&~/fi~,

_we

find,

after _some

algebra

II 54 JOURNAL DE PHYSIQUE I M 8 Notice that

(log

_a

~ _~

la~ equals

the

Lyapunov

exponent ^of

Mj,

whereas

log

_R~ is

the _one for

M~.

Next, a

scaling expansion

has to be

employed

for _{a~. =~} and

R~. Following

reference

[14]

we must use i'

_i'~d~~

~

~,

r,, To

A, and R,, ₌ ± + ^~

~r,

with

A small. To lowest order _in A it is

easily

inferred that

on)y

the

scaling

of R~ matters _in

equation (A

I).

Consequently log

_a

~ ~

/_{a~ =} ^~ Kr. F>, _ii

equation (A

it follows that K

=

whenever

(q~~~

~ l, so that (z~ = I. The

Lyapunov

exponent ^{of both} matrices is determined

by

the average of the stochastic variable _r, and

gives,

_as shown in reference

[14], equation (6).

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