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model. When in the
homogenization regime
the perpendicular
wavelength is set by the
outer radius of the
cladding region so that
the fields are globally in
phase everywhere in
the cladding. The
FourierBessel
coefficients for all
inclusions are therefore
i
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confined between rings
of holes with different
diameters. Indeed the
second mode in the
seven ring structure of
Fig. 8.12 is confined
between the first and
second rings of holes
and has losses around
ten thousand times
larger than that of the
fundamental.
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the third ring (and
probably close to it for
the second ring), we
can expect its losses to
decrease exponentially
with Nr : the fibre is
strictly multimode, but
higher order modes
extend over a wider
spatial region than the
fundamental mode. 8.6
Theory an
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first and second mode
can be so small that
mode conversion
between confined
modes is negligible
[69], so that MOFs can
be operated as if they
were singlemode
fibres. On the contrary,
when strict single
modedness is crucial,
even in the endlessly
singlem
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agreement between
theory and experiment
as far as dispersion
properties are
concerned. 8.2
Chromatic Dispersion
and Losses of
Microstructured Optical
Fibers Authors: B.
Kuhlmey, G. Renversez,
and D. Maystre Date of
submission: 26 July
2002 Published in
Ap
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remuant frquemment
laide dune spatule
en bois. Lorsque le tout
prend une coloration
dore et tend
sagglomrer, arrter la
cuisson. Laisser
refroidir en remuant de
temps en temps. La
ganache P orter la
crme bullition, hors
du feu, ajouter le sucre.
Verser l
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rings, we use
considerations from
Sec. 7.6.2 to locate the
parameters associated
with each ring. From
Eq. (7.52) we know that
the cutoff wavelength
c is shifted towards
larger values with
increasing core size. For
a core consisting of a
missing central
in
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with ultraflat
dispersion close to zero,
it requires Nr 18
(1026 holes) to deliver
losses below 1 dB/km
at = 1.55m. Even
though several
laboratories have
already drawn 7ring
fibers [92] (around 168
holes) or even 11ring
fibers [36] (around 396
holes),
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CF1 model is valid for
the first few rings, the
refractive index of the
core of the equivalent
fibre in this regime is
not nM, but [n] [MOF
(2)], which slightly
modifies the cutoff
wavelength. Fig. 7.35
shows Q as a function
of / for the double
clad MOF [
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towards this collection
of charts. 7.6.4 Impact
on Results of the
Previous Chapters
Extended Modes
Extended modes as we
have defined them in
Sec. 6.3.3 are
resonances of Bloch
waves. The asymptotic
model for the
confinement derived
here can not be applied
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parameters are very
close to the CF1 region
where dispersion
properties dont
converge. In fact, it
appears that the three
upper curves of Fig. 8.9,
for which convergence
is extremely rapid,
correspond in Fig. 8.13
to points below the
cutoffcurve6 , where
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University of Utrecht in
the Netherlands.
Version: 8th March,
2005. Typeset with
LATEX 2 in Computer
Modern Roman, 12pt.
All figures created using
jfig 2.22, Paint Shop Pro
8 and Wolfram
Mathematica 5.
Feynman diagrams
constructed using
feynmf package.
Ab
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du sucre glace dans le
jus dorange et verser la
moiti du jus de citron.
Ajuster le sucre et le jus
de citron au got, la
quantit exacte
dpendant du degr de
maturit des oranges.
Le rsultat doit tre
bien sucr et
lgrement acidul.
Verser en moule
mtallique (mo
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parameters which were
well inside the CF2
region (d/ = 0.775, /
= 0.22), so that modes
were indeed well
confined and the jacket
didnt have much
effect. Our conclusions
would certainly have
been different if the
parameters had been in
the CF1 region.
Desse
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the paper leads to two
conceptually different
MOF designs with very
desirable dispersion
properties. The first
design, a usual
endlessly singlemode
solid core MOF with
identical air holes
forming a triangular
lattice, yields ultraflat
nearzero normal o
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believe to be the first
systematic theoretical
and numerical study of
finite crosssection
solid core MOFs over a
wide range of
parameters. This study
was only possible
through the
formulation and
implementation of an
efficient method well
suited to the s
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number of hole rings Nr
for different
microstructured optical
fiber (MOF) structures.
is the pitch of the
triangular lattice of
holes, and d is the hole
diameter. The points
correspond to the
computed numerical
dispersion, and the
lines to exponential
ba
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than by a sole 7More
precisely, we
mentioned that the
second mode is
confined between the
first and second ring; in
fact the second mode
having a power
distribution in form of
an annulus (cf. e.g. Fig.
7.4), the fact that its
power distribution lies
betwe
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pump injection
efficiency with a good
confinement in MOF
lasers, one can
nonetheless use similar
designs, using higher
order modes of the
inner cladding (region
with small holes)
confined by an outer
cladding (region with
bigger holes). See for
example Re
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approximately until the
cutoff associated with
MOF (1), then fills the
region defined by the
three first rings
(Aeff/) 1/2/ ' 3), and
around the cutoff of
the second equivalent
MOF starts to diverge.
Note that we have
chosen this example for
its didactic
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first rings being of
diameter d1 = 0.15
and the inclusions of
the two outer rings of
diameter d2 = 0.45
[MOF (3) on Fig. 7.34].
From the phase
diagram, we know the
cutoff region associated
with the first few rings
of holes. When well
below cutoff, the CF2
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were aware of the
fundamental mode
cutoff and hence the
existence of an a priori
interesting cutoffregion in parameter
space. Its aim was to
establish the
dependence of
dispersion properties
on all MOF parameters,
including the number
of rings, in connect
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describe all facets of
fundamental
interactions. I.1
Introduction I.1.1 Dual
Models The Early days
of string theory In many
ways, string theory
began from a model
that Gabriele
Veneziano, then at
CERN, wrote down in
1968 to aid in
describing the veritable
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(Aeff/) 1/2/ MOF (3)
Figure 7.35: Designing
double clad MOFs: Q
curves and mode radius
as a function of / for
the double clad MOF
(3) and the two
wavelength dependent
equivalent MOFs (1)
and (2). The geometry
of MOFs (13) is
detailed in Fig. 7.34.
CHAPTE
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considerations, we
expect the fundamental
mode to be slightly
confined by the first
ring of inclusions, then
strongly confined by
the second ring of
inclusions. This is in
agreement with the
effective area of the
fundamental mode
(Aeff ' 10.5 m2 ) being
s
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chosen parameter
values for and d
correspond to the small
limit slope region.
Inset: Crosssection of
the modeled MOF with
3 rings of holes (holes
are in grey), Nr = 3. is
the hole spacing, and
dn is the hole diameter
of the n th ring. The
solid core is f
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8.12). It must be
pointed out that the
MOF design can be
started from other
values of d1: for
example with d1 = 0.6
m, we found d2 = 0.8
m, and d3 = 1.0 m,
and = 2.0 m (data
not shown). Using the
scaling law (8.3), we
can easily derive other
structures ha
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of two dimensional
quantum gravity, X (,
) being considered as
matter fields on (see
I.2.2). Viewed as this
dual picture, we are
effectively solving
problems of (1+1)
quantum gravity. It
serves also as an
alternative description
of topological strings
an
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effective index neff =
/k0 where k0 is the
free space
wavenumber). Due to
the losses resulting
from the finite
transverse extent of the
confining structure, the
effective index is a
complex value, its
imaginary part =(neff)
being related to the
losses L i
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almost no effect on the
dispersion properties of
the MOF [D(4)], but
results in acceptably
low values of geometric
loss for technological
applications: with Nr =
6, the losses are below
10 dB.km1 , and with
Nr = 7 the losses are
below 0.2 dB.km1 .
For Nr