limited by the value zs
of z at which the fibre
starts. Doing so, we
would find an
exponential growth in
the cross-section of the
fibre, but the
exponential growth
would stop at r(zs), and
for larger values of r the
field would be strictly
zero. Further t
ratios. =(neff) decreases
monotonically with
increasing d/, as this
parameter takes the
values 0.40 (1), 0.41,
0.42, 0.43, 0.45, 0.46,
0.48, 0.49, 0.50, 0.55,
0.60, 0.65, 0.70, 0.75
(14). the transition
using all the mentioned
criteria; the results are
su
The model, without the
asymptotic
approximations, has
nevertheless been used
for long wavelengths
(e.g. in Refs. [60, 97]
and many others) and it
was argued that for
long wavelengths nFSM
should converge to nz
given by Eq. (7.8). This
is true for the same
surrounded by a jacket.
3.2.4 Rigorous
Formulation of the
Field Identities In the
vicinity of the l th
cylindrical inclusion
(see Fig. 3.1), we
represent the fields in
the matrix in local
coordinates rl = (rl , l)
= r cl and express the
fields in Fourier-
Folkenberg et al. [100]
strongly corroborates
this cutoff curve.
Further, Mortensen et
al. recently found
results in excellent
agreement with the
above fit using a
different theoretical
approach, adapting the
usual fibre parameter V
to MOFs [101]. 9The
co
to the vector
components A l and
Bl . From Eq. (3.30) we
have A l = X Ni j=1 j6=l
H ljBj + J l0 A 0 ,
(3.40) where H lj =
diag(Hlj , Hlj ), and J l0
= diag(J l0 , J l0 ).
Equation (3.40) is the
representation of the
regular incident field at
cylinder l in
convenient to work
with scaled magnetic
fields: K = ZH, where Z =
(0/0) 1/2 denotes the
impedance of free
space. Each mode is
characterized by its
propagation constant ,
and the transverse
dependence of the
fields E(r, , z, t) = E(r,
)e (zt) , (3.1) K(r,
regular and follows
from Eq. 3.15. The field
reaching the inclusion
will be scattered. The
scattered field radiates
away from the
inclusion: there are
now sources inside the
region delimited by the
inner circle of the
annulus. The scattered
field is hence
number of rings of
holes. In the case of
MOFs with an infinite
number of rings (single
defect in an infinite
photonic crystal) the
analogy with W-fibre
doesnt hold, and it is
difficult to predict a
priori whether the
fundamental mode will
undergo a cutoff
CHAPTER 3. THE
MULTIPOLE METHOD:
FUNDAMENTALS 48
associated with an
outgoing wave
originated from sources
inside inclusion j. In any
annulus not
intersecting or
including inclusion j,
and in particular in an
annulus centered on
inclusion l, this field is
THE MULTIPOLE
METHOD:
FUNDAMENTALS 49
3.2.5 Boundary
Conditions and Field
Coupling While the
field identities of the
previous section apply
individually to each
field component, cross
coupling between them
occurs at boundaries. In
what follows, it is most
for leaky modes has
already been paved.
2.4.5 Spectral
Considerations There is
a discrete infinity of
leaky modes [3, 56]. For
a step index fibre with
infinite cladding,
propagation constants
satisfying nCL < /k0 <
nCO give strictly guided
modes. The
prop
fn(u) = 0 . (3.9) Eq. (3.9)
is the Bessel differential
equation of order n.
The functions fn(u) are
hence linear
combinations of Bessel
functions of the first
and second kind of
order n (Jn(u) and Yn(u)
respectively), or,
equivalently, of Bessel
and Hanke
therefore increases
nearly exponentially
with increasing Nr .
Since CF2 and CF1
converge with
increasing Nr , 26 the
region of / in which
the losses have to go
from those of the CF2
model to those of the
CF1 model keeps a
finite width, whereas
the range c
they depict and how
they were constructed.
All field distribution and
Bloch transform figures
are drawn in a
normalized, linear
colour scale. The
brightest colour
represents the
maximum value of the
distribution within the
depicted frame, the
darkest repr
method. Indeed losses
are already extremely
small at / = CF2 for
small values of Nr : for
Nr > 3 they rapidly
become smaller than
what can numerically
be estimated through
the multipole method.
We could nevertheless
verify directly (through
analysing the
lower than 1014
numerical inaccuracies
in the imaginary part
make the used
algorithm unstable, and
it is then required to set
=(neff) = 0 and find the
minima of the
determinant on the real
axis. 4.4 Computing
Dispersion
Characteristics The
above process o
transition of the
fundamental mode, for
a MOF with d/ = 0.3
used at = 1.55 m.
Curves (1) to (4) are
=(neff) for 3, 4, 6 and 8
rings, curves (5) and (6)
are Reff/ and Q for Nr
= 3 respectively. The
points (a-d) indicate the
position of the field
plots of F
expect in that region.
Convergence of
properties with Nr At
the long wavelengths
end of the transition
region, mode
properties derived from
< nz, so if we consider
the cutoff to occur in
the vicinity of the point
where neff = nz, 33 we
have, c being the
w
test), both evaluation
methods agree up to
the precision of the
numerical integration.
For =(neff) < 1014 we
have no direct means
to check the accuracy
of the results given by
(5.4). Nevertheless we
can get an estimate of
the relative error of the
fields
0.12 | Im(Ez wijn Ez
int )| Figure 4.7: In the
upper and lower parts
of the figure, internal
and Wijngaard
expansions are
compared for
respectively Ez and Kz,
for an air core MOF,
with M = 5 both for the
central air hole and all
other air holes (54 air
ho
define to be the
transition region, mode
properties cannot be
explained in terms of
simple step index
fibres. Width of the
Transition Region
Further, we have seen
that the width of the
negative Q peak keeps
a finite non-zero value
when Nr approaches
infin
minimum, we therefore
also analysed the
behaviour of the loci of
the points at which the
value of Q is half the
value of CHAPTER 7.
MODAL CUTOFF 117 |
Ez| |Hz| < 0.3,
indicating clearly that
the asymptotic
dependence becomes
valid for reasonable
wavelengt
the MOF consisting of
inclusions with
diameter d2 and a core
extended to a whole
missing ring. We
suppose that the values
of 2 and 1 are not
too close, and that 1
< 2. If the wavelength
of the guided light is
smaller than 1, the
light will be confined in
simply results from
considering the balance
of incoming and
outgoing fields. Its aim
is to solve the problem
of scattering from a
system consisting of
multiple CHAPTER 3.
THE MULTIPOLE
METHOD:
FUNDAMENTALS 43 So
O r Si y x Figure 3.2:
Single inclusion in
their influence on the
mode should thus
remain small. Implicitly,
we adapted the truly
guided modes of a fibre
with infinite cladding to
their lossy nature
through letting take a
small imaginary part.
Mathematically, this
would amount to
letting the lossl
McIsaacs [76] eight
mode classes. The
losses in dB/m in
column 3 are obtained
from the imaginary part
of neff by L = 20 ln(10)
2 =(neff) 106 , (4.1)
with in m. In Table
4.1 the losses are large
since more than one
ring is necessary to
achieve losses
compa
largest argument of
Bessel functions on the
boundary of inclusions.
This guarantees that
the cylindrical functions
of largest order in field
expansions behave like
the cylindrical
multipoles of
electrostatics to leading
order, and ensures
rapid convergenc
origin in sources
outside that domain.
2Hankel functions H (1)
m satisfy the outgoing
wave condition and
diverge at 0; their
contribution to the field
in an annulus
surrounding an
inclusion is therefore
associated with fields
originating in sources in
or
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