Biomedical Engineering Reference
In-Depth Information
E
z
¼ AJ
y
ð
k
rÞe
j
yf
H
r
¼
j
k
2
b
vH
z
vr
u3
1
vE
z
v
f
(6.2.11)
(6.2.5)
r
H
z
¼ BJ
y
ð
k
rÞe
j
yf
(6.2.12)
H
f
¼
j
k
2
b
1
r
vH
z
v
f
þ
u3
vE
z
We require that the field in the cladding of the fiber
decay in the
r
direction and be of the form
e
g
r
where g
is the decay constant of the evanescent field, given by
g
2
¼
b
2
n
2
k
0
;
with
n
2
being the index of refraction of
the cladding given by
ð
3
r
2
Þ
1
=
2
.
If we define k
¼ j
g we can choose a modified
Hankel function of the first kind to describe the decaying
behavior of the field in the cladding for large
r.
That is, for
r > a,
(6.2.6)
vr
Because Eqs. (
6.2.1
) and (
6.2.2
) have the same mathe-
matical form, we will solve Eq. (
6.2.1
), understanding
that solutions obtained for it will be valid for Eq. (
6.2.2
).
To obtain Eq. (
6.2.1
) we have already assumed an optical
system with cylindrical symmetry. The longitudinal di-
rection of propagation is the
z
axis and the dependence of
the fields is of the form
e
jð
u
t
b
zÞ
:
The technique of separation of variables will now be
applied to obtain a solution of Eq. (6.2.1). We will
assume that we can obtain independent solutions for
E
z
in f and
r,
that is,
E
z
¼ CH
ð
1
Þ
ðj
g
rÞe
j
yf
(6.2.13)
y
H
z
¼ DH
ð
1
Þ
ðj
g
rÞe
j
yf
(6.2.14)
y
where
A, B,
C, and
D are
unknown constants to be
determined.
Toobtainthetransversefieldsinthecoreandcladding
of the guide, one must use Eqs. (
6.2.3
)through(
6.2.6
).
For example, to obtain
E
r
in Eq. (
6.2.3
)forboth
r < a
and
r > a
one must differentiate the longitudinal fields (i.e.,
E
z
)
of Eqs. (
6.2.11
)and(
6.2.13
), respectively, with respect
to
r
and f and then substitute the results back into
Eq. (
6.2.3
). After some simplification we obtain, for
r < a,
E
z
ð
f
; rÞ¼AFð
f
ÞFðrÞ
(6.2.7)
Since the fiber has circular symmetry, we will choose
a circular function as a trial solution for
F
(f).
Fð
f
Þ¼e
j
yf
(6.2.8)
where
v
is a positive or negative integer. Now we have
E
z
AFðrÞe
j
yf
e
j
yf
(6.2.9)
E
r
¼
j
A
bk
J
0
y
ð
k
rÞe
j
yf
þ Bðj
y
Þð
um
Þ
1
r
J
y
ð
k
rÞe
j
yf
k
2
Taking the second-order derivatives of Eq. (
6.2.9
) with
respect to
r
and f and substituting back into Eq. (
6.2.1
),
we obtain
(6.2.15)
For
r > a
we have
FðrÞ¼
0
d
2
FðrÞ
dr
2
þ
k
2
y
2
r
2
þ
1
r
dFðrÞ
dr
h
i
e
j
yf
ðj
g
rÞþ
um
0
y
(6.2.10)
E
r
¼
1
g
2
bgCH
ð
1
Þ0
r
DH
ð
1
Þ
ðj
g
rÞ
y
y
(6.2.16)
Equation (6.2.10) is a form of Bessel's equation where
k is defined as the wave number and given by the ex-
pression k
2
¼ k
2
b
2
; k
being the complex propagation
constant equal to u
2
m3. This well-known second-order
differential equation has two independent solutions.
Numerous cylinder functions satisfy Bessel's equation.
Energy considerations will dictate the choice of the
functions selected as solutions of Eq. (
6.2.10
); that is,
1.
The field must be finite in the core of the fiber.
Specifically the cylinder function chosen in the core
of the fiber must be finite at
r ¼
0.
2.
The field in the cladding of the fiber must have an
exponentially decaying behavior at large distances
from the center of the fiber.
Because the fields must be finite at the center of the
fiber core, we will choose
J
y
(k
r
) as the form of the so-
lution for
r < a.
Therefore, for
r < a
,
where primed terms (i.e., terms with
0
) mean the first
derivatives with respect to k
r.
Furthermore, for the core
region (
r < a
)
k
2
¼ k
1
b
2
(6.2.17)
k
1
¼
u
2
m
0
3
1
(6.2.18)
and for the cladding region (
r > a
)
g
2
¼
b
2
k
2
(6.2.19)
k
2
¼
u
2
m
0
3
2
(6.2.20)
In a similar manner using Eqs. (
6.2.4
) through (
6.2.6
), we
can obtain for the core region (
r > a
)
