Biomedical Engineering Reference
In-Depth Information
h
i e j yf
E f ¼ j
j b y
The coefficients A, B, C, and D can be written so that A is
the only unknown coefficient. For example, it can be
shown that
r AJ y ð k kum BJ 0 y ð k
(6.2.21)
k 2
h
i e j yf
H r ¼ j
k 2 j u3 1 y
r AJ y ð k rÞþ kb BJ 0 y ð k
J y ð k
H ð 1 Þ
C ¼
A
(6.2.28)
(6.2.22)
ðj g
y
h
i e j yf
H f ¼ j
k 2
ku3 1 AJ 0 y ð k rÞþj b y
J y ð k
H ð 1 Þ
r BJ y ð k
(6.2.23)
D ¼
B
(6.2.29)
ðj g
y
and for the cladding region ( r > a )
where A and B are related to each other via
h
i e j yf
g 2 b y
E f ¼ 1
r CH ð 1 Þ
ðj g gum 0 DH ð 1 Þ0
ðj g
u ð 3 1 3 2 Þ b J y ð k aÞH ð 1 y ðj g
kg a g J 0 y ð k aÞH ð 1 Þ
y
y
B ¼ j y
h
i A
ðj g aÞþj k J y ð k aÞH ð 1 Þ 0
(6.2.24)
ðj g
y
y
h
i e j yf
(6.2.30)
H r ¼ 1
g 2 u3 2 y
r CH ð 1 Þ
ðj g gb DH ð 1 Þ0
y
ðj g
y
In general, the permissible field configurations or
modes that exist in a step-index fiber have six field com-
ponents. For the round fiber, hybridmodes exist as well as
the transverse electric (TE) and transverse magnetic
(TM) modes. The hybrid modes are denoted HE and EH
modes and have both longitudinal electric and magnetic
field components present. In terms of a ray analogy for the
step-index fiber, the hybrid modes correspond to propa-
gating skew rays and the TE and TMmodes correspond to
propagating meridional rays [y ¼ 0 in Eq. ( 6.2.27 )]. For
meridional rays the right-hand side of Eq. (6.2.27) is equal
to zero and one obtains two characteristic equations that
define the TE and TM modes. These equations are
(6.2.25)
h
i e j yf
ðj g rÞþ b y
H f ¼ 1
g 2 gu3 2 CH ð 1 Þ0
r DH ð 1 Þ
ðj g
y
y
(6.2.26)
The constants A , B , C , D ,andb are determined by ap-
plying the boundary conditions for the two tangential
components of the electric and magnetic fields at the core-
cladding interface ( r ¼ a ). The boundary conditions for the
fields at the core-cladding interface can be written as
E z 1 ¼ E z 2
E f 1 ¼ E f 2
"
# ¼ 0
J 0 ð k þ j g a H ð 1 Þ0
J 0 0 ð k
a g 2
k
ðj g
for r ¼ a
0
(6.2.31)
H z 1 ¼ H z 2
H f 1 ¼ H f 2
H ð 1 Þ
0
ðj g
"
# ¼ 0
where subscripts 1 and 2 refer to the fields in the core
and cladding, respectively. Applying these conditions
yields four simultaneous equations for the unknown A , B,
C, and D. This solution yields a determinant. The solu-
tions for A, B, C, and D can be obtained from this de-
terminant provided that the system determinant for the
four equations vanishes. Expansion of this determinant
results in what is known as the ''eigenvalue'' or charac-
teristic equation of the waveguide. This equation defines
the modes in the guide and yields the permissible values
of b , k , and g associated with each mode. The resulting
characteristic equation for the step-index fiber is
ð k þ j g aH ð 1 Þ0
J 0 0 ð k
a g 2
k
3 1
3 2
ðj g
0
(6.2.32)
H ð 1 Þ
0
ðj g
An important parameter for each propagating mode is its
cutoff frequency. A mode is cut off when its field in the
cladding ceases to be evanescent and is detached from
the guide; that is, the field in the cladding does not decay.
The rate of decay of the fields in the cladding is de-
termined by the value of the constant g . For large values
of g , the fields are tightly concentrated inside and close
to the core. With decreasing values of g , the fields reach
farther out into the cladding. Finally, for g ¼ 0, the fields
detach themselves from the guide. The frequency at
which this happens is called the cutoff frequency. The
mode cutoff frequency can be calculated from Eqs.
( 6.2.17 ) through ( 6.2.20 ) for g ¼ 0 and is given by
"
#
J y ð k þ j g a H ð 1 Þ0
a g 2
k
J 0 y ð k
3 1
3 2
ðj g
y
H ð 1 Þ
ðj g
y
"
#
¼ y 3 1
3 2
1 b k 2
k 2
2
þ j g a H ð 1 Þ0
a g 2 J 0 y ð k
k J y ð k
ðj g
y
H ð 1 Þ
k c
m 0 ð 3 1 3 2 Þ
ðj g
y
u c ¼
p
(6.2.33)
(6.2.27)
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