Page 518 - Engineering Electromagnetics, 8th Edition

500 ENGINEERING ELECTROMAGNETICS

1
0.8
0.6
0.4
0.2
0
2 4 6 8 10 12 14
–0.2
–0.4
(a)
4

3

2

1

0
0 1 2 3
(b)

Figure 13.22 (a)Ordinary Bessel
functions of the first kind, of orders 0 and 1,
and of argument β t ρ, where β t is real. (b)
Modified Bessel functions of the second kind,
of orders 0 and 1, and of argument |β t |ρ,
where β t is imaginary.

The Bessel K functions provide this behavior and will apply if β t2 is imaginary.
2 2 1/2
2
Requiring this, we may therefore write |β t2 |= (β − n k ) . The diminishing field
2 0
amplitude with increasing radius within the cladding allows us to neglect the effect
of the outer cladding boundary (at ρ = b), as fields there are presumed too weak for
this boundary to have any effect on the mode field.
−1
Because β t1 and β t2 are in units of m ,itis convenient to normalize these
quantities (while making them dimensionless) by multiplying both by the core radius,
a. Our new normalized parameters become

2 2
u ≡ aβ t1 = a n k − β 2 (150a)
1 0

2
2 2
w ≡ a|β t2 |= a β − n k (150b)
2 0