18
A. Hu et al.
Fig. 1.11 Index profile of
dielectric optical nanofibers.
(Reprinted with permission
from [79]. Copyright 2012.
Elsevier Inc.)
∇
2
+ n
2 k
2
− β
2
E = 0,
∇
2
+ n
2 k
2
− β
2
H = 0
(1.2.14)
where k is the wave vector of the light in vacuum and β is the propagation constant.
Therefore the modes HE vm and EH vm can be expressed by the equations [79]:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(J
U ) v
(U JU) v +
K
v
W K v
⎧
⎨
⎩
(J U ) v
(U JU) v +
n 2
2 K
v
n 2
1 W Kv
=
νβ
kn 1
2 (
V
U W )
4
⎫
⎬
⎭
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(1.2.15)
The TE 0m modes is given by
J 1 (U )
U J 0 (U )
+
K 1 (W )
W K 0 (W )
= 0
(1.2.16)
The TM 0m modes is given by
n
2
1 J 1 (U )
U J 0 (U )
+
n
2
2 K 1 (W )
W K 0 (W )
= 0
(1.2.17)