Page 75 - Fiber Optic Communications Fund

56 Fiber Optic Communications

and J is the total number of modes. Suppose the output of a laser is monochromatic and it is used as fiber
input. The fiber input field can be written as
(x, y, z = 0, t)= f(x, y)e −it . (2.86)

Using Eq. (2.86) in Eq. (2.84), we obtain

J
∑
f(x, y)= A Φ (x, y). (2.87)
j
j
j=1
∗
To determine A , multiply Eq. (2.87) by Φ (x, y) and integrate over the cross-section to obtain
j
k
J
+∞ +∞ ∑ +∞ +∞
∗
∗
f(x, y)Φ (x, y)dxdy = A Φ (x, y)Φ (x, y)dxdy. (2.88)
∫ ∫ k j ∫ ∫ j k
−∞ −∞ j=1 −∞ −∞
Using the orthogonality relation,
+∞ +∞
∗
Φ (x, y)Φ (x, y)dxdy = , (2.89)
∫ ∫ j k jk
−∞ −∞
where is the Kronecker delta function defined as
jk
{
1, if j = k
= . (2.90)
jk
0, otherwise
Eq. (2.88) reduces to
+∞ +∞
∗
A = f(x, y)Φ (x, y)dxdy. (2.91)
k ∫ ∫ k
−∞ −∞
Thus, for the given input field distribution f(x, y), we can find the mode weight factors A using Eq. (2.91)
k
and the total field distribution at any distance z is given by Eq. (2.84). Suppose the output of the laser has
exactly the same transverse distribution as that of the fundamental mode of the step-index fiber, i.e., if f(x, y)=
Φ (x, y)= R (r), from Eq. (2.91), we find A = 1 and A = 0for m > 1. Therefore, from Eq. (2.84), the field
1 01 1 m
distribution at z is
(x, y, z, t)=Φ (x, y)e −i(t− 1 z) . (2.92)
1
Ideally speaking, the fundamental mode LP can be launched to the fiber, which propagates down the fiber
01
without any change in shape and thereby intermodal dispersion can be avoided. However, in practice, the fiber
imperfections and refractive index fluctuations due to temperature and stress can easily transfer power from
the LP mode to higher-order modes. Therefore, the safest way to avoid intermodal dispersion is by ensuring
01
that the fiber is single-moded at the operating wavelength.
Suppose a multi-mode fiber is excited with the Gaussian input
( )
2
x + y 2
f(x, y)= exp − .
2R 2
0
The mode weight factors A can be calculated using Eq. (2.91) and are shown in Fig. 2.24. As can be seen,
p
in this example, most of the power is carried by the LP 01 mode (p = 1).