Page 88 - Fiber Optic Communications Fund

Optical Fiber Transmission 69

2
2 1∕2
For example, if = 1.55 μm, a = 4 μm, and (n − n ) = 0.1, V = 1.62. Therefore, this fiber is
1 2
single-moded at this wavelength. However, if = 0.7 μm corresponding to optical communication in
the visible spectrum, V becomes 3.59 and the fiber is not single-moded at this wavelength. For the given
fiber parameters, the cutoff wavelength is defined as
2 1∕2
2
2a(n − n )
1
2
=
c
2.4048
2aNA
= . (2.169)
2.4048
If the operating wavelength is less than , the fiber will not be single-moded. For fibers with arbitrary index
c
profiles, the Helmholtz equation (2.28) should be solved numerically to find the propagation constants as
n
a function of frequency, from which the conditions for the cutoff of higher-order modes can be established.
Example 2.7
The cutoff wavelength for a step-index fiber is 1.1 μm. The core index n = 1.45 and Δ= 0.005. Find the
1
core radius. Is this fiber single-moded at 1.55 μm?
Solution:
From Eq. (2.8), we find
n = n (1 −Δ) = 1.4428. (2.170)
2 1
Using Eq. (2.169), we find
2.4048 c
a = = 2.907 μm. (2.171)
2
2 1∕2
2(n − n )
1 2
Since the operating wavelength = 1.55 μm > , it is single-moded at this wavelength.
c

2.7.2 Fiber Loss

Before the advent of optical amplifiers, the maximum transmission distance of a fiber-optic system was deter-
mined by the fiber loss, as the optical receivers need a certain amount of optical power to detect the transmitted
signal reliably. Now the optical amplifiers are widely used and yet the maximum reach is affected by the fiber
loss. This is because the optical amplifiers add noise whose power spectral density is proportional to the
amplifier gain, which in turn is proportional to the fiber loss (see Chapter 6). In other words, the amount of
noise in a long-haul communication system is directly related to fiber loss. In addition, if the fiber loss is
small, the amplifier spacing can be increased, which reduces the system cost. So, it is important to design a
fiber with the lowest possible loss.
Let us consider a CW input to the fiber. The optical field distribution is given by Eq. (2.95),
(x, y, z, t)= Φ(x, y,) A() exp (−z∕2)exp [−i(t − ()z)]. (2.172)
r
⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
transverse distribution field envelope optical carrier
The optical power is given by Eq. (2.123),
2
P(z)= |A() exp (−z∕2)| . (2.173)