Page 58 - Fiber Optic Communications Fund

Optical Fiber Transmission 39

Solution:
(a) n = 1.47, n = 1.45. From Eq. (2.7), we find
2
1
2 1∕2
2
NA =(n − n ) = 0.2417. (2.10)
1 2
(b) From Eq. (2.6), the acceptance angle is
−1
i max = sin (NA)= 0.2441 rad. (2.11)
(c) From Eq. (2.8), the refractive index difference Δ is
n − n 2
1
Δ= = 0.0136. (2.12)
n
1
2.3.2 Multi-Mode and Single-Mode Fibers

If the index difference (n − n ) is large or the core radius a is much larger than the wavelength of light, an
2
1
optical fiber supports multiple guided modes. A guided mode can be imagined as a ray that undergoes total
internal reflection. A mathematical description of guided modes is provided in Section 2.4. From ray-optics
theory, it follows that total internal reflection occurs for any angle in the interval [ ,∕2].Thisimpliesan
c
infinite number of guided modes. However, from the wave-optics theory, it follows that not all the angles in the
interval [ ,∕2] are permitted. Light guidance occurs only at discrete angles { , , ··· } in the interval
c 1 2
[ ,∕2], as shown in Fig. 2.6. Each discrete angle in Fig. 2.6 corresponds to a guided mode. Typically, a
c
multi-mode fiber can support thousands of guided modes. As the index difference (n − n ) becomes very
1 2
large and/or the core diameter becomes much larger than the wavelength of light, the fiber supports a very
large number of modes N which approaches infinity, and total internal reflection occurs for nearly any angle in
the interval [ ,∕2]. In this case, the ray-optics theory is valid. As the index difference (n − n ) becomes
c
1
2
smaller and/or the core diameter becomes comparable with the wavelength of light, the number of guided
modes decreases. In fact, by the proper design, a fiber could support only one guided mode (in ray-optics
language, one ray with a specific angle). Such a fiber is called a single-mode fiber, which is of significant
importance for high-speed optical communication.
2.3.3 Dispersion in Multi-Mode Fibers

A light pulse launched into a fiber broadens as it propagates down the fiber because of the different times
taken by different rays or modes to propagate through the fiber. This is known as intermodal dispersion.In
Fig. 2.7, the path length of ray 1 is longer than that of ray 3 and, therefore, the fraction of the incident pulse
carried by ray 3 arrives earlier than that by ray 1, leading to pulse broadening.

n 2

ø c
n 1 ø 1 ø 1 Core diameter = 2a
ø 2
ø 2

Figure 2.6 When the angle of incidence exceeds , total internal reflection occurs only for certain discrete angles.
c