In digital communication systems, information to be sent is first coded in the form of pulses and these pulses of light are then transmitted from the transmitter to the receiver, where the information is decoded. The larger the number of pulses that can be sent per unit time and still be resolvable at the receiver end, the larger will be the transmission capacity of the system. A pulse of light sent into a fiber broadens in time as it propagates through the fiber. This phenomenon is known as pulse dispersion, and it occurs primarily because of the following mechanisms:

1. Different rays take different times to propagate through a given length of the fiber. We will discuss this for a step-index multimode fiber and for a parabolic-index fiber in this and the following sections. In the language of wave optics, this is known as intermodal dispersion because it arises due to different modes traveling with different speeds.
   
   
2. Any given light source emits over a range of wavelengths, and, because of the intrinsic property of the material of the fiber, different wavelengths take different amounts of time to propagate along the same path. This is known as material dispersion and will be discussed in Section IX.
   
   
3. Apart from intermodal and material dispersions, there is yet another mechanism—referred to as waveguide dispersion and important only in single-mode fibers. We will briefly discuss this in Section XI.

In the fiber shown in Figure 7-7, the rays making larger angles with the axis (those shown as dotted rays) have to traverse a longer optical path length and therefore take a longer time to reach the output end. Consequently, the pulse broadens as it propagates through the fiber (see Figure 7-11). Even though two pulses may be well resolved at the input end, because of the broadening of the pulses they may not be so at the output end. Where the output pulses are not resolvable, no information can be retrieved. Thus, the smaller the pulse dispersion, the greater will be the information-carrying capacity of the system.

Figure 7-11 Pulses separated by 100 ns at the input end would be resolvable at the output end of 1 km of the fiber. The same pulses would not be resolvable at the output end of 2 km of the same fiber.

We will now derive an expression for the intermodal dispersion for a step-index fiber. Referring back to Figure 7-7b, for a ray making an angle q with the axis, the distance AB is traversed in time.

(7-13)

or

(7-14)

where c/n₁ represents the speed of light in a medium of refractive index n₁, c being the speed of light in free space. Since the ray path will repeat itself, the time taken by a ray to traverse a length L of the fiber would be

(7-15)

The above expression shows that the time taken by a ray is a function of the angle q made by the ray with the z-axis (fiber axis), which leads to pulse dispersion. If we assume that all rays lying between q = 0 and q = q_c = cos^–1(n₂/n₁) (see Equation 7-8) are present, the time taken by the following extreme rays for a fiber of length L would be given by

		corresponding to rays at q = 0	(7-16)
		corresponding to rays at q = qc = cos–1(n2/n1)	(7-17)

Hence, if all the input rays were excited simultaneously, the rays would occupy a time interval at the output end of duration

or, finally, the intermodal dispersion in a multimode SIF is

(7-18)

where D has been defined earlier [see Equations 7-5 and 7-6] and we have used Equation 7-11. The quantity t_i represents the pulse dispersion due to different rays taking different times in propagating through the fiber, which, in wave optics, is nothing but the intermodal dispersion and hence the subscript i. Note that the pulse dispersion is proportional to the square of NA. Thus, to have a smaller dispersion, one must have a smaller NA, which of course reduces the acceptance angle i_m and hence the light-gathering power. If, at the input end of the fiber, we have a pulse of width t₁, after propagating through a length L of the fiber the pulse will have a width t₂ given approximately by

(7-19)

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Example 7-5

For a typical (multimoded) step-index fiber, if we assume n₁ = 1.5, D = 0.01, L = 1 km, we would get

(7-20)

That is, a pulse traversing through the fiber of length 1 km will be broadened by 50 ns. Thus, two pulses separated by, say, 500 ns at the input end will be quite resolvable at the end of 1 km of the fiber. However, if consecutive pulses were separated by, say, 10 ns at the input end, they would be absolutely unresolvable at the output end. Hence, in a 1-Mbit/s fiber optic system, where we have one pulse every 10^–6 s, a 50-ns/km dispersion would require repeaters to be placed every 3 to 4 km. On the other hand, in a 1-Gbit/s fiber optic communication system, which requires the transmission of one pulse every 10^–9 s, a dispersion of 50 ns/km would result in intolerable broadening even within 50 meters or so. This would be highly inefficient and uneconomical from a system point of view.

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From the discussion in the above example it follows that, for a very-high-information-carrying system, it is necessary to reduce the pulse dispersion. Two alternative solutions exist—one involves the use of near-parabolic-index fibers and the other involves single-mode fibers. We look at these next.