While discussing step-index fibers, we considered light propagation inside the fiber as a set of many rays bouncing back and forth at the core-cladding interface (see Figure 7-7). There the angle q could take a continuum of values lying between 0 and cos^–1(n₂/n₁), i.e.,

0 < q < cos^–1 (n₂/n₁)

For n₂ = 1.5 and D » = 0.01, we would get n₂/n ₁ ^~ and cos ^–1  = 8.1°, so

0 < q < 8.1°

Now, when the core radius (or the quantity D) becomes very small, ray optics does not remain valid and one has to use the more accurate wave theory based on Maxwell's equations. This wave theory is fairly involved [see, e.g., Chapters 7 and 8 of Ghatak and Thyagarajan]. Here we just give some of the important results for a step-index fiber.

In wave theory, one introduces the parameter

(7-28)

where D has been defined earlier (see Equation 7-5) and n₁  ^~  n₂ . The quantity V is often referred to as the "V-number" or the "waveguide parameter" of the fiber. It can be shown that, if

V < 2.4045

only one guided mode (as if there is only one discrete value of q) is possible and the fiber is known as a single-mode fiber. Further, for a step-index single-mode fiber, the corresponding (discrete) value of q is approximately given by the following empirical formula

(7-29)

We may mention here that because of practical considerations the value D of ranges from about 0.002 to about 0.008.

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Example 7-12

Consider a step-index fiber (operating at 1300 nm) with n₂ = 1.447, D = 0.003, and a = 4.2 mm. Thus,



Thus the fiber will be single moded and the corresponding value of q—using Equation 7-29—will be about q = 3.1º. It may be mentioned that for the given fiber we may write



Thus, for

l₀ > 2.958/2.4045 = 1.23 mm

which guarantees that V < 2.4045, the fiber will be single moded. The wavelength for which V = 2.4045 is known as the cutoff wavelength and is denoted by l_c. In this example, l_c = 1.23 mm and the fiber will be single moded for l₀ > 1.23 mm.

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Example 7-13

For reasons that will be discussed later, the fibers used in current optical communication systems (operating at 1.55 mm) have a small value of core radius and a large value of D. A typical fiber (operating at l₀ » 1.55 mm) has n₂ = 1.444, D = 0.0075, and a = 2.3 mm. Thus, at l₀ = 1.55 mm, the V-number is,



The fiber will be single moded (at 1.55 mm) with q = 5.9°. Further, for the given fiber we may write



and therefore the cutoff wavelength will be l_c = 2.556/2.4045 = 1.06 mm.

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