Modal dispersion occurs in multimode fibers that have a larger core diameter or a larger index difference between the core and the cladding than single-mode fibers. Typical core diameters range from 50 to 1 000 |im for multimode fibers and 5 to 10 |im for single-mode fibers. In the ray optics model, different optical modes correspond to light rays propagating at different angles in the core of the waveguide. The derivation of propagation angles in multimode fibers would go beyond the scope of this chapter. Here an approximate calculation will be performed to obtain the modal dispersion.

Consider a fiber waveguide with refractive indices of the core and cladding of n1 and n2, respectively. Assume that the waveguide supports the propagation of more than one optical mode. Two of these modes are shown schematically in Fig. 22.4. Owing to the difference in optical path length, the mode with the smaller propagation angle 9 will arrive earlier at the end of the multimode fiber. The modal dispersion is the time delay between the fastest and the slowest optical mode normalized to the length L of the waveguide.

In the calculation, assume that the phase and group velocity are given by vph = c / n1 ~ vgr. The fastest mode has the smallest propagation angle and we approximate the smallest angle by 9m = 0 ~ 0°. The slowest mode has the largest propagation angle and we approximate the largest angle by 9m « 9c, where 9c is the critical angle of total internal reflection. This approximation can be made without loss of accuracy for multimode fibers which carry many modes.

The propagation times for the fastest and slowest modes per unit length of the fiber are given

by

т _ L T _ L/cos 9c (22 1Л

Tfast _ Tslow _ 73 (221)

c / ni c / ni

where the critical angle for total internal reflection can be derived from Snell’s law and is given by

9c = arccos (n2 / n) . (22.2)

The time delay per unit length, or modal dispersion, is then given by

Ax = Tslow Tfast = n1 С 1 _

cos 9c j

A waveguide supporting many modes has a large time delay between the fastest and slowest modes. Thus modal dispersion increases with the number of optical modes supported by the waveguide.

Exercise: Modal dispersion in waveguides. Calculate the time delay between the slowest and the fastest modes, and the maximum possible bit rate for a 1 km long multimode fiber waveguide with core refractive index n 1 = 1.45 and cladding refractive index n 2 = 1.4.

Solution: Using Snell’s law (Eq. 22.2), one obtains 9c « 15°. The time delay calculated from Eq. (22.3) amounts to At = 170 ns. The minimum time required to transmit one bit of information is given by At. This yields an approximate maximum bit rate of fmax = 1/170 ns = 5.8 Mbit/s. The calculation shows that modal dispersion can be a significant limitation in optical communication. Graded-index multimode fibers or single-mode fibers are therefore required for high-speed communication systems.