Optical Communications, 2nd Edition

7.2 DISPERSION AND PULSE SHAPING IN FIBERS UNDERGOING DIFFUSION

The dispersion discussed in Section 7.1 was purely in terms of isolated ray lines and field modes. A more exact analysis must account for possible diffusion and mode regeneration as the field propagates. To develop this approach, the optical field in the fiber must be treated as undergoing diffusion in which propagation at one ray line can couple, or diffuse, into another ray line. This can be handled by considering the ray angle θ as a continuous variable and allowing diffusion over this angle. This inherently implies a multimode condition—a large number of modes exist so that all angles between θ = 0 (central ray angle) and θ_p (maximum ray angle) are occupied.

The study of fiber power flow based on diffusion theory has been well developed [1]. The analysis begins with the basic diffusion equation in cylindrical coordinates that describes the ray line power in the fiber field P(t, z, θ) at time t, distance z down the fiber, in ray line direction θ. The total power in the fiber at time t and position z is obtained by a circular integration over all propagation angles θ. Hence we denote

Thus Q(t, z) defines the total integrated power that can be collected over the fiber cross-sectional area at distance z and time t. If the initial source power injected into the fiber was an impulse in time, then Q(t, z) is the impulse ...