Bound rays power distribution

We can now calculate the source power carried by bound rays when the fiber is illuminated by the diffuse source. In this part of the chapter we determine the total source power, the total bound-ray power and the radial distribution of bound-ray power over the core cross-section. Later in the chapter we show how to derive the distribution of power among the various bound-ray directions. We assume that the source of Fig. 4-3(a) is placed against the fiber endface in Fig. 4—4, and its surface covers at least the core cross-section. Only the portion of the source within the core cross-section can excite bound rays, so we ignore any effects due to the source outside of this region. The excitation of leaky rays by sources is examined in Chapter 8. 

So far we have assumed a uniform intensity /q over the source. Here we examine the effect of a nonuniform diffuse source on total bound-ray power, using the Gaussian distribution of Eq. (4-3) for intensity [6]. The total power emitted by the source is found by substituting Eq. (4-3) into Eq. (4-9b) with Ps = p, and gives... 

Here we determine the intensity distribution on the core endface for the situation in Fig. 4-8, which enables us to verify the accurai y of the approximate expression for total bound-ray power derived in the previous section. We replace the diffuse source by an equivalent axially symmetric superposition of collimated beams, whose directions of propagation are defined by spherical polar angles 9 and relative to the axial and azimuthal directions, respectively, where 0 < 0 < Jt/2 and 0 < 0 < 2ji. We deduce from Eq. (4-8) that the power density Pj per unit cross-sectional area of beams with directions in the ranges 6 to 6 +dd and 0 to < + d< is given by... 

In Section 4—6 we showed how to determine the total bound-ray power and the radial distribution of bound-ray power within the core of the fiber when illuminated by a uniform diffuse source which abuts the endface. Here we determine the distribution of source power among the bound-ray directions. This distribution can be conveniently described in terms of the ray invariants by defining a distribution function F(fi, I) such that [6]... 

Thus the distribution function for bound-ray power is proportional to the ray half-period. We give examples in Section 4-19 below. 

It is useful to define a normalized distribution function F(j ,T) corresponding to unit total bound-ray power entering the fiber. This quantity is the ratio of F(, T) of Eq. (4-45a) to Fb, of Eq. (4-14), whence... 

The normalized distribution function G( ), corresponding to unit total bound-ray power, is found by analogy with Eq. (4-45b) to be... 

Distribution of bound-ray power 5-22 Example Diffusion equation 5-23 Example Rayleigh scattering... 

Starting from the redistribution of ray power due to a ray incident on an isolated scatterer, we derive an integral equation which governs the distribution of bound-ray power when many scatterers are present in the liber. For convenience we assume a step profile, but the derivation is readily extended to graded-profile fibers [11],... 

Fig. 2-7 Schematic distribution of rays on circular fibers according to the value of the invariants andTfor (a) the step-profile of Eq. (2-8) and (b) the clad power-law fibers of Eq. (2-43) [3]. Shading denotes tunneling rays (TR) and hatching denotes refracting rays (RR). Bound rays (BR) occupy the unshaded regions.

The transit time for an arbitrary profile depends in general on both ray invariants, i.e. t = t(P, 1). Thus a group of rays, each ray having different values of j and T, can all have the same transit time t. In Fig. 4-11 (a) these rays lie along the contour t (]5,7) = t in the jS-Fplane. Rays with common transit time t + dt lie along the neighboring contour t( J) = t+dt. It then follows that the total power arriving at the end of the fiber between times t and t + d is carried by those rays in the shaded area between the two contours, denoted by dA. If we recall the normalized distribution function 7) for bound rays introduced in... 

Chapter 4, describes the distribution of power within a pulse, and this is modified by absorption. If the fiber is weakly guiding and there is little variation in absorption over the core, then, regardless of the profile, all bound rays suffer approximately the same attenuation exp(—a oZ) in propagating distance z, as is clear from Eq. (6-10) when 0 <1 1. Consequently, pulse shape is not significantly affected by core absorption, although total pulse power is reduced by a factor of approximately exp(—a< oZ) [3]. 

The diffuse source of Fig. 4-3 (a) illuminates the endface of a step-profile fiber in Fig. 4-4. This source excites all tunneling and refracting rays, as well as bound rays. In order to determine the power entering the tunneling rays, we must first determine the distribution function. 

In Section 4-17 we introduced the distribution function F (, 1) of Eq. (4-39) to describe the power carried by bound rays, and showed that for diffuse illumination it is proportional to the ray half-period, as expressed by Eq. (4-45a). Thus F 0,T) depends only on the ray path within the core. Since tunneling rays differ from bound rays only in their behavior in the cladding, we deduce that the distribution function for tunneling rays has the same functional form. If we follow the procedure in Section 4-17, this result can be obtained formally, starting from Eq. (4-41) with the limits on and I replaced by those in Eq. (8-4b) below. Thus, for the step profile, we have from Eq. (4-48) that... 

Fig. 8-3 (a) Fraction of initial bound- and tunneling-ray power remaining on a step-profile fiber illuminated by a diffuse source, (b) Distribution of bound-ray (BR) and tunneling-ray (TR) power as a function of P/n where G(p) denotes the normalized function... 

We showed in Section 4-19 that the distribution function F ( , /) for both bound- and tunneling-ray power is given analytically for the clad parabolic profile fiber by Eq. (4-52), where... 

When light propagates along a fiber and impinges on nonuniformities due to imperfections in the fiber, some of its power is scattered, as shown schematically in Fig. 22-1 (a). Part of the scattered power is distributed into forward-and backward-propagating modes, while the remainder is radiated. For multimode fibers, the distribution of scattered power is best treated by the ray methods of Chapter 5. Here we are primarily interested in fibers that propagate only one or a few modes. We treat the nonuniformities of the perturbed fiber as induced current sources within the unperturbed fiber. The results of the previous chapter can then be used to describe excitation of bound modes and the radiation field [1-3]. 

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