Bound rays total power

Since the power of a bound ray is totally reflected back into the core at every reflection, the ray can propagate indefinitely without any loss of power. A... 

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. 

Only part of the total source power is transmitted to bound rays the rest excites leaky rays, which is discussed in Chapter 8. If we denote the amount of source power transmitted to all bound rays by Pj, then the source (Efficiency S is defined by... 

An immediate consequence of this definition is that fibers with equal profile volumes carry identical total bound-ray power when illuminated by a diffuse source. It also follows from Eq. (4-13) that the bound-ray power density per unit area of cross-section at radius r is proportional to the profile shape. If P r) denotes this quantity, then... 

The total bound-ray power excited on a step-profile fiber is obtained by setting v r) = /ijo i i Eq. (4-13). Hence... 

If we substitute the clad power-law profiles of Table 2-1, page 40, for n(r) in Eq. (4-13), the total bound-ray power due to diffuse illumination is readily shown to be [6]... 

The total bound-ray power on each fiber with the modified clad power-law profile is given by Eq. (4-18) with p replaced by p. If we then substitute the above result for p, we verify a result of Section 4-, namely that the modified clad power-law profile fibers carry the same bound-ray power as the step-profile fiber when illuminated by a diffuse source. 

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... 

When the fiber has a step profile, the total bound-ray power is given by Eq. (4-12) with /q replaced by A and a factor exp (—or /p ) introduced into the integrand of the radial integration. On evaluating the integrals we obtain... 

This profile is given by the q = 2 clad parabolic-profile in Table 2-1, page 40. The total bound-ray power and source efficiency are obtained in an analogous manner to the step-profile expressions above. Hence... 

Outside of the circular region only leaky rays are excited, as discussed in Chapter 8. When 00 = 0 and the beam is on axis, bound rays are excited over the whole core and propagate parallel to the fiber axis. If the beam carries uniform power P per unit cross-sectional area, then the total bound-ray power and source efficiency of Eq. (4-11) are given by... 

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]... 

Our starting point is the expression for total bound-ray power of a uniform diffuse source given by Eq. (4—12). We perform the (j) integration and transform... 

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... 

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... 

For convenience we assume excitation by a diffuse source. The total initial bound ray power, is then given by Eq. (4-13) with p and n(r) replaced by p(0) and n (r, 0). We define to be the total power of rays which remain bound... 

To determine P), we note that total bound-ray power due to a diffuse source is given by Eq. (4-18) and, like the step-profile above, is proportional to the square of core radius. Consequently, the maximum fraction of initial bound-ray power that can be radiated is given by Eq. (5-55) and is independent of the exponent q, i.e. independent of profile shape. 

Accordingly the relationship between the total bound-ray power entering the... 

Since total bound-ray power is proportional to the core area in Eq. (4-16), and p /po is the ratio of the core and source cross-sections, it is clear that there is no advantage in the arrangement in Fig. 5-5(a) over placing the source directly against the fiber. Thus, when the source excites rays over a range of directions greater than that for bound rays within the fiber, it is not possible to increase the bound-ray power by use of a taper. This is the identical conclusion reached in Section 4-14 when a lens is used to concentrate diffuse-source power. 

The total scattering cross-section is by definition independent of 0 and 0 and we have assumed independent of position for simplicity. Second, there is a gain from power scattered into direction 0, 0. Ignoring contributions from leaky rays, the total contribution is from bound rays, whose range of 0 values satisfies Eq. (2-6a). Thus the power gained, dPg, is found by integrating Eq. (5-64) over the cross-section... 

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 total tunneling ray power excited on the endface, P,r(0), is found by integrating Eq. (8-3) over the range 0 Hj]. This is facilitated by integrating the second term within the curly brackets by parts in order to remove the inverse sine function, and then amalgamating with the integral of the third term. The substitution = M ., (1 — reduces the combined integral to the form of Eq. (37-116). If we normalize with the total bound-ray power of Eq. (4-16) then... 

The extent of the spatial transient depends on the variation of total ray power along the fiber. We ignore refracting rays, for reasons given above, and define P (z) to be the sum of bound and tunneling ray power at distance z. Bound-ray power is conserved along a nonabsorbing fiber, consequently... 

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