Sources weakly guiding fibers

When a current source is located within a clad fiber of arbitrary profile, the determination of the radiation field is extremely complicated. However, if the fiber is weakly guiding the determination is greatly simplified [2]. It is intuitive that when the variation in the refractive-index profile is small, the source radiates as if it were located in a virtually uniform, infinite medium of refractive index equal to the cladding index n j. The problem is then analogous to the radiation from an antenna in free space. Consequently, we can borrow from standard antenna theory, and couch the solution to radiation from the weakly guiding fiber in terms of the electromagnetic vector potential A [3-5]. 

To determine radiation from sources within weakly-guiding fibers, we must solve Eq. (21-17) for the vector potential. However, if the free-space solution for the particular problem is known, we need only determine the modification due to the fiber profile. The free-space solution is the solution of Eq. (21-17) when n(x, y) = everywhere. The solution to Eq. (21—17)as (x, y) varies, can... 

We showed in Sections 21-8 and 21-11 that when the fiber profile is included the far-field radiation pattern due to sources within a weakly guiding fiber can be described by a correction factor to the free-space pattern. The correction factor is, in turn, expressible as a product of two factors, Cj (0) and (6), as we showed for the step-profile fiber in Section 21-13. By examining the definitions in Eqs. (21-38) and (21-36b), we find that (0) is inversely proportional to G([/) of Eq. (24-31), provided... 

Tubular sources within weakly guiding fibers... 

We are principally interested in determining only the radiation, or far field. As explained in Section 21-8, radiation from sources within weakly guiding fibers is nearly identical to radiation in free space , i.e. in an unbounded medium of uniform refractive... 

In the last five sections we have shown how a lens, on the one hand can increase source efficiency for collimated beam illumination of a fiber, while, on the other hand, it is ineffective at increasing the efficiency of the diffuse source. If we couple these facts with the description of the diffuse source as a superposition of collimated beams, it is evident that the efficiency of a partially diffuse source can be increased by a lens. The situation is illustrated in Fig. 4—8. We assume the step-profile fiber is weakly guiding so that 9 1, and the focal length satisfies... 

In the example above, we established the equivalence of the geometric optics and modal analysis for the total, guided power excited in a weakly guiding, step-profile fiber by a totally incoherent source when the fiber parameter F-> 00. To determine the error in the geometric optics analysis when V is... 

When the fiber is weakly guiding, the approximations of Chapter 13 can be used to simplify Eq. (21-2). If for convenience we assume a circular fiber with current sources parallel to the x-axis in the cross-section, then we deduce from Tables 13-2, page 292, and 14-1, p>age 304, and Eq. (21-3) that the power in each mode is given by... 

In the two examples above, we determined the power radiated from current sources within a fiber by ignoring the variation in profile and assuming an unbounded medium of uniform refractive index n i. Now we determine the correction to the free-space result due to the variation in profile [2], As the fiber is assumed to be weakly guiding, it is intuitive that the correction is small except when the radiation is directed predominantly close to the axis, i.e. 00 = 0. Thus we anticipate that the free-space results are, in general, highly accurate. 

Consider a weakly guiding, step-profile fiber which contains a sinusoidal line source of length 2L on its axis, directed parallel with the x-axis in the fiber cross-section. The magnitude of the distribution is assumed to be given by the tubular source of Eqs. (21-13) and (21-14) with / = 0 and Tq 0. Hence... 

We now examine how radiation from the point dipole and the tubular source is modified by the presence of a fiber. In order to relate the results to the correction factor of Section 21-12, the fiber is assumed to be weakly guiding. 

A tubular source of radius ro is located symmetrically within the core of a weakly guiding, step-profile fiber, i.e. 0 < ro < p, where p is the core radius. To account for the fiber profile, we repeat the analysis of Section 25-13 using the weakly guiding radiation modes of Table 25-4 instead of the free-space modes of Table 25-2. The modal amplitudes of Eq. (25-34a) are replaced by... 

Here we derive the correction factor for a tubular source in the core of a weakly guiding, step-profile fiber. The solution of Eq. (34-25) is expressible as... 

---