Bound rays of fibers

2-2 Construction of ray paths 2-3 Ray invariants 2-4 Ray-path parameters 2-5 Ray transit time 

2-6 Construction of ray paths and ray invariants 2-7 Classification of rays 2-8 Ray-path parameters 2-9 Ray transit time 

2-10 Example Parabolic profile 2-11 Example Clad power-law profiles 

Most of the chapter is devoted to the construction of ray paths and their classification on circular fibers with axisymmetric profiles. However, we also consider noncircular fibers since cross-sections can differ from circular symmetry in practice, e.g. elliptical fibers. Finally, since this chapter parallels Chapter 1 to a large extent, it may be helpful to compare the results of corresponding sections. 

An optical fiber is illustrated in Fig. 2-1. Unless otherwise stated, the core is assumed to have a circularly symmetric cross-section of radius p, surrounded by the cladding, which, for simplicity, is assumed unbounded. The core-cladding interface is the cylindrical surface r = p. Over the core, the axisymmetric refractive-index profile n r) is either uniform or graded, and it takes the uniform value Hd in the cladding. 

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Consider meridional rays incident at angle 0q over the taper cross-section at z = —L. All these rays will become bound rays of the fiber if the extreme ray incident on the taper interface at P in Fig. S-6(a) becomes bound. Given po> P profile, the... 

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.

Fig. 2-8 Rectangular turning-point caustics for fibers with the separable profiles of Eq. (2-58), showing (a) a bound-ray caustic and (b) a tunneling-refracting ray caustic [5].

Initially we assume that the waveguide is composed of materials which are nondispersive. We subsequently determine the modification due to material dispersion. Only bound rays are included in the present chapter, as they characterize the transmission properties of long fibers. The major conclusion of this chapter is that pulse spreading on fibers is minimized if the profile has an approximately parabolic profile. 

Throughout the chapter, we assume that the pulse originates at time t = 0 at the endface z = 0 of a waveguide of arbitrary profile and length z as shown in Fig. 3-1. The pulse is assumed to be composed of bound rays only, all of which are excited simultaneously at z = 0. For convenience, the initial pulse spread is assumed to be zero. In practice, sources excite pulses of finite duration the spread along the fiber is then found by superposition. 

The source is located in a uniform medium of refractive index Mq is assumed to be sufficiently large that it fully illuminates at least the core cross-section of the fiber endface, as is normally the case in practice. A ray from the source is incident on the endface z = 0 at Q in Fig. 4-4, and makes angle 0q with the normal QN, or axial, direction. The polar coordinates of Q on the endface are (r, (f>) relative to the x-axis. We consider only rays incident over the core a ray incident over the cladding cannot become a bound ray in the core. 

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. 

The amount of source power carried by bound rays, Pb, is found by integrating Eq. (4-8) over the complete ranges of values of , r and 0 given below Eq. (4-8), and the range of values of 6q corresponding to bound rays within the fiber, i.e. 0 < where 0nj(r) was derived in Section 4-3. Hence... 

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... 

Thus both Pbr and are smaller than the corresponding expressions for the step profile by a factor of q (q + 2). In particular, the clad parabolic-profile fiber (q = 2) accepts only half as much bound-ray power as the step-profile fiber (q = oo). 

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. 

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... 

When the fiber has a graded profile which decreases monotonically from the axis to the interface, the maximum angle of incidence for exciting bound rays, i.e. (r) of Eq. (4—6),... 

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... 

Consider a source which emits a collimated beam of radius r. The beam can be focused onto the endface of the fiber at Q on the axis by introducing a thin lens of radius equal to the beam radius and focal length/, as shown in Fig. 4-6(a). All of the light from the source excites only bound rays provided the angle 0 subtended by the lens at Q does not exceed the maximum angle of incidence 0 i(O) defined by Eq. (4-6). If for convenience we assume a step profile and set n(0) = /ijo, then 0m(0) = sin ( c<>/ o) c - setting = /tan in the... 

With a view to optimizing bound-ray power, the focal length of the lens, and thus the separation of the source and the fiber, is chosen so that the angle 6 ... 

Under these conditions, maximum bound-ray power results when satisfies Eq. (4-33), i.e. 9 S rtcolf o) c> nd all the light emitted from the source falls on the core endface when f9 p, where 9 is the maximum angle of emission relative to the fiber axis. By combining these two relations, all source power excites bound modes provided (P/ d)( co/ o)9c- Thus the maximum bound-ray power increases by a factor of [rjpy- compared to placing the source directly against the end of the fiber. 

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]... 

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... 

We showed in Section 2-7 that bound rays on clad fibers have values of the invariant which lie in the range of Eq. (2-24a). The corresponding range of values ofTsatisfies 0 where the upper limit is defined by Eq. (4-44). For the step profile... 

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... 

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