Meridional rays

When the angle of incidence 6 is small enough, transmission losses for skew rays are equal in the first-order approximation to those for meridional rays with the same 6 [9]. Therefore, only meridional rays are considered in the present analysis. For the meridional ray, a power attenuation constant 2a (6) is calculated as... 

Figure 1. Meridional ray optic representation of the propagation mechanism in an ideal step-index optical waveguide. From Figure 2-11, G. Kelser [3], Optical Fiber Communications, McGraw-Hill Book Company, New York, NY (1983) with permission.

We should note that g/k(0) is of the order of 10 " to 10 . If m = 0, Eq. (38) is associated with the radially symmetric mode that corresponds to meridional rays. 

CPC critical angle, beyond which the throughput is zero for meridional rays... 

Fig. 5. Raytrace of an inverted compound parabolic concentrator (CPC), with 9 = 30°, for meridional ray incidence angles of 20, 40, 60, and 80°. Note the limited incidence angle on the detector surface for 80° incidence on the CPC.

Fig. 2-2 Ray paths within the core of a step-index fiber showing (a) the zig-zag path of a meridional ray and (b) the helical path of a skew ray, together with their projections onto the core cross-section.

It is convenient to distinguish between rays which cross the fiber axis between reflections known as meridional rays-and rays which never cross the fiber axis-known as skew rays. We see from Fig. 2-2(a) that meridional rays lie in a plane of width 2p through the axis. Consequently, they have properties identical with rays of the corresponding planar waveguide, and Table 1-1, page 19, applies to meridional rays of fibers, if the cartesian coordinate x is replaced by the cylindrical polar coordinate r of Fig. 2-1. Skew rays, on the other hand, follow a helical path, whose projection onto the cross-section is a regular polygon-not necessarily closed-as shown in Fig. 2-2(b). The midpoints between successive reflections all touch a cylindrical surface of radius rj, known as the inner caustic. 

We label meridional rays with the angle 0 between the path and the z-direction, as used in Section 1-2. Accordingly, the ranges of 0 for bound and refracting meridional rays are given by Eq. (1-5), where for the step profile of Eq. (2-2), the complement of the critical angle 0 retains the definition of Eq. (1-3). 

The general shape of the ray paths can be deduced from the paths of Fig. 1-8 for a graded-prolile planar waveguide and the paths of Fig. 2-2 for the step-profile fiber. Assuming that the paths are confined to the core, they have the characteristic forms shown in Fig. 2-4. Meridional rays cross the fiber axis... 

We emphasize that, in general, the transit times depend on both P and /,and reduce to the corresponding expressions for planar waveguides only in the case of meridional rays. However, transit times are independent of /, i.e. independent of skewness, for certain profiles, including the step and clad power-law profiles, as we show below. We also recall from Section 1-9 that graded profiles tend to equalize transit times compared to the step profile. Unlike planar waveguides, though, there is no known profile for which complete equalization of all ray transit times on a fiber is possible. 

It is clear from the discussion at the beginning of Section 3-2 that there is no ray dispersion on the planar waveguide with a hyperbolic secant profile. However, on the fiber with the same profile, ray dispersion is no longer zero. There is no known profile which has zero ray dispersion for both skew and meridional rays. 

The parameters for the linear taper of Fig. 5-6(a) are identical with those of the previous example and the taper angle is SI We can determine very simply the minimum taper length for total power transmission when a meridional ray enters the taper at z = —L and makes angle 0q with the axis. This result apphes to both direct on-axis illumination by a collimated beam, i.e. 0q = 0, and the inclusion of a lens, as discussed in Section 4-11, which transforms the on-axis beam into acone of meridional rays at angle d(, > 0. 

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... 

The approximate ray invariant for a slowly varying step-profile taper is expressed by Eq. (5-60). This relationship is accurate provided that the change 6p z) in taper radius over the local ray half-period is small. If the taper and fiber are weakly guiding then 0 (z) 1, and by generalizing the expression for Zp in Table 2-1, page 40, to slowly varying fibers, we deduce that Zp(z) = 2p(z)/0j(z) for meridional rays. In terms of the local taper angle Q(z), the slow-variation... 

We can quantify the accuracy of Eq. (5-60) for the linear taper of the previous section. At the minimum taper length of Eq. (5-59b) we deduce from Eq. (5-59a) that for meridional rays... 

This result is accurate for all bound rays, with the exception of those few rays whose directions are close to 0, = 0 and 0 <= n/2. For a fixed value of 0 attenuation is a maximum when the ray is meridional, i.e. 0 = n/2, and decreases with increasing skewness. Although a skew ray makes more reflections than a meridional ray over a given fiber length, this is more than offset by the smaller value of the transmission coefficient for skew rays. 

For meridional rays, 6 = nJ2, this reduces to Eq. (7-4) for the planar waveguide. A more rigorous analysis, which takes into account curvature of the interface, shows that the above expression is highly accurate for all refracting-ray directions except extremely close to the critical angle... 

By analogy with Eq. (9-8), the total power at any position around the bend is given by a fourth order integration of the power in each ray over the core cross-section and all leaky-ray directions. The evaluation of this integral is complicated by the fact that, unlike meridional rays, the attenuation coefficient for a skew ray varies along its trajectory. This means the complete trajectory must be determined in order to correctly specify the attenuation. 

Fig. 11-2 The electric field vector is orthogonal to the ray, or local plane-wave direction. On the step-profile fiber, the direction of e for (a) a meridional ray is parallel to a fixed direction, and for (b) a skew ray it changes direction at each reflection. On the parabolic-profile fiber the direction of e changes continuously along the skew-ray path (c).

When a(r) = njl, it follows from Eq. (36-5) that all rays touch the cylindrical surface of radius r- = pv/U in Fig. 36-1 (b). This is the inner caustic, introduced in Section 2-2. We deduce that only the TEo and TMo modes (v = 0) are composed of meridional rays, since = 0, and all HEv and EHv modes are composed of skew rays. We are reminded that all modes are composed of paraxial rays in the weak-guidance approximation. 

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