Skew rays

Schragstrahlen, m.pl. Physics) skew rays. Schragung, /. slope, slant, inclination, bevel, chamfer. 

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... 

Fig.4.48a-d. Trajectories of rays in a confocal FPI (a) incident beam parallel to the FPI axis (b) inclined incident beam (c) perspective view for illustrating the skew angle (d) projection of the skewed rays onto the mirror surfaces... 

Fig. 8.6. Ray diagram for both parallel and skew rays. The parallel rays are focused on the optical axis in the focal plane, while the skew rays are focused in the focal plane, but off the optical axis. The skew in the ray is a direct measure of the ion velocity perpendicular to the optical axis.

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. 

To specify the trajectory of a skew ray, it is clear from Fig. 2-2(b) that, in addition to the inclination 0 to the axial direction, we need a second angle to indicate the skewness. We define 0 to be the angle in the core cross-section between the tangent to the interface and the projection of the ray path, as shown in Fig. 2-2(b). By geometry 0 has the same value at every reflection. 

Pulse spreading in fibers is investigated in exactly the same manner as for planar waveguides. The added complication is that we must include skew rays... 

The collimated beam makes angle 0 with the fiber axis, as shown in Fig. 4-6(b), and the focal point Q is distance rg=f tan 0 along the x-axis, assuming the beam direction is parallel to the x-z plane. Both meridional and skew rays are excited in the core, and all of these rays are bound provided that the largest angle of incidence, corresponding to the ray PQ in Fig. 4-6(b), does not exceed of Eq. (4-6). If the fiber has a step profile, then 0m(> ) = sin ( co/"o) c > nd we can parallel the derivation of Eq. (4-28) to... 

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. 

The refracting-ray transmission coefficient for skew rays on graded-profile fibers is given by Eq. (7-6) within the local plane-wave approximation, except that 0, = njl — a, where a is the angle between the incident, reflected or transmitted rays and the normal at the interface. We deduce from Eqs. (2-14), (2-16) and (2-17) that... 

Fig. 4 Skew-ray path of varying half-period on a bent fiber.

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. 

---