Tunneling-refracting rays

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].

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.

There is also a penalty [2] for achieving reduced pulse spread, since some bound-ray power is inevitably lost to radiation. This occurs because the change in jS(z) over a ray half-period can be sufficient to convert a bound ray into a tunneling or refracting ray-discussed in Section 2-7. The radiation loss is reduced when the nonuniformities result in predominantly forward-direction changes in ray direction, i.e. to higher values of P z). 

The transmission coefficients of Eqs. (7-15) and (7-19) apply to all tunneling rays on graded- and step-profile fibers, except in the limit where a tunneling ray becomes a refracting ray, i.e. when - p. In this limit, the value of the transmission coefficient is relatively large compared to its value for tunneling rays close to the bound-ray limit, i.e. when - oo. Consequently, for certain... 

For situations where the above assumption cannot be adopted, expressions for the transmission coefficient in the transition region between tunneling and refracting rays are available [4,8]. The values of Tare plotted as curve (i) in Fig. 7-2(b) for a skew leaky ray with I = 0.033 on a clad parabolic fiber. To the left of the vertical dashed line, the curve corresponds to tunneling rays and coincides with the local plane-wave expression of Eq. (7-18) as increases [8]. Similarly, to the r ght of the vertical dashed line, the curve corresponds to refracting rays Ind coincides with the local plane-wave expression of Eq. (7-6) as decreases. A similar transition occurs for skew leaky rays on a step-profile fiber [16]. 

The diffuse source of Fig. 4-3 (a) illuminates the endface of a step-profile fiber in Fig. 4-4. This source excites all tunneling and refracting rays, as well as bound rays. In order to determine the power entering the tunneling rays, we must first determine the distribution function. 

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... 

Fig. 8-9 Intensity pattern over the endface of a step-profile fiber, showing (a) a bright circle due to bound rays and (b) a dark band due to refracting rays, and lighter regions due to tunneling rays.

Fig. 8-10 Intensity patterns for the clad power-law profiles as a function of the exponent q and the angle of observation 6q. Black areas denote refracting rays, hatched areas tunneling rays and the white areas within the dashed circles bound rays.

Since 7 of Eq. (9-4) is finite, must bP finite and, therefore, every ray on the bend in leaky. Tunneling rays have > R + p and refracting rays originate on the interface, i.e. = R + p. On a slight bend virtually all rays are tunneling rays. 

The delineation between tunneling and refracting rays on the bend is found by substituting Eq. (9-9) into Eq. (9-5), whence... 

We assume that all power is lost from a refracting ray when it reaches the interface, i.e. T = 1. Tunneling rays lose power only at the turning-point caustic because the inner caustic is convex to the ray path. It is sufficiently accurate to use the linear approximation to the transmission coefficient given by Eq. (7-17), provided we set P = 0,1 = lb and replace p by / + p. On substituting for the profile from Eq. (9-14), we obtain... 

The delineation between tunneling and refracting modes follows from the corresponding delineation between tunneling and refracting rays in Eq. (2-6), i.e. when a =... 

So far in our ray description of light propagation we have concentrated mainly on bound rays, which convey power without loss along nonabsorbing optical waveguides, and have assumed that refracting and tunneling rays are associated with a loss of power by radiation. In this chapter we describe the ... 

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