Leaky rays tunneling

The range of values of invariants for leaky rays is given by Eq. (2-8). However, if we recall the ranges of ray angles for bound, refracting and tunneling rays in... 

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]. 

Snyder, A. W. and Love, J. D. (1976) Attenuation coefficient for tunnelling leaky rays in graded fibers. Electron. Lett., 12, 324-6. 

The disp>ersion due to bound rays on step-profile fibers is given by Eq. (3-3) in the weak-guidance approximation. If we include leaky rays, then only those tunneling rays with effectively zero attenuation are included. Since transit time is independent of skewness, i.e. independent of /, this is equivalent to reducing the lower limit on from to defined by Eq. (8-24b). Thus the difference in transit times between the fastest, on-axis bound ray (p = n ) and the slowest tunneling ray (jS = follows from Table 2-1, page 40, as [7]... 

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. 

In Section 24-18, we derived the power attenuation coefficient for tunneling leaky modes on a. step-profile, weakly guiding fiber. Here we show that, for higher-order modes, Eq. (24-36) is equivalent to the power attenuation coefficient of the corresponding skew tunneling rays. The argument of the Hankel functions in Eq. (24-36) is smaller than the order. Furthermore, we assume that / is sufficiently large that the order of both Hankel functions may be taken to be approximately /. Under these conditions, we can use the approximate forms of Eq. (37-90), and for simplicity we approximate x by the middle expression in Eq. (37-90b). Hence... 

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