Leaky rays refracting

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]. 

Love, J. D. and Winkler, C. (1978) Refracting leaky rays in graded-index fibers. Appl. Opt., 17, 2205-8. 

Fig. 3 Non-TIR-based waveguides, (a) leaky waveguide with light ray being refracted into high-index cladding material,...

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. 

If we substitute for T from Eq. (7-19b) and for Zp from Table 2-1, page 40, and note that p = n in the weak-guidance approximation, it is readily verified that the mode and ray attenuation coefficients are identical [7]. A similar, analytical comparison can be made for refracting leaky modes and rays on the step-profile planar waveguide [8]. 

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