Power transmission coefficients tunneling rays

In this section we generalize the analysis of plane-wave incidence at a planar interface, and consider the incidence of local plane waves at a caustic in a slowly varying graded medium. Our goal is the derivation of the power transmission coefficient for tunneling rays. 

The attenuation of tunneling-ray power in Eq. (7-3) depends on the product y(P, l)z, where the attenuation coefficient is the ratio of the transmission coefficient r to the ray half-period Zp. For the step profile, the latter is given in Table 2-1, page 40, and we use the linear approximation of Eq. (7-21) for T, which is an excellent approximation for all but the most weakly tunneling rays. If we express k in terms of the fiber parameter using the definition inside the front cover, then... 

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 interface and turning-point caustic are curved surfaces. They are defined by two principal radii of curvature which depend on both the core radius p and the bend radius R. Under these conditions we use the localized transmission coefficients of Section 7-14 each time a ray loses power by tunneling. When power is lost by refraction, we employ the Fresnel coefficient of Eq. (35-50) for the step profile, and assume complete power loss for the clad parabolic profile, i.e. T= 1. 

The link between the ray tubes at the caustics is denoted by the hatched region of width 6z in Fig. 35-4(a). Power lost by the incident ray at x,p tunnels through this region and enters the transmitted ray at x j. At and close to the caustics, local plane-wave theory is inadequate, but, provided the caustics are not too close together, it is not necessary to know the fields in these regions in order to determine the transmission coefficient. We can link the solutions in Eq. (35-32) using the connection formulae of WKB theory, which are available in standard texts [7]. Hence... 

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