Refracting rays power transmission coefficient

Consider two semi-infinite media of refractive indices and < n , separated by the planar interface x = 0 in Fig. 35-3(a). A ray, or plane wave, is incident on the interface from the denser medium at angle 6 to the z-direction. Plane wave reflection in this situation is well-known and the power transmission coefficient of Eq. (35-11) is identical to the classical Fresnel coefficient [2]. 

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. 

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