Turning-point caustic

Once this divergence happens, further solution of the differential equation is not possible beyond this point, and we have to reformulate the problem. To clarify our idea, let us consider the ID problem. At the turning point, p q) = 0 and A diverges. If we invert A to A = dq/dp, the divergence is removed and the propagation of A proceeds smoothly through the caustics. This inversion is equivalent to the canonical transformation, (p,q) (—q,p). It can be easily... 

An example of a family of classical trajectories together with the caustics are shown in Fig. 21. The caustics in the asymptotic region appear periodically as turning points of the vibrational motion. In the rearrangement region, the trajectories are no more periodic and the caustics appear rather randomly. As is seen, the present method is demonstrated to work well. 

If we combine curvature of the ray path with the notion of the turning point, then the ray path must qualitatively look like the path in Fig. l-7(a) when Eq. (1-22) has a solution, and like Fig. l-7(b) when Eq. (1-22) has no solution. In the first case the ray path bends continuously until it returns to the axis, and in the second case it reaches the interface, where it is totally transmitted. Since the profile is assumed continuous across the interface, the angles of incidence and transmission are all equal, and 0t = 0 (p). The dashed line at x = x,p is the locus of turning points for all rays with the same value of 0 (0), and is often called the ray caustic or turning-point caustic. 

Fig. 2-6 Section of a tunneling ray path on a graded-profile fiber. In (a) the core path touches the turning-point caustic at P. Radiation originates at Q in the cladding and propagates along QR tangential to the radiation caustic. The projection onto the fiber cross-section is shown in (b).

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. 6-1 The electric field intensity IE decreases exponentially beyond (a) the interface of a step-profile waveguide and (b) the turning-point caustic of a graded-profile waveguide.

To determine the transmission coefficient of Eq. (6-22) for the clad parabolic profile of Table 2-1, page 40, we recall from Eq. (2-19) that the radii of the inner and turning-point caustics are roots of the integrand. Hence the integrand is expressible as... 

We first recall from Section 2-7 the discussion of tunneUng ray paths on monotonic graded-profile fibers with minimum index in the cladding and maximum core index on the axis. Part of the trajectory of a narrow tube of identical tunneling rays is illustrated in Fig. 7-3 (a). Each ray touches the turning-point caustic at radius r, where is the larger root of Eq. (2-19) in... 

Table 7-1 Delineation of ray types according to the value of the radiation caustic radius in the cladding. The corresponding values of the turning-point caustic r,p are...

Rays which have very small values of the transmission coefficient are known as weakly tunneling rays and, following the discussion above, have values of - oo, or, equivalently, according to Eq. (2-23X Thus the radiation caustic is much further from the interface than the turning-point caustic, i.e. r —p>p r,p. In this dtuation, we can approximate r,p by p in the lower limit of integration of Eq. (7-15), and generate an accurate expression for T for all weakly tunneling rays. Since n (r) = in the... 

Most tunneling rays have turning points and radiation caustics which are not far from the interface, i.e. r,p S p, = p. We can then make an expansion about the interface by linearizing the expression within the square root of the integrand in Eq. (17-15) for r < p and for r>p, respectively. This is carried out in Section 35-12 and leads to Eq. (35-48) where [5]... 

There we show that the dependence of the transmission coefficients on the local geometry of a circular fiber involves only the radius of curvature p in the plane of incidence of the ray, i.e. the radius of curvature of the interface or turning-point caustic in the plane defined by the ray path or the tangent to the ray path, respectively, and the normal. We then claim that the locally valid transmission coefficients for interfaces or turning-point caustics of arbitrary curvature have the same functional dependence on p. In these cases p depends on two prindpal radii of curvature instead of the single radius of curvature of the circular fiber. 

A turning-point caustic of arbitrary curvature is illustrated in Fig. 7-5(b), and is defined by the surface n = (r,p), where r,p is the position of the turning point in the radial direction. We assume that the turning point is not too far from the intersection of the radial axis with the interface at r = p, beyond which the profile is uniform, i.e. n — n p) = n i- The tangent to the ray path at P lies in the tangent plane at P and makes angles n/2 — 0 and 0, with the y- and z-axes, respectively. Otherwise the notation is identical with Fig. 7-5(a). In this situation, the local transmission coefficient is given by Eq. (35-55) [19]... 

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