Rays and local plane waves

Local plane-wave description of loss phenomena 671 

When the refractive index n is everywhere constant in an unbounded medium, the electromagnetic fields can be expressed as individual plane waves. If we now suppose 

The electric and magnetic fields E and H of a plane wave in an infinite medium of uniform refractive index n have the forms [S] 

The intensity, or power density, S has magnitude and direction determined by the time-averaged Poynting vector. Thus we deduce from Eq. (35-lb) that 

The reflection and transmission of a plane wave, or ray, which is incident on a planar interface between two semi-infinite, uniform media is determined by Sneirs Laws [2]. In Fig. 35-1, the refractive indices of the medium of incidence and the second medium are and respectively, and the critical angle ot = sin We denote 

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Fig. 11-2 The electric field vector is orthogonal to the ray, or local plane-wave direction. On the step-profile fiber, the direction of e for (a) a meridional ray is parallel to a fixed direction, and for (b) a skew ray it changes direction at each reflection. On the parabolic-profile fiber the direction of e changes continuously along the skew-ray path (c).

Thus the modal and ray transit times are equal only when tj - 1. This condition is satisfied only by those rays belonging to modes well above cutoff, i.e. when Vp U, or, equivalently, when 0 < 0c- Hence is inaccurate for arbitrary values of 9. This inaccuracy arises because the ray transit time ignores diffraction effects, which were discussed in Chapter 10. The step-profile planar waveguide is a special case, however, because all diffraction effects can be accounted for exactly by including the lateral shift at each reflection, together with recognizing the preferred ray directions. TWs was carried out in Section 10-6, and for rays, or local plane waves, whose electric field is polarized in the y-direction in Fig. 10-2, leads to the modified ray transit time of Eq. (10-13). If we use Table 36-1 to express 0, and 0(.in terms of U, Vand Wand substitute rj for TE modes from Table 12-2, we find that Eqs. (10-13) and (12-8) are identical since 0 = 0. It is readily verified that the same conclusion holds for TM modes and local plane waves whose magnetic field is polarized in the y-direction of Fig. 10-2. 

In this part of the chapter, we give an asymptotic description of propagation on weakly guiding waveguides using modes, and demonstrate the equivalence of this description with the ray, or local plane-wave, description of Part I. The specialization to weakly guiding waveguides is done for simplicity of presentation and not for fundamental reasons. 

Here we parallel the two derivations of the eigenvalue equation, given in the previous section, for a fiber with the infinite parabolic profile of Eq. (14-5). We recall from Chapter 2 that the ray, or local plane-wave, trajectory lies between the inner and turning-point caustics of radii and r,p, respectively. Accordingly, the modal fields are... 

Consider the isolated scatterer S in Fig. 5-7(b), and a ray incident on it. Scattering is usually described in terms of a differential scattering cross-section ffj, which determines the distribution of scattered power when a plane wave is incident on the scatterer [12], To relate to scattering in terms of rays, we recall our concept of a ray as a local plane wave, discussed in Section 35-3. We regard the local plane wave associated with the incident ray in Fig. 5-7(b) as part of an infinite plane wave propagating in the same direction. Accordingly we may set... 

The transmission coefficient T is found by using the local plane-wave description of a ray. We regard the local plane wave as part of an infinite plane-wave incident on a planar interface between unbounded media, whose refractive indices coincide with the core and cladding indices and of the waveguide, as shown in Fig. l-3(b). For the step interface, Tis identical to the Fresnel transmission coefficient for plane-wave reflection at a planar dielectric interface [6]. In the weak-guidance approximation, when s n, the transmission coefficient is independent of polarization, and is derived in Section 35-6. From Eq. (35-20) we have [7]... 

The refracting-ray transmission coefficient for skew rays on graded-profile fibers is given by Eq. (7-6) within the local plane-wave approximation, except that 0, = njl — a, where a is the angle between the incident, reflected or transmitted rays and the normal at the interface. We deduce from Eqs. (2-14), (2-16) and (2-17) that... 

The transmission coefficient for tunneling rays on a weakly guiding, step-profile fiber with core and cladding indices and is derived in Section 35-12 within the local plane-wave approximation. Thus Eq. (35-46a) gives [9,14]... 

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]. 

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. 

Consider a medium whose continuous refractive-index profile n(x) varies slowly over a distance equal to a wavelength. A ray path in the region x < 0 of Fig. 35-4(a) touches the turning-point caustic at x = x,p. When there is no path in the region beyond the caustic, the ray is totally reflected from the caustic and no power is lost, i.e. T=0. However, if a transmitted ray originates at the radiation caustic x = x j, then optical tunneling occurs, as described in Chapter 7, and power is lost from the path at x,p to the path at x. If the local plane-wave fields have propagation constant P, then the values of x,p and x are determined by Eq. (35-31). 

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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