Planar waveguides lateral shift

Transit time for the lateral shift 10-6 Lateral shift in planar waveguides 10-7 Preferred ray directions... 

Fig. 10-4 The lateral shift z, of the path in (a) is equivalent to an effective core half width in (b), for the step-profile planar waveguide.

Finally, we emphasize that the lateral shift is not the only manifestation of diffraction on the step-profile planar waveguide. Taken alone, it is insufficient for describing propagation on low- V waveguides. We next investigate another diffraction phenomenon to better appreciate the departure of electromagnetic theory from the ray methods of Part I. 

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. 

Fig. 36-3 (a) Ray path within the core ofwidth2p of an absorbing, step-profile planar waveguide and (b) the effective core width 2p allowing for the lateral shift s. 

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