Bound rays of planar waveguides

1-2 Construction of ray paths 1-3 Ray invariant 1-4 Ray-path parameters 1-5 Ray transit time 

1-6 Construction of ray paths 1-7 Ray invariant 1-8 Ray-path parameters 1-9 Ray transit time 

1-11 Example Parabolic profile 1-12 Example Hyperbolic secant profile 1-13 Example Clad power-law profiles 

Alternative forms for V are given inside the front cover. The ray theory presented here is restricted to multimode waveguides, i.e. waveguides satisfying V I, for reasons discussed in Chapters 10 and 36. 

With reference to Fig. 1-2, the step-index planar waveguide has the refractive-index profile defined by 

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Fig. 1-4 Zig-zag paths within the core of a step-profile planar waveguide for (a) bound rays and (b) refracting rays.

Fig. 1-8 Sinusoidal-like bound-ray path within the core of a graded-profile planar waveguide.

Fig. 1-9 Ray half-period Zp for a bound ray in the core of a graded-proiile planar waveguide. The path length Lp is measured along the path from P and Q.

The geometric optics analysis of the excitation of the planar waveguide by sources assumes that power may enter any bound ray whose direction belongs to the continuum of directions defined by Eq. (I-5a). However, as a consequence of diffraction, power enters certain preferred directions only, and the smaller V, the smaller the number of preferred directions. If V is sufficiently small, then there is only one preferred direction and all bound rays propagate with the same transit time. As V increases, the number of preferred ray directions becomes sufficiently large that it can be accurately approximated by a continuum. 

Fig. 10-5 Section of a bound-ray on a step-profile planar waveguide, showing the complete ray period 2z. ...

The simplest optical waveguide structure is an infinite planar slab of perfectly transparent, isotropic, dielectric material with refractive index rig bounded on both surfaces by similar material of index rio < rig (see Fig. 1). According to SnelFs law, light rays in the slab will be totally internally reflected if they have an incidence angle 6 with a surface satisfying... 

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