Planar waveguides graded profiles

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

This invariant is identical to Eq. (1-25) for graded-profile planar waveguides, and is associated with the translational invariance of the fiber. 

In Section 1-8 we introduced the notion of the complement of the local critical angle for graded-profile planar waveguides. By analogy, the complement of the local critical angle, 0c( ), for fibers is defined in terms of the profile by... 

Example TE modes on graded-profile planar waveguides... 

Table 12-7 TE modes of symmetric graded-profile planar waveguides. Parameters are defined inside the back cover. Prime denotes differentiation with respect to argument, and X = x/p. Even and odd refers to modes with By an even or odd function of x. In profile (b) n is a positive integer or zero, and p of profile (c) is not necessarily an integer. On profiles (a), (d) and (e) all modes are bound.

The general shape of the ray paths can be deduced from the paths of Fig. 1-8 for a graded-prolile planar waveguide and the paths of Fig. 2-2 for the step-profile fiber. Assuming that the paths are confined to the core, they have the characteristic forms shown in Fig. 2-4. Meridional rays cross the fiber axis... 

Fig. 1-1 Nomenclature and coordinates for describing planar waveguides. A representative graded profile varies over the core and is uniform over the cladding, assumed unbounded.

We emphasize that, in general, the transit times depend on both P and /,and reduce to the corresponding expressions for planar waveguides only in the case of meridional rays. However, transit times are independent of /, i.e. independent of skewness, for certain profiles, including the step and clad power-law profiles, as we show below. We also recall from Section 1-9 that graded profiles tend to equalize transit times compared to the step profile. Unlike planar waveguides, though, there is no known profile for which complete equalization of all ray transit times on a fiber is possible. 

The simplest structures for calculating dispersion are the planar waveguides of Chapter 1. We start with the step profile and progress to graded profiles. 

Here we show that all rays on the bent planar waveguide are leaky, regardless of the profile. We regard the bend as part of a circular fiber whose axis is orthogonal to the plane of the bend in Fig. 9-1 and passes through C. There are two interfaces atr = R p, where r is the cylindrical radius from C,and we assume the core profile shape n(r) is unaffected by bending, i.e. n r) = n x), r = x + R. The azimuthal symmetry of the bend ensures that every ray follows a curved sinusoidal-like path. If the core is graded, the path has the... 

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