Profile infinite elliptical parabolic

The circular fiber of infinite parabolic profile was discussed in detail in Section 14-4. If the cross-section is deformed into elliptical shape, the profile is expressible as... 

Fig. 16-1 Contours of constant refractive index for the elliptical fiber of infinite parabolic profile defined in Eq. (16-1).

Table 16-1 Fundamental modes of the elliptical fiber of infinite parabolic profile.

The basis of the Gaussian approximation for circular fibers is the observation that the fundamental-mode field distribution on an arbitrary profile fiber is approximately Gaussian. Coupled with the fact that the same field on an ihfinite parabolic-profile fiber is exactly Gaussian, the approximation fits the field of the arbitrary profile fiber to the field of an infinite parabolic-profile fiber. The optimum fit is found by the variational procedure described in Section 15-1. Now in Chapter 16, we showed that the fundamental-mode field distribution on an elliptical fiber with an infinite parabolic profile has a Gaussian dependence on both spatial variables in the cross-section. Accordingly, we fit the field of such a profile to the unknown field of the noncircular fiber of arbitrary profile by a similar variational procedure, as we show below [1, 2],... 

By analogy with the fundamental-mode solution for the elliptical fiber of infinite parabolic profile given in Table 16-1, page 356, we assume that the solution 4 (x, y) of the scalar wave equation of Eq. (16-3) can be approximated by setting [1, 2]... 

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