Propagation constant Gaussian approximation

The fundamental-mode properties of the weakly guiding, step-profile fiber were given in analytical form in the previous chapter, but, nevertheless, numerical solution of a transcendental eigenvalue equation is required. Within the Gaussian approximation the propagation constant is given explicitly, and all other modal properties have much simpler analytical forms, at the expense of only a slight loss of accuracy [4, 5]. 

The fundamental modes of the infinite parabolic profile fiber have a Gaussian spatial variation it is the exact solution of the scalar wave equation. Thus, the essence of the Gaussian approximation is the approximation of the fundamental-mode fields of an arbitrary profile fiber by the fundamentalmode fields of some parabolic profile fiber, the particular profile being determined from the stationary expression for the propagation constant in Table 15-1. Clearly this approach can be generalized to apply to higher-order modes, by fitting the appropriate solution for the infinite parabolic profile [9]. 

Table 15-4 Low-order modes of the Gaussian-profile fiber. Approximations for the spot size rQ = pR and propagation constant /8, based on the variational solution of Eq. (15-18). 

For the fundamental modes on a circular fiber, is independent of 0. Consequently —tiy and p = p in Eq. (18-21). In other words, the propagation constants for the circular and elliptical fibers are identical for slight eccentricity, provided the core areas are equal [6]. The latter condition is equivalent to requiring equal profile volumes, as is clear from Eq. (17-13). Hence the present result is consistent with the more general result of Section 17-3, which showed that, within the Gaussian approximation, P = on an arbitrary, elliptical-profile fiber of slight eccentricity, provided the profile volumes are equal. 

Here, as mentioned above, we have used (22) to combine the two Gaussians in (19) in a single Gaussian function evaluated at zero time. The symbol in this equation indicates that some irrelevant constants have been dropped, not that there are approximations in this result. Below we replace it with an equality as the constants which normalize the propagators divide out when the appropriately normalized correlation function is computed. The semi-classical amplitude can be conveniently re-expressed by introducing a polar representation of the complex polynomials identifying the initial and final occupied mapping states, thus... 

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