Gaussian approximation for circular fibers

15-1 Gaussian approximation 15-2 Example Gaussian proile 15-3 Example Step profile 15-4 Example Smoothed-out profiles 15-5 Field far from the fiber axis 

The weak-guidance approximation, described in Chapter 13, greatly simplifies the determination of the modal fields of optical waveguides, because it depends on solutions of the scalar wave equation, rather than on vector solutions of Maxwell s equations. For circular fibers, with an arbitrary profile, the scalar wave equation must normally be solved by purely numerical methods. We discussed the few profiles that have analytical solutions in Chapter 14. These solutions, including those for profiles of practical interest such as the step and clad power-law profiles, are given in terms of special functions or by series expansions, which usually necessitate tables or numerical evaluation to reveal the physical attributes of the modes. 

An exception is the infinite parabolic profile of Section 14-4. The fundamental-mode fields of Table 14-2, page 307, have the simple Gaussian dependence exp( — F ) and other modal properties have very elementary forms, from which their physical behavior is immediately apparent. To these facts we add the observation that the fundamental-mode intensity pattern - and hence the field distribution - for step and clad power-law profiles 

In Section 14-10, we introduced the concept of profile volume. We showed in the case of clad power-law profiles of equal volume that some properties, such as the range of single-mode operation and the fundamental-mode intensity distribution are insensitive to profile shape, whereas other properties, such as waveguide dispersion, depend critically on profile shape. Within the Gaussian approximation, we can demonstrate directly the insensitivity of the intensity distribution to profile shape. 

We also discuss generalizations of the Gaussian approximation to other low-order modes. Finally, we briefly describe the equivalent step-profile approximation [4], and compare it with the Gaussian approximation. 

---

Table 15-1 Gaussian approximation for circular fibers. Gaussian approximation for the fundamental-mode fields of weakly guiding, circular fibers. Parameters are defined inside the back cover and coordinates are illustrated in Fig. 14-1...

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],... 

The Gaussian approximation was introduced in Chapter 15 to provide simple, but accurate, analytical expressions for fundamental-mode quantities of interest on circular fibers of arbitrary profile. Here we show how to generalize this approximation and describe fundamental-mode propagation on weakly guiding fibers of arbitrary cross-section. 

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. 

The efficiency with which beams excite the fundamental modes of circular fibers, with the fields of Eq. (13-9), is of particular interest when the fiber is single moded. In order to account for weakly guiding fibers of otherwise arbitrary profile, when analytical solutions of the scalar wave equation for Fo (r) are not available, we use the Gaussian approximation of Chapter 15. The radial dependence of the fundamental-mode transverse fields is approximated in Eq. (15-2) by setting... 

---