Profile infinite parabolic

Our first example is tfie infinite parabolic profile, which corresponds to the g = 2 solid curve and the dotted extension in Fig. 1-10 and is defined by... 

Fig. 1-10 Plots of the power-law profiles of Eq. (1-59) for various values of q. The clad parabolic profile corresponds lo q = 2 and the step profile to q = 00. The hyperbolic secant profile of Eq. (1-51) is shown dashed, and the dotted extension to the q = 2 core profile denotes the (infinite) parabolic profile of Eq. (1-44).

We first consider the infinite parabolic profile defined by... 

All of the profiles appearing in this book fall into one of two classes. The first class consists of clad profiles which have a uniform value n i in the cladding, and an arbitrary variation in the core, such as the representative profile in Fig. 11-1 (b). These profiles are characterized by a discontinuity in the profile or its slope at the core-cladding interface. The second class, as exemplified by the infinite parabolic profile, comprises profiles which vary smoothly over the infinite cross-section, and do not necessarily have a well-defined interface between core and cladding. For these profiles we define p to be a characteristic profile radius, which sets the rate at which the profile changes. We define... 

Table 14-2 Modes of the infinite parabolic-profile fiber. The vector modal fields are found by substitution into Table 14-1. Parameters are defined inside the back cover. 

We discussed the accuracy of the weak-guidance approximation for the infinite parabolic profile in Section 14-4. If we impose the condition j8 = kn we deduce from Eq. (14-9) and Table 14-2 that V p- G q)A Y for the infinite power-law profiles to be weakly guiding. 

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... 

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]. 

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 final example considers a profile which describes the refractive-index distribution for two parallel, infinite parabolic-profile fibers [3]. This profile is illustrated in... 

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]... 

We next consider the fundamental modes of the double parabolic profile waveguide, which has the profile of Eq. (16-30), illustrated in Fig. 16-2(b). Despite the fact that such a waveguide is unphysical for the reasons given in Section 14-4, it provides the siiriplest perturbation problem for any two-fiber waveguide. The unperturbed fiber has the infinite parabolic profile... 

Example Infinite parabolic profile 20-16 Example Gaussian approximation... 

Fig. 20-2 (a) Coordinates for describing the fields of the beam incident on the endface and (b) the fraction of total power of a Gaussian beam entering the modes of an infinite parabolic-profile fiber as a function of the tilt angle 0.. The orientation of the Gaussian beam is shown for (c) on-axis, (d) tilted and (e) offset illumination. 

Table 20-1 Excitation efficiency for the fundamental mode. These expressions are exact for the infinite parabolic-profile fiber when rQ = and, within the...

The spot size, r, depends on the particular profile shape. Examples, including the step and Gaussian profiles are given in Table 15-2, page 340. If we approximate Fo(r) by Eq. (20-24), it follows that all of the results for fundamental-mode excitation of the infinite parabolic-profile fiber, derived earlier in this chapter, apply equally to arbitrary profile fibers provided the appropriate expression for r is substituted into Table 20-1. 

To complement our examples of beam illumination of the infinite parabolic-profile fiber, we now derive expressions for the efficiency of a uniform beam in exciting the modes of a weakly guiding, step-profile fiber. 

Here we parallel the two derivations of the eigenvalue equation, given in the previous section, for a fiber with the infinite parabolic profile of Eq. (14-5). We recall from Chapter 2 that the ray, or local plane-wave, trajectory lies between the inner and turning-point caustics of radii and r,p, respectively. Accordingly, the modal fields are... 

Fig. 36-2 Ray path on an infinite parabolic profile fiber, showing (a) the helical path between successive turning points and (b) the projection of the path onto the fiber cross-section.

We substitute for and Zp from Table 2-1 for the infinite parabolic profile, and express and I in terms of p and v, using the relationships in Table 36-1. Straightforward rearrangement in terms of the fiber and mode parameters leads to Eq. (36-14). A similar analysis of ray paths within the core of a step-profile fiber leads to the asymptotic form of the eigenvalue equation in Table 14-8, page 325 [5]. 

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