Fundamental modes step-profile approximation

If we use the step-profile approximation to determine the values of U for the fundamental modes of the Gaussian-profile fiber, then there is no limitation on the range of V, unlike the Gaussian approximation of Table 15-2. Furthermore, there is so little difference between the approximate and exact values of U, that a plot is indistinguishable from the dashed curve in Fig. 15-1 (b). Plots of the intensity distribution calculated from the step-profile approximation are virtually coincident with the exact solution away from the fiber axis, particularly for small values of V, whereas the Gaussian approximation leads to a significant error, as is evident in the V = 1.592 plot of Fig. 15-l(c). The Gaussian approximation has a far field which decreases too rapidly, as explained in Section 15-5, whereas the choice of field in Eq. (15-22) ensures the correct far field behavior for R > 1. 

In Chapter 11 we discussed the fundamental properties of modes on optical waveguides. The vector fields of these modes are solutions of Maxwell s source-free equations or, equivalently, the homogeneous vector wave equations. However, we found in Chapter 12 that there are few known refractive-index profiles for which Maxwell s equations lead to exact solutions for the modal fields. Of these the step-profile is probably the only one of practical interest. Even for this relatively simple profile the derivation of the vector modal fields on a fiber is cumbersome. The objective of this chapter is to lay the foundations of an approximation method [1,2], which capitalizes on the small... 

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]. 

We show in Table 15-3 that there is only a very small error between the approximate and exact values of U for the fundamental mode, calculated at the exact cutoff of single-mode operation for each profile [7]. The distortion parameter of Table 15-2 is plotted as the solid curves in Fig. 15-2(b) for the m = 2 and step profiles, together with the dashed curves for the exact values [7]. If Fj denotes the values of V for which there is no waveguide dispersion, i.e. D = 0, then... 

With reference to Table 14-3, page 313, we assume that the fundamental-mode fields of an arbitrary profile fiber can be approximated by the fundamentalmode fields of some step-profile fiber, whose radial dependence is expressed by... 

It is readily verified, by repeating the derivation in Table lS-1, page 339, for planar waveguides, that the above expression is the spot-size equation for the Gaussian approximation to the fundamental modes of a step-profile, planar waveguide of core half-width Py. 

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. 

Fig. 20-4 (a) The fraction of total power in a uniform beam that excites modes of a step-profile fiber as a function of the tilt angle 0j, where P includes all modes with the same values of U in Fig. 14-4, and bm is the total excited power [2]. (b) Variation of the excitation efficiency with the fiber parameter for on-axis illumination, where solid curves denote the exact solution of Eq. (20-27c) and the dashed curve is the Gaussian approximation of Eq. (20-28a). (d) The corresponding curves for the fundamental mode for various ratios of beam to core radii calculated from Eqs. (20-27c) and (20-28b). (c) Plots of Pq/Pi for the fundamental mode and different ratios of beam to core radii. 

The expression in Eq. (20-30) for the fundamental-mode efficiency is also the result which we would obtain using the Gaussian approximation of Eq. (20-24) for an arbitrary profile. Thus we have a general expression for lens illumination. For example. Table 15-2, page 340, gives Tq = p/(21n for the step profile, and at K = 2.4 the error between Eq. (20-30) and an exact analysis is less than 1 % [10]. 

Fig. 20-7 (a) The fraction of power, calculated from Eq. (20-34X that enters the fundamental mode as a function of the angular spread 0 of a diffuse source for step (sX Gaussian (g) and infinite parabolic (p) profile fibers, (b) The percentage error in the geometric optics analysis of totally incoherent illumination of a multimode fiber as a function of the fiber parameter. The solid curve is the exact result calculated from Eq. (20-39) and the dashed curve is the approximation of Eq. (20-41) [11]. 

The thin-wire approximation to radiation from the fundamental mode on a bent step-profile fiber is given in terms of the power attenuation coefficient of Eq. (23-12) by setting/ = 0 over 0 < R < 1 and/= 1 elsewhere. The / = 0 integral and eigenvalue equation in Table 14-6, page 319, together with the integral of Eq. (37-91), lead to... 

In Section 22-5 we determined the attenuation of the fundamental mode on a weakly guiding, step-profile fiber due to radiation from a sinusoidal perturbation of the interface, using free-space antenna methods and correction factors. Here we consider the situation when the radiation field is well approximated by a single leaky mode, which can be realized by having an on-axis sinusoidal nonuniformity of the form of Eq. (22-14). The azimuthal symmetry ensures that only HEi leaky modes are excited. Further, the direction of radiation should coincide with the direction of the leaky-mode radiation [23]. If we represent the nonuniformity and the incident fundamental-mode fields by the induced current method, as in Section 22-5, the direction condition is satisfied by setting C = in Eq. (24-43), whence... 

Consider the step-profile tapers of Fig. 28-1, whose ends have fixed core radii Po and p but otherwise are of arbitrary, slowly decreasing radius. We are primarily interested in radiation loss when a local mode approximates the fundamental mode on the taper, but the analysis is easily modified to describe losses from higher-order local modes. This also provides criteria for the accuracy of the local-mode description. 

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