Fundamental modes weakly guiding fibers

It is sometimes useful to know the first few terms in the expansion of the modal fields for A <1 1. As an illustration we consider the two fundamental modes because of their importance in single-mode fibers. Section 32-1 discusses these expansions at length, so it is sufficient to summarize results here. The lowest order fields and values of the modal parameters for the weakly guiding fiber are discussed at length in Chapters 13 and 14. The expansions are in powers of A. Correct to third order, the even HEn mode field expansions are... 

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. 

The two fundamental modes on a weakly guiding fiber of nondrcular cross-... 

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... 

For a given value of the fiber parameter V, there is an optimum inclination which maximizes the power entering a particular mode. In Fig. 20-4(a) these values of 0, correspond to the peak values of P/P. Now it is intuitive that maximum excitation should occur when 0j is similar to the characteristic angle 0, which the local plane-wave vector of the modal field makes with the fiber axis, as discussed in Section 36-2. Since 0 and 02 ttre small on a weakly guiding fiber, the relationship with the modal parameter U in Table 36-1, page 695, shows that 0j/02 = (0i/0c) (VIV). If we insert the values of 0J0 at the peaks in Fig. 20-4(a) and the corresponding values of V/U, from Fig. 14-4, we find that 0j/02 = 1. A more rigorous analysis shows the peaks in Fig. 20-4(a) for all but the fundamental modes are proportional to Jf+iiU) provided K>(/ [2]. For the HE, mode the maximum possible efficiency of 85 % occurs when 0, = 0 and KS3.8. 

We claimed in Section 22-2 that there is negligible coupling between the two polarizations of the fundamental mode due to slight nonuniformities on a weakly guiding fiber. Thus, in the first approximation the incident even, or x-polarized, mode excites power in only the forward- and backward-propagating modes with the same polarization. Hence Eq. (22-21) is replaced by... 

If we substitute Eqs. (23-4) and (23-10) together with the normalization definition of Table 13-2, page 292, into Eq. (23-11), the fundamental-mode attenuation coefficient for an arbitrary, weakly guiding fiber is given by... 

In Section 15-5, we showed ho w to derive the far field of a weakly guiding fiber from a knowledge of the Gaussian approximation to the fundamental mode. As explained below Eq. (15-14), this requires solution of... 

The situation is illustrated in Fig. 36-4. A straight section of a weakly guiding fiber leads into a bend of uniform radius R. The fundamental mode has propagation constant p and has an x-polarized electric field, where the x-axis is parallel to the plane of the bend. Thus, on the straight fiber... 

The fundamental modes of weakly guiding circular fibers are sometimes designated LPq, modes [3] rather than HE,i modes. The nomenclature is intended to emphasize that the modal fields are everywhere polarized in the same direction, i.e. the fields are plane, or uniformly, polarized. 

Modes of weakly guiding waveguides obey the fundamental properties of modes delineated in Chapter 11, and mainly because of the approximate TEM nature of the modal fields, these properties have the simpler forms of TaHe 13—2. The expressions in the first column are in terms of the transverse electric field e, and apply to all weakly guiding waveguides. Those in the second column are for waveguides which are sufficiently noncircular that e, can be replaced by either of the two fields for noncircular waveguides in Table 13-1, while the third column is for circular fibers only, when e, is replaced by any one of the four linear combinations Ct, for circular cross-sections in Table 13-1. We emphasize that Table 13-2 applies to all modes. 

The small polarization correction to the scalar propagation constant due to structural anisotropy is given by Eq. (13-11). For an isotropic fiber of circular cross-section, the corrections for the two fundamental modes are identical, i.e. SPx = Py This is not the case for the anisotropic fiber, since the parameters in Eq. (13-19) depend on polarization. However, SP — SPy is small compared to the difference in propagation constants in Eq. (13-20), Px — Py, since the fiber is weakly guiding, and can be ignored. 

If the principal axes of the anisotropic material are parallel to the optical axes of a fiber of noncircular cross-section, it is intuitive that the two fundamental modes must also be polarized along these axes. We then have a situation identical to the circular cross-section, discussed above, except that (r) in Eq. (13-20) is replaced by P,(x,y) and P,(r) by P,(x,y). Thus, all results for weakly guiding isotropic waveguides apply to weakly guiding anisotropic waveguides by following the simple substitution discussed above. 

Table 14-3 Fundamental and HEi modes of the weakly guiding step profile fiber. Parameters are defined inside the back cover. Modal power is given by where a is the modal amplitude.

On a weakly guiding clad fiber of otherwise arbitrary profile, the fraction of fundamental-mode power residing in the core becomes negligible as F-+ 0. For the step profile, this is clear from the plot of in Fig. 14-3(a). Simultaneously, the fields become nearly uniform over the core, as is evident from the intensity distributions in Fig. 14-3(c), and hence are comparatively insensitive to profile shape. Accordingly, we postulate that the fields depend primarily on the profile volume Q of Eq. (14-42). It then follows from the discussion of the previous section that we can relate the fields of an arbitrary profile, with fiber parameter F, to the known fields of the step profile with fiber parameter Fthrough Eq. (14-46). Thus we replace Fby Fin the small-Fforms for the step-profile fiber listed in Table 14-5. The same conclusion can be derived more formally from the scalar wave equation, as we now show. 

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 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 derivation of the transverse fields of the two fundamental modes on a weakly guiding, noncircular fiber was described in Section 13-5. These fields are given in Table 16-1 in terms of the solution T(x,y) of the scalar wave equation, which in cartesian coordinates has the form... 

To demonstrate the usefulness of the induced-current representation, we determine, through examples, the power radiated from the fundamental mode of a weakly guiding circular fiber due to slight nonuniformities of various kinds. 

Radiation from the sinusoidally perturbed interface of a weakly guiding, step-profile fiber was discussed in Section 22-5 in terms of the induced current of Eq. (22-15). When the fiber is single moded, power is scattered into both forward- and backward-propagating x-polarized fundamental modes from the incident mode. Following the discussion of Section 22-5, we set... 

The fundamental mode on a weakly guiding, bent fiber has been replaced by an antenna, or thin wire, carrying the current distribution of Eq. (23-4). Because the currents are orthogonal to the antenna, it may be helpful to think of them as a continuous distribution of Z-directed current dipoles. To calculate the power radiated, we assume, for simplicity, that the antenna is a closed loop of radius as shown in Fig. 23-2(a). If (s. O, < ) are spherical polar coordinates... 

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 an infinite array of parallel, single-mode fibers which are identical and uniformly spaced. We assume the fibers are weakly guiding and have an otherwise arbitrary refractive-index profile. At the beginning of the array, z = 0, alternate fibers are illuminated with power P, (0) or Pj (0) < Pj (0), denoted by -I- or —, respectively, in the examples of Fig. 29-3. The periodic illumination means that the amplitudes 2>i (z) and biiz) of the fundamental-mode fields, defined in Eq. (29-1), are the same for eveiy... 

Consider an infinite, one-dimensional array of identical, single-mode fibers which are weakly guiding. They are labelled by n = — co,.. ., — 1,0,1. .., oo, as shown in Fig. 29-4(a). Only the center fiber n = 0 is illuminated with unit power at r = 0. As its fundamental mode propagates, power will be coupled to the fundamental modes of the neighboring fibers n = 1. These two fibers in turn couple to then = 2fibersandto the n = 0 fiber, and so on. Thus power is distributed over an increasing number of fibers farther into the array. 

The results of this section apply to all fundamental and higher-order modes on weakly guiding waveguides of arbitrary cross-section, and parallel the results of Sections 13-5 and 13-8 derived by physical arguments. In addition they apply to the fundamental modes of fibers with circular cross-sections, discussed in the next section. 

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