Modes circular fibers

In the absence of material dispersion, pulse spreading on single-mode circular fibers is proportional to the distortion parameter D, as expressed by Eq. (13-18). To calculate this quantity for the clad power-law profiles, we first rearrange the definition as... 

A waveguide is said to be multimoded or overmoded if V> when many bound modes can propagate. At the opposite extreme, when V is sufficiently small so that only the two polarization states of the fundamental mode can propagate, the waveguide is said to be single-moded. For example, the step-profile, circular fiber is single-moded when V < 2.405, as we show in Section 12-9. 

Fundamental modes of waveguides of arbitrary cross-section 285 13-6 Polarization corrections to the scalar propagation constant 286 13-7 Higher-order modes of circular fibers 287... 

Higher-order modes of nearly circular fibers 289... 

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. 

The solution of Eq. (13-8) for the fundamental modes by definition has the largest value of Then, as for the circular fiber there are two modes associated with this solution, one polarized along the x-direction and the other polarized along the y-direction. Both modes have the same scalar propagation constant P, but, because the cross-section is not circular, we know the exact... 

Cj = e , and is expressible in terms of simple integrals in Table 14-1, page 304. Examples of modes on weakly guiding circular fibers are given in the following chapter. 

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. 

Given the solutions f, (R), we then determine the fields e as discussed in Sections 13-4 and 13-7, and summarized in Table 13-1, page 288. We identify the fields of circular fibers in Table 14-1 using the conventional mode nomenclature of Section 11-16. 

To summarize these two sections, the complete modal fields and corrected propagation constants for all bound modes of the weakly guiding circular fiber are given in Table 14-1. Consequently, for each profile considered below, we need only determine the solution of Eq. (14-4). 

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. 

We begin with a brief review of the weak-guidance approximation for fundamental modes on circular fibers. In Sections 13-2 and 13-4, we showed that the two fundamental modes are virtually TEM waves, with transverse fields that are polarized parallel to one of a pair of orthogonal directions. The transverse field components for the x- and y-polarized HEj j modes are given by Eq. (13-9) relative to the axes of Fig. 14-1. The spatial variation Fq (r) is the fundamental-mode solution (/ = 0) of the scalar wave equation in Table 13-1, page 288. Hence... 

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...

Fundamental modes 16-3 Higher-order modes 16-4 Nearly circular fibers... 

We can use the elliptical fiber to quantify the transition from the uniformly polarized modes of the noncircular fiber to the modes of the circular fiber in Table 13-1, page 288, which was discussed qualitatively in Section 13-9. For this purpose we consider the modes corresponding to the 4 21 and 4 12 solutions, which are the successive lowest-order modes after the fundamental modes. Thus Eqs. (16-14) and (37-107) give to within constant multiples... 

When IAI < 1, the modal fields behave like the fields of the circular fiber, i.e. the linear combinations of Table 14-1, page 304, and when A > 1 the modal fields are uniformly polarized on the elliptical fiber. Since A 1 for a practical fiber, it is clear that only the slightest asymmetry is necessary for the modes to be uniformly polarized, and thus the circular fiber is an ideal which requires high precision to be realized. 

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 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],... 

Consider the circular fiber of arbitrary profile n(r) in Fig. 18-l(c) with a small nonuniformity which remains unchanged along the length of the fiber. The nonuniformity has cross-sectional area dA, mean refractive index Hq, and is located at distance Po ftom the fiber axis. For the fundamental modes, the perturbed propagation constant follows from Eq. (18-7) as... 

It is clear from the above discussion that one of the fundamental modes of a circular fiber with an isotropic core is x-polarized with propagation constant x> ttd the other is y-polarized with propagation constant If n(x, y) is the profile of the isotropic fiber, then n = n + Sn for the x-polarized mode and n = J + Sny for the y-polarized mode. If the anisotropy is small and uniform, then 8n and 8ny are constants. Accordingly we set = 0 in Eq. (18-8) and replace 8n by 8n or to obtain [2,3]... 

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. 

We next consider the modes of the elliptical fiber which correspond to the / = 1 modes of the circular fiber. The two solutions of the scalar wave equation for the latter are given by... 

The direction of the fundamental-mode transverse electric field within the core of an arbitrary step-profile fiber is shown schematically in Fig. 12-6. When the nonuniformity of Fig. 18-1 (c) is included, the symmetry of the circular fiber is broken. However, since the nonuniformity is infinitesimal, it is clear that the two fundamentalmode patterns are oriented as shown in Fig. 18-8. The propagation constants associated with Figs. 18-8(a) and 18-8(b) are, and by, respectively. If we substitute Eq. (18-64) into Eq. (18-63) and recall the normalization definition in Table 11-1, page 230, we obtain... 

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... 

Only the HEi modes are excited on the circular fiber, and the power in each mode is determined by substituting Eq. (20-29) into Eq. (20-13) and using the expressions for Fi R) in Table 14-2, page 307. For the fundamental mode the integral in the numerator of Eq. (20-13) is given by Eq. (37-lOOa), whence we deduce the excitation efficiency is [10]... 

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