Circular fibers fundamental modes

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... 

First we consider fundamental modes and then higher-order modes. Like the exact propagation constant P, the scalar propagation constant p is largest for fundamental modes. It is convenient to distinguish between fibers of circular cross-section and waveguides of arbitrary cross-section. 

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... 

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. 

From the above discussion, it is intuitive that the two fundamental modes of an anisotropic fiber of circular cross-section must be polarized along the x and y principal axes, respectively. Hence the anisotropy breaks the geometrical circular symmetry of the fiber, and is analogous to the fiber of noncircular cross-section discussed in Section 13-5. Thus, the x-polarized mode sees a fiber characterized by n x,y), and V, while the y-polarized mode sees a fiber characterized by n,(x,y). A, and V, as illustrated in Fig. 13-1. The 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. 

We ignore the small polarization corrections to P and Py, given by Eq. (13-11), because P f Py for isotropic, noncircular waveguides. This is an accurate approximation, provided the material anisotropy is not so minute as to be comparable to the small contribution of order due to the waveguide structure. The higher-order modes of the noncircular waveguide have the same form as the fundamental modes, except when the fiber is nearly circular, for reasons given in Section 13-9. 

Consider a step profile fiber of circular cross-section whose refractive index profile, , for x-polarized light is shown in Fig. 13-l(b). We assume that is sufficiently small so no higher-order modes propagate. Thus, an x-polarized source will excite the fundamental mode with field given by Eq. (13-20). To... 

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... 

The procedure for constructing higher-order modes of noncircular fibers was established in Section 13-8. For each mode the transverse fields are identical in direction and form to the fundamental-mode fields of Table 16-1, except that F now denotes the appropriate higher-order solution of the scalar wave equation of Eq. (16-3). Only when the fiber cross-section is sufficiently close to circular is this representation inappropriate, as explained in Section 13-9. We quantify this transition in the following section. 

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... 

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],... 

In many of the examples below, the unperturbed fiber has a circularly symmetric cross-section and profile. For perturbations which maintain this symmetry, the polarization of each mode is unchanged, and obeys the rules laid down in Sections 13-4 and 13-7. However, for perturbations which break circular symmetry, the modes are polarized along the optical axes and y of the perturbed fiber. Only the fundamental and HE, modes remain plane polarized on the circular and noncircular fibers. If the perturbation is sufficiently small, the polarization of all other modes lies in the transition region between the circular and noncircular situations, as discussed in Section 13-9. 

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. 

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]... 

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. 

On a circular fiber which propagates only the fundamental modes, scattered power that is not radiated can only be directed into the forward- and backward-propagating, even and odd fundamental modes. If the forward-propagating, even HE,i mode is incident on the nonuniformities in Fig. 22-i(a), then the bound portion of the total electric field is given by... 

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. 

The fundamental and HEi , modes of a fiber of circular cross-section are formed from the scalar wave equation solution with no azimuthal variation. Hence in Eq. (13-8) depends only on the radial position r. There is no perferred axis of symmetry in the circular cross-section. Thus, in this exceptional case, the transverse electric field can be directed so that it is everywhere parallel to one of an arbitrary pair of orthogonal directions. If we denote this pair of directions by x- and y-axes, as in Fig. 12-3, then there are two fundamental or HEi , modes, one with its transverse electric field parallel to the x-direction, and the other parallel to the y-direction. The symmetry also requires that the scalar propagation constants of each pair of modes are equal. 

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