Fundamental modes polarization corrections

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... 

The higher-order modes of waveguides with noncircular cross-sections are constructed from each pair of solutions Pj (x, y) and Pg (x, y) of Eq. (13-8) and their corresponding scalar propagation constants and p. The transverse electric fields of these modes are polarized along the same optical axes as the fundamental modes of Section 13-5. There are two pairs of higher-order modes. Each pair has fields given by Eq. (13-10), with p and P(X) y) replaced by Pg and Pj(x,y) for one pair, and by p and Pj,(x,y) for the other pair. The polarization corrections Sp, Sp, SPy and SPy are obtained from Eq. (13-11) with the appropriate field substituted for e,. 

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. 

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. 

The transverse fields of the fundamental modes in Table 14-1 contain no polarization effects due to the fiber. These effects are included in higher-order corrections through the expansions in Section 32-1. The second-order transverse electric field corrections satisfy the equation in Table 32-1, page 627. If we set f = R, substitute e or e,3 from Table 14-1 fore, and take = 51//A from Table 14-2, it is readily verified that the... 

Fig. 14-3 Fundamental mode quantities for the step-profile fiber, showing (a) the modal parameter U, the fraction of power in the core fj, and the depth of penetration r, (b) the normalized polarization correction SU/AU, (c) the normalized intensity distribution and (d) the normalized variation in group velocity relative to the left ordinate, and the distortion parameter D relative to the right ordinate. Numerical values are given in Table 14-4.

Consider a pulse within which only the two fundamental modes are excited. Waveguide dispersion describes the spread in each mode, but because of elUpticity the spread for each polarization is different. In addition, the sUght difference dpj —SPy between corrected propagation constants implies the respective group velocities are unequal and consequently there will be intermodal dispersion between the two modes. Intermodal dispersion which relies on polarization difference is often referred to as a birefringence effect. 

The polarization corrections, and SPy, to the scalar propagation constant P for the Xq- and yo-polarized modes on the perturbed, noncircular fiber are in general unequal, and their difference describes the anisotropic, or birefringent, nature of propagation. This is of basic interest for the two fundamental modes on single-mode fibers. The calculation of the corrections from the formula in Table 13-1, page 288, requires first-order corrections to the approximation We derive these corrections for the slightly elliptical fiber in Section 18-10. 

By symmetry the fundamental-mode fields of the elliptical fiber are polarized along the X- and y-axes in Fig. 18-2(a). On a step profile, the corrections Sfi, and Sfi, to the scalar propagation constant p can be obtained from Eq. (32-26) by setting e, = 4 x and e, = y, respectively, where x and y are unit vectors parallel to the axes. To second order in eccentricity, the birefringence is given by the line integral. 

As the separation increases, )3+ - )S and the beat length becomes exponentially large. The transfer of power is clearly a consequence of interference, or beating, between the fundamental mode fields in Eq. (18-37), and depends only on the difference between the scalar propagation constants. There is no need to consider polarization corrections to the propagation constants in order to study cross-talk on the composite waveguide. 

The scalar propagation constants P+ and for the fundamental modes of the composite waveguide are given by Eq. (18-35) in terms of the fundamental mode propagation constant for either fiber in isolation and C of Eq. (18-42). We explained in Section 13-5 that polarization corrections are required to correctly distinguish between the propagation constants of each pair of fundamental modes associated with P+ or P-. To determine each correction, we substitute the approximate transverse electric field of Eq. (18-36) into Eq. (13-12), where I now denotes the interface of both fibers. Thus, in the notation of Section 18-12, and with the help of Eqs. (18-36) and (18-33), we obtain 5 by setting... 

The polarization corrections cause interference effects between pairs of fundamental modes with the same propagation constant. For example, if the modes associated with are excited, the difference between and 6Py+ accounts for the apparent rotation of the total transverse fields as they propagate [9]. This was examined in Section 14-7, and can be characterized by a beat length 4n/(SP + —SPy ). We can compare the cross-talk beat length of Eq. (18-40) with the rotation beat length. For large separation, Eqs. (18-40), (18-42) and (37-88) give... 

Thus if Hq and n (po) are sufficiently dissimilar, the perturbed and unperturbed fields can differ greatly within the perturbation region. Even if n s n(po) we need to retain Eq. (18-64) without further approximation in order to describe correctly polarization effects due to the nonuniformity, e.g. the difference in fundamental-mode propagation constants. 

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