Propagation constant 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... 

If we are to account for waveguide polarization properties in the propagation constant, we must add a correction dp to the scalar propagation constant p.To determine Sp exactly we would have to solve the vector wave equation. However, the V, Inn term on the right of Eq. (ll-40a) is small for weakly guiding waveguides, so we use simple perturbation methods in Section 32-4. From Eq. (32-24) we have... 

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 expressions in Table 13-2 for the group velocity and distortion parameter are given in terms of solutions of the scalar wave equation. Given the polarization correction dfi to the scalar propagation constant, we can write down higher-order corrections to these expressions. This is facilitated by first defining the mode parameter U associated with the scalar propagation constant... 

If dU denotes the polarization correction to U, then the exact mode parameter U and propagation constant P are well approximated by... 

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. 

The finite propagation constant corrections SPi discussed above are responsible for interference effects between pairs of modes with the same scalar propagation constant. For example, suppose the odd HE21 and TEqi modes are excited with equal power and all other modes have zero power. If we erroneously ignore all polarization effects, then bPj = 5p = 0, and the total transverse electric field of the fiber follows from Table 14-1 as... 

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. 

For slight perturbations, it is normally sufficient to assume that the transverse, or X, y dependence of the modal fields on the perturbed and unperturbed fibers is similar, i.e. e S e, h = h, and consequently T =. An exception is the calculation of the polarization corrections to the scalar propagation constant, discussed in the following section, for which higher-order corrections to are required. These can be obtained using either eigenfunction expansions, as outlined in Sections 33-9 and 33-10, or Green s functions, as discussed in Chapter 34. 

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. 

The corresponding polarization corrections SPx+ > and SPy- to the scalar propagation constants P+ and P-, respectively, are obtained by substituting the appropriate field into Eq. (13-11). 

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. 

In Chapter 13 we used the polarization properties of the waveguide to determine the direction of e, and the correction S j to the scalar propagation constant Pj. However, the propagation constant p for radiation modes takes any value in the range 0 < jS < kn y and is therefore a continuous variable independent of waveguide polarization. Consequently, higher-order correc-... 

The modal fields depend on the product Pz in Eq. (32-1), where p is the exact propagation constant. Since z can be arbitrarily large, we determine higher-order corrections to the scalar propagation constant p so that our approximate expressions for the fields are accurate over finite distances along the waveguide. Thesr corrections take into account polarization effects due to the waveguide. 

Table 13.4 Comparison of different polarization propagator methods for the calculation of indirect nuclear spin-spin coupling constants J (in Hz) (Vahtras et al, 1992, 1993 Wigglesworth et al, 1998 Enevoldsen et al, 1998 Kaski et al, 1998 Astrand et al, 1999 Jaszuhski and Ruud, 2001 Sauer et al., 2001 Yachmenev et al, 2010). The experimental data are extrapolated to the equilibrium geometry by subtracting calculated ro-vibrational corrections from the experimental values.

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