Corrections waveguide polarization

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... 

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-... 

W. Huang, H. A. Hans, and H. N. Yoon, Ananlysis of buried-channel waveguides and couplers scalar solution and polarization correction. Journal of Lightwave Technology 8, 642-648 (1990). 

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,. 

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 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. 

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 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. 

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