Noncircular waveguides

16-2 Fundamental modes 16-3 Higher-order modes 16-4 Nearly circular fibers 

16-5 Homogeneous function profiles 16-6 Separable profiles 16-7 Example Infinite linear profile 16-8 Example Double parabolic profile 

The simplest example of a noncircular waveguide is the planar waveguide of Chapter 12, whose modes are either TE or TM, as explained in Section 11-16. For each TE mode the electric field lies in the cross-section and is uniformly polarized. Consequently the weak-guidance solution is identical to the exact solution for the field ey and the propagation constant. Both satisfy the scalar wave equation of Eq. (12-16), and examples with analytical solutions are given in Table 12-7, page 264. Within the weak-guidance approximation the 

The circular fiber of infinite parabolic profile was discussed in detail in Section 14-4. If the cross-section is deformed into elliptical shape, the profile is expressible as 

For example, if n = then the definition of A at the back of the book leads to 

---

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. 

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. 

Now we consider other noncircular waveguides with profiles which lead to analytical solutions of the scalar wave equation or to closed-form expressions for other modal properties. 

Love, J. D. and Hussey, C. D. (1984) Variational approximations for modes of noncircular waveguides. Opt. Quant. Elect, (submitted). 

We showed in Section 2-13 that the transit time for a step-profile fiber is independent of the cross-sectional geometry. Consequently Eqs. (3-2) and (3-3) give the ray dispersion for step-profile fibers of arbitrary cross-section. We also found in Section 2-13 that the ray transit time for the noncircular, clad power-law profiles of Eq. (2-55) is identical to the transit time for the symmetric, clad power-law profiles in Table 2-1, page 40, i.e. dependent on only. Thus Eqs. (3-8) and (3-9) also give the optimum profile and minimum pulse spread for those noncircular profiles [5], which includes the clad parabolic-profile fiber of elliptical cross-section. In other words, ray dispersion on step-profilefibers of arbitrary cross-section and clad power-law profilefibers of noncircular cross-section is also given by the corresponding solutions for planar waveguides. 

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,. 

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. 

Waveguides with noncircular cross-sections Fibers with circular cross-sections... 

When the cross-section of the waveguide is noncircular, there is only one solution V of the scalar wave equation for each discrete value of the scalar propagation constant in Eq. (13-8). The direction of e, as expressed by Eq. (13-7), then takes the general form... 

Substituting from Table 13-1, page 288, we deduce that for waveguides of noncircular cross-section and for circular fibers... 

---